Engineering Turbulence Modelling and Experiments 6: ERCOFTAC International Symposium on Engineering Turbulence and Measurements - ETMM6

Engineering Turbulence Modelling and Experiments 6

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Engineering Turbulence Modelling and Experiments 6 Procedings of the ERCOFTAC International Symposium on Engineering Turbulence Modelling and Measurements - ETMM6 Sardinia, Italy, 23-25 May, 2005

Edited by

W. RODI Instimt ftir Hydromechanik Universit~it Karlsruhe Karlsruhe, Germany

M. MULAS Computational Fluid Dynamics CRS4 Pula (Cagliari), Italy

2005

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SYMPOSIUM SCIENTIFIC AND ORGANIZING COMMITTEE Chairman

Co-Chairman

Professor W. Rodi Institut ftir Hydromechanik Universit~it Karlsruhe Kaiserstr. 12 76128 Karlsruhe, Germany

Dr. M. Mulas Computational Fluid Dynamics CRS4 Parco POLARIS, Edificio 1 09010 PULA (CA), Italy Members

B. Aupoix, ONERA, Toulouse, France

N. Kasagi, University of Tokyo, Japan

M. Braza, Institute de Mechanique des Fluides de Toulouse, France

B. E. Launder, University of Manchester, U.K.

I. Castro, University of Southampton, U.K.

D. Laurence, Electricit6 de France, Chatou, France and University of Manchester, U.K.

C. T. Crowe, Washington State University, Pullman, USA

M. A. Leschziner, Imperial College, London, U.K.

E. Dick, University of Gent, Belgium

F. Menter, ANSYS-CFX, Otterfing, Germany

L. Djenidi, University of Newcastle, Callaghon, Australia

O. Metais, IMG-LEGI, Grenoble, France

S. Drobniak, Czestochowa University of Technology, Poland P. A. Durbin, Stanford University, USA B. J. Geurts, University of Twente, Enschede, The Netherlands

Y. Nagano, Nagoya Institute of Technology, Japan R. V. A. Oliemans, Multi-Phase Flow B.V. and TU Delft, The Netherlands P. Orlandi, University of Rome "La Sapienza", Italy N. Peters, RWTH Aachen, Germany

W. Haase, EADS Military Aircraft, Munich, Germany

A. M. Savill, Cranfield University, U.K.

K. Hanjali6, Delft University of Technology, The Netherlands

M. Sommerfeld, Martin-Luther-Universit~it HalleWittenberg, Halle, Germany

A. G. Hutton, QinetiQ, Famborough, U.K.

P. R. Spalart, Boeing Commercial Airplanes, Seattle, USA

W. P. Jones, Imperial College, London, U.K.

R. Verzicco, Polytechnic Institute of Bari, Italy

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vii

CONTENTS

Preface

x/x

1. I n v i t e d L e c t u r e s

Rapid techniques for measuring and modeling turbulent flows in complex geometries G. laccarino and C.J. Elkins, Stanford University, Stanford, USA Large-Eddy-Simulation of complex flows using the immersed boundary method R. Verzicco, Politecnico di Bail, Bail, Italy

17

Transition modelling for general purpose CFD codes F. R. Menter, R. Langtry, ANSYS-CFX Germany, Otterfing, Germany S. V61ker, GE Global Research, Niskayuna, NY, USA P. G. Huang, University of Kentucky, Lexington, Kentucky

31

Possibilities and limitations of computer simulations of industrial turbulent multiphase flows L. M. Portela, R.V. A. Oliemans, Delft University of Technology, Delft, The Netherlands

49

Turbulence Modelling

0

\

(v-7/k) - f Turbulence Model and its application to forced and natural convection K. Hanjalid, Delft University of Technology, Delft, The Netherlands D. R. Laurence, UMIST, Manchester, UK and EDF-DER-LNH, Chatou, France M. Popovac, Delft University of Technology, Delft, The Netherlands J. C. Uribe, UMIST, Manchester, UK

67

Calibrating the length scale equation with an explicit algebraic Reynolds stress constitutive relation H. BJzard, ONERA, Toulouse, France T. Daris, SNECMA Motors Villaroche, Moissy-Cramayel, France

77

Near-wall modification of an explicit algebraic Reynolds stress model using elliptic blending G. Karlatiras, G. Papadakis, King' s College, London, UK

87

viii

Assessment of turbulence models for predicting the interaction region in a wall jet by reference to LES solution and budgets A. Dejoan, C. Wang, M. A. Leschziner, Imperial College, London, UK

97

Eddy collision models for turbulence B. Perot, C. Chartrand, University of Massachusetts, Amherst, USA

107 A stress-strain lag eddy viscosity model for unsteady mean flow A. J. Revell, University of Manchester, Manchester, UK S. Benhamadouche, University of Manchester, Manchester, UK and EDF-DER-LNH, Chatou, France T. Craft, University of Manchester, Manchester, UK D. Laurence, University of Manchester, Manchester, UK and EDF-DER-LNH, Chatou, France K. Yaqobi, EDF-DER-LNH, Chatou, France

117

Turbulence modelling of statistically periodic flows: the case of the synthetic jet S. Carpy, R. Manceau, University of Poitiers, Poitiers, France

127

Behaviour of turbulence models near a turbulent / non-turbulent interface revisited P. Ferrey, B. Aupoix, ONERA / DMAE, Toulouse, France

137

Behaviour of nonlinear two-equation turbulence models at the free-stream edges of turbulent flows A. Hellsten, Helsinki University of Technology, Helsinki, Finland H. Bdzard, ONERA, Toulouse, France

147

Extending an analytical wall-function for turbulent flows over rough walls K. Suga, Toyota Central R & D Labs., Inc., Nagakute Aichi, Japan T. J. Craft, H. lacovides, The University of Manchester, Manchsester, UK

157

Bifurcation of second moment closures in rotating stratified flow P. A. Durbin, M. Ji, Stanford University, Stanford, USA

167

Turbulence Model for wall-bounded flow with arbitrary rotating axes H. Hattori, N. Ohiwa, Y. Nagano, Nagoya Institute of Technology, Nagoya, Japan

175

Application of a new algebraic structure-based model to rotating turbulent flows C. A. Langer, University of Cyprus, Nicosia, Cyprus S. C. Kassinos, University of Cyprus, Nicosia, Cyprus & Stanford University, Stanford, USA S. L. Haire, Lockheed Martin Space Systems Company, Sunnyvale, CA, USA

185

k-e modeling of turbulence in porous media based on a two-scale analysis F. Pinson, O. Grdgoire, CEA Saclay, Gif sur Yvette, France O. Simonin, IMFT, Toulouse, France

195

3. Direct and Large-Eddy Simulations

Effect of a 2-D rough wall on the anisotropy of a turbulent channel flow L. Djenidi, University of Newcastle, Callaghan, NSW, Australia S. Leonarcli, P. Orlandi, Universita Degli Studi di Roma "La Sapienza", Rome, Italy R. A. Antonia, University of Newcastle, Callaghan, NSW, Australia

207

Direct numerical simulation of rotating turbulent flows through concentric annuli M. Okamoto, N. Shima, Shizuoka University, Hamamatsu, Japan

217

Numerical simulation of compressible mixing layers S. Fu, Q. Li, Tsinghua University, Beijing, China

227

LES in a U-bend pipe meshed by polyhedral cells C. Moulinec, UMIST, Manchester, UK S. Benhamadouche, D. Laurence, UMIST, Manchester, UK and EDF-DER-LNH, Chatou, France M. Perik, Computational Dynamics Ltd., London, UK

237

Large eddy simulation of impinging jets in a confined flow D. J. Clayton, W. P. Jones, Imperial College, London, UK

247

LES study of turbulent boundary layer over a smooth and a rough 2D hill model T. Tamura, Tokyo Institute of Technology, Tokyo, Japan Sh. Cao, Tokyo Polytechnic University, Tokyo, Japan A. Okuno, Tokyo Institute of Technology, Tokyo, Japan

257

Flow features in a fully developed ribbed duct flow as a result of LES M. M. Loh6sz, Budapest University of Technology and Economics, Budapest, Hungary and Von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, Belgium P. Rambaud, C. Benocci, Von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, Belgium

267

Coherent structures and mass exchange processes in channel flow with spanwise obstructions A. McCoy, G. Constantinescu, L. Weber, The University of Iowa, Iowa City, USA

277

Large Eddy Simulation of natural convection boundary layer on a vertical cylinder D. G. Barhaghi, L. Davidson, Chalmers University of Technology, G6teborg, Sweden R. Karlsson, Chalmers University of Technology, G~teborg, Sweden and Vattenfall Utveckling AB, .~lvkarleby, Sweden

287

Development of the subgrid-scale models in large eddy simulation for the finite difference schemes M. Tsubokura, The University of Electro-communications, Tokyo, Japan T. Kobayashi, Japan Automobile Research Institute and University of Tokyo, Tokyo, Japan N. Taniguchi, University of Tokyo, Tokyo, Japan

297

Assessment of the digital filter approach for generating large eddy simulation inlet conditions I. Veloudis, Z. Yang, J. J. McGuirk, G. J. Page, Loughborough University, Loughborough, UK

307

4. Hybrid LES/RANS Simulations Hybrid LES-RANS : Computation of the flow around a three-dimensional hill L. Davidson, S. DahlstrOm, Chalmers University of Technology, G/3teborg, Sweden

319

Applications of a renormalization group based hybrid RANS/LES model C. De Langhe, B. Merci, E. Dick, Ghent University, Ghent, Belgium

329

Application of zonal LES/ILES approaches to an unsteady complex geometry flow P. G. Tucker, Y. Liu, University of Wales, Swansea, UK

339

Interface conditions for hybrid RANS/LES calculations A. Keating, G. De Prisco, U. Piomelli, E. Balaras, University of Maryland, College Park, USA

349

Approximate near-wall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers F. Tessicini, Imperial College, London, UK L. Temmerman, Imperial College, London, UK and Numeca International s.a., Brussels, Belgium M. A. Leschziner, Imperial College, London, UK

359

LES, T-RANS and hybrid simulations of thermal convection at high RA numbers S. KenjereL K. HanjaliJ, Delft University of Technology, Delft, The Netherlands

369

5.

A p p l i c a t i o n of T u r b u l e n c e M o d e l s

Industrial practice in turbulence modelling: An evaluation of QNET-CFD A. G. Hutton, QinetiQ Ltd., Famborough, UK 381 Three-dimensional flow computation with Reynolds stress and algebraic stress models G. B. Deng, P. Queutey, M. Visonneau, Ecole Centrale de Nantes, Nantes, France

389

Comparison of turbulence models in case of jet in crossflow using commercial CFD code A. Karvinen, H. Ahlstedt, Tampere University of Technology, Tampere, Finland

399

6. Experimental Techniques and Studies Time resolved PIV measurements for validating LES of the turbulent flow within a PCB enclosure model G. Usera, Universitat Rovira i Virgili, Tarragona, Spain and Universidad de la Repdblica, Montevideo, Uruguay A. Vernet, J. A. Ferrd, Universitat Rovira i Virgili, Tarragona, Spain

411

Skin friction measurements in complex turbulent flows using direct methods J. A. Schetz, Virginia Polytechnic Institute and State University, Blacksburg, USA

421

Reynolds number dependence of elementary vortices in turbulence K. Sassa, Kochi University, Kochi, Japan H. Makita, Toyohashi University of Technology, Toyohashi, Japan

431

Near-wake turbulence properties in the high Reynolds incompressible flow around a circular cylinder by 2C and 3C PIV R. Perrin, M. Braza, E. Cid, S. Cazin, f Moradei, A. Barthet, A. Sevrain, Y. Hoarau, Institut de M6canique des Fluides de Toulouse, Toulouse, France

441

Single- and two-point LDA measurements in the turbulent near wake of a circular cylinder A. Ducci, E. Konstantinidis, E. Balabani, M. Yianneskis, King's College, London, UK

451

Aerodynamics of a half-cylinder in ground effect X. Zhang, S. Mahon, M. Van Den-Berg, C. Williams, University of Southampton, Southampton, UK

461

xii

Turbulent wall jet interaction with a backward facing step N. Nait Bouda, U.S.T.H.B., Alger, Algeria C. Rey, Universit6 Paul C6zanne Aix-Marseille HI, Marseille, France J. M. Rosant, Ecole Centrale de Nantes, Nantes, France T. Benabid, U.S.T.H.B., Alger, Algeria

471

The role of pressure-velocity correlation in oscillatory flow between a pair of bluff bodies S. Obi, N. Tokai, K. Sakai, Keio University, Yokohama, Japan

481

Turbulent structures in a supersonic jet-mixing layer interaction E. Collin, P. Braud, J. Delville, Universit6 de Poitiers ENSMA, Poitiers, France

491

Turbulent properties of twin circular free jets with various nozzle spacing T. Harima, S. Fujita, Tokuyama College of Technology, Shunan, Japan H. Osaka, Yamaguchi University, Ube, Japan

501

LDA-masurements of the turbulence in and around a venturi R. F. Mudcle, L. Deutz, V. A. Nievaart, TU Delft, Delft, The Netherlands H. R. E. van Maanen, Shell E&P, Rijswijk, The Netherlands

511

7.

Transition

Modelling of unsteady transition with a dynamic intermittency equation K. Lodefier, E. Dick, Ghent University, Ghent, Belgium

523

Transition to turbulence and control in the incompressible flow around a NACA0012 wing Y. Hoarau, M. Braza, Institut de M6canique des Fluides de Toulouse, Toulouse, France Y. Ventikos, University of Oxford, Oxford, UK D. Faghani, Euro-American Institute of Technology, Sophia Antipolis, France

533

8.

Turbulence Control

Some observations of the Coanda effect G. Han, M. D. Zhou, I. Wygnanski, The University of Arizona, Tucson, USA

545

Active control of turbulent separated flows by means of large scale vortex excitation A. Brunn, W. Nitsche, Technical University of Berlin, Berlin, Germany

555

xiii

Large-eddy simulation of a controlled flow cavity I. Mary, T.-H. L6, ONERA, Ch~tillon, France

565

Parametric study of surfactant-induced drag-reduction by DNS B. Yu, National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan & National Maritime Research Institute, Tokyo, Japan Y. Kawaguchi, National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

575

Effect of non-affine viscoelasticity on turbulence generation K. Horiuti, S. Abe, Y. Takagi, Tokyo Institute of Technology, Tokyo, Japan

585

Experimental and numerical investigation of flow control on bluff bodies by passive ventilation M. Falchi, G. Provenzano, D. Pietrogiacomi, G. P. Romano, University of Rome "La Sapienza", Rome, Italy

595

9. A e r o d y n a m i c F l o w s

Application of Reynolds stress models to high-lift aerodynamics applications O. Grundestam, Royal Institute of Technology, Stockholm, Sweden S. Wallin, Royal Institute of Technology, Stockholm, Sweden and Swedish Defence Research Agency, Stockholm, Sweden P. Eliasson, Swedish Defence Research Agency, Stockholm, Sweden A. V. Johansson, Royal Institute of Technology, Stockholm, Sweden

607

Turbulence modelling in application to the vortex shedding of stalled airfoils C. Mockett, U. Bunge, F. Thiele, Technische Universit~it Berlin, Berlin, Germany

617

The computational modelling of wing-tip vortices and their near-field decay T. J. Craft, B. E. Launder, C. M. E. Robinson, The University of Manchester, Manchester, UK

627

URANS computations of shock induced oscillations over 2D rigid airfoil: Influence of test section geometry M. Thiery, E. Coustols, ONERA/DMAE, Toulouse, France

637

Zonal multi-domain RANS/LES simulation of airflow over the Ahmed body F. Mathey, Fluent France SA, Montigny Le Bretonneux, France D. Cokljat, Fluent Europe Ltd, Sheffield, UK

647

xiv

Numerical simulation and experimental investigation of the side loading on a high speed train N. Paradot, SNCF (National French Railways), Paris, France B. Angel, Renuda Engineering Computation, Paris, France P.-E. Gautier, L.-M. Cldon, SNCF (National French Railways), Paris, France

657

Large-scale instabilities in a STOVL upwash fountain A. J. Sacldington, P. M. Cabrita, K. Knowles, Cranfield University, Wiltshire, UK

667

10.

Aero-Acoustics

Direct numerical simulation of large-eddy-break-up devices in a boundary layer P. R. Spalart, Boeing Commercial Airplanes, Seattle, WA, USA M. Strelets, A. Travin, Federal Scientific Center "Applied Chemistry", St. Petersburg, Russia

679

Blade tip flow and noise prediction by large-eddy simulation in horizontal axis wind turbines O. Fleig, M. licla, C. Arakawa, The University of Tokyo, Tokyo, Japan

689

A zonal RANS/LES approach for noise sources prediction M. Terracol, ONERA, Ch~tillon, France

699

Aerodynamics and acoustic sources of the exhaust jet in a car air-conditioning system A. Le Duc, N. Peller, M. Manhart, Technische Universit~it Mtinchen, Munich, Germany E.-P. Wachsmann, AUDI AG, Ingolstadt, Germany

709

Characterization of a separated turbulent boundary layer by time-frequency analyses of wall pressure fluctuations R. Camussi, G. Guj, A. Di Marco, University "Roma 3", Rome, Italy A. Ragni, CIRA - Italian Aerospace Research Centre, Capua, Italy

719

11.

Turbomachinery Flows

Study of flow and mixing in a generic GT combustor using LES B. Wegner, B. Janus, A. Sadiki, A. Dreizler, J. Janicka, Darmstadt University of Technology, Darmstadt, Germany

731

An evaluation of turbulence models for the isothermal flow in a gas turbine combustion system K. R. Menzies, Rolls-Royce plc, Bristol, UK

741

Large Eddy Simulations of heat and mass transfers in case of non isothermal blowing G. Brillant, CEA, Grenoble, France and INSA - Centre de Thermique de Lyon, Villeurbanne, France S. Husson, F. Bataille, INSA - Centre de Thermique de Lyon, Villeurbanne, France

751

Turbulence modelling and measurements in a rotor-stator system with throughflow S. Poncet, R. Schiestel, M.-P. Chauve, Univ. Aix-Marseille I & II, Marseille, France

761

12. Heat and Mass Transfer

Impinging jet cooling of wall mounted cubes M. J. Tummers, M. A. Flikweert, K. Hanjalik, R. Rodink, Delft University of Technology, Delft, The Netherlands B. Moshfegh, Link~Jping Institute of Technology, Linki3ping, Sweden

773

Numerical and experimental study of turbulent processes and mixing in jet mixers E. Hassel, S. Jahnke, N. Kornev, I. Tkatchenko, V. Zhdanov, University of Rostock, Rostock, Germany

783

Effects of adverse pressure gradient on heat transfer mechanism in thermal boundary layer T. Houra, Y. Nagano, Nagoya Institute of Technology, Nagoya, Japan

793

Stochastic modelling of conjugate heat transfer in near-wall turbulence J. Pozorski, Polish Academy of Sciences, Gdafisk, Poland J.-P. Minier, Electricit6 de France, Chatou, France

803

Study of the effect of flow pulsation on the flow field and heat transfer over an inline cylinder array using LES Ch. Liang, G. Papaclakis, King's College, London, UK

813

Large eddy simulation of scalar mixing M. Dianat, Z. Yang, J. J. McGuirk, Loughborough University, Loughborough, UK

823

13.

Combustion Systems

Experimental characterization and modelling of inflow conditions for a gas turbine swirl combustor R. Palm, S. Grundmann, M. Weismiiller, S. Saric, S. Jakirli~, C. Tropea, Darmstadt University of Technology, Darmstadt, Germany 835 On the sensitivity of a free annular swirling jet to the level of swirl and a pilot jet M. Garcia-Villalba, J. Fr6hlich, University of Karlsruhe, Karlsruhe, Germany

845

Prediction of pressure oscillations in a premixed swirl combustor flow and comparison to measurements P. Habisreuther, C. Bender, O. Petsch, H. Biichner, H. Bockhorn, University of Karlsruhe, Karlsruhe, Germany

855

Interaction between thermoacoustic oscillations and spray combustion W. A. Chishty, U. Vandsburger, W. R. Saunders, W. T. Baumann, Virginia Polytechnic Institute & State University, Blacksburg, USA

865

Dynamics of lean premixed systems: Measurements for large eddy simulation D. Galley, SNECMA Moteurs, Moissy Cramayel, France and Ecole Centrale Paris, Chatenay-Malabry, France A. P. Melsi6, S. Ducruix, F. Lacas, D. Veynante, Ecole Centrale Paris, ChatenayMalabry, France Y. Sommerer, T. Poinsot, CERFACS, Toulouse, France

875

White in time scalar advection model as a tool for solving joint composition PDF equations: Derivation and application

V. Sabel'nikov, O. Soulard, ONERA, Palaiseau, France

885

The effects of micromixing on combustion extinction limits for micro combustor applications C. Dumand, V. A. Sabel'nikov, ONERA, Palaiseau, France

895

Joint RANS/LES approach to premixd flames modelling in the context of the TFC combustion model V. L. Zimont, V. Battaglia, CRS4 Research Center POLARIS, Pula, Italy

905

Optical observation and discrete vortex analysis of vortex-flame interaction in a plane premixed shear flow N. Ohiwa, Y. Ishino, Nagoya Institute of Technology, Nagoya, Japan

915

xvii

14.

T w o - P h a s e Flows

Simulation of mass-loading effects in gas-solid cyclone separators J. J. Derksen, Delft University of Technology, Delft, The Netherlands

929

On Euler/Euler Modeling of turbulent particle diffusion in dispersed two-phase flows R. Groll, C. Tropea, TU Darmstadt, Darmstadt, Germany

939

Influence of the gravity field on the turbulence seen by heavy discrete particles in an inhomogeneous flow B. Arcen, A. Tanibre, B. Oesterld, Universit6 Henri Poincar6-Nancy I, Vandoeuvre-16s-Nancy, France 949 Modelling turbulent collision rates of inertial particles L. I. Zaichik, V. M. Alipchenkov, Institute for High Temperatures of the Russian Academy of Sciences, Moscow, Russia A. R. Avetissian, All Russian Nuclear Power Engineering Research and Development Institute, Moscow, Russia

959

Large eddy simulation of the dispersion of solid particles and droplets in a turbulent boundary layer flow I. Vinkovic, C. Aguirre, S. Simo6ns, Ecole Centrale de Lyon, Ecully Cedex, France

969

Dynamic self-organization in particle-laden turbulent channel flow B. J. Geurts, University of Twente, Enschede, The Netherlands and Eindhoven University of Technology, Eindhoven, The Netherlands A. W. Vreman, Vreman Research, Hengelo, The Netherlands

979

AUTHOR INDEX

989

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xix

PREFACE These proceedings contain the papers presented at the ERCOFTAC International Symposium on Engineering Turbulence Modelling and Measurements - ETMM6 - held at Villasimius, Sardinia, Italy, in the period May 23-25, 2005. The symposium followed the previous five conferences on the topic of engineering turbulence modelling and measurements held in Dubrovnik, Yugoslavia, in 1990, Florence, Italy, in 1993, Crete, Greece, in 1996, Corsica, France, in 1999 and Mallorca, Spain, in 2002, but was held for the first time under the aegis of ERCOFTAC (European Research Community on Flow, Turbulence and Combustion). The proceedings of the previous conferences were also published by Elsevier. The purpose of this series of symposia is to provide a forum for presenting and discussing new developments in the area of turbulence modelling and measurements, with particular emphasis on engineering-related problems. Turbulence is still one of the key issues in tackling engineering flow problems. As powerful computers and accurate numerical methods are now available for solving the flow equations, and since engineering applications nearly always involve turbulence effects, the reliability of CFD analysis depends more and more on the performance of the turbulence models. Successful simulation of turbulence requires the understanding of the complex physical phenomena involved and suitable models for describing the turbulent momentum, heat and mass transfer. For the understanding of turbulence phenomena, experiments are indispensable, but they are equally important for providing data for the development and testing of turbulence models and hence for CFD software validation. Recently, Direct Numerical Simulations have become an important tool for providing supplementary detailed data. Research in the area of turbulence modelling and measurements continues to be very active worldwide, and altogether 277 abstracts were submitted to the symposium and experts in the field screened the 269 abstracts that arrived in time. 134 abstracts were accepted and 112 final papers were received and each reviewed by two experts. In the end, 90 papers were accepted, and most of these underwent some final revision before they were included in these proceedings. The papers were conveniently grouped in the following sections: - Turbulence modelling -

Direct and large-eddy simulations

- Hybrid LES/RANS simulations - Application of turbulence models -

-

- Aerodynamics flows -

Aero-Acoustics

- Turbomachinery flows -

Heat and mass transfer

Experimental techniques and studies

- Combustion systems

Transition

- Two-phase flows

- Turbulence control

The contributed papers are preceded by a section containing 4 invited papers covering LES and rapid measurement techniques for complex turbulent flows, transition modelling and simulations of multiphase flows. The conference was organised with the support and cooperation of the following institutions and companies:

- ERCOFTAC -

-

KATnet of the European Union Regione Autonoma della Sardegna

- CRS4 -

University of Karlsruhe

- ENEA - European Office of Aerospace Research and Development of the USAF - ANSYS-CFX Germany - CD- Adapco -

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Electricit6 de France Fluent Numeca

We gratefully acknowledge this support and cooperation. We are also grateful to the members of the Scientific and Organizing Committee for their various efforts in making this conference a success. We also acknowledge the help of many Fluid Mechanics experts from all over the world in reviewing abstracts and full papers for the conference. Finally, we express our sincere appreciation for the good cooperation provided by Dr. Arno Schouwenburg and Vicki Wetherell of Elsevier Ltd. in the preparation of the proceedings.

W. Rodi and M. Mulas

1. Invited Lectures

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Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

RAPID TECHNIQUES FOR MEASURING AND MODELING TURBULENT FLOWS IN COMPLEX GEOMETRIES G. Iaccarino and C. J. Elkins Department of Mechanical Engineering Stanford University, Stanford CA 94305

ABSTRACT An approach to measure and model turbulent flows in complex configurations is presented. It is based on the synergistic use of two novel techniques: the experiments are based on magnetic resonance velocimetry, which allows the collection of a large three-dimensional volume of three-component velocity measurements in a short period of time. The numerical predictions are based on the immersed boundary technique that enables simulations to be carried out on Cartesian grids even for realistic, industrial configurations. Computer models of realistic geometries are used without modification in the simulations, and they are accurately reproduced for the experiments using rapid prototyping manufacturing. These two techniques enable analysis of flow systems in great detail by quickly providing a wealth of experimental and numerical data. Moreover, direct comparison between these datasets gives indications of the uncertainties in the data from both methods. Results are presented for the flow in a pipe and in a ribroughened serpentine. In addition, preliminary measurements and simulations of the flow around a coral reef are included.

KEYWORDS Magnetic resonance velocimetry, immersed boundary technique, ribbed serpentine, coral reef.

INTRODUCTION

The analysis of the turbulent flow in complex, industrial configurations is of great importance for improving the design and the performance of a wide variety of engineering devices. Traditionally, such analysis is based on an experimental investigation that involves the construction of the device (typically in a reduced scale) and the direct measure of a few performance parameters. The detailed instrumentation of an industrial device can be extremely time-consuming and expensive. It is common to use only few probes for measuring pressure, flow rates, temperature, etc. and, as a consequence, their position plays a critical role in the significance of the collected data. Although these

data yield valuable information, they do not provide enough information to identify areas of separation or other problematic regions that can cause performance reduction. Several methodologies are available to collect more detailed measurements in flows. Laser Doppler anemometry (LDA) and Particle Image Velocimetry (PIV) (Stanislas Kompenhans and Westerweel, 2000) are two non-invasive techniques. While LDA provides pointwise velocity measurements, PIV provides instantaneous two-dimensional velocity fields, and stereoscopic PIV provides three-component velocities in two-dimensional planes. All of these measurement techniques require optical access in the studied device and this limits their applications to simpler geometries. In addition, these measurements may be available only in selected positions or planes. In many realistic geometries, full coverage of the flow domain by LDA or PIV may be impossible or extremely time consuming. Recently, a technique has been implemented in modern Magnetic Resonance Imaging (MRI) scanners to measure three-component velocity fields in three-dimensional complex geometries. This technique is called Magnetic Resonance Velocimetry (MRV). The method is based on the same principles used in MRI, now routinely employed in medical imaging. MRV is becoming popular in the study of blood flow in vascular medicine, and it has applications in the study of engineering flows as well. The typical fluid used in MRV experiments contains water since medical MRI scanners measure radio frequency signals from excited hydrogen nuclei in the presence of strong magnetic fields. More detailed discussion of MRV can be found in Elkins et al. (2003) and Markl et al. (2003). One advantage of MRV is that it provides detailed three-dimensional data very quickly; a typical scan of a volume of size 32x200x200mm with a resolution of about lmm can be obtained in less than 30 minutes. Another major advantage of MRV is its ability to measure data in complex geometries without the need for optical access. Flow models are typically fabricated using rapid prototyping manufacturing processes (i.e. stereo-lithography). There are several well-documented drawbacks to the MRV technique including signal dephasing due to turbulence and spatial misregistration due to strong accelerations in the flow. In addition, MRV provides only mean velocity measurements and knowledge of turbulence quantities can be important. These shortcomings are being investigated by the authors in an effort to improve MRV. The other existing approach to study and design engineering systems is Computational Fluid Dynamics (CFD). Numerical flow simulations have become a common tool and several software tools are available in the industrial community. CFD calculations are carried out in two steps: the first is the geometry acquisition and mesh generation, and the second is the actual flow simulation. The geometry acquisition requires the transfer of a configuration, typically generated in a Computer-Aided-Design (CAD) environment, into a CFD mesh generation system. This process is very time consuming as the operating principles (geometry definition, tolerances, etc.) of the two software environments can be quite different. Once a watertight definition of the device to be studied is available in the mesh generator, the air-solid has to be defined. The air-solid represents the volume that is effectively occupied by the fluid (typically it is just the negative of the real device). A computational grid, i.e. a collection of small Computational Volumes (CV) covering the entire air-solid, is then generated in a semi-automatic way. Control on the resolution and the quality - the size and shape of the CVs, respectively - of the grid requires substantial user-intervention and might be very time-consuming. Once a grid is available, the solution of the equations governing the flow can be carried out. For complex applications, the use of CFD is still challenging as the first phase of the process described above can be quite difficult and time consuming. Techniques that simplify and automate the grid generation have great potential in sustaining the widespread use of CFD. The Immersed Boundary (IB) method eliminates the need for the construction of the air-solid thus simplifying substantially both the

geometry acquisition and the mesh generation phases. The IB method (Mittal and Iaccarino, 2005) uses a mesh that covers the entire computational domain (typically a large box) without the device of interest; the effect of this on the flow is then accounted for by modifying the governing equations through source terms that mimic the presence of the solid boundaries. Cartesian mesh techniques were introduced for fluid flow simulation in the 70s (Peskin, 1972) but only recently have been applied to complex, industrial flows and in the turbulent regime (Iaccarino and Verzicco, 2003). The availability of the MRV and IB techniques to study turbulent flows in realistic configurations creates an opportunity for a new paradigm in engineering design as measurements and simulations can be used together. The combined MRV-IB approach provides a wealth of information for the designer at a resolution that is well above what is usually available. The data are typically complementary as the two techniques have different strengths and weaknesses but provide enough overlap to create confidence in the results. In this paper some applications of the MRV-IB approach are presented with the objective of illustrating the advantages of the techniques. Comparisons of the results obtained using the two methods to more conventional PIV measurements are presented to evaluate their accuracy. As the two techniques are relatively new, further research is currently ongoing to fully evaluate their capabilities; this aspect is discussed at the end of the paper.

MEASURING TECHNIQUE: MAGNETIC RESONANCE VELOCIMETRY MRV is a non-invasive experimental method for measuring mean velocities using modern medical Magnetic Resonance Imaging (MRI) systems. All of the measurements presented in this paper were made using a 1.5 T GE Signa CV/I system (Gmax=40mT/m, rise time=268 microsecs). For a discussion of the principles of MRI, the reader is referred to Stark and Bradley (1999), yon Schultess and Hennig (1998), and Haacke et al. (1999). In addition, a brief discussion of MRV principles is found in Elkins et al. (2003) where the MRV technique is described in detail. Most MRI systems image hydrogen protons which are abundant in the fluids and tissues in living things. Protons have magnetic moments (spins) that align with the direction of a strong magnetic field. If knocked out of alignment with the external magnetic field, the spins will relax back into alignment and precess about the field direction with a frequency proportional to the strength of the magnetic field. Hence, when a spatial magnetic field gradient is applied to create a continuously varying magnetic field, the spins along the direction of the gradient have different precession frequencies. This principle can be exploited to image an object. In imaging, the spins are knocked out of alignment with a strong, constant field. Then a magnetic field gradient is applied. As the spins relax back into alignment with the constant field, they broadcast RF signals, each with its own precession frequency dependent on its position in the gradient. The signals are measured with an RF coil, and the output from the coil is the combination of the signals from all of the spins. The inverse FFT of this signal is used to calculate the density and position of the spins, which is represented by the MR image of the object. Magnetic field gradients can be applied in all three dimensions to produce three-dimensional images. The MRV sequence is based on phase-contrast MRI in which velocity is encoded in the phase of the emitted RF signal. Here, too, the inverse FFT is used to convert the sampled RF signals into an object image and velocity fields. Water doped with a gadoliniumcontrast agent is used for the MRV experiments. The flow models and flow loops must be MR compatible and contain no metal. Rapid prototyping materials are typically plastic and, therefore, well suited for MR experiments. In addition, steady flow centrifugal pumps driven by induction motors are also MR compatible and can be used inside the magnet room if placed several

meters from the magnet. MRV is possible in opaque models and opaque fluids although the allowance for some visual inspection is recommended in order to purge bubbles from the model. For measurements made in large flow models or passages, MRV measurements are typically carried out usingtwo successive scans, corresponding to flow on and flow off conditions, respectively. There are inherent asymmetries and imperfections in the magnetic field in the magnet bore. The subtraction of the flow off scan from the flow on scan helps correct for errors related to these imperfections, eddy currents and other sources of off-resonance effects. A procedure to measure time-dependent flows using MRV has also been developed (Markl et al, 2003). In this case scans are phase locked to an appropriate trigger signal (an ECG signal for physiologic scans). For turbulent flows the collection of successive datasets allows more accurate mean velocity information; typically, if there is time, 3 to 10 dataset are collected and averaged.

M O D E L I N G TECHNIQUE: IMMERSED BOUNDARY A P P R O A C H The Immersed Boundary approach belongs to the family of Cartesian methods. An underlying regular grid is used and modifications to the algorithm or to the governing equations are introduced to represent boundaries that are not aligned with grid lines. There are several variants of the IB methods (Mittal and Iaccarino, 2005); the main differences are related to the way the no-slip wall conditions are enforced. In the present approach, an interpolation scheme is applied to enforce the boundary condition off the wall in the first computational cell. A least squares approach is used. The method is based on the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations using a finite-volume second-order discretization. Turbulence is modeled using a linear eddy viscosity approach based on the two-equation k-g model (Iaccarino et al., 2003). This model has been specifically developed for use with the IB approach, and it does not require the computation of the wall distance or the use of complex boundary conditions at solid walls. Other models have been implemented in the present solver, but only results with the k-g models are included here (Iaccarino and Verzicco, 2003). Another remarkable aspect of the present solver is its ability to handle locally refined grids with hanging nodes. An important component of the IB approach is the grid generator. As discussed, the starting point is an STL model of the configuration of interest (an unconnected triangulation generated in a CAD environment). Initially, a uniform Cartesian mesh is generated with a resolution corresponding to the bulk resolution required by the user away from the solid walls. In the second phase, this grid is refined in the vicinity of the IB until a user-specified mesh spacing is reached everywhere on the immersed boundary. The grid refinement is based on the localization of the STL triangles on the grid. This is accomplished using a ray tracing technique. The cells are split in each Cartesian direction separately with the objective to minimize the distance between the cell center and the closest STL triangle. The resulting grid has strong non-isotropy in the regions where the IB is aligned with the Cartesian directions with obvious savings in terms of overall mesh size. Once the required resolution is reached, a final ray-tracing step is carried out to separate the computational cells in fluid, solid and interface cells. The interface cells are cut by the STL surface and have their cell center laying in the fluid part. Note that this distinction is only meaningful when the original STL file is the representation of a watertight surface. Finally, for the interface cells, the coefficients of the least-square interpolation are pre-computed and stored.

For typical applications the mesh is generated in a few minutes for a few million grid cells. The flow solution is then carried out on this grid, and given the high quality of the mesh (all the cells are hexahedral) the convergence to steady state is typically very good. Several applications of the present approach to a number of problems have been published in the literature (Iaccarino and Verzicco, 2003; Mittal and Iaccarino, 2005; Moreau et al, 2004).

APPLICATIONS Three problems are presented to show the predictive capabilities of the MRV and IB approaches. The first one is a simple, canonical flow, whereas the other two introduce substantial geometrical complexity. Flow in a pipe

The fully developed, turbulent flow in a pipe is considered. This problem has been used to evaluate the accuracy of MRV in comparison with data collected using laser Doppler anemometry. Flow in a straight, rigid pipe with a 19 mm inner diameter is imaged. The Reynolds number is Re=6,400 (based on the bulk velocity). The imaging volume with its dimensions is shown in Fig. 1.

Figure 1: Pipe flow imaging field of view (left) and immersed boundary grid (right) A long inflow tube is used to ensure that the flow is fully developed as it enters in the measurement areas. A total of 256x32x256 measurement points were collected. Three complete datasets were acquired in 26 minutes; the final results are averaged to reduce the errors. Three velocity profiles have been measured at different locations inside the pipe, and the comparisons to LDA show reasonable agreement (Elkins et al, 2003). Simulations have been carried out on the same geometrical model. Periodicity is enforced in the streamwise direction and the flow rate is specified. A view of the computational grid in a cross section of the pipe is reported in Fig. 1. The grid consists of 300,000 cells with a wall-normal resolution at the boundary of 0.1mm; it was generated in less than a minute on a SGI workstation. Calculations were carried out using the k-g turbulence model and steady-state was achieved after ~500 iterations in ~25 minutes.

Figure 2: Mean velocity profile for the flow in the pipe The comparisons of the velocity profiles as a function of the radius are reported in Fig. 2. Three sets of MRV data are reported at different streamwise locations (numerical results show negligible variability in the streamwise direction and only one profile is shown). The overall agreement is within 6%.

Flow serpentine with oblique ribs The second example is the flow in a serpentine duct; this configuration is typical of the cooling flow passages within turbine blades. A sketch of the geometry is reported in Fig. 3.

Figure 3: Ribbed serpentine model. The model was drawn in SolidWorks and fabricated using a stereo-lithography machine. The serpentine has a square cross section height of H = 20mm and ten staggered oblique ribs on the top and bottom walls. The rib height is 0.1H and the pitch (distance between two successive ribs) is 0.6H; the rib angle is 45 degrees. In the experiment, fully developed pipe flow enters the first leg of the serpentine through a converging section yielding a uniform velocity profile. The flow is investigated for a Reynolds number of Re=10,000, based on the passage height and the bulk velocity. A total of 36x256x256 measurement points were collected (corresponding to a resolution of 36 x 280 x 280 mm). Six complete datasets were acquired in 27 minutes; the final results are averaged to reduce the errors.

Figure 4: Horizontal and vertical cross-sections of the immersed boundary grid for the ribbed serpentine model. The simulations have been performed using the same geometrical definition of the serpentine. A series of locally refined grids with successively increased resolution was generated; only results obtained on a grid with 3 million grid points are reported here. Two cross sections of the computational mesh are reported in Fig. 4; the resolution at solid walls is 0.005mm (H/400) and in the bulk of the duct is ~0.1mm. The mesh was generated automatically in about two minutes. Steady state flow calculations were carried out assuming a uniform velocity profile at the inflow in accordance with experimental observations. The calculations required about 3 hours of CPU time on an SGI workstation.

Figure 5: Comparison of MRV, IB and PIV in-plane velocity vectors in the U-bend (region 1 in Fig. 3). PIV: open arrows; MRV: filled arrows; IB: hollow arrows. An additional set of experimental measurements was collected using PIV to identify the overall quality of both MRV and IB data. Visual access was available in the area corresponding to the first U-bend in a plane located at mid-height of the duct. The comparison is shown in Fig. 5; the overall agreement is remarkable, given the complexity and three-dimensionality of the flow at this location. The flow approaches the bend at an angle of about 45 degrees (somewhat lower according to the MRV data) and a large separation occurs as it leaves the bend. Comparing to the PIV dataset which is the most reliable, it is

evident that both the MRV and IB results are capturing the separation correctly in terms of size and the overall speed and direction with reasonable accuracy. It is also interesting that the IB method appears to be inaccurate in predicting the flow direction after the bend whereas the MRV shows a discrepancy in the flow upstream of the bend. The flow downstream of the bend is characterized by strong turbulence non-equilibrium but otherwise low levels of turbulence intensity (Iacovides and Raisee, 1999). This situation poses a challenge to linear eddy viscosity models, and it is likely that only a Reynolds-stress model would provide better predictions in this area. On the other hand, the flow in the region approaching the bend is dominated by two strong streamwise vortices (generated by the presence of the oblique fibs) and very high levels of turbulence. In this case, uncertainty in the MRV data associated with strong turbulence and flow acceleration is the likely cause of the differences with respect to the PIV measurements.

Figure 6: Comparison of MRV and IB velocity magnitude in the plane at rib mid-height. Top: IB simulations; bottom: MRV measurements. Two additional sets of comparisons of MRV and IB results are reported in Fig. 6 and 7. In Fig. 6, a plane at rib mid-height is reported to show the level of detail that the two techniques are providing; note that the MRV resolution is -1.1mm and the IB resolution is -0.1mm. In Fig. 7, the streamwise velocity in two planes (upstream and downstream of the first U-bend) is reported. The flow approaching the bend is characterized by high speed toward the inner part of the bend and a characteristic reverse C shape; this is a result of the two strong counter-rotating streamwise vortices present in the duct. The flow leaving the bend, on the other hand, is substantially more uniform although the peak velocity is located off the center towards the outer part of the bend. This is a consequence of the presence of massive separation on the inner side of the bend (as illustrated in Fig. 5). The overall agreement between MRV and IB is remarkable in spite of the limitations observed above. A more complete set of comparisons between the MRV and IB results is reported in Iaccarino et al. 2003.

Figure 7: Comparison of MRV and IB velocity magnitude in before (left, region 2 in Fig. 3) and after the U-bend (fight, region 3 in Fig. 3). Top: IB simulations; bottom: MRV measurements. Flow around a coral colony

The final example is the flow around a coral. The study is motivated by the observation that hydrodynamics directly affects coral growth, energetics, and health (Chang et al, 2004). Understanding the flow in scales important to the coral proves difficult because of the complex geometry and the turbulent structures resulting from the interaction between the geometry and the water motion. In order to analyze such interactions using the MRV and IB methods, first a computer model of the geometry of a real coral skeleton was created. A coral skeleton has been digitized using computed tomography (CT) and converted into an STL file using a software package called Mimics (Materialise NV). This CAD model was then used to rapid-prototype a scaled-down version of the original coral; the model height is 12cm).

Figure 8: Coral skeleton (left) and rapid prototyped model (right). In Fig. 8 the original skeleton and the model are shown. The model is placed inside a channel designed to fit into the MR scanner; its main dimensions are 19cm x 17cm. In Fig. 9 the experimental set-up with the

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coral mounted in the test section is shown. The average streamwise velocity is 5.2cm/sec corresponding to a Reynolds number (based on the coral height) of 8,000. Preliminary measurements have been carried out but only one set of MRV data has been collected so far. In Fig. 10, the MRV streamwise velocities are presented. Flow off, flow on and the actual velocity data (flow-off subtracted out of the flow-on measurements) are reported.

Figure 9: Coral experimental set-up

Figure 10: MRV measurements at 1/3 coral height. The images are scans for flow-off (left), flow on (center) and the flow off subtracted from the flow on respectively (right). Flow is from bottom to top. The color gradation is an indication of streamwise velocity (mm/s). Wakes corresponding to the single branches and the overall low-speed area downstream of the coral are clearly shown; the lack of side-to-side symmetry is an indication of the complex, three-dimensional structure of the flow. An unexpected low speed region fairly far upstream of the coral is observed in Fig. 10. This is likely due to slight differences between the flow on and flow off scans, and the MRV scans are being repeated to eliminate these errors. Initial measurements of the volumetric flow rate at different

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streamwise sections show substantial discrepancy (more than 10%), especially in the regions downstream of the coral. This error has not been observed in previous results. Current work is devoted to the determination of the causes of the lack of strict mass conservation in the coral flow. The IB method is used to perform simulations in the same configuration. A preliminary, coarse computational grid was generated. It consists of 1 million grid cells with a resolution of 0.05cm and 0.5cm close to the coral and in the wake, respectively. Two cross sections of the mesh are reported in Fig. 10. This mesh was generated in less than a minute on a SGI workstation. Steady state calculations were carried out Reynolds numbers ranging from 5,000 to 50,000.

Figure 11: Immersed boundary grid for the coral. View of a plane in the streamwise direction (left) and plane across the coral 1/3 of the height (right). Given the somewhat low Reynolds numbers, the calculations are carried out with and without the turbulence model. The latter in particular, show strong unsteadiness and appear to be more consistent with the experimental observations. In Fig. 12 a horizontal plane at 1/3 of the coral height is shown to illustrate the flow structure in the streamwise direction. As expected, we observe the presence of small branchwakes within the overall coral wake. More interestingly, pockets of high speed are present between the branches; this can potentially have a strong effect on mass transfer (in this case nutrient transport from the water to the coral) and will be analyzed in detail when more reliable MRV measurements are available.

CONCLUSIONS AND PERSPECTIVES The use of a combined experimental and computational approach has been illustrated; the measurements are based on magnetic resonance velocimetry and the numerical simulations on the immersed boundary approach. Although relatively new, both methods appear very promising, and when combined together, offer potential for several innovative applications in the study of engineering turbulent flows. Three applications have been presented: the flow in a simple straight pipe, the flow in a complex three-legged rib-roughened serpentine and the flow around a coral. The results have been compared to "conventional" measurements obtained using PIV with satisfactory accuracy. In particular, it has been shown that the wealth of information provided by MRV allows the identification of specific areas of inadequacy of simple eddy-viscosity models used in the simulations.

Figure 12: IB predictions at 1/3 coral height. Streamwise velocity. Flow is from left to right. The grey scale gradation is an indication of streamwise velocity (m/s). Top: MRV data; middle: a snapshot of an unsteady laminar IB solution; bottom: steady state turbulent IB solution. Right column represents the same data reported in the left column without the geometry. Three major areas of application of the MRV-IB approach are envisioned: 1) validation and verification of computational tools, 2) design optimization and 3) reverse engineering. The validation of computational, predictive tools remains one of the pacing items in the use of computeraided engineering. Common practice is to perform preliminary studies on configurations that are somewhat simpler than the application of interest and for which experimental information is available. This step is necessary in building credibility in the numerical tools, but, certainly, it is not a sufficient warranty of reliability in the real-world scenario. The ability of both IB and MRV techniques to use

computer designed and rapid-prototyped models, respectively, allows for a testing mockup consistent with the device of interest and to generate data that can afterward be used to closely verify specific numerical tools or assumptions. The second area of interest is design optimization. For this problem, experiments are too expensive and time consuming and, therefore, CFD methods are the tools of choice. Conventional body-fitted CFD approaches have inherent difficulties in handling families of geometry which present large geometrical variability or have some parameters that can vary discontinuously: as an example, high-lift airfoil optimization when the number of components can be changed. The IB method represents an obvious solution for shape optimization problem from a geometrical modeling point of view. However, as discussed above, the accuracy of numerical predictions should be always verified, especially when, during an optimization procedure, unusual configurations arise. For this problem, MRV could be used as a posteriori tool that allows the investigation of a few selected configurations in more detail. By comparing MRV and IB, it is also possible to evaluate if certain configurations have been selected or discarded for the wrong reasons, i.e. due to inaccuracies in the modeling and not to lower figures of merit. The last application of the MRV-IB approach is in reverse engineering. In the third example showed in the paper, the flow around the coral, the starting point is a "device" which has certain features whose function requires further investigation. In the example presented, the construction of the RP model is based on a three-dimensional scanning of an existing coral skeleton. Once a digital representation (CAD model) is available, a rapid prototyping manufacturing process can be used to build a physical model, and the same CAD representation can be used to perform the simulations. The MRV and IB approaches are promising techniques to study flows in complex geometries: However, further work is required to fully establish their level of accuracy. Current research in MRV is related to improving the mean velocity measurements in turbulent flows and extending MRV capabilities to measure turbulent quantities (Saetran and Elkins, 2004). Another area that has received attention, although not discussed in this paper, is the use of MRV for unsteady, periodic flows (Markl et al. 2003). Current research on the numerical side has focused on the development of wall models for accurately predicting boundary layers on immersed boundaries not aligned with grid lines (Kalitzin et al. 2004). In addition, work is in progress to extend the approach for simulating conjugate heat transfer. In this case, the IB provides another direct advantage over body-fitted approaches, as the mesh within solid bodies, is automatically available. Initial results are very promising (Moreau et al. 2004).

REFERENCES Chang, S., Iaccarino G., Elkins C. J. (2004) Towards the study of hydrodynamics of coral reefs. CTR Annual Research Briefs, 121-132. Elkins, C.J, Markl, M., Pelc N., Eaton, J.K. (2003), "4D Magnetic Resonance Velocimetry for Mean Velocity Measurements in Complex Turbulent Flows," Experiments in Fluids 34:494-503. Fukushima E (1999) Nuclear magnetic resonance as a tool to study flow. Ann Rev Fluid Mech 31, 95-123 Kalitzin G., Medic G., Iaccarino G., Durbin P. (2004) Near wall behaviour of turbulence models and implications on adaptive wall functions. J. Comp. Physics, to appear. Haacke M; Brown R; Thompson M; Venkatesan R. (1999). Magnetic Resonance Imaging. New York: Wiley-Liss.

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Iaccarino G. Kalitzin G., Elkins C. J. (2003) Numerical and experimental investigation of the turbulent flow in a fibbed serpentine passage. CTR Annual Research Briefs, 118-128. Iaccarino G., Kalitzin G., Khalighi B. (2003) Towards and immersed boundary RANS flow solver. AIAA Paper 2003-0770 Iaccarino G. and Verzicco R. (2003) Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56-3, 331-347. Iacovides H. and Raisee M., (1999) Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blades. Int. J. Heat and Fluid Flow, 20, 320-328. Mittal R. and Iaccarino G. (2005) Immersed boundary method. Ann Rev Fluid Mech 37, 239-261. Markl M., Chan F., Alley M., Wedding K., Draney M., Elkins C. (2003) Time resolved three-dimensional phase contrast MRI (4D-Flow). J. Magn. Reson. Imaging 17, 499-506. Moreau S., Iaccarino G., Kalitzin G., (1004) Toward Conjugate Heat Transfer in Complex Geometries with an Immersed Boundary Cartesian Solver, ASME Paper HT-FED-2004-56834. Peskin C. S. (1972). Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis. Physiol., Albert Einstein Coll. Med., Univ. Microfilms. 378:72-30 Saetran, L. and Elkins, C.J. (2004). Private communication. Stanislas M, Kompenhans J and Westerweel J. (2000) Particle Image Velocimetry, Kluwer Academic Publishers, The Nederlands. Stark D; Bradley W. (1999). Magnetic Resonance Imaging. St. Louis, Mosby-Year Book. von Schulthess G; Hennig J. (1998). Functional Imaging. Philadelphia, Lippincott-Raven; pp. 261-390.

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Engineering TurbulenceModelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

LARGE-EDDY-SIMULATION USING THE IMMERSED

OF C O M P L E X BOUNDARY

FLOWS

METHOD.

R. Verzicco P o l i t e c n i c o di Bari, D I M e G a n d C E M e C , V i a Re D a v i d 200, 70125, Bari, Italia.

ABSTRACT

In this paper we will consider recent advances in the simulation of moderately high Reynolds number flows in complex geometric configurations. Although modern computers are experiencing an unprecedented growth in computing power, the numerical simulation of the above flows is still challenging owing to the handling of complex geometries and turbulence modeling that are the classical bottlenecks for the application of computational fluid dynamics (CFD) to industrially relevant problems. In this respect the immersed boundary (IB) method has shown to be a valid alternative for the treatment of complex geometries although additional issues must be addressed. This paper aims at describing the main techniques, showing some illustrative examples and discussing the main drawbacks and possible solutions.

KEYWORDS

Immersed boundary method, Complex geometries, Large-Eddy-Simulation, Turbulent flows, Industrial CFD.

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INTRODUCTION In computational fluid dynamics a lot of effort is made to simulate high Reynolds number flows with the aim of understanding the dynamics of turbulence that affects countless aspects of our everyday life. This is usually done by considering simple model problems prescribed in trivial geometries since this brings to a straightforward domain discretization into a regular grid and to accurate and efficient numerical algorithms. On the other hand when a flow is bounded by a complex geometry the computation becomes more involved and less efficient. This is due to the computational overhead introduced by the body fitted mesh which leads to a trade-off between high Reynolds numbers and geometrical complexity. In fact, the generation of a body-fitted mesh requires that starting from some geometrical description of the body, a surface grid is first produced and this is used as a boundary condition for a volume grid covering the domain of interest. Although several techniques are available for the mesh generation (structured or unstructured grids) some common features are that on the regularity properties of the grid (like skewness, aspect ratio and size distribution of the cells) depend the quality of the flow solution. It must be noted, in addition, that the use of curvilinear grids introduces a significant overhead in the per-node operation count and that the generation of a grid fitted to a complex three-dimensional object requires a highly trained operator for a number of hours that can easily exceed the time needed to obtain the flow solution. Unfortunately the majority of flows encountered in practical applications involve complex geometric configurations whose adequate treatment is one of the main obstacles for the application of CFD as a standard design tool. Within this scenario the advantages of a method capable of simulating flows in complex geometries using simple non-body-conformal meshes are evident and this consideration is the main motivation for the immersed boundary method. Since the seminal work by Peskin (1972) (and a less notorious one by Vieceli, 1969) numerical simulations of flows inside or around complex geometric configurations without resorting to curvilinear body conformal grids have become feasible and a valid alternative to standard methods. The original procedure consisted of approximating the boundary of the i m m e r s e d body by a sequence of elastic fibers whose endpoints were tracked in a Lagrangian way according to the fluid velocity computed on a Cartesian grid. Given the stiffness of each element it is possible to compute the force locally exchanged between fluid and boundary and therefore the forcing term to apply to the momentum equation for the fluid. This procedure was successfully employed for the simulation of heart beating of mammals (Peskin & McQueen, 1989) and, more in general, to the flow around deformable bodies (Zhu & Peskin, 2003). The same method could be extended to flow/rigid body interactions by making the boundary elements stiffer. This extension, however, introduced additional problems like the appropriate choice of the elastic properties of the elements that, when too soft, yield anyway a deformable boundary, when too rigid, make the system of equations stiff. Since the original papers several amendments have been proposed with the aim of improving the stability and the applicability of the immersed boundary method. These improvements range from the forcing of the governing equations (Goldstein, Handler & Sirovich 1993, Saiki & Biringen 1996, Mohd-Yusof 1997, Fadlun et al. 2000, Kim, Kim & Choi 2001) to the spatial distribution of the forcing (Peskin 1972, Beyer & Leveque 1992, Lai & Peskin 2000). Detailed descriptions of the different techniques are given in Iaccarino & Verzicco (2003) and Mittal & Iaccarino (2005) here

is suffices to note that most of the techniques lead to a spatial second-order-accurate solution. An important drawback of the immersed boundary method is that, as the Reynolds number increases it becomes more difficult to fulfill the near-wall resolution requirements. In fact, while using a body conformal grid it is relatively easy to cluster grid nodes in the wall normal direction, the same clustering by a Cartesian grid requires refinements in all directions. We will show, in particular, that within the Large-Eddy-Simulation (LES) modeling this implies that if NBF and NIB are, respectively, the number of nodes needed by a three-dimensional body-fitted LES at Reynolds number Re and NIB the same quantity for the IB method it results NIB/NBp ": Re~ Although IB simulations are considerably less expensive than the body-fitted counterparts this increase of resolution can not always be accounted by "brute force" grid refinements. This brings to the need for near-wall modeling (Piomelli & Balaras, 2002) which still is an open issue for standard body-fitted-grid simulations and even less consolidated for IB methods.

LES TURBULENCE

MODELING

For moderately high Reynolds number flows the governing equations are the filtered Navier-Stokes equations that for incompressible flows read: 0--7 + V . (tiff) = - V / ~ - V . ~ +

V2fi + f,

and

V . fi = 0.

(1)

is the filtered velocity, ~ = f i ~ - tiff the sub-grid-scale (SGS) stress tensor, /~ the filtered pressure and f the forcing term needed to impose the boundary conditions at the fluid/immersed body interface (see Iaccarino & Verzicco, 2003 and Mittal & Iaccarino, 2005 for specific expressions of f). The tensor ~ must be modeled and following the Smagorinsky procedure its deviatoric part can be parametrized as - Tr(~) = 2uTS

with

~z -- C L f I ~ I

and

I s I--- (2S" ~)1/2

(2)

A is the filter width and S is the large-scale rate-of-strain tensor that can be computed from the filtered velocity S = (Vfi + v~T)/2. The trace of the sub-grid-scale stress tensor does not need to be explicitly modeled since it can be incorporated in a modified pressure ~* = ~- Tr(~). The model parameter C can be either externally assigned according to the original Smagorinsky model (Smagorinsky, 1963) or computed as part of the solution following a dynamic procedure (Germano et al., 1991). The first possibility is less expensive in terms of CPU time even if the resulting SGS model turns out to be too dissipative close to the walls and needs to be damped by a "ad hoc" function (for example the van Driest damping). The dynamic procedure, on the other hand, implies an additional filtering over a "test filter" and has a computational cost of the order of 30% of the total CPU time; it yields, however, a number of advantages highly desirable in a numerical simulation. First of all the turbulence model does not rely on user defined parameters and C is computed from the resolved field. The turbulence model automatically switches off in the near-wall region or within laminar flow patches without the need of damping functions. In addition, the dynamic determination of C allows for some energy backscatter (negative values of C) at least as long as the total viscosity (molecular + turbulent) does not become negative.

20

The computation of the coefficient C usually requires a spatial averaging before it can be used for the computation of the turbulent viscosity. In simple flows (homogeneous turbulence or plane channels) there is always one or more homogeneous directions that can be used for the averaging of C. In contrast, for flows in complex three-dimensional geometries homogeneous directions are not available and an alternative procedure is needed. One possibility is to use a filtered C instead of its raw counterpart which is equivalent to the pointwise averaging of C among the closest neighbors. This makes sense if one considers that according to the Kolmogorov hypotheses, in the inertial range turbulence is always locally homogeneous. Results obtained by this procedure are those by Verzicco et al., (2000) and Verzicco et al., (2002) which are briefly described in the successive sections. A more elegant and effective way for averaging C is by a Lagrangian algorithm. A complete discussion of the model is given in Meneveau, Lund & Cabot (1996) here we only note that since the dissipation is advected according to the fluid velocity, a natural way for averaging C (which parametrizes the dissipation) is along a streamline. This is performed by a weighting function which gives more weight to recent times, smooths out sharp fluctuations and preserves the spatially local nature of the model. This procedure has been successfully used by Balaras (2004) for the turbulent flow over a wavy wall; some of his results will be commented later in this paper.

BOUNDARY

RECONSTRUCTION

A fundamental issue of the IB method is the imposition of the forcing f at the fluid/body interface. The need for an explicit computation of f depends on the particular technique, even if every IB method eventually needs a reconstruction of the solution around the immersed surface. In fact, consider figure 1 where the shaded surface is the immersed boundary; it is immediately evident that while the boundary condition must be imposed over the surface, the flow unknowns are located somewhere in the computational cell and their position does not coincide with that of the surface; this is true for colocated methods as well as for staggered methods in which every flow variable is discretized in a different position. Simple mono- two- and full three-dimensional reconstruction schemes have been proposed (Fadlun et al. 2000, Gilmanov, Sotiropoulos & Balaras 2003, Tseng & Ferziger 2003, Kim et al. 2001) with different degrees of precision and efficiency although all of them yielding a second-order accuracy. One general procedure consists of a preliminary tagging of the computational cells which classifies the nodes into external, internal and interface points. The tagging can be performed by a ray-tracing algorithm (O'Rourke, 1998) as discussed in Iaccarino & Verzicco (2003) which is a standard in computer graphics. From each interface node (point A in figure 1) the normal to the immersed surface is drawn and the intersection W is computed. A tetrahedron is then constructed with A and the three closest external nodes (B, C and D) such that the point W is contained inside the tetrahedron. Every flow variable q (velocity component, density, temperature, scalar concentration, etc.) is then computed in A in such a way that the same variable assumes the values in B, C and D computed from the governing equations without any forcing and the prescribed boundary condition in W. Within a linear reconstruction q assumes the form: q = a x + by + cz + d

(3)

21

with a, b, c and d determined by the conditions q = qB, qc, qD and qw, respectively, at the points B, C, D and W. Once the coefficients a, b, c and d are known from the above conditions the value of q in A is simply given by qA -- aXA + byA + CZA nt- d. If, instead of Dirichlet conditions, Neumann boundary conditions (Oq/On = Cw) are to be applied the wall condition q = qw is replaced by V q . n = Oq/Oxc~ + Oq/Oy~ + O q / O z " / = ac~ + b~ + c~ = Cw, where c~, ~ and ~ are the components of the normal n in W.

Figure 1: Sketch of one possible interpolation scheme for the velocity reconstruction at the immersed boundary.

We wish to stress that according to equation (3) the solution behavior in the near-wall region is linear by definition which implies, in other words, that the first external points must be within the viscous sublayer. This is certainly true for a laminar flow or for direct numerical simulations (DNS) of turbulent flows but it is unlikely to happen even for moderately high Reynolds number LES simulations. If the immersed boundaries are largely aligned with coordinate grid lines a brute force grid refinement can be sufficient to get enough wall resolution. This was done in Verzicco et al., (2002) although this strategy can not be pursued for any three-dimensional geometry. A possible alternative was proposed by Majumdar, Iaccarino & Durbin (2001) which used a quadratic interpolation instead of the linear one of equation (3). This avoided the linear flow behavior close to the immersed boundary but, on the other hand, needed a larger stencil for the solution reconstruction. Additional difficulties are introduced by quantities whose near-wall behavior is intrinsically nonlinear even with enough wall resolution. In LES this is the case of the turbulent viscosity UT that, for equilibrium flows, decreases with (n+) a power law, being n the wall-normal direction. In this case the correct flow behavior could be recovered by using in equation (3) q = (pT) 1/3 although, as noted by Balaras (2004), the evaluation of the test-filtered quantities in the vicinity of the immersed boundary is not straightforward since it requires either modified stencils or the inclusion of internal body points. On the other hand, even if the velocity components are reconstructed from the surrounding points, the turbulent viscosity is needed to compute the viscous fluxes and inaccurate boundary values can affect the external solution. This suggests that a near-wall modeling procedure can avoid possible inaccuracies and give substantial improvements in the affordable Reynolds numbers.

22

NEAR-WALL

MODELING

One problem in the application of LES to wall-bounded flows at realistic Reynolds numbers is that, since the most energetic flow structures must be explicitly solved, the near wall resolution is comparable to that of a direct numerical simulation (DNS). In particular, if 5+ is the wall lengthscale, assuming a structured body-fitted grid covering an object of dimension L in which r/ is the wall-normal direction and X and ~ the other two directions we have A X ,.~ 5+, A~ ~ 5+ and At] ~ r/, the latter indicating that a grid stretching is performed in order to refine the near-wall region and save computational points far from the object. Following Chapman (1979) we have L/5 + ,.., Ra ~ therefore the number of points per cubic L must scale as NBF "' Re 1"~ ln Re (Pope, 2000). On the other hand, if the grid is not body-fitted there is not a wall normal coordinate line and the grid spacing must be ~ 5+ in all three directions. In other words in the IB method every direction can be normal or parallel to some part of the body and the grid must be, on average, equally fine in every direction. This implies that the number of nodes covering a cubic L now behaves as N~B ~ Re 2~4 and the ratio with the body-fitted counterpart is NxB/NBF "' Re~ Re. It must be observed that this estimate applies only to the number of nodes and not to the cost of the simulation since LES on simple Cartesian meshes is considerably less expensive than on curvilinear grids. In addition, in the IB method a fraction of the nodes (typically 10-30%) falls inside the immersed body where the flow needs not be computed, thus reducing the count of the dynamically active nodes. Nevertheless it is clear that since NIB/NBF increases with Re, LES simulations which are already challenging over body-fitted meshes become eventually unfeasible for IB methods. One way to overcome this difficulty is to replace the near wall region with an appropriate wall model feeding the LES with approximate wall boundary conditions, thus avoiding the direct computation of the near-wall region (Piomelli & Balaras, 2002). The advantages of this approach are evident if one considers that far from the wall an adequate spatial resolution for LES requires that the grid spacing A be only a fraction of the integral length scale s (Jimenez, 2003). The reason is essentially that the resolved scales are required to carry most of the flow stresses and they are produced at most at dimensions one order of magnitude smalled than s Assuming then A ~ s implies that order of thousands nodes per cubic s would be enough to adequately simulate by LES a turbulent flow. It is worth noting that the total number of nodes needed by the simulation, either NBF or NIB, depends on the ratio L/s which in turn is a problem-dependent value; nevertheless even if very large it is independent of the Reynolds number thus making feasible the simulation of industrial flows. Details and comparisons for the most popular wall models currently used in LES can be found in the review by Piomelli & Balaras (2002), in the present paper we only describe one of them which has been used in combination with IB methods. Following Balaras, Benocci & Piomelli (1996) a boundary layer equation for the tangential velocity components can be written as:

c9--~ (~ + ~T) 0rlJ = Fi

with

Fi = - ~ +

Oxj + Ox-----~'

(4)

being r/the wall normal direction. This equation can be solved within a "layer" between the solid boundary and the "external" LES solution in such a way that the external solution of equation (4) becomes the new wall boundary condition for LES.

23

Wang ~: Moin (2002) applied several versions of the above model and among them Fi = 0 yields the simplest: the equilibrium stress balance model. The eddy viscosity YT is obtained by a mixing length model with near wall damping ~T -- Pnrl+(l- e-~+/A) 2, with n --0.4 and A - 19. rl+ - U/5 + is the distance from the wall in viscous units computed from the instantaneous local friction velocity. It must be noted that the calculation of ~T needs U+ which relies on the friction velocity ur. The latter, in turn, is obtained from equation (4) which contains PT. An iterative procedure is than required that stating from a tentative value of ur (usually the value at the previous time step) solves simultaneously for equation (4) and the definition of PT. 0.3

0.25 0.2

rl 0.15 0.1 0.05 0

~

0

i

0.2

f

0.4 U

i

0.6

Figure 2: Wall velocity profile as function of wall distance. U+ - 30,

i

0.8

1

Re = 3900 first g r i d p o i n t a t

R e - 300 first g r i d p o i n t a t r/+ - 5.

One difficulty in the application of the above model to the IB method is that equation (4) is solved for the tangential velocity components along the wall normal coordinate; if from one external node (say point B in figure I) the wall normal is drawn, this will not intersect any other computational node. In addition each node will have a velocity which is neither tangential nor parallel to the wall; additional interpolations are therefore required to apply equation (4) in the IB context. This has been done by Tessicini et al., (2002) where further details of this procedure can be found. The authors report that the computational cost of this model, including the additional interpolations is about I0~ of the total CPU time. Some results obtained by this model are given in the section of the results, here we want to comment on an interesting property of this model which is summarized by figure 2. In particular, the equilibrium stress balance model yields the logarithmic law of the wall for rl+ >> 1 and a linear profile for rl+ ~ I. This implies that if the first external node is within a small multiple of 6 + (the Reynolds number is small or if the flow has locally a reduced turbulence level) the model automatically returns a linear velocity profile which is the same as if the model were absent. If instead the first external node is at tens or hundreds of wall units the model returns a logarithmic velocity profile. In other words, this model has the advantage of automatically switching off in flow regions where it is not needed similarly to the dynamic sub-grid-scale model for LES. In the original application (Tessicini et al., 2002) equations (I) were solved down to the second external node while equation (4) was used to determine the velocity components at the first external point. An improvement to this procedure is to solve equation (I) down to the first external

24

Figure 3: Flow in the IC piston-cylinder assembly at Re = 2000, and 65 x 65 x 151 (0 x r x z) grid, dynamic Smagorinsky subgrid-scale turbulence model, a) Sketch of the grid and of the geometry, b) t = 7r/2, (crank angle 90 ~ projected velocity vectors, meridional plane, c) t = 7r/2, projected velocity vectors, 15 mm below the head. nodes and use equation (4) to compute the wall stress. This is equivalent to assign a slip-velocity at the immersed boundary which provides the LES simulation with the "correct" wall shear stress. This procedure seems to give better results with respect to the previous implementation although a full validation within the IB context has not been performed yet (F. Tessicini, Personal

Communication). EXAMPLES

This part of the paper is devoted to the description of some illustrative applications of the above mentioned techniques; in the next two sections the Reynolds number of the flows is low enough to allow for sufficient spatial resolution near the wall. The successive example describes simulations where the resolution is marginal and only the large alignment of the immersed boundaries with the grid makes the flow computation possible. In the last example the Reynolds number is so large that the numerical simulation becomes unfeasible without a near-wall turbulence model.

Flow in a Model IC P i s t o n - C y l i n d e r Assembly In this section the LES of the three-dimensional flow in an axisymmetric piston/cylinder assembly with a fixed valve is illustrated. The configuration is reported in figure 3a and experimental measurements (phase averaged mean and RMS radial profiles of axial velocity) are available (Morse, Whitelaw & Yianneskis, 1978) for the validation of the numerical results. In the experiment, the piston was externally driven so that the fluid flowed into the cylinder from outside during the downward piston motion and vice-versa when the piston moved up. Since the valve was fixed and a tiny annular gap was left open between the valve and the cylinder head, the compression phase is not included in the flow dynamics. The piston was driven by a simple harmonic motion at a speed of 200rpm ~ 21rad/s which for the present geometry yields a mean piston speed of Vp = 0.4m/s

25

Figure 4: Radial profiles of averaged axial velocity components at t = 0.2x (crank angle 36~ a), b) and c), respectively at sections 10 mm, 20 mm and 30 mm below the head. Symbols: Experiments Morse et al. (1978), ~ " present LES Simulation. (when averaged over half cycle). The Reynolds number of the flow based on Vp and on the piston radius is Re = 2000 in air. It is worth noting that although the piston has an half-cycle mean velocity of only V--p- 0.4m/s the fluid driven through the valve gap has velocities up to 20Vp and for the adequate description of this flow a sub-grid-scale turbulence model is mandatory. In figure 3bc snapshots during one instant of the oscillating cycle are given and the high threedimensionality of the flow can be appreciated from the vector plots in orthogonal sections. Radial profiles of axial velocity were obtained by phase averaging the fields over four cycles and then averaged in the azimuthal direction. Three profiles at different axial locations are shown in figure 4. The comparison with the experimental data shows that the LES results are always in agreement with the experiments. In Verzicco et al. (2000) velocity profiles in additional sections and RMS profiles of axial velocity are also reported consistently showing a very good agreement with the measurements. The same flow was LES simulated by Haworth & Jansen (2000) and they used an unstructured boundary fitted, deformable mesh; the quality of the results is comparable even though the immersed boundary technique resulted much less expensive. Flow in a Wavy Channel This application (Balaras, 2004) considers the fully developed turbulent channel flow with the bottom wall having the shape of a sinusoidal wave. The problem geometry is that of figure 5a with a value of 2a/,~ = 0.1 and a spanwise dimension of the domain equal to the streamwise length. Several grids have been considered by Balaras (2004) the finest of which uses 288 • 64 • 130 nodes in the streamwise, spanwise and cross-stream directions. The Reynolds number based on the length H - ~ and the bulk velocity Ub is Re = 6760. The flow dynamics is governed by the unsteady separation downstream of the hill top and the successive reattachment in the throat. An instantaneous snapshot through a vertical plane is shown in figure 5b where the spanwise vorticity evidences the formation of the thin shear layer in the ascending part of the bottom and a recirculation in the descending half. Mean profiles of streamwise and wall-normal velocity components and resolved turbulent kinetic energy are given in figure 6 for one representative section showing a substantial agreement with reference DNS data (Maa/J & Shumann, 1996). The same kind of agreement is observed for other sections as shown by Balaras (2004) where further details on the simulation technique and

26

Figure 5: a) Sketch of~the problem. ~b) Instantaneous snapshot of spanwise vorticity through a vertical cross section (y/h = 1.2). Figures adapted from Balaras (2004) reproduced with permission.

Figure 6: Mean vertical profiles at the streamwise location x/h = 1. a) stramwise velocity, b) wall-normal velocity, c) resolved turbulent kinetic energy, o DNS by Maa# & Schumann (1996), coarse (192 x 64 x 88) LES by Balaras ( 2 0 0 4 ) , fine (288 x 64 x 130) LES by Balaras (2004). Figures adapted from Balaras (2004) reproduced with permission. flow dynamics can be found. Similar results were obtained by Tseng & Ferziger (2003) on a similar problem with slightly different geometrical parameters and turbulence modeling. Both examples demonstrate that when the near wall resolution is sufficient a LES with an IB method yields results of the same quality as the standard body-fitted methods.

Flow Around a Model Road Vehicle The objective of this study is to investigate by LES the dynamics of the wake past a model road vehicle for which experimental data are available for comparison (Khalighi et al., 2001).

Figure 7: a~ Road-Vehicle Configuration and Computational Grid in the Symmetry Plane (only one every 4 grid-points are shown), b) Averaged streamwise velocity through the symmetry mid-plane at Re- I00000.

27

!

1.5

1.0

1.ot

0.5

o.5t ,

0.0

0'~. 5 ufUinlet

0.0

0.5 U~inlet

1.0

Figure 8: Streamwise velocity profiles in the wake. o 9Experiments Khalighi et al., (2001); . . . . LES at Re = 20000; 9LES at Re = 100000. The configuration is reported in figure 7; the simulations are performed on a Cartesian grid made up of 220•215 points over a domain 13.5H • 4.2H • 3.77H in the streamwise, vertical and spanwise direction respectively, being H the height of the body. The experimental Reynolds number based on the free-stream velocity and the model height is Re = 170000. Preliminary simulations were carried out assuming that the main features of the flow and the corresponding trends in the flow dynamics at the back of the body were independent of Reynolds number if this was sufficiently high. Accordingly, the Reynolds number of the numerical simulations was fixed at Re = 20000; it was observed that indeed the numerical simulations showed all the trends and the flow features observed in the experiments. However, some quantitative differences were present. For this reason, additional simulations have been performed at Re = 100000 showing a much better quantitative agreement with the experimental data. Quantitative results are reported in figure 8 in terms of time-averaged streamwise velocity profiles in two sections downstream of the base. The measurements are compared with two LES simulations performed at Re = 20000 and Re = 100000; the high Reynolds number simulations agree well with the experiments. The defect velocity as well as the length of the recirculation region are accurately captured. The low Reynolds number simulations agree qualitatively with the measurements but strongly overpredict the thickness of the bottom-wall jet. The high Reynolds number results have also been compared to the experiments in terms of drag coefficients; a value of Co = 0.291 was computed from the LES simulations, whereas Co = 0.3 was the corresponding measurement. Additional analysis were carried out to study the unsteady dynamics of the flow and the effects of drag reduction devices; they are reported and discussed in Verzicco et al., (2002). Flow A r o u n d a Hydrofoil In the present example (Tessicini et al., 2002) is it considered the flow around a hydrofoil of chord C and maximum thickness H. The geometry is that used by Blake (1975) which investigated the flow experimentally. The Reynolds number based on the hydrofoil chord is R e c = 2.15 x 106 while based on the maximum thickness is ReH = 1.02 • 105. Following Wang & Moin (2000) the simulation is performed over the rear 38% of the hydrofoil chord (figure 9), on a domain 0.5H • 41H • 16.5H respectively in the spanwise, cross-stream and streamwise directions with

28

Y

-2

-8

0

X

8

Figure 9: Instantaneous streamwise velocity contours in the wake of the hydrofoil trailing edge.

(Au p 0.08'

H

.

Ue

Ue

:I

'

Urm U---e

Figure I0: Mean and rms U = v/u~ + u~ profiles normalized by the local external velocity Ue. Yw is the local y coordinate of the immersed surface, a) full LES on body-fitted mesh by Wang ~: Moin (2000), present IB solution without wall model. Sections at x/H - -3.125, -2.125, -1.625, -1.12, -0.625, 0. b) o experiments by Blake (1975), full LES by Wang & Moin (2000), LES with wall model by Wang & Moin (2000) present IB results with wall model. Sections as in a). c) the same as b) for the rms profiles. Sections as in a). a grid of 49 x 206 x 418 nodes. It is worth noting that although the grid is stretched around the hydrofoil surface the average wall grid spacing is about 60 wall units. This spacing does not allow the assumption of linear velocity profile in the region between the first external node and the immersed surface. Consistently, the simulation without wall model (although a dynamic sub-grid-scale model is activated) completely mispredicts the separation point and the velocity profiles (figure 10a). This test is particularly severe for the IB method since the flow separates over the curved surface in a point which is determined by the balance of the wall viscous stresses and the external pressure gradient. Since none of the coordinate lines is aligned with the hydrofoil surface the computation of the correct boundary layer dynamics is particularly challenging. In figures 10b the average velocity profiles are reported together with the experimental results and analogous data produced by body-fitted LES simulations. It can be noted that the introduction of a wall model yields a substantial improvement to the results whose agreement with the experiments is of the same quality as the body fitted results. The agreement with the experiments is less satisfactory for the rms profiles (figure 10c) although also the body fitted simulations are affected by some mismatch with experiments.

29

CONCLUSIONS The aim of the present paper was to show, by describing some illustrative examples, the recent advances in the field of LES in the context of IB methods. Owing to lack of space the paper could not discuss related topics like Cartesian Methods or applications to compressible flows. The reader is referred to the reviews by Iaccarino &: Verzicco (2003) and Mittal ~: Iaccarino (2005) for further reading. Before concluding the paper, however, we would like to shortly discuss one more point about the near-wall refinement. As shown by Iaccarino &: Verzicco (2003) some additional wall refinement can be obtained by immersed grids, i.e. by local fine grid patches applied over a coarser grid, the former being placed at the immersed body/fluid interface. This has shown to work very well in the context of RANS simulations since local near-wall refinements were possible without the node count increase typical of the structured grids. Unfortunately the same procedure could not be applied to LES simulations since preliminary tests have shown that the discontinuous grid spacing produced by the immersed grids yielded spurious stresses which in turn disrupted the turbulent viscosity computation (see figure 7 of Iaccarino ~ Verzicco, 2003). A possible solution was indicated in a recent paper by Mahesh, Costantinescu ~ MoAn (2004) who showed a conservative method for LES on unstructured grids; although this context is basically different from that of the structured grids used in the IB methods this might allow to combine immersed grids and IB methods in such a way that the conservation requirements of LES can be satisfied.

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30

JIMI~NEZ, J. (2003) Turbulence. Perspectives in Fluid Dynamics, Batchelor, Moffatt & Worster Eds. LAI, M.C. AND PESKIN C.S. (2000) An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys., 160, 705-719. MAA~, C. AND SCHUMANN, U. (1996) Direct numerical simulation of separated turbulent flow over a wavy boundary Flow simulations with high performance computers. Notes on numerical fluid mechanics., Hirschel Ed., 52, 227-241. MAHESH, K., COSTANTINESCU, G. AND MOIN, P. (2004) A numerical method for large-eddy simulation in complex geometries. J. Comput. Phys., 197, 215-240. MAJUMDAR, S., IACCARINO, G. AND DURBIN, P.A. (2001) RANS solver with adaptive structured non-conforming grids. Annual Res. Briefs, Center for Turbulence Research, 353-364. MCQUEEN, D.M. AND PESKIN, C.S. (1989) A three-dimensional computational method for blood flow in the heart: (II) contractile fibers. J. Comput. Phys., 82, 289-297. MENEVEAU C., LUND, T.S. AND CABOT, W.H. (1996) A Lagrangian dynamic sub-grid-scale model of turbulence. J. Fluid Mech. 319, 353-385. MITTAL, R. AND IACCARINO, G. (2005) Immersed Boundary Methods. Annu. Rev. Fluid Mech. 37. MOHD-YOSUF J. (1997) Combined immersed-boundary/B-spline methods for Simulations of flow in complex geometries. Annual Research Briefs, Center for Turbulence Research, 317-328. MORSE, n. P., WHITELAW, J. H. ~ YIANNESKIS, M. (1978) Turbulent flow measurements by Laser Doppler Anemometry in a motored reciprocating engine. Report FS/78/24 Imperial College, Dept. Mesh. Eng. O'ROURKE, J. (1998) Computational Geometry in C Cambridge University Press. PESKIN, C.S. (1972) Flow Patterns Around Heart Valves: A Digital Computer Method for Solving the Equations of Motion. Ph.D. thesis, Physiology, Albert Einstein College of Medicine. University Microfilms 72-30, 378. PIOMELLI, U. AND BALARAS, E. (2002) Wall-layer models for Large-Eddy-Simulations. Annu. Rev. Fluid Mech. 34, 349-374. POPE, S.B. (2000) Turbulent Flows Cambridge University Press. SAIKI, E. M. AND BIRINGEN S. (1996) Numerical Simulation of a Cylinder in Uniform flow: Application of a Virtual Boundary Method. J. Comput. Phys., 123, 450-465. SMAGORINSKY, J. (1963) General circulation experiments with the primitive equations. Monthly Weather Rev., 91 (3), 99-164. TESSICINI, F, IACCARINO, G., FATICA, M., WANG, M. AND VERZICCO R. (2002) Wall modeling for large-eddy-simulation using an immersed boundary method CTR, Ann. Res. Briefs, 2002, 181-187. TSENC, Y. H. AND FERZIGER J.H. (2003) A ghost-cell immersed boundary method for flow in complex geometry. J. Comput. Phys., 192, 593-623. VERZICCO, R., MOHD-YUSOF, J., ORLANDI, P. AND HAWORTH, D. (2000) Large-Eddy Simulation in Complex Geometric Configurations Using Boundary Body Forces. AIAA J., 38, 427-433. VERZICCO, R. FATICA, M., G. IACCARINO, P. MOIN AND B. KHALIGHI (2002) Large Eddy Simulation of a Road Vehicle with Drag Reduction Devices. AIAA J., 40, 2447-2455. VIECELI, J.A., (1969) A method for including arbitrary external boundaries in the MAC incompressible fluid computing technique. J. of Comp. Phys., 4, 543-551. WANG, M. AND MOIN, P., (2002) Dynamic wall modeling for LES of complex turbulent flows. Phys. of Fluids, 14, 2043-2051. ZHU, L. AND PESKIN, C.S., (2003) Interactions of two filaments in a flowing soap film. Phys. of Fluids, 15, 128-136.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

Transition Modelling for General Purpose CFD Codes

F.R. Menter l, R. Langtry l, S. V61ker2 and P.G. Huang 3 1ANSYS-CFX Germany, Software Development Dept., 83624 Otterfing, Germany 2 GE Global Research, One Research Circle, Niskayuna, NY 12309, USA 3 University of Kentucky, Lexington, Kentucky 40506-010

ABSTRACT The paper addresses modelling concepts based on the RANS equations for laminar-turbulent transition prediction in general-purpose CFD codes. Available models are reviewed, with emphasis on their compatibility with modem CFD methods. Requirements for engineering transition models suitable for industrial CFD codes are specified. A transition model, which satisfies most of the specified requirements is described, including results for a variety of different complex applications. KEYWORDS Laminar-turbulent transition, intermittency, local formulation, turbulence modelling, transport equation, SST model.

INTRODUCTION In the last decades, significant progress has been made in the development of reliable turbulence models, which allow the accurate simulation of a wide range of fully turbulent engineering flows. The efforts by different groups have resulted in a spectrum of models, which can be used for different applications, while balancing the accuracy requirements and the computational resources available to a CFD user. However, the important effect of laminar-turbulent transition is not included in the large majority of today's engineering CFD simulations. The reason is that transition modelling does not offer the same wide spectrum of CFD-compatible model formulations as available for turbulent flows, although a large body of publications is available on the subject. There are several reasons for this unsatisfactory situation. The first is that transition occurs through different mechanisms in different applications. In aerodynamic flows, transition is typically a result of a flow instability (Tollmien-Schlichting waves or cross-flow instability), where the resulting exponential growth eventually results in a non-linear breakdown to turbulence. In turbomachinery applications, the main transition mechanism is bypass transition (Morkovin, 1969, Mayle & Schulz, 1997) imposed on the boundary layer by high levels of

32 turbulence in the freestream, coming from the upstream blade rows. Another important transition mechanism is separation-induced transition (Mayle, 1996), where a laminar boundary layer separates under the influence of a pressure gradient and transition develops within the separated shear layer (which may or may not reattach). Finally, an already turbulent boundary layer can re-laminarize under a strong favourable pressure gradient (Mayle, 1991). While the importance of transition phenomena for aerodynamic and heat transfer simulations is widely accepted, it is difficult to include all of these effects in a single model. The second complication arises from the fact that the conventional (RANS) averaging procedures do not lend themselves easily to the description of transitional flows, where both, linear and non-linear effects are relevant. RANS averaging eliminates the effects of linear disturbance growth and is therefore difficult to apply to the transition process. While methods based on the stability equations, like the e n method of Smith & Camberoni (1956) and van Ingen (1956) avoid this limitation, they are not compatible with general-purpose CFD methods as typically applied in complex geometries. The reason is that these methods require a priori knowledge of the geometry and the grid topology. In addition, they involve numerous non-local operations, which are not easily implemented into today's CFD methods (Stock & Haase, 2000). This is not to argue against these models, which are an essential part of the desired "spectrum" of transition models required for the vastly different application areas and accuracy requirements. Much like in turbulence modelling, it is important to develop engineering models, which can be applied in the day-to-day operation by design engineers on varying geometries. Closer inspection shows that hardly any of the current transition models is CFD-compatible. Most formulations suffer from non-local operations, which cannot be carried out (with reasonable effort) in a general-purpose CFD code. It has to be understood that modem CFD codes do not provide the infrastructure of computing integral boundary layer parameters, or allow the integration of quantities along the direction of external streamlines. Even if structured boundary layer grids are used (typically hexahedra or prism layers), the codes are based on data structures for unstructured meshes. The information on a body-normal grid direction is therefore not easily available. In addition, most industrial CFD simulations are carried out on parallel computers using a domain decomposition methodology. This means in the most general case that boundary layers are typically split and computed on different processors, prohibiting any search or integration algorithms. Furthermore, for general purpose CFD applications, the grid topology relative to the surfaces is not known a priori, as the user has the freedom to freely choose both, geometry and grid topologies. The main requirements for a fully CFD-compatible transition model are therefore: 1. Allow the calibrated prediction of the onset and the length of transition. 2. Allow the inclusion of the different transition mechanisms. 3. Be formulated locally (no search or line-integration operations). 4. Avoid multiple solutions (same solution for initially laminar or turbulent boundary layer). 5. Do not affect the underlying turbulence model in fully turbulent regimes. 6. Allow a robust integration with similar convergence as underlying turbulence model. 7. Be formulated independent of the coordinate system. Considering the main classes of engineering transition models (stability analysis, correlation based models, low-Re models) one finds that most of these methods lack one or the other of the above requirements. The only formulations, which have historically been compatible with CFD methods, are low-Re models (Jones & Launder, 1973, Rodi & Scheuerer 1984). However, they typically suffer from the close interaction of the transition capability and the viscous sublayer modelling, which prevents an independent calibration of both phenomena (Savill 1993, 1996). In addition, low-Re models can at best

33 be expected to simulate bypass transition, which is dominated by diffusion effects. From a global perspective (without accounting for the differences between different models in the same group), standard low-Re models rely on the ability of the wall damping terms to also capture some of the effects of transition. Realistically, it would be surprising if models calibrated for viscous sublayer damping would faithfully reproduce the many effects of transitional flows. It is understandable that models using damping functions based on the turbulent Reynolds number have some transition characteristics. Nevertheless, the effect is best described as "pseudo-transition", as it was never actually built into the model. However, there are several models, where transition prediction was considered during model calibration (Wilcox, 1994, Langtry & Sjolander, 2002, Waiters & Leylek, 2002). It is interesting to note that several of these models use the vorticity (or strain-rate) Reynolds number Rev as an indicator for estimating the state of the laminar boundary layer. Nevertheless, these model formulations are based on a close connection of the sublayer and the transition calibration. Recalibration of one functionality also changes the performance of the other. It is therefore not possible to introduce additional experimental information, without a substantial re-formulation of the entire model. This operation can only be performed reliably by the model developer (or experts on model formulation). More complex models for transitional flows have been developed by Steeland & Dick (1996) and Lardeau et al. (2004). These models do however require a separate transition onset criteria, which is typically not formulated locally. The engineering alternative to low-Re models are correlation-based formulations like those of AbuGhannam & Shaw (1980), Mayle (1991) and Suzen et al. (2000). They typically correlate the transitional (momentum thickness) Reynolds number to local freestream conditions, like turbulence intensity and pressure gradient. These models allow for an easy calibration, even by non-experts in turbulence modelling and are often sufficiently accurate to capture the major effects. In addition, correlations can be developed for the different transition mechanisms, ranging from bypass to natural transition to crossflow instability to separation induced transition. The main shortcoming of these models lies in their inherently non-local formulation. They typically require information on the integral thickness of the boundary layer and the state of the flow outside the boundary layer. While these models have been used successfully in special-purpose turbomachinery codes, the non-local operations have precluded their implementation into general-purpose CFD methods. Transition simulations based on linear stability analysis, like the e n method, are the lowest closure level, where the actual instability of the flow is simulated. In the simpler models described above, the physics is introduced through the calibration of the model constants. However, even the e n method is not free of empiricism, as the n-factor is not universal and depends on the wind tunnel or freestream environment. The main obstacle to the use of the e n model is however that the required infrastructure is typically very complex. The stability analysis is often based on velocity profiles obtained from highly resolved boundary layer codes, which are coupled to the pressure distribution of the RANS solver. The output of the boundary layer method is then transferred to a stability method, which then provides information back to the turbulence model in the RANS solver. The complexity of this set-up is mainly justified for special applications where the flow is designed to remain close to the stability limit for drag reduction, like laminar airplane wing design. LES and DNS are suitable tools for transition prediction (e.g. Durbin et al. 2002), although even there, the proper specification of the external disturbance level and structure poses substantial challenges. These methods are far too costly for engineering applications, and are currently used mainly as research tools and substitutes for controlled experiments. Despite its complexity, transition should not be viewed as outside the range of RANS methods. In many applications, transition is enforced within a narrow area of the flow, by strong geometric disturbances, pressure gradients and/or flow separation. Even relatively simple models can capture these effects with sufficient engineering accuracy. The challenge to a proper engineering model is

34 therefore mainly the formulation of models, which are suitable for implementation into a general RANS environment. The present authors have recently developed a correlation-based transition model, built on transport equations, using only local variables. The central idea behind the model has been described in Menter et al. (2002). The major numerical and modelling deficiencies associated with that prototype model have been eliminated by Menter et al. (2004) and a wide range of turbomachinery-related flow problems has been computed by Langtry et al. (2004). The model has since been extended to aerodynamic flows (Langtry & Menter, 2005) and is now run within the software package CFX-5, as well as the GE in-house code Tacoma on numerous industrial applications. The model satisfies all requirements given above, except for the last o n e - coordinate independence. This is a consequence of the fact that transition correlations are based on non-Galilean invariant parameters, like the turbulence intensity Tu. As boundary layer transition is always relative to walls, this is only an issue if multiple moving walls exist in a single computational domain. Efforts are underway for eliminating this restriction. The model given in Menter et al. (2004) and Langtry et al. (2004) has been developed in a joint project between GE Global Research, ANSYS-CFX and the University of Kentucky. The model consists of two components. The first is the genetic infrastructure provided by two transport equations, which link the CFD code to experimental correlations. The second component are the correlations themselves. The innovation lies in the genetic infrastructure, which allows a direct coupling of general purpose CFD method with experimental transition data. The entire infrastructure of the formulation is given in Menter et al. (2004). The correlations were partly built on internal data and are not public domain. As the interfaces for the transitional correlations are clearly defined, other groups can use their own correlations as available for their application.

THE STRAIN

RATE

REYNOLDS

NUMBER

One of the central variables in the formulation of transport equations for transition prediction is the strain-rate (or sometimes vorticity) Reynolds number: Re v (x , y) : py2 S #

(1)

where y is the wall distance, p is the density,/t is the molecular viscosity and S is the absolute value of the strain rate. The importance of Rev lies in the relation of its maximum value inside the boundary layer to the momentum thickness Reynolds number Reo, of Blasius (or more generally Falkner-Skan) profiles (Menter et al. 2002): Re v (x,

Y)max

~"

Re o ( x )

(2)

where y is the location where Rev has its maximum. The function Re,., can be used on physical reasoning, by arguing that the combination of y2S is responsible for the growth of disturbances inside the boundary layer, whereas v : At/p is responsible for their damping. As y2S grows with the thickness of the boundary layer and v stays constant, transition will take place once a critical value of Re v is reached. The connection between the growth of disturbances and the function Re v was shown by Van Driest and Blumer (1963) in comparison with experimental data. The models proposed by Wilcox (1994), Langtry & Sjolander (2002) and Walters & Leylek, (2002) use Revin physics-based formulations of transition models. These models appear superior to conventional low-Re models, as they implicitly contain information of the thickness of the boundary layer. Nevertheless, the close integration of viscous sublayer damping and transition prediction does not allow an independent calibration of both sub-models.

35 In an alternative approach, Menter et al. (2002), Menter et al. (2004), Langtry et al. (2004) and Langtry & Menter (2005) proposed a combination of the strain-rate Reynolds number with experimental transition correlations using standard transport equations. Due to the separation of viscous sublayer damping and transition prediction, the new method has provided the flexibility for introducing additional transition effects with relative ease. Currently, the main missing extensions are cross-flow instabilities and high-speed flow correlations, which do not pose any principle obstacle. The concept of linking the transition model with experimental data, has proven an essential strength of the model, which is difficult to achieve with closures based on a physical modelling of these diverse phenomena.

THE 7-Reo MODEL As the model solves a transport equation for the intermittency, 7, and the transitional momentum thickness Reynolds number, Re| the model was named 7-Re| model. As it is described in detail in Menter et al. (2004), the equations are given here in compact form for completeness. The transport equation for the intermittency, 7; reads:

O(pr___))+ at

~(pU jr) Oxj

=

py, _ Ey, -I- ey2 -- E~,2 q- ~xj

~ q-

(a)

The transition sources are defined as follows:

eyl -- Flength Ca l P a [~tFonset ]ca;

E T1 - cel eyl ~r

(4)

where S is the strain rate magnitude. Flengthis an empirical correlation that controls the length of the transition region. The destruction/relaminarization sources are defined as follows"

Pr2 =

C a 2 P~"~ ~l~'turb "

E 7"2 = C e 2 PT2 7

(5)

where f~ is the vorticity magnitude. The transition onset is controlled by the following functions: Rev = PY2----~S; RT= pk /a /.tO m

Fonset 1 m

Re

v

.

....

2.193 9Re ec

,2 = min (max

(6) .....

.....

Fo.... 3 = m a x ( 1 - ( FRor ~\ -3~, 0 - ./.;. ~ j .... =max(Fo .... 2 - Fo.... 3,0); F,,rb = e - ( ~ l '

(8)

36 Re0c is the critical Reynolds number where the intermittency first starts to increase in the boundary layer. This occurs upstream of the transition Reynolds number, Re0~, and the difference between the two must be obtained from an empirical correlation.

Both the

Flength and

Re0c correlations are

functions of R e ~ . The constants for the intermittency equation are:

Cel : 1.0;

Ca1 = 2.0;

Ce2 :

C,~ ----0.5;

Ca2 = 0.06;

50;

Crr = 1.0;

(9)

A modification for separation-induced transition is:

/ L Re)J

Ysep = min 2. max

v 3.235 Reoc

-- 1 , 0

F r...... h ,2

/

F~" Freattach : e

(Td-;

"7"ee : max (?', ?'sep)

(lO)

The boundary condition for 7'at a wall is zero normal flux while for an inlet 7'is equal to 1.0. ,.,.,

The transport equation for the transition momentum thickness Reynolds number, R e ~ , reads:

cq~('o ~tff" e at ) +

OXg

-:Pot+

~x j

O'~ (fl "~ fit ) Ox j

(l l)

The source term is defined as follows" Pot = Cot 7P

(Re ot -P-,e ot X1 9O- Fo,);,

: minlmaxlkee''41~

o 1,,2

t =

500 /2 pU

2

1o

15 0 BL" 8 = 5 0 ~ y 98eL 0BL = Re~________~B. 8BL = -7pU

(12)

(13) (14)

U

pO)y 2 Re o) = ~ " /z

_( R% ) 2 ~.lE+5)

Fwa~e = e

(15)

era-2.0

(16)

The model constants for the Re~ equation are: ca - 0 . 0 3 "

The boundary condition for R e ~ at a wall is zero flux. The boundary condition for Reo~ at an inlet should be calculated from the empirical correlation based on the inlet turbulence intensity. The model contains three empirical correlations. Re|

is the transition onset as observed in

experiments. It is used in Eq. 12. Flength is the length of the transition zone and goes into Eq. 4. Re| the point where the model is activated in order to match both, Re|

and Flength, it goes into Eq. 7.

is

37 Reot =

f(Tu,/],);

Flength =

f(Reot ~

Re~ =

f(ff.eot )

(17)

The first empirical correlation is a function of the local turbulence intensity, Tu, and the Thwaites' pressure gradient coefficient ~,0 defined as: ~0 = (02/v)dU/ds

(18)

where dU/ds is the acceleration in the streamwise direction. The transition model interacts with the SST turbulence model (Menter, 1994), as follows: (19)

= Y~#Pk" /gk = min(max(7'~g,0.1),l.0)Dk

(20)

-(',/' Ry = ,oy~,,~ ; F3 =e /t

k~2o) . F~ -max(Florig,F3)

(21)

where Pk and Dk are the original production and destruction terms for the SST model and Florig is the original SST blending function. Note that the production term in the o~equation is not modified. The rational behind the above model formulation is given in detail by Menter et al. (2004). In order to capture the laminar and transitional boundary layers correctly, the grid must have a y+ of approximately one. All simulations have been performed using CFX-5 with a bounded second order upwind biased discretisation for the mean flow, turbulence and transition equations, except for some of the flat plate cases, which have been computed with the boundary layer code of the University of Kentucky.

TESTCASES The remaining part of the paper will give an overview of some of the public-domain testcase which have been computed with the model described above. This naturally requires a compact representation of the simulations. The cases are described in more detail in Menter et al. (2004), Langtry et al. (2004) and Langtry & Menter (2005), including grid refinement and sensitivity studies.

Flat Plate Testcases Testcases presented are the ERCOFI'AC T3 series of flat plate experiments and the Schubauer and Klebanof flat plate experiment, all of which are commonly used as benchmarks for transition models. The three cases (T3A-, T3A, and T3B) have zero pressure gradients with different freestream turbulence levels corresponding to transition in the bypass regime. The Schubauer and Klebanof (S&K) test case has a low free-stream turbulence intensity and corresponds to natural transition. The T3C4 test case consists of a flat plate with a favourable and adverse pressure gradient imposed by the opposite converging/diverging wall. It is used to demonstrates the transition models ability of

38 predicting separation induced transition and the subsequent reattachment of the turbulent boundary layer.

Figure 1: Results for flat plate test cases with different freestream turbulence levels and pressure gradients (FSTI- Freestream Turbulence Intensity). Figure 1 shows the comparison of the model prediction with experimental data for all computed cases. In all simulations, inlet turbulence levels were specified to match the experimental turbulence intensity and its decay rate. The agreement with the data is generally good, considering the diverse nature of the physical phenomena computed, ranging from bypass transition to natural transition to separationinduced transition. Turbomachinery Test Cases Zierke & Deutsch compressor cascade

For the present test case (Zierke & Deutsch, 1989), transition on the suction side occurs at the leading edge due to a small leading edge separation bubble on the suction side. On the pressure side, transition

39 occurs at about mid-chord. shown in Figure 2.

The intermittency contours and the wall shear stress (cf) distribution are

Figure 2: Intermittency contours (left) and cf-distribution against experimental data (fight for the Zierke & Deutsch compressor. There appears to be a significant amount of scatter in the experimental data, however, in principal the transition model is predicting the major flow features correctly (i.e. fully turbulent suction side, transition at mid-chord on the pressure side). One important issue to note is the effect of stream-wise grid resolution on resolving the leading edge laminar separation and subsequent transition on the suction side. If the number of stream-wise nodes clustered around the leading edge is too low, the model cannot resolve the rapid transition and a laminar boundary layer on the suction side is the result. For the present study 60 streamwise nodes where used in between the leading edge and the x/C equal to 0.1 location. Von-Karman Institute turbine cascade

The surface heat transfer for the transonic VKI MUR 241 (FSTI = 6.0%) and MUR 116 (FSTI = 1.0%) test cases (Arts et al. 1990) is shown in Figure 3. The strong acceleration on the suction side for the MUR 241 case keeps the flow laminar until a weak shock at mid chord, whereas for the MUR 116 case the flow is laminar until fight before the trailing edge. Downstream of transition there appears to be a significant amount of error between the predicted turbulent heat transfer and the measured value. It is possible that this is the result of a Mach number (inlet Mach number Mainlet=0.15, Maoutlet=l.089) effect on the transition length (Steelant and Dick, 2001). At present, no attempt has been made to account for this effect in the model. It can be incorporated in future correlations, if found consistently important. The pressure side heat transfer is of particular interest for this case. For both cases transition did not occur on the pressure side, however, the heat transfer was significantly increased for the high turbulence intensity case. This is a result of the large freestream levels of turbulence which diffuse the laminar boundary layer and increase the heat transfer and skin friction. From a modeling standpoint the effect was caused by the large freestream viscosity ratio necessary for MUR 241 to keep the turbulence intensity from decaying below 6%, which is the freestream value quoted in the experiment. The enhanced heat transfer on the pressure side was also present in the experiment and the effect appears to be physical. The model can predict this effect, as the intermittency does not multiply the eddy-viscosity but only the production term of the k-equation.

40

Figure 3: Heat transfer for the VKI MUR241 (FSTI = 6.0%) and MUR116 (FSTI = 1.0%) test cases

RGW Compressor Cascade The RGW annular compressor (Schulz and Gallus, 1988) features a fully three-dimensional flow, including sidewall boundary layers originating upstream of the blade. This flow topology poses a major challenge to standard correlation-based transition models, as complex logic would be required to distinguish between the different boundary layers. Figure 4 shows a comparison of the simulations on the suction side of the blade with an experimental oil-flow picture. For comparison, a fully turbulent flow simulation is also included. It can be seen that the transition model captures the complex flow topology of the experiments in good qualitative agreement with the data. A comparison between the transition model and the fully turbulent simulation shows the strong influence of the laminar flow separation on the sidewall boundary layer separation. The flow separation on the shroud is significantly reduced by the displacement effect of the separation bubble in the transitional simulation. As a result, the loss coefficient, Yp=0.19, in the fully turbulent simulation is much higher than the experimental value of Yp=0.097. The simulation with the transition model gives a value of Yp=0.11 in much closer agreement with the experiment.

Figure 4: Fully turbulent (left) and transitional (fight) skin friction on the suction side of the 3D RGW compressor cascade compared to experimental oil flow visualization (middle). ( Y p - loss coefficient).

41 More turbomachinery related testcases can be found in Langtry et al. (2004), including an unsteady rotor-stator interaction simulation. Wind Turbine Test Case

The testcase geometry is a 2D airfoil section, as typically used for GE wind-turbine blades. It operates in a low FSTI environment with a turbulence intensity of only around 0.1% at the leading edge. As a result, natural transition occurs on both the suction and pressure surfaces. The inlet value for the co in this application was choosen to match the experimental transition location at 0 ~ All other angles of attack have been computed with the identical settings. For a detailed discussion see Langtry et al. (2004).

Figure 5: Predicted transition location (left) and drag coefficient (fight) as a function of angle of attack for a wind turbine airfoil in comparison with experiments and the en-method used in XFOIL. The transition locations vs. angle of attack for the present transition model are shown in Figure 5 (fight). Wind tunnel results and predictions XFOIL (v6.8) based on an en method are plotted for comparison. The experimental data were obtained using a stethoscope method. The current model captures the dependence of the transition location on the angle of attack in very good agreement with the data.The effct of the transition model is clearly visible also in the drag coefficient Figure 5 (fight). Numbers could not be provided on the y-axis, due to data confidentiality. 3D NREL Wind Turbine Simulation

Simulations have been carried out for the NREL wind turbine (Simms et al., 2001). This is a notoriously difficult testcase to compute with CFD and no attempt can be made to cover the complexity of a comparison of CFD and experimental data. However, in the simulation of this flow, it turned out that substantial differences were observed between fully turbulent and transitional results at severe stall conditions. Figure 6 shows the shaft torque in comparison with experiments (left) and the flow topologies computed for fully turbulent and transitional settings.

42

Figure 6: Shaft torque at different wind speeds (left). Flow topology on suction side for fully turbulent and transitional simulations (fight)

At a wind speed of 20 m/s, the flow topology computed with the fully turbulent and the transitional approaches are very different. This results in a 80% change in output torque. The lower output torque appears to be the result of a laminar separation in the leading edge region of the suction side of the blade. The transitional simulation is in much closer agreement with the experimental data.

Aeronautical Test Cases Transition in aeronautical flows is typically a result of Tollmien-Schlichting waves or a crossflow instability. The current model does presently not include correlations for crossflow instabilities. It does however account for natural transition including pressure gradients. For more details on the testcases see Langtry and Menter (2005).

McDonald Douglas 30P-30N Flap The McDonald Douglas 30P-30N flap configuration was originally a test case for the High-Lift Workshop/CFD Challenge that was held at the NASA Langley Research Center in 1993 (Klausmeyer and Lin, 1997). It is a very complex test case for a transition model because of the large changes in pressure gradient and the local freestream turbulence intensity around the various lifting surfaces. The experiment was performed in the Langley Low Turbulence Pressure Tunnel and the transition locations were measured using hot films on the upper surface of the slat and flap and on both the upper and lower surfaces of the main element. The skin friction was also measured at various locations using a Preston tube (Klausmeyer and Lin, 1997). For the present comparison the Reynolds number Re=9xl06 and an angle of attack ct=8 ~ was selected. The freestream conditions for k and co were selected to match the transition location at the suction side of the slat. The other transition locations are an outcome of the simulation.

43

Figure 7: Contour of turbulence intensity (Tu) around the McDonald Douglas 30P-30N flap as well as the measured (Exp.) and predicted (CFD) transition locations (x/c) as a function of the cruise-airfoil chord (c = 0.5588 m). Also indicated is the relative error between the experiment and the predictions.

A contour plot of the predicted turbulence intensity around the flap is shown in Figure 7. Also indicated are the various transition locations that were measured in the experiment (Exp.) as well as the locations predicted by the present transition model (CFD). In the computations the onset of transition was judged as the location were the skin friction first started to increase due to the production of turbulent kinetic energy in the boundary layer. In general the agreement between the measured and predicted transition locations is very good. The largest error was observed on the lower surface of the main element were the predicted transition location was too far downstream by approximately 6% of the cruise-airfoil chord. DLR F-5 wing The DLR F-5 geometry is a 20 ~ swept wing with a symmetrical airfoil section that is supercritical at a freestream Mach number of 0.82. The experiment was performed at the DLR by Sobieczky (1994) and consists of a wing mounted to the tunnel sidewall (which is assumed to have transitioned far upstream of the wing). At the root the wing was designedto blend smoothly into the wall thus eliminating the horseshow vortex that usually develops there. The experimental measurements consist of wing mounted static taps at various span wise locations and flow visualization of the surface shear using a sublimation technique.

44

Figure 8:DLR-F5 wing with transition. Simulations (left); Experiment (middle and fight) The experimental flow visualization is shown in Figure 8 (fight). Based on the flow visualization and the pressure measurements a diagram of the flow field around the wing was constructed and can be seen in Figure 8 (middle). From the measurements the boundary layer is laminar until about 60% chord where a shock causes the laminar boundary layer to separate and reattach as a turbulent boundary layer. The contours of skin friction and the surface streamlines predicted by the transition model are shown in Figure 8 (left). From the skin friction the laminar separation and turbulent reattachment can be clearly seen and both appear to be in good agreement with the experimental diagram from about 20% span out to the wing tip. Near the wing-body intersection, the experiments indicate earlier transition than the simulations. This might be due to the omission of the cross-flow instability in the transition model.

Eurocopter Cabin The helicopter testcase was investigates to demonstrate the models ability of solving flows around complex geometries including multiple physical effects. There are no experimental data available in the public domain, which precludes the use of the testcase for model validation. Nevertheless, interesting transitional phenomena can be observed. In addition, the numerical performance of the transition model can be investigated. The grid for this case consisted of about 6 million nodes and each solution was run overnight in parallel on a 16 CPU Linux cluster. The convergence of lift and drag is shown in Figure 9 for the fully turbulent (top) and transitional (bottom) solutions. The transition model does not appear to have any adverse effects on the convergence and converges similar to the fully turbulent solution. This was also observed in most of the other cases. Typically the convergence is somewhat reduced in the transitional simulations, as the transition location has to settle down before convergence can be reached. The overall increase (additional equations and convergence) of the model is typically --20%. The

45 transitional flow on the fuselage and tails resulted in a 5% drag reduction compared to the fully turbulent solution.

Figure 9: Convergence of lift and drag for the fully turbulent (left) and transitional (fight) for Eurocopter simulation

Figure 1O: Contour plot of skin friction for a fully turbulent (left) and transitional (fight) for Eurocopter cabin. Isosurface indicates reverse flow.

The predicted skin friction for a fully turbulent and transitional solution is shown in Figure 10. The main differences in the transitional solution are that the front part of the fuselage, the two outside vertical tail surfaces and the outer half of the horizontal tail surface are laminar. The fact that the transition model predicted turbulent flow on the middle vertical stabilizer and the inner part of the horizontal stabilizer was unexpected. Further investigation revealed that this was caused by the turbulent wake that was shed from the fuselage upstream of the tail. This is best illustrated in Figure

46 11. The left picture shows an iso-surface of the turbulent flow. The turbulent wake is clearly visible and can be seen passing over the middle vertical stabilizer and the inner part of the horizontal stabilizer. Consequently, the transition model predicts Bypass transition on these surfaces due to the high local freestream turbulence intensity from the wake.

Figure 11: Iso-surface of turbulent flow (left) and surface value of intermittency (fight) indicating the laminar (blue) and turbulent (red) regions on the Eurocopter airframe The Eurocopter testcase demonstrates the potential of the transition model for solving complex aerodynamic flow problems, with the inclusion of 1st order transitional effects. Further model refinements are required for calibration of the model for such flows, including a model extension for crossflow instability.

CONCLUSIONS Methods for transition prediction in general purpose CFD codes have been discussed. The requirements, which a model has to satisfy to be suitable for implementation into such a code, have been listed. The main criterion is that non-local operations should be avoided. The T-Re0 transition model was built on these requirements. The model solves two transport equations and can be applied to any arbitrary geometry. Current limitations of the model are that crossflow instability is not included in the correlations and that the transition correlations are formulated nonGalilean invariant. Both limitations are currently investigated and can in principle be removed. An overview of testcases computed with the new model has been given. Due to the nature of the paper, the presentation of each individual testcase had to be brief. More details on the testcase set-up, boundary conditions grid resolutions etc. can be found in the cited papers. The purpose of the overview was to show that the model can handle a wide variety of geometries and physicaly diverse problems. The authors belief that the central concept of combining transition correlations with locally formulated transport equations has a strong potential for including 1st order transitional effects into today's industrial CFD simulations.

47 ACKNOWLEDGEMENT The model development and validation at ANSYS CFX was funded by GE Aircraft Engines and GE Global Research. Prof. G. Huang and Dr. B. Suzen from the University of Kentucky have supported the original model development with their extensive know-how and their in-house codes. REFERENCES

Abu-Ghannam, B.J. and Shaw, R., (1980), Natural Transition of Boundary Layers - the Effects of Turbulence, Pressure Gradient and Flow History. J. of Mechanical Engineering Science, Vol. 22, pp. 213 - 228. Arts, T., Lambert de Rouvroit, M. and Rutherford, A.W. (1990), Aero-thermal Investigation of a Highly Loaded Transonic Linear Turbine Guide Vane Cascade", von Karman Institute for Fluid Dynamics, Technical Note 174. Durbin, P.A., Jacobs, R.G. and Wu, X., 2002, DNS of Bypass Transition, Closure Strategies for Turbulent and Transitional Flows, edited by B.E. Launder and N.D. Sandsam, Cambridge University press, pp. 449-463. Jones, W. P., and Launder, B. E., (1973). The Calculation of Low Reynolds Number Phenomena with a Two-Equation Model of Turbulence," Int. J. Heat Mass Transfer, Vol. 15, pp. 301-314. Klausmeyer, S.M. and Lin, J.C., (1997). Comparative Results From a CFD Challenge Over a 2D Three-element High-Lift Airfoil. NASA Technical Memorandum 112858. Langtry, R.B., Menter, F.R., Likki, S.R., Suzen, Y.B., Huang, P.G., and Vrlker, S. (2004). A Correlation based Transition Model using Local Variables Part 2 - Test Cases and Industrial Applications. ASME-GT2004-53454, ASME TURBO EXPO 2004, Vienna, Austria. Langtry, R.B., and Menter, F.R., (2005). Transition Modeling for General CFD Applications in Aeronautics, AIAA Paper 2005-522, Reno, Nevada. Langtry, R.B., and Sjolander, S.A. (2002). Prediction of Transition for Attached and Separated Shear Layers in Turbomachinery", AIAA-2002-3643, 38 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. Lardeau, S., Leschziner, M.A. and Li, N. (2004). Modelling Bypass Transition with Low-ReynoldsNumber Nonlinear Eddy-Viscosity Closure, Flow, Turbulence and Combustion, Kluwer Academic Publishers. Mayle, R.E. (1991). The Role of Laminar-Turbulent Transition in Gas Turbine Engines, Journal of Turbomachinery, Vol. 113, pp. 509-537. Mayle, R.E. and Schulz, A. (1997), The Path to Predicting Bypass Transition, ASME Journal of Turbomachinery, Vol. 119, pp. 405-411. Mayle, R.E. (1996), Transition in a Separation Bubble. Journal of Turbomachinery, Vol. 118, pp. 752759. Menter, F.R. (1994). Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol. 32, No. 8, pp. 1598-1605. Menter, F. R., Esch, T. and Kubacki, S. (2002). Transition Modelling Based on Local Variables, 5 th International Symposium on Turbulence Modeling and Measurements, Mallorca Spain. Menter, F.R., Langtry, R.B., Likki, S.R., Suzen, Y.B., Huang, P.G., and Vrlker, S., (2004). A Correlation based Transition Model using Local Variables Part 1- Model Formulation. ASMEGT2004-53452, ASME TURBO EXPO 2004, Vienna, Austria. Morkovin, M.V. (1969). On the Many Faces of Transition, Viscous Drag Reduction, C.S. Wells, ed., Plenum Press, New York, pp 1-31. Rodi, W. and Scheuerer, G. (1984). Calculation of Laminar-Turbulent Boundary Layer Transition on Turbine Blades. AGARD CP 390 on Heat transfer and colloing in gas turbines, 18-1.

48 Savill, A.M., (1993). Some Recent Progress in the Turbulence Modelling of By-pass Transition, In: R.M.C. So, C.G. Speziale and B.E. Launder, Eds.: Near-Wall Turbulent Flows, Elsevier, p. 829. Savill, A.M., (1996), One-point Closures applied to Transition. Turbulence and Transition Modelling, M. Hallbiick et al., eds., Kluwer, pp. 233-268. Schulz, H.D., Gallus, H.D., (1988), Experimental Investigation of the Three-Dimensional Flow in an Annular Compressor Cascade. ASME Journal of Turbomachinery, Vol. 110, October. Simms, D., Schreck, S., Hand, M, and Fingersh, L.J. (2001). NREL Unsteady Aerodynamics Experiment in the NASA-Ames Wind Tunnel: A Comparison of Predictions to Measurements. NREL Technical report, NREZ/FP-500-29494. Smith, A.M.O. and Gamberoni, N. (1956). Transition, Pressure Gradient and Stability Theory, Douglas Aircraft Company, Long Beach, Calif. Rep. ES 26388,. Sobieczky, H. (1994), DLR - F5: Test Wing for CFD and Applied Aerodynamics, Test Case B-5 in AGARD FDP Advisory Report AR 303: Test Cases for CFD Validation. Steelant, J., and Dick, E. (2001), Modeling of Laminar-Turbulent Transition for High Freestream Turbulence. Journal of Fluids Engineering, Vol. 123, pp. 22-30 Stock, H.W. and Haase, W. (2000). Navier-Stokes Airfoil Computations with eN Transition Prediction Including Transitional Flow Regions, AIAA Journal, Vol. 38, No. 11, pp. 2059 - 2066. Suzen, Y.B., Xiong, G., Huang, P.G. (2000), Predictions of Transitional Flows in Low-Pressure Turbines Using an Intermittency Transport Equation. AIAA-2000-2654, AIAA Fluids 2000 Conference. Van Driest, E.R. and Blumer, C.B. (1963). Boundary Layer Transition: Freestream Turbulence and Pressure Gradient Effects, AIAA Journal, Vol. 1, No. 6, pp. 1303-1306. Van Ingen, J.L., (1956). A suggested Semi-Empirical Method for the Calculation of the Boundary Layer Transition Region, Univ. of Delft, Dept. Aerospace Engineering, Delft, The Netherlands, Rep. VTH- 74. Walters, D.K and Leylek, J.H. (2002). A New Model for Boundary-Layer Transition Using a SinglePoint Rans Approach. ASME IMECE'02, IMECE2002-HT-32740. Wilcox, D.C.W. (1994), Simulation of transition with a two-equation turbulence model, AIAA J. Vol. 32, No. 2. Zierke, W.C. and Deutsch, S., (1989), The Measurement of Boundary Layers on a Compressor Blade in Cascade-Vols. 1 and 2, NASA CR 185118.

Engineering TurbulenceModelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

49

P O S S I B I L I T I E S A N D L I M I T A T I O N S OF C O M P U T E R S I M U L A T I O N S OF I N D U S T R I A L MULTIPHASE

TURBULENT

FLOWS

L. M. Portela t and R. V. A. Oliemans Kramers Laboratorium voor Fysische Technologie J. M. Burgerscentrum for Fluid Mechanics Delft University of Technology Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands tCorresponding author, email: [email protected]

ABSTRACT We give an overview on the usage of computer simulations in industrial turbulent dispersed multiphase flows. We present a few examples of industrial flows: bubble columns and bubbly pipe flows, stirred tanks, cyclones, and a fluid catalytic cracking unit. The fluid catalytic cracking unit is used to illustrate the complexity of the physical phenomena involved, and the possibilities and limitations of the different approaches used: Eulerian-Lagrangian (particle-tracking) and Eulerian-Eulerian (two-fluid). In the first approach, the continuous phase is solved using either RANS or DNS/LES, and the individual particles are tracked. In the second approach, the dispersed phase is averaged, leading to two sets equations, which are quite similar to the RANS equations of single-phase flows. The Eulerian-Eulerian approach is the most commonly used in industrial applications, however, it requires a significant amount of modelling. Eulerian-Lagrangian RANS can be simpler to use; in particular in situations involving complex boundary conditions, polydisperse flows and agglomeration/breakup. The key issue for the success of the simulations is to have good models for the complex physics involved. A major weakness is the lack of good models for: the turbulence modification promoted by the particles, the inter-particle interactions, and the near-wall effects. Eulerian-Lagrangian DNS/LES can play an important role as a research tool, in order to get a better physical understanding, and to improve the models used in the RANS simulations (either Eulerian-Eulerian or Eulerian-Lagrangian). KEYWORDS Multiphase flows, turbulent flows, industrial flows, numerical simulation, Eulerian-Lagrangian, Eulerian-Eulerian, particle tracking, two-fluid, DNS, LES, RANS, dispersed flows.

50

INTRODUCTION Turbulent multiphase flows occur in numerous industrial processes, covering a wide range of phenomena and scales. Contrary to the computation of single-phase flows, where the physics is well understood, and relatively general methods can be employed, in multiphase flows the fundamental understanding of the physical phenomena is still rather poor, and it differs significantly in different types of flows. Therefore, the appropriate simulation technique will depend on the type of flow. Multiphase flows can be divided into two categories: (i) continuous multiphase flows, and (ii) dispersed multiphase flows. In the first category, (e.g., formation of waves in the flow of air over water), each phase is treated as a continuous medium, together with a method to track the interface; the modelling and numerical issues are quite similar to single-phase flows, with the additional complexity of the interface tracking and interface resolution. In the second category (e.g., pneumatic transport of solid particles), one of the phases is considered as a continuum and the other phase is considered as composed of discrete components, the particles (the term "particles" can denote solid particles, drops or bubbles). Here we will consider only dispersed multiphase flows. In principle, the balance equations together with the constitutive equations of each phase and the interfacial conditions, can be used to obtain a fully-resolved simulation of dispersed multiphase flows (i.e., a dispersed multiphase flow is nothing more than a continuous multiphase flow with a large number of different phases). However, this is extremely costly, and is only used in fundamental studies involving a relatively small number of particles. For example, this approach has been used in fundamental studies of bubbly flows (Esmaeeli and Tryggvason, 1998 and 1999) and in the study of the interaction of colliding particles in homogeneous isotropic turbulence (Ten Care et al., 2004). In industrial simulations, or in fundamental studies involving a large number of particles, some type of model for the interaction between the continuous phase and the particles, and for the inter-particle interactions, is used. The modelling of the interactions depends strongly on the type and number of particles. In Figure 1 are shown schematically the different coupling mechanisms that can occur in dispersed multiphase flows. For very dilute flows, the influence of the particles on the continuous phase can be neglected (one-way coupling). As one progresses towards more dense flows, the turbulence modification by the particles needs to be taken into account, and both the forcing of the particles by the flow and the forcing of the flow by the particles need to be computed simultaneously (twoway coupling). For even more dense flows, inter-particle interactions need also to be taken into account. Inter-particle interactions can involve hydrodynamic coupling (i.e., the force between a particle and the surrounding fluid depends also on the velocity and position of the neighboring particles) and collisions. The collisions can lead to coalescence and breakup, which might also need to be considered. For extremely dense flows (e.g., dense fluidized beds) the continuous phase does not play a significant role, and the flow is completely dominated by the inter-particle interactions. The type of particles, and in particular their size and inertia, also plays an important role on how the interactions are modeled. Typically, for gas-solid flows the particles are small (size of the same order of the smallest turbulence scales) but their inertia is quite important, whereas for bubbly flows the size of the bubbles is of the same order of the intermediate turbulence scales, making it more difficult to model the interaction between the bubbles and the turbulence. Ideally, the simulations should be able to predict the particle-distribution, and the velocities and velocity fluctuations of both the continuous and dispersed phases. However, the interaction between the turbulence and the particles, and the presence of walls, leads to strong inhomogeneities and to the clustering of the particles. The understanding of these phenomena is still rather poor, and the success of the simulations is strongly dependent on how the physics involved is incorporated into the models. Regardless of the physical models used, there are two broad types of approaches for the com-

I BOUNDARYCONDITIONSI INITIALCONDITIONS TOKES I

~ T i P . t

~_~ ~

t

HYDRODYNAMICI COUPLING I

INITIALCONDITIONS

Figure 1: Coupling mechanisms. putation of dispersed multiphase flows: (i) Eulerian-Lagrangian, also known as particle-tracking, and (ii) Eulerian-Eulerian, also known as two-fluid. In the first approach, the continuous-phase is solved using the Navier-Stokes and continuity equations, using either DNS/LES or RANS, and the individual particles are tracked. In the second approach, the dispersed phase is averaged, leading to two sets of equations, which are quite similar to the RANS equations of single-phase flows, together with some extra coupling-terms. The Eulerian-Eulerian approach is somehow akin to the RANS approach of single-phase flows, and the computational effort is quite similar to it, therefore it is usually preferred in industrial simulations. The major inconvenient is that it requires more levels of modelling; e.g., regarding the wall boundary conditions for the dispersed phase. Also, it is not straightforward to implement more complex phenomena, like agglomeration and breakup, and can become quite cumbersome for polydisperse flows (flows with several particle sizes). Due to these problems, in numerous industrial flows Eulerian-Lagrangian RANS simulations are used, particularly when breakup and/or agglomeration are involved. Eulerian-Lagrangian DNS/LES is used mostly as a research tool, in order to improve the models used in the RANS simulations (either Eulerian-Eulerian or EulerianLagrangian). In the paper we present some examples of industrial flows, and explain briefly the EulerianEulerian and Eulerian-Lagrangian approaches, the role they play and their limitations. INDUSTRIAL FLOWS We present a few examples of industrial turbulent multiphase flows, to illustrate the wide range of situations that occur, and the major issues involved. Figure 2 shows the picture of a bubble column, which is widely used in the chemical and biochemical industries. The bubbles are injected at the bottom, and the buoyancy force induces a strong circulation in the bubble column. Even though there is no net liquid-flow-rate, the flow can be quite turbulent, with the liquid moving up in the center and down near the walls. Depending on the process, the diameter can reach a few meters, and the volume-fraction occupied by the bubbles can be quite high (up to more than 50%). The bubbles can have a wide range of sizes, from sub-millimeter to centimeter. Typically, the bubbles and their wakes can have sizes of the same order of the intermediate turbulence scales of the flow. This and the high volume-fraction make the process quite complex and challenging to model. To complicate the situation, quite often small catalyst particles, with a size of the order of 100 microns, are often added to the liquid. Due to the wake of the bubbles and their interaction with the turbulent flow, the concentration of these catalyst particles can be highly non-uniform and extremely difficult to simulate (Fan and

52 Tsuchiya, 1990).

Figure 2: Picture and schematic representation of the flow in a bubble column (Chen et al., 1994) In a bubble column there is no net liquid-flow. However, in numerous situations gas bubbles are injected in order to induce a net liquid-flow, a process known as gas-lift. This is commonly used in the oil industry (currently, about 40% of the oil production is through gas-lift). Essentially, a gas-lift oil-well consists of a long vertical pipe (diameter of the order of 10 cm and length up to more than 1 km), with gas injected at the bottom. The gas reduces the weight of the column, therefore "pumping" the oil up. Typically, one has a turbulent flow with a low Reynolds number (of the order of 10,000), with a bubble volume-fraction of the order of 10%. Similarly to the bubble column, the high volume-fraction, the complex bubble-turbulence interactions, and the occurrence of bubble breakup and coalescence, make it quite challenging to obtain accurate simulations. For example, it has been observed experimentally that the injection of smaller bubbles leads to a more uniform bubble concentration and a more efficient gas-lift (Guet et al., 2003), however, it is not possible to predict this phenomenon without the use of experimentally-fitted values for the transverse lift-force acting on the bubbles (Guet, 2004). Actually, even for a single bubble in simple laminar shear flows, the mechanisms that promote the transverse lift-force are still not well understood, and are subject of current research (e.g., Tomiyama et al., 2002). An overview of gravity-driven flows for both bubble columns and pipe flows can be found in Mudde (2005). Figure 3 shows a stirred tank. Essentially, it consists of a tank filled with liquid and an impeller to stir the liquid. This device is used in numerous processes, like crystallization and agglomeration of particles, mixing of different liquids, and chemical reactions with catalyst particles. Even in single-phase flows, the simulation is already quite challenging for RANS models, due to the strong circulation and unsteadiness involved. In multiphase flows, with the presence of particles, droplets, or bubbles, the situation is strongly complicated by the interaction of the turbulence with the dispersed phase; e.g., Hollander et al. (2001) found that the small turbulence scales can strongly affect the agglomeration/breakup of particles in crystallizers. Current industrial practice uses RANS or TRANS simulations, but due to the unsteady nature of the flow and the influence that the different turbulence scales exert on the dispersed phase, LES can be extremely useful in revealing the details of these phenomena (Derksen, 2003b). The bubble column and the stirred tank are, essentially, "mixing devices"; however, in numerous industrial processes one wants to separate the different phases. In gas-solid flows a common device is the cyclone separator. A typical design is shown schematically in Figure 4. Essentially, it consists of a conic tube, where the particle-laden gas is fed radially. Due to the centrifugal force, the particles are thrown to the wall and fall to the bottom part, where they are collected in a bin,

53

Figure 3: Schematic representation and snapshot of the LES simulation of a stirred tank filled with water and loaded with small glass beads (Derksen, 2003b); center: cross-section midway between two baffles; right: cross-section just below the impeller disk whereas the clean gas leaves the separator at the top. The basic principle is very simple, and one is mainly interested in determining the pressure drop and the separation efficiency (percentage of particles collected as a function of their size). Current industrial simulations typically use RANS for the continuous phase, together with the tracking of the particles. However, the high circulation and the presence of a precession vortex in the central region of the cyclone make it quite a challenging problem for RANS models (Hoekstra, 2000), and considerable insight can be gained using LES. For example, in Figure 4 are shown snapshots of the particles at five at five instants in time, obtained from LES simulations (Derksen, 2003a). The left-most snapshot shows the moment where the feeding of the particles to the cyclone was interrupted, and the other snapshots show the time-evolution of the particles inside the cyclone and collector bin at the bottom. One can see that the particles have a tendency to become trapped at the top of the cyclone, and might escape together with the clean gas.

Figure 4: Schematic representation and snapshots of the LES simulation of a cyclone (Derksen, 2003a), showing the time-evolution (from left to right) of the particles, after their feeding into the cyclone was interrupted, at the left-most snapshot.

54

Often, multiphase flow devices are used not separately, but as part of a complex process. A nice example is the Fluid Catalytic Cracking (FCC) unit. FCC is a trillion dollar world-wide operation and a crucial component of modern oil refineries. It converts heavy residual hydrocarbons into lower molecular-weight products, such as gasoline (world-wide, installed production capacity is more than 500 million tons per year). The basic elements of an FCC unit are shown schematically in Figure 5. Essentially, the heavy-oil feed is introduced at the bottom of the riser, together with some steam, where it meets the hot catalyst particles. The oil is vaporised by the catalyst particles, and the cracking reaction starts, breaking the heavy-oil into lighter products. At the exit of the riser, the catalyst particles are separated from the products by means of a series of cyclones. The catalyst particles are partially deactivated-activated by deposited coke, and are sent to the regenerator, which operates as a bubbling fluidized bed, where the coke is burnt by injecting air from below. The combustion products (flue gas) leave the regenerator through a series of high-efficiency cyclones, which collect the small dust resulting from the attrition of the catalyst particles. The regenerated catalyst particles are returned to the riser entrance where they join the oil feed, re-initiating the process. The whole process involves a large combination of multiphase flow devices, operating under very different conditions, from a dense fluidized bed in the regenerator to a dilute flow in the riser. The riser looks quite a simple device. Essentially, it is a tube with a height of about 50 meters and a diameter of about 1.5 meters, with a gas velocity of about 10-20 m/s and catalyst particles with a size of about 50 microns. However, the simplicity can be quite deceptive. Typically, the particle volume-fraction is of the order of 1% and the Reynolds number is quite high. Under these conditions, essentially, all the coupling mechanisms play an important role. The particle volumefraction is quite small, but it is not negligible, and since the density of the catalyst particles is much higher than the density of the gas, the particle mass-fraction can be rather high (the mass of the particles can be up to about ten times the mass of the gas), therefore the influence of the particles on the turbulence is very important. Also, due to the high mass-fraction of particles, collisions and hydrodynamic interactions play a crucial role. Another important aspect is associated with the relaxation time of the particles. Even though the particles are small, they are quite heavy, and the particle relaxation time is of the same order of the large turbulence-time-scales, therefore the particles do not behave as tracers. They are pushed by the turbulence in quite complex ways, leading to particle-clustering and non-uniform concentration, with a high particle-concentration near the wall.

Figure 5: Schematic representation of the basic elements of a FCC unit (left), and of some of the multiphase flow devices that are used (right).

55

EULERIAN-EULERIAN SIMULATIONS In this approach both phases are averaged, leading to the following equations for the conservation of mass and momentum:

O(~ p~) + b-~ 0 (~p~j)

- 0

0 -0 0 cO -~(~p~v~,)+-~j(~p~u~,v~j)~g,+-~xj(~-J~,j)--~(~p~ < v'~,v'~j>~)+

(1) < 7, >~ (2)

The index k denotes the phase (e.g., k = 1 for the continuous phase and k = 2 for the dispersed phase), and the indices i and j are associated with the usual Cartesian tensor notation. The volume-fraction is denoted by a, the density by p, the velocity by U, the stress by a, and the gravitational acceleration by g. The average stress and velocity are denoted, respectively, by and U, and the velocity fluctuation is denoted by U'. An average evaluated at the position of the phase k is denoted by < >k. The fist term on the right-hand-side of the momentum equation represents the gravitational force, the second the surface stresses, the third the kinetic stresses and the fourth the interfacial stresses. For the continuous phase, the surface stresses are the usual pressure and viscous stresses, whereas for the dispersed phase they are closely related to the inter-particle interactions. For the continuous phase, the kinetic stresses are the usual Reynolds stresses, and for the dispersed phase the interpretation is similar: the flux of momentum due to the velocity fluctuation of the particles. The interfacial stresses incorporate the transfer of momentum between both phases, due to the force between the particles and the surrounding fluid. The equations above are quite similar to the RANS equations of single-phase flows, and can be solved using the same type of numerical methods. The key issue is how to model the "closure terms" that are required for the surface stresses, kinetic stresses and interfacial stresses. Depending on the flow, this may require models for the: (i) Reynolds stresses of the continuous phase, (ii) kinetic stresses ("Reynolds stresses") of the dispersed phase, (iii) interfacial forces, (iv) interparticle interactions (collisions and hydrodynamic coupling), and (v) boundary conditions (e.g., wall-particle interaction). The interfacial stresses are usually modeled in a straightforward way, using knowledge about the force acting on a single particle immersed in an infinite homogeneous medium (e.g., Stokes drag for small particles). Depending on the situation, the model for the force may be modified to taken into account: flow inhomogeneity, the presence of other particles, the presence of walls, etc.. The modelling of the surface and kinetic stresses depends strongly on the type of flow and is less straightforward; in a way it resembles the "closure models" of single-phase RANS. The standard approach is to use a simple single-phase RANS model for the continuous phase (usually, the standard k - e model), and transport equations for the surface and kinetic stresses of the dispersed phase. For very dilute flows with small solid particles, the surface stresses of the dispersed phase, which are closely associated with the inter-particle interactions, can be neglected and the kinetic stresses of the dispersed phase can be determined using a simple "local equilibrium" model, which assumes that the kinetic stresses of the dispersed phase are determined by the Reynolds stresses of the continuous phase. In general, the "local equilibrium" model works well, except very close to walls (Portela et al., 2002), and it gives reasonably good results for dilute flows of small solid particles. For more dense flows, more sophisticated models are used for the surface and kinetic stresses of the dispersed phase, taking into account the effect of inter-particle interactions. Following previous work by Reeks (1991 and 1992), Simonin and co-workers developed a model based on the particle kinetic pdf equations, which has been quite successful in dealing with a wide range of particle concentrations (Simonin et al., 1995). For example, Wang (2001) used Simonin's approach for the

56 simulation of the flow in a riser, and obtained a good agreement with available experimental data, as shown in Figure 6. 0.10 0.08

-

0.06

-

0.04

-

0.02

-

0.00

0.0

modeling in this work, e=0.98 9 measured,Nieuwland

.j I

I

I

I

0.2

0.4

0.6

0.8

r/R[-]

1.0

Figure 6" Radial profile of the particle volume-fraction in a riser flow (comparison between Eulerian-Eulerian simulations using Simonin's correlated collision model and experiments), from Wang (2001). For very dense flows (e.g., fluidized beds), a model commonly used is based on the kinetic theory of granular flows (Gidaspow, 1994). However, a major problem of this model is the unrealistic highsensitivity it has to the choice of the restitution coefficient used in the inter-particle collisions (Pita and Sundaresan, 1991). Essentially, the model does not take properly into account the particleturbulence interactions, and this is "corrected" by somewhat arbitrary choices of the restitution coefficient. For very dense flows, the particle-turbulence interactions are not very relevant, and the model works reasonably well. However, for less dense flows, like risers, the results can be quite poor (Wang, 2001). The Eulerian-Eulerian approach is widely used in industrial simulations, not only in two-phase flows (gas-solid, liquid-solid, and gas-liquid) but also in more complex three-phase flows (slurries), involving bubbly flows and small catalyst particles (Oey et al. 2001); if properly used, it can give very useful information, in terms of design and prediction of the performance of multiphase flow equipment. However, a major weakness is that it strongly depends on the models used, which are based on situations far simpler than the ones to which the simulation is usually applied. In particular, the models that are commonly used for the interfacial stresses are based on the force acting on a single particle in an infinite homogeneous medium, and the situations in which they are applied can be rather different. For example, Eulerian-Eulerian simulations have been used to improve the injection system in FCC risers (Patureaux and Barthod, 2000), but in order to obtain good results, the equations for the drag force acting on the particles had to be fitted to experimental data. The major difficulty of Eulerian-Eulerian simulations is not in dealing with the complex geometries often found in industrial flows, for which they can give reasonably good qualitative results, but in dealing with the complex interactions occurring in what it may look a deceptively simple flow, which cannot be properly captured with the models currently used. A good example is the FCC riser. Due to the gradient in the turbulence intensity, the particles tend to be pushed towards the wall (turbophoresis), which can lead to very-high particle-concentrations near the wall. Due to the high particle-concentration, both the influence of the particles on the turbulence and interparticle collisions become important, and accurate models for the interplay of all these effects are needed. Also, the particles tend to cluster, and their distribution is far from uniform (Eaton and Fessler 1994); by its nature (one-point closure) this cannot be predicted with the models currently used in Eulerian-Eulerian simulations.

57 EULERIAN-LAGRANGIAN

SIMULATIONS

In this approach, the continuous phase is solved using either DNS/LES or RANS, and the individual particles are tracked. For the RANS simulations, the equations of the continuous phase are essentially the same as in the Eulerian-Eulerian approach above, but now each particle is tracked using a model for the force between the particle and the fluid, and; if needed, models for interparticle collisions and particle-wall collisions are also used. Since the particles are tracked, it is rather straightforward to deal with polydisperse flows and to consider particle rotation and wallroughness. Also, it is relatively easy to include phenomena such as breakup and agglomeration, as done, e.g., by Sommerfeld and co-workers (Ho and Sommerfeld, 2002). Two major difficulties are: (i) the large number of particles that might be needed, (ii) how to incorporate the turbulence effects on the particle trajectory and dispersion. The first difficulty is usually alleviated by introducing "representative particles" and "fictitious collisional particles" (Sommerfeld, 2001). The second issue is usually considered by adding a velocity fluctuation, using the turbulence kinetic energy and dissipation, obtained from the RANS simulation. However, in order to deal with these two issues an extra-level of modelling is introduced, which might not represent adequately the complex particle-turbulence interactions. Eulerian-Lagrangian RANS has been used in numerous industrial flows, from rather dilute, like cyclones (Hoekstra, 2000) and pneumatic conveying (Huber and Sommerfeld, 1998), to quite dense, like fluidized beds (Goldschmidt et al., 2004). Even though it is more flexible and requires less modelling, it still suffers from one of the major weaknesses of the Eulerian-Eulerian approach: the use of RANS for the continuous phase, which makes it quite difficult to predict accurately the complex interactions between the particles and the turbulence. In particular, both the EulerianEulerian and Eulerian-Lagrangian approaches do not take properly into account the modification of the turbulence promoted by the particles. Typically, a standard k - e model is used, together with an extra particle-forcing term in the k and e equations: Dk = P k - c + Tk - cp Dt De = 7)~ - V~ + T~ - 7)p Dt

(3)

(4)

where k is the turbulence kinetic energy, e is the dissipation of k, and 7)~ is the dissipation of e. 7) denotes the production and T the transport (of k and e). However, the extra particle-forcing terms, % and 7)p, are small, and do not take into account the large disruption in the turbulence dynamics promoted by the particles (Li et al. 2001, Bijlard et al. 2002). In Eulerian-Lagrangian DNS/LES simulations, the continuous phase is simulated using either DNS or LES for the continuous phase, together with the tracking of the individual particles. In this approach the particles are considered as point-particles, and the interaction between a particle and the surrounding fluid is represented through a force located at the position of the center of the particle. This approach keeps the level of modelling to a minimum, provided that the interparticle distance is large, and the particles are small when compared with the smallest relevant flow scales (Portela and Oliemans, 2001). Since the particle-concentration is small, the influence of the particles on the continuity equation can be neglected. The influence of the particles on the fluid is felt through an extra-force in the Navier-Stokes equation. The equations being solved are"

v.u=0 OU

~{-07 + ( v u ) . ~7) = - v p + ~v~O + v . ~ + ~

(5) (6)

together with the equation of motion of each particle"

dVp

Mp --~ = Fp

(7)

58

The extra term in the Navier-Stokes equations, $', is the force per unit of volume due to the particies. The force acting on a particle, Fp, is obtained using the velocity of the continuous-phase at the center of the particle, together with some model for the particle-fluid force (e.g., Stokes drag for small heavy particles). Usually, the subgrid stress-tensor, 5s, is modeled in the same way as for single-phase flow LES (PorteIa and Oliemans, 2002). Provided the particles are small and the inter-particle distance is large, Eulerian-Lagrangian DNS/LES simulations are extremely useful in getting a better fundamental understanding of the particle-turbulence interactions, and they have been used by several groups in fundamental studies of turbulent flows laden with small particles: solid particles, droplets or bubbles (e.g., Squires and Eaton 1991, Uijttewaal and Oliemans 1996, Li et al. 2001, Marchioli and Soldati 2002, Portela and Oliemans 2003, Mazzitelli et al. 2003). In particular, in our group we have been using this approach to get a better fundamental understanding of the dynamics of wall-bounded flows laden with small particles, and to provide support for the development of better models for the EulerianEulerian and Eulerian-Lagrangian RANS simulations. In Figure 7 is shown an example of how the turbulence structure can lead to the clustering of particles into elongated streamwise streaks near the wall, in a channel-flow laden with small heavy particles, and how, in turn, the particles can promote a large disruption of the turbulence structure near the wall. The disruption of the turbulence dynamics by the particles can produce large changes in the turbulence kinetic energy balance, as shown in Figure 8, where we can see that the particles promote a large reduction in the production and dissipation of turbulence kinetic energy. A topic of current research is how to incorporate these effects into the turbulence models (Bijlard et al., 2002). For example, in Figure 9 is shown how the particles change the value of the "constant" C, used in the standard k - ~ model.

Figure 7: Turbulence and particle structure very near the wall of a channel flow; left" fluid streaks for the flow without particles, center: modification of the fluid streaks promoted by the presence of small heavy particles, right" particle streaks.

59

Figure 8: Effect of the particles on the turbulence kinetic energy budget (Equation 3), obtained from the DNS simulations of a turbulent channel flow laden with small heavy particles (Reynolds number based on the wall shear-stress and channel height: Re~- = 360, particle relaxation time in wall-units: 7+ = 58, particle mass-fraction: Cm = 0.65). In order to get a better fundamental understanding and to develop physically-based models, it is very useful to isolate the different phenomena, and for this Eulerian-Lagrangian DNS/LES is an ideal tool. On purpose, for the results shown in Figures 8 and 9, inter-particle collisions were not considered. However, they can strongly affect the particle-concentration, and in turn the particle-turbulence interactions, as observed in the DNS simulations of Li et al. (2001). For example, in Figure 10 is shown how the inter-particle collisions can dramatically reduce the particle-concentration close to the wall. Note, however, that the differences between elastic and inelastic collisions are very small. All these changes can have important consequences, in terms of predicting the hydrodynamics and the particle-distribution, which in turn has large consequences in terms of designing and predicting the performance of the equipment. For example, the output of gasoline in a FCC unit can be strongly affected by the turbulence dynamics and the distribution of the catalyst particles in the riser. Actually, the particle-turbulence interaction can influence not only the turbulence, but also produce quite unexpected changes in the mean flow itself. For example, in the case of particle-laden horizontal flow, the non-uniform modification of the turbulence promoted by the particles can lead to a secondary flow, as observed in the LES simulations of Belt et a1.(2004). Another aspect that currently is not considered in Eulerian-Eulerian models, but which can have important consequences, is the effect of the wall-roughness. For example, using EulerianLagrangian RANS simulations and experimental data, Sommerfeld and co-workers found that the wall-roughness can strongly decrease the particle-concentration near the wall (Huber and Sommerfeld, 1998). The wall-roughness can also have quite unexpected effects, as observed by Van't Westende et al. (2004) using Eulerian-Lagrangian LES simulations: they found that the secondary flow due to non-uniform roughness in a pipe (as it occurs in the core region of horizontal annular flow) can lead to secondary flow, which, in turn, can strongly affect the distribution of the particles in the pipe (or of the droplets entrained in the gas core region of the annular flow).

6O

0.14

iii:t 0

,.,.,.-- ......... -;:-_o ....... 20

40

60

80

100

120

140

t

160

Z+ Figure 9: Effect of different mass-loadings and particle relaxation times on the "constant" Cu used in the standard k - ~ model, obtained from the DNS simulations of a turbulent channel flow laden with small heavy particles (Re~. = 360). CONCLUSION Compared with single-phase, multiphase flow simulations still have a long way to go. The major difficulty is the multitude of phenomena involved, and the still rather poor understanding of the physics, which sometimes can have rather unexpected and important consequences. Due to this, different approaches tend to be used for different types of flows, depending on the particular phenomena involved and the type of models needed/used. Due to the fact that it can easily deal with complex geometries, large Reynolds numbers, and a large number of particles, the Eulerian-Eulerian approach is the most commonly used in industrial applications. The structure of the equations and the computational cost is somewhat similar to single-phase RANS. However, it requires a significant amount of modelling, and EulerianLagrangian RANS simulations can be simpler to use, in particular in situations involving complex boundary conditions (e.g., rough walls), polydisperse flows, and agglomeration/breakup. Regardless of the approach, the key issue for the success of the simulations is to have good models for the phenomena involved. In particular, a major weakness is the lack of good models for: the turbulence modification promoted by the particles, the inter-particle interactions, and the near-wall effects. Eulerian-Lagrangian DNS/LES can play a very important role in developing a better physical understanding, and therefore better models. However, a major problem is the lack of well-controlled detailed experiments, where the different phenomena are isolated and the models can be tested. These experiments would play a very useful role in validating the DNS/LES simulations, which can then be used for systematic parametric studies. In terms of the industrial user, a critical need is the understanding of the different phenomena involved, when they are (or are not) important, and how they can (or cannot) be taken into account. In this respect, perhaps it would be useful to develop a "Best Practice Guidelines for Multiphase Flows", along the lines of what has been done for single-phase flows (Casey and Wintergerste, 2000).

30 0 . . . . 28

,. . . .

'~

~

,

_

50 b

..,

' .... '

",

100 .....

'

I

0 9

~ .

.

.

.

.

.

2.5

50 I

,

r "

'

2

24"

c,ot|iiion& R u n 6 collisions, R u n 3

............

too '

'

9

.... =

I

r162

- - ~

2.5

2

7

no

lg

g

I

elastic ~xdllskms, R u n 4 |Mlmmtk~ Run

22

2O

9 '

1.5

1.5

16

1o 8 6

2

-

O.b ..

, . . . , - - ,7_' "~ ~ - ~ . _ , ~ . ~ v . - _~ _T.. _ . . . _

00 . . . . . . . .

50

y

10.5

L _ . _ _ ~ - . _ ~ ~

. . . . . .

100

00

'

'

'

~'

501

,,

y

L

t ............

1001

,,

,

0

Figure 10: Influence of the inter-particle collisions on the particle-concentration profile, obtained from DNS simulations of a channel flow laden with small heavy particles (Re~ = 250); left: comparison between collisions and no-collisions; right: comparison between elastic and inelastic collisions (Li et al., 2001). Acknowledgments We would like to thank our colleagues and Ph.D. students at the Kramers Laboratory for some of the results presented here, and for the numerous discussions that we had on multiphase flows. Our work has been partially financed by the Dutch Foundation for Fundamental Research on Matter (FOM), through its program on "Dispersed Multiphase Flows", and by the Dutch Foundation for Technological Research (STW). References

Belt, R.J., Van't Westende J.M.C., Portela, L.M., Mudde, R.F. and Oliemans, R.V.A. (2004). Particle-Driven Secondary Flow in Turbulent Horizontal Pipe Flows. Proceedings of the Third International Symposium on Two-Phase Flow Modelling and Experimentation, September 2224, Pisa, Italy. Bijlard, M.J., Portela, L.M. and Oliemans, R.V.A. (2002). Effect of the Particle-Induced TurbulenceModification on Two-Equation Models for Particle-Laden Wall-Bounded Turbulent Flows. In:Rodi, W. and Fueyo, N. (eds.), Engineering Turbulence Modelling and Experiments 5, 949-958. Elsevier, Amsterdam. Casey, M. and Wintergerste, T. (2000). Best Practice Guidelines for Industrial CFD. ERCOFTAC. Chen, R.C., Reese, J. and Fan, L.S. (1994). Flow Structure in a 3-Dimensional Bubble-Column and 3-Phase Fluidized-Bed. AIChE J. 40:7, 1093-1104. Derksen, J.J. (2003a). Separation Performance Predictions of a Stairmand High-Efiqciency Cyclone. AIChE J. 49:6, 1359-1371. Derksen, J..]. (2003b). Numerical Simulation of Solids Suspension in a Stirred Tank. AIChE J. 49:11, 2700-2714. Eaton, J.K. and Fessler, J.R. (1994). Preferential Concentration of Particles by Turbulence. Int. J. Multiphase Flow 20, 169-209 (supplement). Esmaeli A. and Tryggvason, G. (1998). Direct Numerical Simulations of Bubbly Flows. Part 1. Low Reynolds Number Arrays. J. Fluid Mech. 377, 313-345.

62 gsmaeli A. and Tryggvason, G. (1999). Direct Numerical Simulations of Bubbly Flows. Part 2. Moderate Reynolds Number Arrays..]. Fluid Mech. 385, 325-358 Fan, L.S. and Tsuchiya, K. (1990). Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions. Butterworth-Heinemann, Boston. Gidaspow, D. (1994). Multiphase Flow and Fluidization. Academic Press, San Diego. Goldschmidt, M.J.V., Beetstra, R. and Kuipers J.A.M. (2004). Hydrodynamic Modelling of Dense Gas-Fluidised Beds: Comparison and Validation of 3D Discrete Particle and Continuum Models. Powder Technology 142:1, 23-47. Guet, S. (2004). Bubble Size Effect on the Gas-Lift Technique. Ph.D. Thesis, Delft University of Technology, The Netherlands. Guet, S., Ooms, G., Oliemans, R.V.A. and Mudde, R.F. (2003). Bubble Injector Effect on the Gaslift Efficiency. AIChE J. 49:9, 2242-2252. Hoekstra, A.J. (2000). Gas Flow Field and Collection Efficiency of Cyclone Separators. Ph.D. Thesis, Delft University of Technology, The Netherlands. Ho, C.A. and Sommerfeld, M. (2002). Modelling of Micro-Particle Agglomeration in Turbulent Flows. Chem. Eng. Science 57, 3073-3084. Hollander E.D., Derksen, J.J., Portela, L.M. and Van den Akker, H.E.A. (2001). Numerical ScaleUp Study for Orthokinetic Agglomeration in Stirred Vessels. AIChE J. 47:11, 2425-2440. Huber, N. and Sommerfeld, M. (1998). Modelling and Numerical Calculation of Dilute-Phase Pneumatic Conveying in Pipe Systems. Powder Technology 99, 90-101. Li, Y.M., McLaughlin, J.B., Kontomaris, K. and Portela L. (2001). Numerical Simulation of Particle-Laden Turbulent Channel Flow. Phys. Fluids 13:10, 2957-2967. Marchioli, C. and Soldati, A. (2002). Mechanisms for Particle Transfer and Segregation in a Turbulent Boundary Layer. J. Fluid Mech. 468, 283-315. Mazzitelli, I.M., Lohse, D. and Toschi, F. (2003). On the Relevance of the Lift Force in Bubbly Turbulence. J. Fluid Mech. 488, 283-313. Mudde, R.F. (2005). Gravity-Driven Bubbly Flows. Annual Rev. Fluid Mech. 37 (in press). Oey, R.S., Mudde, R.F., Portela, L.M. and Van den Akker, H.E.A. (2001). Simulation of a Slurry Airlift Using a Two-Fluid Model. Chem. Eng. Science 56:2, 673-681. Patureaux, T. and Barthod, D. (2000). Usage of CFD Modelling for Improving an FCC Riser Operation. Oil ~ Gas Science and Technology 55:2, 219-225. Pits, J.A. and Sundaresan, S. (1991). Gas-Solid Flow in Vertical Tubes. AIChE J. 37:7, 10091018. Portela, L.M., Cota, P. and Oliemans, R.V.A. (2002). Numerical Study of the Near-Wall Behaviour of Particles in Turbulent Pipe Flows. Powder Technology 125, 149-157. Portela, L.M. and Oliemans, R.V.A. (2001). Direct and Large-Eddy Simulation of Particle-Laden Flows Using the Point-Particle Approach. In: Geurts, B.J., Friedrich, R. and M~tais, O. (eds.), Di'Fect and Large-Eddy Simulation IV, 453-460. Kluwer, Dordrecht. Portela, L.M. and Oliemans, R.V.A. (2002). Subgrid Particle-Fluid Coupling Evaluation in LargeEddy Simulations of Particle-Laden Flows. Proceedings of the ASME International Mechanical Engineering Con/~rence and Exposition, November 17-22, New Orleans, USA (ASME paper IMECE2002-33113). Portela, L.M. and Oliemans, R.V.A. (2003). Eulerian-Lagrangian DNS/LES of Particle-Turbulence Interactions in Wall-Bounded Flows. Int..I. Num. Meth. Fluids 43:9, 1045-1065. Reeks, M.W. (1991). On a Kinetic Equation for the Transport of Particles in Turbulent Flows. Phys. Fluids A 3..a, 446-456. Reeks, M.W. (1992). On the Continuum Equations for Dispersed Particles in Nonuniform Flows. Phys. Fluids A 4:6, 1290-1303. Simonin, O. Deutsch, E. and Boivin, M. (1995). Large Eddy Simulation and Second-Moment Closure Model of Particle Fluctuating Motion in Two-Phase Turbulent Shear Flow. In: Durst, F.

63 (ed.), Selected Papers from the Ninth Symposium on Turbulent Shear Flows, 85-115. Springer, Berlin. Sommerfeld, M. (2001). Validation of a Stochastic Lagrangian Modelling Approach for InterParticle Collisions in Homogeneous Isotropic Turbulence. Int. J. Multiphase Flow 27', 18291858. Squires, K.D. and Eaton J.K. (1991). Measurements of Particle Dispersion Obtained from Direct Numerical Simulations of Isotropic Turbulence. J. Fluid Mech. 226, 1-35. Ten Cate, A., Derksen, J.J., Portela, L.M. and Van den Akker, H.E.A. (2004). Fully Resolved Simulations of Colliding Monodisperse Spheres in Forced Isotropic Turbulence. J. Fluid Mech. 519, 233-271. Toiniyama, A., Tamai, H., Zun, I. and Hosokawa, S. (2002). Transverse Migration of Single Bubbles in Simple Shear Flows. Chem. Eng. Science 57, 1849-1858. Uijttewaal, W.S.J. and Oliemans, R.V.A. (1996). Particle Dispersion and Deposition in Direct Numerical and Large Eddy Simulations of Vertical Pipe Flows. Phys. Fluids 8:10, 2590-2604. Van't Westende, J.M.C., Belt, R.J., Portela, L.M., Mudde, R.F. and Oliemans, R.V.A. (2004). Interaction of Particles with Secondary Flow in High Reynolds Number Horizontal Pipe Flow. Proceedings of the Third International Symposium on Two-Phase Flow Modelling and Experimentation, September 22-24, Pisa, Italy. Wang, ,!. (2001). On the Hydrodynamics of Gas-Solid Flows in Downers. Ph.D. Thesis, Twente University, The Netherlands.

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2. Turbulence Modelling

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

67

T U R B U L E N C E MODEL AND ITS APPLICATION TO F O R C ED AND NATURAL C O N V E C T I O N

(v2/k) - f

K. H a n j a l i 6 1 , D. R. L a u r e n c e z,3, M. P o p o v a c I a n d J.C. U r i b e 2, 1Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, N1 2UMIST, PO Box 88, Manchester M60 1QD, UK 3 EDF-DER-LNH, 6 quai Watier, 78401 Chatou, France

ABSTRACT We present the rationale and some validation of a version of Durbin's elliptic relaxation eddy-viscosity model, which solves a transport equation for the velocity scale ratio v 2/ k instead of v 2. The new model, developed independently at TU Delft and UMIST (in two variants, shows improved robustness, faster convergence and less sensitivity to grid nonuniformities. The two variants differ insignificantly in the formulation of the v2/k and f equations: the UMIST model endeavours to remain close to Durbin's original one, while TUD variant introduces quasi-linear pressure strain formulation but with some further numerically beneficial simplifications. The model validation in a range of attached, separating and impinging flows with heat transfer, as well as in natural convection in a tall cavity, showed satisfactory predictions in all cases considered.

INTRODUCTION w

Since its appearance, the v 2 - f model of Durbin (1991) has attracted substantial interest both among academia and industry because of its simple formulation, plausible physical rationale and a range of successful applications. The introduction of an additional ("wall-normal") velocity scale 'v2 and of elliptic relaxation to sensitize v 2 to the inviscid wall blocking effect are the main features of the model. Whilst the v 2 - f model is far from providing a panacea for all situations (highly anisotropic three-dimensional flows may still need a full second-moment or higher-order nonlinear eddy-viscosity approach), it offers an interesting model option, superior to most common eddy-viscosity models. It seems especially suited for predicting near-wall phenomena - friction and heat transfer in not-so-complex flows. This has been demonstrated in successful predictions of several wall-bounded flows with heat transfer, featuring impingement, separation and buoyancy. However, the v 2 - f model poses some numerical difficulties arising from the stiffness of the boundary condition, which make it less attractive for industrial applications. The source of stiffness is associated with the wall boundary condition fw = limv~0 -20v--2u2/(ey 4), which makes the computations sensitive *Authors names are listed alphabetically

68 to the near-wall grid clustering especially when the first grid point is placed at too small y+. The problem can be obviated by solving simultaneously the v 2 and f equation, but most commercial as well as inhouse codes use more convenient segregated solvers. Alternative formulations of the v 2 and f equations have been proposed by Lien and Durbin [ 14] (hereafter LDM), which permit fw = 0, but these perform less satisfactorily than the original model [13], [9]. In order to obviate the numerical stiffness whilst still retaining the major features of the Durbin's elliptic relaxation concept, a new model formulation was recently proposed independently by Hanjalic [12] and Laurence [ 13], which solves a transport equation for the velocity scale ratio instead of the equation for v 2. The two model versions differ slightly in some details and coefficients, but their performances are very similar: both versions predict a range of generic flows with a quality comparable with - and in some cases better than - the original Durbin's model. Because of a more convenient formulation of the equation for and especially of the wall boundary condition for the elliptic function the model is less sensitive to small values of y+, what makes it more robust and less sensitive to nonuniformities and clustering of the computational grid. Other advantages of this formulation is the absence of e from the transport equation for This makes this equation uncoupled from the usually troubling e equation, what further contributes to the model robustness. It is also noted that the is nondimensional and bounded between the zero value (at a solid wall) and its isotropic value of 2/3.

et al.

v2/k

et al.

v2/k

f,

v2,/k.

v2/k

The new model has so far been validated in a boundary layer on a flat plate, fully developed channel flow at several Re numbers, pulsating channel flow, impinging jets with heat transfer at different Re numbers, flow and heat transfer behind a backward-facing step, flow in an asymmetric diffusor and over a periodic hill and natural convection in a tall cavity. We present here the rationale and a summary of the two model variants, and illustrate their performance with a series of new validations.

THE RATIONALE

vt

In Durbin's v 2 - f model, the eddy viscosity is defined as = C~, v2r (where r is the turbulence time scale), and evaluated by solving the conventional transport equations for the turbulence kinetic energy k and its dissipation rate e in the form

Dr-= p - e + ~

v+

(1)

plus the transport equations for v 2 and a Helmholz-type elliptic equation for the relaxation function f, which introduces the effects of pressure-strain correlations as in a second-moment closure.

(v2/k)

A transport equation for the ratio can be derived directly from the v 2 and k equations of Durbin (1991). The direct transformation yields:

D(v--~/k)Dt= f

(v-g/k)k"P + ~xk cO(

v+

pt)O(~;2/k) cr(~/k) Oxk

+X

(3)

where the "cross diffusion" X is a consequence of transformation

x=~

(v2/k)

2 (

~,+

vt ) O(-vS/k) Ok (7(~/k) - cOxk Oxk

(4)

The solution of the equation (3) instead of v 2 should produce the same results. However, from the computational point of view, the following advantages can be identified:

69 In the original v 2 equation, the sink is represented by ev2/k, which is difficult to reproduce correctly in the near-wall layer as e becomes large and v 2 / k tends to zero as the wall is approached; instead, in the (vg/k) the sink term contains the turbulence energy production T', which goes smoothly to zero at the wall leaving only the stable viscous diffusion as dominant near wall mechanism. Because (v2/k) c< y2 when y ~ 0, the wall boundary condition for (v2/k) deduced from the budged of (v 2/k) equation in the limit when the wall is approached, reduces to the balance of only two terms (with neglect of X) with a finite value at the wall, the elliptic relaxation function f and the viscous diffusion D"(v--~/k), whereas P(v--~/k)/k varying with ya (in fact with y4 when eddy viscosity is used) goes to zero at the wall: fw = y---,O lim

-2.(.~/k)

(5)

y2

- The above boundary condition is more convenient and easier reproducible as compared with fw in the v 2 - f model. In fact, the boundary condition for fw (eqn. 5) has the identical form as that for ew and can be treated in analogous manner in the computational procedure. The mere fact that both the nominator and the denominator of rio are proportional to y2 instead of/14 as in the original v ~- - f model (with y = 0 being a singular point in both cases), brings improved stability of the computational scheme.

THE ~ - f MODEL (UMIST) m

The first model variant, introduced in [ 13], with the new variable denoted as 99 v 2 / k - hence hereafter labelled as the ~p - f model, uses a reformulation of the elliptic function f in order to make it tend to zero at the wall: =

-

f = f +

2v(v~vk)

+ vV2qo k The corresponding ~ and f equations, to be solved together with the k and e equations (1, 2) are:

D~ ~ 2 ut Oqo Ok 0 D t = -f - "P-k -{ k a k Ox j Ox j "+" ~ L2V2f - f = ~(C~ - 1) ~ -

(6)

[ I/t O_~ ] I. ~'~J

(7)

- C2-~ - 2 u- Ocp O___k_k_ uV2cp

(8)

, L = CL max

(9)

k OZj OZj

Where ut, r and L are defined as:

ut = C.qok'r, "r = max

, C~-

v

, C,7 ~ 1

This modification ensures the correct behavior of f far from the wall in contrast to the LDM formulation of [14]. The coefficients used in this model are summarized in the table below: In Figure 1 the budgets

.~1,~

Cel

Ce2

cr~

ak

C1

C2

C~

CL

C,7

0.22

1.4['1+0.05,/!] \ V~/

1.85

1.3

1.0

1.4

0.3

6.0

0.25

110

f

of the terms in equation (7) are presented. The production of cp comes from f, while the destruction is represented by T'qD/k. The diffusion term on the right hand side of (7) compensates for the misalignment of the maximum between the production and destruction by transporting q~ this into the near-wall region.

70

Figure 1" Budgets of ~ in a channel flow

Figure 2: Friction coefficient on a flat plate

The cross-gradient term is mainly a sink term only positive in the viscous sub-region and it could be neglected if f was decreased by altering the coefficients in the equations, as discussed below. The model has been implemented in an unstructured finite-volume code [2] and tested in different configurations as reported in [13]. As an illustration of the basic model verification, we show in Fig. 2 the wall friction factor in a boundary layer on a fiat plate computed with the ~ - f model, LDM version of the v 2 - f and Menter's SST model [15], compared with the experiments of Wieghardt and Tillman [19]. It can be seen that the ~ - f model outperforms both the LDM and SST model. The computation was done with the same CFL number for all models on a mesh clustered towards the wall, with 80 cells in the wall normal direction. Other tests lead to similar conclusions: in the flow over periodic hills the results for all the formulations of the model were similar, whereas in an asymmetric diffuser the predictions were improved, as illustrated by profiles of the mean velocity and the streamwise turbulent stress at two locations in Fig. 3. Here below we present recent application of the ~ - f model to natural convection in a tall cavity.

Figure 3: Mean velocity and streamwise turbulent stress at two locations in an asymmetric diffusor. Symbols: experiments, Buice & Eaton (1997); lines: computations

THE ~ - f MODEL (TUD)

This model variant proposed in [ 12] denotes the new variable as ~ -- v 2 / k - hence hereafter denoted as - f model - differs in two features from the ~ - f model. First, in order to reduce the ~ equation to a simple source-sink-diffusion form, the term X (equation 4) is omitted. As shown in Fig. 1, this term is not significant, though close to the wall it has some influence. In order to compensate for the omission of X one can re-tune some of the coefficients. Another novelty is the application of a quasilinear pressure-strain model in the f-equation, based on the formulation of Speziale, Sarkar and Gatski (1991) (SSG), which brings additional improvements for non-equilibrium wall flows. It was shown by Wizman et al. [20] that the elliptic relaxation model based on SSG requires significantly less reduction of pressure strain than the original model based on IP. This, together with Fig. 1 explains why the cross diffusion term X could be omitted in the ~ - f model. The computations of flow and heat transfer in a plane channel, behind a backward facing step and in a round impinging jet show in all cases satisfactory agreement with experiments and direct numerical simulations. The incorporation of the quasi-linear SSG model for the pressure-scrambling tclTn 2

Hij,2 -- -C~TPaij %-C3kSij %-C4k(aikSjk + ajkSik -- -~(SijaklSkl) %"C5k(aik~jk %"ajk~ik)

(10)

into the wall normal stress component, with the neglect of 7")22(~ 0), yields the following form of the f equations in conjunction with the ~ equation (3) (with X = 0)"

L2V2/-f = ~1

(C1

--1) %"C z

( -

~

- C5

~-

(11)

Adopting the coefficients for the SSG pressure-strain model, with C~ = 0.65 and C5 = 0.2 (for arguments see [12]) and noting that the last term in equation (1 l) can be neglected as compared with the first term because ( C 4 / 3 - Cs) ~ 0.008, we arrive to the following set of model equation constituting together with the the k and ~ equations (1, 2)- the ~ - f model: (12)

vt = C , ~ k 7

1 L e v 2 f _ f -- -~ (C1 - 1.0) + C;~-

Dt = f -

o[(

T) + -~x k

u + -~<

( -

(13)

l

~

(14)

with wall boundary conditions ffw = 0 and fw = l i m y - ~ o ( - 2 ~ / Y 2) (eqn. 5 ). The equations set is completed by imposing the Kolmogorov time and length scale as the lower bounds, combined with Durbin's (1996) realizability constraints: "r = m a x

rain

, x/~CuISI~.

[ (2 L = CL m a x

rain

k~

(15)

, C~- u_e

]r ~ (_~)1/4] (16)

, x / 6 - ~ u i S l ~ ] ' Co

where a <_ 1 (recommended a = 0.6). The following coefficient are recommended:

let,

Io.22

eel

ca2

el

e;

(7k

0-6

o-~

C-r

eL

Col

1.4(1+0.012/~)

1.9

1.4

0.65

1

1.3

1.2

6.0

0.36

85 I

72 It is noted that instead of using equation (5) for f~, one can make further simplifications to satisfy zero wall boundary condition for f (in analogy with the original Jones-Launder (1972) formulation of the low-Re-number dissipation equation) by solving equation (13) but for f with JTw = 0 and getting f from:

1/2~ 2

f=

f + 2u

(0(

-~z~J

(17)

which is then used in ( equation. The viscous terms in equations (17) and (6) have the same near-wall asymptotic behaviour. However, as also experienced with the e equation, this is not necessary since the boundary condition f,,, = - 2 ( / y 2 is very robust. Basic validation and tuning of the ( - f model was performed with respect to DNS of a plane channel at several Re numbers (Re~ = 390, 590 and 800). In Fig. 4 profiles of velocity and turbulent quantities (including ( and f) are shown, nondimensionalised with the inner-wall scales for a channel flow at Re~ = 800, and compared with the DNS of Tanahashi et al. (2004). Note that results with ( - f model are obtained with the mesh with the wall-nearest cell-center at y+ -- 0.01, which corresponds to the first y+ used for DNS. The agreement for U +, k + and uv + is excellent over the whole cross-section, and so is for ( in the wall region where the elliptic effects are influential. A further DNS-based model validation is illustrated in Fig. 5 for a pulsating turbulent channel flow generated by an imposed sinusoidal pressure variation for the Stokes length scale 1~=14 (where l~ = x/2/a; +, a~+ -- wu/U~ and a~ is the forcing frequency), for which Scotti and Piomelli reported recently large eddy simulations. No special remedy was introduced to capture high-frequency unsteadiness, yet the reproduction of the cyclic variation of the wall shear stress and of velocity profiles at different phases is quite satisfactory, providing support of adequate response of the ~ model to the imposed unsteadiness and its applicability to unsteady flows.

Figure 4: Velocity and turbulent quantities in channel flow Re~ -- 800. Symbols: DNS data of Tanahashi et al. (2004). Full line: ( - f

73

symbolsi LES 2-

30-

~=

1-

20v

0-

"

lines: 44 model 0~/8 1~/8 2ff8 3d8 4~/8 5tl8 6t/8 " " 7z/8

) 10-

I~. -1' 0

'

o'.2'

~'fmodel o'.,

[

'

o'.6

'

o'.8

'

o'"0'.1

........

i

'

' 'y;'"'l'o

.......

1';o'

'

'

Figure 5: Pulsating Channel, Re~- - 350. Cycle variation of the wall shear stress (left) and velocity profiles at different phase angles. Symbols: LES of Scotti and Piomelli (2001). Lines: ~-f model SOME APPLICATIONS TO HEAT TRANSFER We present now some results for heat transfer obtained with the ~ - f and ~ - f models. First, we consider forced convection and present some results of computation of velocity fields and heat transfer in two generic test cases: a separating flow behind a backward facing step and in a round impinging jet. The temperature field was obtained by solving the RANS energy equation with constant fluid properties, using the isotropic eddy diffusivity z4/aT where ut is given by equation (12) and ~r = 0.9

Backward-facing step flow and heat transfer. Figure 6 presents the mean velocity profiles at several cross-sections, as well as the friction factor and Stanton number along the bottom wall behind a step in a backward facing step flow. Computations are performed with the standard Durbin's v 2 - f and the ~ - f models. Velocity profiles obtained with the two models at several locations within the separation bubble and around reattachment are practically indistinguishable and both in good agreement with experiments of Vogel and Eaton (1985). The same can be concluded for the friction factor and Stanton number in the recirculation bubble and a few step height downstream from reattachment, but a more significant difference appear further downstream in the recovery region, where the ( - f model captures better the trend of the friction factor, and slightly worse for the Stanton number. The difference between the two models could be attributed to the difference in the pressure strain models. The agreement with the experiments in general can be regarded as fully satisfactory for both models. Round impinging jests. Figure 7 shows the mean velocity profiles and Nusselt numbers in a normally impinging round jet issuing from a fully developed pipe flow for two Reynolds numbers, 23000 and 70000. The distance between the pipe exit and the plate is H / D = 2. The results are compared with the available experimental results of Cooper et al. [8] for the flow properties and with Baughn and Shimizu (1989) and Baughn et al. (1991) for heat transfer, as well as with the v 2 - f computations using the coefficients of Behnia et al. (1998). The velocity profiles (left) for both Re numbers are similar with some improvements retumed by the ( - f model, especially for r / D - 3.0. Similar quality of predictions has been obtained for the Nusselt number (right), especially for Re=23000, where the ( - f model shows some slight overprediction, but a more realistic shape of the curve. For Re=70000, both models show visible overpredictions in the stagnation region, though much better than reported by most other turbulence models. Here to, the ( - f model shows some improvement over the v 2 - f model. Natural convection in a tall cavity. The natural convection cavity studied by Betts and Bokhari [6] has been computed using the LDM, Launder-Sharma k-e and the qo - f model. The flow inside the cavity is driven by the temperature difference between the vertical walls creating boundary layers where the shear stress changes rapidly close to the wall, thus requiring a model capable of resolving correctly

74

Figure 6: Backward-facing step, Re = 28000. Velocity profiles, friction factor and Stanton number. Symbols: experiments of Vogel and Eaton (1985). Full line: (-f; dotted line: v2-f model

Figure 7: Impinging jets, Re=23000 and 70000. Velocity profiles and Nusselt number. Symbols" experiments of Baughn and Shimizu (1989), Baughn et al. (1991). Full line" ~-f; dotted line: v2-f the near wall region. The experiments were done at two different Rayleigh numbers, Ra - 0.86x106 and 1.43x106 in a cavity with an aspect ratio of H / W = 28.68. In Figure 8 the velocity profiles are plotted for two Rayleigh numbers at selected locations. The predictions of vertical velocity by the qD- f model are in much better agreement with the experiment than other two models, especially in the middle and the top portion of the cavity for both Rayleigh numbers considered. The LDM model over predicts the velocity peak in the middle of the cavity, with an overprediction at higher Rayleigh number of 40% and 31%. Near the horizontal walls, the ~ - f model slightly underpredicts the velocity on the decelerating side, but the slope of the velocity profile in the mid-width is closer to the experiment, compared with the LDM. The temperature profiles for the high Rayleigh number can be seen in Figure 9. In terms of temperature profiles there is not much difference between the two formulations, both give good predictions at the different stations of the cavity.

75

Figure 8" Velocity profiles at selected cavity heights for Ra = 0.86x106 (left) and Ra = 1.43x106 (right).

Figure 9: Temperature profiles at two characteristic cavity heights for Ra = 1.43x106. CONCLUSIONS

A new version of the Durbin's elliptic relaxation eddy-viscosity model, based on solving the time scale ratio v2/k instead of v 2, has brought significant improvements in the model robustness and in quality of predictions, the major features of the new model are the new boundary conditions for the elliptic relaxation function, and fixed sign source terms. Two model variants, developed independently at UMIST and TU Delft, follow the same rationale, but differ in some minor details. Each model variant has been implemented in an in-house unstructured CFD codes, both using uncoupled solvers, and tested in a variety of attached, separating, impinging and buoyant flows and heat transfer with very different grids. In all cases considered, the model confirmed the already known advantages of the elliptic relaxation approach and yielded good agreement with the available experiments and DNS data. The new model is envisaged as a prospective model for industrial CFD, especially when wall friction and heat transfer are in focus. Acknowledgement. The work at TU Delft has been supported by the EU CEC Research Directorate-General through project MinNOx, Contract No. ENK6-CT-2001-00530. The development and isothermal applications of UMIST's model has been supported by the FLOMANIA project, sponsored by EU Commission, Growth Programme, Contract No. G4RD-CT2001-00613.

76

References [1] 7th ERCOFTAC/IAHR Workshop on refined flow modelling, UMIST, Manchester, UK, 1998. www.ercoftac.org/databases [2] Archambeau F., Mechitoua N. and Sakiz M. A finite volume method for the computation of turbulent incompressible flows- industrial applications. International Journal on Finite Volumes, To appear. [3] Baughn, J. and Shimizu, S. 1989. Heat transfer measurements from a surface with uniform heat flux and an impinging jet. ASME Journal of Heat Transfer 111 1096-1098. [4] Baughn, J.W., Hechanova, A.E. and Yan, X. 1991. An Experimental Study of Entrainment Effects on the Heat Transfer From a Flat Surface to a Heated Circular Impinging Jet. Journal of Heat Transfer 113, 1023-1025. [5] Behnia, M., Parniex, S. and Durbin, P.A. 1998. Prediction of heat transfer in an axisymmetric turbulent jet impinging on a flat plate. Int. J. heat Mass Transfer 41 1845-1855. [6] Betts EL.and Bokhari H. New experiments on natural convection of air in a tall cavity. In 4th UK National conference on heat transfer, pages 213-217. IMechE Conference transactions, 1995. [7] Buice C.U. and Eaton J.K., Experimental investigation of flow through an asymmetric plane diffuser, Dept. Mech. Eng., Stanford University, TSD-107, 1997. [8] Cooper D., Jackson D.C., Launder B.E. and Liao G.X. Impinging jet studies for turbulence model assessment- I. Flow field experiments. Int./J. Heat Mass Transfer, 36(10); 2675-2684, 1993 [9] Davidson L., Nielsen P.V. and Sveningsson A. Modifications of the v 2 - f model for computing the flow in a 3d wall jet. In K. Hanjali6, Y. Nagano, and M. Tummers, editors, Turbulence, heat and mass tranfer, pages 577-584, 2003. [ 10] Durbin, P.A. 1991. Near-wall turbulence closure modelling without 'damping functions'. Theoret. Comput. Fluid Dynamics, 3, 1-13. [ 11] Durbin, P.A. 1996. On the k - r stagnation point anomaly. Int. J. Heat Fluid Flow 17 89-90. [12] Hanjali6 K., Popovac M. and Had~iabdi6 M. A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD. Int. J. Heat Fluid Flow, (in press, 2004) [13] Laurence D.R., Uribe J.C. and Utyuzhnikov S.V. A robust formulation of the v 2 - f model. Flow, turbulence and combustion, (in press, 2004), www.kluweronline.com/issn/1386-6164/contents, prepublication date: 04/14/2004. [ 14] Lien F.S.and Durbin P.A. Non-linear k - E - v 2 modeling with application to high-lift. Center for Turbulence Reasearch, Proceedings of the summer school program, pages 5-25, 1996. [15] Menter F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, pages 1598-1605, 1994. [ 16] Speziale, C.G., Sarkar, S. and Gatski, T. 1991. Modelling the pressure-strain correlation of turbulence: an invariant system dynamic approach. J. Fluid. Mech. 227, 245-272. [ 17] Tanahashi M., Kang S.-J., Miyamoto T., Shiokawa S. and Miyauchi T, 2004. Scaling law of fine scale ediiesin turbulent channel flow up to ReT = 800. Int. J. Heat Fluid Flow 25, 331-340. [18] Vogel, J.C., and J.K. Eaton, 1985. Combined Heat Transfer and Fluid Dynamic Measurements Downstream of a Backward-Facing Step, ASME, J. Heat Transfer, 107, 922-929. [ 19] Wieghardt K. and Tillman W. On the turbulent friction layer for rising pressure. Technical Report TM-1314, NACA, 1951. [20] Wizman V., Laurence D., Kanniche M., Durbin P., Demuren A. Modelling near-wall effects in second-moment closure by elliptic relaxation Int. J. Heat Fluid Flow 17, 255-266, 1996.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

77

CALIBRATING THE LENGTH SCALE EQUATION WITH AN EXPLICIT ALGEBRAIC REYNOLDS STRESS CONSTITUTIVE RELATION H. B6zard 1 and T. Daris 2 10NERA, Department of Modelling for Aerodynamics and Energetics, BP 4025, 31055 Toulouse, FRANCE E-mail: [email protected] 9 SNECMA Motors Villaroche, Rond Point Ren6 Ravaud, 77550 Moissy-Cramayel, FRANCE E-mail: thomas.daris @snecma.fr

ABSTRACT The way of calibrating a two-equation model associated to an explicit algebraic Reynolds stress (EARSM) constitutive relation is shown. The model is first expressed in a very generic way so that any type of turbulent scale can be considered. Analytical relations involving the model constants are derived from basic physical behaviours that the model has to fulfill. A more restrictive constraint coming from the EARSM expression arises compared to a classical eddy viscosity model. As too many degrees of freedom appear, choices are made to simplify the relations. A new k - k L model is proposed that fulfills all constraints. A near-wall model is developed that gives a linear evolution of the second scale in the inner region of the boundary layer. The model is applied in simple boundary layer flows and in more complex engineering flows involving Navier-Stokes computations.

KEYWORDS Turbulence models, algebraic Reynolds stress models, EARSM, length scale, boundary layer, APG, RANS, CFD.

INTRODUCTION Explicit Algebraic Reynolds Stress Models (EARSM) need the knowledge of two turbulent scales, e.g. the turbulent kinetic energy k and a second turbulent scale which are obtained from their transport

78

equations. Usually the EARSM expression is calibrated independently considering homogeneous flows and the underlying two-equation model is taken from an existing one where the constitutive relation is changed from the linear Boussinesq expression to the non-linear EARSM expression. However modifying the constitutive relation modifies also the behaviour of the two turbulent scales and the model needs a new calibration. Moreover the choice of the two-equation model and especially of the second determining scale and of its transport equation is very important and can greatly influence the overall behaviour of the model, often more that the change of the constitutive relation itself. To calibrate a turbulence model, it is important to check that a number of basic flow characteristics can be reproduced. For example it is a well known behaviour, as seen in experiments, that the logarithmic law of the wall for a boundary layer velocity profile is not influenced by the pressure gradient. However most existing two-equation models are not able to reproduce this basic behaviour, as shown by Huang & Bradshaw (1995). Another important feature to achieve is that the computed solution of a turbulent flow should not be sensitive to the boundary conditions, e.g. to the value prescribed to the turbulent scales outside the flow. The solution should have smooth velocity and turbulence profiles at the edge of the turbulent flow, as seen in experiments. However some two-equation models are not independent to the free-stream turbulence or predict too sharp profiles at the edge, as shown by Cazalbou et al. (1994). The EARSM relation can even worsen this behaviour. The procedure to calibrate a two-equation model associated to a non-linear EARSM constitutive relation will be presented. The method is applied to provide a new k - k L model, associated to the EARSM constitutive relation of Wallin & Johansson (2000), which is able to reproduce a number of basic flow features encountered in zero (ZPG) and adverse (APG) pressure gradient flows.

GENERIC EQUATIONS We follow the approach of Catris & Aupoix (2000) extended for a non-linear constitutive relation. The two-equation model is expressed as a k - r model where the second turbulent scale r is sought as a general combination of k and e, say r = kme n, where m and n are not yet prescribed. r can represent any turbulent scale such as e, ~ = elk, L = k3/2/e, etc. Developments are made in the context of two-dimensional incompressible high Reynolds-number flows. The k - r equations are sought as:

Dt = Pk - e + div De r (Cr Pk - Cr Dt = k + Cr162162 q)

+ div

9grade + Cck

gradk grade

(1)

+ div

(

u~' r \aOk ~gradk

grade 9gradk + Ckk

) (2)

gradk, gradk

The constitutive EARSM relation from Wallin & Johansson (2000) giving the expression between the t t Reynolds stresses u~uj and the velocity gradients ~Oui reads"

2 ( Oui - ~ ' ~ + 5 k6~ = ~; o~j

OUj) Oxi

~(ex) - k'~ij

(3)

The first right hand side term of Eqn. (3) is a linear term equivalent to the classical eddy viscosity relation. The equivalent eddy viscosity reads u~ = C~kT-, with 7- = k/e, where the coefficient C~ is ~(ex) non constant and depends on the turbulent scales and the velocity gradients. The second term '~ij is a non-linear term responsible for anisotropy of the Reynolds stresses and is a complex expression of the

79

turbulent scales and the velocity gradients. The complete expression of C~ and a}~.x) will not be given here and can be found in Wallin & Johansson (2000). Eqns. (1) and (2) are generic and can represent any type of turbulent models. The Cr and Cr coefficients of the production and destruction terms of the C-equation are related to the classical C~1 and C~2 coefficients of a standard c transport equation through: Cr = n C ~ + m and C~2 = nC~ 2 + m.

CONSTRAINTS Relations between the eight model c o n s t a n t s Cdpl, Cdp2, Cdpdp, Cdpk, Ckk, ok, O'dp, O'dpIcand the constants of the EARSM expression can be obtained by forcing the model to reproduce a number of basic flow behaviours, following Catris & Aupoix (2000).

Homogeneous Flows The first flow to address is the decay of isotropic turbulence. This homogeneous flow has no velocity gradient and no diffusion. By representing the energy spectrum by two power laws and by assuming a self-similar decay of isotropic turbulence, it can be shown that the destruction coefficient of the cequation takes the form C~2 = (3s + 5)/(2s + 2) with s being the spectrum exponent, s is bounded between 1 and 4 so that C~2 lies between 1.7 and 2, see Aupoix (1987). Usually a value of 1.92 is retained, based on experiments. Another interesting flow is homogeneous shear where there is no diffusion. From the k and c transport equation, it can be deduced a transport equation for the length scale L = k3/2/c which reads: d L / d t = (3/2 - C~I)PkL/k - (3/2 - C~2)r

(4)

As the mean flow does not impose any length scale in an homogeneous flow, the first source term linked to the velocity gradient in the length scale equation should be null, e.g. C~1 = 3/2. ZPG and Weak A P G Boundary Layers

The analysis of the behaviour of turbulence models in the logarithmic region of a weak APG boundary layer is based on the work of Huang & Bradshaw (1995). To study the logarithmic region, the momentum and transport equations are written in wall variables: y+ = yu~ , u+ = u

p+ =

v dp

k+ =__k

r

e u 2m-4~

_- u'v'

where y is the wall distance, u~- the friction velocity, p the fluid density, v the fluid viscosity, dx dp the longitudinal pressure gradient and u'v' the shear stress. By neglecting the convection and the diffusion, it can be shown that: -u-7-V-Tc+ - 1 + p+y+. For a ZPG boundary layer, dimensional analysis yields the logarithmic law for the velocity profile: u + -- ln(y+)/n + C with the yon K~irm~n constant n ~ 0.41. For weak APG, i.e. at low values of p+y+, the quantities are sought as Taylor expansion of p+y+. In particular rc is sought at first order as: ~c = n0 + r;lp+y + where no is the classical value for ZPG (~o = 0.41) and eCl represents the effect of the pressure gradient on the logarithmic law slope. By developing the transport equations in terms of p+y+ and equalling the terms of same order, the values of ~0 and ~1 can be obtained as functions of the model constants. At order 0, i.e. for ZPG, it reads:

~0 =

-- 2b12o(Cr -- Cr ) n~(c~ + 1/~)

(6)

80

where b12o -- u ' v ' / k is the value of the anisotropy component in the ZPG log region. It is only a function of the EARSM model constants and experiments show that it is close to -0.15, which is the value found with Wallin & Johansson (2000) EARSM model. At order 1, i.e. for weak APG, ~1 is a complex function of all the model constants and will be not reproduced in the present paper. It can be shown from experiments that this slope does not change with the pressure gradient, even for strong APG, see Sk~re & Krogstad (1994). It means ~;1 -- 0, which provides a new relation that the model constants have to fulfill. S t r o n g A P G B o u n d a r y Layers

For strong APG, Townsend (1961) found the so-called "half-power" region located above the logarithmic region where the velocity profile evolves as the square-root of the wall distance. The analysis is performed by using the "pressure" variables: (pdp)

~=

1/3

y Up

' ~)= I/

~x

~t -- ~ ~

~ __ _k_

Up

"~" t ?.try

~) _.~ ~)~t 2pm - 4n

?_t2p

ut v!

l In

?.tp2

Assuming a strong pressure gradient (p+y+ >> 1) and assuming the Bradshaw relation - u ' v ' = k x / ~ 0.3k is valid in this region, it can be shown that:

(7) =

h

- ~ , v , = ~, ~ = 2 / ~ ,

~ = ~/~,

~ = ~+n/~o;~/~-"

(8)

The solutions of the equations are sought as power functions of the wall distance: ti = Au$ "~ , k = Ak~:TM , q~ = ACq~'~ with A ~ = 2/~, Ar = t~,"~-m/2~-'~ , ru = l / 2 , rk = l , r e = r e + n ~ 2

Ak

=

1/~-O~,

(9)

Introducing the expressions (9) into the transport equations where convection is neglected and identifying the powers of ~ leads to the correct values for the exponents ru, rk and re. However this only means the model is dimensionally correct but not that it is able to reproduce the half-power law for the velocity profile. Through the transport equations the expressions of the coefficients Au, Ak, and Ar are found as functions of the model constants, but they are too complex to be reproduced here. Specifying that these coefficients take the values presented in (9) leads to new relations that the model constants have to fulfill. E d g e o f a T u r b u l e n t Flow

The analysis of the behaviour of a turbulent model at the edge of a turbulent flow is based on the work of Cazalbou et al. (1994). The main result is that the transported turbulent scales have to go to zero in the outer flow, which ensures the independence of the inner flow to the residual turbulent values prescribed at the boundary of the computational domain outside the turbulent region. The analysis for a boundary layer is performed using the outer variables:

Y

A = 1 - ~-,

u(~)= ~-

u~-

~

'

K(~)= u--~~ k E(~)=c6 ~(~)= ' u--~~ '

r

%_2~-3"5"

(10)

where 5 is the boundary layer thickness and u~ is the outer flow velocity. The solutions of the quantities are sought as power functions of A, say:

{

u ( ~ ) = Uoa ~o , K ( a ) =

Koa ~ , E(a)=

9 (A) = (I)oA~ with (I)o = K ~ E ~

Eoa ~

and er = reek + ne~

(11)

81

The edge of the turbulent flow is approached when y ~ (5, i.e. when A --~ 0 and the quantities go to zero in a smooth way if the exponents of the power functions follow: eu > 1 , ek > 1 , er > 1. It must be pointed out that these constraints are made on the transported scales which means that all turbulent scales are not equivalent from this point of view. Introducing expressions (11) into the momentum and transport equations and identifying the same powers of A gives relations between the exponents and their expression as functions of the model constants. Particularly it can be shown that e~ = 2ek - 1, which implies er -- ek(m + 2n) - n. It can also be shown that ek = akeu. Other constraints have to be added to the previous ones as the edge problem is a convection-diffusion problem where production and dissipation are negligible. It can be shown that the convection and diffusion leading term exponent is ek - 1 and that the production leading term exponent is 2eu - 1, so that production is negligible if 2eu - 1 > ek - 1, i.e. ek < 2eu, which implies crk < 2. This is a classical constraint for linear eddy viscosity models. However associating the EARSM constitutive relation leads to a more restrictive constraint. Indeed in EARSM models, the production to dissipation ratio Pk/e is a solution of the algebraic equation for the Reynolds stress tensor. As a consequence the effective C~ is a function of Pk/c and decreases when the ratio Pk/c increases, e.g. in strong shears. However at the edge the shear goes to zero and one would expect Pk/c to go to zero as well, which implies that the effective C~ reaches asymptotically a finite value at the edge which depends only on the values of the constants of the EARSM expression. They are no strong physical argument for the behaviour of Pk/c at the edge. However it can be evidenced by the mean of DNS as in Spalart (1988) for a flat plate boundary layer. The fact that Pk/e goes to zero at the edge should thus be considered as a new constraint for an EARSM model compared to a classical linear eddy viscosity model where C~, is a constant and does not depends on Pk/c. It is straightforward to show that Pk/c has the power eu - ek, so that it goes to zero if ek < eu, which implies crk < 1. The violation of this constraint leads to a non-physical behaviour of the flow at the edge, as shown by Hellsten (2004) for a k - w EARSM model. Finally the solution for the behaviour at the edge is given through a second degree equation, which means two possible solutions. But it is not possible from the analytical expressions to know which one is the physical solution and a numerical resolution is only possible in this case. This equation reads:

{ (Cr162 [Cr r _~_~~1)-]-ek(Cck -'1---)--Cu] -+'Ckke2k'-Oo'r

(12)

with er = ek(m + 2n) - n and ek = crkeu

MODEL CALIBRATION As too many degrees of freedom appear in the previous expressions, the model is simplified by cancelling the third cross term (Ckk = 0) because it does not appear for the behaviour in the logarithmic region and the extra-diffusion term (ack = c~). In this case Eqn. (12) for the behaviour at the edge simplifies and one of the two solutions is ek = n / ( m + 2n). It is possible to set this solution unphysical and to keep the other as the physical one by choosing n < 0 and m + 2n > 0. A solution is m = 5/2 and n -- - 1 , which provides a k - kL model. The kL-scale is quite interesting as it goes naturally to zero at the wall and at the edge of the turbulent region and evolves linearly will the wall distance in the logarithmic region. Finally the model constants resulting from all constraints are: Cr

; Cr

= 0 . 5 8 ; Cr162= - 1 . 3 8

; C c k = 1.52 ; ak = 0 . 9

; ar

(13)

The values of the coefficients that characterize the different behaviours seen previously are deduced from the relations and are shown in Table 1. The new k - kL EARSM-WJ model is the only existing

82

model that fulfills almost all the prescribed constraints. Only for the half-power region an improvement could be obtained as the predicted slope Au for the velocity profile is lower than the theoretical value. However it should be pointed out that none of existing models could even predict real values for Au. Moreover all existing models predict non-zero value for ha, which means they all predict a dependence of the logarithmic law slope to the pressure gradient, which is unphysical. The exponents eu and ek for the behaviour at the edge are quite high to ensure very smooth profiles for the velocity and the kinetic energy, as it can be seen in experiments. TABLE 1 CHARACTERISTICS PREDICTED BY THE K-KL EARSM-WJ MODEL Coefficient Exp. orTheory Present model

n0 0.41 0.41

/,i;1 Au 0 4.8 (2/n) 0 3.1

Ak 3.3 ( 1 / ~ C , , ) 4.4

Ar 8.3 6.5

eu >l 21

ek >l 18

er >l l0

NEAR-WALL MODEL To be used in wall bounded-flows, a near wall model has to be developed. It is possible to build a linear length scale L from the wall to the logarithmic region. As the turbulent kinetic energy behaves as y2 close to the wall, the/eL-scale would behave also as y3, which is not recommended for grid dependency purpose. However it is possible to build directly a linear/eL-scale by adding a suitable term in the ~eL transport equation. The behaviour of the model in the near-wall region of the boundary layer is analyzed using the momentum and transport equations written in wall variables where convection is neglected, following Cousteix et al. (1997). To obtain the missing terms in the/e and ~eL transport equations to equilibrate the model at the wall, an analytical model for the velocity is used. It is based on the mixing length model of van Driest and reads:

du + dy +

v/1 + 4r12- 1 with r / = ny+ [ 1 - exp ( - ~ 6 ) 2rl2

(14)

The eddy viscosity profile is obtained by integrating the momentum equation which is simply (1 + l,/t+) du+ = 1. The diffusion of the/e-equation is equilibrated at the wall by adding the classical viscous destruction term -2u/e/y 2, which provides the y2 behaviour of/e at the wall. It is then assumed that the /eL-scale is linear and reads r = kL = n y / C y 4. The/e and ~eL equations can be solved using the van Driest analytical velocity and eddy viscosity profiles and the prescribed form of the kL-scale, so that the missing term needed to equilibrate the/eL-equation can be obtained numerically. Different analytical forms for this term have been proposed but most of them have exhibited numerical instabilities. Finally one term exhibits good robustness and reads: r - c~f~

.~y

y~

with f w - exp(-CfwRy),

Ry-

, c~

= 0.00075,

cs~ = 0.068

(15)

The fw wall damping function is necessary in the logarithmic region as the variable r behaves as ya/2 instead of going to zero. Now the equations can be solved without prescribing the eddy viscosity but using the EARSM constitutive relation. A near-wall model is also needed for the EARSM expression. It follows the model of Wallin & Johansson (2000) which is based on a bounding of the turbulent time scale and on a damping function multiplying the Reynolds stress anisotropy components calculated at equilibrium. As a consequence the equivalent eddy viscosity takes the form:

83

{ t*~ -- flC~kT" with ~- = max ({, C~X/~) , c = k5/2/r + 2 u k / y 2 , C.,- = 6 fl - 1 - e x p ( - C y t ~ / ~ - Cy~Ry) with Cul = 0.0105 and Cu2 = 0.025

(16)

Finally the k and r = k L transport equations including near-wall terms read:

D t = Pk

r

y2

Pk - Cr

t- div

u+

gradk

crk /

- C ~ fw u-)~(~ + div

u+

(17) grade

Dt

(18) + 6r162 grade 9grade + Cck--~gradr 9gradk

The boundary conditions associated to the k - k L model are simple and natural as both scales go to zero at the wall. In the outer flow a turbulence level Tu is prescribed to give the turbulent kinetic energy value koo -- ~.%o~u. 3 ~ 2 ,7-,2 The k L value in the outer flow is given through a prescribed level of the eddy viscosity: ( k L ) ~ = ut~ ~ / C ~ . Usually a value of ut~ = 0.01uoo is taken but the result is not sensitive to it.

APPLICATIONS

ZPG Boundary Layer The first flow to address is the flat plate boundary layer. As the kL-scale is built to behave linear in the near-wall region to give a relative grid independency, it is important to check the effect of the grid refinement at the wall on the predicted friction coefficient. To perform tests in boundary layers, the k - k L EARSM-WJ model has been implemented in a parabolic boundary layer code developed at ONERA. Figure 1 shows the evolution of the friction coefficient Cf with the Reynolds number based on the momentum thickness Re for different distances to the wall of the first grid point y+. 0.003

i[Oo

z

Io Cf theo

i~~ o00

--

i",~Oo_

,-

.- - -

,.:-,,~

. . . . . . .

"--...."".,,.,.

o , o.0o

y+,=0.1

l I

y,'=2Y ~ t

...... y",=5

-

/ !

0.0015-

0.001

:

0

I 25000

i I 50000

I 75000

100000

R0

Figure 1: Flat plate friction coefficient predicted by the k - kL EARSM-WJ model. Effect of the first grid point height. The calculated evolutions are compared to the K~rm~in-Schoenherr correlation. It can be seen that up to y~- -- 2 the effect is negligible with less than 1.5% variation on Cf. For y+ = 5 and y+ = 10 the Cf decreases respectively by about 4.5% and 10%, which is not so much in relation to these high values of y+. It proves that the k - k L model is not very much sensitive to the wall grid refinement, which is a consequence of the linear kL-scale.

84

I

I

I

__-~" k-kL EARSM-WJ

30-

I

4

~ / "

I

3

I

I

106

I

k-kL EARSM-WJ

"m

10'

20-

~

U§

k+2

_

kL +

10-

0

1

i 10'

10 ~

I 102

0

y"

103

104

102 y§

] 0~

Figure 2: Velocity, turbulent kinetic energy and k - k L EARSM-WJ model.

kL

I

I

10'

..............................

io2 103

10'

~

10 s

I

[ ~'k-kL EARSM-WJ // o'-I-~ .... ~y'/c. . . . . . . . . . . . . . . . . . / .......

i

I 0~

I O'

102

y*

103

104

profiles for the fiat plate boundary layer at

Ro

105

= 105.

The velocity, turbulent kinetic energy and k L profiles are shown in Figure 2 for R o = 10 5 in semilogarithmic coordinates. The logarithmic law for the velocity is perfectly matched by the model. The k-profile exhibits a too low peak in the buffer region (k + ~ 3.4) compared to experiments or DNS which give k + ~ 4.5. However this characteristic should not be an inconvenient as the model would not need a high grid refinement to capture the peak of k. The kL-profile is very close to the theoretical profile /(-y5/4 ~fl+/~, in all the wall region, which is the expected behaviour.

APG Boundary Layer The k - k L model should exhibit a full independence of the logarithmic law to the pressure gradient as its /~1 value is null. The experiment of Sk~e & Krogstad (1994) is a good case to test the response of turbulence models to the pressure gradient. Figure 3 presents the evolution of the friction coefficient (left part) and of the velocity profile at the last experimental station (right part). The present model is compared to the experiments and to the Chien k - c, the Menter SST k - co and the Spalart-Allmaras (SA) models. !

0.003

O Exp. Skare

I

I

!

I

60-

k-kL + EARSM-WJ --0.002 -

k-~ Chien

........ k-6~ SST

~.

,

...... S/~ f - ' S ' S . . . . ~"'~ /z f , i , I , / , I / . ........... 0.001 I ; .........................

~. U" ,

I

i

I

i

J

k-kk + EARSM-WJ

~,

k-~: Chien

f~.~'

50-

---

40-

...... SA:

........ k-f#: SST

'

............ Log!law 30-

1

',, :

:8/./

/'_~//:

./ ,," ..'"/....:>'~: ............ /!.1f.'

2010-

0.0 2.8

i 3.3

318 x(m)

I 4.3

I 4.8

0 5.2

I

10 ~

l

10'

,

I

102 y+

t

t

103

0'

Figure 3" SkSre and Krogstad strong APG boundary layer experiment. Evolution of the friction coefficient (left) and velocity profile in wall variables (right). Comparison of k - k L EARSM-WJ, Chien k - e, SST k - w and Spalart-Allmaras (SA) models with experiments. The k - k L model gives good comparisons with the experiments on both the friction coefficient and the velocity profile. The best result is obtained here with the SA model, but the k - k L model gives better results than the SST model. As already known the k - e model is unable to predict adverse pressure

85

gradient flows. It can be seen on the velocity profile that the k - k L model gives the independence of the logarithmic law to the pressure gradient, as expected, on the contrary to all other models. Low Speed Airfoil Near Stall

The k - k L EARSM-WJ model has been implemented in the ONERA Navier-Stokes code ( e l s A ) and a RANS application of the model has been performed on the AIRBUS "A-airfoil" near stall conditions. This case has been experimented in ONERA F1 and F2 wind tunnel, see Gleyzes & Capbern (2003). It has been used for validation of turbulence models in ECARP and FLOMANIA European projects. The experimental conditions are U~ = 5 0 m . s -~ ( M ~ ~ 0.15), c~ = 13 ~ R e ~ = 2.1 106. In the computation the angle of attack is set to 13.3 ~ to take into account the wall interference effects. The transition is fixed at 12% chord on the upper side to simulate the natural laminar transition occurring in the experiments and at 30% chord on the lower side as in the experiments. The computations are performed with an implicit scheme in multigrid approach and the convergence for the present non-linear model is similar to what is obtained with a classical eddy viscosity model. Figure 4 shows the comparison for the pressure and friction coefficient distributions on the airfoil surface between the k - k L EARSM-WJ and SpalartAllmaras (SA) models and the experiments. _~

L

4-~, 3-

1

:

L

oExp . . . . . --k-kL

~ ....

.1.

1-

O.6

. . . . L"

EARSM:WJ SA

1 !

,xx \

0.4

:

o x o

"

i

I o ....

I Exp.

'~'!

k-kL EARSM-WJ

0.015-

-Cp

0-

-o.~

-10.0

0.2

0.4

0.6 X/C

0.8

1.0

-0.4

" ~ ,

0.6

-

. . . .

I 0.7

I

:-:-k-kL EARSI~I-~WJ

-

0.01Cf

oo

_t. . . . .

I

Exp. SA

-Cp

1-

I

oc

SA

o :x 2-

I f

I 08 X/C

i 0.9

.0

0.005

,, ,,I

o.o

~l~

-0.005

I

0.0

i

0.2

"

I

014

0.6

01.8

1.0

X/C

Figure 4" Pressure and friction coefficient distributions on A-airfoil, M ~ -- 0.15, ~ - 13.3 ~ R e = 2.1 106. Comparison of k - k L EARSM-WJ and Spalart-Allmaras models with experiments. The k - k L model underestimates the experimental velocity levels at the leading edge and on most of the upper surface, on the contrary to the SA model which overestimates them. The consequence is a lower lift coefficient predicted by the k - k L model ( C c ~ 1.44) compared to the experimental value (CL ~ 1.52) and a higher value for the SA model ( C c ~ 1.65). The pressure plateau seen on the upper surface in the trailing edge region with the k - k L model and in the experiments indicates a separation, which is confirmed by the negative Cf. Usually on this configuration most turbulence models do not succeed in predicting the trailing edge separation, as with the SA model where the flow remains attached except just at the trailing edge. However the k - k L model gives a too large separated region (~30% chord) compared to the experiments (,,~20% chord). This point is still in consideration for improvement.

CONCLUSION A new k - k L two-equation turbulence model associated with the EARSM constitutive relation of Wallin & Johansson (2000) has been developed. It fulfills a number of basic but essential flow features encountered in homogeneous flows and APG boundary layer flows. It has been shown how to translate these features into relations involving the model constants, which can be used to calibrate any other turbulence

86

model. The kL-scale is interesting as it goes naturally to zero at the wall and at the edge and is linear in the logarithmic region of a boundary layer. A near-wall model has been developed to give also a linear behaviour of the kL-scale down to the wall, which has been shown when applying the model in a boundary layer code. The new model has been calibrated to give fully independence of the logarithmic law slope to the pressure gradient, which has been demonstrated numerically in a strong APG boundary layer case. Moreover the new model is able to predict the half-power law for the velocity profile. Except the k - kL model, none of existing models is able to reproduce perfectly these features. Finally the new model has been validated in a near stall airfoil case involving RANS computations, where it has shown improvements compared to the widely used Spalart-Allmaras model. The proposed k - kL EARSM-WJ model is a promising model to be employed for CFD applications.

ACKNOWLEDGMENTS This work has partly been carried out within the HiAer Project (High Level Modelling of High Lift Aerodynamics). The HiAer project is a collaboration between DLR, ONERA, KTH, HUT, TUB, Alenia, EADS Airbus, QinetiQ and FOI. The project is managed by FOI and is partly funded by the European Union (project ref: G4RD-CT-2001-00448). This work is also supported by the French Ministry of Defence (DGA/SPA6/ST/STA) in collaboration with Dassault-Aviation.

REFERENCES Aupoix B. (1987). Homogeneous turbulence: Two-point closures and applications to one-point closures. In Special Course on Modern Theoretical and Experimental Approaches to Turbulent Flow Structure and its Modelling. AGARD-FDP VKI Lecture Series, AGARD Report 755. Catris S. and Aupoix B. (2000). Towards a calibration of the length-scale equation. International Journal of Heat and Fluid Flows 21:5, 606--613. Cazalbou J.B., Spalart P.R. and Bradshaw P. (1994). On the behavior of two-equation models at the edge of a turbulent region. The Physics of Fluids 6:5, 1797-1804. Cousteix J., Saint-Martin V., Messing R., B6zard H. and Aupoix B. (1997). Development of the k-~ turbulence model, 11 t h Symposium on Turbulent Shear Flows, Grenoble, France. Gleyzes C. and Capbern P. (2003). Experimental study of two AIRBUS/ONERA airfoils in near stall conditions. Part I: Boundary layers. Aerospace Science and Technology 7, 439-449. Hellsten A. (2004). New advanced k - ~ turbulence model for high lift aerodynamics. In 42 nd AIAA Aerospace Sciences Meeting, Reno, USA, AIAA paper 2004-1120. Huang P.G. and Bradshaw E (1995). The law of the wall for turbulent flows in pressure gradients. AIAA Journal 33:4, 624-632. Skgtre EE. and Krogstad E-A. (1994). A turbulent equilibrium boundary layer near separation. Journal of Fluid Mechanics 272, 319-348. Spalart ER. (1988). Direct simulation of a turbulent boundary layer up to Ro = 1410. Journal of Fluid Mechanics 187, 61-98. Townsend A.A. (1961). Equilibrium layers and wall turbulence. Journal of Fluid Mechanics 11, 97-120. Wallin S. and Johansson A.V. (2000). An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. Journal of Fluid Mechanics 403, 89-132.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

87

NEAR-WALL MODIFICATION OF AN EXPLICIT ALGEBRAIC REYNOLDS STRESS MODEL USING ELLIPTIC BLENDING

G Karlatiras and G Papadakis Department of Mechanical Engineering, King's College London, Strand, London, WC2R 2LS, UK

ABSTRACT

The elliptic blending approach is used in order to modify an Explicit Algebraic Reynolds Stress Model so as to reproduce the correct near wall behaviour of the turbulent stresses. The modification introduces one additional equation compared to the standard k-~ model, namely the equation for the elliptic blending parameter 7. The model was used to simulate the turbulent fluid flow and heat transfer between parallel flat plates at 3 Reynolds numbers based on the friction velocity Re,=180, 395 and 590 and the flow in a backward facing step at ReH=28,000. The model does not use the distance from the wall thus it can be easily applied to complex geometries. The comparison with DNS data or experimental measurements is quite satisfactory.

KEYWORDS Eddy viscosity models, Heat transfer, Elliptic relaxation, Explicit algebraic Reynolds stress models, Wall blockage effect.

1. INTRODUCTION Two-equation, linear eddy viscosity turbulence models are still widely used for modelling industrial flow problems. However for complex flows, for example when strong streamline curvature, adverse pressure gradient, flow separation or system rotation are present, the Boussinesq approximation is not accurate and the linear models do not give satisfactory results especially for the turbulence quantities. One of the strategies to tackle this problem is to solve transport equations for each Reynolds stress component as to resolve fully the turbulence anisotropy. However there is a need to reduce the complexity of such models while retaining these features that allow the reproduction of dynamically important phenomena, such as the stress anisotropy, the near-wall blocking effect etc. This requirement has led to the development of Algebraic Reynolds Stress Models. These models are based on the assumption, proposed by Rodi (1976), that the convection minus diffusion in the dynamic equation of each Reynolds stress component is equal to the product of the same quantity for the kinetic energy k multiplied by the individual stress normalised by k. This assumption reduces the original set of partial differential equations into a set of algebraic, non-linear

88 equations, which link (implicitly) the Reynolds stresses with the mean velocity gradients. However, the resulting system is stiff and this can lead to stability and convergence problems. A way to alleviate these problems is the development of the so-called Explicit Algebraic Reynolds Stress Models (EARSM) in which the Reynolds stress components are related explicitly to the strain rate and vorticity tensors and their invariants. Various models have been developed, for example by Taulbee (1992), Gatski and Speziale (1993) and Wallin and Johansson (2000), which provide very encouraging results and significantly reduced the instability problems. These models are still under development. Their predictive accuracy for quantities like skin friction coefficient or Nusselt number (in case heat transfer simulations are also needed) will depend on their ability to reproduce the correct near wall behaviour of the Reynolds stresses. The present paper introduces a modification to account for near-wall effects on an explicit algebraic Reynolds stress model and is based on the idea of elliptic blending of Manceau and Hanjalic (2002) and Manceau (2003). The results are validated against DNS data for the flow between parallel flat plates and experiments in a backward facing step.

2. M O D E L I N G A P P R O A C H

2.1. The Explicit Algebraic Reynolds Stress Model away from the wall Following the assumption proposed by Rodi (1976), the Reynolds stress transport equations are simplified to the following equation:

u;u;

(1)

where q0~j is the redistribution term, P0 is the production term of ulu I , ~ij is the dissipation rate and Pk is the production of turbulent kinetic energy. Each one of the above terms can be written in terms r r

of the anisotropy tensor a with componentsa~j =

uiuj

2

5;j. Different models for q0~ and e~j can be k 3 used. The choices for these two terms and the coefficients are identical to the ones used by Wallin and Johansson (2000). The LRR model proposed by Launder, Reece and Rodi (1975) was used for the redistribution term q0~j far from the wall i.e.: , r --[-

2 4 9C2 + 6 =-C~eaij + oeSij + 11 ((~ikSkj q- SikOf, kj ---~-O{,klSlkSij ) 7C 2 - 1 0

11

(2) ((~ik~'-'~kj -- ~"~ikC~kj)

where S , ~ are the mean strain rate and vorticity tensors respectively scaled with the turbulent time 2 scale 1: = k/~. For the dissipation tensor e~j the isotropic assumption was used i.e. e~j = ~e60. The production P0 can be written in tensorial form as P = . 4~S. ~(aS . + S.a ) + ~(a~ ~ a ) and Pk as 3 Pk = -~" tr(aS). Substituting these expressions in equation 1 and setting the coefficient C2 - 5/9 the following simplified expression is obtained (Wallin and Johansson (2000)):

89

l C, - 1 + ~P,)a = - ~ 15S8 + -49( a K ~ - K ~ a )

(3)

In EARSM, the anisotropy tensor a is expressed explicitly in terms of a finite number of symmetric and traceless tensor bases T (i), which are formed from the strain rate (S) and vorticity (f~) tensors i.e.

M

a = '~-' 13iT(~ . The number M of tensor bases used depends on the dimensionality of the flow; 3 for 2D

i=1 flows and 10 for 3D flows. The scalar coefficients 13iare functions of the invariants of the tensors T (i). Since the present study examines only 2D flows, the anisotropy tensor a can be written as:

{1 __ ]~IS .+.]~2 (S, ~,.,~-~,~, S).+. ~3 (S 2 --~1 HsI) where

IIs

(4)

is the trace of S 2. Substituting equation 4 into 3 and equating the expressions that multiply 1

each of the basis tensors T (i) (i.e. S, S. f ~ - f ~ . S, S 2 --~IIsI ), a linear 3x3 system for the coefficients [3~, [32, [33 can be obtained. Solving the system the following expressions are obtained:

13,=

where

IIc~is the trace

6 N , 5 N 2 - 2Hn

132=-

6 1 , 5 N 2 - 2Hta

(5)

of ~22 and N is the solution of the following cubic equation:

N3-C[N 2 -127Hs + 2 / / n ] N where C[ - 9 / 4 ( G equation (6).

f~3=0

+ 2C;Hn = 0

(6)

1). Cardano's formula provides an analytic expression for the positive root of

2.2. Near Wall Treatment The coefficients [3/(i = 1,2,3)must be modified close to the wall in order to reproduce the correct asymptotic near wall behaviour. Following the ideas of Manceau and Hanjalic (2002) and Manceau (2003), an elliptic blending parameter ~, is introduced whose value is 0 at the wall and 1 away from it and thus quantifies the blockage effect of the wall. The equation for ~, is: y - L 2 g 2 y =1

(7)

I 3

3

11

where L is the turbulent length scale defined by L = 0.161. max k ~ / ~, 80- v ~ / e ~ . The above investigators developed an elliptic blending model in which they use this parameter to blend the values close to and far from the wall for the components of the redistribution term (q0~) as well as the dissipation tensor (eij). In the present investigation, this parameter is used instead to blend the values of the coefficients [3i. Wallin and Johansson (2000) modify the coefficients close to the wall as:

90

~llowRe = f i l l ,

~2lowRe = f 1 2 ~ 2

--

B2 (1 -2 max(Hs , H eq )

fl 2 ),

~3lowRe"--

3B2 - 4

(1 - 3,-2)

(8)

m a x ( I I s , H eq )

where B 2 = 1.8 and HS q ~ 5.74 f o r C1 = 1.8. Note that 133 =0 away from the wall (equation 5) but not close to it. The function fl should be equal to 0 at the wall and 1 away from it so in principle it can be evaluated as a function of 7. Wallin and Johansson (2000) use a van-Driest function of the form f~ = 1 - e -y*/26 but since this expression uses the dimensionless distance from the wall y+ it makes the model difficult to apply in complex geometries since the calculation of y+ is not a trivial task. An alternative formulation is proposed here, which is based on the elliptic blending approach and eliminates the use of the distance from the wall. For the turbulent flow between parallel plates, the coefficients 13, (i = 1,2,3) can be evaluated from the elements of the anisotropy tensor. The following relations can be easily obtained:

[31 ._. ( ~ 1 2 ,

~2

_ (~11 -- 0~22 -,

cr

4or 2

[33_._ -

3a 3_.___~3 2or 2 1

(9)

dU

The parameter cr is the non-dimensional shear rate, defined as ~ = - z where 1: is the turbulence 2 dy g

t--------'N

time scale~ = max{ k , 6 . 0 ~ v ] , which is the usual timescale with a lower viscous bound given by the \

Kolmororov scale (Durbin (1993)). It is well known that the near wall behaviour of the shear stress in wall units is u--~+ = a , u ( y +)3 +...and of turbulent kinetic energy isk § = a k ( y +)2 +...so the component ~+

uv

(112

of the anisotropy tensor varies asa12 = k §

auv y§ + .... The non-dimensional shear rate close ak

to the wall is cr w+ =-~1 r + ddY U §+ ~-2rw 1 + because U+ ~ y § and l:w § is the time scale at the wall in wall units. From equation (9), the asymptotic trend for the coefficient 131 (in wall units) is: Otuv y+ +... ot~___L= ~

= cy+

Gtk

=--2

1 --X+w 2

auv y+ +... ak Xw +

(10)

thus this coefficient should approach 0 with a rate O(y+). The asymptotic behaviour of the elliptic blending function 7 is now examined. The one-dimensional form of equation (7) is" Y L2 d27 =1 ay ~

(11)

This equation is subject to the Dirichlet boundary condition 7 = 0 at the wall and the Neumann condition d7/dy = 0 at the half height of the channel y=c. For constant L, the analytical solution of equation (9) using these boundary conditions is:

91 y __

y-2c

_(eL+e ~'=

L )

2c l+e L

+1

(12)

Using Taylor series to approximate the exponential function close to the wall (ie for small y) the following asymptotic behaviour is obtained in wall units' 7+

=

y+ L+

y+2 --+ 2L+ 2

....

(13)

In practice L + is not constant. However, this expression, using the value of L + at the wall, was checked against the numerical solution of equation (11) in which L + was evaluated using the DNS values of k +, e+ and was found to approximate it very well close to the wall. The function fl is now constructed in such a way as to reproduce the asymptotic behaviour of equation (10). Assuming that fl is a quadratic function of "f and using the DNS values of auv,ak, Vw+,/3~W,L+ the following expression is obtained: f~ = a s ( y - y : ) + y :

(14)

where the constant af is evaluated from: as =-

2auv L+ ~

(15)

The DNS values for all parameters used in equation 15 are shown in the following table 1 for 3 Reynolds numbers. TABLE 1 DNS DATA FOR THE EVALUATION OF a s 131w "iTw ~uv ~k ~;w(=2Rk) L+ ai +

Re~= 180 -0.04014 14.49 7.2* 10-4 0.08538 0.17078 20.00 0.58

Re~=395 -0.045485 12.88 9.3153"10 "4 0.10878 0.21756 18.85 0.55

. . . . . . . . . . . . .

Re~=590 -0.04697 12.50 9.71785" 10-4 0.11556 0.23113 18.50 0.53

It can be seen that the coefficient a s does depend on the Reynolds number, although not strongly. The value 0.55 for Re~=395 has been selected for the calculations reported in the following section. The described approach allows the evaluation of fl as a function of the blending parameter ~/without using the normal distance from the wall. Although there are efficient methods for computing this distance for complex geometries, it is certainly advantageous if the wall blocking effect can be accounted for without the use of this distance. Moreover, the model allows the evaluation of the turbulent viscosity as v, = C e f K r where C~sr = --~1 1 ~ ~

i.e. this coefficient instead of being constant (equal to 0.09) as

in the standard k-e model is now a function of the local flow parameters.

92 3. TEST CACES 3.1. C h a n n e l F l o w Predictions

In this section results for 2 two cases obtained using the previously described algebraic model are presented. The first part of this test case refers to turbulent flow between parallel flat plates for three different Reynolds numbers (based on the friction velocity and half width of the channel): 180, 395 and 590. The second part of this test case refers to heat transfer predictions for the same flow configuration for Prandtl numbers 0.4, 0.6, 0.71, 1 (for Re~=180), as well as Prandtl number 0.71 (for Re~=395). The predictions for both cases are compared against available DNS data (Moser et al 1999, Kawamura et al, 1998 & 1999). In figure 1, predictions of the mean velocity against DNS data for the three different Reynolds number are shown. The number of cells used in the wall normal direction was 40, 10 of which were in the area 0
fact that no optimization on the other model coefficients has been attempted. Note that the v'v' component is accurately predicted i.e. the model can capture the wall blockage effect. uK

30 u+

~95

L5

..... :w

15 10

o

5

,',

~

~"~ ~

.......

A" ~ ' = = -

~ ,. ; ;

...... ,;0 ..... 1;=00!

y,,

Figure.2" k, u'u .,.v. .v. ,. . w

Figure.l" Umean for Re~=180,395,590

,

!:=

f\l

,1

li

w , u v (Re~ = 1 8 0 )

........

.....

~ T

-~-

~00

~0:0!,

Figure.3" k,u'u .,v. v, . .w. w. ,u v' (Re~=395)

[

i

....... ~

.......

~ .... ~ U % 0

~

;00~]o0

Figure.4: k,u 'u .,v. v, . .w. w. ,u . v (Re~=590)

i

93

0:

0.4 P

0.31

0.3-

I

0 O0 ~

0.21

O

0.2

0.1]

0.1-

I

O

0 0.01

,

O

OOO-

0.1

1

y+

,

,

10

100

~

, 1000

0 0.1

0.01

1

y+

10

100

1000

Figure.6: Dissipation (Re~=590)

Figure.5" Production (Re~=590)

Figure 5 and 6 show the production and dissipation of the turbulent kinetic energy respectively for Re~=590. The predictions are satisfactory. The heat transfer calculations have been carried out using the same grid configuration. The eddy diffusivity approach was used for the turbulent heat fluxes (i.e. u~|

- v--c-~c~| ) and for all computations the turbulent Prandtl number was equal to 1. O"T ~ X i 0.901

2018 ~

0.801

16 ~

Pr=0.71

14 ~

Pr=0.6 A~n ~

1210

.. "nz .:~x%

o~,.o9"~__

0.701 0.601

Pr:0.4

0.501

!

o~/ ~/

[]

Pr=1.0

~,

Pr=0.71

0

Pr=0.4

0.401

81

0.301

61

~

0.201

4~

0.101

2 ~ 0

-VQ+

8

0.001 10

y+

100

10

1000

Figure.7" 0+ at Pr=l, 0.71, 0.6 and 0.4.

y+

100

io00

Figure.8:- v'| at Pr=l, 0.71, 0.6 and 0.4.

In figure 7, the mean temperature profiles for Re~=180 and four Prandtl numbers (from 0.4 to 1) are shown. The horizontal axis is the logarithmic scale of the distance y+ from the wall. The predictions are quite satisfactory although there is slight over prediction on | for the higher Prandtl numbers. Looking at the turbulent fluxes in the direction normal to the wall (figure 8), the agreement between the computational and the DNS data is again very satisfactory. Furthermore, in figure 9 the variation of Nusselt against the Prandtl number is presented and good agreement is obtained. 22

.......

21 .u 20 19 18 17 16 15 14 0.2

0.4

0.6

Pr

0.8

1

112

Figure.9: Nusselt against Prandtl number for Re~=180.

94

1 7 16

0.9

|

0.8 t

4

0.7

12

0.6 ]

10

o

0.5

8

0.4

6

0.3 t

4

0.2 t

2 0

, 0.1

o

,

1

10

Y+

,

t

100

1000

~

o,

i

1

0.1

10

y+

1000

at Pr=0.71.

Figure.11" - v'|

Figure. 10: 0+ at Pr=0.71.

100

For Re~=395, the results from the computations match well the DNS data for the mean temperature and the turbulent heat flux as can be seen from figures 10 and 11 respectively. Again there is slight overprediction to |

3.2. Flow in a backward Facing Step

In this section results from the flow field over backward facing step case at ReH=28,000 are presented. The experiments have been carried out by Vogel and Eaton (1985). The expansion ratio is 1.25. For the numerical simulation 16093 cells were used and the mesh was twice locally refined close and around the recirculation zone. Figures 12-14 show comparison of the mean velocity profiles against experiments at various locations X*=(X-XR)/XR with reference to the reattachment point XR. F4

4

-

() 35

3.5 ]

i

3

2!

() (

C

(

C

( ( C O

C 0 0

o

-0.5

()

v.

0

UlUref

0.5

1.51

1.5

1

Figure. 12:U/Uref at X*=-0.67

0'it

1.5 /

L

-0.5

U

5

1

Figure. 13 :U/Uref at X*=-0.55

0 I -0.5

v

.

0

UlUref 0.5

1

Figure. 14:U/Uref at X*=-0.44

The agreement of the computed velocity field against the experimental measurement is again quite satisfactory. In figures 15-17 the temperature distribution at locations X*=-0.75, 0.45 and 1.25 is presented respectively. The agreement is quite good. Furthermore, figure 18 shows comparison with experiments of the variation of the Stanton number along the flat wall boundary. The agreement is reasonably good but there is scope for improvement. The observed discrepancies can be attributed to

95

several reasons such as the details of the inlet flow velocity profile, value of the turbulent Prandtl number etc. 2.5

2.5

(X-Xr)/Xr=0.45

, 2

2

I

1.5

2.5

..........................

(X-Xr)/Xr=-0.75

g

(X-Xr)/Xr=1.25

2

C

1.5

1.5 (

1 ( (

0.5

0.5

0.5 (

0 -5

q i

o~

' T-Tre 0 5

10

Figure. 15:T-Tref at X*=-0.75

-5

I

-5

O

5

Figure. 17:T-Tref at X * = 1.25

Figure. 16:T-Tref at X*=-0.45

St

0.004 0.003 / / ~ 0

0

0.002 0"0010 I

-1

0

1

2 (X-Xr)/Xr3

4

5J

Figure.18: Stanton number. 4. CONCLUSIONS This paper presented a near wall modification of an EARSM that uses the elliptic blending model ideas to merge the near wall region with the rest of the flow. The model can capture the near wall blockage effect, without making use of the distance from the wall, thus it can be applied easily to very complex geometries. The model was used to predict the turbulent flow and heat transfer between parallel flat plates and backward facing step. The computations were compared against DNS results and experimental data and the agreement was satisfactory.

5. A C K N O W L E G E M E N T

The research work described in this publication relates to the MinNOx project, contract ENK6CT-2001-00530, which was supported by the European Community. The authors take sole responsibility for their findings and recommendations. The Community is not accountable for the outcome and future use of the publication and the opinions may not represent that of the Community.

96 6. R E F E R E N C E S

Durbin P. A. and Petterson Reif B. A. (2001), Statistical Theory and Modelling for Turbulence Flows, John Wiley and Sons, Ltd. Durbin P. A. (1993), A Reynolds stress model for the near-wall turbulence, J. Fluid Mech. 249, 465-498. Gatski T. B. and Speziale C. G. (1993) On explicit algebraic stress model for complex turbulent flows, J. Fluid Mech. 254, 59-78. Gatski T. B. and Jongen T. (2000) Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows, Progress in Aerospace Sciences, 36 655-682. Hadzic I. (1999). Second-Moment Closure Modelling of Transitional and Unsteady Turbulent Flows, PHD Thesis Technical University of Delft Kawamura, H., Abe, H. and Matsuo, Y. (1999) DNS of turbulent heat transfer in channel flow with respect to Reynolds and Prandtl number effect. Int. J. Heat and Fluid Flow, 20, 196-207. Launder B. E., Reece G. J. and Rodi W. (1975). Progress in the development of Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537-566. Manceau R. (2003) Accounting for wall-induced Reynolds stress anisotropy in Explicit Algebraic stress model. Third International Symposium on Turbulence and Shear Flow Phenomena Sendai, Japan. Manceau R. and Hanjalic K. (2002) Elliptic blending model: A new near-wall Reynolds-stress turbulence closure, Physics of Fluids, 14, 744-754. Moser, R.D., Kim, J. and Mansour, N.N. (1999) Direct numerical simulation of turbulent channel flow up to Ret=590. Physics of Fluids, 11:4, 943-945 Pope S B. (2000), Turbulent Flows, Cambridge University Press. Pope S.B. (1975) A more generalised effective-viscosity hypothesis J. Fluid Mech. 72, 331-340. Rodi W. (1976) A new algebraic relation for calculating the Reynolds stresses, ZAMM, 56, 219221. Speziale C. G., Sarkar S. and Gatski T. B. (1991). Modelling the pressure-strain correlation of turbulence:an invariant system dynamics approach, J. Fluid Mech. 227, 345-272 Taulbee D. B. (1992). An improved algebraic Reynolds stress model and corresponding nonlinear stress model, Phys. Fluids A 4 2555-2561. Vogel J. C., Eaton J. K. (1985) Combined Heat Transfer and Fluid Dynamic Measurements Downstream of a Backward-Facing Step, Journal of Heat Tansfer 107, 922-929. Wallin S., Johansson A. (2000). An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows, J. Fluid Mech. 403, 89-132.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

97

A S S E S S M E N T OF T U R B U L E N C E M O D E L S F O R PREDICTING THE INTERACTION REGION IN A W A L L J E T B Y R E F E R E N C E T O LES S O L U T I O N A N D BUDGETS A. Dejoan, C. Wang and M. A. Leschziner Department of Aeronautics, Imperial College London, London SW7 2AZ, UK

ABSTRACT

Highly-resolved LES and experimental data for a plane wall jet are used to study the characteristics of three turbulence models, two based on second-moment closure and the third being a non-linear eddy-viscosity model. The study is motivated by the observed importance of Reynoldsstress transport in the interaction region between the outer shear layer and the near-wall layer of the wall jet. Comparisons are presented for mean-flow quantities, second moments and budgets. Also included is an a-priori study of two approximations for the turbulent transport of the Reynolds stresses, a process that contributes substantially to the stress budgets. The study reveals major defects in the closure approximations for the modelled terms in the stress-transport equations, including that for stress diffusion. In particular, stress diffusion by pressure-velocity correlations, which is much more important than generally assumed, is not realistically modelled by the approximations considered.

KEYWORDS

Turbulent wall jet; LES; Reynolds-stress models; Second-moment budgets.

1

Introduction

Most fluid flows may be regarded, in principle, as consisting of a collection of interacting shear layers and regions subjected to normal straining. While these flow components may be relatively simple, when existing in isolation, it is their interaction that introduces 'complexity' and that is the main cause for predictive failures, especially with statistical models. In most circumstances, the component flows are too indistinct and the interaction too complex or ill-defined to attempt to separate the flow components and to identify the nature of the interaction among them. However, there are important exceptions, and these provide a route to a fruitful investigation of the mechanisms involved and allow questions to be answered on whether particular closure assumptions

98

represent the interaction realistically. One such flow is the wall jet, in which a near-wall shear layer interacts with an outer shear layer evolving from a separation line at the edge of the jet-discharge nozzle. This flow shares some interesting generic features with the post-reattachment region of a separated flow, which consists of a near-wall shear layer that develops from the reattachment point and a free shear layer above it, originating from the curved shear layer on the outer edge of the recirculation bubble. In both flows the two principal components have different structural features and interact strongly across an overlap region as the flow evolves. In the case of post-reattachment recovery, it may be surmised that the observed inability of most turbulence models to return correctly the recovery from separation reflects defects of the models in representing the interaction between the two sets of scales in the flow forming the post-reattachment wake. Thus, the wall jet allows some fundamental interaction mechanisms to be studied in well-defined conditions, which may then be applicable to practically more interesting separated flows.

Figure 1: Flow geometry and instantaneous U-velocity iso-contours.

The present study takes advantage of the highly-resolved LES solutions obtained by Dejoan & Leschziner [1] for a plane, statistically two-dimensional wall jet at conditions also examined experimentally by Eriksson [2]. Especially useful are budgets for second moments covering the overlap region between the near-wall and free shear layers. The budgets reveal, in particular, the important role played by Reynolds-stress transport over a broad region of the shear flow spanning major portions of the boundary layer and the outer shear layer. A further important observation relates to the migration of large scales from the outer layers towards the wall, causing a substantial elevation of the integral length scale throughout the boundary layer, right down to wall. The implication is, therefore, that any model likely to represent correctly the above processes will have to be based on second-moment closure and may, furthermore, requires significant modifications to its length-scale equation which is almost invariably calibrated by reference to relatively simple, isolated shear layers. Attention is focused in this study on the predictive characteristics of three models, two based on second-moment closure (Speziale et al. [3] and Shima [4]), the third being a non-linear eddyviscosity model (Abe et al. [5]). By its very nature, the last obviously does not account for stress transport, and its inclusion herein aims to provide a reference level against which to compare the second-moment models. The particular choice of this model was motivated by its exceptionally favourable performance, within its class, in both attached and separated flow, including postreattachment recovery (Jang et al. [6]). This model therefore provides an appropriate yardstick. The study has two strands: (i) a comparison of RANS predictions with the LES solution; and (ii) an a-priori study of alternative closure proposals for Reynolds-stress diffusion which, as noted above, is observed to be an important contributor to the budget in the interaction (overlap) region of the wall jet.

99

2

T h e g e o m e t r y and its s i m u l a t i o n

The geometry under consideration is shown in Fig. 1. The jet is discharged from a wall slot into a semi-infinite, stagnant environment. The Reynolds number, Re = Uob/u, is 9600, Uo being the maximum inlet velocity, b the slot height and u the fluid viscosity. The simulation has been performed over a domain extending from the wall to 10b above it and to 22b in the streamwise direction. The choice of the latter is based on the experimental observation of Eriksson et al. [2], for the same conditions as simulated here, that the flow essentially reaches a self-similar state at 20b. The homogeneous spanwise domain size is 5.5b. The numerical grid contains 420x208x96 (8.4 million) nodes, and has been carefully chosen to give very low subgrid-scale contributions, approximated with Germano's dynamic Smagorinski model. The ratio of cell size to the Kolmogorov length is, typically, 5-10, and spectra show the cut-off to lie in the dissipative range. The near-wall cell-aspect ratio is, typically, A y + / A x + / A z + = 1.2/24/23, the wall-nearest grid node lying at y+ = 0.6. The complete description of the simulation is given in Dejoan & Leschziner [1].

3

The models

Three models are examined herein. Two are second-moment closures, one being the model of Speziale et al. [3], extended by Chen et al. [7] to low-Re conditions, and the other the model of Shima [4]. The choice of these particular variants is somewhat arbitrary, and others are being considered in further studies in train. However, there is some rationale in preferring these two formulations. First, both models allow integration to the wall. Second, neither model involves "wall reflection corrections" to the pressure-strain approximation. Third, the former model, albeit in its high-Re form, has gained particular popularity, not only in its own right, but also as a basis for the derivation of the explicit algebraic Reynolds-stress model of Gatski & Speziale [8]. Finally, the latter model is a low-Re extension of the closure of Launder et al. [9] - another popular and influential model, as well as one that forms the basis for Wallin & Johansson's [10] explicit algebraic Reynolds-stress model. A distinctive feature of the model of Speziale et al. (denoted by SSG, henceforth) is the non-linear form of the pressure-strain approximation: 1 ~ i j -- -- (C1c + C 1 P k k ) bij + C2e(bikbkj - 5bmnbmn(~ij) + (C3 - C~*IIb/2) k S i j

2 +C4k(bikSjk + bjkSik - 5bm,~Smr~Sij) + Csk (bikWjk + bykWik)

(1)

where bij = u-~/izmum - }Sij, &j = 1/2(OUi/Oxj + OUj/Oxi), Wij = 1/2(OUi/Oxj - OUj/Oxi), Pij = -(u--j~OUi/Oxj +u-i-~OUj/Oxk) and I I = bijbji; k is the turbulence energy and e the dissipation. The above is the high-Re form. As noted earlier, its low-Re extension by Chen [7], not given herein, has actually been used in this study. The pressure-strain approximation used in Shima's model is:

C1 pc

(~ ij "--

(

-

15iju--K~) - pC2 (Pij - 1 -5 -5

) -

(ov

(

ov,

2 - 5

1 )

(2)

where Dij = -(u-Tg-ZOUk/Oxi + u-N-ZOUk/Oxj). Although this appears to be a linear proposal, the coefficients C 1 - C4 are functions of the first and second stress invariants, and this makes the model quasi-linear in the Reynolds-stress. The benefit is to obviate the need to include wall-reflection

100

fragments. The model by Jakirlic & Hanjalic [11] is the most elaborate member of the group of closures involving such quasi-linear approximations, but this has been found to give serious numerical problems in the present geometry. Both second-moment models incorporate Daly & Harlow's generalised-gradient-diffusion hypothesis [12]- see equation (3) below. The non-linear eddy-viscosity model of Abe et al. [5] (denoted later by AJL) rests on a quadratic constitutive stress-strain-vorticity equation derived from an algebraic simplification of a secondmoment closure in which the pressure-strain term is represented by the most general, tensorially linear approximation in the anisotropy tensor (Abe et al. [13]). The basic explicit algebraic approximation is then augmented by two additive fragments intended to account, respectively, for high normal straining and strong near-wall anisotropy, the latter returning correctly the asymptotic decay of all Reynolds-stresses towards the state of two-component turbulence at the wall that is observed from DNS. This decay cannot be represented solely by the use of terms combining the strain and vorticity, and the approach taken by Abe et al. has thus been to add a tensorially correct wall-related term to the constitutive stress-strain/vorticity relation, aij -- ~ k --g(~ij 2 = f ( S i j , f~ij...), which takes into account the wall orientation. In the model variant used here, the wall-direction indicator is: dij = Ni/~/NkNk (with Ni = Old/Oxi and ld = Yn, the wall distance), which is then used in the additive wall-anisotropy correction of the form: Waij = - f w (didj -- ~ d k d k ) x f(SikSkj, Sikf~kj, S k j ~ i k , S 2, f~2...), where fw is a viscosity-related damping function. Alternative wall-orientation indicators, independent of wall distance, may readily be used. In the above damping function, a composite time scale is used, which combines the macro-scale k/e with the Kolmogorov scale y/~/e. The damping function fw then provides a smooth transition between the two scales across the near-wall layer. Two variants of the model exist" one operating in conjunction with the dissipation-rate (c) equation and the other with an equation for the specific dissipation rate (w). The former variant is used herein. The model is fully described in Abe et al. [5] and Jang et al. [6]. The latter publication demonstrates that the model does indeed return the correct wall-asymptotic behaviour of the stresses both for attached and separated flows. ]

For reasons explained earlier, an a-priori examination of two approximations for the diffusion of the Reynolds stresses forms the second part of the present study. The proposals considered are those of Daly & Harlow [12] and Hanjali5 & Launder [14], respectively" dij - ~Xk

0.22--ukulc Oxz J a n d dij - ~Xk

0.1 C

~

Oxl

+ uju----[ OXl

+ UkUl OXl

(3)

These are by far the most frequently used approximations in second-moment-closure models. Although stress diffusion is usually held to be of subordinate importance, relative to other processes contributing to the budgets, this is not the case for the wall jet. Hence, the validity the above approximations is likely to reveal itself especially clearly for this flow.

4

Computational procedure

Computations were performed with a non-orthogonal, collocated, cell-centred finite-volume approach implemented in the code 'STREAM' (Lien & Leschziner [15], Apsley & Leschziner [16]). Convection of both mean-flow and turbulence quantities is approximated by the 'UMIST' scheme (Lien & Leschziner [17]) - a second-order TVD approximation of the QUICK scheme. Mass conservation is enforced indirectly by way of a pressure-correction algorithm. Within this scheme,

101

Figure 2: Left: growth rate of the wall jet. Right: evolution of the wall thickness, (~ = the streamwise direction.

p/u~, along

Figure 3: Profiles of the mean streamwise velocity, RMS of normal Reynolds stresses and turbulence energy, scaled with wall variables.

the transport and the pressure-correction equations are solved sequentially and iterated to convergence. The computational domains extends to 35b above the lower wall and 150b in the streamwise direction. The grid contains 150 x 222 non-uniformly distributed nodes, and has been carefully constructed to ensure grid-independence. At the upper boundary, fixed-pressure and zero-streamwisevelocity conditions are prescribed, allowing free entrainment normal to the boundary. In the actual flow, the inlet conditions (at the nozzle-exit plane) are essentially laminar, with the turbulence intensity being less than 2%. These are the conditions used in the RANS computations presented here. However, other inflow conditions, prescribed at locations downstream of the nozzle and taken from the LES solution, have been tested. Moreover, different levels of turbulence intensity, within the range 2% to 20%, have been tested to examine the influence on the self-similar behaviour of the wall jet. These tests yielded virtually no sensitivity of the RANS solutions far downstream to the details of the inflow conditions. The aim of using a very long domain for the RANS computations was to ensure, unequivocally, that a self-similar state would be reached - the condition on which the study focuses - and to avoid any uncertainties arising from transition.

102

5

5.1

Results

Model performance

In this first part of the presentation of results, model predictions are contrasted against experimental data and LES solutions. The global spreading rate of the jet, in terms of Yl/2/b, and the streamwise evolution of the wall friction, represented by the wall thickness, 5~ = v/u~-, along the streamwise direction are given in Fig. 2. Because of uncertainties associated with transition, the initial streamwise portion x/b < 10 needs to be ignored. Both second-moment closures, especially the SSG model, are seen to predict excessive spreading rates and very slow approach towards self-similarity, but give a broadly correct evolution of the wall-shear stress. In contrast, the AJL model returns a spreading rate closely matching the experiment, while the wall-shear stress is too high and, more importantly, diminishes too slowly. Results for velocity and second moments are given in Fig. 3. Except for the LES, all other profiles relate to x/b = 75. This location is well within the self-similar region. In contrast, the LES profile apply to x/b = 20. This is the furthest location for which comparisons with LES data is possible. Reference to the experimental and LES-predicted spreading rate (Fig. 2) suggests that the jet has effectively reached the self-similar region at this location, although some minor adjustments do occur further downstream. Hence, comparisons at this location may be taken to reflect, essentially, the ability of the models to predict the self-similar characteristics. The results in Fig. 3 show both second-moment models, especially the SSG formulation, to perform poorly. Both predict insufficient near-wall anisotropy, partly associated with inadequacies in the low-Re fragments of the models, and both give surprisingly high turbulence-energy and Reynolds-stress levels in the outer shear layer, consistent with the excessive spreading rate seen in Fig. 2. The SSG model also gives an especially poor representation of the mean velocity in the interaction region and its fringes. Because of the strong effect of the outer shear layer with the near-wall layer and the relatively low Reynolds number, there is no distinctive log-law region. In fact, the analysis of the LES data provided by Dejoan & Leschziner [1] demonstrates, on the basis of budgets, that the interaction region lies within 30 < y+ < 90. The SSG model thus performs especially poorly in this region, in which it also predicts too low levels of shear stress. The AJL model, although not accounting for anisotropy transport, gives a far better representation of both mean velocity and stress fields. This includes the near-wall anisotropy, reflecting the favourable performance of the model fragment (4). The results given in Dejoan & Leschziner [1] show that the non-alignment of the zero-stress and zero-strain locations reported by Eriksson et al. is well reproduced by the LES and is unambiguously caused by turbulent transport of the Reynolds stresses. This distinctive behaviour of the wall jet, reflecting the strong interaction between the outer and inner shear layers, can obviously not be predicted by the AJL model. However, on the basis of the results included above, this interaction, if at all realistically represented by the stress-diffusion fragments within the second-moment models, is less influential than other processes, which appear to be poorly represented by the second-moment closures examined.

5.2

Analysis of budgets

In what follows in this section, budgets for turbulence energy and, where appropriate, stresses are given in an effort to gain further insight into the characteristics given in the previous section. First, budgets for the turbulence energy derived from the AJL and SSG models are compared in Fig. 4 to that obtained from the LES at x/b = 20, with attention confined to the near-wall and interaction regions. Close similarity is observed between the AJL and the LES budgets, while the SSG budget shows a poor representation of dissipation and diffusion. Production is also too high, with the peak

103

too close to the wall. To some degree, the poor performance below y+=20 may be attributed to the low-Re extension of the basic SSG model, but the defects are clearly more deep-rooted. Budgets for the individual stress components are only applicable to second-moment closure, and comparisons between the SSG and LES budgets for v~ and ~-V are given in Fig. 5. These stresses are the most interesting, in so far as they control the mean flow and also display most clearly the substantial contribution of stress transport. In the budgets, the stress diffusion, Tij, includes the effect of the triple correlations as well as pressure diffusion, while Oij represents the pressure-strain correlation (see model approximation (1)). The comparisons reveal some major differences between the SSG and LES budgets. First, both LES budgets show that stress transport is extremely influential in the region y+ < 100. Very close to the wall, pressure-strain and pressure diffusion (which largely dominates the turbulent transport in this region) are large and of opposite sign, but this particular feature may be set aside, as it is of no significant consequence to the turbulent stresses. Away from the viscous sublayer, however, the major differences are important and influential. The SSG model gives an entirely wrong representation of the production, pressure-strain process and diffusion of the shear stress. The production is too high, as the opposing diffusion is far too low, and the pressure strain is positive, as observed in a standard boundary layer (see Mansour et al. [18]), rather than the negative value derived from the LES. An even poorer representation is provided for the wall-normal stress V~. In the LES budget, the balance is mainly between the negative pressure-strain correlation and the positive diffusion, while the SSG model implies a balance between the much too high level of dissipation, suggesting that the approximation cvv = 2/3ck is not appropriate, and positive pressure-strain, again as observed in the standard boundary layer. Thus, the implication is that closure of the stress-transport equations contains some major defects, among them the approximation of stress diffusion, which is clearly badly represented.

5.3

A-priori study of stress diffusion

A conclusion emerging from the LES results is that the turbulent transport of the stresses plays an important role in the interaction between the inner and outer shear layers. In the previous section, the LES data have been used to give a view on how the various fragments of SSG model represent the associated processes. However, this view is clouded by the interaction among these processes, which arises when the model operates within an actual computation. A better view of the realism of any one fragment can be gained by performing a-priori studies of the individual fragments by reference to the simulation data. This is done here for the stress-diffusion terms, in view of their importance in the interaction region. The particular approximations considered are those of Daly & Harlow and Hanjalid & Launder (eq. 3), the former used in both the SSG and Shima models. Results are presented in two ways in Fig. 6. First, LES-predicted and measured triple correlations are compared to corresponding RANS approximations, assuming that the latter represent these correlation and that the contribution of pressure diffusion is negligible. The alternative presentation compares the diffusive contributions to the stress budgets - i.e. the gradients of the sum of viscous, triple and pressure-velocity fragments. In the former, data for ~ extracted from the LES and measured by Eriksson et al. are compared to the model approximating the transverse diffusion of ~--~, rather than the longitudinal diffusion of ~-v. Similarly, uuv is compared to the transverse diffusion of ~-g. The comparison relating to the triple correlations suggests that both RANS approximations give reasonable representations of the LES and experimental data, except for uuu, which is the least influential term. However, it is the derivatives of the sum of these correlations and the pressurevelocity terms that enter the stress-transport equations, and Fig. 6 shows both RANS approximations to give a poor representation of the real transport. To separate the contributions arising from the triple and pressure-velocity correlations, LES-derived distributions are given for the total

104

transport and that total less the pressure-velocity contribution. As seen, pressure diffusion plays a major role in respect of ~ and ~-~. Thus, while the RANS models may be reasonable approximations of the transport associated with the triple-correlations, they do not capture pressure diffusion which is very important in the present flow. This is one reason, but by no means the only one, for the failure of the SSG model in the present flow. 0.3

I~'~0.

. . . . . . . . . . . .

0.3

............

:

0.2

I;z,':

k

'l ' t

1"-* D,,+ TkI-I

~

, ' i ,~,

~c~ /I I~Ek

r

9-

,~-0.1e,

i

_0.'z

25

§ Y

0.2

.~-0. 0

.

2

,3

1

50

I ~ 1 , t , 1 , I , 100 125 150 175 200 225 250

75

Figure 4: Turbulence-energy budgets normalized by the wall scales. Production: Pij; viscous diffusion: D~; turbulent transport including pressure diffusion:

~0.1

-

i

SSG + lowReynoldsext.

-0.2 -0"I -0.

i],

~*-* D r + Tk[/

k

~

Tij----(uiujuk ),k -- (p/p(uiSjk + UjSik ) ),k; 0

25

50

75

100

125 150 175 200 225 250 4Y

............. ~

0.1

convection: Cij.

t~v~P'~ 9

.._, D v + T v

o

-0.1

LES

SSG + lowReynoldsext.

-0o1~ lull

0 ' ~5' 5'0' ~'5' 1;0'1;5' l;O'1~5'2;0'2~'250 +

0

1

25

,

I

50

,

I

75

Y

i ............. ~ "~ 0.1 -/ ~

t

r

.

]~dv tick" ]' I*-'D:Y"T I -I t~> C

~

uv

Figure r

5:

- p/p(ui,j

,

i

25

,

i

50

,

i

75

,

i

,

I

,

I

,

I

uv

I o.,L~.

,

I

I

I

I

I

I

100 125 150 175 200 225 250 + Y

No v

J

,-.D +L,. J

-~

LES

-0.2 II

I

g uv

~-0.1

0

,

,

i

,

i

,

1

,

i

,

.0.2F

100 125 150 175 200 225 250 + Y

Reynolds-stress budgets, scaled + uj,i) is the pressure-strain.

1 0

with

25

50

wall

75

100 125 150 175 200 225 250 + Y

variables;

see

Fig.4

for

labels;

105

10 ~ t-

,'

~ IO

~~ F

nun

0'

I

'

] ~~~~~1" 9

._J[ ~ 9 x/b=2oX/b=20 (LEs)(Erikss~et al.)

" ~

9 x/b= 70 (Eriksson et al. ) Hanjalic and Launder model Daly and Harlow model

0

0

50

I 2 ]---

VVV

-,

'

y+

'I h

100

, '

150

~I - ] ' l

-

-

*

x/b=20 (Eriksson el al.) x/b=20 (LES) 9 x/b=70 ( Eriksson et al. )

'

0

Hanjalic and Launder model Daly and Harlow model

-2

0 ' UUV

50

y+

100

I

'

I

|

150 '

"

1

0

i

i

- - Dr+ T u- DPu (LES) - - Dv+ T u (LES) o--o Dr+ T u (Daly and Harlow) [o---oDr+ T u (Hanjalic and Launder)

-- -- Hanjalic and Launder model x/b=70 (Eriksson et al.) Daly and Harlow model

-2

0 2 U VV

'

50

y+

I00

1

'

1

150 '

Hanjalic and Launder model

0

x/b=70 ( Eriksson et al. )

-0"1f

Daly and Harlow model

-2

5'0

-0.20 0

50

0.06~ |

y+

'

100

150

,~o

+ Y

,;o

2~o

250

~ ,' ' ' .. . . . [ - - Ov+T - D:~ (LES) 0.1

I~ 0.04 ~ " - - - , , i

~

o--,o Dv+ r (Daly and Harlow) [o---oD + T~ (Hanjalic a n d Launder) 9 v vv

#

-0.1

- - D r + T - DPv (LES)

__ Dv+rv (LBS) -0.2

0"02

~ D v+ Tuv (DMy a n d Harlow) ,---, Dr+ T (Hanjolic a n d L a u n d e r )

-0.3 UV I

0

,

I

50

,

I

100

,

I

+ 150 Y

,

1

200

,

250

-0"40 ' 215 ' 510 ' 75 ' 100' 125' + 1}0 '175 '200' 225'250 Y

Figure 6" A - p r i o r i study of turbulent transport terms; Di% - --(P/P(Ui~jk + Uj~ik)),k

6

Conclusions

This study started from the premise that a realistic approximation of Reynolds-stress diffusion, only possible within the framework of second-moment closure, was likely to be an important element in modelling correctly the details of the interaction between the outer shear layer and the boundary layer within a wall jet. This supposition was derived from the examination of LESpredicted budgets which show that stress diffusion is a very influential process, especially for the shear stress and the cross-flow normal stress. Surprisingly, the two second-moment closures examined return predictions which are considerably inferior to that from a reference model which cannot account for stress transport. The former do not only give a poor representation in the near-wall and interaction regions, but also in the outer shear layer. Consideration of the budgets for one of the two closures indicates some major defects in the approximation of pressure-strain, stress-diffusion and dissipation terms, while the non-linear eddy-viscosity model gives, in contrast, a turbulence-energy budget that agrees well with the LES solution. Evidently, therefore, stress diffusion is not the principal mechanism dictating the overall predictive quality of a model; other, more influential mechanisms, need to be approximated well, alongside diffusion, for the model to give realistic predictions, at least for the wall jet. A detailed a-priori examination of two approximations for stress diffusion showed both to give credible representations of the processes associated with the triple correlations of velocity fluctuations, but not of the total diffusion, which includes pressure-fluctuation-driven diffusion. Other elements in the closure can obviously not perform

106

realistically within a framework in which all terms operate interactively. Acknowledgments The authors are grateful for the financial support of the research by the UK's Physical Sciences and Engineering Research Council (EPSRC)

REFERENCES

[1] A. Dejoan and M.A. Leschziner. Large eddy simulation of a plane turbulent wall jet. Phys. Fluids, To appear, 2004. [2] J. Eriksson. Experimental studies of the turbulent wall jet. PhD Thesis, KTH, 2003. [3] C.G. Speziale, S. Sakar, and T.B. Gatski. Modelling the pressure-strain correlation of turbulence: an invariant dynamical system approach. J. Fluid Mech., 227:245-272, 1991. [4] N. Shima. Low-Reynolds-number second-moment closure without wall-reflection redistribution terms. Int. J. of Heat and Fluid Flow, 19:549-555, 1998. [5] K. Abe, Y-J. Jang, and M.A. Leschziner. An investigation of wall-anisotropy expressions and lengthscale equations for non-linear eddy-viscosity models. Int. J. Heat and Fluid Flow, 24:181-198, 2003. [6] Y.J. Jang, M.A. Leschziner, K. Abe, and L. Temmerman. An investigation of anisotropy-resolving turbulence models by reference to highly-resolved les data for separated flow. Flow, Turbulence and Combustion, 69:161-203, 2002. [7] H.C. Chen, Y.-J Jang, and J.C. Han. Computation of heat transfer in rotating two-pass square channels by a second-moment closure model. Int. J. Numer. Methods Fluids, 43:1603-1616, 2000. [8] T.B. Gatski and C.G. Speziale. On explicit algebric stress models for complex turbulent flows. J. Fluid Mech., 254:59-78, 1993. [9] B.E Launder, G. J. Reece and W. Rodi. Progress in the development of a Reynolds stress model turbulence closure. J. Fluid Mech., 68:537-566, 1975. [10] S. Wallin and A.V. Johansson. An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flow. J. Fluid Mech., 403:89-132, 2000. [11] S. Jakirlic and K. Hanjalid. A second-moment closure for non-equilibrium and separating high- and low-Re-number flows. Proc. 10 th Symp. On Turbulent Shear Flows, Pennsylvania State University, 23:23.25-23.30, 1995. [12] B.J. Daly and F.H. Harlow. Transport equations in turbulence. Phys. Fluids, 13:2634-2649, 1970. [13] K. Abe, T. Kondoh, and Y. Nagano. On Reynolds-stress expressions and near-wall scaling parameters for predicting wall and homogeneous turbulent shear flows. Int. J. Heat and Fluid Flow, 18:266-282, 1997. [14] K. Hanjalid and B.E Launder. A Reynolds stress model of turbulence and its application to thin shear flows. J. Fluid Mech., 74:593-610, 1972. [15] F.S. Lien and M.A. Leschziner. A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, part I: Computational implementation. Comp. Meth. Appl. Mech. Eng., 114:123-148, 1994. [16] D.D. Apsley and M.A. Leschziner. A new low-Reynolds-number nonlinear two-equation turbulence model for complex flows. Int. J. Heat and Fluid Flow, 19:209-222, 1998. [17] F.S. Lien and M.A. Leschziner. Upstream monotonic interpolation for scalar transport with application to complex turbulent flows. Int. J. Num. Meths. Fluids, 19:527-548, 1994. [18] N.N. Mansour, J. Kim, and P. Moin. Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech., 194:15-44, 1988.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

107

EDDY COLLISION MODELS FOR T U R B U L E N C E Blair Perot and Chris Chartrand Department of Mechanical and Industrial Engineering University of Massachusetts, Amherst, MA 01003, USA

ABSTRACT Simple fluids such as gases and liquids are the result of collisions between molecules. More complex fluids, such as granular flows and colloidal suspensions (non-Newtonian fluids), result from the more complex collision (or interaction) behaviors of their constituent particles. In this paper it is demonstrated that collision rules can be constructed for large chunks of fluid material (eddies) such that the resulting collective system behaves like the mean (RANS) flow of a turbulent fluid. The collision model approach has a number of advantages over classic Reynolds stress transport (RST) models. For example, turbulent transport does not require a model and mathematical constraints like realizability are automatically satisfied. Using some ideas from lattice-Boltzmann methods and adaptive moving mesh algorithms for CFD it is shown that this modeling approach can be made computationally efficient and comparable in cost to classic Reynolds stress transport (RST) models. Finally, it is shown that the collisional approach to turbulence modeling can lead to some insights into turbulence and turbulence modeling that would probably not have been achieved via the traditional RST approach. KEYWORDS Turbulence, Modeling, RANS, Reynolds Stress Transport, Collision Model. INTRODUCTION The traditional approach to modeling turbulence or non-Newtonian fluids is to hypothesize equations for the unknown stress tensor (in turbulence this is the Reynolds stress tensor). Because, the eddies making up the flow are roughly the same size as the gradients in the mean flow these eddies respond on similar timescales as the mean flow. This means that algebraic models are rarely predictive, and evolution equations for the stress tensor must be hypothesized. In turbulence, these are the Reynolds stress transport (RST) equations. Simpler turbulence models, such as the k-e model or algebraic Reynolds stress models, are simplifications of the RST equations.

108

There is a strong analogy between turbulent fluid flow and Non-Newtonian or granular flows. Very similar to turbulent flows, transport equations are very often developed for non-Newtonian stress tensors (the Oldroyd-B model and FENE-P models are examples). In fact, we note that many important turbulence modeling concepts (realizability, material frame indifference, tensor consistency) actually find their origins in the non-Newtonian literature at this transport equation level. However, it has long been recognized in the non-Newtonian fluid community that transport equation models have serious limitations. An alternative approach is to model the fluid at the particle collision level rather than using a transport equation for the stress. This approach is more versatile, and in many ways, more fundamental. For example, modeling a gas as particles with binary elastic hard sphere collisions gives the Navier-Stokes equations and the perfect gas law when the density is high, as well as the correct gas behavior even when the density is low (when Navier-Stokes is not valid). In this work we investigate the possibility of modeling turbulence as a collection of interacting particles. NUMERICAL SOLUTION OF COLLISION MODELS Once a certain collision behavior has been hypothesized there are three very different ways to solve the particle system numerically and obtain a prediction of the fluid behavior. The most straightforward technique is the 'molecular dynamics' approach where one numerically tracks all the particles in the domain, and performs collisions when they occur. This approach has a computational cost equivalent to large eddy simulation (LES) and is not considered further. The other two approaches note that one does not really care what happens to individual particles but only what happens to particles on average. The quantity of interest then becomes the probability density function that describes the probability that a particle (at a certain place and time) has a certain velocity. The evolution of the probability distribution function, f obeys the exact equation

of +v, af 0-7

of

df I

(1)

-~x, + a~ --~ = --~ co,,,s,.....

where a, is the acceleration due to external forces (like gravity) and the right-hand side describes the average affect of the collisions on the PDF. It is this average collision behavior that we now wish the models to predict. There are two different ways to solve this PDF equation. One way is to assume the collision model has a Fokker-Planck form (see equations 2 through 4). Then using the equivalence between the Fokker-Planck equation and the Langevin equation (Brownian motion), it is possible to construct a Lagrangian particle method. This is the approach extensively researched by Pope (1994, 2000) and coworkers. The Lagrangian particles move like Brownian dust particles. They move with the mean flow and are randomly perturbed using a prescription given by the model. In this way each particle is independent from all the others, and simply interacts with the average of all the other particles. This is less expensive than tracking and implementing individual collisions ('molecular dynamics' approach) but is still expensive because a large statistical sample of particles is required. The numerical approach used in this project was to solve the PDF equation using a standard Eulerian mesh in physical space, x, as well as in velocity space, v. Normally, this approach would be rejected outright since 10 mesh points in each direction then requires a million mesh points (106) to mesh all six variables (x and v) and is too expensive. The resolution to this problem is to use an extremely coarse mesh in the velocity space (3 points in each direction). This means we are solving 27 equations for each point in space. For comparison, a RST model solves 3 velocity,

109

1 pressure, and 6 stress equations (10 equations) per point in space. However, since the RST equations are highly coupled and nonlinear, and the PDF equations are not, the solution times are very comparable. A very coarse mesh in velocity space is an idea borrowed from Lattice-Boltzmann methods for solving the Navier-Stokes equations. These methods solve a PDF equation with a very simple collision term that is intended to give Navier-Stokes (Newtonian) fluid behavior. The difference here is that we solve a PDF equation with a much more complex collision term, which results in RANS behavior for the fluid. The coarse mesh is acceptable in both cases because the interest is not in the PDF itself but in its lowest order moments - the mean flow and the stresses. These low order moments can be reasonably extracted from a very coarse approximation of the PDF. Note that the Langevin approach is equivalent to approximating the PDF with a random sample, and a large Collision Models sample is needed even to approximate the low order moments A..,11 reasonably well. The Langevin PDF Methods Particle Tracking approach is slower because it ..~..~ ('molecular dynamics ') provides more information (about the higher order moments). Unfortunately, we have little Coarse Discretization Langevin Equation ('lattice methods ') (Lagrangian particle methods) interest, in engineering turbulence models, in the extra information the Figure 1. Taxonomy of collision model approaches. Langevin solution method provides.

~

While the approach taken in this work is inspired by the success of lattice-Boltzmann numerical methods, the approach is significantly different. This is because the PDF governing molecular interactions (Lattice-Boltmann) has a variance (width) that is much larger than the mean and which is essentially constant (related to the speed of sound). In contrast, the PDF for turbulence has a variance which is much smaller than the mean (turbulence intensities are measured in percent), and which can vary significantly (in time or space). This is illustrated in Figure 2. To capture the turbulence PDF with only three points it is necessary to have a moving adaptive

0

0

Figure 2: Left: typical PDF for molecules. Right: typical PDF for turbulence. mesh in velocity space. In order to avoid losses due to interpolating one mesh to another as the mesh moves, we implemented a fully conservative scheme in which the mesh moves continuously in time (during the timestep). This uses technology previously developed by Perot & Nallapati (2003) for moving meshes in physical space. In actual practice the PDF is three-dimensional. An isosurface of an actual PDF (the 50% value) is shown in Figure 3. This PDF is modeling the behavior of the Le Penven et al (1985) return-to-isotropy Case III > 0 experiment. Note the fairly

110

large changes in the shape and size of the distribution even for this simple experiment. It can also be seen in this figure that a spherical PDF corresponds to isotropic turbulence.

Figure 3" Evolution of the 50% isosurface of the PDF for the return-to-isotropy experiment of Le Penven et al. (case III> 0).

T H E O R E T I C A L ANALYSIS Lundgren (1967) first derived the exact expression for the collision term in the PDF evolution equation for turbulence. As might be expected, this collision term can not be expressed solely in terms of the PDF, and solution of the PDF evolution equation requires a model for the collision term. In this work we have focused on generalizations of the Fokker-Plank collision term. In its simplest form this collision term has the form, df

I

--~ co,....... where u, =

Ivffdv

_

0

av,

a2f [~(7il, i~)f ]+bav._..7

(2)

is the mean velocity and a and b are model constants. For turbulence this needs

to be generalized. Pope and coworkers use the form,

dsI

......= o%.

,

s] + b--gS- + v Ox,2

(3)

!

where v j = vj - u j is the fluctuating velocity and the first term (the drift term) now has a matrix model parameter G,j, and a viscous term has been added for near wall

(low Re number)

calculations. The conversion of these Fokker-Planck models to a Langevin equation for numerical solution dictates that the diffusion term (with b) be isotropic and not have a tensor coefficient. In this paper we analyzed the following even more generalized Fokker Plank model.

-

"av,/

(4)

The last term on the right hand side accounts (exactly) for the mesh motion. The first three terms involve model tensors. Sometimes, these tensors are isotropic and governed by a single parameter.

111

The viscous terms account for low Reynolds number effects and strong inhomogeneity. They do not involve any additional parameters and were derived via analysis and the condition that the model be exact as it approaches a wall (in the laminar sub layer). The zeroth moment of the PDF equation (Eqn 4) is the mass conservation equation. velocity moment of the PDF equation gives the momentum equation,

ou. ~-+

O(u,u. ~,+ ~.)_a.

0 vu,,,] _-~[

The first

(5)

This implies that the acceleration is given by a, = - p , , + (flu,.,),. The viscous contribution to this acceleration is necessary only if the viscosity is not constant. Taking the moment of the modeled PDF equation with respect to v', V'm gives the Reynolds stress transport equation, ~ .

OR,,, ~_O(u,R,,r,,)+ at

ax,

OT,,,,

+(u,,,,R;,, +u,,/R,m)=(G,,,jR;, , +G,,,R,m)+(H,,r, , +Hmn )

ox,

.

-(Jm;U"J+J"jum')'

.

.

.

.

+ a [ aR~m]

(6)

0(Rm~

~~L --~--,_j-2vK,~ , O.x.j

where Tnmi -- ~V' nvlm V'i f d v and K = 89 is the turbulent kinetic energy. The tensors G,j, H,j, and J~ determine the model. Complex dissipation and pressure-strain models can be implemented via these tensors. The equation for the total resolved (or mean) kinetic energy, E r = ~-~v;v, f d v --~ t~.,, is

oE,o+--~ ~['e0x,,~ +u~(~-~,.~)]=4p.,), +u.,;R,.-~,,,(.,,, +u,,,)+~, L ~ ]

(7)

The resolved kinetic energy correctly loses energy as a result of large scale dissipation, and via turbulence production. It is completely specified and does not depend on the model coefficients. The details of these derivations can be found in Chartrand (2004). When implementing the Fokker-Planck collision model (Eqn. 4) on a coarse mesh, it is attractive to make the change of variables j~ - In(f). I f f is close to Gaussian (which is expected) then j~ will be close to parabolic. This parabola can be accurately resolved and interpolated by the three points available in our scheme. The evolution equation for j" is, C3)C+ v, Of ~-7

.+

+

a

( a, - ar,,e~,h ) ~-~: - G,, G;ff 'j Of'~ + - ~ i(J,j+vu, j)

Ov;

'

+ vK,. ~

0% J

+(jo+vu,;)

' +'-K-~,

0)]

H ,j - - ~ + H o

^ ~Of + ~ a v

Ov;

Ox;

0) 0) Ov;Ovj

(8) +v

Ox , Ox,

112

While there are more terms to compute in this version, the equation for j? is much more accurate to solve numerically. In addition, low order methods and simple (3 point) difference stencils suffice because )7 is expected to be very close to quadratic. The models for the tensors G,j, H,/, and J,; require a time scale to be dimensionally correct. For this reason an additional transport equation for the timescale must be included in the model. We have used the standard epsilon transport equation for this purpose since it is very commonly used in RST models as well. SUMMARY OF THE M O D E L The collision model used in this paper is given by,

G,j = C ~, S,j + C ,~ W ,j -F ~ C p 2 K

()()1 /

where ~ = c l+10v ~

/K

,i

'j

RnmRmn

R.

~ G4

(9a)

---K

H. = TCd R ,j

(9b)

4 =-~KC~24;

(9c)

is the modified dissipation that goes to zero in regions of

strong inhomogeneity such as near walls, and P = - R , mu,, m is the standard turbulent production rate.

1

The frame invariant strain-rate and rotation-rate tensors are respectively S,j = ~(u,,j + u),,) 1

and W,j - w(u;,j - U;,,) + c # ~ k ' where ~k is the rotation rate of a non-inertial frame of reference. 2

For comparison with classic RST models, the equivalent Reynolds stress transport equation would be, +

Ot

+

Ox; P ~R, Rm,.R,., (C;,S,,,j +C:~Wm,)R,, , +(C:,S,,, +C:;W,,)Rjm +Cpz-~Rm,,-2R, , ' 00Rmn + 4 KC*pzgm. + ~ v

Ox,

(lo)

2vOK(amnl

8x,

Oxt ~ K ),l

Note that the model constant Cj does not effect the Reynolds stress transport equation. However, it does have an effect on the higher order moments (such as T,m.) and the turbulent transport term. This constant can be related to the Kolmorgorov constant (Pope, 2000). The other model constants are actually parameters and are given by, Cp2 =

Vt

V + Vt

.2F,

_

Cp2 -

Vt

V + Vt

.4F,

Cp2 = - 0 . 2 F 2 + .006 s '~

(11)

113

where the eddy viscosity is given by v, =.12F K2 and F =-~det(R 0/k) is the standard twoe component parameter that is 1 in isotropic turbulence and 0 for two-component turbulence. The transport model for the epsilon equation is standard and is given by,

c~e

0e

o,

-fix,

--+u, where C~, = 1.43,

C , 2 --

=

~

(C,,P-Che)

+A

Ox,

0e

(v+C,3vr)

(12)

-i-gx,

11/6, C~3 = 0.83, and fairly standard values.

SUMMARY OF RESULTS The model was tested on anisotropic decaying turbulence. This is essentially a test of the models ability to correctly predict slow pressure-strain or return-to-isotropy. The eddy collision model has no model constants associated with return to isotropy. In classic RST models, return to isotropy is parameterized by at least the Rotta constant (Rotta, 1951), and quite frequently by an additional return constant to parameterize nonlinear return effects. The Sarkar & Speziale (1990) model is an example of a two parameter nonlinear return model. The derivation of the parameter-free eddy collision return model is found in Perot & Chartrand (2004). This parameter-free model is nonlinear, and strongly realizable, and was discovered as a direct result of the collisional model framework. Two different experiments (Choi & Lumley (2001) and Le Penven et al (1985)) and five different data sets were used to evaluate the performance of the model in figures 4 and 5. Except for Le Penven case III<0 (where all models show difficulty) the parameter-free model agrees well with experimental data. 0.42

r----,F---~---]

..... l---!

0.3

.... ]

E

0.056

-

--

I- - - [ ..... ' ..... I-

0.04 ~

-t

x-

"

~,

'"

0.032

0.032[-

-:--

I - -t

0.028

! -

0.024

~"%,~<~..

0.02 !-

--1--

~

" " ~ .....

-

-4

0.012

0.06 l____~ 0

(a)

I _ 4_-T~zJ 0.03

time

0.06

0.008 L 0.09

0

(b)

i

~ 0.06

L ..... t........ iL ~'_

time

0.12

O.

0.008

--~--~-: - ~ 0

0.02

0.04

time

(c)

Figure 4: Reynolds stresses for Choi and Lumley. (a) Case A, (b) Case B, (c) Case C-2. Symbols are the experimental data, lines are the Rotta model predictions (CR = 0.8), the dashed lines are the SS model predictions (CR=0.8, CN=0.8), and large dashed lines are the current model

0.06

114

Figure 5: Reynolds stresses from Le Penven, Gence and Comte-Bellot. (a) case III>0, (b) case III<0. See Figure 4 for Legend. Next the model was tested in a variety of homogeneous shear flows. They key to predicting these flows correctly is in the modeling of the fast pressure-strain. In the current model three parameters are devoted to the modeling of the fast pressure-strain. The performance of the model is shown in figure 6. In the absence of rotation, the current model performs well. The final figure shows the turbulent kinetic energy in a shear flow as a function of time at three different rotation rates. Only the zero rotation case (upper curve) is well predicted.

Figure 6: Eddy collision model applied to homogeneous shear flows. Symbols are experimental or DNS data. Lines are the model predictions.

Finally, the model was implemented and tested in fully developed channel flow at Re=590. The results are shown in figure 7. The issue in channel flow is to correctly account for inhomogeneity and low Reynolds number effects. In this situation, the modeling of the dissipation tensor requires close attention. This term dominates near the wall and balances viscous diffusion. Details of the dissipation model are found in Perot & Natu (2003). The model for the dissipation

115

tensor is exact in regions of strong inhomogeneity and involves no model parameters. The second to last term in Eqn. 4 is due to this dissipation model. The fact that the model is exact in this limit is important. It means that the diffusion is exactly balanced at the wall, and therefore that the Reynolds stresses always have the correct asymptotic limits near a wall. This means that elliptic relaxation approaches are not required. In addition, computational stability is significantly enhanced since this is the region where Reynolds stresses are close to becoming unrealizable. 24

r

r

-

r

F-+--

ss I -

21

-r-

T---,--

7-

-7--T

;

--7 ....

F--

1

'~I'~

'!

m

t

i+

o. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

01

'++~+~++]

. . . . . . .

02

03

04

05

06

07

oa

09

Y

(a)

~

1

o[~

0

.

.

o~

(b)

.

.

02

.

.

.

03

.

.

04

.

.

os

.

.

.

06

.

.

07

.

08

09

0

o1

(c)

Y

] 02

03

04

os

06

o?

08

09

1

Y

o o oNs~ti

i

o,

Y

(d)

Y

(e)

Y

(f)

o2

o~

o~

o~

~

o~

om

o9

Y

(g)

Figure 7: Turbulent channel flow at Re=590. Symbols are the DNS data of Moser et al., lines are the model predictions. (a) Mean velocity, (b) Turbulent kinetic energy, (c) Dissipation rate, (d) R11, (e) R22, (f) R33, (g) R12.

CONCLUSIONS This paper demonstrates that collisional models are a viable alternative to RST models. In one instance, we have even been able to remove a model parameter due to insights gained from this viewpoint. However, it is also clear that this approach, as it stands, has most of the same difficulties and limitations of RST models. In particular, the fast pressure-strain model largely dictates the model's performance in flows with mean flow gradients (most flows). Fast pressurestrain models have many constants and a great deal of predictive uncertainty associated with them. In addition, the scale (or epsilon) transport equation remains (as with RST models) a source of significant error and parameterization (many constants). Finally, although we have used LatticeBoltzmann discretization ideas, the implementation of these collision models is not as computationally efficient as classic Lattice-Boltzmann methods. A moving adaptive mesh is required making the method computationally comparable to RST models. ACKNOWLEDGEMENTS This work was sponsored by the Office of Naval Research under grant number N00014-99-1-0194.

116

REFERENCES

C. Chartrand (2004), Eddy collision models for turbulence, Masters Thesis, University of Massachusetts, Amherst. K. S. Choi and J. L. Lumley (2001), The return to isotropy of homogeneous turbulence, J. Fluid Mech. 436, pp 57-84. Le Penven (1985), On the Approach to Isotropy of Homogeneous Turbulence: Effect of the Partition of Kinetic Energy Among the Velocity Components, Frontiers in Fluid Mechanics, (ed. S. H. Davis & J. L. Lumley), 1-21, Springer. T. S. Lungren (1967), Distribution functions in the statistical theory of turbulence, Physics of

Fluids, 10, 5. R. D. Moser, J. Kim and N. Mansour (1999), Direct numerical simulation of turbulent channel flow up to Re=590, Phys. Fluids. 11,943-945. J. B. Perot & C. Chartrand (July, 2004), Modeling return-to-isotropy using kinetic equations, Submitted to Physics of Fluids. J. B. Perot & R. Nallapati (2003), A Moving Unstructured Staggered Mesh Method for the Simulation of Incompressible Free-Surface Flows, Journal of Computational Physics, 184, 192-214. J. B. Perot & S. Natu (May 2003), A model for the dissipation tensor in inhomogeneous and anisotropic turbulence, Submitted to Phys. of Fluids. S. B. Pope (1994), Lagrangian PDF methods for turbulent flows. Annual Rev. Fluid Mech. 26, 23-63. S. B. Pope (2000), Turbulent Flows, Cambridge University Press. J. Rotta (1951), Stastische theorie nichthomegener turbulenz I, Z fiir Physik 129, 547-572. S. Sarkar and C. G. Speziale (1990), A Simple Nonlinear Model for the Return to Isotropy inn Turbulence, Physics of Fluids, A2:l, pp. 84-93

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

117

A STRESS-STRAIN LAG EDDY VISCOSITY MODEL FOR UNSTEADY MEAN FLOW A. J. R e v e l l 1, S. B e n h a m a d o u c h e 1'2, T. C r a f t 1, D . L a u r e n c e l ' 2 a n d K. Y a q o b i 2 1University of Manchester, PO Box 88, Manchester M60 1QD, UK 2 EDF-DER-LNH, 6 quai Watier, 78401 Chatou, France ABSTRACT A new eddy viscosity model is proposed to include stress-strain lag effects in the modelling of unsteady mean flows. A transport equation for the lag parameter, hereby denoted Gas is derived from a full Reynolds Stress Model (RSM), to be solved in conjunction with a standard two equation low Reynolds number Eddy Viscosity Model (EVM). The performance of the new k - c - Gas model is compared to the flow in a channel driven by a pressure gradient oscillating around a non-zero mean. Results are compared with Large Eddy Simulations (LES) of the same flow and the addition of the lag parameter equation is shown to give improved results when compared to the standard EVM.

KEYWORDS

LES, DES, URANS, EVM, Oscillating channel, Stress-strain alignment

INTRODUCTION Many everyday turbulent flows are inherently unsteady, both industrial (e.g. flows in the cylinders of internal combustion engines, in tubines, wakes of bluff bodies... ) and natural (e.g. the blood flow in arteries). These flows can be a result of imposed fluctuating boundary conditions or geometry induced oscillations, or a combination of both. The presence of such unsteadiness in a flow can significantly alter the behaviour of important parameters such as the Reynolds stresses, turbulent kinetic energy and dissipation rate. Despite the existence and derivation of progressively more complex modelling, simulation or hybrid approaches in the field of turbulence modelling, models for the Unsteady Reynolds Averaged Navier Stokes equations (URANS) are still widely used for the computation of complex flows. However the accuracy of such standard eddy viscosity models have not been carefully established and several fundamental assumptions in their derivations for steady flows in near turbulent equilibrium, no longer hold in the application to transient cases. Through the Boussinesq approximation [ 1], the EVM provides a direct link between the turbulent stress tensor uiuj in its dimensionless form (aij = uiuj/k - 2/3(5~j, where k is the turbulent kinetic energy, ~5~j is the Kronecker delta function) and the mean rate-of-strain tensor S~j, erroneously forcing them to

118

be directly in phase. The phase lag between the stress and strain tensors is essential in the case when the flow is subjected to periodic strain and the lack of this feature leads to continuous over-generation of turbulent kinetic energy [2]. It is well documented that standard EVMs significantly overpredict the production of turbulent kinetic energy in presence of strong strain. This has led to various limiters as in the v 2 - f model [3], the k - ~ SST model [4] and the Linear Production model [5] among others. The need for such bounds comes from the fact that E V M s yield a production, Pk, proportional to the square of the strain rate S~j. In contrast, the full Reynolds Stress Model calculates an exact production which is linear in the strain rate. Indeed, one can write exactly Pk = Cask []S]] where Cas = - a i j S i j / I l s I I is a non-dimensional parameter representing the degree of alignment between stresses and strains. Jakirlid, [6] solved the full stress transport equation to include stresses in Pk, as a means of improving k levels used in a E V M representation of the stresses subsequently applied to the momentum equation. The results show some good improvements for a range of testcases, but the computational cost is very high. The R S M has been shown to act like an LES Sub Grid Scale model in some inherently transient flows [7]. Indeed, in homogeneous cyclic strains, the R S M shows that turbulence intensity grows until inertial effects are large enough such that the stress tensor no longer follows the strain tensor, causing turbulence production to be blocked [2]. The use of the full R S M in combination with LES is appealing for hybrid approaches (e.g. D E S [8]), but is still often perceived as impractical for industrial applications. It is also interesting to note that the reduction of eddy diffusion coefficient in the two equation modelling, [9] calibrated on the basis of R S M modelling can lead to the appearance of unsteady vortex structures in agreement with the physical experiment. The aim of this work is therefore to develop a model for the stress-strain lag, which appears to be a key parameter in rapidly evolving flows. This is obtained by assuming an R S M model for the time derivative of the tensor aij as well as terms including the rate of change of the strain tensor Sij. The model will be derived and validated for homogenous shear and channel flows, before being applied to the oscillating channel case. Its performance will be compared to LES results for the flow in a channel driven by a pressure gradient oscillating around a nonzero mean. Recent LES calculations of this flow are repeated, and compared to the results provided by Scotti and Piomelli [10], [ 11 ] with the same parameters. The objective is to use the LES database for the validation of URANS models in the future, but the current paper is limited to an introduction of this work.

STRESS-STRAIN LAG

The tensors of both stress anisotropy aij and strain rate Sij are 3 • 3 symmetrical tensors. This is important since it can be shown that the eigenvectors of a symmetric tensor are real and orthogonal. The anisotropy tensor has zero t r a c e ctijc~ij -- 0 and is dimensionless by definition, whereas the strain rate tensor is an inverse time scale and has zero trace only in the condition of incompressibility, SijSij = O, which is assumed for this work. As previously stated, the eddy viscosity model assumes that these two tensors are aligned. DNS data from a channel flow and from a homogeneous shear flow can be used to see that even in these cases, this assumption is not true. Invariants are values of a tensor that remain constant regardless of the coordinate system and they can be used to describe the tensor. Invariants used here to characterise aij and Sij are defined in Eqn. 1.

Ila[I = x/aijaij

and

IlSl[ = v/2SijSij

(1)

The alignment of two 2 • 2 symmetric tensors is trivial since there is only one angle between the sets of orthogonal principle axes. This is important as it implies that the alignment for all 2 dimensional flows is representable with a single dimensionless scalar. For a 3 • 3 tensor, the orientation is defined

119

by the 3 Euler angles, which can be obtained through calculation of eigenvectors. The orientation of each tensor allows the misalignment to be given as the difference between each Euler angle, and so three scalar values are necessary to define the full stress-strain lag in a fully 3 dimensional flow. The calculation of the eigenvalues of both tensors, per node, per timestep, is too expensive for large calculations. A simpler approach would instead consider the inner product of the two original tensors, to give a scalar measure of alignment rather than an angle. This parameter, (i.e. C'~s = a#Sij/[]a]l ]S 1), will return a value of 1 when aij is aligned with Sij, and a value of 0 when tensors are mutually perpendicular. The alignment parameter must fit into the 2-equation modelling framework and so the term Gas is defined in Eqn. 2 which is limited by + ]Jail. The case of Ca~ = [Jail corresponds to the linear EVM, so in effect this parameter quantifies the degree of non-linearity of the relationship. C~ =

DERIVATION

OF k - e -

C~

aiySij

(2)

IIsll

MODEL

The transport equation for Gas will now be derived. This is necessary in order to capture the effects of stress-strain lag in transient flows. The aim is to correct ut in unsteady flows, whilst unaffecting E V M results in steady state. Given the definition of Cas from Eqn. 2, the time derivative is proposed using the product rule (where D e ~ D r = Or + UkOr

DOes

1

Dt

IlSll

(

Daij DSij s ~ j - - ~ + a i j - - - ~ -[- Gas

DISlI)

(3)

Dt

It is proposed that the total derivative for aij in Eqn. 3 can be calculated from the full Reynolds Stress Model (Eqn. 4) where small letters denote fluctuating quantities and large letters denote mean quantities. The terms for production Pij and viscous diffusion, DlIIij "-". . . . i..... are exact, but the remaining terms must be modelled. The production can be rewritten as shown in Eqn. 5, where the influence of the rotation tensor f2ij can be seen. The tensorial dissipation in the R S M can be linked to the dissipation equation by assuming local isotropy eij = ~2edij . This is accurate as long as the Reynolds number is high, however when the Reynolds number is lower or when approaching a wall, this is not exact. The pressure strain term 17# will be approximated using the SSG model shown in Eqn. 6, derived by Speziale et al. [ 12]. Duiuj

Dt

= P i j -f- I-Iij - s

Pij -- - k

-[-

T.. op viscous

131IIij

+ C4k

a~a~j - 5~jA~

a~kSjk + ajkS~k -- 5 a ~ & ~ 6 ~ j

u uj k

(4)

(5)

-~ ij -Jr- a i k S k j + S i k a k j -+- a i k ~ k j + ~ i k a j k

n~j = - (Cl~ + c;P~) a~j - c ~

aij =

ro turbulent -1- ~-.. l_)llIij

j 3

- C3

+ Csk (a~kfijk + a ~ k )

sij 1(o = -2 ~

C1 1.7

+

1(o + Oxi J

C63

f~ij = -~

C4

c5

0.625

0.2

Oxj

Oxi

(7)

120

Using the definition of the anisotropy tensor (Eqn. 7) and the full RSM(Eqn. 4), the transport equation for the anisotropy tensor is given in Eqn. 8, where Diff ~ij contains all required diffusion terms. ] Daij = --[Ply + II{j - cij + (Pk - c)(aij + 6ij)] + Diff a" Dt k

(8)

Rodi [13][ 14] developed the Algebraic Stress Model (ASM), in which the neglection of transport of aij is referred to as the weak equilibrium assumption (Daij/Dt - Diffusion ~'~ = 0). This is necessary to construct an algebraic approximation for the anisotropies, based on a specific RSM. In this case the anisotropy relation is implicit and a coupled, nonlinear system of equations must be solved. The approach was shown to retain some features from RSMs but numerical difficulties resulting from a lack of diffusion or viscous damping lead to problems particularly for complex flows. An alternative approach, known as Explicit Algebraic Reynolds Stress Models (EARSM e.g. [ 15]) instead aim to obtain the anisotropy relation explicitly in terms of the mean flow field. Some of these have been shown to be accurate in rotating and sheared flows, although they remain unable to incorporate history and evolution effects of the stresses into their predictions. The present approach is different from the ASM approach in that the entire RHS ofEqn. 4 is used. Given that these terms are balanced to give the advection of the stresses, this approach will be inherently able to incorporate evolution effects. To simplify the implementation, the diffusion terms from the RSM will be dropped, and their effect incorporated within a simple gradient diffusion term in the general transport equation of C~. The general transport equation for Ca~ can therefore be rewritten after substitution of Eqn. 4 into Eqn. 3 as shown.

DC~, Dt -

1

IlSll (SijPij + SijIIij + aijSij - a~jSijc)

(9)

DSij DISI) aij~ + C.~- Dt + Diffc"~

1( IlSl

The SSG variant of the full transport equation for Gas is given by Eqn. 10. The strain rate parameter is = k IlSll/~ the second invariant of anisotropy is A2 = aijaij.

,1

and

Dt

=

0.266 + 0.325

S l l - 2.7~Ca~ + 0.1

IIS co~

(10)

-k- 1.05 S i j a i k a k j - t - 0 . 7 5 SijaikSj-k nt- 1.6 S i j a i k ~ j k

,~

IlSll

__ aij J~Sij

IlSll

Gas D

IlSll

Dt

IlSll

IlSll § Diffc~ Dt

The turbulent viscosity ut is redefined using an updated value of C~ according to Eqn. 12 and the new diffusion term is defined by Eqn. 13 with the constant Ocas = 2.

]s

g-mew __ C

Chew-max

(11)

min (O.09, Cas ) ,0.01] 0

ut

(12)

(13)

If the equation is to correctly model the log-layer region, then it is a requirement that it returns the standard value of C~ in equilibrium conditions (i.e. when production to dissipation ratio is unity and

121

r/ -- 3.333). This is tested by taking equation 10, setting all time derivatives to zero, and neglecting convection and diffusion. Since for the linear EVM there is no normal anisotropy, a i i = 0, it became necessary to take C~ as zero in order to achieve the correct balance, giving a value Cas = 0.32 which is equivalent to Cu -- Cas/rl -- 0.096 and hence close to the standard value. It should be noted that future work will employ a Non-Linear Eddy Viscosity Model (NLEVM e.g. [ 16]) to provide the ai# values in Eqn. 10, which will avoid this drawback. The correctional effects of the Ca, term on the parameter Cu are expected to be even greater since normal anisotropy is non-zero and so more of the terms in Eqn. 10 are in play. For the results in this paper, the Low Reynolds number k - e model proposed by Launder and Sharma (LS) [17] is used as the baseline model, with the straightforward extention described in Eqns. 11 and 12.

NUMERICS Calculations for the present flow were performed using the EDF in-house CFD code, Code_Saturne [ 18]. It is an unstructured finite-volume code based on a collocated discretization for cells of any shape. It solves turbulent Navier-Stokes equations for Newtonian incompressible flows with a fractional step method based on a prediction-correction algorithm for pressure/velocity coupling (SIMPLEC) and a Rhie and Chow interpolation to avoid pressure oscillations. Many turbulence models are available (among them the standard and low Reynolds number version of k - e, SSG, v 2 - f, LES . . . . ). Several LES calculations have been recently computed successfully and validated with Code_Saturne[ 19], [20]. Both the standard Smagorinsky model with a constant set to 0.065 and the classical dynamic model have been used to compute the oscillating channel flow. The time advancing scheme is second order with a combination of Crank-Nicolson and Adams-Bashforth schemes. A fully centred scheme is used for the convection term without any upwinding. In the case of non-orthogonal cells, a gradient reconstruction technique is used to maintain the second order accuracy in space.

O S C I L A T I N G CHANNEL

LES results are available to describe the periodic variation of the flow field. An oscillating pressure gradient is applied as the periodic boundary conditions to a similar mesh to that used in the previous section. The flow is forced by a mean pressure gradient given in Eqn. 14 following the notations given in Scotti and Piomelli [ 10]. 71"

Pl (x,t) = APo [l + acos (f~t + -~) ] x/L~

(14)

In the following, the frequency f~ = 3.57s -1 and the amplitude a = 50, which is refered to in [10] as an intermediate frequency. This is an interesting frequency which causes the turbulent quantities to become out of phase and is therefore suitable to the present study (whereas the higher frequencies influence only the flow very close to the wall). The pressure is lagged in order to start the acceleration phase at ~2t = 27rt/T where T is the period time. The channel half height 5 and the mean pressure gradient APo/L~ are set to give a Reynolds number of 350, based on the mean friction velocity ur -- v/SAPo/2pLx. A set of 100 nodes is used to discretize the 1-D domain for the URANS calculations with a near wall refinement. The domain size in the LES calculations is 65~" x 45 x 3/25~- and the number of cells is 64 x 64 x 65 with wall refinement. The maximum Courant number in all the calculation did not exceed 1. In all the computations, a phase averaging is used to compute mean values. Phase averaging is denoted by (-). It

122

is defined for a physical variable r as follows.

(r

1 (V, t) -- --~

(r

(v, t) = -~ ~=~ ~

n=l

f

V,~+n

(15) I

~, v, z ~ + ~ eze~

(16)

RESULTS Initial work was carried out to validate the LES for the oscillating channel flow. Figure 1 shows the periodic variation of several quantities for two versions of our LES, Scotti's LES and the URANS models. Overall the agreement between the 3 LES calculations is very reasonable, although the dynamic model seems to overpredict both the velocity and kinetic energy in comparison to the other two. There is a slight discrepancy in the variation of the wall shear stress between the new results and the reference data, where a bump described between t i t = 0.6 and t i t = 0.9 is not picked up. This is mirrored by discrepancies in the periodic variation of kinetic energy, for the same time bracket, which also corresponds to deceleration region of the oscillation ( t i t = 0.25 and tiT -- 0.75). This behaviour is an indication of the strong influence of near wall phenomena on the rest of the flow.

4o

,

,

,

_ 30 - -

,

,

:

:

9

REFERENCE

! ~ t. ~-

i

0

LES smagorinsky

+

LES dynamic model

j.'/

Ucl 20 - _

.//

,

,

i

"

~

/ , t.

~ ~/ip;

10

: 10

-

_ 5

+ + + + ! ~ " ~

0 5

(Scotti)

k - v. m o d e l L S k - e m o d e l L S w i t h Cas

+++++:+

+++ 9 : :_++ ,~oOO..~,~ :-1^D.v-.--~ : qa~ i

kmax

,j,

"

~

~

--[ -t

~ ~..~..

~- '" ~

.

.

.

mnmm .mi 9

'l~wall 0 0

0.125

0.25

0.375

0.5

0.625

0.75

0.875

1

t/T Figure 1: Time series over period of oscillation for centreline velocity Uct, maximum streamwise kinetic energy kmax and wall shear stress rwau. Comparison to LES reference data [10] for validation of LES dynamic and LES Smagorinksy models. Results from Low Reynolds number k - e model [ 17], with and without C,s modified viscosity.

Given the decent agreement between the Low Reynolds number k - ~ [ 17] and the reference data, it can

123

be said the selection of a good near wall model or damping function, together with an adequate near wall refinement is enough to capture the main features of this flow. Although the reference profile of kinetic energy suffers from lack of data points, the asymmetric bump during deceleration shown by all 3 LES is a well documented feature of this flow and should not be ignored. The k - e - C~s model is in very good

Figure 2: Streamwise profiles at Ct/T for r = 0 to 7. Columns from left to right: Velocity U (scale 0 -+ 40), turbulent kinetic energy k (scale 0 -+ 12) and alignment, C~s (scale 0 ~ 0.4) vs y/3. Results shown for LES dynamic model, LES Smagofinksy model, Low Reynolds number k - ~ model [ 17], with and without Ca~ modified viscosity. For key refer to Figure 1 The same general conclusions can be drawn through cross-reference to Figure 2 which shows streamwise profiles of velocity, kinetic energy and Cas at eight different phase times, corresponding to the x-axis labels of Figure 1. It can be seen that the shape of the velocity and kinetic energy profiles are improved when using the k - c - C a s model particularly in the acceleration phase. Moving back into the deceleration region, the near wall maxima of kinetic energy and velocity fall slower than the LES while the results with the unmodified URANS model are decent, but this is only by chance i.e. because they are falling from lower maximum values. Tracing the maximum of the LES profiles of Cas reveals an elliptic path over the period, which represents the movement of the phase lag between production and dissipation of k.

124

This flow is certainly influenced to a large extent by its near wall features and there is an apparent phase switch between best predictions being offered by the k - c model and then by the k - r - Gas model. This is likely due to the fact that the terms of the C~, transport equation are not valid near the wall. Since they come directly from the high Reynolds SSG model, there is no damping of the pressure strain. The Low Reynolds k - c model has improved behaviour in the decelleration region and it is reasonable to expect that a low Reynolds number version of the k - c - C~s model, or even a version coupled with the v 2 - f , would retum similar improvements.

C O N C L U S I O N S AND F U T U R E W O R K The degree of alignment of stresses and strains is expressed as a dimensionless parameter and a transport equation is derived to account for the time evolution of this term. The equation is derived from a full RSMmodel so as to incorporate phase lag information and to automatically limit the production of kinetic energy in regions of strong strain. The eddy viscosity is globally modified so there is no discrepancy between the turbulent transport and the mean flow equation. LES results for a channel flow oscillating at an intermediate frequency are validated and used to show the improvements in the new k - c - Gas model over the standard model.

Figure 3: Preliminary results for flow around NACA0012 at max Lift at an incidence of 60 ~ using k - c - Gas model. TOP: Contour plot o f Pressure overlaid with flow streamlines. BOTTOM" Contour lines of modified C . with labels. Both at same location and time.

The motivation behind the model development stems from the observation that in rapidly varying mean flows, such as bluff body wakes or staggered tube bundles, RSMs can produce large unsteady structures similar to LES. In homogenous turbulence subject to cyclic straining it was shown that the stresses increasingly lag behind the strains until production is shut off. Therefore RSMs seem an attractive idea for DES, but is too computationally expensive. The present model is very economical since it is has almost no additional expense over a standard k - c in terms of convergence time, and only slightly

125

higher storage requirements. It is also very easy to implement in contrast with the RSM. Figure 3 shows contour plots from some early calculations on the NACA0012 airfoil at an incidence angle of 60 ~ This is a massively separated flow and provides a more suitable test for the k - c - Cas model. From Figure 3 it can be seen that C u is reduced in the regions corresponding to the shedded structures (shown by the pressure contours and flow streamlines). This behaviour is in agreement with the prediction of strongly detached flows by modified two-equation models that suggest the C , reduction based on the behaviour of the R S M modelling for the same class of flows [9], in the context of the Organised Eddy Simulation, also used to improve the R A N S length scale in D E S for the same NACA0012 configuration [21]. It appears that the response of the k - ~ - Cas model to regions of stress-strain lag could be used to automatically reduce k levels in the U R A N S ~ D E S transition when unsteady structures start to emerge, instead of the ad-hoc switch based on grid characteristics as commonly used in DES; in which mesh dependency is one of the current breakdowns. A detailed study of the k - c - C ~ model is under way using the experimental study around a circular cylinder [22].

Acknowledgements Authors are grateful to A. Scotti and U. Piomelli for fast and friendly provision of their LES data and clarifications for its use. Financial support via the FLOMANIA and DESider projects is also acknowledged. The FLOMANIA project (Flow Physics Modelling - An Integrated Approach) is a collaboration between Alenia, AEA, Bombardier, Dassault, EADS-CASA, EADS-Military Aircraft, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and St. Petersburg State University. The project is funded by the European Union and administrated by the CEC, Research Directorate-General, Growth Programme, under Contract No.G4RD-CT2001-00613. The DESider project (Detached Eddy Simulation for Industrial Aerodynamics) is a collaboration between Alenia, ANSYS-AEA, Chalmers University, CNRS-Lille, Dassault, DLR, EADS Military Aircraft, EUROCOPTER Germany, EDF, FOI-FFA, IMFT, Imperial College London, NLR, NTS, NUMECA, ONERA, TU Berlin, and UMIST. The project is funded by the European Community represented by the CEC, Research Directorate-General, in the 6th Framework Programme, under Contract No. AST3-CT-2003-502842. This work was also aided in part by the Marie Curie Fellowship Program, contract number HPMT-CT-2000-00079.

References [1] J. Boussinesq. Theorie de l'6coulement tourbillant. MOm. PrOs. par div. savant a lacad, sci. Paris, 23:46, 1877.

[2] I. Had~i6, K. Hanjali6, and D. Laurence. Modeling the response of turbulence subjected to cyclic irrotational strain. Physics offluids, 13(6):1740-1747, 2001.

[3] R Durbin. Near-wall turbulence closure modelling without 'damping functions'. Theoretical and computational fluid dynamics, 3:1-13, 1991.

[4] F. R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications.

AIAA J.,

32:1598-1605, 1992.

[5] V. Guimet and D. Laurence. In A linearised turbulent production in the k - E model for engineering applications, 2002. ETMM5, Mallorca, Spain.

[6] B. Basara and S. Jakirli6. A new hybrid turbulence modelling strategy for industrial cfd. Int. J. Numer. Meth. Fluids, 42:89-116, 2003.

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[7] S. Benhamadouche and D. Laurence. Les, course les, and transient rans comparisons on the flow across a tube bundle. International Journal of Heat and Fluid Flow, 2003. [8] P. R. Spalart, W.-H. Jou, M. Strelets, and S. R. Allmaras. In C. Liu and Z. Liu, editors, Comments on the Feasability of LES for Wings, and on a Hybrid RANS/LES Approach, Rushton, LA, 1997. First AFOSR International Conference on DNS/LES 4-8 August. [9] Y. Hoarau, M. Braza, P. Rodes, G. Tzabiras, C. Allain, D. Favier, E. Berton, and M. Maresca. Turbulence modelling of unsteady flows with a pronounced periodic character around an airfoil. In Proceedings, IUTAM symposium "Unsteady Separated Flows ", Toulouse, France, 2002. [10] A. Scotti and U. Piomelli. Numerical simulation of pulsating turbulent channel flow. Physics of Fluids, 13(5):1367-1384, 2001. [11] A. Scotti and U. Piomelli. Turbulence models in pulsating flows. AIAA Paper, (2001-0729), 2001. [12] C. G. Speziale, S. Sarkar, and T. B. Gatski. Modeling the pressure strain correlation of turbulence: an invariant dynamical systems approach. Journal of Fluid Mechanics, 227:245-272, 1991. [13] W. Rodi. Theprediction offree turbulent boundary layers by use of a two equation model of turbulence. PhD thesis, University of London, 1975. [ 14] W. Rodi. A new algebraic relation for calculating the Reynolds stresses. Z. A ngew. Math. Mech, 56:219-221, 1976. [15] S. Wallin and A. V. Johansson. An explicit algebraic stress model for incompressible and compressible turbulent flows. Journal of Fluid Mechanics, 403:89-132, 2000. [16] T. J. Craft, B. E. Launder, and K. Suga. Development and application of cubic eddy-viscosity model of turbulence. International Journal of Heat and Fluid Flow, 17:108-115, 1996. [ 17] B.E. Launder and B. I. Sharma. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disk. Letters in Heat and Mass Transfer, 1:131-138, 1974. [18] F. Archambeau, N. Mechitoua, and M. Sakiz. A finite volume method for the computation of turbulent incompressible flows - industrial applications. International Journal on Finite Volumes, to be published. [19] N. Jarrin, S. Benhamadouche, Y. Addad, and D. Laurence. Synthetic turbulent inflow conditions for large eddy simulation. In Turbulence, Heat and Mass Transfer, Volume 4, Beggel House, Inc., 2003. [20] S. Benhamadouche, K. Mahesh, and G. Constantinescu. Collocated finite volume schemes for les on unstructured meshes. In CTR. Proceedings of the 2002 Summer Program, Stanford, 2002. [21] M. Braza and Y. Hoarau. Prediction of the unsteady 3d flow past a naca0012 wing beyond stall by des-oes approach. In Notes on Numerical Fluid Mechanics and Interdisciplinary Design - FLOMANIA European program, 2005. [22] R. Perrin, M. Braza, E. Cid, S. Cazin, F. Moradei, A. Barthet, Y. Sevrain, and Y. Hoarau. Near-wake turbulence properties in the high reynolds incompressible flow around a circular cylinder by 2c and 3c piv. In Procs. ETMM6 Conf., 2005.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

127

TURBULENCE M O D E L L I N G OF S T A T I S T I C A L L Y P E R I O D I C FLOWS: T H E C A S E OF T H E S Y N T H E T I C JET S. Carpy and R. Manceau Laboratoire d'Etudes A~rodynamiques UMR 6609, University of Poitiers/CNRS/ENSMA, France

ABSTRACT

Computations of a synthetic jet are performed with usual RANS-equations solved in a timeaccurate mode (URANS), with the standard k-c model and the Rotta+IP second moment closure. The purpose of the present work is to investigate the ability of these standard turbulence models to close the phase-averaged Navier-Stokes equations. Results are compared with recent experiments by Yao et al. provided to the CFD Validation of Synthetic Jets and Turbulent Separation Control workshop held in 2004. Comparisons of the performance of the models with experimental data show that the models are globally able to reproduce the dynamics of the flow. However, several characteristics are not well predicted, e.g., convection velocities and spreading rates. Moreover, results suggest that second moment closure is the minimum modelling level suitable for pulsed flows: regions of negative production play a significant role, and their prediction requires to solve transport equation for the Reynolds stresses. KEYWORDS

Turbulence modelling, URANS, Synthetic Jet, Statistical Unsteadiness, Control 1

INTRODUCTION

Although statistical turbulence modelling (RANS) represents the current industrial standard, there is a considerable interest in the possibility of obtaining informations on the large-scale unsteadiness. As long as Large Eddy Simulation will remain too expensive for overnight simulations, some room exists for the development of less expensive methods able to predict the very large scales of the flow. This type of approaches have important applications in different domains: among others, aeronautics [11, 17], control [12, 15], aeroacoustics [4], power generation [7] and environment [19] can be cited. In many statistically steady flows, as soon as the time derivatives are included in the equations, unsteady RANS solutions appear naturally [6, 8, 14, 16-18, 20]. In this case, long-time-averaged solutions are often better than solutions obtained by steady-state computations [10, 16]. However, there is no clear consensus on the definition of the mean operator for this type of computations: the definition of the turbulent scales resolved by the method is not a priori known. Recent approaches, like Semi-Deterministic Modelling (SDM) [3, 4], Very-Large-Eddy Simulation (VLES) [25], Detached-Eddy Simulation (DES) [24], Limited Scales simulations (LNS) [5] or Partially-averaged Navier-Stokes modelling (PANS) [13], are designed in order to ensure unsteady solutions, but do not address the theoretical problem of the significance of the solution. In the present study, the flow belongs to the class of statistically periodic flows, i.e., flows in which the boundary conditions are periodic in time. The use of RANS models does not raise any theoretical issue in this case, since Reynolds average is equivalent to phase average. However, obtaining

128

a periodic solution is not guaranteed: some frequencies can appear that are not harmonics of the forcing frequency. Moreover, while the relative performance of the numerous turbulence models available have been investigated extensively in statistically steady flows, their ability to close successfully the phaseaveraged equations in periodic flows have received much less attention, and generally for modified versions of the models [9, 26, 27]. Therefore, a workshop dedicated to the evaluation of modelling strategies in statistically periodic flows was recently organized [28]. The present study focuses on the comparison of the performance of first and second moment closures" the standard k-e model and the R o t t a + I P second moment closure are used, without any case-specific modification. Standard wall functions are applied since near-wall regions do not play a significant role. The flow studied in the present paper is one of the test cases selected for the workshop [28]. The flow consists in an oscillatory jet with zero-net-mass flow issuing into quiescent air. This flow is of high industrial interest, in particular in the frame of turbulence control, since it is representative of MEMS piezoelectric technology, as developed by Steifert e t al. [23] and Amitay e t al. [1].The experiments used as reference in the present study were performed at NASA LaRC by Yao e t al. [29] and are available on the web site of the workshop [28]. Notation

In such a problem, it is convenient to introduce two averaging operators: long-time averaging, denoted by . and phase averaging, denoted by <. >. Instantaneous quantities are denoted by stars (u~, p*) and the following definitions are used" Long-time-averaged velocity:

Ui = u i N

Phase-averaged velocity:

(1)

*

Ui = <

ui

>

(2)

In order to avoid confusion with usual notation in RANS modelling, the phase-to-phase fluctuating velocities are denoted by u i l/.

II

*

ui = ui-

*

< ui >

(3)

Thus, in the present periodic case, the transport equations to be solved are those of the phaseaveraged velocity components Ui, those of the Reynolds stresses < u i" u j" > (or of k" = ~1 < u ittu itt > in the frame of the eddy-viscosity model), and of e", the dissipation rate of k". 2

DESCRIPTION DATA

OF THE

GEOMETRY

AND

AVAILABLE

EXPERIMENTAL

The oscillating jet of air emanates from a slot in the floor. The slot is 1.27 mm wide (h) and 28h long, in order to produce a 2-D plane jet at the central plane of the floor. The slot is at the bottom of a cubic enclosed box 480h large, at the centre of the floor and parallel to the sides of the outer walls. The geometry and the coordinate system are shown in Fig. 1. The jet flow is produced by a piezo-electric diaphragm on the side of a narrow cavity under the floor, which is driven at 444.7 Hz. The maximum jet velocity generated could reach 28 m s -1. Although the actuator operating parameters including diaphragm displacement, internal cavity pressure, and internal cavity temperature were monitored to provide boundary conditions for CFD simulations, we have chosen not to model the internal cavity: the present work focuses on the performance of turbulence models, rather than on the other difficult issue of representing a vibrating diaphragm with simple boundary conditions. Computational results in the present paper are compared with the PIV measurements performed by Yao e t al. [29]. Phase-averaged (every 5 deg) and long-time-averaged velocities and Reynoldsstresses (only components in the x - y plane) are available in the window -2.15 < x / h < 3.1, 0.1 < y / h < 6.6, z = 0.

129

Figure i: Geometry, domain size and coordinate system.

Figure 2: Measured vertical velocity over the centre of the slot as a function of phase (x=O, y=O.lmm).

Figure 3: Computational grid. 3

NUMERICAL

METHOD

AND SENSITIVITY STUDIES

Computations are performed using Code_Saturne, a finite volume solver on unstructured grids developed at EDF [2], for vectorial and parallel computing. Space discretization is based on a collocation of all the variables at the centre of gravity of the cells. Velocity/pressure coupling is ensured by the SIMPLEC algorithm, with the Rhie & Chow interpolation in the pressurecorrection step. The Poisson equation is solved using a conjugate gradient method, with diagonal preconditioning. Computations are performed using 4 processors of a PC cluster. This study is performed using usual turbulence models: the standard k-~ model [22] and the R o t t a + I P second moment closure [21].

3.1

Boundary conditions

Since the flow inside the cavity is not solved, unsteady Dirichlet boundary conditions are specified at the slot exit (Fig. 2), which are functions of x and phase: profiles at y/h = 0.08 of the I1_ I I available phase-averaged velocities Ui and Reynolds-stresses are extracted from PIV data and used as inlet conditions. The missing component < w"2> is reconstructed by assuming w "2 ~ - - ~ u .2 ~. Concerning the dissipation rate s", prescribing an inlet value is already an issue. Indeed, 6'~ obviously varies with time, as well as all the other usual turbulent scales (time scale, length scale, eddy-viscosity). A possible way is to arbitrarily assume that one of them is constant, e.g., the length scale L. A periodic e" can thus be obtained from #' = k"3/2/L. Preliminary computations

130 (not shown here) have indicated that a value of L of the order of magnitude of what can be expected in a fully developed turbulent channel flow (typically one-tenth of the half-width of the slot) gives too much turbulent dissipation (abrupt relaminarization). Therefore, the value of the length scale prescribed in the computations presented here is half the slot width. The necessity of using such a large value is to be linked to the fact that, despite a turbulence rate of 5%, the flow issuing from the cavity is far from being a fully developed turbulent flow: the Reynolds number based on the slot width and the maximum inlet velocity is only 2400, and the geometry of the cavity does not allow the turbulence to reach a spectral equilibrium before issuing from the slot. Solid-wall boundary conditions cannot be used at the remaining boundaries of the domain, since the fluid is considered incompressible: when fluid is blown from the slot, it is necessary to enable an equivalent volume of fluid to go out of the calculation domain in order to satisfy incompressibility. Since the experimental box is huge compared to the size of the slot (480h), a smaller domain was used, as shown in Fig. 1. Outlet boundary conditions are applied at the open boundaries of the domain: as shown in section 3.3, the results are not influenced by this restriction of the domain. During the suction phase of the cycle, fluid enters the domain through these boundaries: Dirichlet boundary conditions are then used for turbulent quantities, using values identical to those used for the initial conditions described just below.

3.2

Initial conditions

Computations are started from rest state Ui = 0, with a low residual turbulence energy ( x / ~ / V m a x =0.05%) and a dissipation rate corresponding to the ratio ~t/~ = 10. Computations are performed from the beginning with central differencing with large time steps (At corresponding to 2.5 deg), which shows the stability of the code (for convenience, time steps are given in degrees throughout the paper, i.e., the value corresponds to 360fAt, where f is the frequency of the jet). When a satisfactory periodic state is reached (after about 15 cycles), the time step is reduced and the computation is run until a new periodic state is obtained. Note that periodicity was only evaluated in the region where the flow is of interest (basically the region in which the results can be compared with experiments): indeed it would take hundreds of cycles to reach a periodic solution in the entire domain.

3.3

S e n s i t i v i t y studies

In order to investigate the influence of the size of the domain, the time-step and the grid, seven different computations were performed with the standard k-c model and the Rotta+IP second moment closure. Conclusions are given below.

Influence of the domain size k"-~" computations were performed using two different domain sizes (small and large, Fig. 1) with the same grid resolution (i.e., for the large domain, cells were added around the grid used for the small domain). The grid covering the small domain is shown in Fig. 3a. Computations were carried out with the same time step corresponding to 2.5 deg. Fig. 4 clearly shows that the restriction of the domain has no effect on the solution, at least in the region of interest. Influence of the time step The influence of the time-step on the solution has been carefully investigated. First, using the coarse grid in the small domain, a progressive reduction of the time step from 2.5 deg to 0.25 deg showed that the solution becomes independent of the time step below a value of 0.5 deg. These conclusion was confirmed using the fine grid, as shows in Fig. 5.

Grid refinement Two different grids (coarse grid: 32900 cells, fine grid: 47610 cells), shown in Fig. 3, are used. Fig. 5 shows that the effect of grid refinement is relatively small: although it is clear that grid convergence is not fully achieved yet, the error is probably not significant enough to influence the

131

Figure 4: Influence of the size of the domain. Velocity Figure 5: Influence of the grid and time step. Velocity profiles in the symmetry plane, profiles in the symmetry plane. main conclusions. Computations that this is indeed the case.

4

on a finer grid will be carried out in the near future to confirm

RESULTS AND DISCUSSION

Figs. 6 and 7 show comparisons of the isocontours of the vertical phase-averaged velocity V from PIV measurements, k'-z" and RSM computations, for both blowing and suction phases of the cycle (see Fig. 2). It can be seen that the global topology of the phase-averaged flow is well reproduced by both models. The pair of large vortices (Fig. 6) that appears at the beginning of the blowing phase are directly inherited from Navier-Stokes dynamics: the turbulence models do not play a role in their creation. However, the models influence the evolution of these vortices, and the Reynolds-stress model produces a more realistic picture. It is seen that the turbulence models play the role that is expected in this type of flow. The resolved velocity field Vi is indeed free from turbulent eddies" Vi represents only the oscillating motion induced by the diaphragm. During the suction phase of the cycle, Fig. 7 shows that the fluid is mainly sucked from the sides. This explains why in Fig. 2, the integral of the curve is not zero: what is plotted is the velocity at the centre of the slot, at the elevation y - 0.08h. Since fluid is mainly blown at the centre of the slot and sucked from the sides, a very significant part of the fluid that enters the cavity during the suction phase is not seen at that point. Therefore, computations cannot be performed using uniform profiles (independent of x): it is necessary to base the inlet conditions on profiles extracted from the PIV velocity fields, as is done in the present study. The global mass flux in the computations are thus nearly zero. This characteristic behaviour of the flow is confirmed by Fig. 8, in which long-time-averaged velocity fields are shown. It clearly appears that the fluid tends to issue from the cavity at the centre of the slot and to enter from the sides. Both models (eddy-viscosity and Reynolds stress models) correctly reproduce this behaviour. Fig. 9 shows profiles in the symmetry plane of the phase-averaged velocity V. It appears that the k'-~" model predicts a much too rapid decrease of the peak velocity, and, in particular, is unable to reproduce the acceleration above the slot which is shown in the P IV data: the velocity at the slot exit reaches its maximum 28.3 m s -1 at phase 45 deg (see Fig. 2) and, at phase 90 deg, the peak velocity around 4 mm (3.15h) is close to 35 m s -1. The "fluid particle" with the maximum velocity at phase 90 deg has been experiencing an acceleration since it issued from the slot. Fig. 9 shows that this behaviour is reproduced, but underestimated, by the Reynolds stress model. The convection velocity of the peak is also underestimated. In order to investigate this major difference between the two models, it is convenient to focus on

132

Figure 6" Contours of phase-averaged velocity V at phase 90 (dashed lines = negative contours).

Figure 7: Contours of phase-averaged velocity V at phase 225 (dashed lines = negative contours).

Figure 8: Contours, vectors and streamlines of long-time-averaged velocity.

133

30

\ ~

i .%. 90deg 135deg 45deg ,.'." ~ ~ 180d ,, : , . , - ' , : eg

,,:

20

i

',-,, :', :./--~, J .-" ' , ,/.. ', .~270deg V'/ ',/ ", ," ,~L-, ,~l , i

~' ,, ~

~," ,^

PIV

___

-

co Im r

,

9~

315deg 0 'o 1

.

.

.

........

P/I

30000

20000 10000

o -...........

-10000 0.005 '

.

40000

~7 *,

.

5OOOO

RSM

'~ ,'', , ;, ~,(, , , , .

.

6OO0O

k'-e

0 -10

,.:

i +

0.015 '

-20000

0.o 2

y (m)

Figure 9: Comparisons of velocity profiles in the symmetry plane,

i

O.OOl

i

0.002

i

0.003

i

0.004

i

0.005

y (m)

i

0.006

i

0.007

i

0.008

i

0.009

0.01

Figure10: Production tensor and Pk" along the centreline (x=O) at phase 60, RSM.

the material derivative of the velocity in the symmetry plane x = O" dV

013

dt =

Oy + ~ OxkOx-----~

02V

0 < v "2 >

Oy

(4)

Acceleration results from a competition between pressure gradient, viscous diffusion and Reynolds stress. The main difference between the two models obviously lies in the Reynolds stress tensor. Indeed, Figs. 10 and 11 show that the Reynolds stress model predicts a region of negative production right above the slot. In the present 2-D case, using the incompressibility condition, the production of k" reduces to:

OV =<

>

v,,2 <

0V > -fly

u"v" <

(OU >

OV) +

(5)

In the symmetry plane, the last term is zero because < u"v" > = 0. Fig. 10 shows that the first term is positive and the second term is negative. This is a direct consequence of the unsteadiness of the boundary conditions. Indeed, since the velocity at the slot exit decreases rapidly after phase 45 deg, the partial derivative OV/Oy at phases > 45 deg is positive between the slot exit (y - 0 mm) and the location of the peak velocity: for instance, it is seen in Fig. 9 that at phase 90 deg, OV/Oy is positive between y = 0 mm and y ~ 4 mm. Therefore, in or close to the symmetry plane, the second term of Eq. (5) is necessarily negative, and leads to a decrease of < v ''2 >, since this term is actually ~P22. 1 On the contrary, the first term, which is 1Pll, is positive. The production of k" is negative because < v"2> is larger than < u"2> in the jet. It is thus clear that in such a flow, the combination of anisotropy in the jet and decrease of the velocity at the slot exit after the maximum necessarily implies negative production. Kinetic energy is thus transfered from turbulence to the oscillating motion through the component < v 'a >. The other components < u "2 > and < w "2 > also lose their energy via redistribution to < v ''2 >. The k"-e" model is unable to predict negative production, which has two direct consequences. Firstly, instead of providing energy to the oscillating motion, turbulence increases rapidly and changes the sign of the momentum budget Eq. (4). The previously quoted increase of the peak velocity between phases 45 deg and 90 deg is then missed" the peak velocity is much too weak, as seen in Fig. 9, which leads to too short a penetration of the jet into the quiescent air. The second consequence is a strong overestimation of the turbulence kinetic energy, as shown in Fig. 12. The RSM also predicts regions of negative production at the periphery of the large eddies (Fig. 1 la). A detailed analysis shows that production in these regions are dominated by the first two terms in Eq. (5)" < u"2> - < v"~> is negative and OV/Oy is positive. Anisotropy is transported here

134

Figure II: Contours of Pk" and velocity vectors at phase=6Odeg.

Figure

12:

Contours of

kn

at phase=6Odeg.

from the region of the symmetry plane where it is produced. The k"-c" model is of course unable to capture such a mechanism. Finally, Fig. 13 shows profiles of long-time averaged velocity V. Surprisingly, despite the strong overestimation of k", the k " - ~ '' model reproduces the spreading of the jet better than the RSM. This is a good example of the beneficial effect of the compensation of errors. 5

CONCLUSIONS

The synthetic jet studied in the present work is a very challenging test case for statistical modelling. The work was dedicated to the investigation of the relative performance of two closure levels: eddyviscosity and Reynolds-stress models. The results show that the global dynamics of the flow is reproduced: the generation of large-scale periodic contra-rotating vortices, and the fact that fluid is mainly blown close to the symmetry plane and sucked from the sides. This is not surprising, since these mechanisms are directly inherited by the phase-averaged equations from Navier-Stokes dynamics. The solutions are free from turbulent eddies (i.e.,represent only the oscillating motion),

135 i

~,

_ y=8mm

.

i~'~,~~_

+

PIV

. ..

RSM

............

__y-6~m__

.

.

y=2mm

.

.

.

.

t -0.01

-0.005

0.005

0.01

Figure 13: Long-time-averaged V-velocity profiles for different values of y.

which are completely accounted for by the models: this shows that the standard models derived for statistically steady flows are suitable for flows treated with phase averaging (at least for the present case). However, the models are far from predicting correctly all the features of the flow. The k"-#' model predicts a much too high turbulence energy, a rapid decrease of the peak velocity, and, consequently, too short a penetration of the jet into quiescent air. The Reynolds-stress model gives a much more realistic solution, but slightly underestimates the peak velocities, the convection velocity and the spreading rate of the jet. The analysis of the turbulence production mechanisms shows that the k"-# ~ model is not suitable for this type of flows (for unsteady flows?): in the region just above the slot, when the inlet velocity decreases, turbulence production is necessarily negative, which is reproduced by the Reynoldsstress model. Using the linear Boussinesq equation yields a strong positive turbulence production, which is sufficient to change the sign of the momentum budget. Moreover, regions of negative turbulence production appear at the periphery of the large-scale eddies, which are due to the transport of anisotropy from the symmetry plane to the eddies. This mechanism suggests that second moment closure is the minimum level of modelling suitable for this flow, since transport of anisotropy cannot be reproduced by nonlinear eddy-viscosity and algebraic models. However, the origin of the deficiencies of the Reynolds-stress model are not clarified yet: they could be linked to the particular form of the Rotta+IP model (redistribution, turbulent diffusion), but they could also suggest that nonequilibrium models, such as two-scale models [9, 26], are necessary. REFERENCES

[1] M. Amitay, D.R. Smith, D.E. Kibens Vand Parekh, and A. Glezer. Modification of the aerodynamics characteristics of an unconventional airfoil using synthetic jet actuators. AIAA Journal, 39:3:361370, 2001. [2] F. Archambeau, N. Mehitoua, and M. Sakiz. Code Saturne: A finite volume code for the computation of turbulent incompressible flows- Industrial applications. Int. J. on Finite Volume, Electronical edition: http ://averoes. math. univ-paris13, ff /html, ISSN 1634(0655), 2004. [3] S. Aubrun, P. L. Kao, H. Ha Minh, and H. Boisson. The semi-deterministic approach as way to study coherent structures. Case of a turbulent flow behind a backward-facing step. In Proc. fourth Int. Syrup. Engng. Turb. Modelling and Measurements, Ajaccio, Corsica, France, pages 491-499, 1999. [4] F. Bastin, P. Lafon, and S. Candel. Computation of jet mixing noise due to coherent structures: the plane jet case. J. Fluid Mech., 335:261-304, 1997. [5] P. Batten, U. Goldberg, and S. Chakravarthy. LNS-An approach towards embedded LES. AIAA paper 2002-0427, AIAA 40th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2002. [6] S. Benhamadouche and D. Laurence. LES, coarse LES, and transient RANS comparisons on the flow across a tube bundle. In W. Rodi and N. Fueyo, editors, Proc. 5th Intl. Syrup. Engng. Turb. Modelling and Measurements, Mallorca, Spain. Elsevier, 2002.

136 [7] S. Benhamadouche and D. Laurence. LES, coarse LES, and transient RANS. Comparisons on the flow across a tube bundle. Int. J. Heat Fluid Flow, 24:470-479, 2003. [8] G. Bosch and W. Rodi. Simulation of vortex shedding past a square cylinder near a wall. Int. J.

Heat Fluid Flow, 17:267-275, 1996. [9] K. Bremhorst, T. J. Craft, and B. E. Launder. Two-time-scale turbulence modelling of a fully-pulsed axisymmetric air jet. In Proc. 3rd Int. Syrnp. Turb. Shear Flows and Phenomena, Sendai, Japan, pages 711-716, 2003. [10] P. A. Durbin. Separated flow computations with the k-e-v: model. AIAA J., 33"659-664, 1995. [11] T. B. Gatski. DNS/LES for NASA aerodynamic needs and engineering applications. In C.L. Liu, L. Sakell, and T. Beutner, editors, DNS/LES-Progress and Challenges, pages 25-34. Greyden Press, Columbus, Ohio, 2001. [12] N. Getin. Simulation nurndrique du contr6le actif par jet pulsds de l'dcoulernent turbulent autour d'un cylindre circulaire. PhD thesis, Ecole Centrale de Lyon, 2000. [13] S. S. Girimaji, R. Srinivasan, and E. Jeong. PANS turbulence model for seamless transition between RANS and LES: Fixed-point analysis and preliminary results. In Proc. 4th ASME_JSME Joint Fluids Engineering Conference, Honolulu, Hawaii, USA, 2003. [14] H. Ha Minh and A. Kourta. Semi-deterministic modelling for flows dominated by strong organized structures. In Proc. 9th Int. Syrnp. Turbulent Shear Flows, Kyoto, Japan, 1993. [15] A.A. Hassan and R.D Janakiram. Effects of zero-mass synthetic jets on the aerodynamics of the NACA-0012 airfoil. AIAA Paper 97-2326, 1997. [16] G. Iaccarino and P. Durbin. Unsteady 3D RANS simulations using the v2-f model. In Ann. Res. Briefs, pages 263-269. Center for Turbulence Research, Stanford University, 2000. [17] G. Jin and M. Braza. Two-equation turbulence model for unsteady separated flows around airfoils. AIAA J., 32(11):2316-2322, 1994. [18] S. Johansson, L. Davidson, and E. Olsson. Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a k-e turbulence model. Intl J. Nurner. Meth. in Fluids, 16(10):859-878, 1993. [19] S. Kenjere~, K. Hanjali6, and G. Krstovi6. Combined effets of terrain orography and thermal stratification on pollutant distribution in a town valley: a T-RANS simulation. In Proc. Second Int. Syrnp. Turb. Shear Flow Phenomena, Stockholm, Sweden, volume 1, pages 103-108, 2001. [20] W. C. Lasher and D. B. Taulbee. On the computation of turbulent backstep flow. Int. J. Heat Fluid Flow, 13:30-40, 1992. [21] B. E. Launder, G. J. Reece, and W. Rodi. Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech., 68(3):537-566, 1975. [22] B. E. Launder and D. B. Spalding. The numerical computation of turbulent flows. Cornp. Meth. Appl. Mech. Engng., 3(2):269-289, 1974. [23] A. Seifert, T. Bachar, D. Koss, M. Shepshelovich, and I. Wygnanski. Oscillatory blowing, a tool to delay boundary-layer separation. AIAA Paper 31-2052, 1993. [24] P.R. Spalart, W.-H. Jou, M. Strelets, and S.R. Allmaras. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In C. Liu and Z. Liu, editors, First AFOSR International Conference on DNS/LES, 4-8 August, Ruston, LA, Advances in DNS/LES, Greyden Press, Columbus, OH, USA, 1997. [25] C. G. Speziale. Turbulence modeling for time-dependent RANS and VLES: a review. AIAA J., 36(2):173, 1998. [26] K. Stawiarski and K. Hanjalic. A two-scale second-moment one-point turbulence closure. In Proc. 5th Int. Syrnp. Engng Turb. Modelling and Measurements, Mallorca, Spain, 2002. [27] S. Tardu and P. Da Costa. Modeling of unsteady turbulent wall flows with and without adverse pressure gradient by a k-omega/rapid distorsion model. In Proc. Second Int. Syrnp. Turb. Shear Flow Phenomena, Stockholm, Sweden, volume 1, pages 205-210, 2001. [28] Workshop on CFD validation of synthetic jets and turbulent separation control. http://cfdval2004.1arc.nasa.gov/. [29] C. Yao, F.J. Chen, D. Neuhart, and J. Harris. Synthetic jet flow field database for CFD validation. AIAA Paper 2004-2218, 2004.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

137

B E H A V I O U R OF T U R B U L E N C E M O D E L S N E A R A TURBULENT/NON-TURBULENT INTERFACE REVISITED P. Ferrey a and B. Aupoix ~ aONERA/DMAE Centre d'I~tudes et de Recherches de Toulouse B.P. 4025, 2, Avenue l~douard Belin 31055 Toulouse CEDEX 4, France e-mail: [email protected]

ABSTRACT The behaviour of turbulence models near a turbulent/non-turbulent interface is investigated. The analysis holds as well for two-equation as for Reynolds stress turbulence models using Daly and Harlow diffusion model. The behaviour near the interface is shown not to be a power law, as usually considered, but a more complex parametric solution. Why previous works seemed to numerically confirm the power law solution is explained. Constraints for turbulence modelling, i.e. for ensuring that models have a good behaviour near a turbulent/non-turbulent interface so that the solution is not sensitive to small turbulence levels imposed in the irrotational flow, are drawn.

KEYWORDS Eddy viscosity models, Model constraints, Reynolds stress models, Turbulence Modelling

1

INTRODUCTION

A well-known drawback of some turbulence models such as the Wilcox (1988) k - w model, the Smith (1995) k - L model or the Baldwin and Barth (1991) one-equation model is that the solution is sensitive to the small level of transported turbulent quantities imposed outside of the turbulent regions. Indeed, for low turbulence levels outside of the boundary or free shear layers, turbulence should propagate from the turbulent regions towards the non-turbulent regions, a condition which is violated by these models. The solution thus depends upon the small level of eddy viscosity outside of the turbulent region, i.e. upon the external turbulence length scale. Such an unphysical behaviour cannot be accepted as the numerical solutions are thus unreliable. The real physical behaviour near a turbulent/non-turbulent interface is rather complex, with an

138

interface which is highly corrugated and induces fluctuations in the irrotational, non-turbulent field. Describing it is out of the capabilities of standard turbulence models which all more or less rely upon turbulence equilibrium assumptions. Therefore, all what is required is that the turbulence model correctly propagates information from the turbulent towards the non-turbulent region to avoid undesirable free-stream sensitivity.

2

STANDARD

ANALYSIS

For the sake of simplicity, only incompressible flows will be addressed. The behaviour near a turbulent/non-turbulent interface was first addressed by Saffman (1970) and deeper investigated by Cazalbou et al. (1994) for eddy-viscosity models. The analysis has been extended to Reynolds stress models by Cazalbou and Chassaing (2002). They showed that, near the interface, the problem should reduce to a convection/diffusion equilibrium and that source terms in the turbulence transport equation should be negligible. The problem is investigated in a reference frame linked to the interface. The ordinate y is along the normal to the average interface and the relevant length is )~ = Y e - y, i.e. the distance to the turbulent/non-turbulent interface. A defect form for the velocity profile is considered as W = u e - u, ue being the velocity in the non-turbulent region. In the turbulent region, the solution of the simplified transport equations for mean defect velocity W, turbulent stresses u i~u j~, turbulent kinetic energy k and e.g. turbulent dissipation rate c is sought for as power laws as: W = Au~ ~

u~u~ = Auu)~ ~k

k = Ak)~ ~k

c = A~)~ ~

The A are undetermined scaling factors and the exponents c~ should be positive for the turbulence to propagate towards the non-turbulent region and larger than unity to achieve a smooth matching with the external flow. These power-law behaviours as well as the constraints on the coefficients were validated by numerical computations. Moreover, for Reynolds stress models, Flachard (2000) and Cazalbou and Chassaing (2002) pointed out the importance of the diffusion model in the behaviour near the interface. For example, Hanjalid and Launder (1972) diffusion model induces a strong and unrealistic turbulence anisotropy near the interface while a good behaviour is obtained with Daly and Harlow (1970) model.

3 3.1

PRESENT Turbulence

ANALYSIS models

However, it turned out that the above analysis is too simple and has to be revisited. The present analysis holds as well for two-equation models, whatever the constitutive relation (eddy viscosity, non-linear eddy viscosity or explicit algebraic Reynolds stress model) as for full Reynolds stress models. For that, generic models, following Catris and Aupoix (2000), are introduced. Two-equation models solve two transport equations, generally for the turbulent kinetic energy k and an arbitrary length scale determining variable (I) = k m c n. Following Catris and Aupoix (2000), a generic model reads:

139

Dk O( Ok) D---t= Pk - e + ~ Dkkut-~z k

(1) (2)

+C,~r

O~ 0,~ ut Ok O~ ut'~ Ok Ok (~ Oxk Oxk + Cck k Oxk Oxk + C~k k 2 0 x k Oxk

This is a generic form as the transport equation for any combination of k and (I), deduced from the above equations, has the same form as equation (2), its constants being linked to those of equations (1) and (2). Compared to Catris and Aupoix (2000), a diffusion term has been dropped in the turbulent kinetic energy transport equation for the sake of simplicity. These transport equations have to be coupled with a constitutive relation, of the eddy viscosity form, either linear or non-linear. Similarly, for Reynolds stress models, the generic transport equations read:

DuiUJDt = Pij + IIij - cij + ~

Dkk~uku z Oxi

(3)

(4) O~ . .-7~_., 0 ~

+Cee 5 u2

C

1 Ok _.-v-~.., O,~

,~ 9 Ok ~

Ok

-5-~x~'~ + ~ - i - ~ x ~ ' ~ + c;~-5-~x~ Ox~

where Piy, IIij and eij respectively stand for the Reynolds stress production, redistribution and destruction terms the form of which is not of concern here. Following Cazalbou and Chassaing (2002) conclusions, the analysis is restricted to Daly and Harlow (1970) diffusion model.

3.2

E q u a t i o n s in the vicinity of the turbulent region edge

A complete derivation of the equations can be found in Ferrey (2004). Either a time evolving flow, as in Cazalbou et al. (1994) or a spatially evolving flow, as here, can be considered. For the analysis, it is more convenient to use, as length scale determining variable, the eddy viscosity ut or v'gA for Reynolds stress models. Considering a two-equation model, a two-dimensional steady E flow, neglecting viscosity and introducing g = - Dkk-~, Vo where V0 is the velocity component normal to the interface, the equation system for the momentum and transport equations reduces to:

dW d(dW) Dkk dA = d-"~ g ~

(5)

dk d(dk) d---~= d---~ g--~

D~~ = d(

(6)

dg

g~dk)

( dg) ~

g dg dk

(~ dk)

(7)

In the above equations, the problem has been assumed to reduce to an advection/diffusion equilibrium, so that source terms have been dropped in the turbulence transport equation. This has however to be checked and leads to other constraints for the model (see, e.g. Cazalbou et al. (1994)), not to be discussed here.

140

The boundary conditions at the interface are: lim W = 0 lim k = 0 limg = 0 A--~0

A--~0

(8)

A---,0

Integrating equation (6) with the above boundary conditions yields: Dkk g=--~o ut=

k dk

(9)

dA

Introducing the above relation into equation (7) leads to the following relation" ag-~+b

+c~-~ + d =

-~

(~o)

0

where the coefficients a, b, c and d are linked to the diffusions coefficients C and D as: a :

b -- --(Dut~,t + Cvtpt)

-Dutut

C :

--(Cutk

+ Dutk -- D k k )

d = -Cs

(11)

As model coefficients are interrelated when the length scale determining variable is changed, these coefficients can be expressed, referring to a k - e model form, as: a = - D~

b = D~ + C~

c = - 4 ( D ~ + C ~ ) + Dkk -- D~k -- C~k

(12)

d = 4 (D~ + C~) + 2(D~k + C~k - Dkk) + C~k The same analysis holds for a Reynolds stress model, using Daly and Harlow diffusion. This is due to the fact, pointed out by Cazalbou and Chassaing (2002) that, using this diffusion model, the anisotropy levels are constants in the turbulent region below the turbulent/non-turbulent interface. Therefore, the same expressions for the a, b, c and d coefficients are retrieved. The analysis is also similar for non-linear eddy viscosity models or explicit algebraic Reynolds stress models since they predict an isotropic state near the interface where the velocity gradient tend towards zero and reduce to a modified eddy viscosity model for the turbulent shear stress --U/V/.

It must be pointed out that, to prevent counter-gradient diffusion, a has to be negative. Moreover, writing the model balance in the logarithmic region, it is easily checked that b is positive. 3.3

Solutions

near the turbulent

r e g i o n edge

Equation (10) has two obvious solutions such that ~dA2 = 0, which correspond to a linear profile for the eddy viscosity. From relation (9), these solutions correspond to power law solutions for the turbulent kinetic energy profile. They read: A

1

u

m

(13)

ozk

~k

ct k+ -

A

g=~

c+x/'--A ~

1

2b

ak

c-

x/~

A = c 2 - 4bd

2b

The power law solution found by Cazalbou et al. (1994) is retrieved together with a second power law solution which has already been pointed out by Catris (1999) and Catris and Aupoix (2000). As the transformed eddy viscosity g is linear, it must be pointed out that the length scale deter-

141

mining variable and turbulent kinetic energy evolutions are linked, as pointed out by Daris (2002) and Aupoix et al. (2003), as: 2a~ m OL~+ --" 1 (14) The new and important point is that equation (10) has a third solution, which can only be expressed in parametric form, the parameter being rn = ~ , as"

I

l

fl+ I

a(rn)=clm--G_ with

1

il-

{ __11~-( _ ~__ff)l (

g ( m ) = Go m - -2-4

F ( x ) = Hypergeom

m - --

F

fl+ =

a

a

/3- =

v/-~a k

(15) (16)

rn

+

O~k Ot k X

/3-, 1 - fl+, 1 +/3-; a~- - a k

)

where Gauss' hypergeometric function is defined as: +~C(a+k) C(b+k) V(c) x k Hypergeom(a, b, c; z) = y~ r (a) r (b) r (c + k) k--~

(17)

k=0

i 11

Similarly, the turbulent kinetic energy profile can be expressed as: a

k(~) = Ko

(18)

1k m

a~-

The velocity profile can be linked to the turbulent kinetic energy profile. For eddy viscosity models, equations (5) and (6) can be integrated, using boundary conditions (8), as: dW D k k W = g ---d-A

dk k = g --d-f

(19)

so that, eliminating g, W = k~

(20)

For Reynolds stress models, the momentum equation reduces to: dW Vo---dA

du'v'

(21)

dA

so that, as the anisotropy levels tend towards constants near the interface:

(22)

W oc u'v' cx k 3.4

Behavior

of the parametric

solution

This parametric solution is the only relevant solution as a small perturbation of the linear solutions of equation (10) will lead to ~d)~ 2 7~ 0 and thus to the parametric solution. The boundary conditions at the interface (8) and the above solutions for k (18) or A (16) show that the interface corresponds to m = --1_. As, near the interface, V0 is negative and therefore g o~k

is positive and tends towards zero at the interface, m = ~ must be positive so that a~- must be positive.

142

14 1

3"

Vt

o,..:l.i. . . . . . . . . . . . . . . . . . o

1

z

3

4

5

6

,,

7

oz

o4

o6

Decreasing rn

oo

1

lz

14

1.5

1B

Z

Increasing rn

Squares: a k- solution

-

Dashed-dotted line: a k+solution

Figure 1" Solutions for the eddy viscosity profile near the interface when both exponents are positive J 0 li] V t

m [] [] [] o m

vt 0Z 0

t4

--'''Z

~

....

4 ....

; ....

; ....

1'0 . . . .

1'2 . . . .

I~I""

i

-0

Increasing rn Decreasing rn Squares: c~[ solution

Dashed-dotted line: c~k+solution

Figure 2: Solutions for the eddy viscosity profile near the interface when only one exponent is positive Two cases must be considered, whether c~k+ is positive or negative, its sign being that of d. If + c~k+ is positive, it is obvious from its definition (13) t h a t c~k > c~[. Two behaviours are possible according to the sign of dd-"~m _- - ~dA 2" If it is negative, rn decreases from 1a k to ~--ff' 1 for which the transformed eddy viscosity g is infinite. As m = ~ , the g evolution asymptotes the two power law solutions (13) for )~ ~ 0 and )~ ~ co. This is the situation depicted in the left part of figure 1. If d___~ is positive, g blows out rapidly and only has an a s y m p t o t e for A ~ 0. dA Similar behaviours are observed if c~k+ is negative. If rn decreases the g profile asymptotes both power law solutions, so t h a t g rapidly becomes negative while it blows out if rn increases, as shown in figure 2. Unfortunately, no way to determine the sign of -aT dm has been found. 3.5

Numerical

solutions

The occurrence of this parametric solution have been checked using a code solving the selfsimilarity equations for various simple flows such as the outer region of the b o u n d a r y layer, the

143

wake, the mixing layer and the plane or round jets (B~zard, 2000). As self-similarity reduces the equation set to a one-dimensional problem, which is solved using a time marching technique, grid convergence is easily achieved. An example of result is provided in figure 3 where the "eddy vis-

012 011

~

--

01 0.09 0 08 007 >'006 0 05 0.04 003

-

0 02

~ ..

/

001 0 0

005

011

0.~5

n

Figure 3: Solution of the "eddy viscosity" profile in the vicinity of the interface when the exponents c~ are positive m

v12 k cosity" --7is plotted for the outer region of the boundary layer, using a Reynolds stress model for which both c~k are positive. The lower figure shows the eddy viscosity profile in self similar coordinates ( uezx ~t versus r/ = ~, where A is here the Clauser's thickness) while the upper figure shows an enlargement of the solution near the interface. The two linear solutions for the eddy viscosity are also plotted. It can be checked that the parametric solution is retrieved and that it asymptotes both linear laws, as in figure 1 (left). As the c~- solution is reached only in the very vicinity of the interface, it explains why previous works concluded that the c~k+ solution was obtained.

As in Cazalbou et al. (1994) analysis, fluid viscosity has been neglected. When it is accounted for, or when a small level of turbulence is present in the irrotational flow, the c~- solution is difficult to observe as it is superseded by the viscous effects or the free-stream condition. Moreover, the c~- solution generally extends over a very restricted (if not null) number of cells with usual grids. For models for which the cross diffusion coefficient C~k is null, which involves most of the classical models, Cazalbou's solution corresponds to the c~+ solution. The other power law solution, which gives the tangent for ~ = 0, corresponds to a ; = 0 and c~- = ~ n (e.g. c~- = 1 for standard k - e models, c~- = 1 for k - w models). Therefore, the length scale determining variable (I) = kme n smoothly tends towards zero.

4

CONSEQUENCES FOR TURBULENCE MODELS

The first important point is that not only c~k+ has to be considered, as previously done by Cazalbou et al. (1994) but also c~- and that c~- also must be positive. This explains why Cazalbou et al. (1994) were unable to apply their analysis to the Ng and Spalding model: among the models they considered, it was the only one for which c~- is negative.

144

0.14 0.13

012

0.12

011

0.11

01

0.1

0 09

0.09

008

0.08

ooOO,

007

!!!

;>" 0.07

>006 0 05

0.04

0

0.03

0 04 0 03 0 02

0.02

001

0.01 ....

i

....

0.2

i

04

.....

- i - , - , - ' - , -

0 6

i-

0.8

'-

r'"

o

'-

.

.

.

.

.

o~,5

.

.

.

.

.

o',

.

.

.

o',5

n

n Figure 4" Sensitivity to free-stream values of the eddy viscosity of a model such that all a are positive

Figure 5: Sensitivity to free-stream values of the eddy viscosity of a model such that all a are not positive

Cazalbou et al. (1994) recommended that a k+ be positive to have the correct information propagation. Among the four possible behaviours depicted in figures 1 and 2, it seems that the case where a + is positive and m is decreasing, where the g evolution asymptotes the a k+ power law, is the only acceptable case. In all other cases, g blows out and a matching with the g profile in the turbulent region, where production and destruction terms are no longer negligible, seems more problematic. Indeed, this is this behaviour that is retrieved in all the numerical simulations. Therefore, it seems that the constraint should be that both ak must be positive, i.e." 1

1

--=_ > 0

~---;>0

Ol k

Ol k

(23)

where inverses are used to discard infinite values for the ak, as suggested by Cazalbou (private communication). Numerical checks tend to support the above conclusion. As an example, "eddy viscosity" \(v'2k) E profiles are plotted for the outer region of a boundary layer, using Reynolds stress models. Different solutions are obtained, always imposing a very small turbulence level outside of the boundary layer but varying the "eddy viscosity" level. In figure 4, both a k are positive and the solution in the boundary layer is insensitive to the imposed "eddy viscosity" level outside. Turbulence propagates from the turbulent region towards the non-turbulent region so that the eddy viscosity first falls to a very small value near the interface before rising again to the value imposed in the external flow. In figure 5, a k+ is negative and the solution is deeply affected by the external "eddy viscosity" level. Turbulence propagates from the external flow into the turbulent region, which is not wanted. It should be pointed out that the above analysis can no longer be strictly applied since the "eddy viscosity" thus no longer tends towards zero at the boundary layer edge. Moreover, for each transported quantity (turbulent kinetic energy, Reynolds stress tensor component, length scale determining variable) the exponent a should be such that source terms are negligible compared to the advection and diffusion terms. This leads to very different constraints according to the constitutive relation. Provided the parametric solution is bounded by the two power law solutions, the analysis is thus similar to the one proposed by Cazalbou et al. (1994).

145

5

s

0)

kx

1

-1

-1

0

1

2

3

I-- ' ' 4

I 5

Of,k

Figure 6: Behaviours of the exponents c~ for several length-scale determining variables

A last point to be mentioned is that the generic character is in some sense lost near the interface. In other words, the choice of the length scale determining variable must be done carefully since, rewriting a model to change the length scale determining variable may affect its behaviour. Indeed, from equation (14), the exponents of the turbulent kinetic energy and the length scale determining variable are interrelated. This link between the exponents c~ is shown in figure 6 for some popular k L e(-~) If only a positive value of length scale determining variables (a~ o( g, ~ o( ~~, T (X ~, the exponents c~k and c~r for the length scale determining variable is looked for, which ensures a correct information propagation, any length scale determining variable can be used, with caution. If at least c~k+ and c~+ are imposed to be higher than unity, to ensure a smooth matching with the _ k 3/2 small external level, ~- c< k or l c< ~ E are forbidden.

5

CONCLUSIONS

The above analysis shows that the behaviour of a turbulence model near a turbulent/non-turbulent interface is more complex than previously considered and that the real solution is not given by power laws but is a parametric solution, which generally seems to asymptote the power laws. This explains why the numerical solutions were confused with the power law solutions. The above analysis holds as well for two-equation models, whatever the constitutive relation, as for Reynolds stress models. It leads to a more complex constraint, imposing that both c~k coefficients must be positive. The analysis still requires to be extended, on the one hand to be able to determine which solution, i.e. increasing or decreasing m is obtained and why and, on the other hand, to account for viscosity and small external turbulence levels.

146

5.1

Acknowledgments

The authors wish to thank DGA (French Ministry of Defence) which granted P. Ferrey's thesis. Part of this work took place within the FLOMANIA project (Flow Physics Modelling- An Integrated Approach) which is a collaboration between Alenia, Ansys-CFX, Bombardier, Dassault, EADS-CASA, EADS-Military Aircraft, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and St. Petersburg State University. This project is funded by the European Union and administrated by the CEC, Research Directorate-general, Growth Programme, under Contract No. G4RD-CT2001-00613.

REFERENCES

B. Aupoix, H. B~zard, S. Catris, and T. Daris. Towards a calibration of the length-scale equation. In J. P~riaux, M. Champion, J.J. Gagnepain, O. Pironneau, B. StouiIlet, and P. Thomas, editors, Fluid Dynamics and Aeronautics New Challenges, pages 327-349. CIMNE, September 2003. B.S. Baldwin and T.J. Barth. A one-equation turbulence transport model for high Reynolds number wall-bounded flows. AIAA Paper 91-0610 29th Aerospace Science Meeting, Reno, Nevada, January 7-10 1991. H. B~zard. Optimisation of two-equation turbulence models. In C. Dopazo, editor, Advances in Turbulence - Proceedings of the Eighth European Turbulence Conference, page 978, Barcelona, June 27-30 2000. S. Catris. Etudes de Contraintes et Qualification de Moddes it Viscosit~ Turbulente. PhD thesis, SUPAERO, Toulouse, 14 Octobre 1999. S. Catris and B. Aupoix. Towards a calibration of the length-scale equation. International Journal of Heat and Fluid Flow, 21(5):606-613, October 2000. J.B. Cazalbou and P. Chassaing. The structure of the solution obtained with Reynolds-stresstransport models at the free-stream edges of turbulent flows. Physics of Fluids, 14(2):597-611, February 2002. J.B. Cazalbou, P.R. Spalart, and P. Bradshaw. On the behavior of two-equation models at the edge of a turbulent region. Physics of Fluids A, 6(5):1797-1804, May 1994. J.B. Daly and F.H. Harlow. Transport equations in turbulence. The Physics of Fluids, 13(11): 2634-2649, November 1970. T. Daris. Etude dc ModUles de Turbulence it Quatre Equations de Transport pour la Pr~vision des Ecoulements Turbulents Faiblement Chauffds. PhD thesis, SUPAERO, Toulouse, France, 12 D~cembre 2002. P. Ferrey. Moddles aux Tensions de Reynolds avec Prise en Compte de l'Intermittence de Fronti~re. PhD thesis, l~cole Nationale Sup~rieure de M~canique A~rothechnique, Poitiers, France, 6 D~cembre 2004. L. Flachard. Etude de Contraintes et Qualification de ModUles aux Tensions de Reynolds. PhD thesis, SUPAERO, Toulouse, 8 Novembre 2000. K. Hanjali5 and B.E. Launder. A Reynolds stress model of turbulence and its application to thin shear flows. Journal of Fluids Mechanics, 52(4):109-638, 1972. P.G. Saffman. A model for inhomogeneous turbulent flows. Proceedings of the Royal Society of London, A 317:417-433, 1970. B.R. Smith. Prediction of hypersonic shock wave turbulent boundary layer interactions with the k - 1 two equation turbulence model. AIAA Paper 95-0232 33Td Aerospace Sciences Meeting & Exhibit, Reno, Nevada, January 9-12 1995. D.C. Wilcox. Reassessment of the scale-determining equation for advanced turbulence models. AIAA Journal, 26(11):1299-1310, November 1988.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

147

BEHAVIOUR OF NONLINEAR TWO-EQUATION TURBULENCE MODELS AT THE FREE-STREAM EDGES OF TURBULENT FLOWS A. Hellsten 1 and H. B6zard 2 1 Helsinki University of Technology, Laboratory of Aerodynamics, E O.Box 4400 FIN-02015 HUT, Finland 20NERA - Department of Modelling for Aerodynamics and Energetics B.E 4025, F-31055 Toulouse - CEDEX 4, France

ABSTRACT The behaviour of two-equation turbulence models at the outer edges of turbulent flows is studied in this paper. The focus is on nonlinear models with simple scalar-diffusivity gradient-diffusion model for turbulent transport. Conditions to obtain physically correct smooth solutions at the edge are presented and discussed. A constraint is derived for the diffusion model coefficient for turbulent kinetic energy to guarantee smooth solutions also when nonlinear constitutive models are employed. Finally, the analysis is extended to a case of a generalized gradient-diffusion model.

KEYWORDS Turbulence modelling, Nonlinear two-equation models, Free-stream edge, Turbulent/nonturbulent interface, Model constraints

INTRODUCTION Practical engineering applications of the computational fluid dynamics (CFD) are still largely based on the numerical solution of the Reynolds averaged Navier-Stokes equations (RANS). Turbulent flow problems of engineering interest often feature high Reynolds numbers and are thus difficult if not impossible to solve by means of large eddy simulation, not to mention the direct numerical simulation. Reynolds stress transport modelling is, in principle, the most sophisticated level of RANS modelling available today. In practical work, however, Reynolds stress transport closures are rather seldom used as principal tools. Two-equation eddy-viscosity modelling is still by far the most popular level of RANS

148

turbulence modelling. Two-equation models can be considered to consist of a scale-determining model and of a constitutive model. The former provides two scalar variables, e.g. k and c, solved from their modelled transport equations. The constitutive model provides the Reynolds stress tensor as a function of the turbulent scales and the mean-velocity gradient. The constitutive models range from the simple linear Boussinesq-relation up to high-order tensor-polynomial expressions which can be derived from some underlying stress-transport closure by invoking the weak-equilibrium assumption (explicit algebraic Reynolds stress models, EARSM), or it may be derived on an a d h o c basis (nonlinear eddyviscosity models). Currently, a move from standard linear eddy-viscosity modelling towards nonlinear two-equation models, especially EARSMs, seems to be going on, at least in some areas of fluids-related engineering. This paper deals with the behaviour of scale-determining models near the free-stream edges of turbulent flows especially in connection with nonlinear constitutive models. The turbulent/nonturbulent interface, or in other words the edge between vortical and irrotational fluctuations on the outer edge of turbulent flows is known to be very sharp in reality. Its thickness is determined by the viscosity. The instantaneous shape of this edge is highly irregular and evolves in time along the motion and evolution of the large turbulent structures. This motion induces irrotational potential fluctuations outside the edge (Bisset, Hunt & Rogers 2002). The fact that the real turbulent/nonturbulent interface is very thin may have led to some misinterpretations in the context of RANS-modelling. The Reynolds averaged velocity field shows no sharp edges. Instead, the mean velocity approaches smoothly its free-stream value with no sudden changes in its gradient. This is primarily because the location and position of the actual sharp edge varies in time (which also means that turbulence is observed as intermittent in a fixed-poinI measurement). Moreover, the potential fluctuations outside the actual interface also influence the Reynolds averaged velocity. Therefore, the Reynolds averaged velocity field features a smooth profile with continuous and bounded gradient around the edge. Naturally, RANS turbulence models should be designed keeping this in mind. In contrast to this, models have been proposed that provide sharp outer edges with the velocity gradient tending towards a discontinuity at the edge. Not much attention has been paid to this as the numerical dissipation tends to smear out the edge in practical computations. Some of those scale-determining models that show proper smooth outer-edge behaviour with the linear constitutive model, may predict sharp or even a highly unphysical "hook-shaped" velocity profiles when combined with a nonlinear constitutive model. The theoretical analysis is based on the work of Cazalbou, Spalart & Bradshaw (1994). They found that the model behaviour in the outer-edge region can be analysed using a simplified one-dimensional nonlinear front-propagation problem. This is arrived at by assuming that the turbulent transport governs the time evolution of fronts, while the production and destruction terms are negligible on the edge regions. Cazalbou et al. (1994) mainly analysed the k - c formulation, but they discussed also some k - k L and k - co models. In this study, we reformulated the problem as a steady convection-diffusion problem similarly as Catris & Aupoix (2000). This form can be obtained from the problem of Cazalbou et al. through a simple Galilean transformation. Kok (2000) applied Cazalbou's analysis to design a new k - co model with a cross-diffusion term, i.e. the inner product of the gradients of k and co, added to the co-equation. Cazalbou et al. (1994), Catris & Aupoix (2000) and Kok (2000) studied only linear two-equation modelling. Recently, Cazalbou & Chassaing (2002) have extended this analysis to stresstransport models. To the authors' knowledge, however, this kind of analysis has not been applied to nonlinear two-equation models by other researchers. The present authors have so far presented some special cases. Bdzard & Daris (2005) found, using the original analysis by Cazalbou et al. (1994), that in cases of nonlinear constitutive model, a much stricter constraint must be applied to the diffusion coefficient of turbulent kinetic energy k than in the case of linear constitutive model. This is to ensure that the ratio of production and dissipation of turbulent kinetic energy k remains bounded. Hellsten (2004b) extended the analysis for k - co models with a special case of variable C,. These new theoretical results were then exploited in the development of a new k - co model employing the EARSM developed

149

by Wallin & Johansson (2000) as its constitutive model (Hellsten 2004a, Hellsten 2004b), and in a new k - k L model employing the same constitutive model (B6zard & Daris 2005). In this paper, the analysis is generalized to arbitrary second scale variable r ~ kme n, and applied to a special case of variable C.-coefficient assuming that also C. follows a power function. This is because variable C. is a central feature of nonlinear constitutive models, and essentially the only difference between linear and nonlinear models that has to be taken into account in this level of approximation. The main part of the analysis assumes simple scalar-diffusivity (eddy-viscosity) gradient-diffusion model, but the analysis is also extended to the popular generalized gradient-diffusion model by Daly & Harlow (1970). A constraint is also formulated for the Daly-Harlow model coefficient. The solutions of the idealized edge problem are of a power form, and for certain values of the model coefficients, the solutions are not sufficiently differentiable on the edge. Cazalbou et al. (1994) proved that in such cases the equations have a weak solution. As Cazalbou and Chassaing stated, this (possibly weak) solution cannot be considered as unique, so that its validity must be checked by comparing with numerical solutions. Such comparisons have been made against numerical solutions of self-similar problems as well as those of the full Navier-Stokes equations. In fact, Ferrey and Aupoix found very recently that there, indeed, are two power solutions and also a third solution that generally asymptotes the power solutions (B. Aupoix, private communication 2004). The power solution of the present and the previous work has been shown to adequately follow the numerical solutions, and thus it is believed to reflect the model behaviour in real problems. However, as pointed out by Aupoix, more work must be done in order to really understand the role of the other solutions and their implications to the modelling.

THE PROBLEM FORMULATION

A generalformulation of two-equation models At this point, we restrict our attention to two-equation models employing gradient-diffusion models with scalar-diffusivities for the turbulent transport terms. This, for instance for turbulent kinetic energy k, is k 2 Ok Ok TJ k) ~ a k C . T Oz----j = gkvT 8zj

(I)

where T~k) is the turbulent flux vector of k,/iT is the eddy viscosity, and crk is a model coefficient. In the end of this section, we will extend the study to the case of the generalized gradient-diffusion model of Daly & Harlow (1970) employing tensor-valued diffusivity. The scale-determining part of arbitrary two-equation turbulence model employing the above given simple gradient-diffusion model for turbulent transport terms can be written in the following quite general form (Catris & Aupoix 2000) Dk

= P - e+

Dr

_

m

r (Cr

(2)

(u + okuT)

- Cr

+

0[

(u + o-eL,r)

0r

+ (u + cr,k~'r)

r

(3)

ok 0r r ok ok ~ 00 00 nt-OdlliT k OXj OXj --t-Crd2L/Tk20Xj OXj ~ Crd3l/Tr OXj OXj where r ~ kme ~ is an arbitrary second scale variable. In addition to (2) and (3), some constitutive model is needed for the Reynolds stress tensor and for the eddy viscosity. Most of the actual models

150

available for practical applications are much simpler than (2) and (3) not including all the inner products of the gradients in the C-equation. Also, the secondary diffusive term in the C-equation depending on the gradient of k is almost always zero. One motivation behind this general formulation is that it allows, in principle, arbitrary transformations between different choices of the definition of r i.e. the parameters m and n, without adding any further terms. This means that transformation from one second scale variable to another reduces to a change of the values of the model coefficients as noted by Catris & Aupoix (2000). See also Hellsten (2004b) for more details. It turned out, however, that retaining this full form of the generalized model complicated the analysis quite much. This problem was avoided easily by dropping the second gradient-product term, i.e. the term depending on the coefficient era2, from (3). As a matter of fact, also the secondary diffusive term could have been omitted, and still the resulting analysis would have covered most, if not all, of the interesting models. We kept that term, however, because it did not complicate our analysis considerably. We wanted to keep the other cross-gradient terms because several interesting k - co models include the oral-term, and the new k - k L model developed by B6zard & Daris (2005) utilizes also the Crd3-term.

The idealized edge problem Cazalbou et al. (1994) proposed that behaviour of turbulence models near outer edges of shear layers can be understood by studying a simplified one-dimensional problem involving only ordinary differential equations. For general two-equation models of the form of (2) and (3) (but without the cra2-term), the corresponding equations can be written as dU

d (d~yU)

Vd---y

=

dy

vdk d-y

=

dy

V~~

= dy ~162

(4)

uT

d ( d ~ )

(5)

crk uT

-~-Or163

-~-Odl

ur dk de uT (d_~) 2 k dy dy t- ~d3-~-

(6)

Note that the source and sink terms have been omitted as negligibly small in the edge region. The convective velocity V must be assumed constant around the edge. The Reynolds number is assumed high, so that viscosity can be omitted. In this problem, all the variables U, k, and r are defined as positive quantities that go to zero on the edge. Thus U can be considered either as a velocity defect of a wake-like flow or a velocity excess of a jet-like flow. Note that Cazalbou et al. (1994) and Kok (2000) formulated the problem in a moving coordinate frame to make V zero. In such a frame, the problem takes the form of an unsteady nonlinear diffusion problem. This problem may have a solution of a power form U(y)

=

Uof a

(7)

k(y)

=

kof b

(8)

r

=

r

(9)

c

where f ( y ) = max \

60

;0

(10)

and U0, k0, r and ~50are the characteristic scales of the problem. The exponents a, b, and c are functions of the a-coefficients and the parameters m and n. It is assumed that U, k, and r are zero outside the edge. Cazalbou et al. (1994) assumed that UT = C , k2,/c with constant C,. The power solution is possible

151

provided that UT ~ f , which in the generalized formulation becomes uT = C # ( ~ 2 n + m / r 1In c,,a f. Cazalbou et al. (1994) presented the necessary conditions for such solution to exist. The power functions (7) - (9) are not necessarily differentiable on the edge point (y = 50) with all values of the model coefficients. Cazalbou et al. (1994) showed that there is a corresponding weak solution in such cases. Note that all existing models do not obey this power solution as the exponents may become negative or unbounded with some calibrations.

ANALYSIS AND DISCUSSION

A solution for particular case of variable Cu As stated above, the power solution is possible only if UT = C # ( k 2 n + m / r 1In "': f . Cazalbou et al. (1994) ensured this by requiting that 2b - c = 1 in the case of k - e models, and that C# must be constant. We can also allow variable C , if we assume that it follows a power function, say C , ~ C , of ~ with/3 < 1. In such cases, we must require that

(]s

1In fl-Z

:::>

m

c

n

n

(2 + - - ) b

= 1 -/3

(11)

Inserting the trials (7) - (9) into the system (4) - (6) and using the relation (11), we can solve the unknown exponents, a, b, and c and the convective velocity V. The solution reads

a -b -c

=

V

:

n(1 - / 3 ) ( G r + trek + Oda)Crk D n(1 -/3)(0-r + or + ~d3) D n(1 - / 3 ) ( c r k - Gdl ) D n(1 - / 3 ) ( a r + trek + Cra3)CrkC.o D

(12) (13) (14)

'~o

(15)

r

with V -- ( 2 n + 7Tt) (O'q~ + O'r

--[- Od3 ) -- O"k + O-dl

(16)

We must require that all the exponents are positive and that D -r 0. In many cases cr~ + trek + Od3 > 0 and crk - Odl > 0 while both n and D must have the same sign. In a majority of existing models both D and n are positive, but in some models they are both negative. Bdzard's k - kL model deviates from the main stream in that it has n / D < 0 but also ar + aek + Od3 < 0 and ok - ad~ < 0 rendering the exponents positive.

Discussion Behaviour of wide variety of different models around the outer edges of turbulent flows can be studied employing the solution (12) - (16). However, ,2 is still unknown and we must consider also the behaviour of constitutive models in this problem before proceeding. The simplest constitutive model is the linear Boussinesq model with constant C,. This equals to setting fl = 0 in the above solution. In the nonlinear constitutive models as well as in the SST-model by Menter (1994), C , is typically approximately inversely proportional to the nondimensional strain parameter S at large values of S. At smaller values of S, these models give a constant or nearly a constant distribution. Fig. 1 shows this for two constitutive models, the E A R S M by Wallin & Johansson (2000) and the SST by Menter (1994). In this

152

:zk f,..)

0.12 [ 0 . 1 |0.08 0.06 0.04 0.02 0 0

I

,

' ~

2

4

IEARSM: ] SST .............

6

8

10

Figure 1" C , in simple parallel flow as a function of 5' according to the E A R S M and the SST models. simplified problem, 5' is reduced to 7- dU/dy, where 7- = k/c is the turbulent time scale. Thus, the solution (12) - (16) with 0 < / 3 < 1 can be thought to asymptote the real E A R S M - or SST-model solution when 5' -+ ec. In fact, it holds quite well when 5' is larger than, say 3 or 3.5. On the other hand, the solution with/3 = 0 can be though to asymptote the real solution as 5' ~ 0. It is immediately seen from (12) - (14) that if C , decreases towards the edge as with 0 < /3 < 1, the edges in the solutions become sharper. This may lead to a situation where the velocity exponent a, originally larger than one with constant C , , now becomes less than one. The edge will be perfectly sharp with dU/dy being indefinite always when a _< 1 (a weak solution). This is an unphysical situation and may also cause numerical troubles. In practice, the numerical dissipation and the molecular viscosity may alleviate the situation to some extent, but also the numerical solutions will be qualitatively wrong at least when a < 1. Now, knowing that C , distributions decreasing towards the edge may spoil the solution even qualitatively, this kind of situation must be avoided. Fig. 1 shows that increasingly large values of S lead to decreasing C , while small values of S should keep C , roughly constant. We can now require S to approach zero towards the edge. As we are requiting S --+ 0, we may assume that = 0 as explained above. To determine the exponent of S, we assume that the dissipation-rate solution at the edge follows the same power form as before, say c ~ e0f d. From the momentum equation (4), it can be shown that the leading order terms (lowest powers of f ) are a - 1 for the convection part and 2b - d + a - 2 for the diffusion part because we assumed constant C , , i.e./3 - 0. Equaling these powers leads to d = 2b - 1. The exponent of S then becomes a - 1 + b - d = a - b, and requiting S to go to zero at the edge means that its exponent has to be positive. Using the expressions of a and b given in (12) and (13) we obtain a constraint for crk

a-b>O

~

crk> 1

(17)

An alternative route to the same conclusion is to study the behaviour of the production to dissipation ratio Pie. In E A R S M constitutive models, Pie is a solution of an algebraic equation for the Reynolds stress tensor. In the E A R S M expression of Wallin & Johansson (2000), this ratio is obtained as a solution of a third-degree equation, as a function of the velocity gradient invariants and the turbulent scales. As a consequence, the effective C~ can also be taken as a function of P/c. The shape of this curve is not much different from C~(S)-curve, shown in Fig. 1, since P/c = C~S2 in this case. At the edge, both S and P/c should go to zero, which implies that the effective Cu reaches asymptotically a finite value at the edge which depends only on the values of the constants of the E A R S M expression. These are not strong physical arguments for the behaviour of S and PIe at the edge. However, the DNS-data by Spalart (1988) for a flat-plate boundary layer enforces this, see Fig. 2. The model predicted behaviour of P/c can be analysed similarly as that of S. The turbulent production P has the power 2b - d + 2(a - 1) which becomes 2a - 1 after substitution of d = 2b - 1. The production to dissipation ratio has thus the power 2a - 1 - d which becomes 2(a - b). With (12) and (13) this again implies crk > 1.

153

1.6 " ~ '",,,........ ' ' 1.4 : : - . ~ ",,",.... 1.2 "-.'.?,.~'-,",, 1 .... .... "';~:-,"~',)::..... .......@ ~ i i " . 0.8 ~ ~ i 0.6 0.4 0.2 0 0.6 0.7 0.8 0.9

' S Reo=300 S Re0=670 S Reo=1410 P/~ Re0=300 P/~ Reo=670 Re0 =1410

1

1.1

1.2

: ............. .............. .................. ........ .......

1.3

1.4

y/8

Figure 2: S and

P/c around the outer edge of a boundary layer from the DNS-data I

1.0

I

I

2.0

I

I

0,=1.1

,

o,=1.1

ok=1.0

0.75-

by Spalart (1988).

1.5-

........... crk=O.9

i

cr,=a .0

]

.............. a,=0.9

/

,J]

,.-..

z) ~o 0.5-

to

~- 1.0--

I

..,

',

,

II

g

N 0.25-

0.0

0.5-

I

0.0

0.5

I

1.0

"r//'r/w=o,

0.0

I

1.5

2.0

."s

I

0.0

0.5

I

1.0

i I

1.5

I

2.0

Figure 3" Numerical solutions of the velocity defect and the production to dissipation ratio of a selfsimilar wake using k - e models associated to EARSM expression. Effect of the crk value for a given edge behaviour. In addition to S and P/c, we can study the shear-stress anisotropy a12 -- u'vt/k. It is evident from several experimental and DNS-studies, e.g. Bradshaw (1967), Spalart (1988), Sk~re & Krogstad (1994), that a12 approaches zero towards the edge. This fact is yet another reason to require crk > 1, because a12 ~ fb(ok-1) As this is completely independent of/3, we do not have to assume anything about the behaviour of the constitutive model. Cazalbou et al. (1994) found also a constraint on crk for the production to go to zero more rapidly than the diffusion, which was o.k > 0.5. The constraint o.k > 1 found here for non-linear models is more restrictive. Some models, such as most k - e models, have their o.k value falling just on the limit o.k = 1. However one should give a value above the constraint to avoid the PIe ratio going to infinity and the effective C , going to zero. Violating the constraint could lead to numerical problems at the edge. This is illustrated in Figs. 3 and 4. Fig. 3 shows the computed evolution of the non-dimensional velocity defect and of the production to dissipation ratio in a self similar wake with three k - e models associated to the E A R S M expression of Wallin & Johansson (2000): the standard Launder-Sharma model with crk = 1 (dashed line) and two modifications of this model, one which fulfils the constraint ~rk = 1.1 (continuous line) and one which violates the constraint o.k = 0.9 (dotted line). The o-~ and C~ 1 constants have been recalibrated for each model to respect the same evolution at the edge for the velocity (a = 1.43) and the same log-law slope value (~c = 0.43) as given by the Launder-Sharma model, C~2 keeping its classical value of 1.92. The model with o.k -- 1 gives a constant value of PIe at the edge, which is almost found numerically. However, even a small violation of the constraint (crk = 0.9 instead of 1) gives a strong

154

0.96 -2

2

0.92 ...x-- - "

-3

0.88

-4

0.84

-5

a

..... .~... -~"

-

.

.

.

.

.

0.8 0.5

0.6

0.7

0.8

0.9

1

1.1

0.5

y/5

0.6

0.7

0.8

0.9

1

y/5

Figure 4: Left: numerical solutions of the velocity defect near the outer edge of a self-similar equilibrium boundary layer using three different k - w models. Right: corresponding velocity profiles from full Navier-Stokes solutions at Reo ~ 13,000. peak of P/c and a "hook-shaped" velocity profile close to the edge, which is unphysical. On the other hand, ok = 1.1 gives smooth and physically feasible distributions of P/~ and the velocity defect at the edge. It may also be pointed out that lower values of c~k could even lead to a complete divergence of the numerical solution. Fig. 4 (left) shows three different numerical k - w solutions near the free-stream edge of a self-similar boundary layer. The first model (Hellsten 2004a, Hellsten 2004b) gives a qualitatively correct smooth solution at the edge, the second model (Kok 2000) predicts a sharp edge, and the third case is a genetic combination of Kok's model and the EARSM, which predicts a sharp "hook-shaped" velocity profile at the edge. The qualities of these numerical solutions are as expected based on the theoretical results. It must be understood that the edge behaviour shown here for some k - c and k - w models is not a specific feature of these models, but a general property of all two-equation models equipped with nonlinear constitutive models and gradient-diffusion models similar to (1) for turbulent transport terms. We also performed a full Navier-Stokes computation of a flat-plate boundary layer using a grid that is relatively coarse around the edge. The grid resolution around the edge is considered comparable with the grids used in high-quality practical CFD-simulations. This simulation was done in order to verify that the conclusions made based on the simplified self-similar problems are valid also in full Navier-Stokes simulations. The same k - w models were tested as in Fig. 4 (left). The velocity profiles are shown in Fig. 4 (fight). Now, the numerical truncation error largely smears out the sharp solutions, but the unphysical "hook shape" is still clearly visible even though the grid spacing was Ay ~ 0.086, which is roughly eight times as large as than in the self-similar case.

Extension to a case of generalized gradient-diffusion model We have so far studied only models employing the simple scalar-diffusivity gradient-diffusion model (SGDM) given by (1) for turbulent transport terms. Are the above conclusions valid also for generalized gradient-diffusion models (GGDM) with tensor-valued diffusivities? The popular model by Daly & Harlow (1970) (DH)

T~k) ~ c k k ~

Ok ~ujuk 8x-~'

(18)

155

written here for k, is studied next to shed some light onto this question. With the assumptions made in the edge analysis, the DH-model can be written as

Ck v'v' k 2 dk dk T(k) "~ C u k Cu-~-@y = cr;UTd--y

(19)

with or; = Ckv'v'/(C,k) being an effective crk. In this case, (19) is of the same form as the SGDM except that cr~ is no more constant but a function of the velocity-gradient invariants scaled by turbulent time scale. The constraint crk > I must now be reformulated as lim o-~ >

S--,0

1

(20)

Generally, cr~ depends on the vorticity, but in simple shear flows also vorticity approaches zero with vanishing strain rate, and we obtain lima,= s~o

~2/3C k ~ 7.41Ck

(21)

thus Ck > 0.135. In cases with nonzero vorticity at the limit of vanishing shear, e.g. frame rotation, the Wallin-Johansson EARSM gives limit values higher than 7.41 for v'v'/(C,k) leading to a less restrictive constraint for Ck. Hence, it may be reasonable to require Ck > 0.135 in case of Wallin-Johansson EARSM. However, it must be remembered that the limit value of v'v'/(C,k) may be different with some other constitutive model. We checked that numerical computations support this constraint. Other generalized gradient-diffusion models can be analysed in a similar fashion.

CONCLUSIONS This paper discusses the behaviour of two-equation turbulence models at the outer edges of turbulent flows. The focus is especially on nonlinear EARSM-based models with simple scalar-diffusivity gradient-diffusion model for turbulent transport. The analysis is presented in terms of a generalized twoequation formulation with its second scale variable in a parametric form r ~ kmc '~. Owing to this, the results can easily be applied to any given two-equation model. It is shown that the solution may become unphysically sharp or "hook-shaped" if the mean shear rate scaled by the turbulent time scale does not approach zero towards the edge. A constraint is derived for the model coefficient crk of the turbulent transport model of k to avoid such anomalous solution. This is crk > 1, and it ensures physically meaningful smooth solution at the edge also when a nonlinear constitutive model is employed. In addition to the nondimensional mean-shear rate, also the ratio of turbulent production to dissipation is considered as well as the shear-stress anisotropy. All these quantities should vanish at the edge, and they will do so if the new constraint is satisfied. We extended this analysis also to the generalized gradient-diffusion model of Daly & Harlow (1970), and obtained a similar kind of constraint for its model coefficient Ck > 0.135. Similar analysis can be easily carried out also for other generalized gradient-diffusion models. The theoretical results provide understanding of why some model combinations tend to give unphysical solutions around the outer edges. Moreover, the analysis provides constraints for the model coefficients to avoid such unphysical solutions.

156

ACKNOWLEDGEMENTS Part of this work was carried out within the HiAer Project (High Level Modelling of High Lift Aerodynamics, 2001-2004). The HiAer project was a collaboration between DLR, ONERA, Royal Institute of Technology in Sweden, Helsinki University of Technology (HUT), Technical University of Berlin, Alenia, European Aeronautic Defence and Space Company EADS Airbus, QinetiQ and Swedish Defence Research Agency (FOI). The project was managed by FOI and partly funded by the European Union (Project Ref: G4RD-CT-2001-00448). This work was also supported by the National Technology Agency of Finland (Tekes). The support from the European Union and Tekes is gratefully acknowledged.

REFERENCES

B6zard, H. & Daris, T. (2005), Calibrating the length scale equation with an explicit algebraic Reynolds stress constitutive relation, in W. Rodi & M. Mulas, eds, 'Engineering Turbulence Modelling and Experiments 6', Elsevier. Bisset, D., Hunt, J. & Rogers, M. (2002), 'The turbulent/non-turbulent interface bounding a wake flow', Journal of Fluid Mechanics 451,383-410. Bradshaw, P. (1967), 'The turbulent structure of equilibrium turbulent boundary layers', Journal of Fluid Mechanics 29, 625-645. Catris, S. & Aupoix, B. (2000), 'Towards a calibration of the length-scale equation', International Journal of Heat and Fluid Flow 21(5), 606-613. Cazalbou, J.-B. & Chassaing, E (2002), 'The structure of the solution obtained with reynolds-stresstransport models at the free-stream edges of turbulent flows', Physics of Fluids 14(2), 597-611. Cazalbou, J.-B., Spalart, E & Bradshaw, E (1994), 'On the behavior of the two-equation models at the edge of a turbulent region', Physics of Fluids 6(5), 1797-1804. Daly, B. & Harlow, E (1970), 'Transport equations of turbulence', Physics of Fluids 13, 2634-2649. Hellsten, A. (2004a), New advanced k - cJ turbulence model for high-lift aerodynamics, in '42nd AIAA Aerospace Sciences Meeting', AIAA. AIAA paper 2004-1120. Hellsten, A. (2004b), New Two-Equation Turbulence Model for Aerodynamics Applications, PhD thesis, Helsinki University of Technology, Espoo, Finland. ISBN 951-22-6933-3 (print), 951-22-6934-1 (pdf, available at http://lib.hut.fi/Diss/). Kok, J. (2000), 'Resolving the dependence on freestream values for the k - cJ turbulence model', AIAA Journal 38(7), 1292-1295. Menter, E (1994), 'Two-equation eddy-viscosity turbulence models for engineering applications', AIAA Journal 32(8), 1598-1605. Skfire, E & Krogstad, E (1994), 'A turbulent equilibrium boundary layer near separation', Journal of Fluid Mechanics 272, 319-347. Spalart, E (1988), 'Direct simulation of a turbulent boundary layer up to Ro = 1410', Journal of Fluid Mechanics 187, 61-98. Wallin, S. & Johansson, A. (2000), 'A complete explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows', Journal of Fluid Mechanics 403, 89-132.

Engineering TurbulenceModellingand Experiments6 W. Rodi (Editor) 6) 2005 ElsevierLtd. All rights reserved.

157

E X T E N D I N G AN ANALYTICAL WALL-FUNCTION F O R TURBULENT FLOWS O V E R R O U G H WALLS K. Suga ~, T.J. Craft 2 and H. Iacovides 3 ~Computational Physics Lab., Toyota Central R & D Labs., Inc., Nagakute, Aichi, 480-1192, Japan 2ZSchool of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester, M60 1QD, U.K.

ABSTRACT This paper reports the development of a refined wall-function strategy for the modelling of turbulent flow over rough surfaces. In order to include the effects of fine-grain surface roughness, the present study extends a more fundamental work previously carried out at UMIST on the development of advanced wall functions of general applicability. The presently proposed extension is validated through comparisons with data available for internal flows through rough pipes, channels and for external flows over rough surfaces. Then, its further validation in separating flows over a sand dune and a sandroughened ramp is discussed. The validation results suggest that the presently proposed form is successfully applicable to a wide range of attached and separated turbulent flows over fine-grain rough surfaces. KEYWORDS RANS, Turbulence modelling, Eddy viscosity, Wall-function, Rough wall, Separating flow INTRODUCTION

Although many recent low-Reynolds-number (LRN) turbulence models perform satisfactorily, industrial engineers still routinely make use of classical wall-function approaches for representing near-wall turbulence. One reason for this is that, despite advances in computing power, their near-wall resolution requirements make LRN models prohibitively expensive in complex three-dimensional industrial flows. This is particularly true for flows over rough surfaces, where one cannot hope to resolve the details of small wall-roughness elements, and the wall-function approach is thus the only practical strategy for industrial applications. Despite the above comments, in contrast to many other modelling issues, wall-functions have received little attention over the last few decades. Consequently, the strategies in use have, for the most part, been those proposed in the 1970's that assume a semi-logarithmic variation of the near-wall velocity and either a constant, or at most a linearly varying, total shear stress between the wall and the near-wall node (e.g. Launder and Spalding, 1974). However, it is well known that they do not apply in flows with strong pressure gradients and separation.

158 There have been several attempts to replace the wall-function approach for rough wall turbulence. Patel and Yoon (1995) tested the rough wall extensions of the k - t o and the two-layer k-e. models. Durbin et al. (2001) and Aupoix and Spalart (2003) respectively modified the two-layer k - e model and a one-equation model. Although these extensions are rather simple and can overcome some of the defects of the classical wall-function, they require a fine grid resolution near the walls. In order to address industrial requirements, the UMIST group recently proposed an alternative wall-function strategy for flow over smooth walls which, while still semi-empirical in nature, makes assumptions at a deeper, more general level than the log-law based schemes. The approach is called the analytical wall-function (AWF) and integrates simplified mean flow equations analytically over the near-wall control volume, assuming a near-wall variation of the turbulent viscosity (Craft et al., 2002). This analytical integration then produces the value of the wall shear stress and other quantities which are required over the near wall cell. The objective of this study is to propose and validate an extension to the AWF approach that allows for the effects of fine-grain surface roughness. It will be shown that relatively simple adaptations of the method result in an approach that does reproduce many of the features of flows over rough surfaces, whilst retaining the computational efficiency associated with wall-functions. The flow test cases chosen for the validation are internal flows through rough pipes and channels, external flows over rough surfaces and separating flows over a sand dune and a sand-roughened ramp. ANALYTICAL WALL-FUNCTION STRATEGY AWF for Smooth Walls Although the detailed model expressions of the AWF can be seen in Craft et al. (2002), a brief summary is described below. In the AWF, the wall shear stress is obtained through the analytical solution of a simplified near-wall version of the transport equation for the wall-parallel momentum. The main assumption required for the analytical integration of the transport equations is a prescribed variation of the turbulent viscosity/tt over a wall-adjacent computational-cell. This is done as shown in Figure 1, using yv~ as the thickness of the viscosity dominated sub-layer, and assuming that/tt is zero for y
It, =max{0, oq.t(y*-y;)},

.LI/2 / v , a=c~%, c~=2.55 and %=0.09, and/t, v, y, and ke are respectively the molecular where y * - yKe viscosity, the kinematic viscosity, the wall normal distance and the turbulence energy at the node P. In the context of Figure 1, the near-wall simplified form of the tangential momentum equation becomes ,nnY" I-~ y

I y"

p

OAr

U

.4-

Smoo~

9~,

i ! i

Nt T

= max[O, Otla(y* - y~" )]

log(y+) y "- y

4~2"~ z

"-

v

Figure 1: Near wall cells.

Figure 2: Effects of surface roughness on near wall velocity.

159

'ay* (g't'Ixt)

=k~LOx(puu)+-g~x =c~,

(2)

where P, P, x and U are the fluid density, pressure, the tangential coordinate direction and the mean tangential velocity component, respectively. Further assumptions are that convective transport and the wall-parallel pressure gradient do not change across the wall-adjacent cell, and can thus be evaluated from nodal values. The former is evaluated using pointwise, non-conservative, discretisation. A boundary condition needs to be prescribed at the far side of the near-wall control volume, side n in Figure 1, and this is obtainedby linearly interpolating the nodal values of U on either side of the control volume face. For a uniform grid, this leads to U, = 0.5(Up + U N). Equation (2) can then be integrated analytically over the two regions 0 < y < Yv and yv < y < y,, resulting in

dU [ (Cry•+ Av)/IX' if y ' < y~ dy*=i CuY*+-~--- ,if y*>_y ~ [gll+oc(y'-y~)t

I1

. . . -~gCvy,~ +~1 Avy. +By, Ify


(3)

U= iCv y,+I ~_..~__ - Cv (1-(xy:)lln[l+(x(y*-Y;)]+/~v, if y*> y; t'~ l ~ ,~ j L

where Au , Bv , A~ and fly are integration constants which are obtained by applying the above boundary condition and ensuring that U and its derivative are continuous at yr. The wall shear stress can then be obtained using the results of the above integration:

xw=ix

=Ix

~

=

y---o

(4) V

The analytical velocity variation and prescribed turbulent viscosity also lead to an expression for the local generation rate of k, which can be integrated over the wall-adjacent cell, to produce an cell-averaged value for solving the k equation. To evaluate the cell-averaged dissipation rate, t is assumed to vary according to t=kr 3/2/(c~y), away from the wall and to be constant very close to the wall as in conventional wall-functions. The location of the interface between the two regions, y~, is, however, now determined by requiring continuity of t across the two regions. This leads to

e=kJ/21(cty~)=2vkply~.

(5)

Thus y~ = 2ct = 5.1. The value of the constant y*v is taken to be 10.7, arrived at by considering fully developed flows in straight pipes.

Extension for Rough Walls Conventional wall functions are made sensitive to the effects of fine-grain surface roughness through the introduction of the dimensionless roughness height (e.g. Cebeci and Bradshaw, 1977). If the average roughness height is denoted by h, then a dimensionless roughness height h + can be defined in terms of the friction velocity u~, by h+-huJv. The constant B in the log-law expression, U § = ~:-~In y+ + B, is then taken to be a function of h § so that as h§ increases B is reduced. This has the effect of displacing the log-law to the fight, as shown in Figure 2, which increases the wall shear stress. A more generalized approach, still within the framework of wall functions, is to replace h + with h ~ where h ~ - h kpV2/v. Nevertheless, such extensions of conventional wall functions for rough surfaces bring with them all the original assumptions of these strategies and consequently tend to suffer similar limitations to their range of applicability.

160

Figure 3: Near wall cells over a rough wall. The development of the AWF provides, perhaps for the first time, an opportunity to take into account the effects of fine-grain surface roughness using a near-wall modelling strategy that does not include the log-law assumption and all the predictive limitations that it entails. In common with conventional wall functions, the extension of the analytical strategy to flows over rough surfaces involves the use of the dimensionless height, h*. In this case, however, h ~ is used to modify the near-wall variation of the turbulent viscosity. More specifically, the dimensionless thickness of the zero-viscosity layer yv'is no longer fixed at 10.7, but instead becomes a function of h ~ The proposed expression for the dimensionless thickness of the zero-viscosity layer is

y*~ = y~, ~-(h"/70)'}

with m = max{(0.5-0.3(h"/70) ~ exp(0.287)),(1-0.7(h"/70) -0.2' exp(0.1208))} (6)

and yvs* =10.7. The function for the index, m, has been determined through a series of numerical experiments on fully-developed flows through straight pipes and comparisons with data from the Moody chart (Moody, 1944). For yv* > 0, corresponding to transitional roughness with h~ the analytical solutions derived for smooth walls can still be used, but with the above modified value foryv*. When y,,*<0, corresponding to a fully-rough surface with h*>70, the viscosity-dominated sub-layer is destroyed. The analytical solution of the mean flow then needs to be changed, since integration is performed over the single region 0
by" - ~gy" (Ix + Ix,)-~-r

kpL~x'P~)+~

(7)

where z'is the total shear stress. In a developed turbulent boundary layer over a smooth wall (where F D =0) with constant pressure gradient, the total shear stress is obtained from ~9x

v~ FSP]

~y----r= -~eL-~x_]= const.

(8)

Thus, z" varies linearly in the boundary layer. In a zero pressure gradient case, r is constant near the wall. However, measured data reported by Krogstad et al. (1992) showed that in some rough wall boundary layers, the near-wall peak value of the turbulent shear stress exceeded the wall shear stress. The measurements by Tachie et al. (2003) also show this tendency. Unlike in a sublayer over a smooth wall, r now includes the drag force from the roughness elements in the inner layer which is proportional to the local velocity squared and becomes dominant away from the wall, compared to the viscous force. This implies that the convective and pressure gradient contributions should be represented somewhat differently across the inner layer, below the roughness element height, and hence

161

the quantity Fz~ has been introduced in Eqn.(7). In the present study a simple approach has been taken, by assuming that the total shear stress remains constant across the roughness element height, which is in line with the discussions ofKrogstad et al. (1992). Consequently, one is led to 3 r / b y * = 0 ( y < h ) and ~ / Oy* = v 2 / k~,[O(pUU) / ~x - ~ P / ~x] over the rest of the wall-adjacent cell 0' > h), and thus 0 +0P Fo=~xx (pUU) ~ x ' y < h =0

(9)

, y>h.

Using this form, the momentum equation (Eqn. 7) is integrated analytically over the wall-adjacent cell. As shown in Figure 3, separate integration results are obtained for the four cases: (a)Yv < 0 , (b) 0 < Yv < h, (c) h < yv < y, and (d) y, < y~. Note that the form of the dissipation rate does not have any modification for wall roughness. RESULTS AND DISCUSSIONS The CFD codes used in this study are in-house finite-volume codes: TEAM (Huang and Leschziner, 1983) and STREAM (Lien and Leschziner, 1994). The former is used to compute turbulent pipe, channel and boundary layer flows and the latter is to compute turbulent separating flows. Both codes employ the SIMPLE pressure-correction algorithm. TEAM uses an orthogonal staggered grid arrangement whilst STREAM uses a non orthogonal collocated one employing Rhie and Chow's (1983) interpolation and the third order MUSCL type scheme for convection terms. The AWF has been implemented with the "standard" linear k-E and also with a cubic non-linear k-e model (CraR, Launder and Suga, 1996: CLS). For comparison, flow predictions have also been obtained using the LRN k-e model (Launder and Sharma, 1974: LS). (Note that it has been confirmed that both codes produce consistent results, at least in turbulent channel flows.) Wall Parallel F l o w s Pipe f l o w s

Figure 4 compares the presently predicted friction coefficient and the experimental correlation for turbulent pipe flows, known as the Moody chart (Moody, 1944). The turbulence is modelled by the high Reynolds number (HRN) k-e model with the AWE In the range of h / D = 0 - 0.05 (D: pipe diameter) and Re =8,000- l0 s , it is confirmed that the AWF performs reasonably well over a wide range of Reynolds numbers and roughness heights. (The number of computational grid nodes ranged from 12 to 200 across the pipe radius for Reynolds numbers ranging from 8,000 to l0 s, whilst the value of y* at the wall-adjacent node ranged from 50 to 500.) 0.1

........

9 ......

.

....

hk'D "

RL-~e. . . . . . . . . . . . . . . . . . . . . .

o 5xlO,,,,.~.,

~

'.~-"~--~"

~

~

3

o

^

,,

o~

u 2)<10-

~

,~

., I x l 0 -2,

9

v

. 5 x 1 0 -3

~

o .3,~,1u

"~3~~_ "

~

-,~.........~=

-" z x l o

. o

25 20

LRNLS

ay/n=o.o]5 ---o-. ay/o=o.o45 -.-a- a~r

~

~

"~i

.,jj~,,.- o

1. ~..../

-3-

,, l x l 0 - J , -, 5 x l 0-"

10

oo,

ooo

.

1000

I. A W F cal. ~ .

.i

.

IXlO-

~ .

.

,

0

.

.

.

,

.

1000(9 1(90(900 l e + 0 6 Re

.

.

,

.

1e+07

.

.1

.

]e+08

5

og law

i

.

lO

100

1000

]0000

y+

Figure 4: Friction factors in pipe flows (Moody chart). Figure 5: Mean velocity in channel flows.

162

o.o|" O.Oll

\ o.olo \

Cf

Profiles at R e , :

::::::::::::::::::::::::::::::::: ....\

o.ooe ":,\

o.~

.................

m-~ hU,/v=O.OOE+O0 ~ hO,/v,,O.12E+04 ,.-* hU,h,-O.21E+04 ~hO./~'O.30E+04. :

-

5.00E+06

:::::::::::::::::::::::::

-.

9 U"

::::::::::::::::::::::::::: ~ h = O.OE+O0 h'= 0.0 y*= 217.8 e--eh" 0.4E-03 '~", 56.1 ' ~ " 140.2 h" 0.7E-03 h " 104. ' ~,, 297.2 ~ -~-h= 0.1E-02 h'= 153 9 ' ~:= 307 9

= e~:

k

0.008 0.11~ 0.004 : 0.

i

8.0

I"............................. ~ . .

8.25

e.5

8.711

.

7.0

.!i

725 7,8 Log,(Re,)

Figure6: Predicted Cf for smooth and rough wall boundary layers; lines with symbols: AWF; broken lines: exp. correlations.

0

I

2

3

4

L,g,,O0

5

0.0

0./3

0.5

0.75

1.0

y/~

125

Figure 7: Mean velocity and turbulence energy profiles at Rex=5x 10"; lines with symbols: main grid solutions; solid lines without symbols: nearwall analytical wall-function profiles.

Channelflows Figure 5 shows mean velocity distributions in turbulent channel flows at Re=10 s. In the case of smooth walls, h / D = 0 (D: channel height), the HRN k-e model with the AWF reproduces the result of the LRN LS k-e model regardless of the near-wall cell size. The meshes used have 49, 19 and 14 nodes across the channel, resulting in near-wall cell heights of Ay1/D =0.015,0.045,0.09. In the rough wall cases with h/D=O.O1 and 0.03, it is clear that the AWF reproduces the log law distribution for rough walls:

U+ =llny+ +B-AB

(10)

K"

where AB=(1/~)lnh+-3.3 with ~c=0.42 andB=5.2.

Flat plate boundary layerflows A further test of the new AWF approach has been to apply it to a fiat plate boundary layer flow. Tests with different near-wall cell sizes again showed the present treatment to be relatively insensitive to the size of the near-wall cell. Figure 6 shows the resulting predictions of the skin friction coefficient, Cf, plotted as a function of non-dimensional downstream distance, Rex, for a number of roughness element sizes, h (nondimensionalized with the free-stream velocity Uo and kinematic viscosity v). Also shown are the correlations for smooth and fully rough boundary layers, given by

C/= 0.0271" ~e x-1/7 ,

for smooth walls

=[2.87+l.581n(x/h)~ -2"5, forroughwalls.

(11)

For the smooth plate, the predictions are seen to lie slightly below the correlation. In the rough wall cases, at lower values of Re,, the flow is still in the transitional stage, but as Rex increases the predictions do approach the fully rough correlation. Corresponding profiles of the mean velocity and turbulence energy across the boundary layers at Re x = 5• are shown in Figure 7. As in the pipe flow cases, the predicted effect of the roughness is to displace the log-law to the right, whilst the level of the normalized turbulence energy is hardly affected, except in the immediate wall vicinity.

163

Figure 8: Dune profile and computational grid. Separating Flows Sand dune flow

Figure 8 shows the geometry and a computational mesh used for a water flow over a sand dune of van Mierlo and de Ruiter (1988). Their experimental rig consisted of a row of 33 identical two-dimensional sand dunes covered with sand paper, whose averaged sand grain height was 2.5mm. Since they measured the flow field around one of the central dunes of the row, streamwise periodic boundary conditions are imposed in the computation. In the experiments, the free surface was located at y=292mm and the bulk mean velocity was Ub = 0.633m / s. Thus, the Reynolds number based on Ub and the surface height was 175,000. In this flow, a large recirculation zone appears behind the dune. Due to the flow geometry, the separation point is fixed at x=0 as in a back-step flow. Figure 9(a) compares the predicted friction velocity distribution on meshes of 330 • 50, 165 • 50 and 165 • 25. In the finest mesh, the height of the first cell facing the wall is 4mm which is still larger than the grain height of 2.5mm. The turbulence model applied is the cubic CLS k-~ model with the differential length-scale correction term for the ~ equation (Iacovides and Raisee, 1999). The lines of the predicted profiles are almost identical to one another, proving that the AWF is rather insensitive to the computational mesh. (In the following discussions, results on the finest mesh of 330 • 50 are used.) Figure 9(b) indicates the performance of the present AWF compared with results obtained without the FD term, and those from treating the wall as smooth. Obviously, the present rough wall extension of the AWF is the closest to the experiments, confirming the importance of the FD term. Note that the predicted reattachment length appears slightly longer than that of the experiments even with the residual term. Figure 10 compares flow field quantities predicted by the CLS and the LS models with the AWF. In the distribution of the mean velocity and the Reynolds shear stress, both models agree reasonably well with the experiments as shown in Figures 10(a),(b) while the CLS model predicts the streamwise normal stress better than the LS model (Figure 10(c)). These predictive trends of the models are consistent with those in separating flows by the original LRN versions reported by Craft et al. (1999) and thus it is confirmed that coupling with the AWF preserves the original capabilities of the LRN models. :ooexA

0.1

"-'-

(a)

Grid Grid

.

.

.

.

~

.

0.05

0.05

0.0

0.0

-0.05

-0.05

0

0.2

0.4

0.6

0.8 x(m)

1

1.2

1.4

1.6

(b)

'

'

'

'

oooExpt.

0.1

329><50 165><50

-- CLS-A WF(rough ) .... C L S - A W F ( r o u g h w i t h o u t FD). -'- C L S - A W F ( s m o o ~

0

0.2

0.4

0.6

0.8 x(m)

Figure 9: Friction velocity distribution over a sand dune.

1

1.2

1.4

r.6

164

0.3

x.=0.060.21

0.3.7

0.60

0.82

0.97

1.12

1.27

1..42

1.58

O.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1

,

i

..

0

0.2

~

~ 1

l

0.4

0.6

0.8

(a)

. 1.2

1.4

1.6

1.8

x(m) 0.3

,x=0:060.21

0.;,_3._7

0.60

,0.82

0.97

~

","

1.12

1.27,1:42

1.58

O.25 0.2 0.15

o.1

o

;

0.05

~'o

-0.05 -0.1

r

~

~

0

(b)

0.2

..

~

0.4

0.6

0.8

'

1

9 ,

....

- a - - - : - -

1.2

~

1.4

1.6

7

1.8

x(m)

-

0.3

o

~

_

_

~

~

~

~

~

0.25 0.2 0.15 0.1

o ,0%

0.05 0

(c)

~ io

'"

-0.05 -0.1

~ ,

ooExt.x/p~

.......

~

.

CL,o-A

W F

~ 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x(m)

Figure 10: Mean velocity and Reynolds stress distribution of the sand dune flow.

Ramp flow Figure 11 illustrates the flow geometry and computational mesh used for the computations of the channel flows with a ramp on the bottom wall by Song and Eaton (2002). A two-dimensional wind tunnel whose height was 152mm with a ramp (height H=2 lmm, length L=70mm, radius r=127mm) was used in their experiments. Air flowed from the left at a free stream velocity Ue=20m/s with developed turbulence (Reo=3400 at x=0mm for the smooth wall case; Reo=3900 for the rough wall case). Since for the rough wall case sand paper with an averaged grain height of 1.2mm covered the ramp part, the height of the first computational cell from the wall is set as 1.5mm. This flow field includes an adverse pressure gradient along the wall and a recirculating flow whose separation point is not fixed, unlike in the sand dune flow. The experiments showed that the recirculation region extended between 0.74 < x'(: x / L) < 1.36 and 0.74 < x" < 1.76 in the smooth and rough wall cases respectively.

Figure 11" Ramp geometry and computational grid.

165

Figure 12(a) compares the mean velocity profiles for both the smooth and rough wall cases. For both the cases, the agreement between the predictions and the experiments is reasonably good. The predicted recirculation zones are 0.69 <x" <1.40 and 0.57 <x' < 1.60, in the smooth and rough wall cases, respectively. Figures 12(b)-(d) compare the distribution of the Reynolds stresses. (The values are normalized by the friction velocity U,.,,: at x ' = - 2 . 0 .) The agreement between the prediction and the experiments is again reasonably good in each quantity. 2.5 2 r

1.5

!il

1 0.5

L

0

4

.

-0.5 -1

1'.0

0.0

(a)

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~XT~t. f s m o o t h )

-

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o

CES-A WF(rough)

""~-~ --~

:i

0.0 3 . 0 --u '/Ur

(b) 2.5 2

c'=-2.00

ZOO

o

o

o

o

1.5 1 0.5 0 -0.5 -1 O.O '

(c)

3'.0

~u,/Ur162 2.5 2

,~

:" = - 2 . OO

~

ZOO

o ~~ ~~

1.5 1 0.5 0 -0.5 -1

(d)

.-0

C L S tl~W F ( s m o o t h )

c~

I~.

~..

o.o '

3'.0

Figure 12" Mean velocity and Reynolds stress distribution of the ramp flow.

q,_

~ l

-

166 CONCLUSIONS The analytical wall function, which had been developed for application to problems with smooth walls, has been extended to account for the effects of fine-grain surface roughness. The overall strategy of the wall function derivation and application has been retained, with the wall roughness accounted for by modifying the non-dimensional thickness of the modelled viscous layer immediately adjacent to the wall. The extended scheme has been validated in fully developed channel, pipe and developing fiat plate boundary layer flows over a wide range of Reynolds numbers and roughness element heights as well as separating flows over a sand dune and a round ramp. The proposed model performs well with both linear and nonlinear k-e models. The predictive accuracy depends on the turbulence model employed in the main flow field, while the sensitivity to near wall grid resolution is low.

REFERENCES

Aupoix, B., Spalart, P.R. (2002). Extensions of the Spalart-Allmaras turbulence model to account for wall roughness. Int. J. Heat Fluid Flow, 24, 454-462. Cebeci, T., Bradshaw, P. (1977). Momentum Transfer in Boundary Layers, Hemisphere Pub. Co. Crat~, T.J., Gerasimov, A.V., Iacovides, H., Launder, B.E. (2002). Progress in the generalization of wall-function treatments, lnt. J. Heat Fluid Flow, 23, 148-160. Crate, T.J., Iacovides, H., Yoon, J.H. (1999). Progress in the use ofnonlinear two-equation models in the computation of convective heat-transfer in impinging and separating flows. Flow, Turb. Comb., 63, 59-80. Crafi,T.J.,Launder.B.E., Suga, K. (1996). Development and application of a cubic eddy-viscosity model of turbulence. Int. J. Heat Fluid Flow, 17, 108-115. Durbin, P.A., Medic, G., Seo, J.M., Eaton, J.K., Song, S. (2001), Rough wall modification of two-layer k-e. ASME Journal of Fluids Engineering, 123, 16-21. Huang, P.G., Leschziner, M.A. (1983). An introduction and guide to the computer code TEAM. UMIST Mech. Eng. Rep. TF/83/9. Iacovides,H., Raisee, M.(1999). Recent progress in the computation of flow and heat transfer in internal cooling passages of gas-turbine blades. Int. J. Heat Fluid Flow, 20, 320-328. Krogstad, P.A., Antonia, R.A., Browne, L.W.B.(1992). Comparison between rough- and smooth-wall turbulent boundary layers. J.Fluid Mech., 245, 599-617. Launder, B.E., Sharma, B.I. (1974). Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer, 1, 131-138. Launder, B.E., Spalding, D.B. (1974). The numerical computation of turbulent flows. Comput. Meth. Appli. Mech. Engrg., 3, 269-289. Lien, F-S., Leschziner, M.A.(1994). A general non-orthogonal finite-volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part l: Numerical Implementation. Comp. Meth. Appl. Mech. Engng., 114, 123-148. Moody, L.F. (1944). Friction factors for Pipe Flow. Trans. ASME, 66, 671-678. Patel, V.C., Yoon, J.Y.(1995). Application of turbulence models to separated flow over rough surfaces. ASME J. Fluids Eng., 117, 234-241. Rhie, C.M., Chow, W.L.(1983). Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J., 21, 1525-1532. Song, S., Eaton, J.K.(2002). The effects of wall roughness on the separated flow over a smoothly contoured ramp. Experiments in Fluids, 33, 38-46. Tachie, M.F., Bergstrom, D.J., Balachandar, R. (2003). Roughness effects in low-Reo open-channel turbulent boundary layers. Experiments in Fluids, 35, 338-346. van Mierlo, M.C.L.M., de Ruiter, J.C.C. (1988). Turbulence measurements above artificial dunes. Delft Hydraulics Lab., Rep. Q789, Delt~, The Netherlands.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

167

BIFURCATION OF SECOND MOMENT CLOSURES IN ROTATING STRATIFIED FLOW P. A. Durbin and M. Ji Mechanical Engineering Department Stanford University Stanford, CA 94305-3030, U.S.A.

ABSTRACT Equilibria of second moment closure models respond to rotation and to stratification by bifurcating between solution branches. For some region of the rotation-Richardson number plane the bifurcation surface corresponds to sustained turbulence. Outside of this region, the solution converts to decaying turbulence. Strictly speaking, the bifurcation is initially to an algebraically growing solution, which subsequently becomes the decaying solution branch. A closed form solution for stratified, rotating shear flow shows that equilibria very easily can fail to exist. This is the case for some models in the literature. Unphysical solutions are then seen in temporal integrations. The analytical solution provides a well-posedness constraint on model coefficients. A set of constants that satisfy well-posedness, and agree with certain experimental data, is found.

KEYWORDS turbulence modeling, rotation, stratification, closure modeling, RANS

PREFACE In the course of attempting to apply standard SMC's to rotating, stratified flow, we encountered a difficulty. In some regions of Richardson number-rotation number parameter space our computations diverged. This was a persistent and perplexing problem. Ultimately, a full equilibrium analysis uncovered the cause: the models had no real, stable equilibria. Herein, we will describe the problem, our analysis, and a recalibrated model that is better behaved. This work is motivated by the desire to improve predictions of heat transfer in turbines. At the present stage, it is very much on the side of fundamentals of modeling. We hope that by the time of the conference some practical aspects will have been explored.

168

ROTATING

STRATIFIED

FLOW

Equilibrium analysis has proved to be a powerful tool for understanding and developing Second Moment Closures (SMC). The equilibrium criterion was introduced by Rodi, 1976): it is dbi~/dt = 0 where b is the anisotropy tensor. Full equilibrium analyses were poineered by Speziale & Mac Giolla Mhuiris (1989). The most telling aspect of their approach is the changes in number and stability of solution branches as a parameter varies. This is referred to as 'bifurcation' of solution branches. (That terminology is a bit misleading because one branch does not merge into the other; however, it is the established terminology.) 0.8

I (P+G)/E- 1

branch1

..........................

/ 0.4 t [ g/Sk

()

b--

Rig

" ..........

*~

....

".

6~.~,

".. ~ ". O/2 "'... @ ~'"-..

-

0:1

branchl=healthy turbulence branch2=unhealthy~decaying turbulence

?' ? br nch2:'stable

branch 1

/

"......

0:3

1~1
....

; ...... i~,1

i_1

i.

St Figure 1: Bifurcation of equilibria in stratified flow (Durbin, 2002). ~ represents a parameter, like Ri. rb and rc are bifurcation and critical vMues. The basic idea of bifurcation analysis stems from the k - r system in homogeneous turbulence, combined into an equation for 6/k. Invoking equilbrium, the time-derivative of that equation is set to 0: (r - 1)Pie - (C~2 - 1)1 = dt[c/k] = 0 This has 2 solutions: PIe = (C~2- 1)/(C~1 - 1) and r = 0. Forcing effects that appear in the SMC equations can cause 'bifurcation' between these two solutions. A full analysis leads to the solution branches illustrated in figure 1 for the case of stratified, non-rotating, flow. They correspond to either exponential (branch 1) or to power-law (branch 2) time-dependence. On the second branch, the solution changes from power-law growth to powerlaw decay at a critical value of Richardson number. There are two notable values of Ri, the bifurcation point and the stabilization point. At this stage, theory says the two solutions are

169

1) 2)

73 = c'~2-1, o.-~,

k ,'., eat;

=

k ~ tA;

~

0;

A = c~2-c~1 ~.~-1 ( ~ ) ; A =

~-~ depends on constitutive model

T)-I Ce2_l_~P(Cel_1) ; 73 depends on constitutive model

where 73 - Pie. The dependence on constitutive model determines where, and whether, bifurcation between these solutions occurs. It can be shown that if the constitutive relation for the Reynolds stress does n o t satisfy lim

ISk/61-~o~

bijSijk/e = O(1)

the model cannot bifurcate (Durbin & Reif, 1999). Hence, simple scalar viscosity models have no chance to bifurcate if C~ - const. But SMC models can meet the above bifurcation criterion. Combined rotation and stratification arises in flow internal to turbine blades. Buoyancy is driven by centrifugal accelerations. Now there are two paramaters, the Richardson number and the rotation number. The bifurcation point becomes a bifurcation curve. We will explain how the General Linear Model, with usual constants, produces unphysical equilibria: stratification seems to be the culprit. Pathologies include: no stable equilibrium; no real valued equilibria; and two real, positive equilibria (for k/e) that bifurcate into the complex plane, leaving no real solution. From the standpoint of mathematical well-posedness, these are undesirable properties; conversely, model coefficients should be constrained to avoid these properties. In time integrations, the effect of ill-posedness can appear either as blow up at finite time, or as nearly chaotic, relaxation o s c i l l a t i o n s - in other words, as unphysical behaviors. New constants have been found, based on empirical data, and on the constraint that the equilibrium surface in the Richardson number-Rotation parameter plane should be physical. The form of model is quite standard. We consider the RST equations in homogeneous turbulence

dtuiuj = Pi3 + P~j + Gij - 2/3eSij + r dtuiO = P~o + P~o + G~o + r dtO'-~ = - 2uj OOj(~ - CR ~ ~ Ir

where P,j

=

- (u--~0k vj + u--~0k v,)

I~j Gij

-- - 2 ~ F (Qktu-:-~ -t- eimu--~) = -~(giujO + gjui---O)

f~F is the frame rotation vector. Closure of r r

=

and r

is by the General Linear formulation:

-C1bi:ie T k(V2 + C3) (bikSkj + bjkSki- 35ijbmnSnm )

1~ 4 k -- k(C2 - C3)(b,k~tA + bjkf~A) - 64 (Gij - "5 ,jGkk) + -gS, j

(1)

e ~

f~A is the absolute rotation tensor, relative to fixed axes. A closed form solution to the General Linear Model was derived for stratified shear flow rotating

170

Bifurcation surface for the IP model: (a) contours of the real part of a positive root of Eq. (2), (b) shows different regimes of the two positive roots of Eq. (2): 1, two real roots; 2, one real and one imaginary root; 3, two imaginary roots; 4, two complex roots. Figure 2:

about a spanwise axis. fl is the magnitude of the frame rotation rate. On branch 1, the solution has the form

[.

+.

1/211/2

2

+ .(S)+.Ri(~)+.Ri(~)+.Ri2+.Ri+.]

(2)

where each 9 is a known function the model constants. Four roots are found as a function of rotation (Ro = $2/S) and stratification (Ri). For a properly posed model, only one should be real, positive and stable. The first criterion means that the quantities under the square roots in (2) must be non-negative. The latter requires that eigenvalues be computed at the equilibria. That was done by computing the Jacobian matrix numerically. At some value of Ro and Ri the solution (2) may become complex. At that point the model should switch to branch 2. The solution for that branch is described in Ji & Durbin (2004). When equilibrium and bifurcation analysis was applied to the General Linear Model, with constants from the literature (e.g., Gibson & Launder, 1978), we obtained curious results: for some combinations of rotation and stratification no real solutions existed, or no stable equilibria existed. These are decidedly not the correct mathematical properties for a useful RANS closure. The first two time-histories in figure 3 illustrate non-physical behaviors. The corresponding regions are labeled on figure 2. Regions 1 and 2 are suitable for branch 1. Region 3 is where branch 2 should take over. For some sets of constants, branch 2 is not stable over the entire region 3; that is the case where zones with no real, stable solutions exist. Table 1: Model coefficients for the IP and the new model.

IP New

C~ C2 C3 C4

Clc C2c C3c

C4c C5c CR

1.8 1.8

2.9 2.5

0.0 0.0

0.6 0.6

0.0 0.0

0.6 0.32

0.4 0.4

0.0 0.13

0.4 0.4

1.6 1.5

171 0.15 0.1 0.05 0 Z

2

.o~ -0.05 -0.1

J -o

~

1~

1'5

2'0 st =;~ ~

-0.15 -0.2

~

(a)

'~

'5

2~o

-0.250

400

600

800

St

1000

1200

(b)

I

o1 , o2 0.05

0t ~176176

-0.1~ ]

0.1

i

0.05 0

t' t'

/ "

,.

z

t;

"

i

li

V

-0.1

~, / /

w,

':

~

"-"

;

-,., "

_o, It -~ _0.25 s 0

, 5

, 10 Nt

, 15

,,

-0.250

20

o

10

Nt

15

20

(d) 0"15 I

::J o

',~ ,7i

//

,~ -0.05

~

',~

s

---

-0.1 -0.15 -0.2 -0"250

;

1'0 f(Q,S)t

15

20

F i g u r e 3: Time histories of b12 in various regions, a) the stable r o o t is complex ~ blow-up, b) no stable roots exist, c) stable stratification but no rotation: solid line is for Ri = 0.37 and dashed line for Ri = 0.48. d) combined rotation and stratification: Ri = 0 . 3 7 for solid and Ri = 0.48 for dashed line; f~/S = 0.9 for both. e) rotation but no stratification: solid line is for f~/S = 0.7 and dashed line for f~/S = 0.9. Time scale f ( f 2 , S ) = 2[(1 - C2 + C3)(f~ - S/2) + f~]

I72

Figure 4: Bifurcation surface for the new model: [a) contours of the real part of a positive root of Eq. (21, (b) shows different regimes of the two positive roots of Eq. (2): I, two real roots; 2, one r ed and one imaginary roots; 3, two imaginary roots. We have found a set of coefficients, cited in table 1, that is consistent with experimental data and produces an acceptable bifurcation surface. Figure 4 alludes to the analysis. No region 4 exists and branch 2 is stable over all of region 3. Various time-histories, representative of the regions are displayed in figure 3. Panels c, d and e show damped, oscillatory solutions. With pure stratification(c), the period scales on the BruntVaisda frequency. For rotation plus shear (e), the equilibrium solution depends on a combination of the absolute and relative vorticity 2[(1- Cz C,)(Q - S/2) Q] that can be derived from (1). This does not entirely collapse the time-dependence, but it is more suitable than either S or R individually. The combination of rotation and stratification ( d ) oscillates with a frequency that we have not analyzed.

+

1.8 1.6 1.4

. . .

+

173

Acknowledgement: Sponsorship by General Electric aircraft engines, and the Department of Energy ASC program is gratefully acknowledged.

References Durbin, P. A. (2002). A perspective on recent developments in RANS modeling ETMM5 Elsevier Press, 3-16. Durbin, P. A. & Pettersson Reif, B. A. (1999). On algebraic second moment models Flow, Turbulence and Combustion 63, 23-37

Gibson, M. M. & Launder, B. E. (1978). Ground effects on pressure fluctuations in the atmospheric boundary layer, J. Fluid Mech. 86, 491-511. Ji M. & Durbin, P.A. (2004). On the equilibrium states predicted by second moment models in rotating, stably stratified homogeneous shear flow, Phys. Fluids 16, 3540-3556. Rodi, W. (1976). A new algebraic relation for calculating the Reynolds stresses, ZAMM 56, T219-221. Speziale, C. G. & Mac Giolla Mhuiris, N. (1989). On the prediction of equilibrium states in homogeneous turbulence, J. Fluid Mech. 209, 591-615.

This Page Intentionally Left Blank

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

175

TURBULENCE MODEL FOR WALL-BOUNDED FLOW WITH ARBITRARY ROTATING AXES

H. Hattori 1, N. Ohiwa I and Y. Nagano 1 1Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan ABSTRACT Rotating flows have been encountered in many engineering relevant applications such as a flow in turbomachinery. Recently, rotating channel flows with arbitrary rotating axes have been investigated by direct numerical simulation (DNS). Although numerical studies of spanwise rotating channel flow have been reported, there are few reports on streamwise and wall-normal rotating channel flows. In the present study, first, we conducted DNS of various rotational channel flows. It is found from the DNS results that cases of streamwise rotating channel flow which have a mean spanwise velocity caused by rotation involve the counter gradient turbulent diffusion. Thus, since it is well-known that the conventional eddy viscosity turbulence model can not predict this case, we have to consider the reconstruction of a nonlinear two-equation turbulence model based on a cubic model. Consequently, we evaluated existing nonlinear two-equation turbulence models based on our DNS of streamwise and wall-normal rotating channel flows. Using the results of evaluation, we have improved the modeled expression for Reynolds stress, in which a new cubic-type nonlinear model has been proposed. The proposed nonlinear twoequation turbulence model can accurately predict rotating channel flows with arbitrary rotating axes. KEYWORDS Turbulence Modelling, Turbulence, Rotational flow, Arbitrary Rotating Axes, Direct Numerical Simulation

INTRODUCTION Rotating flows have been encountered in many engineering relevant applications such as a flow in turbomachinery. Recently, a turbulence model has been improved for analysis of a turbulent flow with rotation. In particular, in order to conduct more precise analysis of a rotating turbulent flow, a nonlinear two-equation model is required (Nagano & Hattori 2002), in which rotating channel flows with spanwise rotations are predicted accurately. On the other hand, rotating channel flows with arbitrary rotating axes have been investigated by direct numerical simulation (DNS). Although DNS studies on spanwise rotating channel flow have been reported (Nagano & Hattori 2002; Kristoffersen & Andersson 1993; Lamballais et al. 1996), there are few reports dealing with streamwise and wall-normal rotating channel flows (e.g., Oberlack et al. 1999; Wu & Kasagi 2004). In the present study, in order to obtain fundamental statistics on rotating channel flows with arbitrary rotating axes, DNS has been conducted using the spectral method. At the same time, we have made a DNS database for the evaluation of turbulence models. DNSs provide detailed turbulence quantities which are difficult to obtain from measurements, and also provide the budget of the turbulent transport equations.

176 TABLE I METHODS

FOR

DIRECT

NUMERICAL

Grid Time viscosity term advancement other terms Spatial scheme Grid points (x~ x :r2 x x3) Computational volume

SIMULATION

Regular grid Crank-Nicolson method Adams-Bashforth method Spectral method 64 x 65 x 64 47r6 x 25 x 27r6

In particular, turbulent diffusion and dissipation terms in the turbulent transport equation can be obtained from DNSs, which are essential to construct a turbulence model. Based on the present DNS of streamwise and wall-normal rotating channel flows, existing nonlinear two-equation turbulence models have been evaluated. Using the results of evaluation, we improve the modeled expression for Reynolds stress by proposing a new nonlinear two-equation model for rotating channel flows with arbitrary rotating axes. G O V E R N I N G E Q U A T I O N S IN R O T A T I O N A L C O O R D I N A T E S Y S T E M We consider a fully-developed turbulent channel flow with rotation at a constant angular velocity as shown in Fig. 1. The governing equations for an incompressible rotating channel flow in reference frame rotational coordinates can be described in the following dimensionless forms:

~l,/+,i = 0 D u~/Dt+

(1)

+ i + ( 1 / Re~ )u<jj . + - ~ik~Ro~-ku~+ = -Peff,

(2)

where D / D t implies the substantial derivative, the Reynolds number Re~ is based on the mean friction velocity of both walls 'u~ and channel half-width 6, and the rotation number Ro~k is defined as 2~k6/u~-. The Einstein summation convention is used, and a comma followed by an index indicates differentiation with respect to the indexed spatial coordinate. The centrifugal force can be included in the effective pres+ __ P~ + - H~ 1 2 sure Peff(if fluid properties are constant, where p+ is the normalized static pressure, H~o = (ROy t{oj ) 1/'2 is the absolute rotation number, and ro is the dimensionless distance from the rotating axis (Wu & Kasagi 2004). The spectral method is used for the DNS (see Nagano & Hattori 2003). The computational conditions are listed in Table I. The boundary conditions are non-slip conditions for the walls as well as periodic conditions in the streamwise and spanwise directions. The Reynolds-averaged equations for the turbulence model can be written as follows:

I-Ji,i ---

(3)

0

DI)~/DI = -(1/p)Peff.i + (v,O.,,i-- ~ ) . j ttiaj

--

(2/3)k6.~9

-

2ColJtSij

-Jr-

- 2eiktf~.D~

High Order Terms

t/t = (7~,fu(k2/~-)

(4) (5) (6)

where the quadratic modeled expression for Reynolds stress applying the rotational turbulent flow in Eq. (5) was proposed by Nagano & Hattori (2002) as follows: uiuj =

(2/3)kS,:j-(2/fn)2ut.%j + ( 4 C D / . f R ) k ' r ~ o [(S~kWkj - ll]kSki) - (S~kSkj -- S , . , , S ..... 6
j ~

47rb

'

_ . . . .

..........

C

2a t~ ...... ,I, ~ .

LfJ2~ ~ 3

1

Figure 1: Rotating channel flow and coordinate system

(7)

177 TABLE 2 COMPUTATIONAL CONDITIONS

Re~ I

ROT1

150 Case l: Streamwise Rotation STR l STR2 STR3 STR4 1.0 2.5 7.5 10.0

STR5 15.0

Rot2

Case 2: Wall-Normal Rotation WNRI WNR2 WNR3 WNR4 0.01 0.02 0.05 O1.

WNR5 0.3

ROsa

Case 3: Spanwise Rotation SPR 1 SPR2 SPR3 SPR4 0.0 0.75 1.5 3.0

SPR5 5.0

where TRo = u t / k is the characteristic time-scale, C o is the model constant, fR is the model function, S~j = ((J.z,j + (Jj,~)/2 is the strain-rate tensor, and IA'~j = ~ij - ~i~m~,,~ = ((fi,j - ~)j.i)/2 - ~ijm~tm is the absolute vorticity tensor. The last term of the right-hand side is introduced to reproduce the wall-limiting behavior and the anisotropy of Reynolds normal stress components (Hattori & Nagano 2004) given as follows: Q,,, = ( 4 C D / f R ) k T ~

[(S~kI4~j -- IV~kSkj) - (S~kSkj - S.,~nS..,,.6~j/3)]

(8)

where 7-R,L,is the characteristic time-scale for reproducing the wall-limiting behavior and the anisotropy of Reynolds normal stress components. The transport equations of turbulence quantities, i.e., turbulence energy, k, and dissipation-rate of k, c, which compose the eddy diffusivities, can be given as follows: Dk/Dt

(9)

= uk,jj + 7~ + Hk + Pk - c

D e / D r = ue,jj + T~ + 1-I~ + c (C~aPk - C~2fEz) + E + R

(10)

where Tk and T~ are turbulent diffusion terms, Hk and H~ are pressure diffusion terms, and Pk (= - u~[~?~,j) is production term. E is an extra term, and R is a rotation-influenced addition term (Nagano & Hattori 2002). RESULTS AND DISCUSSION Discussion of DNS results in view of turbulence modelling In order to determine improvements in the expression of rotating, wall-bounded turbulent shear flows, DNSs of fully developed channel flows with streamwise, wall-normal and spanwise rotation are carried out using the spectral method (Nagano & Hattori 2003) for the 15 cases indicated in Table 2. DNS results of cases for STRI,-~STR5 and WNR1 ~ W N R 5 are shown in Figs. 2 and 3, respectively (Case 3 is not shown here). In these cases, the spanwise mean velocity appears to be caused by a rotational effect, which increases with the increase in rotation number in both cases. In particular, with increase in the spanwise mean velocity of Case 2, a streamwise mean velocity decreases due to the exchange momentum between the streamwise and spanwise velocity as in the following equations obtained by Eq. (4): 0 = -(llp)P.x

+ (uCiy9 - g77) . y -

2Q~"

~

0 = (z,,l~v - v--g.,), y + 2fl(r

(1 1)

Therefore, in order to predict the flow of Case 2 exactly, the spanwise mean velocity should be reproduced by a turbulence model. Note that the wall-normal rotating DNS (Case 2) of Wu & Kasagi (2004) was carried out on constant streamwise mean velocity condition, but the present DNS does not maintain the constant velocity. Thus, the Reynolds shear stress, ~ tends to decrease remarkably with the increase

178

in rotational number as shown in Fig. 3(b). On the other hand, it is found from DNS results that cases of streamwise rotating channel flow (Case 1) involve the counter gradient turbulent diffusion in the spanwise direction shown in Fig. 2. In view of turbulence modelling, this fact clearly demonstrates that the linear and the quadratic nonlinear two20

.

1

iL

10

0 "17o; =1.0

- "

Ro~

=7.5 ----- Ho=10 -

....

i

o

'

k.i"

Ro~ = 15

i

'

- - - Ro,.

1

0.3

Ro,=li0 ,~. /~,o'~

=2.5

....

RoT =7.5

-----

R o ~ = 10

l,"

0

~Ro,=l.0

/

t

---Uo,=2.5 . . . . nor=7.5

/

|

|

ollll "

V/6

0

/

-----/?,o,-=15

, 9 ",\!4

. ~ ' ~ . _ - ~ - - ~ . f ~ " ...... ~ ........ "-" . . . . . .

9"t/-

I

1

t

-i

-----Ro,=10

//, 0 l I-I/"

0

2

y/6

/

/ o2F + "l

-1

Ro~ = 15

(b) R e y n o l d s shear stresses, u v . , .

(a) Streamwise mean velocities '

17o~ =

....

1

-10

yl~

-'--

.............

J

I

~

1

~

y/6

2

(d) Reynolds shear stresses, v w

(c) Spanwise mean velocities

Figure 2: DNS results of streamwise rotating flows (Case l) '

~

Ro~'=O.Ol

=0.02 9 - "ROT=0.04

- - - Ro~

20

---....

Ro:

=0.1

Ro:

=0.1

Ii +~

/.-

.......

" - - -

o

'

[

t

'
15 - -

-

Ho~

=0.01

- - - - - ROT =0.1

Ro~

=0.02

....

,/" . - - - " - / ,9;

:/-il"

.

.

.

.

.

.

(b) Reynolds shear stresses, u v

o ~..

4 "\

"\\

.

;,.- ,...-~.r

, ~,..~ ,.....~-

:

-0.2

"

-

~

i

Ro~

'

=0.01

Ro~=0.02 9 " " H o - =0.04-

l

--

H,o~ =0.

Ro~

v/,~

(d) Reynolds shear stresses, v,u'

Figure 3" DNS results of wall-normal rotating flows (Case 2)

I0

=0.15

'

v/a

(c) Spanwise mean velocities

t

- -....

-0"40

"~

~. ~

~" " - ' -~,~-"7.'" '- ~ \

---

'

2

0.z

o.;

- - - : ' 7 . : - ~ :\.

.. . . . . . . . . . . . . .

~/a

=0.1

Ro~

" " "HOT=0.04

10

i

i

(a) S t r e a m w i s e m e a n v e l o c i t i e s

-.....

=0.01 RoT =0.02 Ro, =0.04 RoT = 0 . 1 0 Ho~ = 0 . 1 5 RoT

2

179

equation models can not be applied to calculate the case of a streamwise rotating channel flow. Thus, the following modeled equation (12) employed in the linear and the quadratic nonlinear two-equation models can not clearly express a counter gradient turbulent diffusion: -vw

= ut lfV,u

(12)

where the Reynolds shear stresses of a quadratic nonlinear two-equation model are expressed identical to a linear model. Also, it is noted that the rotational term does not appear in the momentum equation of a fully-developed streamwise rotating channel flow indicated as Eq. (13), in which the rotational effect is included implicitly in the Reynolds shear stress in the same manner as the spanwise rotational flow (Nagano & Hattori 2002). Therefore, a cubic nonlinear two-equation model or Reynolds stress equation model should be used for the calculation of streamwise rotating channel flows. 0 = -(1/p)/Sx + (u/-f' y - '~,') , y Evaluation

and

0 = - ( 1 / p ) / S y - v--7,y, ,Y

modelling

Figure 4 shows the assessment result using the present DNS of nonlinear two-equation models in the case of streamwise rotating channel flow. The evaluated models are a cubic model by Craft et al. (1996) (hereinafter referred to as NLCLS) and a quadratic model by Nagano & Hattori (2002) (NLHN). Obviously, the quadratic model cannot reproduce this flow as mentioned in the previous section. The cubic model indicates overprediction of mean streamwise velocity, (7, and underpredicts the Reynolds shear stress, vw, in the case of a higher rotating number. The wall-limiting behaviour of Reynolds stress components is shown in Fig. 4 (f). Although the NLHN model is modelled to satisfy the wall-limiting behaviour of the Reynolds stress component in the wail-normal direction in spanwise rotational flow, it can be seen that the model does not reproduce the wall-limiting behaviour. Evaluations of wall-normal rotating flows predicted by the NLCLS and the NLHN models are indicated in Fig. 5. In this case, it can be seen that the NLHN accurately reproduces turbulent quantities. Thus, 3O

iL2o 10 0 0 0 Ro~ - I .(I

- -

0

O I (~

,

i

,

1

.y/~

2

(a) Streamwise mean velocities

'

I '"--

-i 0

~

"-

,

,

l

-

NL('LS

-;---NLHN

:q/~

- -

I

I

0

2

I

DNS -

NL('L%

,q/a

2

(c) Spanwise mean velocities

(b) Reynolds shear stresses, uv

~

!

T

10 ~

~i 10-2 +~,f 1 0 4 ~

1 0 -6

Lmes

NLHN

10 -~ 10

0c 9

I

R o : = .0

0 D N S 7" - - - NLCLSJ ~ NLiI~7

t

(d) Reynolds shear stresses, vu,

u/,s

(e) Turbulence energy

~_

-I,

0-2

10-1

10 ~

l0 t

y~

102

(f) Wall-limiting behaviour of Reynolds stress components

Figure 4: Evaluations of predicted streamwise rotating flows (Case 1)

180

one may consider that a streamwise rotating flows can be predicted using a quadratic nonlinear model. However, the wall-limiting behaviour of the Reynolds stress component in the wall-normal direction is not predicted in the vicinity of the wall shown in Fig. 5 (f). From these results, we propose a cubic expression of Reynolds stress in a two-equation turbulence model which can adequately predict rotational channel flows with arbitrary rotating axes, in which a modelling of wall-limiting behaviour of Reynolds stress components is also considered. In this study, in order to construct a cubic nonlinear model, the Reynolds stress expression proposed by Pope (1975), the explicit algebraic stress model by Gatski & Speziale (1993) and our previous nonlinear model (Nagano & Hattori 2002) are adopted for the proposed model. The nondimensional form of the proposed cubic model is given as follows: b~j = - 3/(3 - 2r/2 + 6~2){5ij + (S~k|l/~.J -- Wi#~ kj) - 2 ~_,ik~ kj - ~'.,.... ~.,-~

+ 6/(2- ~ +

r

( w ; ~ . ~ v ; 5 - 14*

~*

~

~4. . . . . .

-

. . . .

+ ( IVi*k I V~*t S"e*j + S~*kI4 k*t I V~ - S~*,~IV,*~,,~H ;,*t2 (5ij / 3) ] } 1

.,

,

1

,

,

(14)

,

where .q = (Si~/SiS)5 and ~- = (14 ij WO)5, and bij, S.ij and W 0 are nondimensional quantities respectively defined as follows: b~ = C;~b~2.

.% = C;~-RoS~,

(15)

H ~, : 2C;~-Ro W, j

Since the proposed model in Eq. (14) can be written as the following equation for the Reynolds shear stress, ~ (or b23) in streamwise rotating flow, in which Sa2(= S'21), W12( = -W21), $23( = $32) and IV23(= -I432) exist, the Reynolds shear stress, vw, can be reproduced by the present model" b23 = 3 / ( 3 -

2r/2 + 6~2){S~3 + 6/(2 -712 + ~2)[(W,2*aS~2S~3-

$21S;2IV2",3)

u T, -qt.- (14 .2 1 W. ; 2 S. 2 3 . -j- W'23 w ' ' *32 ~-5~* 23 -'[- $219 1Vi,214-~3 + ~-5~, 23'" 32 14-, 23 )] }

Figure 5" Evaluations of predicted wall-normal rotating flows (Case 2)

(16)

181 In Eq. (14), the functions 3/(3 - 2 r / - 2 + 6r ~) and 6/(2 - r/~ + r may be taken a negative value or0 with the increase in r/. Thus, in order to avoid taking a negative value or 0, we modeled these functions taking into account rotational effect as follows"

3/(3 - 2r/2 + 6~~) "~ 1 / [ 1 + (22/3) (I4 .... 2/4) + (2/3) ( W ' 2 / 4 - S .2 - f~,2) fB + ( 2 / 3 ) s 2]

(17)

6/(2 - 7/~ + r

(18)

~ 3/[1 + (3/2)(W*~/4)

+ (1/2)(W'2/4

- S .2 - . f . ; 2 ) ft~ + ( 1 / 2 ) f *~]

where

fB = 1 +

f~. = (CDrno)2 {ft,.. (2emjiWij - 12.,)} ~

G (W*~/4 - `5'*2 - f.~2),

(19)

The model function in Eq. (19) is introduced to avoid an inappropriate value of (W'2/4 - S *~) with the increase in a rotational number. Since f,9 _ 0 is consistently kept in non-rotational flows, inadequate Reynolds stresses predicted by the proposed model are not given in non-rotational flows. Finally, in order to construct a precise model for rotational flow, a cubic term introduced in an NLCLS model (Craft et al., 1996) is added in the proposed model. Consequently, the final form of the present cubic nonlinear model is given as follows" uiuj

=(2/3)kSq -

CllJt~ij + C2k(T2o

-~

"F~w)(~ik14/kj

-- ['VikSkj )

+ ~.~k(~o + ~ ; ) [ s ~ & j -(a/3)6,~SmnSm.] + C4~o [n4~W~ -- (a/3)6~W,.nn%,,,] (20)

- (2/3)5,,jSe,,~IV,,~,~I,V,,e] + cTk'r3o ( S . o S ~ S ~ , - SoW~t14"~t ) 2

3O 20

,

,

l h , : ~

,L

0

.' .- oooo-

~ooO~._..-

0 0

H.-=7 5 -~ _ ~ o . ~ O

0 ~

~

0

. . . . . . . . . . vuuo,a. . . . .

o '~2s 0 f

--- -

N I ( [,.N( 1 9 9 6 1 Prc,,cnl nl~cl

I

~

"

g/i,

,,,_'is

.,t~oomx~Oooo~

1[.. = I0

0' ~ =

o DNS . . . . NLCLS(1906)

Present model q,.~,

I

1r =I.o

.....

....

(b) Reynolds shear stresses, uv

ooooo

/ Ro~='~.0' ' (% 6~..~ Symbols: {+~i

tl'~A

4~ "~

0

-.r.:.,.v,,r-,,~m~

91

2

(c) Spanwise mean velocities

L, . . . .

'Ro;=16.0/ I'~ ~~176 t '

DNS P . . . . . . t model

zx~ -. . .-.

~-~2-t '2~

]

~1

10-2 I ~1 I'Z" 10 -4

4

~

2' ~

''

y+2

]+~ 10 "6 ~t

0

a I""

i/

,y+4/~,""

22I.r ~ > . ' ~ ]/

r

o

---NLCLS( 19961 i ~ Present,model 1 q/',5

-2 0

'

I~ - U - - - s 0

4~'-

-I 0

(a) Streamwise mean velocities 0.2

~

'1

o DNS -[ . . . . NLCLS(1996) .~ , .... ,.0 m p r .......... d e ' /

"g'="="~ ....... 7. . . . . .

~ :,/~ (d) Reynolds shear stresses, vw

! _.~~. ~.~~

10-,,'t

," ! 0 -2

0

1

g/6

(e) Turbulent intensities

ROT = I 0 . 0

o

~

" ..... I 0 -~

I 0~

k-~2~

v2* 1 01 f

2

(f) Wall-limiting behaviour of Reynolds stress components

Figure 6: Distributions of predicted streamwise rotating flows (Case 1)

I'0 2

182

where vt = Ct,,f~zk2/g and C~, = 0.12, and the model functions and constants are given as follows:

(q = - - 2 / f m , c2=--4CD/fm, C3=4CD/.&I, c 4 = - S C ~ S 2 / ( f R l f ~ 2 ) cs - - 4C~fm~/(fm,fR2) , cs - - - 2C])/(fmfR2) C7 - - 1O C t , 9~2 . , C D, CD -0.8 fna = 1 + (CD'rRo) 2 [(22/3)14 ~ + (2/3)(I,V ~ - S ~ - f~) f~] .f~ -

1 + ( C D r ~ o ) '~ [ ( 3 / 2 ) W ~ + ( 1 / 2 ) ( W

.fm~= 1 2 , 1 -

(21)

2 - S ~ - f~) f~]

exp

L

The characteristic time-scale rR~, is introduced in an NLHN model (Nagano & Hattori 2002) for an anisotropy and wall-limiting behaviour of turbulent intensity defined as: 7R,. - V / ( 1 / 6 ) ( f R o / C D ) / f s w (1 - 3(~,vlfv2/8 )

f21

(22)

f~,, f~w = [ ( ~ -

f s w = W 2 1 2 + S'2/3 -

V/H'212)fw(1)]~

(23)

where the wall reflection function is defined by fw({) = exp[-(Rtm*/{)'~], and the corrected Reynolds :/-- D1/4 number for the rotating flow is given as Rt,, = (Ctmn* R~/4)/t'-'t,n,4 + n*) (Nagano & Hattori 2002). Moreover, it is found from the evaluation shown in Figs. 4 (f) and 5 (f) that the wall-limiting behaviour of Reynolds stress component is not satisfied for the tested rotating flows. Thus, the characteristic timescale "rR~, is slightly modified as follows: (24) where S*

--

S ..... fb, W*

(:,~mn

---

(-g,m,n 1/1",~,,~2:. ,

211

Ir

o

R,,

~

..............

oL

- o I

~ - - ......................... ----:_~ o

q~

Ro, =0.15

Ro -o.I 5 . . . . . . . . . . . . . . . . . . . . . . . _. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. . . . . . . . . . . . .

....... :::; .............

=0.04

o

o

o

......

r:z'_- 5 - ---------"z-'2 ~. -- - . . . . .

1% =0.02

o Ho.

o

~

Present model

-1 [

1

--

,

i

0

(a) Streamwise mean velocities

- -

1

Present

model

0

,.._-~--

...........

0,01

.... NL( ~Prescm

~ ..........

,

u/~

L S ( I096 inodel ---L..~

,,l/'

(b) Reynolds shear stresses, ,u~,

(c) Spanwise mean velocities

0.5

........

-, . . . . . .

no•---0.15 ,- .....

....... _ -

"" e , o , ~ ~ .

~

,-"

[,~

' Ro-

5['~ A . f l "~

'

t~t

-ok

LI

%

'

'

=0.01

10 ~

Ro, =0.1

I~ 10-2

Symbols : DNS L i n e s : Prcscnt modcl

I'~i 10 -4 0" 6

.....

.::

10-~ ~ : 4 / ~

o

,,:

,0.,,

//

lO-I

I0

....

~2 =~176

o DNS - -

0

~ O

~

~-...-'''---0.5

,,

0

NLCLS(1996) P r e s e n t model

~/~ (d) Reynolds shear stresses,

2

'vu,

1

;q/6

(e) Turbulent intensities

i

10 I

10 ~

101

10

y+

2

(f) Wall-limiting behaviour of Reynolds stress components

Figure 7" Distributions of predicted wall-normal rotating flows (Case 2)

183

On the other hand, concerning the modeled transport equations of k and c as indicated in Eqs. (9) and (10), the identical equations of the NLHN model (Nagano & Hattori 2002) are adopted. However, a rotation-influenced addition term, R, in the c-equation (10) is generalized as follows:

R

=

C~f~ke~ieWiidef2e

(25)

where de is the unit vector in the spanwise direction.

E v a l u a t i o n o f proposed m o d e l The predictions of fully-developed streamwise rotating channel flows calculated by the present model are shown in Fig. 6. The results with the NLCLS model are also included in the figure for comparison. It can be seen that the present model gives accurate predictions of all cases, and the test case of the highest rotational number is adequately reproduced by the present model. Also, the Reynolds normal stresses are indicated in Figs. 6 (e) and (f). Now, the present model adequately reproduces redistribution of Reynolds normal stresses, and can predict wall-limiting behaviour exactly. Also, cases of wall-normal rotating channel flows are shown in Fig. 7. The present model can predict similar to the prediction of the NLHN model, but the wall-limiting behaviour is satisfied exactly as indicated in Fig. 7 (f). In order to confirm the performance of the present model, cases of rotating channel flow with combined rotational axes are calculated, in which DNS databases of these cases are provided by Wu & Kasagi (2004). The rotation numbers of Case STSP are given as ROT:,: = 2.5 ~ 15 and Ro~ - 2.5, of Case WNSPI is RoTy = 0.04 and /~o~-~ = 2.5 ~ 11 and of Case WNSP2 is RoTy = 0.01 ~ 0.04 and ROT,. -- 2.5, respectively. The predicted mean velocities are shown in Fig. 8 with DNS data (Wu & Kasagi 2004). It can be seen that the proposed model gives accurate predictions of all cases, since

I0

0

'

20 IL

~

I0 0

0

0

0

Prc.~ntmodcl 0

I

0

"~

(a) Case STSP: Streamwise mean velocities 20

-I .50

I !t,i~~

(b) Case STSP: Spanwise mean velocities 20,

2

1

0

(c) Case WNSPI" Streamwise mean velocities

I'= I0

10 I

I0

01

h'.,, 0.02 .Ho 2.5

Ro.,=0 0 1 .

0 ()

,

i ~

Prc~/t/s~odel

(d) Case WNSPI" Spanwise mean velocities

1

0

~

1

Prc~cnlmodel '

0

'

Ib, = 2.5

I

2!/b

y,'t'

(e) Case WNSP2" Streamwise mean velocities

(f) Case WNSP2: Spanwise mean velocities

Figure 8: Distributions of mean velocities in an arbitrary axis rotating channel flow

184

the proper modelling for rotating flows with arbitrary rotating axes is introduced in the present cubic nonlinear two-equation model. CONCLUSIONS We have carried out DNSs of various rotation-number channel flows with arbitrary rotating axes in order to obtain fundamental statistics on these flows. It is found from DNS results that cases of streamwise rotating channel flow involve a counter gradient turbulent diffusion regarding spanwise turbulent quantities, so that the linear and the quadratic nonlinear two-equation models can not be applied to calculate the case of streamwise rotating channel flow. Using the present DNS data, both the cubic and the quadratic two-equation models are then evaluated. It is found that the existing models can not accurately predict turbulent quantities in a streamwise rotating channel flow. Therefore, using the assessment results, we propose a new cubic expression of Reynolds stress in a two-equation turbulence model which can satisfactorily predict rotational channel flows with arbitrary rotating axes. The proposed model can also reproduce anisotropy of turbulent intensity near the wall, and thereby satisfies the wall-limiting behaviour of turbulent quantities in the cases under study.

Part of this work was supported by Chubu Electronic Power Co. Inc., and through the research project on "Micro Gas Turbine/Solid Oxide Fuel Cell Hybrid Cycle for Distributed Energy System" by the Department of Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST). REFERENCES

K. Abe, T. Kondoh, and Y. Nagano, 1997, "On Reynolds-Stress Expressions and Near-Wall Scaling Parameters for Predicting Wall and Homogeneous Turbulent Shear Flows," Int. J. Heat and Fluid Flow, 18, pp.266-282. T. J. Craft, B. E. Launder, and K. Suga, 1996, "Development and Application of a Cubic Eddy-Viscosity Model of Turbulence," Int. J. Heat and Fluid Flow, 17, pp. 108-115. T. B. Gatski and C. G.Speziale, 1993, "On Explicit Algebraic Stress Models for Complex Turbulent Flows," J.Fluid Mech., 254, pp.59-78. H. Hattori and Y. Nagano, 2004, "Non-Linear Two-Equation Model Taking into Account the WallLimiting Behavior and Redistribution of Stress Components, " Theoretical and Computational Fluid Dynamics, in press. R. Kristoffersen and H. I. Andersson, 1993, "Direct Simulations of Low-Reynolds-Number Turbulent Flow in a Rotating Channel," J. Fluid Mech., 256, pp. 163-195. E. Lamballais, M. Lesieur and O. M6tais, 1996, "Effects of Spanwise Rotation on the Vorticity Stretching in Transitional and Turbulent Channel Flow," Int. J. Heat and Fluid Flow, 17 pp. 324-332. Y. Nagano and H. Hattori, 2002, "An Improved Turbulence Model for Rotating Shear Flows," Journal of Turbulence, Institute of Physics, 3-006, pp. 1-13. Y. Nagano and H. Hattori, 2003, "DNS and Modelling of Spanwise Rotating Channel Flow with Heat Transfer," Journal of Turbulence, Institute of Physics, 4-010, pp. 1- 15. M. Oberlack, W. Cabot, and M. M. Rogers, 1999, "Turbulent Channel Flow with Streamwise Rotation: Lie Group Analysis, DNS and Modeling," Proc. 1st Int. Syrup. on Turbulence and Shear Flow Phenomena, pp.85-90. S. B. Pope, 1975, "A More General Effective-Viscosity Hypothesis," J. Fluid Mech., 72, pp.331-340. H. Wu and N. Kasagi, 2004, "Effects of Arbitrary Directional System Rotation on Turbulent Channel Flow," Physics of Fluids, 16, pp. 979-990.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

185

APPLICATION OF A NEW ALGEBRAIC STRUCTURE-BASED MODEL TO ROTATING TURBULENT FLOWS C. A. Langer 1, S. C. Kassinos 1,2, and S. L. Haire 3 1 Department of Mechanical and Manufacturing Engineering University of Cyprus, Nicosia, Cyprus 2 Center for Turbulence Research, Stanford University, Stanford, CA 94305 3 Advanced Technology Center, Lockheed Martin Space Systems Company, Sunnyvale, CA 94089

ABSTRACT A primary goal of RANS based modeling is to determine the Reynolds stress tensor in order to close the turbulence problem at the mean velocity level. However, the Reynolds stresses alone do not characterize adequately the turbulence, especially in presence of rotation; the structure of the turbulence is also important. Here hypothetical turbulent eddies are used to bring awareness of turbulence structure into the turbulence model. Averaging over an ensemble of eddies produces a set of one-point statistics, representative of the eddy field, and a set of equations of state relating the Reynolds stresses to these statistics. An algebraic model for the eddy statistics is constructed in terms of the local mean deformation and two turbulence scales; the turbulent kinetic energy and the large-scale enstrophy (LSE). Contrary to existing ad-hoc definitions of the second scale equation, the LSE equation has a fundamental background; it is derived from the large-scale vorticity equation. The algebraic model is further sensitized to the presence of walls, ensuring proper asymptotic behavior. The complete model has been found to produce very good results for a set of channel flows in fixed frames and in spanwise-rotating frames of reference.

KEYWORDS

rotation, turbulence modeling, structure based, large scale enstrophy, algebraic model, near wall

INTRODUCTION Flow predictions have become a standard feature of modem flow system design. Where turbulence is important there is need to have a good model for the turbulent transport and for the turbulent stress, in order to produce adequate predictions of skin friction, flow separation, heat and mass transfer, and other flow features. As a result of the efforts of many contributors, turbulence models are now quite adequate for simple flows, but there remain important engineering problems where improved models are needed. For example, improved models for turbulence in rotating systems would enable better turbomachinery design. The work developed by the structure-based turbulence modeling group at Stanford has been

186

motivated by this need. Reynolds and coworkers (e.g. 1991, 1994, 2001) have repeatedly argued that for rotating turbulence, information about the turbulence structure is crucial. The Reynolds stresses only characterize the componentality of the turbulence, i.e., which velocity components are more energetic. The turbulent field has much more information than that contained in the Reynolds stresses, which is important in presence of rotation, and which is described by the turbulence structure. For instance, the dimensionality of the flow is important. This carries information about which directions are favored by the more energetic turbulent eddies: if the turbulent eddies are preferentially aligned with a given direction, then the dimensionality is smaller along that direction. Important contributions for spanwise rotating flows have been introduced by Pettersson Reif et al. (1999), Wallin and Johansson (2002) among others. These explore the traditional componentality approach. Here we aim at exploring a structure formulation approach. This paper describes the ideas behind, the form of, and the results obtained with a new algebraic structure-based turbulence model (ASBM) developed for engineering use (see Haire and Reynolds, 2003; Langer and Reynolds, 2003). This model employs three differential equations: the equation for the turbulent kinetic energy, a variation of the recently developed equation for the large-scale turbulent enstrophy, and an elliptic relaxation equation for a blocking parameter, used for sensitizing the turbulence structure to the proximity of surfaces. The algebraic model relates the turbulence stresses and structure to these scales and the mean deformation tensors.

THE TURBULENCE STRUCTURE TENSORS As in Kassinos and Reynolds (1994) and Kassinos et al. (2001), the turbulent stream-function vector, gt~, is introduced to explore and elaborate concepts of turbulence structure. From the g/k definition it follows that the vector stream function at one point is determined by the vorticity at all points through a Poisson equation. ui = e~jk axj '

axe

axkaxk -- --m~.

The turbulence structure tensors are defined in terms of one point correlations of vector stream function gradients, and hence they contain non-local information about the turbulence.

,,

av; ava

a% av ,

Rij -- uiu j -- EistEjpq OXs OXp '

=

Dij = Oxi Oxj '

aq OXk OXk '

(2)

where Rij, Dij, and F/j are respectively the Reynolds stress, the structure dimensionality, and the structure circulicity tensors. Dij and F/j carry information about the large-scale, energy-bearing, structure of the turbulence not conveyed by Rij. Rij measures the componentality of the turbulence. If the turbulence has one zero component (say u~ = 0), then it is two-component (2C), but it is not necessarily two-dimensional (2D). If none of the vector stream function components varies with Xl, then Dll = 0, indicating that the turbulence is 2D and independent of Xl. It need not be 2C when it is 2D. Similarly, if the large-scale vorticity is entirely aligned with the Xl axis, then all F/j other than F11 are zero. For homogeneous turbulence, the contractions of the structure tensors are all twice the turbulent kinetic energy, Rii -- Dii = Fii = q2 = 2k. Normalized structure tensors are then defined as

rij = Rij/q 2,

dij - D i j / q 2,

fij = Fij/q 2.

(3)

187

Moreover, for homogeneous turbulence, there is a constitutive relationship among the tensors,

rij -[- dij -+"f ij = ~)ij.

(4)

Thus only two of the tensors are linearly independent. This suggests that it could be difficult to model turbulence in terms of a single one of them as one hopes to do in Reynolds stress transport modeling.

THE STRUCTURE-BASED ALGEBRAIC STRESS MODEL

The eddy-axis concept (Kassinos and Reynolds, 1994) is used to relate the Reynolds stress and the structure tensors to parameters of a hypothetical turbulent eddy field. Each eddy represents a twodimensional turbulence field, and is characterized by an eddy-axis vector, ai. The turbulent motion associated with this eddy is decomposed in a component along the eddy axis, the jetal component, and a component perpendicular to the eddy axis, the vortical component. This motion can be further allowed to be flattened in a direction normal to the eddy axis (a round eddy being characterized by a random distribution of kinetic energy around its axis). Averaging over an ensemble of turbulent eddies gives statistical quantities representative of the eddy field, along with constitutive equations relating the normalized Reynolds stresses and turbulence structure to statistics of the eddy ensemble.

(5)

rij = ( 1 - , ) 8 9 + (1 - * ) Z [ 89

-anmbmn)Sij- 89 +anmbmn)aij- bij+ainbnj+ajnbni]

-'t- (--'~;/~'~T)(Eiprapj %- Ejprapi)

{ 111 -- ~ ( 1 --

anmbmn)]~)kr -+ ~,bkr -- ~aknbnr },

dij = 89

(6)

fij = ~)ij - rij - dij.

(7)

The eddy-axis tensor, aij = < V2aiaj >, is the energy-weighted average direction cosine tensor of the eddy axes. The eddy-axis tensor is determined by the kinematics of the mean deformation. Eddies tend to become aligned with the direction of positive strain rate, and they are rotated kinematically by mean or frame rotation. Motion around the eddy is called vortical, and motion along the axis is called jetal. The eddy jetting parameter ~ is the fraction of the eddy energy in the jetal mode, and (1 - ~) is the fraction in the vortical mode. Under irrotational mean deformation, eddies remains purely vortical (r = 0). Shear produces jetal eddies, and in the limit of infinite rapid distortion ~ --, 1 for shear in a non-rotating frame. For shear in a rotating frame, r ranges from 1 for zero frame rotation to 0 for frame rotation that exactly cancels the mean rotation in the frame, for which the mean deformation in an inertial frame is irrotational. The eddy helix vector 7k arises from the correlation between the vortical and jetal components. Hence 7k = 0 for purely vortical turbulence (~ = 0) or for purely jetal turbulence (~ - 1). Typically 7k is aligned with the total rotation vector f~k. The eddy-helix vector is the key factor in setting the shear stress in turbulent fields. Flattening is used to describe the degree of asymmetry in the turbulent kinetic energy distribution around an eddy. A round eddy has no preferential distribution. If the motion is not axisymmetric around the eddy axis, the eddy is called flattened. The eddy-flattening tensor, bij, is the energy-weighted average direction cosine tensor of the flattening vector. The intensity of the flattening is given by the flattening parameter, ~;. Under rapid irrotational deformation in a fixed frame eddies remain axisymmetric. Rotation tends to flatten the eddies in planes perpendicular to the rotation direction.

188

Eddy-Axis Tensor Model Following Reynolds et al. (2000), the eddy-axis tensor, aij, is computed on the analysis frame, where the turbulence might be at equilibrium or very close to it. The eddy-axis tensor is computed with no reference to the frame rotation, as it is only kinematically rotated by it (Kassinos and Reynolds, 1994; Haire and Reynolds, 2003). The evaluation is divided in two parts. Initially a strained eddy-axis tensor, ~j, is evaluated based on the irrotational part of the mean deformation. Next a rotation operation is applied, sensitizing the eddy-axis tensor to mean rotation. This procedure produces eddy-axis tensor states that mimic the limiting states produced under RDT for different combinations of mean strain with on-plane mean rotation, while guaranteeing realizability of the eddy-axis tensor. The strained ~ j is given by * * 2 * (S~kaSkj at-Sjk4i~SmnaSnm~ij) T"

1 = 58v +

,

9 s ao + 2 V / 2x, S l,9pS~,qa~pq

(8)

where S~j = S i j - Skkfij/3 is the traceless strain-rate tensor, x is a time scale (Eq. 28), and a0 = 1.6 is a "slow" constant. This gives realizable states for the eddy-axis tensor under irrotational deformations. The final expression for the homogeneous eddy-axis tensor, aij (for near-wall regions see Equation 19), is obtained by applying a rotation transformation to the strained eddy-axis tensor, @,

aij = HikHjlaSkl, where f22p

=

~-'~ij ~"~ik~"~kj Hij = ~)ij + hi V/~..~d2p + h2 ~2----7 ,

(9)

~"2pq~"~pq.The orthonormality conditions HikHjk = 8ij and HkiHkj = 8ij require hi - v/2h2 - h22/2.

(10)

h2 is determined with reference to RDT for combined homogeneous plane strain and rotation (see Reynolds et al., 2000), 2-2( 89

+ vq-r)

if r < 1 r=

h2 = 2 - 2V/ 89 - v/1 - l/r)

if r > 1

apq~"~qrS*p S*knSnmamk

(11)

Flattening Tensor Model The flattening tensor bij is modeled in terms of the mean rotation rate vector, f2i, and the frame rotation rate vector, f2f,

bU =

(a, + c a() (aj +c a S) (~'~k -+-Cb~"~f) (~'~k -k- Cb ~'2f)

,

C6 = -1.0.

(12)

Eddy-Helix Vector Model The helix vector 7k is taken as aligned with the total rotation vector,

+z

V

k k

(13)

189

Structure Scalars Modeling ~, [3 (see Eq. 13), and ~ is a crucial part in the construction of the model. The equations for these scalars are found by comparing target turbulent states, with the result obtained with the constitutive equations (5,6,7), given a local mean deformation. Throughout the model development there is a strong effort to make it consistent with RDT solutions, aiming to improve model dependability and realizability for a wide range of mean deformations, as well as to obtain guidance in the functional shape chosen for the structure parameters. Tentative functional forms for the structure parameters are thus chosen with reference to RDT. A set of parameter values is chosen to mimic the isotropic turbulent state (the eddy structure is expected to consist of axisymmetric (~ - 0 ) , vortical ( ~ - 0) eddies). Finally interpolation functions (along with model constants) are chosen to bridge these limiting states (isotropy and RDT). They are selected specially to match a canonical state of sheared turbulence, observed in the log region of a boundary layer. The structure scalars are parameterized in terms of 1]m , TIf , and a 2, representatives of the ratio of mean rotation to mean strain, frame rotation to mean strain, and a measure of anisotropy respectively. These in turn are defined in terms of ~2x2, ~2"c2, and $2x2; measures of the strength of the mean rotation, total rotation, and mean strain respectively. ~ represents a time scale of the turbulence (Eq. 28).

Ylm~/~f~-~, ^2

~-~m = --aij~ik~'~kj,

rlf~TIm--sign(X)~O~T, =

r

r

--aij~"~ik~-~kj,

a 2 =--apqapq,

(14)

r

=--aijSikSkj,

X ~ aij~"~ikSkj.

(15)

The structure parameters are then defined with the help of auxiliary functions (see Table 1).

((l"lm--lqf)2)('l"lm--.lqf[V/3(a2- 89

d?--t~* X (Tim_,qf)2_k(l_a2) 2 x [rlm_rlflv/3(aZ 89

)

13= I]*, Z=Z*x

(17) [~(a2-~)l

p'.

(18)

WALL BLOCKAGE As a no-slip wall is approached, the velocity is driven to zero through the action of viscous forces. Furthermore, the velocity vector is reoriented into planes parallel to the wall through an inviscid mechanism (wall blocking) which acts over distances far larger than the viscous length scale. Thus the velocity component normal to the wall is driven to zero faster than the tangential components. In the structure-based model it is postulated that the eddy orientation shall also become parallel to the wall as it is approached. A wall-blocking procedure is then introduced to reorient the eddies into planes parallel to the wall. The structure parameters are also sensitized to wall blocking, such that the modeled Reynolds stresses are consistent with the expected near wall asymptotic behavior. Following Reynolds et al. (2000), the homogeneous eddy-axis tensor, ~j, is computed based on the homogeneous algebraic procedure, Equations 8 and 9 (note that the superscript "h" has been added in the current section). It is then partially projected onto planes parallel to the wall,

aij = I-IikHjla ,l,

I4ik .

1. (5ik.

Bik), .

. 02

1

(2

h Bkk)amnBnm ,

(19)

190 Tim-- 1

T/f<0 0<Tif
TiS > 1

I~l

~1 TiS 3rlf (1

1 1

I1

(1

T/f2) (1 + v/(a2

j

1

2 ,~1

89189

(1

1

E

TiS 1 3Tif -1

(Tif-l,Tifv/(a2 1+ b 2 ~

~)0

0.

1/3))

rlf)

Tim : 0 Tif _< x/~/4

b0

Zl

-1

1451 (2Tif)2 ~ 3/4

1/3)

~0 ((2Tif)2/93/4

_

]-'

(1 b3(1

a 2)

131)(Tif 1) a2)+(Tif 1)

~0

(2Tif) 2 0.342 3/4 +(1

1

TIT)2

l+blTifl(1

0.342)

(2n f)2 63/4

_

_

-1 ns > x/~/4

(1 +Xo)/3

Xo

i

m

Tim

(~*

~*

< 1 ,0(n,) + [r (n,) - r

> 1

1 +b4 (TIf x/~/4)Tifv/(a2 (1 a 2)

1/3)

X*

~0(n,) + [~ (n,) - ~0(n,)]n~m Z0(n,) + [Z~ (n,) - Z0(n,)]n~m

(r (T/f, a2) - 1/3) 1 / 3 + 1 +(Tim- 1 ) / ( 1 - a 2)

131(T/f, a 2) 1 - + ( T i m - 1 ) / ( 1 - a 2)

•1 (T/f, a 2) 1 + ( T i m - 1 ) / ( 1 - a 2)

Table 1' Structure scalars" auxiliary functions in Eqns. 16-18. b0 - 1.0, bl - 100, b2 = 0.8, b3 -- 1.0, 64 = 1.0, 1], =--Tim+[4/x/~+(Z-4/V/-3)qm]qf. where Hik is the partial-projection operator, and D 2 is such that the trace of aij remains unity. The blockage tensor Bij gives the strength and the direction of the projection. If the wall-normal direction is x2, then B22 is the sole nonzero component, and varies between 0 (no blocking) far enough from the wall, to 1 (full blocking) at the wall. Bij is computed by d~),i~,j

Bij-- ~----~,kCI) if

~,k~,k > 0.

(20)

If all gradients of 9 vanish, Bij is computed from an average over surrounding points. The blocking parameter, ~, is computed by an elliptic relaxation equation

L 2 ~2r Oxk~x------~k= ~'

U'c L--v -- 23,

(21)

with 9 -- 1 at solid boundaries, and ~,n - O~/OXn = 0 at open boundaries, where Xn is the direction normal to the boundary. To recover proper asymptotic behavior of the Reynolds stresses, r12 o,: O(x2) and r22 o,: O(x2), as the wall at x2 - 0 is approached, the homogeneous jetal, ~h, and helix, ~ , parameters are modified using - 1 + (~h _ 1) (1 - Bkk) 2 ,

7 = ~ (1 - Bkk).

(22)

A consequence of this approach is that realizability is automatically satisfied for rij, dij and J~j. Furthermore the fundamental constitutive equation (Eq. 4) relating the three tensors remains satisfied.

191 SCALE EQUATIONS The algebraic stress and structure model can in principle be used with any two-equation turbulence model (e.g. k-e) that produces the time scale x. The scalar equations chosen here are an extension, for inhomogeneous wall-bounded flows, of the model equations of Reynolds et al. (2002) (RLK02) for homogeneous flows. They proposed equations for the turbulent kinetic energy k and for the large-scale enstrophy, (02 based on the large-scale energy-bearing motion. This large-scale motion is responsible for large-scale mixing and ultimately sets the dissipation rate which happens at the small (dissipative) scales. Contrary to existing ad-hoc definitions of a second scale equation, the (02 equation has a fundamental background: it is based on the large-scale vorticity equation. Its terms represent large-scale processes, and their exact form provides valuable guidance when making choices for their closure. The model extension is built such that different terms approach a solid wall with proper asymptotic behavior, while returning the homogeneous form proposed in RLK02 when appropriate. Further discussion is omitted for space considerations, but details can be found in Langer and Reynolds (2003). The evolution equation for the turbulent kinetic energy is

Oko__+t

vjN

Ok =

_ _~gi -

O

+

V~jk-l-

,

(23)

where e, the dissipation rate of turbulent kinetic energy, has been modified from RLK02 for improved predictions,

e = FEk-~ + v(02,

Fe =

CE(3fudji).

(24)

The large-scale turbulent enstrophy equation is

~)~2/2

O~ 2/2 _ ~2jf}j a/if/

+ Vj axj -

~

a

- [Cc~ -(9rijdjkfki)Cc~

vSj~+ ~

(02T ~ - Q~ v-fi (02-

(25)

9

The length scale is evaluated from l2 -

12

1+

l+ct,

1+ (1 - ~ k k ) 2

Cl2

UxkUxkJ '

l~-

(26)

-- ~ "

The model constants for the scale equations were optimized as in RLK02, and for matching the channel DNS database of Moser et al. (1999). CE = 0.3,

CCo2T/Ce= 2.56,

Cll = 0.1,

Cl2

=

20,

C~p/Ce = 1.73,

o~k - 0.25,

~k = 1,

C~v = 5/6,

(27)

C ~ -- 1.45.

The algebraic equations for the turbulence structure respond to the mean deformation normalized with a turbulence time scale, x, defined as

The boundary conditions at a solid wall are

(Xn is the direction normal to the wall), Ok

k-0,

bx~

=0.

(29)

192

M O D E L RESULTS

Results shown here correspond to a family of pressure-driven fully-developed channel flows, in presence of spanwise frame rotation, aligned with the mean flow vorticity. The mean flow is given by Ui = {UCy), 0, 0} in a coordinate system xi = {x,y,z} where y is the wall normal direction (the sole direction of inhomogeneity), x is the streamwise direction with associated velocity U, and z is the spanwise direction. The frame rotation rate vector is given by f ~ f - {0, 0,f~f}. The wall-normal mean velocity necessarily vanishes by continuity for a fully developed channel flow with zero velocity at the walls. The solutions depend on two parameters; the friction Reynolds number, Re.c - u.ch/v, and the rotation number, Ro = ~ f 2 h / U b , where ~'~f is the magnitude of the frame rotation rate, h is the half height of the channel and Ub is the bulk velocity in the channel. For the fully developed rotating channel flow the friction velocity can be defined in terms of the streamwise pressure gradient, u2 = - h d ( P / p )/dx. Figure 1 corresponds to a set of fixed frame channel flows. Figure l(a) shows mean velocity profiles in wall coordinates (normalized by the wall shear stress and viscosity) for a series of friction Reynolds numbers. Comparisons are made with the DNS of Moser et al. (1999) at Re.c = { 180,395,590}. Two distinct log laws are also shown. The traditional one given by Coles and Hirst (1969), and the revised one given by Osterlund et al. (2000). Figure l(b) shows the turbulence intensities for Re.c - 590. The anisotropy predicted in the log region is a testament to the accuracy of the ASBM in this case. Figure 1(c) shows the structure-dimensionality components for this case. Of note is the dll component. It is the smaller component indicating structures preferentially aligned with the x-direction. Furthermore it shows a minimum near the wall, where near-wall streaks aligned with the flow direction have come to be expected. 30 I :__ DNS (Moseretal., i999) " " -/ l .......

l~

ASBM

In(y*)/041 +5

~ ..... 0~ 1

..~

~ S y mDNS,b Moser o l sei al.: (1999) ]

AsBM

.,-~1

(b)

1

(a)

1 00

Y

1000

. . . . . .

(c)

d~j 0.4

u' 10

1 |

0.8 [

0

200

+ 400

Y

600

0

.

,

100

.

200 +

Y

.

.

.

300

400

Figure 1" Fixed frame channel flow. (a) Mean velocity. (b) Turbulence intensities. (c) Normalized structure dimensionality. Figure 2 corresponds to a set of channel flows in a rotating frame, where ~ f is aligned with the mean flow vorticity. The ASBM is compared against the DNS of Alvelius (1999). Figure 2(a) shows mean velocity profiles normalized by the bulk velocity at Re.c = 360. With frame rotation, Ro = 0.22, the velocity profile becomes asymmetric about the centerline of the channel. In spanwise frame rotation, the Coriolis terms drop out of the mean flow equations making them insensitive to direct effects of frame rotation. The mean flow asymmetry is then a secondary effect due to the effect of the frame rotation on the Reynolds stresses and turbulence structure. In Figure 2(b) the ASBM turbulence intensities are compared to the DNS results. The ASBM captures the anisotropy of the Reynolds stresses and its dependence on frame rotation. Notice in particular the fact that the wall-normal intensity, v~, outgrows the streamwise intensity u~ in the core region of the channel. Figure 2(c) shows profiles of the structure dimensionality tensor. Comparing with Figure 1(c) it is clear that the dimensionality is little affected by the frame rotation. It does display an asymmetry, but this results directly from the asymmetry in the mean velocity gradient. There are no dramatic changes as in the Reynolds stresses.

193

1.5 1 0.5 0 0,

.

.

.

.

.

43 L

.

(b)

021 '

(c)

o.

O. dij 0.0.

'~

". . . . . .

33

/ ,/

p

,

,

.

(

,"), o

y/h

.

.

]

o

y/h

,

-o.

-1

-0.5

0

y/h

0.5

Figure 2: Spanwise-rotating channel flow at Rez = 360, Ro - 0.22. Symbols: DNS (Alvelius, 1999).

Other sets of scale equations can be used to complement the ASBM. Implementations of k-e and k-o3 models are widely available in CFD packages. These are not typically sensitive to rotation. However, when enhanced with the ASBM, the models become an attractive engineering solution. Kassinos et al. (2004) are evaluating different combinations, turning rotationally-challenged eddy-viscosity models into alternatives capable of computing flows with frame rotation. Figure 3 shows results obtained when standard k-co and V2F equations were coupled to the ASBM procedure (the ASBM stresses are used in the mean momentum equations, while the simpler eddy viscosity transport is kept in the transport equations for the turbulent scales). It is well known that linear eddy-viscosity models cannot capture the distribution of the energy intensities in channel flow. However, when coupled with the ASBM procedure, these models are able to predict quite satisfactorily the distribution of turbulence intensities for all rotation numbers.

CONCLUSIONS A new algebraic structure-based model has been presented as an attractive altemative for the engineering analysis of complex flows. Despite the simplicity of the fully developed channel flow, the results presented here demonstrate (i) the capability of the model to be integrated directly to the wall, (ii) the appropriate response of the model to situations where strong rotation is present, and where the turbulence is the sole responsible for the secondary effects observed, and (iii) the improvement of predictions of standard eddy-viscosity models, readily available in CFD packages, after coupling them with the ASBM procedure. In fact, there is an ongoing effort towards the model evaluation in more complex flows (see Kassinos et al., 2004).

ACKNOWLEDGMENT This work is dedicated to the memory of Professor William C. Reynolds. We also would like to acknowledge the support of the US AFOSR and of the European Commission under a Marie Curie grant.

References

Alvelius, K. (1999). Studies of turbulence and its modelling through large eddy and direct numerical simulation. PhD thesis, Department of Mechanics, KTH, Stockholm, Sweden. Coles, D. E. and Hirst, E. A. (1969). Memorandum on data selection. In Coles, D. E. and Hirst, E. A., editors, Proc. 1968 AFOSR-IFP-Stanford Conf., volume II, pages 47-54, Stanford, CA.

194 3

Re=395,

Re=180, Ro=0.77

Ro=0.

1.5

1 U' 0o1' ' " :0'.5

0

....

015 "

~I .

-!

2 1 -0.5

o

0.5

-0.5

0

0.5

z

U'V'

'~ 1.5

]"

DNS

o

1

. .

I

V2F+llsbm k-(~ .asbm

0.5

.............. -1

~

-.

0

.

1

-0.5

0

0.5

1

-0.05

v.q -0.5 0 6.5

Figure 3" Eddy-viscosity models coupled with the ASBM. Left: Non-rotating case, Symbols: DNS (Moser et al.), lines: models. Right: Spanwise rotating case. Symbols: DNS (Alvelius), lines: models. Haire, S. L. and Reynolds, W. C. (2003). Toward an affordable two-equation, structure-based turbulence model. Technical Report TF-84, Mech. Engng. Dept., Stanford Univ. Kassinos, S. C., Langer, C. A., Kalitzin, G., and Iaccarino, G. (2004). Application of a new algebraic structure-based turbulence model to complex flows. Proceedings of the 2004 Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ. Kassinos, S. C. and Reynolds, W. C. (1994). A structure-based model for the rapid distortion of homogeneous turbulence. Technical Report TF-61, Mech. Engng. Dept., Stanford Univ. Kassinos, S. C., Reynolds, W. C., and Rogers, M. M. (2001). One-point turbulence structure tensors. J. Fluid Mech., 428:213-248. Langer, C. A. and Reynolds, W. C. (2003). A new algebraic structure-based turbulence model for rotating wall-bounded flows. Technical Report TF-85, Mech. Engng. Dept., Stanford Univ. Moser, R. D., Kim, J., and Mansour, N. N. (1999). Direct numerical simulation of turbulent channel flow up to Rez = 590. Phys. Fluids, 11(4):943-945. Osterlund, J. M., Johansson, A. V., Nagib, H. M., and Hites, M. H. (2000). A note on the overlap region in turbulent boundary layers. Phys. Fluids, 12(1): 1--4. Pettersson Reif, B. A., Durbin, P. A., and Ooi, A. (1999). Modeling rotational effects in eddy-viscosity closures. Int. J. Heat Fluid Flow, 20:563-573. Reynolds, W. C. (1991). Towards a structure-based turbulence model. In Studies in Turbulence, Lumley 60th birthday Symposium, pages 76-80. Springer-Verlag, New York. Reynolds, W. C., Kassinos, S. C., Langer, C. A., and Haire, S. L. (2000). New directions in turbulence modeling. Presented at the Third lnt. Symp. on Turbulence, Heat, and Mass Transfer. Nagoya, Japan. Reynolds, W. C., Langer, C. A., and Kassinos, S. C. (2002). Structure and scales in turbulence modeling. Phys. Fluids, 14(7):2485-2492. Wallin, S. and Johansson, A. V. (2002). Modelling streamline curvature effects in explicit algebraic Reynolds stress turbulence models. Int. J. Heat Fluid Flow, 23:721-730.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

195

k - c M O D E L I N G OF T U R B U L E N C E IN P O R O U S M E D I A B A S E D O N A TWO-SCALE ANALYSIS E PINSON

1, O. GRI~GOIRE 1, O. SIMONIN 2

1 CEA Saclay, DEN/DM2S/SFME, 91191 GIF SUR YVETTE Cedex, France 2 IMFT, UMR CNRS/INP/UPS, All6e du Professeur Camille Soula, 31400 TOULOUSE, France

ABSTRACT In this paper, turbulent flows in porous media are considered. Following previous studies, we apply both statistical and spatial averages. The solid matrix action on turbulence is then put forward as a sub-filter production. To modelize this term, we perform a two-scale analysis that highlights energy transfers between the mean motion, the macroscopic and the sub-filter turbulent kinetic energies. Within this framework, we show that the sub-filter production is an energy transfer between the mean motion kinetic energy and the turbulent kinetic energy. We propose to modelize this sub-filter production using the wake dissipation and the work performed by the mean macroscopic flow against the mean specific drag. We also show that the work performed by the macroscopic fluctuating motion against the fluctuating drag is a supplementary energy transfer from macroscopic turbulence to sub-filter turbulence. It is added to the classical turbulent cascade. From this analysis, a macroscopic k - e model is then derived for a stratified porous media. A comparison between this model and fine scale simulations is carded out. KEYWORDS Porous media, multiscale analysis, wake dissipation, drag, k - e model, channel flow, volume average.

INTRODUCTION The macroscopic modeling of turbulent flows passing through porous media concerns many practical applications such as nuclear reactors, heat exchangers or canopy flows. In such flows, various study scales coexist. The challenge of the macroscopic modeling is not to reproduce the fine structure dynamics of the flow but to take into account information embedded in smaller scale for large scale modelization. With this aim, we choose to use two average operators: the statistical average that is practical for turbulence study and the spatial average, well adapted for the porous media approach. This latter operator choice generally depends on the geometrical characteristics of the media under study (Quintard and Whitaker (1994a), Quintard and Whitaker (1994b)). If macroscopic quantities length scales are large with respect to the filter size then the spatial average is assumed idempotent. The brackets (')z and the symbol ~ respectively denote a spatially averaged quantity and its deviation from this averaged value. The statistical

196

average and the fluctuation of some quantity ( are respectively denoted ( and ~'. In a strict mathematical way, both averages commute (Pedras and De Lemos (2001)). However, every modelization step, related to an average application, involves simplifications. Hence, the macroscopic turbulence modelization necessarily depends on the order of application of these two averages (Nield (2001), Travkin (2001)). Following Pedras and De Lemos (2001) and Nakayama and Kuwahara (1996), we choose first to apply the statistical average in order to get a structured view of turbulent flows and to benefit from the amount of knowledge available in the literature about RANS modeling. The spatial average is then applied. A practical way to highlight the solid matrix action on turbulence at a macroscopic scale is to study the double averaged balance equation of the turbulent kinetic energy (TKE) k for an incompressible fluid m

Dr

=-

__0

ax--:r

k)r

,---o10

7 ~.-~,,r162

+

~xj(9( ~,Or ]

r

5<),

dispersion of --

0~/,

- v

spatially filtered shear production

D 9~Dr

where r is the porosity, and is the macroscopic lagrangian differential operator with respect to the bulk flow velocity. The spatial filter use lets the dispersion term appear. This term describes, at a macro-scale, the convective transport of the studied quantity by the local velocity heterogeneities. It is generally modeled by an anisotropic diffusivity tensor (Whitaker (1967)). Using the spatial averaging theorem (Whitaker (1967)) and the filter idempotence, the spatially filtered shear production can be shared into two contributions _

o < ,

-

Ox, )' =

_

~

)~"

(2)

The first contribution is the production term induced by the macro-scale shear. The second term, that we call sub-filter production, is induced by the microscopic velocity gradients. Therefore, it is directly related to boundary layers at solid surfaces. In this paper, we focus on the modelization of this term. The analysis of eddies dynamic in porous media led some authors to take into account a supplementary transfer between scales (Finnigan (2000), Vu, Ashie, and Asaeda (2002)). Indeed, large eddies are partially or totally broken by the porous matrix to create small eddies. At macro-scale, this additional transfer is in competition with the turbulent cascade transfer at a macro-scale. In such a situation, it is difficult to derive a single-scale model accounting for both behaviors. Finnigan (2000) underlines that turbulence spectra in forest canopies exhibits a spectral cut-off. Hence a two-scale approach is a natural way to analyse such a turbulent flow if the filter length scale is in accordance with the cut-off frequency. Following this last remark, we develop in the first section a two-scale analysis in order to improve the understanding of energy transfers between scales in porous media. The balance equations for the turbulent kinetic energies and mean motion kinetic energies are introduced. It is shown that, according to the porous medium properties, various transfers compete. In the last section, we derive a macroscopic k - e model that is able to reproduce the evolution of the micro-scale turbulent quantities for the particular case of the turbulent channel flow. Encouraging results are shown and possible improvements are proposed.

F O R M A L T W O - S C A L E ANALYSIS

Introduction of the

two-scaleanalysisquantities

Both statistical and spatial averages allow to split up any physical quantity into a mean value and a fluctuation (for the statistical average) or a deviation (for the spatial average). According to the double

197

decomposition concept (Pedras and De Lemos (2001)), based on the formal mathematical commutativity of both average operators, one can write for the i th component of the velocity

u~ = (~)~ + (<), + ~

+ ~u',.

(3)

Various kinetic energies can then be built, depending on the scale under interest. The macro-scale and micro-scale TKE, respectively defined by I I / I (Ui)f(lti)y/2 , F" = ~..~..12 .

-~M

(4)

represent the energy which is contained in eddies of characteristic length respectively higher or smaller than the space filter. A cross-scale TKE may also be defined

~c = ( < ) ~ a < / 2

(5)

Let us emphasize that, due to the idempotence property of the spatial average, any macro-scale quantity is equal to its spatial averaged value. We obviously have (6)

(k)I - (kM)I ~- ( F ) I .

In the same way, one can introduce the macro-scale, micro-scale and cross-scale Reynolds tensors, ream ~e spectively denoted hereafter R~a, ~ and R~a R.j = ( u . ) , % ) , .

R.j

.-&~ = ~M

Some authors previously focused on the modelization of the macro-scale quantities ~M and R~ (Antohe and Lage (1997), Getachew, Minkowycz, and Lage (2000)). Lastly, two kinetic energies for the mean velocities are defined, say the macro-scale and micro-scale mean kinetic energies ~ M ___ (~.)y(~//)i/2 ' ~

___ 6 ~ 6 ~ / 2 .

(8)

Derivation of the (-~M),, (-~),, (-~M), and (-g"), balance equationsfor a constant porosity medium In order to highlight kinetic energy transfers, we derive balance equations for (kM)z, (k"~)z, (EU}z and (E"~)I. Nevertheless, for the sake of simplicity, the balance equation of the double averaged velocity is the sole shown balance equation of the four velocities in Eqn. 3 D(<)z _

D----~-

1 0(P)z

0

p Ox, t - ~

( 0(<),.~

0

0 (~< ~ ) z + Fr

(9)

u Oxj ] - - ~ x j ( ~ ' j ) ' - ~

where Fr is the i th component of the specific mean drag (N/kg) of the solid inclusions. The balance equation for {~M), is obtained by multiplying the {u{)z balance equation by {u{)I and by applying the statistical average. This equation can be rewritten to make appear classical terms such as

198

(uO(-kM).~)

turbulent and molecular diffusions, viscous dissipation and pressure-velocity correlation

O (k M u~)z

O

,

10(u~)~,(P')z

0

Dt turbulent diffusion of

turbulent diffusion of k c pressure-velocity correlation

kM

~"

C

molecular diffusion of (~M)f 0

(k~)' -R,j

- - M C')( ~ / / ) f

Oxj

-(-R'~ -b-~x~)'

~-(-#)' macro-scale viscous

0~'/

Oxj

Y

"-

-"-

dispersion of ~c

macro-scale shear

micro-scale shear

production

production

dissipation

(10) work performed by the fluctuant

Y

macro-scale velocity against F~

turbulent cascade through the cut off frequency

In (10), there are two production mechanisms: the shear production in the macro-scale flow and a part of the micro-scale production. Multiplying the 3u~ balance equation by ~u~ and applying both averages, we derive the (k'n)i balance equation

D(-k~)~

0

0

,,

~,

10(au:~SP')i

,,

turbulent diffusion of k m

,, dispersion of ~

,, p

Oxi

~,

0 { O(-k'~)i) ~ u Ox3

pressure-velocity correlation molecular diffusion of (~-m)f

~/~77,_\

0

macro-scale shear

micro-scale shear

production

production

micro-scale

"

viscous dissipation

(11)

$

.~

.,-

~,

Oxj

work performed by the fluctuant macro-scale velocity against F~,

~"

turbulent cascade through the cut off frequency

The classical turbulence transfer induced by the turbulent cascade is recovered. For clear flows, Reynolds, Langer, and Kassinos (2002) derived a similar term for the spectral transfer through the cut-off frequency. The work performed by the fluctuating macro-scale velocity against F~ is shown to act as a supplementary transfer between both TKE. This spectral transfer is specific to flows in porous media. Nevertheless, this formal work does not allow us to link the micro-scale production to the work performed by the macro-scale mean flow against the specific mean drag. This link is established by the the mean flow energy balance equations. These equations read

D(EM)I D-------i--=

-

10(P)i(~)i

p -

Ox, o(<),

+

0 (o/~/, / ~

\ o

Ox, /

-

u

0/
+ (Rio)z Ox~ - (~),~. (~5~ ~5~-j)z- (N)zFr

0Ox~ ,<<>, (12)

199

Figure 1: Schematic description of energy transfers between considered kinetic energies.

D<E'>I Dt

-

oa<,

o<<>,

o <~

aN>,+ <<>,~7,.

(13)

Two-scale analysis contribution The transfer and production terms have been formally derived in the previous section and they are summarized on Figure 1. Now, in order to simplify kinetic energy exchanges, we shall assume that the micro-scale and macro-scale turbulent motions are weakly correlated. Hence the cross-scale Reynolds tensor and TKE shall be neglected. The macro-scale production is then the sole energy source of (kM>i from the mean flow. Let us emphasize that the micro-scale production is the sum of two contributions -

o~<>

=

_-~

o<<>,

-
(14)

coming from both macro-scale and micro-scale mean flow energies. Assuming the separation of scales, we neglect the large eddy creation from this micro-scale production. Various scenarii of energy transfers from (EM)I to (k-~)i can be considered. For low laden media, the situation is almost the same as for clear flow. The shear of the mean macroscopic flow creates macroscale turbulence. This turbulence is for a part transfered to smaller eddies and then dissipated (turbulent cascade). The other part is transfered to (k"~)~ through the work performed by the fluctuant macroscale motion against the fluctuating specific drag (see Figure 1). For high laden media, energy transfer

200

AY 9 9

...........................................................

_e 2

"~""~ ) " ' " ~ Z

- 5e

Figure 2: Sketch of the considered plane channel configuration. follows an opposite way. The macro-scale mean flow quickly yields its energy to the micro-scale one. This transfer is due to the work performed by the mean macro-scale motion against the mean specific drag. Turbulence is then reduced to small structures, supplied by the micro-scale production contribution coming from (E'm)i. Figure 1 shows that the influence of the solid inclusions on turbulence is restricted to a sole energy transfer from large scales to small scales and this transfer is related to the drag. However other tranfers supply the micro-scale turbulence. The micro-scale energy production is then not equal to the work performed by the macroscopic flow against the mean specific drag. Assuming a nearly homogeneous macroscopic flow, Eqn. 13 reduces to -

-

0 ~

~ F r(<)~

- (t' 0~<0xjO~<),Oxj "

work performed by the mean macroscopic flow

wake dissipation (g~o)f

(~R,j -~zj) J. = Y sub-filter production

(15)

against the mean specific drag

In canopy turbulence models, some authors considered that the sub-filter production is conventionaly the work performed by the mean macroscopic flow against the drag force minus a sink term. This term is supposed to represent the accelerated cascade of TKE due to the plant foliage elements (Green (1992)). We derived formally this supplementary dissipation, hereafter called "wake dissipation" and denoted (gw)~. Let us remark that for a steady laminar flow the wake dissipation is strictly equal to the work performed by the mean macroscopic flow against the drag force such that the microscopic production is zero. In general, (gw)i depends on the sub-filter flow imbalance compared with its steady state. DERIVATION OF A k - e MODEL FOR A STRATIFIED POROUS MEDIUM In this section, we approach the stratified medium by a plane channel. The turbulent flow is homogeneous in the spanwise x direction (see Figure 2). No eddies larger than the clearance e between the two plates can thus exist. Hence, we simply have (k-~)i = (k)i.

Balance equation for (-~)r In the configuration under study, there is no macro-scale production. Therefore, (k~)i has a sole production mechanism, given by Eqn. 15. The (k-m)ibalance equation then reads

- Ox----~(kin u))z

p

Ox~

Ozj (~-m 5~JJ)z.

(16)

The specific drag force can be related to the friction velocity uf and to the channel width

Fr = 2u}/e.

(17)

201

Assuming that the boundary layer is close to equilibrium, we can use wall functions to approach the local velocity profile, for example the Reichards law (Laurence and Boyer (2002)) ~ ( Y w ) = uf g(y+) with y+ = uf Yw,

1

(18)

//

g(Y+) = 0 - ~ In (1 + 0.41y +) + 7.8

( 1 - e x p ( - y + / l l ) - Ti-exp(-y+/3))

,

where yw is the distance to the wall. Calculating the velocity gradient from Eqn.18 and using its expression in Eqn. 15, we get (-gw)~ = Cw(Rer) -~e with Rer = ~

and Cw(Re,-) = 4

~y+

dy + .

(19)

The function Cw is related to micro-scale velocity gradient (which highest values are concentrated in a thin region closed to the wall). The integrand in the Cw definition is then very low for large y+ values. Splitting the integral into two parts, we numerically deduce that for Rer > 50, Cw nearly reaches a constant value. The turbulent diffusion of ~-m and the pressure-velocity correlation are modeled by introducing the first gradient approximation

-Ox--7
p

Ox,

- Oxj

Ox, ] '

(20)

where ak is a constant (hereafter assumed equal to 1) and ut~, is modeled, as a first proposal, using spatially filtered quantities

l/t, --- Cl,<~'n)2/<Em)i.

(21)

We choose C~, equal to 0.09 in order to recover the classical k - ~ turbulent model in the clear flow limit. Finally, leaning on averaged microscopic results for turbulent channel flows, the dispersion contribution is neglected. Modeled balance equation for (~m >, The (~'~)i balance equation is derived by analogy with the (k-~)i balance equation and by introducing a characteristic time scale for each production/sink phenomenon. The modeled balance equation thus reads D(Y"~)I C'" Dt - rp

,-. <~>z

rt

<Em>z

Tw

<E,~>z+

~

u+

ae ]

Oxj ] '

(22)

where C~, C~w and C~2 are constants. Indeed, the drag is directly related to the velocity gradient at the wall and this gradient depends on the viscous sublayer growth. The representative thickness of the sublayer is approximated by y+ ~ 5 (Schlichting and Gersten (2000)). The characteristic velocity in this region is clearly the friction velocity u I. Hence, we propose rp ~ u/u} .

(23)

The wake dissipation depends on the velocity profile within the whole boundary layer. Since this region is dominated by turbulent transport, we propose to build the characteristic time scale as a function of a turbulent representative velocity (~-~>~/2 b'

% cx

"

(24)

202

Finally, in order to recover the clear flow limit, we choose

~-~ <~>:1<:>:.

(25)

=

The model constants should be calculated in order to recover the asymptotic state values. Turbulent flows in distinct geometries and for different Reynolds numbers correspond to different asymptotic turbulent values. This asymptotic state is unavoidably a data for the macroscopic model in order to fit its coefficients. The model thus must be based on results obtained from local numerical simulations or from measurements. For the asymptotic state, denoted by the supscript ~ , the (~-m): balance equation reads 2u~o~

u~

e (~7):- C~(Rer)'-2e

(go~):= 0 ,

(26)

For high Reynolds number flows with boundary layers close to equilibrium, we assume Cw to be constant (see Eqn. 19) and we deduce

m-2e u?=

Cw =
4

<~>:

(27)

uyo~

The C, 2 constant is classically chosen equal to 1.9. According to Eqn. 22 and Eqn. 26, the other constants are related to the asymptotic values and to C~2 C, p T-t~

I

Fr

(~-7):-

1 (g~):=O.

(28)

Two conditions on C~ and C,,, permit to satisfy Eqn. 28 and we elect the simplest one, for which both contributions in Eqn. 28 are zero. C,~ and C~,, are then independent from each other Csp -~- C s z T p • / T t •

,

Caw -- Ce2Tw=/Ttc~

9

(29)

Results The macroscopic model is compared with fine scale simulations carried out with the k - e model (Mohammadi and Pironneau (1994)) implemented in the CAST3M CFD code. Classical wall laws are used. Velocity, TKE and viscous dissipation profiles at the entrance of the channel are fiat except near walls where they follow laws prescribed by Laurence and Boyer (2002). These local results are then spatially filtered to get a macroscopic evolution of the turbulent quantities as reference results. Furthermore, the macroscopic turbulence model is implemented in a 1D code. Simulations are performed for a Reynolds number (Re = (~zz2e/u):) equal to 105.For the entrance values of the macroscopic turbulent quantities a first case with ko = 5 ( ~ ) : , eo = 2 5 ( ~ ) : and a second case with k0 = 1 0 ( ~ ) : , e0 = 1 0 0 ( ~ ) : are considered. Our aim is to reproduce the macroscopic evolution of spatially averaged turbulent quantities while referring the least possible to microscopic data. In further developments, a model to predict the equilibrium turbulent quantities values shall be derived. Moreover, we assume that the friction velocity is close to its asymptotic value, which implies that the sub-filter production (given by Eqn. 13) is constant. Figure 3 and 4 show that the modeled macroscopic TKE and viscous dissipation rapidly reach their asymptotic values. Nevertheless, the reference simulations exhibit an oscillating behaviour around these asymptotic values. By observing the reference evolution of the sub-filter production, we note that this quantity has also an oscillating behavior that is not recovered by our model. We deduced that the hypothesis u / = u/~ does not apply at the channel entrance (see Figure 5). The nearly homogeneous macroscopic flow assumption (Eqn. 13) is also certainly questionable near the entrance section. In order to efficiently approach the sub-filter production, we suggest to add a non-equilibrium factor .A to the wake dissipation. Since the friction velocity evolution should not be a data for the macroscopic model, we define

-~j > I<~o~>:.

(30)

203

In further developments, we will focus on the modelization of this term, which should take into account the friction velocity imbalance (Barrett and Keith Hollingsworth (2003)). 5

I

I

4.5

I

I

I

I

2

I

reference evolution

macroscopic model

~

1.8

4 IX v

1.6

83.5

'I

I

'

f'

.I

I

i

9 x

1.4

3

I ~ 2.5 2 1.5 1 0.5

i

reference sub-filter produc_~ion rezerence le)f_ modeled sub-filteruroduction m o d e l e d (~)!

1.2

0.8 I

0

i0

20

30

1

I

I,

I

40

50

60

70

x/e

x

~-

0,6

0

80

I

"

I

210

10

I

30

40

510

610

710

80

x/e

Figure 3: Evolution of the spatially filtered TKE (graph on the left) and balance between the sub-filter production and the spatially filtered viscous dissipation, ko = 5
i0

i

I

I

.

i

macroscopic model

8

6 ~" 5 ~'~ 4 3 2 1 0

2

i

reference evolutmn

9

9

I

1.s

I

I

modeled

1.6

v

I.

I

I

reference sub-filter produ, c_~ion . . . . r_eterence l e ) ~ modeled sub-filtervroduction

~u

I

9

x

.

1.4

1.2

0.8

0

I

10

I

20

310

410

510

610

710

0"6 0

80

110210310410:061071080

x/e

x/e

Figure 4: Evolution of the spatially filtered TKE (graph on the left) and balance between the sub-filter production and the spatially filtered viscous dissipation, ko = 1 0 ( ~ > z, ~o = 100(~>z, Re = 105.

1.25

. . . . . . . ' I koL, '-- 10~, ~ I~o> ~ . ',oo<~>,
1.2

,/~f

8

--

9

,/Lie ~

1.15 I.I

1.05

i II

I.

0.95

1,11

I

I0

II.Ii.ii.i ii. I I I I 9 I I 4 I ~ "

I

20

I

30

I

40

-- --

I

50

I

60

I

70

80

Figure 5: Evolution of the friction velocity. Comparison with its asymptotic value.

204

CONCLUSION A formal two-scale analysis has been carried out to put forward each transfer existing within the turbulent motion and the mean motion in a porous medium. The sub-filter production modelization is based on a quasi-equilibrium assumption in the micro-scale mean energy balance equation. This production is function of the wake dissipation and the work performed by the mean macroscopic motion against the mean specific drag. A three time scales k - e model is then derived for a stratified porous media. The wake dissipation and the drag can be formally derived in such a flow. Encouraging results for a given Reynolds number are obtained. The definition of a non-equilibrium factor and its study in further developments introduce possible macroscopic model improvements. References

Antohe, B. V., & Lage, J. L. (1997). A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Tran., 40, 3013-3024. Barrett, M. J., & Keith Hollingsworth, D. (2003). Correlating friction velocity in turbulent boundary layers subjected to freestream turbulence. AIAA J., 41(8), 1444-1451. Finnigan, J. (2000). Turbulence in plant canopies. Annu. Rev. Fluid Mech., 32, 519-571. Getachew, D., Minkowycz, W. J., & Lage, J. L. (2000). A modified form of the k - e model for turbulent flows of an incompressible fluid in porous media. Int. J. Heat Mass Tran., 43, 2909-2915. Green, S. R. (1992). Modelling turbulent air flow in a stand of widely-spaced trees. PHOENICS J. Comp. Fluid Dyn. and Applic., 5, 294-312. Laurence, D., & Boyer, V. (2002). A shape function approach for high- and low-reynolds near-wall turbulence model. Int. J. Num. Meth. in Fluids, 40, 241-251. Mohammadi, B., & Pironneau, O. (1994). Analysis of the k-epsilon turbulence model. Masson, John Wiley & Sons. Nakayama, A., & Kuwahara, E (1996). A macroscopic turbulence model for flow in a porous media. J. Fluid Eng.-T ASME, 121, 427-433. Nield, D. A. (2001). Alternative models of turbulence in a porous medium, and related matters. J. Fluid Eng.-T ASME, 123, 928-931. Pedras, M. H. J., & De Lemos, M. J. S. (2001). Macroscopic turbulence modeling for incompressible flow through undeformable porous media. Int. J. Heat Mass Tran., 44, 1081-1093. Quintard, M., & Whitaker, S. (1994a). Transport in ordered and disordered porous media II: generalized volume averaging. Transport Porous Med., 14, 179-206. Quintard, M., & Whitaker, S. (1994b). Transport in ordered and disordered porous media I: the cellular average and the use of weighting functions. Transport Porous Med., 14, 163-177. Reynolds, W. C., Langer, C. A., & Kassinos, S. C. (2002). Structure and scales in turbulence modeling. Phys. Fluids, 14(7), 2485-2492. Schlichting, H., & Gersten, K. (2000). Boundary layer theory (8th revised and enlarged edition ed.). Springer. Travkin, V. S. (2001). Discussion : "Alternative models of turbulence in a porous medium, and related matters (D.A. Nield, 2001, J. Fluid Eng.-T ASME 123, pp. 928-931)". J. Fluid Eng.-TASME, 123, 931-934. Vu, T. C., Ashie, Y., & Asaeda, T. (2002). A k - e turbulence closure for the atmospheric boundary layer including urban canopy. Bound.-Layer Meteorol., 102, 459-490. Whitaker, S. (1967). Diffusion and dispersion in porous media. AIChE, 13(3), 420--427. Whitaker, S. (1999). Theory and applications of transport in porous media : the method of volume averaging. Kluwer Academic Publishers.

3. Direct and Large-Eddy Simulations

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

207

EFFECT OF A 2-D R O U G H W A L L ON THE A N I S O T R O P Y OF A TURBULENT CHANNEL FLOW L. Djenidi l, S. Leonardi 2, P. Orlandi 2, and R.A. Antonia 1 1 Discipline of Mechanical Engineering, University of Newcastle, NSW 2308, Australia 2 Dipartimento di Meccanica e Aeronautica, Universita Degli Studi di Roma "La Spienza", 00184 Rome, Italy

ABSTRACT

The effects of 2-D roughness elements on the Reynolds stress anisotropy tensor, b U (= uiuj 1 6 q --3 ~, where uiu j is the Reynolds stress tensor, q - uiu i and 6/j the Kronecker delta symbol) in a turbulent channel flow are investigated using data obtained from direct numerical simulations (DNSs). The roughness elements consist of transverse square rods with a size k and are placed on one wall of the channel only. While k is kept constant (k/h = 0.2, h is the half-width of the channel), the spacing, w, between the rods is varied from w/k = 1 to w / k - 39. The results show that the 2-D roughness dramatically changes the structure of the wall region flow, and that the changes are controlled by w/k. In terms of turbulence modelling, they indicate that the modelling of the near-wall region needs to reflect both the structural changes and the influence of w/k on these changes. Possible forms of models are discussed.

KEYWORDS Turbulent channel flow, rough wall, Reynolds stress tensor anisotropy, turbulence modelling

INTRODUCTION Rough wall flows are important types of flows, which present a challenge to both engineering applications and fundamental research. In the former case, an understanding of the physical phenomena involved in rough wall flows would be of fundamental interest to momentum and heat transfer problems. The study of different rough surface geometries are attractive since it allows to the response of the flow to various surface conditions to be investigated ; The overall objective would be

208

to gain some insight into the near-wall turbulence production mechanism(s). Such information is of paramount importance for the modelling of near-wall turbulence over rough walls. A relatively growing amount of work on rough walls (see Jimenez, 2004, for a detailed review) is appearing in the literature. While these studies are interesting, the results presented are not readily exploitable from a turbulence modelling point of view. The present paper reports DNS data for a twodimensional rough wall turbulent channel flow. The main aim is to determine how this particular roughness geometry affects the Reynolds stress anisotropy with the the longer term view of improving the development of turbulence models for rough walls.

NUMERICAL DETAILS A DNS of the incompressible non-dimensional Navier-Stokes and continuity equations is performed on a Cartesian computational domain. The equations have been discretized using a staggered central second-order finite difference approximation. Here, we recall only the main features since details of the numerical method can be found in Orlandi (1999). The discretized system is advanced in time using a fractional step method with viscous terms treated implicitly and convective terms explicitly. At each time step, the momentum equations are advanced with the pressure at the previous step, yielding an intermediate non-solenoidal velocity field. A scalar quantity r projects the non-solenoidal field onto a solenoidal velocity one. A hybrid low-storage third-order Runge-Kutta scheme is used to advance the equations in time. The roughness is treated by the immersed boundary technique described in detail by Fadlun et al. (2000). It consists of imposing a zero velocity on the body surface, which does not necessarily coincide with the grid points.

2h

,~ w

k

Ik

Figure 1. Configurationof the channel flow simulation

The flow configuration (a fully developed turbulent channel flow with square bars at the bottom wall) is shown in Figure 1. This configuration (one rough wall and one smooth wall) is chosen to remove the symmetry effect when both surfaces are identical and for allowing a comparison between smooth and rough walls. Several values of w/k have been investigated, with k = 0.2h, but only few are shown here (1, 3, 7, 19 and 39). Periodic boundary conditions apply in the streamwise (x) and spanwise (z) directions, and there is a no-slip condition at the wall. The computational domain is 8h x 2h x nh in the x, y and z directions (y is the direction normal to the wall), respectively. The Reynolds number is Re 4200. It was verified that the results are not affected by blockage and the convergence of the calculations is satisfactory. For more details on the conditions of the simulation, see Leonardi et al. (2003).

209

RESULTS AND DISCUSSION Since from a modelling point o f view, the p r i m a r y interest is in the averaged quantities, the Reynolds stresses have been time and spaced averaged; the space averaging was done in the spanwise direction and over one w a v e l e n g t h A, ( 2 = w + k) in the longitudinal direction.

0.6

0

.

.

.

.

.

.

.

.

.

.

~,1 "-~----- - . . . . r'd /I

.

.

.

.

.

.

.

~-~-'~~- -~ ~_~_~_--2_'57J-

',},'31

-0.:3

b/1 -0.6

-1.0

-0.5

0

0.5

1.0

y/h

0.50

;I

I

i ,1 ',

0.25

b22

i i

'

3

\ -0.25

-0.50

.

-1.0

.

.

.

.

.

.

.

.

.

.

.

-0.5

.

0

.

.

0.5

y/h

Figure 2. Components of the bUtensor for several wave length 2 = w/k (1, 3, 7, 19 and 39). The number on the curve indicates the case ofw/k.

1.0

210

0.30

.

.

.

.

.

.

.

.

b33 0.15

_o.~o

/ \-~i, -1.0

-0.5

0

0,5

1.0

y/h

b12

0.1

l~ ~: ~ !

"t,,,'~

"\ "\

"','~

t

-0.1

-1.0

-0.5

0

0.5

1.0

y/h

Figure 2. (Continued) Components of the bo.tensor for several wave length 2 = w/k (1, 3, 7, 19 and 39). The number on the curve indicates the case of w/k. Figure 2 shows the four (non-zero) components of bij. (bll, b22, b33 and b12) for w/k - 1, 3, 7, 19 and 39 across the channel (the region -1.2 < y/h < -1 represents the rough wall- the bottom wall-, and y/h = 1 is the smooth wall -top wall). Clearly, the rough wall dramatically alters the distributions from those of the smooth wall with the biggest changes occurring within the height of roughness elements. Notice also that the roughness shifts the "symmetrical point" toward the smooth wall (the shift being minimum for w/k = 1 and maximum for w / k - 7); this is clearly visible on the b12 distributions. On the the rough wall side, one can distinguish roughly two main regions, 0 < y < k and y > k. However, the distinction becomes less clear as w/k increases.

211

0.20 1 st point a b o v e the baseline of the roughr

0.15 []

[]

[]

[]

0 0

.

0.10

o

l

o o2s tl

o oso

[] 0

39 19 7 3 1

0.05

0

0.02

0.04

III

Figure 3. Anisotropy invariant map for the Reynolds stress tensor for several values of the wavelength/1, = w/k (1, 3, 7,19 and 39) within the region -1.2 y/h. < -1" The inset is the AIM for the entire channel)

The geometry w/k = 7 appears to be a limiting case, beyond which this distinction is quite weak. It is interesting to point out that w/k = 7 is the geometry for which the form drag is maximum and the skin frictional drag minimum (Leonardi et al., 2003). Thus, w/k = 7 would appear to be a critical value for this type of 2-D roughness geometry. This is well highlighted in the anisotropic invariant map (AIM, Lumley and Newman, 1977) in Figure 3, where the data are shown for 0 __y _< k (the inset shows the AIM across the entire channel.) For 0 __y <_k, the AIM shows strong variations. Since the spanwise direction is a principal axis for the Reynolds stress tensor, the changes in sign of III reflect significant variations in the u32 (the subscript 1, 2, and 3 represent the longitudinal; normal-to-the-wall and spanwise directions), which in turn indicate structural changes in the near-waU region of the flow. Quite remarkably, when w = 7k, the data follow the asymmetric limit for which u32 is larger than u1 andu22 As w/k increases beyond 7, the data start to return to the smooth wall signature; however, the process is quite slow since even for w/k = 39, the recovery is not complete. Clearly, the region 0 []yDk for w/k < 7 presents too formidable a challenge for modelling. However, since the data have been space averaged in the spanwise direction over one wavelength, the observed behaviour of the data in the region 0 _
212

0.75

0.50

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i

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.

0

.

I , ' ,,':" '

~?./...

/',,\\

~/

0.25 r~,,,, \

0

/'

.

.

.

.

0.5

.

.

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.

1.0

.

.

.

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.

.

.

1.5

.

.

.

.

.

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2.5

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197"~- --~___ ~

-0.1

.~

.

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.

__i

.

.

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.

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__._ ~__.,.,

.

.

.

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.

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-0.2

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0

. . . . . . . . . . . . . . . . . . . . . 0.5 1.0 1.5 2.0 (Y-Ymax)lh

0 3,

,

- ~ ....~.., - --.---

~

-~ 1,,/.,/

\,, \, ,,

\i!,

W2o

0

0.5

1.0

1.5

2.0

(Y'Ymax)/h

Figure 4. Components b, for several values of 2 = w/k (1, 3, 7, 19 and 39), shifted by Ymax.The number on the curve indicates the value of w/k.

2.5

213

0.2

0.1

,

'19'

'

"

\",,

\

"

x

x\

zl

-0.1 "-~_

-0.2

. . . .

0

,

0.5

.

.

.

.

.

.

.

.

.

1.0

,

1.5

i . . . .

,

. . . .

2.0

2.5

(Y'Ymax)/h

Figure 4 (Continue). Componentsb!2 for severalvalues of 2 = w/k (1, 3, 7, 19 and 39)m shifted byy.... The number on the curve indicates the value of w/k.

turbulent kinetic energy dissipation rate equations (Finnigan, 2000), which need to be modelled. The relatively complex behaviours of the b# distributions as w/k varies reflect the contributions of these terms. Thus, in order to develop a turbulence model, which faithfully mimics the roughness effects on the flow, one is required to evaluate the contributions of these extra terms in all the equations used for the computation. This task is not easy and is certainly not practical for more complex roughness geometries (3D rough wall for examples). In addition, the problem is compounded by the difficulty of determing the location of the equivalent smooth wall, which varies with the roughness geometry. In that respect, one may need to resort to a simpler method. Classically, this is done by carrying out the calculation down to a location above a roughness located in the log region; empirical results are then used as boundary conditions. It is tempting to extend this approach here. For instance, by taking the point where b ll is maximum, as the location at wich to apply the boundary contion, an interesting behaviour emerges (Figure 4, the data below y/h = 0 have been omitted). The b o. have been y-shifted by an amount equal to Ymax/h corresponding to the local positive peak in b ss on the rough wall side. While this shift is somewhat artificial, it can help discern a trend of interest from a turbulence modelling point of view. One can notice that for (y-Ymax)/h > 0, all the b Udistributions present a relatively simpler variation with w/k than when the complete distributions are considered. Note though that the bs2 distributions evolve in an opposite manner compared to the variations of the normal Reynolds stress distributions. Relative to the smooth wall distributions, the level of the magnitude of the rough wall distributions drop for w/k = 1, increases for 3 and 7 then returns progressively to that of the smooth wall; bs2 first increases then decreases. The behaviour suggests a non-negligible correlation between the streamwise and normal-to-the-wall velocity fluctuations for w/k = 3 and 7. This would imply that, for 3 < w/k < 7, the flow on the rough wall side is more anisotropic than on the smooth wall side. One may argue that for these cases the interaction between the elements and the flow is strongest than for the other cases, which is likely to reflect important fluid ejections from the fluid between the elements into the overlaying flow. This may not be too surprising since it is for these values of w/k that the form drag is most important (Leonardi, 2003). Taking the above consideration, it may be enough, to a first approximation, to modify current modelling schemes for smooth walls such that they can be used for rough walls. Since the present

214

study focuses on the b O. terms, a discussion on the turbulence modelling modifications using the present results may be appropriate. The b/j components enter the modelling process via the return-toisotropy term in the Reynolds stress transport equations. For simplicity sake, the simplest model, that of Rotta, is used here: (1)

FI o. = -CIcbo. ,

where c is the mean turbulent kinetic energy dissipation rate and C1 a constant. It is clear from the figures that the rough wall affects the flow differently than the smooth wall. In particular, the (normalto-the-wall) blocking effect of the wall on the Reynolds stresses is altered; this alteration depending, at least in the present case, on 2. One may simply modify Rotta's model in a manner similar to that adopted in low Reynolds number second-order closures. Classically, such modelling is as follows: s n~ = n,j +n,j,

(2)

where I-IS/j is a corrective term "activated" in the near-wall region (see for example Launder et a l . , 1975; Launder and Shima, 1989; Shima, 1988). Following such an approach, one may form a rough wall flow model as follows:

(3)

1-I~ = FI U + FI 0

The term 1-IRushould account for the rough wall effects; in other words, it would represent the extra terms discussed above. Thus, considering the comments made with regard to Figure 4, I-IRumay; to a first approximation, have a form similar to FlS/j with, of course, different coefficients that would reflect the rough wall effects on bij observed in Figure 4. Also, FlSu may be developed so as to recover the form of I-lSu when the rough wall is replaced by a smooth one. However, the simple addition of a term may not be the best way to model the near-wall behaviour. It would be perhaps better to used a matching expression where I-I*/j= I-IRij near the wall and FI*ij = FI/j in the region where the roughness effects are the least felt. Such a model can have the following form: YI~ = (1- f ) F I o + f I I ~

,

(4)

where f is a "matching" function equal to 1 in the region where the rough wall effects are felt most strongly, and 0 elsewhere. The expression for I-IR09could be a function of the invariants II and III, the turbulence Reynolds number and the roughness parameters, such as the density and size, or 2. The dependence of 1-IS/jon A can be supported by the data in Figure 5 showing the value of bll at (y-ymax)/h = 0 as a function of 2. Expression (4) is similar to the low Reynolds number form of the model for eg (Hanjalic and Launder, 1976; Antonia et al, 1994). It should be clear that the expressions of FIRg in (3) and (4) may not be the same. Further DNS data analysis should be carried out to develop an adequate expression for I-IRij. A word of caution needs to be given regarding expressions (3) and (4). Firstly, the forms are only here to illustrate two possibilities where the genetic expression for Fig is linear (expression (1)). One may for example use non-linear expression such as that of Lumley and Newman (1977):

-$8o.) ;

(5)

215

0.6

0.4 x

0.2

0

10

20

30

40

w/k

Figure 5. Distribution of bllmax for several values of 2. = w/k (1, 3, 7, 19 and 39).

a and b can be either constans or functions oflI and III. Secondly, expressions (3) and (4) are based on the discussion of the data in Figure 4.

CONCLUSIONS The DNS results show that the 2-D roughness dramatically changes the structure of the wall region flow, and that the changes are controlled by w/k. As a result, the anisotropy of the Reynolds stresses is strongly altered, at least in the near-wall region. In terms of turbulence modelling, they indicate that the modelling of the near-wall region needs to reflect both the structural changes and the influence of w/k on these changes. Possible forms of models for the retum-to-isotropy are discussed. They are both based on existing models and use the same approach as that used for low Reynolds number turbulence modelling. However, further DNS data analysis is required to complete the present study in order to provide final expressions for these models. In particular, the DNS data should be used to extract information on the extra terms in the equations of motion resulting from a space averaging process.

ACKNOWLEDGEMENTS L. Djenidi and R. Antonia acknowledge the support of the Australian Research Council.

216

REFERENCES

Antonia, R.S., Djenidi, L. and Spalart, P.R, (1994). Anisotropy of the dissipation tensor in a turbulent boundary layer, Phys. Fluids, 6, 2475-2479. Fadlum, E.A., Verzicco, R., Orlandi, P. and Mohd-Yusof, J., (2000). Combined immersed boundary finite-difference methods for three-dimensional complex flow simulation. J. comput. Phys, 161, 35-60. Finnigan, J. (2000). Turbulence in plant canopies, Ann. Rev. of Fluid Mech., 32, 519-571. Hanjalic, K. and Launder, B.E., (1976). Contribution toward a Reynolds stress closure for low Reynolds number turbulence. J. Fluid Mech., 74, 593-610 Jim6nez, J. (2004). Turbulent flows over rough walls. Ann. Rev. of Fluid Mech., 36, 173-197. Launder, B.E., Reece, G.J. and Rodi, W. (1975). Progress in the development of a Reynolds stress turbulence closure. J. Fluid Mech., 68, 537-566. Launder, B.E. and Shima, N. (1989). Second moment closure for the near-wall sublayer: Development and application. AIAA Jnl., 27, 1319-1325. Leonardi, S., Orlandi, P., Smalley, R.J., Djenidi, L. and Antonia, R.A., (2003). Direct numerical simulations of turbulent channel flow with square bars on the wall, J. Fluid Mech., 491,229-238. Lumley, J.L. and Newman, G. (1977). The retum to isotropy of homogeneous turbulence, J. Fluid Mech., 82, 161-178. Orlandi, P. (1999). Fluid Flow Phenomena, A Numerical Tool Kit, Kluwer. Shima, N. (1988). A Reynolds stress model for near-wall and low Reynolds number regions. Jnl. Fluids Eng. ASME, 110, 38-44.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

217

DIRECT NUMERICAL SIMULATION OF ROTATING TURBULENT FLOWS THROUGH CONCENTRIC ANNULI

M. Okamoto and N. Shima

Department of Mechanical Engineering, Shizuoka University, Johoku 3-5-1, Hamamatsu, 432-8561, Japan

ABSTRACT Rotating turbulent flows through concentric annuli with several radius ratios and rotation numbers were systematically investigated by means of direct numerical simulation (DNS). In the outer-wall rotation, the flow is stabilized as expected from an existing DNS of an axially rotating pipe flow without an inner rod. The profiles of the mean azimuthal velocity and Reynolds shear stress in the large radius ratio case are different from those in the small radius ratio case. In the inner-wall rotation, the flow is unstable near the inner rod and the mean quantities are strongly dependent on the radius ratio and rotation number. The instantaneous fluctuating structures are distributed along the inner-wall motion and become helical.

KEYWORDS Direct numerical simulation, Concentric annuli, Outer-wall rotation, Inner-wall rotation, Turbulent flow, Helical structure.

INTRODUCTION Turbulent flow through concentric annuli is important in various engineering applications and is investigated by many researchers. Rehme (1974) studied the concentric annular flow experimentally. From the viewpoints of numerical investigation, Satake & Kawamura (1993) performed the large eddy

218

simulation of the flow and Chung, Rhee & Sung (2001) and the authors (2002) simulated the flow by the direct numerical simulation (DNS). In this flow, due to the curvature effect the distributions of mean quantities are asymmetric unlike symmetric ones of plane channel flow. In the present paper, we study the concentric annular flow with the inner-wall and outer-wall rotations. The wall-rotation effects induce the azimuthal motion and have similarities with three-dimensional boundary layers. In the laminar flow, the outer-wall rotation produces an approximately forced vortex and the inner-wall one produces an approximately free vortex. However, the effects of both wall-rotations in the turbulent flow are complicated. In this work, we systematically perform DNS of the rotating turbulent flows through concentric annuli with various radius ratios and rotation numbers and investigate the mean quantities including the high-order statistics and instantaneous fields.

N U M E R I C A L P R O C E D U R E AND F U N D A M E N T A L EQUATION The flow configuration and cylindrical coordinate system are given in Figure 1. The numerical scheme of the present DNS is the second-order center difference and interpolation, and the pressure Poisson equation is solved by the direct method with the fast Fourier transformation. The time advancement is done by the second-order Adams-Bashforth method. The computational domain size is 14 6 • 6 •

in

the axial, radial and azimuthal directions, respectively. The variables are nondimensionalized by the radial half width ~ and the axial global friction velocity uTg defined by u~g - - ~ [ - & t P / d x

and the

corresponding intrinsic dimensionless parameters in the present flow are the Reynolds number

8U~g Re - - ~ , v

(1)

the rotation number by the rotating wall velocity W0 N = W~ ,

(2)

UTg and the radius ratios

a=

Rin Rout

(3)

In the present DNS, the fixed Re is set to 150, N is 0, 5 and I0 and a is 0.05, 0.I, 0.2 and 0.5. The computational mesh is listed in TABLE I. In order to resolve all essential scales of the turbulence motion, the axial grid for the inner-wall rotation case and azimuthal one become sufficiently fine. The axial grid-resolution A x + near the inner wall, which is normalized by the local friction velocity, is 6.2 ~ 26, while A x + near the outer wall is 3.9 ~ 16. The radial one Ar + near the inner wall is 0.39 ~ 0.70, while that near the outer wall is about 0.36. The azimuthal

one

(ginAO) + is 1.0 ~ 9.0 and (RoutAO) § is

14.1 ~ 18.1. The governing Navier-Stokes equations with the incompressible condition in the cylindrical coordinate system are expressed as

219

Ou

Ou

Ou 1 0 u

Op

02 u V ~r ( Ou)

~ = - u ~ - v - - - - w ~ - ~ + v ~ + Ot Ox 3r r O0 Ox Ox2 Ov

Ov

Ov

w2

10v

019

r-Or

02V

+

V 02 u

(4)

r 2 O0 2

V O2V V..~.V

V O(rOV ~

n

Ot

=-u--

-v--

Ox

Ow

Or

Ow "-

- - U ~

Ot

10rv + - ~

Ox

r

O0

Ow - - V - -

Ox

Ou

- - w ~

1 --

Or

--W

r

+ ~ -

~

r

Or

Ow

vw

O0

r

. . . . . .

+ V ~

Ox 2

10p

r Or

Or

+

r2 0

02

r2

2 v Ow

r 2 00

,

0 2W + - -V .~F( r O__~r) + V O2W - VW + 2 V OV

+

r 00

+--~

v-~

r

~-2-0~ 7"

(5)

(6)

"~"~'

10w + - ~

r Or

= 0.

(7)

r O0

The velocity components u, v, w are the axial, radial and azimuthal velocities, respectively. In the fullydeveloped turbulent flow, the mean quantities are stationary and homogeneous in the axial and azimuthal directions. The equations of the mean velocity fields are written by - -

+---

dx 1

r v

-

r Or

,

(8)

Or

0

~

Here, u'v' and v'w' are the shear components of the Reynolds stress. In the case of the laminar flow vanishing the Reynolds stress in (8) and (9), the solution of U is U = R~2t dP

4V-~

{( )2 r

G

l- a

-1+

log a

2 t r

log

~o,t

,

(lo)

and those of W are W_

Wo

(

1-a 2

r

_a 2R

Rout

t t '

(11)

in the outer-wall rotation and 1- a 2

r

Rout ,

(12)

in the inner-wall rotation. The velocities U and W in the laminar rotating turbulent flow through concentric annuli are independent of each other. When a is small, the first linear term in Eqn. 11 is dominant and the outer-wall rotation in the laminar flow produces an approximately forced vortex. On the other hand, the laminar solution in the inner-wall rotation (12) almost represents a free vortex near the inner wall.

RESULTS AND DISCUSSION Outer- Wall R o t a t i o n

The mean velocities are shown in Figure 2. The axial velocity increases and the asymmetry is enhanced as N increases. The fact suggests that owing to the outer-wall rotation the flow becomes

220

stable as expected from the DNS of an axially rotating pipe flow without an inner rod by OrlandiFatica (1997). The azimuthal velocity profiles of the small radius ratio differ from the linear shape (11) in the laminar flow and are concave shapes. However, the profiles for ct=0.5 become close to that in the turbulent Couette flow including the strong mean velocity-gradient near the walls. There are no differences between the W / W0 profiles of N=5 and 10. We give the wall frictions ( ~'x and z o ) at the outer wall, wall-friction angles (0 =arctan(v 0 /~'x)) and bulk velocity in Table 2. The axial wall friction "rx decreases slightly by increasing N, while the azimuthal one 7:o is not sensitive to N and is almost 0 except ct=0.5. The decreasing behavior of v x agrees with that of turbulent flow in a rotating pipe. The outer wall-friction angle is smaller than the inner one. In the axial mean velocity profiles of Figure 1(a), the bulk velocity increases as N becomes larger. The Reynolds shear stress profiles are given in Figure 3. The positive peak of u'v' becomes smaller by the outer-wall rotation and this tendency is independent of ct. The negative peak of v'w' near the inner wall increases by increasing ct and there is a reverse of sign between the results of the small radius ratio and those of the large one. In Figure 3(c) of w'u', the peak near the inner wall is negative at the small radius ratio, while that is positive at ct=0.5. The Reynolds normal stresses are shown in Figure 4. Near the outer wall y>l.5, u'u' and w'w' increase and v'v' decreases. However, the normal stresses near the inner wall increase as N becomes larger. In Figure 5, we show the production term of v'w' as

__0(w)

(13)

Pvw =-v'v'rort, 7 , 4

in N=10 near the inner wall. The budget components are normalized by uz / v. Due to the increment of

v'v' and azimuthal mean velocity-gradient, the negative production grows as the increase of t~. In particular, Pvw of the small radius ratio is almost equal to 0 under the buffer region. The negative increment of Pvw causes the reverse of v'w' between the small and large radius ratios in Figure 3(b). The production and pressure strain terms of the normal stresses near the outer wall are given in Figure 6. The production terms are ,0U

P.u = -2u'v ~ , Or

) P~ = O, Pww = - 2 v ' w ' r O [ W Or ~,--7" '

(14)

and the pressure strain ones are .......

Ou' 2 p, Orv' 2 p, Ow' ~uu-2P'"~z, ~w--r -'~'r' ~wW=-r 00 "

(15)

The contribution of Pww vanishes and the pressure strain terms have influence on the enhancement of w'w' and diminution of v--~v'. In the two-point correlations of Figure 7 and the fluctuating velocity contours in the (x, 0) plane of Figure 8, there are not large differences between the rotation and non-rotation cases. This indicates that the turbulence structure near the outer wall is insensitive to the outer-wall rotation. The profiles of the ,

,

3

helicity ( H - u ~ o i ) near the outer wall normalized by ur / v are shown in Figure 9. In the buffer region the positive peak is observed and H becomes negative in the center region.

221 Inner-Wall Rotation Next, the results of the inner-wall rotating flow are shown in this subsection. In the Figure 10, we plot the mean velocities. The axial mean velocity decreases as N increases and in particular U for the small radius ratio varies near the inner wall. In the azimuthal mean velocity of Figure 10(b), the strong gradient exists near the inner wall and in the case of the large radius ratio the weak gradient appears near the outer wall. The normalized azimuthal velocity W / W0 depends on the rotation number unlike that of the outer-wall rotation in Figure 2(b). In the case of the small rotation number and radius ratio

(N=5 and ct=0.05, 0.1), the negative azimuthal velocity appears near the inner wall. The formal solution of the azimuthal mean velocity (9) is expressed as W= l _ ar 2

{(Ri 2 2)(R~n. n+0f ~r n d r -v'w') ( 2 v'w'} . "TT-ot Vr - 1 - Ri-----cn~fR~ r2 ) r Vr

(16)

When v'w' is a positive quantity as can be seen lately, the first term is positive and the second one is negative. Due to the contribution of the second term, the azimuthal mean velocity becomes negative. We give the wall frictions at the inner wall, wall-friction angles and bulk velocity in Table 3. The wall frictions monotonously increase as N becomes larger and the increase ratio is large for the small radius ratio. The result of the wall-friction angle indicates that the inner-wall rotation for the small radius ratio has no influence on the flow field near the outer wall and that the inner-wall rotation varies the friction angle significantly in comparison with the outer-wall rotation. As can be seen in the axial mean velocity profiles of Figure 10(a), the bulk velocity decreases as N increases. These facts indicate that owing to the inner-wall rotation the flow becomes unstable. The Reynolds shear stresses near the inner wall enhance as N increases in Figure 11. In particular, the growth ratio of the negative peak of u'v' with respect to N is very large in small or. Figure 12 shows the Reynolds normal stress profiles. Due to the strong production of w'w' (14) constituted by the large azimuthal mean velocity-gradient and shear stress v'w' in contrast with that in the outer-wall rotation, the enhancement of w'w' is observed near the inner wall. The high-order statistics of ct=0.1 are shown in Figure 13. In Figure 13(a), the skewness result of N=0 agrees with that of Chung, Rhee & Sung (2001) and the skewness in large y§ increases by increasing N. In the viscous layer, the skewness is negative for the rotation effect. The flatness near the inner wall drops as N increases. The negative peak of the helicity appears in the viscous layer and the positive one arises in the buffer layer. This behavior of the helicity is different from that in the outer-wall rotation case. The two-point correlations near the inner wall are given in Figure 14. Due to the rotation effect, the correlation distance in the axial direction is shortened. The rotation effect does not vary the profiles of the azimuthal two-point correlation. Figure 15 displays the instantaneous distribution of the axial fluctuating velocity in the (x, 0) plane near the inner wall. The fluctuating streaky structures are distributed along the inner-wall motion and become fine and helical in accordance with the axial twopoint correlations in Figure 14.

222

CONCLUSION In this study, we have performed the DNS for rotating turbulent flows through concentric annuli. Stabilization takes place in the outer-wall rotating flow. The azimuthal mean velocity normalized by the outer-wall velocity strongly depends on the radius ratio. The rotation effect varies the profiles of the Reynolds stress. However, the fluctuating structures do not change in contrast to those in the innerwall rotation. On the other hand, flow destabilization is observed in the inner-wall rotation and the azimuthal mean velocity is dependent upon the rotation number and radius ratio. The Reynolds stress increases near the inner wall as the rotation number augments. The helical fluctuating structures arise near the inner wall.

REFERENCES Chung S.Y. Rhee G.H. and Sung H.J. (2001). Direct numerical simulation of turbulent concentric annular pipe flow. Proc. 2nd Int. Symp. on Turbulence and Shear Flow Phenomena 3, 377-382. Okamoto M. and Shima N. (2002). Direct Numerical and Large Eddy Simulations of Turbulent Flows Through Concentric Annuli. Proc. 5th Int. Symp. on Engineering Turbulence Modelling and Measurements, 219-228. Orlandi E and Fatica M. (1997). Direct simulations of a turbulent pipe rotating along the axis. J. Fluid Mech. 343, 43-72. Rehme K. (1974). Turbulent flow in smooth concentric annuli with small radius ratios. J. Fluid Mech. 64, 263-287. Satake S. and Kawamura H. (1993). Large eddy simulation of turbulent flow in concentric annuli with a thin inner rod. Turbulent Shear Flows 9, 259-281.

TABLE 1 COMPUTATIONALMESH Outer-Wall Rotation 0.05

0.1

0.2

0.5

Inner-Wall Rotation

N

x

r

0

0 5 l0 0 5 10 0 5 10 0 5 10

128 128 128 128 128 128 128 128 128 128 128 128

128 128 128 128 128 128 128 128 128 128 128 128

256 256 256 256 256 256 256 256 256 512 512 512

a 0.05

0.1

0.2

0.5

N 0 5 10 0 5 10 0 5 10 0 5 10

x 128 256 512 128 256 512 128 256 512 128 256 256

r 128 128 128 128 128 128 128 128 128 128 128 128

0 256 256 256 256 256 256 256 256 256 512 512 512

223

20

1

x

1 .

.

~l 10

.

.

.

1

'":~'"

ot--0.5...~.,~.. ~

.,-.--.~.

O _ ,~.-

.,,-

,~---o.z....."2"~/" 0

II I

I"-.'--"

"-..., -.,

/". . . . . .

0

....

" ,,.

"~'~

/.'"

2~ O : - " " ~ "

" "" ""

.......... $ 7 " cz=O.s ......... " . -....... J " '~:~' ....... . , ~ - " ~ Laminar

1

",,

0-

,a':~--"-

. .........

solution.,

........ : j ' Outer-wall rotation

Y ~

0

r

~

N=0

. . . . .

N=5

Or=0.05 .......... . I - - ....... . I ~'

-

0 = .,a'-:~ ",

0 0

0.5

1

1.5

2

0

''''l''''l'''' 0.5

Y

Figure 1" Flow configuration and coordinate system.

I '''' 1

1.5

2

Y

Figure 2: Mean velocities in the outer-wall rotation, (a) U, (b) W.

TABLE 2 OUTER WALL-FRICTION, WALL-FRICTION ANGLE AND BULK VELOCITY IN THE OUTER-WALL ROTATION 0.5 0.2 0 5 10 0 5 10 0.964 0.954 0.932 0.958 0.947 0.913 0.962 0.944 0.919 0.965 0.965 0.935 0.001 0.001-0.001 0.001 0.006 0.002 -0.001 0.001 0.003 0.000 0.028 0.041 0.002 0.006 0.006 0.003 0.010 0.014 -0.001 0.018 0.035 -0.001 0.053 0.093

cz N

0

~'x Z"0

0.05 5

10

0.1 5

0

10

Oin Oout 0.001 0.001-0.001 0.001 0.006 0.002 -0.001 0.001 0.003 0.000 0.029 0.044 U B 1 5 . 1 15.4 16.6 15.0 15.4 16.3 15.0 15.2 16.1 15.0 14.9 15.6

1

0

).5

-r! ~176.......

i".'.-.... ., -~'J'~'\

I

a--u.z

i . . . I " . --"

0

ct=0.5

".r..z : ~ "

..7 1' 9

I

";'

f.'x

-~

_

-~

-~ , ~=0.1

I,)~./~i,

1'q--o.2

a--005

-o-o21 0.5

1 Y

1.5

2

~'"

! i

S"'!

L'/""

.t0.--0.1

/

-l

;" ~

--,l ~ t

1

0

0

~-,,~

",zl

.... 0

J .... 0.5

./'"./

~ .... 1 Y

t .... 1.5

_'," /:

""~.--- z'"

t'

l

.... 2

0

i .... 0.5

i .... 1

i .... 1.5

Y

Figure 3: Reynolds shear stresses in the outer-wall rotation, (a) u'v', (b) v'w', (c) w'u'.

2

224

Figure 4: Reynolds normal stresses in the outer-wall rotation, (a) u'u', (b) v'v' (c) w'w'.

Figure 5: Production of v'w' in N=10 near the inner wall.

Figure 6: Production and pressure strain of ct=0.1 near the outer wall, (a) P/i'

(b) t~//.

Figure 7: Two-point correlations of ct=0.1 at y=1.936, (a) axial direction, (b) azimuthal direction.

Figure 8: Contours of instantaneous axial fluctuating velocity of et=0.1 at y= 1.936.

225

20 0.04

1 :[ .......

(b)

10-

0.03 0.02 -

0

.....

_

....

0.01 +~

0"

N=5

0

-0.01 -

:...

-0.02

..... N=5 ] ..... N=10 .... I .... I .... I .... 0 0.5 1 1.5 2 y

-0.03 -0.04 0

"'1 .... I .... I .... I .... I .... I .... I .... 10 20 30 40 50 60 70 80 y+

Figure 9: Helicity of a=0.10 near the outer wall.

0-

a---0.1

,'~:~-" :a.-..,~. . . . . . . cz=0.05 .... I .... I .... I .... 0 0.5 1 1.5 2 y

Figure 10: Mean velocities in the inner-wall rotation, (a) U, (b) W.

TABLE 3 INNER WALL-FRICTION, WALL-FRICTION ANGLE AND BULK VELOCITY IN THE INNER-WALL ROTATION a

0.05

N

0

0.1

5

10

0

0.2

5

10

0

5

10

0

0.5 5

10

1.50 2.01 2.50 1.24 1.58 1.88 1.06 1.19 1.31 Tx 1.88 2.71 3.26 TO 0.003 1.30 3.28 0.005 0.886 1.94 -0.001 0.600 1.45 -0.007 0.306 0.729 Oin 0.002 0.448 0.789 0.003 0.415 0.731 -0.001 0.363 0.659 -0.001 0.252 0.509 Oout 0.001 0.005 0.009 -0.001 0.007 0.018 -0.001 0.032 0.072 0.000 0.061 0.19 U B 15.1 15.0 14.9 15.0 14.9 14.5 15.0 14.7 13.7 15.0 14.7 13.0

1

1

2~

I (b) I .-. a=0.5

0.5 0

0

iI ...... ""'"

l

,

....

0

/'

J'' ,2 ,,.. . . . - . - - - ~ ' ' ~ " - ' ' " - ' ' " "" ~

=0.5

i x. \.X a=0.2

0

(c)

...,

} .,:--: ~//'~=0.2

-~

0 ~'~. 0

t!c~=0.1 0-{.4 f,-,---'--""

i l . "\.~=0.1 0~i~3 "'" ~ ' ~ " ~ " """ . . . . . .

i/~ 9

I ....

0

0.5

1

y

1.5

2

0

,.-

. . . . . :" '"--: '":: "-" '

0.5

1

1.5

2

0

y

I ....

0.5

I ....

1

I ....

1.5

y m

Figure 11: Reynolds shear stresses in the inner-wall rotation, (a) u'v', (b) v'w', (c) w'u'.

226

m

Figure 12: Reynolds normal stresses in the inner-wall rotation, (a) u'u' (b) v'v', (c) w'w'.

Figure 13: Skewness, flatness and helicity of ct=0.1 near the inner wall, (a) S, (b) F, (c) H.

Figure 14: Two-point correlations of ct=0.1 at y=0.064, (a) axial direction, (b) azimuthal direction.

Figure 15: Contours of instantaneous axial fluctuating velocity of ct=0.1 at y=0.064.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

227

NUMERICAL SIMULATION OF COMPRESSIBLE MIXING LAYERS Song Fu and Qibing Li Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

ABSTRACT Three-dimensional spatially developing compressible planar mixing layers are studied numerically for convective Mach number Mc -- 0.4, 0.8 and 1.2. The present results for the flow-field structure, the mean velocity profiles, the mixing-layer growth rate, and Reynolds stresses agree well with those of experiments and other numerical studies. The normalized growth rate decreases with increasing Mc. Shocklets are found to exist in the mixing layer at Mc = 1.2 and their formation mechanism shows good agreement with the model of flow around a bluff body. The effect of compressibility on the large-scale structures is stronger than that on the small-scale ones. The budget of the Reynolds-stress transport equations agree well with that from the temporal developing results. The magnitudes of most of the contributing terms in the budget reduce with increased compressibility effect except the pressure-dilatation term although it is still very small.

KEYWORDS Numerical simulation, Spatially developing mixing layer, Compressibility, Shocklets, Budget of transport equation

INTRODUCTION The compressible mixing layer has been a research focal point in recent years for its distinct turbulent characteristics unknown in incompressible case (Vreman et al. 1996, Pantano and Sarkar 2002, Kourta and Sauvage 2002, Goebel and Dutton 1991, Debisschop et al. 1994, Barre et al. 1997, Rossmann et al. 2002, Gutmark et al. 1995). A well-known feature is that the normalized growth rate of the mixing layer decreases rapidly with increasing convective Mach number M c defined as the ratio of the freestream velocity difference to the sum of the sound speeds. The mixing layer also becomes highly three-dimensional in contrary to the Brown-Roshko type vortices (Brown & Roshko 1974) seen in low speed mixing layer. The normalized turbulent quantities also decrease in magnitude with the increase in M c . In the high M c mixing layers there exist unsteady shocklets that require high fidelity numerical scheme to capture.

228 Due to the limitation of computer capability, many of the existing numerical simulations of the compressible mixing layer are in two-dimensional (2-D) or of temporal development in which the periodic conditions are applied in the streamwise direction. While the temporal simulations are computationally efficient, spatially evolving simulations can resolve more flow physics. In the present work the three-dimensional (3-D) spatially developing compressible mixing layers, with M c ranging from 0.4 to 1.2, are investigated numerically with the simplified gas-kinetic BGK scheme (Li & Fu 2003). The aim is to further elucidate the fascinating flow structures as M c increases.

COMPUTATIONAL PARAMETERS In the present study, the same grid system of uniformly spaced in the x and z direction and stretched in the y direction is employed for the three cases. The computational domain size is defined as L x x Ly x Lz = 350x 120x 30 with N x x Ny x Nz = 875 x 200x 80 grid points. The ratio of the maximal cell size to the minimal size ( Aymin = 0.2 ) is 10.5. All variables are nondimensionalized by the initial vorticity thickness 6o,(0 ) , the free stream parameters p~, a 1, Aa21 at the high-speed side. The initial momentum thickness is given as d m ( 0 ) - 0.25. The free stream sound speed is set to a I - a : = 1, and pressure p~ = P2 = 1/y, where gas constant y = 1.4. The kinematic viscosity v = 0.001 is adopted for all of the computations. Table 1 provides the parameters of three computational cases where Re c and Re are the Reynolds numbers based on U c - (UI + U2)/2 and AU - U~ - U 2 respectively. The present computational domains are large enough for the flow to achieve fully developed state, as when x = L x ,

Xeff /6ml are all larger than 1500, where Xeff - - x ( 1 - U 2/U1) , 6ml is the momentum thickness of the high-speed stream set to half of the total thickness here. The criterion for xetr / 8m~ > 500 was proposed by Papamoschou & Roshko (1988). The last four columns in table 1 summarizes the values of the key non-dimensional parameters calculated at the downstream location of the mixing layer center (x = 320). The Kolmogorov scale L, and the micro-scale Reynolds number R e x . are defined as Re x =q2~/5/vo~ , where q2 =u,2 +v,2 +w,2 is twice the turbulent kinetic energy. The

L,7 = ( v 3 / o ~ 25,

Reynolds numbers are large enough for turbulence to occur. In all the three cases, the minimal cell size 0.2 is somewhat larger than the Kolmogorov scale, turbulent dissipation may not be well resolved. However, the quantities associated with large-scale structures are of the major concern here. TABLE 1 : COMPUTATIONALPARAMETERS Case

M 1

M2

Mc

Re c

Re

Rex

M,

L~

6m

1

1.9

1.1

0.4

375

200

478

0.211

0.069

3.14

2

2.9

1.3

0.8

525

400

790

0.365

0.050

3.35

3

3.5

1.1

1.2

575

600

1197

0.514

0.042

3.51 ,,,

At the inflow boundary (i = 1), the flow quantities, such as density, velocity, and pressure are prescribed, a broadband forcing (Stanley & Sarkar 1997) with random phase shift in the spanwise direction is also superimposed at the inlet to generate the early development of the shear layer. For the outflow and the two lateral side boundaries Thompson's non-reflecting boundary conditions are employed and the pressure correction proposed by Poinsot & Lele (1992) is also applied at the outflow boundary. The streamwise velocity profile of the inflow is given by the hyperbolic tangential function. The simulation

229 is started with an initialized flow field in which the streamwise velocity is determined by the inflow profile and the transverse velocity is fixed at zero value. The density and pressure fields are uniform throughout the flow field. A test case is calculated with the same parameters as case 1 except for the coarser computational mesh to examine the level of grid dependency. The cell size is about one and a half times larger than that in case 1. Figure 1 shows the comparisons of the coarse and fine grid results at different streamwise locations. The mean streamwise results with different computational mesh vary little showing the present grid mesh adequate in resolving the flow features. 1 (~~I~. ~"! ~ ~ , ~ 0.8 " ~ . .~ . % 0.6

........... o A •

x=O.6L, (FG) x=O.rL, (FG) x=0.8L~ (FG) Cerf x=O.6L, (CG) x=0.7Lx (CG) x=O.SL=(CG) 1997

0.2 " / h i '"J 0.15 ~ 0.1

CA

to

I~ 0.4

~~,

x=0.6L x (FG) x=0.7L, (FG) x=0.8~ (FG) x=0.6L=(CG) x=0.rL x (CG) x=0.8L=(CG) Pantano 2002 Rogers 1994 Bell 1990

. ~x x 9 9

0.05

0.2 0

a~

.x~,~x~ ~,,9 ~.,~ ~ " 9 J :~'~ ~ -s':~ ,,~ .~" ~, ,11' 9 .~

I

-5

,

,

0.2[(c)

,

,

i

,

0 (Y'Yc)l~m(x)

i

,',~,3L~-~

5

:

0

0.2

-5I

J

,

~

,

0I

i

(Y'Yc)/~m(X)

,

,

a I

5

(d)

o 0.15

0.15

o[o.1L t % . .

~ o.1 o

o 0,.~

o ....

~ .... (Y'Yc)/Sm(X)

~,

o

-5

0 (Y'Yc)/Sm(X)

5

Figure 1: Profiles of the mean streamwise velocity (a) and the velocity fluctuation intensity (b, c, d) at different locations (Case 1). 'Cerf' is the error function profile which is the first order approximation to the mean streamwise velocity in incompressible mixing layer. The symbol 'FG' represents the results calculated on the fine grid and 'CG' on the coarse grid. The legends are the same in (b,c,d).

RESULTS Flow Structures and Shocklets

In a lower M c mixing layer such as the case 1, the typical vortex pairing can be observed from the isosurface of the pressure p , the second invariant Q of the velocity gradient tensor (Vu) as well we the second largest eigenvalue, 22 , of $2+ f~2 where S and [~ are the symmetric and antisymmetric components of Vu. The present results verified the conclusion of (Jeong & Hussain 1995) that in most cases the Q- and 22-definitions of a vortex are similar. Figure 2 shows the isosurfaces of 22 in the mixing layer with M c = 0.4 and M~ = 1.2. It is seen there that the mixing at M~ = 0.4 has similar vortex evolution patterns to the incompressible flow, changing from spanwise roller to the so called 'helical pairing' and then to turbulence. The 'ribs' can be clearly seen between the rollers. For the

230

mixing layer with M c = 1.2, oblique structures occur in the upstream region of instability. The structures related to pressure can be better identified through the isosurfaces of pressure at high M~, although they can hardly describe the small structures such as the 'ribs' for the low M~ cases.

Figure 2: Isosurfaces of 22 (22 =-0.01 )

Figure 3: Spectra of the streamwise velocity in the inner region of the mixing layers (x = 0.8Lx, =0.2,-0.7,-0.4, respectively) for different M~. For clarity, the spectra are divided by the

y" = ( y - y . ) ~ 8.

factors: 10 for M~ = 0.8 and 100 for M

c

= 0.4.

The information of vortex evolution can be further found from the energy spectrum. Figure 3 shows the power spectrum of streamwise velocity along the geometric centerline (y=0). Here the Strouhal number is defined as S t = f S , , ( 0 ) / U c . Wide range spectrum, including the spectral slope -5/3, can be seen in the figure suggesting that the mixing layer has become fully developed turbulent flows. In the mixing layers with different Me, the streamwise velocity spectra are similar except for the peak positions in the low-frequency part. That is, compressibility mainly affects large-scale fluctuations. From the distribution of the density and streamwise velocity shown in figure 4, the strong discontinuity around the low-speed fluid engulfed into the high-speed can be clearly seen. These discontinuities show good similarity with the flow around a bluff body. The present numerical images also agree well with Rossmann's experimental results, thus verifying the model of shock formation.

231 To further study the shocklets in the mixing layer, one must identify the location of a shock. A good method is to check the entropy of the fluid moving across the discontinuity through tracking the fluid particles. In view of the complex flow structure and large amount of computational data for the threedimensional flow, a much simpler method is adopted here. Since a shock corresponds to a large pressure difference in the direction perpendicular to the shock, the existence of shock is verified in three ways: (1) the pressure gradient field Vp is examined to find the locations with high peaks; (2) the dilatation V. u is checked for large negative values as shocks lead to a strong compression of the fluid; (3) the densities on both sides of these jump interface with three or four cells size are checked with Rankine-Hugoniot (R-H) relations. If an interface satisfies these three verifications, it is deemed as a shock and the maximal pressure gradient direction is the shock orientation. Figure 5 shows two of these shocks whose orientations are nearly perpendicular to the spanwise direction. The pressures besides them are (p~,p~)=(0.55809,0.90895) and (0.62775,0.79097) , and the densities (p~,p2) =(0.75623, 1.1688) and(0.91172, 1.0748). The pressure jumps are thus 1.63 and 1.26 and are identified with dashed circles in the figure. The deviations from the R-H relation are 9% and nearly zero, respectively. Furthermore, the velocity vectors shown in figure 5 indicate that they are oblique shocks, corresponding to stationary inviscid shocks with Mach numbers M a = 1.3 and 1.6, with oblique angles fl = 750 and 430 , respectively. It is further noted that the shocklets in the mixing layer have complex structures in three dimensions. The effects of the numerical scheme, the boundary and initial conditions, the size of the computational

Figure 4" Instantaneous flow fields at different streamwise and transverse planes. (a) density contours at x - Lx / 3 , x = 2Lx/3 and x = Lx, respectively; (b) Contours at y = - 1 4 , top picture denotes to density and the bottom for the streamwise velocity.

Figure 5" The instantaneous density fields (left), divergence V. ~/(AU / 6~, (0)) fields with velocity fields (right) on a spanwise plane ( M c = 1.2,

z = 15 ). The mean convective velocity Uci is

subtracted from the velocity fields.

232 domain, and the lack of effective criterion to identify the shock, as well as the complexity of threedimensional flow structure lead to the difficulties in the shocklets identification among different studies (Vreman e t al. 1996). For the experimental investigations on the existence of shocklets very powerful visualization and measurement techniques are required (Papamoschou 1995; Alvi et al. 1996; Rossmann et al. 2002). Statistics and Compressibility

The turbulence statistics is obtained in about six times of the maximal time scale T = L x / U c . Figure 6 shows the development of the mixing layers where one can see that the mixing layer reaches linear region after some distance from the upstream and the center of the mixing layer leans to the low speed side. This feature is observed in experiments but can not be captured in the numerical simulations of temporally developing mixing layer. With increasing convective Mach number, the mixing layer shows more stable, thus much longer streamwise distance is required for the flow to develop turbulence. The normalized growth rates (shown in Figure 7) calculated from the momentum thickness in the fully developed region decrease evidently with increasing M c which agree well with existing data (Papamoschou & Roshko 1988; Rossmann et al. 2002; Goebel & Dutton 1991; Debisschop et al. 1994) and the numerical results (Freund e t al. 2000; Kourta & Sauvage 2002). The normalizing factor 6'nO, the growth rate for incompressible mixing layer, is set 0.04 indicating the corresponding vorticity

Figure 6: Development of mixing layers with different M c . Left is the normalized momentum thickness and right the center of mixing layer.

Figure 7: Normalized growth rates of different M c mixing layer.

233

thickness growth rate to be 6',oo = 0.18 or 0.16 if the mean streamwise velocity profile is an error function or a hyperbolic tangent, respectively. The peak linear mode amplification rate, which has been shown to be highly correlated to the spreading rates (Ragab & Wu 1989; Sandham & Reynolds 1990), calculated by Day et al. (1998) for spatially developing plane mixing layers, is also shown in this figure and is suppressed more rapidly than the present results. The data in this figure exhibit significant scattering that is partly attributed to the different experimental conditions and partly to the different mixing-layer thickness definitions adopted by various authors, calculating the growth from the momentum, the vorticity thickness, or the mean pressure fields. 0.25f

0.25

0.15

-9. = , . ~ . . . . . ~,:,,--~.......~ 9

o.o5 [ - . ~ " !~_--"

." /

- - - ........

or---

....

......... 100

X

200

015 -".

r

o.o5 tim=

.=:/,,u ,~::,':AiJ

auv .x/AU 300

o

////.

-.-

~ . / . .

~ _

--

,,"

_-

...

-

.......

~..!~5.... r

%,,oxlAU

%.JaU

a,,.,o,/aU ......

100

i

t

7"i-':-':-', X

200

~i uv~Hi/,',ui 300

Figure 8: Evolution of the maximum velocity fluctuation compared with the momentum thickness and the integrated turbulent energy in the mixing layer with M c = 0.4 (left) and M~ = 1.2 (right) Different developing regions can also be found in figure 8, where k, is the integration of the fluctuation energy along the transverse direction, o-,v is the mean square root of the shear stress. The locations where k, changes significantly are similar to that of 8 m, but behind that of the maximum Reynolds stresses. In low M c mixing layer, the shear stress achieves its maximum earlier than the other Reynolds-stress components. In high-M c mixing layer, three evolution stages can be observed more clearly and the streamwise Reynolds stress component reaches its maximum value first but then decrease quickly. These agree with the fact that turbulent energy is generated mainly in the streamwise direction through the interaction of the shear stress and the streamwise velocity gradient, -uvOU/Oy. The streamwise component of the turbulence energy is then redistributed to the other two components. In high-Mc flow, the compressibility inhibits turbulent energy redistribution from streamwise to transverse and spanwise directions, thus the streamwise component remains large for a significant distance. These trends agree with the experimental results (Elloit & Samimy 1990; Goebel & Dutton 1991). B u d g e t o f Reynolds Stress Transport

To further understand the compressibility effects on turbulence, the budgets of Reynolds-stress transport are evaluated. Here the Favre averaged transport equations are considered, 0r:i/0t = C,j + P,j + Dy + I-I0 + e~ + M 0

(1)

where the Reynolds stresses r0, the convection term C~j, the production term P,j, the pressure-strain term FI,j, the dissipation term co, and the mass flux term My are defined as,

234

tt

t? , ' ~

-'-77" . ~ -7" "--7- 5 --,---7,. ~ --'g-ii,j = p u,,j + p uj., , ~,j = -o'++uj. k -o'j+u,.+ , M ~ = u+p.j + ujp.+

v,'-,r

]

v

t

p_

--7--7"

~

~

r

~

-"g--

u,o)k.+

-

r

,

-

(2)

upi+.,

r

I.D,.,+ = D;, + D',_,.+ D~ - (,,.,o-,,+ + ~,o-.,+).,+ - (,~,u.,.~+).,, - ( p u,, G + p ",+4,+)., The diffusion term D~ includes three parts, the viscous diffusion, the turbulence diffusion and the pressure diffusion.

4.0 -

:Ca) ._c

/

2.0

~

Ct2 PI2

--"--

\

I

13 .o.

4.0

(b)

~,

-.-~-.-. o,,

\

J

~

+

n,,

~

~,,

.,_

o

Cl2

~ -- --~.- ----~----.

Vi2 D~2 D~

~"

Dtr2

+o

mx

.

" ~ 0.0

~-

oo

.

m o

-2.0

-2.0 I

-50

a

,

~

~

I

0.0

~

(Y'Yc)/Sm(X)

~

~

~

I

50

I

-5.0

~

~

,

~

I

0.0

J

(Y-Yc)/8,.(x)

~

~

~

I

6.0

Figure 9: Budgets of Reynolds stress transport at the location about x = 0 . 6 L x of the mixing layers with M c = 0.4 (a), and 1.2 (b) Figure 9(a) shows the budget of shear stress transport in the fully developed region of the mixing layer with Mc = 0.4, where all the terms are normalized with AU 3/ 8 m(x). The distributions of most of the terms agree with the temporal developing simulations of Pantano & Sarkar (2002) and the budget error can be seen very small indeed. Figure 9 also tell us that the budget of the shear stress is dominated by the production term and the pressure-strain term. The pressure diffusion and turbulence diffusion are much smaller. Different to the production term, the pressure-strain correlation term is negative throughout the mixing layer, acting as the dissipation term. The dissipation term can also be seen inherent small. With increasing M+, the value of each term decreases (Fig.9b) , so does that in the turbulent energy budget except for the pressure-dilatation term although it is still very small. The decrease of each term in the budget with increasing M c is consistent with the previous studies that the reduction of the pressure-strain correlation not only inhibits the transport of the turbulence energy from the streamwise direction to the transverse direction, but also decreases the production of the turbulence energy, resulting in the decrease of the growth rate of the mixing layer (Pantano & Sarkar 2002).

CONCLUSION Three-dimensional spatially developing planar compressible mixing layer at M c = 0.4,0.8 and 1.2 are studied numerically. The present results provide the flow-field structures, the characteristics of velocity fluctuation, which are all in good agreement with experimental and other numerical results. Shocklets are found in the mixing layer at M c, = 1.2 and their formation mechanism shows good agreement with the model of flow around a bluff body. The effect of compressibility on the large-scale structures is stronger than that on the small-scale ones. The magnitudes of most of the contributing

235 terms in the budget reduce with increased compressibility effect. REFERENCE

Alvi, F.S., Krothapalli, A. & Washington, D. (1996). Experimental study of a compressible countercurrent turbulent shear layer. AIAA J. 34:728-735. Barre, S., Braud, P., Chambres, O. & Bonnet, J.P. (1997). Influence of inlet pressure conditions on supersonic turbulent mixing layers. Exp. Therm. Fluid Sci. 14:68-74. Brown, G.L. & Roshko, A. (1974). On density effects and large structures in turbulent mixing layers. J. Fluid Mech. 64:775-816. Day, M.J., Reynolds, W.C. & Mansour, N.N. (1998). The structure of the compressible reacting mixing layer: insights from linear stability analysis. Phys. Fluids 10:993-1007. Debisschop, J.R., Chambres, O. & Bonnet, J.P. (1994). Velocity field characteristics in supersonic mixing layer. Exp. Therm. Fluid Sci. 9:147-155. Elloit, G.S. & Samimy, M. (1990). Compressibility effects in free shear layers. Phys. Fluids A 2:12311240.

Freund, J.B., Lele, S.K. & Moin, P. (2000). Compressibility effects in an turbulent annular mixing layer: Part 1. turbulence and growth rate. J. Fluid Mech. 421:229-267. Goebel, S.G. & Dutton, J.C. (1991). Experimental study of compressible turbulent mixing layers. AIAA J. 29:538-546. Gutmark, E.J., Schadow, K.C. & Yu, K.H. (1995). Mixing enhancement in supersonic free shear flows. Annu. Rev. Fluid Mech. 27:375-471. Jeong, J. & Hussain, F. (1995). On the identification of a vortex. J. Fluid Mech. 285:69-94. Kourta, A. & Sauvage, R. (2002). Computation of supersonic mixing layers. Phys. Fluids 14:37903797. Li, Q. & Fu, S. (2003). Numerical simulation of high-speed planar mixing layer. Comput. Fluids 32:1357-1377. Pantano, C. & Sarkar, S. (2002). A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. 3'. Fluid Mech. 451:329-371. Papamoschou, D. (1995). Evidence of shocklets in a counterflow supersonic shear layer. Phys. Fluids 7:233-235. Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimental study. 3'. Fluid Mech. 197:453-477. Poinsot, T.J. & Lele, S.K. (1992). Boundary conditions for direct simulations of compressible viscous flows. Jr. Comput. Phys. 101:104-129. Ragab, S. A. & Wu, J. L. (1989). Linear instability in two-dimensional compressible mixing layers. Phys. Fluids A 1:957-966. Rossmann, T., Mungal, M.G. & Hanson, R.K. (2002). Evolution and growth of large-scale structures in high compressibility mixing layers. J. Turbulence 3,009. Sandham, N.D. & Reynolds, W.C. (1990). Compressible mixing layer: linear theory and direct simulation. AIAA J. 28:618-624. Stanley, S. & Sarkar, S. (1997). Simulations of spatially developing two-dimensional shear layers and jets. Theoret. Comput. Fluid Dynamics 9:121-147. Vreman, A.W., Sandham, N.D. & Luo, K.H. (1996). Compressible mixing layer growth rate and turbulence characteristics. Jr. Fluid Mech. 320:235-258. Xu, K. (2001). A gas-kinetic BGK scheme for the Navier-Stokes equations, and its connection with artificial dissipation and Godunov method. 3'. Comput. Phys. 171:289-335.

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237

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

LES IN A U-BEND PIPE MESHED BY POLYHEDRAL CELLS C. M o u l i n e c 1, S. B e n h a m a d o u c h e

1,2 D.

L a u r e n c e 1,2 a n d M. Peri6 3

1UMIST, PO Box 88, Manchester M60 1QD, UK 2 EDF-DER-LNH, 6 quai Watier, 78401 Chatou, France a Computational Dynamics Ltd., 200 Shepherd Bush Road, London, W6 7NY, UK ABSTRACT Large Eddy Simulation of an incompressible fluid in a straight pipe with a circular cross-section is investigated with a Finite Volume (FV) method based on polyhedral cells, using synthetic turbulence at the inlet. Results of this non-periodic simulation are quite accurate after 2 diameters from the inlet, showing that the structures are self-sustainable. The method is then extended to a flow in a 180 ~ U bend pipe with circular cross-section, using an original automatic and boundary-layer-adapting meshing technique. This configuration, or a more convoluted version of it, is often encountered in industry (e.g. in car engines, air-conditioning, etc) and could not be regularly and smoothly meshed by cylindrical grids. The Reynolds number based on the bulk velocity and the hydraulic diameter is in both cases equal to 54,700. Comparisons are made with the experimental results from Azzola et al. (1986). Success of the simulation is mostly due to the excellent properties of the FV scheme on polyhedral cells. The fact that lines connecting cell centres are nearly orthogonal to the cell faces ensures that the numerical scheme is virtually free from any numerical diffusion. This was demonstrated on an array of 2-D inviscid vortices which are self sustained without loss of intensity.

KEYWORDS Large Eddy Simulation, Unstructured Finite Volumes, Polyhedral Cells, Synthetic Turbulence, Vortex Method, U Bend Pipe, Circular Cross-Section

INTRODUCTION Large Eddy Simulation (LES) with unstructured meshes for industrial cases is becoming a reality. LES of tube bundles using Finite Elements (tetrahedral cells) were presented in Rollet-Miet et al. (1999), then using unstructured Finite Volumes in Benhamadouche et al. (2002 & 2003) with similar energy conserving properties of unstructured meshes as investigated by Perot (2000). While the previous findings where based on the Code_Saturne software of EDF, the present paper is based on the commercial STAR-CCM code from CD adapco Group. This code goes one step further in using polyhedral cells with a large number of faces. Instead of pure dualization of a tetrahedral mesh, not optimal due to a very large number of faces (50 to 60 on average in 3-D - and concavity), it is obvious from Figure 1 that reconstructing

238

a polyhedral cell and new cell centered topology offer more possibilities for grid quality optimization than any other shape with a restricted number of faces (see Peri6 (2004) for a wider description of the advantages of polyhedral cells in complex geometries). Cell faces are almost orthogonal to the cell centre connections along which the fluxes have to be approximated. Centroids are used as control volumes in this full cell-centred approach, approximation of volume integrals by the midpoint rule is second order accurate when polyhedral cells are used as no interpolation is required. Using this kind of cells is also a very interesting alternative to non-conform grids for local refinement, the non-conformity introducing strong non-orthogonalities that may considerably affect kinetic energy conservation, which is a crucial issue for LES.

Figure 1: Polyhedral CV in a node-based FV method and a tetrahedral mesh (left). Equivalent polyhedral CV in a cell-centered FV method and optimized CV properties (fight).

Another main issue in industrial LES concerns the generation of fluctuating inlet boundary conditions. Following the work of Sergent (2002) and Jarrin et al. (2003), where random 2-D vortices are generated in the inlet plane normal to the streamwise direction while the streamwise fluctuations are obtained by solving a Langevin equation. Synthetic inlet turbulence is adapted in the present work to polyhedral meshes with a special focus on the boundary conditions for the random vortices. Large Eddy Simulation of an incompressible fluid in a straight pipe with a circular cross-section first shows the ability of the present method, based on polyhedral cells with a finite volume technique and synthetic turbulence at the inlet, to yield a developed flow after only a couple of diameters in the streamwise direction. The method is then extended to a flow in a 180 ~ U bend pipe with circular cross-section. This configuration is often encountered in industry (e.g. in car engines, air-conditioning, etc) and relies on a simple geometry, but cannot be regularly and smoothly meshed by hexahedral cells (generating a mesh by cylindrical transformation is not general enough for our purpose and is not considered here). The usual practice of having near-wall cells elongated in the streamwise direction is very much limited to channel flows. On the other hand the present wall normal-flat but wall parallel-isotropic cells is well suited for complex boundary layers where the mean flow direction is not known a priori. For the present case, in the curved section with strong secondary motions, isotropic polygons are more adapted than axially elongated prismatic cells. Note also that a structured grid would have resulted in a stretched grid on the outer side of the bend. The outline of this paper is as follows: the numerical method is described in the second section and its properties related to the kinetic energy conservation are studied in the case of the 2-D laminar TaylorGreen vortex case in the third section. The fourth section describes briefly the principle of synthetic turbulence while the fifth and the sixth sections respectively present its applicability when used in combination with polyhedral cells for a flow in a cylindrical pipe and for a flow in a 180 ~ U bend pipe both with circular cross-section.

NUMERICAL M E T H O D The fluid is considered as Newtonian and the flow as incompressible and isothermal. The filtered NavierStokes equations are discretized by a Finite Volume (FV) method (Demirdzic & Muzaferija, (1995)) on unstructured meshes with cells of any shape. All the variables are stored at the center of gravity of the

239

cells and midpoint rule is used to approximate all surface and volume integrals. Pressure-velocity coupling is ensured by a SIMPLE algorithm and the Poisson equation is solved by an algebraic multigrid solver. The Rhie & Chow (1983) reconstruction at the cell faces is activated to prevent from possible spurious oscillations in the pressure field due to colocated variable arrangement. The time advancing scheme is fully implicit using three-time levels and quadratic interpolation in time, while the convection and diffusion terms are discretized by a fully centered scheme. Special attention is made for the calculation of the gradients (deferred correction) as they play a major role in the accuracy of the results when unstructured meshes are used. Both spatial and temporal discretization are formally second-order accurate; for more details see Ferziger & Perid (1999). LES is performed with the standard Smagorinsky model for which the subgrid scale viscosity reads:

7ij -- (CsA)2v/2SijSij

(1)

where Cs is the Smagorinsky constant, A is the filter width and Sij is the filtered rate of strain tensor. A general expression of the filtered width suitable for Finite Volume method and unstructured grids is given by A = 2f2 89where gt is the volume of a cell. The near-wall treatment is achieved by activating the Van Driest damping function and the turbulent viscosity then reads:

n+ ut = ((1 - e--~)CsA)2v/2SijSij

(2)

with A + = 26 and n + being the non-dimensional distance to the wall.

KINETIC ENERGY CONSERVATION. 2-D LAMINAR TAYLOR-GREEN VORTICES A key requirement in the use of LES is kinetic energy conservation. In order to evaluate it with the numerical schemes used in STAR-CCM, the Euler equations are applied to the Taylor-Green vortex testcase. This case concerns counter-rotating vortices whose number is managed by the parameter m equal to 1 (see Eqn. 3). The domain is a two-dimensional square whose side is equal to 27r which implies that four vortices are in the computational domain. The solution of the 2-D Euler equations is then given by:

I ~ = +~i~(.~)co~(mV) v = -cos(rex)sin(my)

(3)

p = - ~1 (cos(2mx) + cos(2my)) u and v are the 2-D velocity components, p is the pressure and (z, y) are the Cartesian coordinates. The first tests with the TaYlor-Green vortex test-case including viscosity (not shown here) demonstrated second order accuracy for both time and space, because of the use of the three-time level method as temporal scheme combined with the Finite Volume approach and a centered scheme for the spatial discretisation. The results presented here concern the INVISCID case where the vortices should be selfsustained indefinitely. Although the solution is steady state, solving it with a time marching scheme is very demanding as the effects of advection and pressure must balance exactly. The kinetic energy k on an arbitrary mesh is defined as follows: 1 k -" 2--~tot ~/(u2

-'l'- v2)~-'~I

(4)

where I stands for the cell index, fti is the cell volume and ~'~tot is the total volume of the computational domain. The evolution of k in time is studied on various types of grids with density of faces of the

240

same order, the first one being Cartesian (with 4,096 cells), the second one generated by tetrahedral cells (5,702 cells) and the last one by polyhedral cells of arbitrary shape (2,952 cells). Symmetric boundary conditions are prescribed in all directions and the simulations are performed during 12 physical seconds with a time step of 0.01s. The mean velocity being the square root of k and the side of vortex L = 7r, the mean vortex tum-around time is T = L/U, i.e. the simulation corresponds to 6 tum-around times. Figure 2 shows the tetrahedral and polyhedral meshes (the polyhedral mesh is generated from the tetrahedral one). In both cases, the control volumes are reasonably regular. The structured boundary layer inset is not used here. Figure 3 shows the velocity vectors on both meshes after the 12 s. In the tetrahedral case (Figure 3 (left)), the symmetry in the velocity patterns is already broken, whereas it is not visible yet for the grid with polyhedral cells (Figure 3 (fight)). The theoretical maximum of velocity magnitude is equal to one. For the mesh with tetrahedral cells, the error is equal to 12% whereas it is only 4% for the mesh with polyhedral cells. Note that in both cases, the maximum of velocity magnitude is overestimated.

Figure 2: The two conjugated meshes with tetrahedral cells (left) and polyhedral cells (fight). A zoom of the comer is overlayed and lower par cropped.

Figure 3: Velocity vectors after 12 seconds. Left: Simulation with tetrahedral cells. Right: Simulation with polyhedral cells. Figure 4 shows kinetic energy versus evolution in time for the three meshes. Whereas the conservation looks obvious when hexahedral and arbitrary polyhedral cells are used (although a small decay can be noticed), the decay is very pronounced when tetrahedral cells are used. The relative error on the energy after 12 s is of 0.05% for the Cartesian mesh, of 0.08% for the mesh with polyhedral cells and of 0.3% for the mesh with tetrahedral cells. As a conclusion, polyhedral cells clearly have better properties of kinetic energy conservation than tetrahedral ones without any special treatment, probably because of the general shape of the polyhedra, which are close to spherical, thus avoiding adding too much correction in the gradient estimation. Moreover the polyhedral mesh has only half the number of control volumes of the original tetrahedral grid, and besides, the CPU time per node and time-step is smaller than for the tetrahedral grid since the better orthogonality leads to improved conditioning of the linear systems to be solved.

241

0.25 T.,- ~

-

--.

\ 0.248

N

a< 02.

~ ---

Polyhedra

\

\\

Teb'ahedra

'~

Hexahedra

0.244 o ....

;~'

' ' 'Tim. ~ ....

; ....

112

Figure 4: Kinetic energy evolution during 12 s for the Cartesian mesh, the mesh with tetrahedral cells and the mesh with polyhedral cells. SYNTHETIC INLET TURBULENCE FOR U N S T R U C T U R E D MESHES An important issue in industrial LES concerns the inlet boundary conditions to be prescribed, because of their unsteadiness. The method used here is based on a method developed by Sergent (2002) and further adapted by Jarrin et al. (2003) in the EDF software Code_Saturne. Its principle lays on the generation of 2-D vortices in the inlet cross-section combined with a Langevin equation for the direction normal to the inlet cross-section.

LES IN A STRAIGHT PIPE WITH CIRCULAR CROSS-SECTION MESHED BY POLYHEDRAL CELLS The flow in a straight pipe with circular cross-section is investigated to check the potentiality of the polyhedral cells (hexahedral cells are not the best choice of meshing because of cells with sharp angles, where the discretization is usually not accurate). As our final goal is the calculation in a U bend pipe, the flow in a straight pipe is simulated at the same Reynolds number as for the U bend pipe, i.e. Re = 54,700 to match the experimental data of Azzola et al. (1986). The near-wall layer is meshed by 10 prisms (shallow extruded 2-D skin polyhedra) so that n + is equal to 2 for the nearest prisms to the wall. Note that the layer of prisms reaches n + -- 70, which ensures to cover the viscous and the buffer layers. The domain length is 20R, where R is the radius of the cylinder.

0.25 F 1.2~ .

~

Z=2 ....

~ 0.2 f

~ z . 5

i-

....

~0.151~.75 "i 0.5

~

z.2

:--

"~

~: z.,

k"

.... Z=6 .......... Z=7

r

l:

.,-

Z=3

.......

[

m

|

9

Z=4

Z=6

z=e

Azzola etaL

(Z=2)

-

9

[ 0, ~

-.~_

__~"'_"-,

9 9

......

9

0.25 0

. . . . . 0.2

0.4

Radius

0.6

0.8

1

0/

0 L

i

i

I

0.2

. . . . . .

0.4

Radius

i

0.6

,

,

,

i

0.8

,

,

, ,k

Y

Figure 5" Profiles of the mean streamwise velocity W (left), of w' (fight). Synthetic turbulence is used as inlet boundary conditions to get the mean velocity, the turbulent kinetic energy and the dissipation rate. A precursor calculation with a statistical model is ran in a cross-section

242

of the U bend pipe in the same conditions as the ones described before (same Reynolds number, same radius) on a fine grid meshed by hexahedral cells. The statistical model is the Speziale-Sarkar-Gatski (SSG, 1991) model and the results of this simulation are used as input data for the vortex method. The synthetic turbulence is prescribed in the inlet plane, outlet boundary conditions are used for the outflow and no-slip conditions at the wall. Figure 5 (left) shows the streamwise velocity profiles and Figure 5 (right), the Reynolds stresses in the streamwise direction for cross-sections located between Z-2R to Z=8R, Z being the streamwise coordinate. The streamwise velocity profiles obtained by LES almost coincide as of Z-2R. They also match the experimental data quite well. On the other hand, the Reynolds stress in the streamwise direction reaches homogeneity in Z slightly further from Z=3R. The peak in the near wall region is well predicted compared to Azzola et al. (1986) data. One can conclude from that test that 1.5 diameters are sufficient for the synthetic turbulence to develop realistic structures. Note that this is a reasonably high enough Reynolds number Re=57,400, to be considered as industrial, and to demonstrate also the potential of polyhedral cells for LES in the STAR-CCM code.

LES IN A 180 ~ U-BEND

PIPE

We now consider the flow in a 180 ~ U bend pipe with circular cross-section studied experimentally by Azzola et al. (1986) using LDA. Figure 6 shows the configuration and the main dimensions. Azzola et al. (1986) observed complex secondary motions and separation at the start of the bend. The study here focuses on the dynamics in the bend, and not on the downstream region, therefore the length (see Figure 6) of the pipe located downstream to the bend is relatively short and taken equal to 1OR. The length of the region upstream to the bend is equal to 6R. Synthetic inlet turbulence is generated in the same manner as described for the straight pipe is used. Outlet botmdary conditions are prescribed for the outflow and no-slip conditions for the walls, the Van Driest damping function being then activated.

~ S

0=90

,0=180~

'.

.

.

_

.

.

.

.

.

.

.

____~_~_ .

.

.

.

.

.

.............

.

.

. . . . . . .

.

~:______x__

/

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . .

oi-

1

Figure 6" Sketch of the 180 ~ U bend pipe geometry. The polyhedral mesh is generated automatically using a meshing tool from the CD adapco Group. It has 671,455 polyhedral cells. A layer of 10 prisms is built in the wall normal direction allowing n + -- 2 for the first layer. Figure 7 (left) shows that polyhedral cells in the core of the domain have an arbitrary shape and Figure 7 (right) the inlet cross-section and the layer of prisms near the wall. Figure 8 presents the comparison between the LES profiles and the experimental data of Azzola et al. (1986) available in the bend for the axial streamwise mean velocity, for the circumferential velocity, and the Reynolds stresses in the axial and the circumferential direction. The LES predicts well the mean flow and relatively well the Reynolds stress in the axial direction, but underestimates the Reynolds stress

243

Figure 7: Left: Section of the domain showing the polygonal cells at the surface and the polyhedral cells in the core of the mesh. Right: Inlet cross-section where can be seen the layer of 10 prisms in the wall normal direction. in the circumferential direction in the near-wall region. As the same tendency is observed in simple channel flows with coarse (i.e. not quasi DNS) meshes, this does not come from the use of polyhedral cells. More advanced subgrid scale models than the standard Smagorinsky one used herein might further improve results, not to mention possible uncertainties in the 18 years old experimental results. Thus, the validation of STAR-CCM for LES can be considered as quite satisfactory. Figure 9 shows the secondary motions present in four different cross-sections of the U-bend pipe. The axis of symmetry in the y direction is used to plot the results; a longer averaging period should be used to obtain a perfect symmetry (here the period was 100 physical seconds after having ensured that the flow was fully turbulent). Two symmetrical Dean vortices are present at 45 ~ close to the inside of the pipe, between them the high momentum fluid from the inner side of the bend (bottom) is pushed toward the outer bend (top) in a plume like manner. Complex "Coriolis like" forces are at play, and at 90 ~ the former large vortices start to break down creating new ones above them. At 135 ~, four main Dean vortices are clearly present in the flow. Their intensity is smaller than for the vortices observed at 45 ~. At 177 ~ two of the four vortices have disappeared and only two small ones are still visible on these time averaged plots, but the instantaneous fields are much more convoluted. Experimental data from Sudo et al. (2000) for a pipe for which the curvature radius Rc = 4 instead of Rc = 6.75 in this paper show a similar behavior for the Dean vortices.

CONCLUSION LES with polyhedral cells have been carried out for the first time. This type of control volumes present the same meshing flexibility as Finite Elements with tetrahedral cells, but were revealed to be much more economical and accurate. Almost perfect kinetic energy conservation was demonstrated for the polyhedral FV method on the inviscid 2-D Taylor-Green vortices test-case (also illustrating perfect stability, perhaps even puzzling for a second order centred convection scheme). Next a LES of a pipe flow without periodic conditions was performed, adapting the synthetic vortex method of Jarrin et al. (2003), which only uses input from a standard Reynolds stress transport model. Equilibrium results for the mean

244

velocity and Reynolds stress tensor were achieved after only 2 diameters at Re=57,400, which makes the method applicable to complex industrial flows. As a demonstration the LES in a 180 ~ U bend pipe has been carried out using polyhedral cells in the centre and extruded flat polyhedral cells in the boundary layers. The results compare well with the experimental data of Azzola et al. (1986) despite the high Reynolds number and show the potential of the STAR-CCM code for complex geometries. Acknowledgement. This work has been supported by the Teaching Company Scheme number 4209.

References [ 1] Azzola J., Humphrey J.A.C., Iacovides H. And Launder B.E. 1986. Developing Turbulent Flow in a U-Bend of Circular Cross-Section: Measurement and Computation. J. of Fluid Eng., 108 214-221. [2] Benhamadouche S. and Laurence D.R.P. (2002). Global Kinetic Energy Conservation with Unstructured Meshes. Int. J. Num. Meth. Fluids, 40 561-572. [3] Benhamadouche S. and Laurence D.R.P. (2003). LES, Coarse LES and Transient RANS Comparisons on the Flow across a Tube Bundle. Int. J. Heat and Fluid Flow, 24 470-479. [4] Benhamadouche S., Mahesh K. and Constantinescu G. (2002). Colocated Finite Volume Schemes for LES on Unstructured Meshes. Summer Program Stanford, CTR Stanford. [5] Demirdzi6 I. and Muzaferija S. (1995). Numerical Method for Coupled Fluid Flow, Heat Transfer and Stress Analysis using Unstructured Moving Meshes with Cells of Arbitrary Topology. Comput. Methods AppL Mech. Eng., 125, 235-255. [6] Eggels J.G.M., Unger F., Weiss M.H., Westerweel J., Adrian R.J., Friedriech R. and Nieuwstadt ET.M. (1994). Fully Developed Pipe Flow: a Comparison between Direct Numerical Simulation and Experiment. J. Fluid Mechanics, 268 175-209. [7] Ferziger J.H. and Peri6 (1999). Computational Methods for Fluid Dynamics (Second Edition), Springer, Berlin. [8] Jarrin N., Benhamadouche S., Addad Y. and Laurence D.R.E (2003). Synthetic Turbulent Inflow Conditions for Large Eddy Simulation. Turbulence, Heat and Mass Transfer 4 , Begell House, inc. Eds. Hanjalic, Nagano and Tummers, 467-474. [9] Peri6 M. (2004). Flow Simulation using Control Volumes of Arbitrary Polyhedral Shape. Ercofiac Bulletin, 62, 25-29. [10] Perot B. (2000). Conservation Properties of Unstructured Staggered Mesh Schemes. J. Comp. Phys., 159 58-89. [11] Rhie C.M. and Chow W.L. (1983). Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J., 21, 1525-1532. [12] Rollet-Miet P., Laurence D. and Ferziger J. (1999). LES and RANS of Turbulent Flow in Tube Bundles. Int. J. Heat Fluid and Flow, 20, 241-254. [13] Sergent E. (2002). Vers une MOthodologie de Couplage entre la Simulation des Grandes Echelles et les ModOles Statistiques, PhD Thesis, Ecole Centrale de Lyon. [ 14] Speziale C.G., Sarkar S. and Gatski T.B. (1991). Modeling the Pressure-Strain Correlation of Turbulence, an Invariant Dynamical Systems Approach. J. Fluid Mechanics, 227 245-272. [15] Sudo K., Sumida M. and Hibara H. (2000). Experimental Investigation on Turbulent Flow through a Circular-Sectioned 180 ~ Bend. Experiments in Fluids, 28 51-57.

245

Figure 8: Profiles of LES results compared to the database of Azzola et al. (1986) using the symmetry plane (see Figure 6), in various angles of the bend 3 ~ 45 ~ 90 ~ 135 ~ and 177 ~ starting from the bottom of the image. Left: Streamwise velocity vs radius. Middle-left: Circumferential velocity vs radius. Middle-fight: Streamwise Reynolds stress vs radius. Right: Circumferential Reynolds stress vs radius.

246

Figure 9: Secondary motions in four cross-sections of the U-bend pipe, at 45 ~ 90 ~ 135 ~ and 177 ~

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

247

LARGE EDDY SIMULATION OF IMPINGING JETS IN A CONFINED FLOW D. J. Clayton and W. P. Jones Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ.

ABSTRACT Large Eddy Simulation (LES) has been applied to a representative primary combustion zone in an isothermal constant density simulation. The geometry consists of an annular passage that feeds a row of six port-holes. The resulting radial jets impinge strongly within a confined core cross-flow. Both uncoupled (core only) and coupled (core and annulus) simulations are considered. The uncoupled simulation applies detailed experimental data as port boundary conditions. The findings conclude that the coupled LES can adequately reproduce port characteristics resulting in a good description of the core combustor flow-field, potentially superior to that given by the uncoupled case and far superior to that given by RANS predictions.

KEYWORDS LES, primary combustion region, impinging jets, jets in cross-flow, coupled simulation.

INTRODUCTION

Much work reported in the open literature aims to improve the understanding of gas turbine combustor flow-fields. The use of impinging jets is common in the primary combustion zone, providing a region of highly recirculating flow and a stable initial combustion. This zone is considered to be one of the most challenging to model computationally. Within the primary region, although an accurate description of burning is ultimately sought, an underlying understanding of the fluid mechanics is required. In the present work, LES is applied to an isothermal constant density configuration. Both uncoupled (core only) and coupled (core and annulus) simulations are considered. Historically, uncoupled simulations have often applied uniform velocity profiles (derived from one-dimensional correlations) as port boundary conditions, e.g. Adkins et al (1986). The use of flat inlet profiles has been shown to give rise to noticeable differences in terms of jet spreading and upstream recirculation characteristics, Manners (1987), and it has been argued by many authors that for jet in cross-flow configurations it is insufficient to represent the incoming jets via simple boundary conditions to the

248 cross-flow. The uncoupled LES described in the following sections improves upon the use of empirical uniform boundary conditions by applying detailed measured mean and turbulence profiles at the port inlets. Regardless of the success of this method the experimental cost of acquiring realistic port inlet conditions for all combustors of interest makes the uncoupled approach prohibitive. This promotes the development of fully coupled simulations in which the core/annulus interactions must be adequately captured, hence resulting in an accurate simulation of the port velocity profiles. The majority of similar computational studies have adopted a RANS approach, where two-equation eddy-viscosity models are commonly used for closure. While the results obtained with such methods can be useful there are a number of well-known inadequacies particularly in highly curved and recirculating flow. It has been shown that combustor core and annulus fields have a highly unsteady nature and consequently RANS methods invariably under-predict the jet strength and backflow size, e.g. McGuirk et al (2001). These unsteady features can potentially be captured by LES. In terms of LES application much of the literature is limited to the use of single and non-impinging multiple jets, e.g. Wille (1997) and Gicquel et al (2002) respectively. The mechanism of the through-port core/annulus interaction has recently been analysed with both RANS and LES methods by Spencer (1998) and Spencer et al (2003).

GOVERNING EQUATIONS AND MODELLING The equations describing the motion of the large eddy structures in a turbulent flow are obtained by applying a spatial filter to the continuity and Navier-Stokes equations. Assuming that spatial filtering commutes with spatial differentiation allows the filtered equations to be written as: 0/9+

=0

Ot

---~

at

. . . .

axj

(1)

~X i

+

c3xi OXj

/a

OXj

+

OXi

OXj

(2)

where p is the density, b/i is the velocity component in the Xi direction, p is the pressure and/a is the assumed constant molecular viscosity. The overbar denotes the application of the spatial filter, r0. is the unknown sub-grid scale stress that in this work is closed by the standard Smagorinsky (1963) model:

r,j - -~1r~ 4 = 2 ,p(Cs2 )a ~2~JL, S~

(3)

,Usgs

where S,j is the filtered rate-of-strain tensor and A is the filter width corresponding to the smallest resolved motion, commonly equated to the cube root of the local cell volume. In the present work the dynamic version of the closure, Germano et al (1991), is adopted where the local value of the Cs parameter is determined via the application of both a grid and test filter. Assuming the model parameter is invariant of filter width allows these two filter levels to be related by introduction of the resolved stress tensor thereby resulting in a solvable set of equations for the local length scale. In some circumstances the dynamic model gives rise to unphysical values of Cs and hence its use is often not recommended. It does however automatically modify the Cs parameter at walls, negating the need for wall damping by directly accounting for the loss of turbulent production in the unresolved scales.

249 Scalar mixing is described in terms of the mixture fraction ~', which is strictly conserved. With appropriate boundary conditions the spatially filtered evolution equation for ~' (Eqn 4) can be used to track the position of a chosen inlet stream.

(4)

O. gs OX i

(5)

Closure is obtained by the introduction of a gradient approximation for the sub-grid scalar flux, Eqn 5. cr is the Prandtl/Schmidt number and O'sgs is assigned a fixed value of 0.7 in this work. The LES code employed utilizes Cartesian velocity components and a boundary-conforming general curvilinear coordinate system with co-located variable storage. The formulation is implicit and an approximate factorisation technique is adopted with applied pressure smoothing to determine the pressure field. With the exception of the convection terms in the mixture fraction equation all spatial derivatives are approximated by central differences and all time derivatives are evaluated using a second-order three-point backwards difference scheme. Preconditioned conjugate gradient solvers are used to solve the assembled sets of algebraic equations. The resulting method is second-order accurate in space and time. The mixture fraction must remain bounded and so a TVD scheme is adopted based on a flux limited central difference approximation, Jones (1994). Various forms ofTVD limiter exist, however that proposed by Van Leer (1974) is used here.

MODELLED GEOMETRY The modelled geometry is representative of a primary combustor zone with six radially impinging jets, stemming from six annulus port-holes. The configuration corresponds to that studied experimentally by Spencer (1998). The extensive LDA data available as a result of this study makes it ideal for comparison to both uncoupled and fully coupled simulations. From this point forward U, V and W represent the axial, radial and tangential mean velocity components respectively, u, v and w represent the corresponding velocity fluctuations and x, r and 0 the associated spatial components. Uc is the core bulk axial velocity (= 0.201m/s) and Vj is the bulk jet radial velocity. The configuration represents a primary combustion region with a velocity ratio ( V / U c ) of 5.0, bleed ratio (annulus inlet mass flow rate/jet mass flow rate) of 0.5 and a jet Reynolds number of2xl 04.

COMPUTATIONAL PARAMETERS To model the physical geometry appropriately a cylindrical mesh is used with constant circumferential lines representing the core/annulus dividing walls. Solution convergence on the cylindrical grid is slow due to the high cell aspect ratio and skewness at the centreline. This is particularly the case in the pressure correction solver where the (large) magnitude of the coefficients of the centreline nodes make solution difficult. Closure of the Cartesian form of the governing equations requires implementation of both centreline and periodic boundary conditions. The two presented cases are defined below. Further setup information is given in Table 1.

250 Case I - In the core only uncoupled simulation, experimentally measured mean and turbulence profiles are applied directly as boundary conditions to the jet inlets. The measured data exhibits negative radial velocities within the port (see Figure 5), which in the uncoupled case are simply set to zero. Experimental data is only available on the main port diameter and so this profile is scaled and assumed to apply across the entire opening. Case 1 1 - The coupled core and annulus geometry allows full core/annulus interaction and a fully developed and correlated turbulent port flow. The grid used in the uncoupled case is extended in the radial direction to fit the annulus and dividing walls, which are implemented with immersed boundary conditions. Integration times are slightly larger in this case due to the fine mesh required to define correctly the position and width of the embedded walls.

TABLE 1 SETUP CONDITIONSFOR CASE I AND CASE II Case I 146, 49, 105 Grid size (x, r, 0) 1.04x103m AXmin 3.01x10Bm Axmax 4.49xl 04m Armin 1.00xl03m Armax 1.80x 10Sm s rA0max 3.17x 10"3xn 0.2/0.3 CFL/CFLmax 2.34x10 rsec Ataverage Nsteps ~652,500

Case II 146, 77, 105 Grid size (x, r, 0) 1.04x10-3m AXmin 3.01x10-3m AXmax Armin 4.49X 10-4m Armax 1.00xl0 -3m rAOmin 1.80xl 0-Sm rA0max 5.17x103m 0.2/0.3 CFL/CFLmax 2.18x 10rsec Ataverage Nsteps ~692,000

The flow-through-time based on the bulk core velocity is approximately 0.5 seconds. Both simulations were run for 1.5 seconds after initial conditions were washed from the system. The port centres are defined at x=0mm with the axial extents of the domain at x=-150mm and x=100mm. The radii of the core and annulus are 45mm and 70mm respectively with a dividing wall thickness of 5mm. To achieve adequate resolution of the ports the grid is refined in this region. Axially a growth parameter of 1.09 is used upstream and downstream of the port edges and circumferentially the same ratio is applied to the regions between the ports. Large grid expansion ratios should be avoided in LES due to the commutation error, which generally affects the small scales where spatial derivatives are large. It is argued by Wille (1997) that where expansion ratios are very large, i.e. in adaptive or cylindrical meshes, so too are numerical errors in the large scales and explicit treatment of the commutation error is not necessary. The port openings are castellated with each open area defined to within 1.2% of its true area. Grid resolution is not altered close to the walls with most Y+ values (based on core, annulus and jet Reynolds numbers) appropriate for application of the mean wall-law. This is not the case for the through-port walls, which are not well resolved, resulting in a lack of turbulent production. Figure 1 shows the final mesh structure, the shaded regions indicate the castellated port openings. With multiple exits from the domain the correct mass flow split 1 (50% bleed ratio) must be ensured. A fixed velocity is applied at the annulus exit, indirectly enforcing the flow split between the core and annulus, and a non-reflective outflow boundary condition is applied at the core exit. Results show that the flow approaching the annulus exit plane is heavily biased towards the inner wall. This is a consequence of the application of an assumed uniform outflow velocity profile resulting in an unrealistically strong recirculation forming close to the annulus outlet plane. Although a more sophisticated outflow condition could potentially prevent these recirculations from forming, e.g. a constant static pressure In the experimentalstudy the outflowrate fromthe annulus is controlledby a systemof valves.

251

Figure 1: Uncoupled (core only) and coupled (core and annulus) mesh. (,4 section has been removed to show the castellated port region)

condition, their presence suggests that the computational domain may be too short (extending only 5 annulus heights beyond the port centreline). All inlet conditions are specified by application of a Gaussian-distributed perturbation to the mean flow. This method, which is applied to the main core and annulus inlets in both cases and to the port inlets in the uncoupled case, is known to produce a non-coherent and uncorrelated turbulent field. Preliminary simulations indicate the solution is minimally affected by the chosen resolution and modelling parameters.

RESULTS Mean velocity vectors on a plane through port centres (averaged over all six ports) are shown in Figures 2 and 3 for the uncoupled and coupled cases respectively. The enclosed areas of the vector plots indicate regions of reversed axial flow. The instantaneous passive scalar field at a single instant of time is also shown for the coupled case, clearly illustrating the high degree of core mixing present. Unfortunately there is no data available for comparison purposes, as scalar properties were not measured in the experimental study. The strongly impinging jets result in highly unsteady instantaneous backflow bursts which are known to reach an axial location of x=-ll0mm in a typical experimental realisation. LES gives a similar result with x=-100mm in the coupled realisation shown. Experimentally, the time-averaged stagnation point at the head of the recirculation zone is at -57mm. LES predicts a stagnation point of-56mm for Case I compared to -67mm for Case II; the coupled LES over-predicting the size of the backflow region. This is in contrast to most coupled RANS cases which generally under-predict the backflow size due to an inability to simulate the strong backflow bursts. Experimentally, the impingement point is at x=13mm. The uncoupled LES predicts a position of x=15mm and the coupled LES a position of x=12.5mm. The jets are deflected closer to the axial direction in the uncoupled case causing the forward location of the impingement point. This is likely to be due to the presumed shape of the port profiles away from the main diameter, which serves to over-predict the size of the recirculation downstream of the ports on the core wall. Experimental data indicates the onset of separation at the outer annulus wall above each port at a bleed ratio of 50%. The formation of these 'bubbles' is known to be one of the most sensitive regions to annulus flow

252

Figure 2: Uncoupled mean vector plot (Enclosed regions signify U < 0).

Figure 3" Coupled mean vector plot (Enclosed regions signify U < 0) and instantaneous mixture fraction contours.

253 asymmetry, Daly (1994). Averaged LES data shows no separation, however instantaneously separation is intermittently present over some of the ports. Selected mean and RMS velocity profiles are shown in Figure 4. Both cases are in good agreement to experiment. It is notable how well the coupled case predicts mean values through the port (x=-5mm and x=5mm), implying the interaction between the core and annulus is well captured. At x=-10mm the experimental and computational annulus profiles show the flow migrating to the inner wall, anticipating the presence of the ports. At x=40mm the annulus axial velocity field correctly reestablishes itself to approximately half of its inlet value, whereas in the core, jet merger results in a peak axial velocity of 8Uc at the centreline. This value is over-predicted in the uncoupled case potentially due to a slight lack of turbulent intensity downstream of impingement preventing sufficient mixing of the jet shear layers. Many of these experimental characteristics were originally noted by Spencer (1998). In both uncoupled and coupled cases turbulence levels are well predicted although underestimated upstream of the ports, thought to be due to the growing grid size in this region. The high levels of turbulent anisotropy within the core is captured, with v = 2 • u at impingement. The slightly higher-intensity correlated turbulence stemming from the coupled annulus is in keeping with the increased mixing observed from the mean velocity profiles. Towards the outflow, turbulence levels in the annulus become excessively high. This is due to the production in the shear layers of the recirculations present at the outflow plane (identifiable from the coupled vector plot). Figure 4 also shows RANS data taken from Spencer (1998). Mean velocity profiles are only available at two axial locations (x=-5mm and x=5mm) and an RMS velocity profile only at one axial location (x=-5mm). In terms of mean profiles, RANS performs similarly to LES close to the port centreline with both underestimating the strength of the backflow. Major discrepancies can be seen in the turbulence field predicted by RANS ~articularly in the axial direction. The maximum value of normalised turbulent kinetic energy (k/Uc) at impingement is 16, with RANS predicting a peak value of 5.5 and LES a peak value of 12. In LES, instantaneous bursts transport fluid upstream beyond the mean stagnation point. This is not the case for RANS, where the mean and turbulence fields are severely underpredicted in this region. A good prediction of the port velocity profile is largely dependent on a well-captured through-port flow history. Figure 5 compares the experimentally measured jet exit profiles and jet angles (defined as V/U such that 0 ~ axial flow and 90 ~ = radial flow) with those emanating from the annulus in the coupled case. At the upstream edge of the port, separation within the opening experimentally blocks the flow. This blockage is not captured computationally perhaps due to a lack of grid resolution of the port walls. Predicted flow through the front 3mm of the port is responsible for the over-predicted backflow size in the coupled case, with an excessive percentage of the flow splitting rearwards at impingement. This through-flow is also responsible for an over-prediction of the port discharge coefficient (Cd - a method of computationally calculating Cd for this geometry has been presented by Spencer (1998)) [Cd, Experimental- 0.639, Cd, LES = 0.731]. Negative radial velocity components within the primary port are a significant flow feature that are known to dramatically affect the core flow-field. However, their omission in both the uncoupled (where negative velocities are clipped to zero) and coupled cases do not appear to have dramatic consequences under the particular flow conditions prevailing in these simulations. Experimentally, at the rear of the port opening a small recirculation forms on the annulus wall as flow is pulled rearwards to feed the ports. This recirculation is predicted to be slightly too strong, with the reversed flow retaining too much of its axial momentum. Generally, apart from the front 3mm of the port, the flow angle is in excellent agreement to experimental data with variations of flow angle being predominately due to variations in the axial velocity component. Figure 5 also shows the port turbulence profile in the radial direction (which is representative of all three directions). The over-predicted turbulence through the port centre is due to the high levels of turbulence within the annulus. The port leading edge value is under-predicted due to the omission of the through-port vortex and both front and back port edges appear to suffer from a lack of grid

254

V/Uc ( M e a n Radial)

U/Uc ( M e a n Axial) 5

~

x = -10mm

,_.....

~

0

/ o o"~.,

~

-5

3;

4;

50

60

~.o.oooo..

~~

#

70

,

;

o,

,o

-10 0 0/

x = -5mm

, i...~'-

!

-5 ~

~

o

f

5]J

,

-

v

20

,

,

,

,

30

40

50

60

70

.oLe'~

,,~,j .

0

~ e~0 5o/,.-~;'s"?o

-10

-10

x = 5ram

--'7o--20

-5

.

20

I~~"

30

.

. . . 40 50

-5 ~ o o O -

"',~ 0

-5

70

60

30

40

50.r

760

,-"

~\

o'~, -,.-..,,.

-10 1 0.5

x = 40mm

x ~ Z : . 7 -Oooooo " ~- 10

20

30

, ~ 40

, ' - ~ 1 7, 6 1 7 -.~ ~ 6

50

60

70

~ r G ~ ~ . ~ m _ ~ ,~_o ,

0

,

-0.5

-~'

N

^

' "~ - - ~ , k . O , ~' ' 20 ~176176 '

10

3; 1

0

x = -5ram

oooooOOoo _. -_._,_

Radius (mm)

32~ O o o

_

~ 0

10

20

30

,o_o_o.o oo~. ....

~

.

.

,

0

40

20

d"o--'6--~ ', , Q

50

60

70

I I

0

0

,

40

0

10

,

20

o

o -a'-~

50

60

----r-

,

,

--~

,

0

70

30

-~'-~'-o--u-'. ,

40

,

,

50

O

60

70

~

0

~~

x = 40turn 10

20

--'"

30

10

30

40

50

,

20

40

o

o

60

Radius (ram)

0

60

70

_o o

,

30

,

40

,

50

,

60

-r

70

Oo ~

,

10

o

~

20

,

30

,

40

...... , 50

b'--6"" t~'-;, , 60

70

o

..0__6......

o

70

'(;'-~-~

50

o~

~?.0o

/

9 b

0

20

~o~2&-..

'Oooo o "'-..._ ,

~

10

o

- .....

'~176176

30

~oo

X = 5111111

Oo

,"

o.o o o . ""-------o ,

,

10

o

v/Uc ( R M S Radial)

u/Uc ( R M S Axial)

2

o

-1

Radius ( r a m )

x = -10ram

o

~ ' ~ ' " " . . . . . "r 50 60 70

10

20

30

40

50

60

70

Radius (mm)

Figure 4: Mean (top) and RMS (bottom) velocity profiles. - Case I, - Case II, - RANS)

(o = Experimental,

(The experimental and RANS data are from Spencer (1998))

255

Figure 5: Port mean velocity, jet inlet angle and radial RMS velocity profiles. (o A [] = Experimental, - Case II)

resolution. It appears the nature of the port inlet turbulence field (uncorrelated in the uncoupled simulation and correlated but incorrectly distributed in the coupled simulation) has little affect on the resulting core field where turbulence levels appear to be dominated by that generated at impingement.

CONCLUSIONS Large Eddy Simulation has been applied to a non-reacting uncoupled and coupled primary combustion zone. In the uncoupled case the use of measured mean and turbulence profiles, directly providing the port boundary conditions, appears to greatly improve upon the use of uniform inlet profiles (wellknown to be deficient, e.g. McGuirk et al (2001)) resulting in a good description of the mean and turbulence fields within the combustor core. The main disadvantage of this uncoupled method is the expense of the detailed experimental data required. The coupled LES appears to be capable of simulating the mean port velocity profiles to an accuracy sufficient to produce a well-predicted core impingement location and strength, though it is arguable whether the uncoupled or coupled simulation gives the best overall results. The coupled case does not predict the through-port vortex leading to an over-predicted discharge coefficient and a slightly larger than expected core backflow region. The port vortex is generally considered a major flow feature, however its omission in both the uncoupled (where the port is simply blocked) and coupled cases does not appear to have a dramatic affect on the core flow-field for the particular flow conditions applied in these simulations. The benefits of the fully developed and correlated turbulence stemming from the ports in the coupled case seems limited, as it appears the core turbulence field is dominated by that generated at impingement. Comparison of LES results with RANS data shows major improvements in the turbulence field, however both methods

256 give a similar mean velocity distribution close to the jet inlet region. The main discrepancies in RANS results are in upstream regions where the method fails to predict the strong upstream bursts beyond the mean stagnation point. This is not the case for LES, with both uncoupled and coupled simulations capturing these bursts and the subsequent upstream characteristics.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support provided by the Commission of the European Union under contract number G4RD-CT-2000-00402 (MOLECULES).

REFERENCES

Adkins, R.C. and Gueroui, D. (1986). An improved method for accurate prediction of mass flows through combustor liner holes. ASME, 86-GT- 149. Daly, J. (1994). Annulus/Port Flows. Department of Transport Technology, Loughborough University. Germano, M., Piomelli, U., Moin, P. and Cabot, W.H. (1991). A dynamic sub-grid scale eddy viscosity model. Physics of Fluids 3:1790-1765. Gicquel, C.P., Schonfeld, T. and Poinsot, T. (2002). LES of apposed jets in cross-flow. Proceedings and Trends in Numerical and Physical Modeling of Turbulent Processes in Gas Turbine Combustors, Darmstadt University of Technology. Jones, W.P. (1994). Turbulence modeling and numerical solution methods for variable density flows. Libby, P. and Williams, F. (Editors). Turbulent Reacting Flow. Academic Press: 309-347. Manners, A.P. (1987). The calculation of the flow through primary and dilution holes. Rolls-Royce report CRR00290. McGuirk, J.J. and Spencer, A. (2001). Coupled and uncoupled CFD prediction of the characteristics of jets from combustor air admission port. Journal of Engineering for Gas Turbines and Power, 123: 327-332. Smagorinsky, J. (1963). General circulation experiments with the primitive equations. Monthly Weather Review, 91:99-164. Spencer, A. (1998). Gas turbine combustor port flows. PhD Thesis, Department Transport Technology, Loughborough University. Spencer, A. and Adumitroaie, V. (2003). Large eddy simulation of impinging jets in cross-flow. The 48th ASME Gas Turbine and Aeroengine Technical Congress, Atlanta, GT2003-38754. Van Leer, B. (1974). Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second order scheme. Journal of Computational Physics, 14: 361370. Wille, M. (1997). Large eddy simulation of jets in cross-flow. PhD Thesis, Department of Mechanical Engineering, Imperial College London.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

257

LES study of turbulent boundary layer over a smooth and a rough 2D hill model

Tetsuro Tamura 1 , Shuyang Cao 2 and Azuma Okuno 3 1,3

Department of Environmental Science and Technology, Tokyo Institute of Technology G5-7, 4259 Nagatsuta, Yokohama 226-8502, Japan 2 Wind Engineering Research Center, Tokyo Polytechnic University 1583 Iiyama, Atsugi 243-0297, Japan

ABSTRACT Large eddy simulation (LES) is carried out to investigate the turbulent boundary-layer type of flows over a hill-shaped model with a steep or a relatively moderate slope. Also, we focus on the surface condition of a hill, such as roughness effects as well as curvature effects. For the Sub-grid Scale (SGS) modeling of LES, the dynamic Smagorinsky model is used. Instantaneous velocity profile of turbulent boundary layer on the corresponding surface is imposed at inflow boundary. In order to generate time-sequential data of inflow turbulence, the method by Lund (1998) or Nozawa and Tamura (2002) is employed for a smooth- or a rough-wall turbulent boundary layer. According to the preceding paper (Tamura and Cao, 2002), the wake flow of a steep hill forms a separation region and generates unsteady large-scaled vortices. Also, it was shown that the separated shear layers behavior and the vortex motions are sensitively affected by the oncoming turbulence, such that the shear layer comes close to the ground surface, or the size of a separation region becomes small. This paper discusses unsteady phenomena of the wake flows over a smooth and a rough 2D hill-shaped obstacle and aims to clarify roughness effects on the flow patterns and the turbulence structures distorted by the hill. For the numerical validation, we also carry out the wind tunnel experiment for the same model at Re=4300-5000, which is defined by the hill height and the velocity at the hill height. The comparison between LES and experimental data are performed.

KEYWORDS inflow turbulence, LES, curvature effect, turbulent boundary layer, hill wake, rough surface

INTRODUCTION There have been thus far a great number of studies regarding turbulent flows over a hill-shaped

258 obstacle from an engineering point of view, such as prediction of atmospheric diffusion affected by topography, site selection for wind energy or wind-resistant design of structures to strong wind. This kind of non-equilibrium flow has been mainly examined by analytical method, wind tunnel experiment or field measurement. Jackson and Hunt(1975) showed the linear theory for prediction of turbulent flow over a low hill. Britter et al.(1981) carried out wind tunnel measurements for mean velocities and turbulence intensity over a rough, bell-shaped two-dimensional hill and compared them with the results by the linear theory and the finite difference method with mixing-length turbulence model. It is found that these theoretical approaches are limited in applicability to hills of low slope. Also, Mason and King(1985) presented field measurements the mean flow and turbulence statistics over an isolated, roughly circular hill as well as the results from a model which makes a careful application of linear theory to real terrain. They show the linear theory fails to predict the extent of the velocity reduction in the lee of the hill. Athanassiadou and Castro(2001) presented laboratory experiments of aerodynamically fully rough, neutral flow over a series of sinusoidal hills. They compared the experimental data with the linear theory predictions for various components of the turbulence statistics. Recently the technique of computational fluid dynamics (CFD) is also often applied to above problems, basically using the turbulence modeling. Ying and Canuto(1997) studied turbulent flow over two-dimensional hills using a second-order closure model. Ross et a1.(2004) carried out numerical simulations using a range of one-and-a-half order and second-order closure scheme. Comparing with wind-tunnel measurements, they show second-order turbulence scheme generally provides the best agreement with the experimental data, however none of scheme succeeded in providing a good prediction of the components of turbulence quantities in the separation region. As far as the time-averaged values such as mean velocity, sufficiently accurate data can be certainly obtained. But now the required items increase so rapidly that time has come to use even timesequential data of turbulent flows for estimating various kinds of peak values as well as fluctuating characteristics. Establishment of the safety index is expected for a concentration of hazard material gasses or strong wind gust acting on structures. So, the time-dependent computation such as direct numerical simulation (DNS) or large eddy simulation (LES) is expected to be used for the prediction of turbulent flows over a hill model. Henn and Sykes(1999) carried out LES of separated turbulent flow over wavy smooth surfaces in neutral channel. Gong et a1.(1996) performed LES study for aerodynamically rough turbulent boundary-layer flow over a sinusoidal surface. Brown et al.(2001) also carried out LES of turbulent flow over rough sinusoidal ridge. A distributed drag canopy formulation was introduced to model aerodynamically rough surface with physically high accuracy. It has been shown that LES results are in better agreement with the experimental data than the results from a mixing-length closure model. Recently, Allen and Brown(2002) performed LES of turbulent flows over two-dimensional periodic and isolated ridges, which are steep enough to cause separation. Comparing with wind-tunnel measurements and various turbulence closure models, it is shown that the LES results for an isolated ridge are almost reasonable. But for such a computation, most important issue is how to set the inflow condition of the computational domain for the oncoming turbulent boundary layer. For this issue, we have performed the unsteady numerical simulation, mainly DNS, to clarify the effect of oncoming turbulence on the hill wake (Tamura & Cao, 2002) (Tamura, Cao & Shindou, 2003). For the inflow condition of the computational domain, instantaneous velocities of turbulent boundary layer were imposed at each computational time step based on the method by Lund, Wu & Squires (1998). In comparison with the computed flows over a hill with a steep and a relatively moderated slope, we found that the wake flows over a steep obstacle with the slope exceeding the critical angle generally form a separation region and generate unsteady large-scaled vortices. Also, we showed that the separated shear layers behavior and the vortex motions are sensitively affected by the oncoming turbulence, such that the shear layer comes close to the ground surface, or the size of a separation region becomes small. Otherwise, in view of the turbulence modeling of the complicated flows with velocity acceleration or deceleration, favorable or adverse pressure gradient and surface curvature effects etc., we provided DNS data of turbulence structures including variation of turbulent energy budget by the hill shape.

259

The present paper, as a next step of this study, focuses on the surface condition of a hill. Large eddy simulation (LES) is carried out to investigate the turbulent boundary-layer type of flows over a 2D hillshaped model with smooth or rough surface. This paper discusses unsteady phenomena of the wake flows affected by the surface roughness and aims to clarify the turbulence structures distorted by a smooth or a rough hill. In order to generate inflow turbulence, the method by Lund et al. (1998) or Nozawa & Tamura (2002) is employed for a smooth- or a rough-wall turbulent boundary layer. For the SGS modeling of LES, the dynamic Smagorinsky model is used. Computational model for LES of wake flows over a hill shape is given by the sinusoidal curve. The critical slope for the existence of separation is thought to be 16 degree, so we provide two examples which have maximum slopes of 12 and 32 degrees. For the spatially developing turbulent boundary layer on the smooth flat plate, the momentum thickness Reynolds number is set to 1760, while 1430 for the experimental data by DeGraaff et al. (2000). For the LES of the flow over a hill, the Reynolds number defined by the hill height and the velocity at the hill height is equal to 1930 and 1740-3240 for the case of smooth and rough surface. For the numerical validation, we also carry out the wind tunnel experiment for the same model at Re-4300-5000, which is defined by the hill height and the velocity at the hill height. Rough surface is given by actually distributing the roughness rectangular blocks on a flat plate, because the roughness parameters such as roughness height and roughness density can be explicitly and accurately estimated. Maliska's type (1984) of staggered mesh is employed for finite difference approximation of the incompressible Navier-Stokes equations in curvilinear coordinate system.

PROBLEM FORMULATION

In this paper two kinds of hill shapes are considered, subjected to incident turbulent flows of boundary-layer type. The geometrical shapes of the hills are defined by the following equation of sinusoidal curve, y = H cos ~(~-L-) where H is the height of the hill and L is the length as shown in Figure 1. Taking into consideration that the critical slope for the existence of separation is 16 degree, we provide two examples which have maximum slopes of 32 ~ (L/H=I.O/0.4) and 12 ~ (L/H=3.0/0.4). In this study we shall denote the stream-wise, wallnormal, and spanwise velocity components by u, v, and w, with the corresponding coordinates being x, y, and z. The numerical model consists of two computational domains, one is for the main simulation of the turbulent flow over the hill, and the other is for the auxiliary simulation for generation of inflow turbulence. Both simulations

Figure 1 Computational model for LES of wake flows over a hill shape given by the sinusoidal curve

Figure 2 Schematic of numerical model for flow over a hill to oncomin~ turbulence

(1)

260 are carried out simultaneously and the data numerically obtained in the auxiliary simulation are provided at every time step as the inflow boundary condition for the main simulation (Figure 2). Cartesian coordinate and generalized coordinate systems are adopted for the auxiliary simulation for inflow generation and the main simulation of hill flow respectively. The method by Lund (1998) or Nozawa & Tamura (2002) is used for the auxiliary simulation of spatially developing smooth- or rough-wall turbulent boundary layer, where the velocity field at a downstream station is rescaled, and then reintroduced as a boundary condition at the inlet. Details of numerical method for inflow turbulence can be referred to the preceding paper (Tamura & Cao, 2002), so we focus on explanation of the numerical method for the hill flow in the main computational domain.

GOVERNING EQUA TIONS I N THE GENERALIZED COORDINATE SYSTEM The numerical model for the flow over the hill is formulated by using the generalized curvilinear coordinate system. As the filtered governing equations for LES, the continuity and the Navier-Stokes equations are given as follows, Olgi - OU m Ox i -- J - ~ m = 0 Jr"

J ~t

(UmU,)=

a~m

---~m~7-~X, p

(2)

.-1---~ R e a~m

a mn

_ a~n

)

.

--~m

Z'#

(3)

where 1 a~, n

U m -- - -

J axj

Uj

(4)

and

G mn - 1 a~m a L

(5)

~'0" --" ltillJ -- HiUj

(6)

J axj axj

where all variables are given as a filtered component, and Ug, p, z"U, t and Re denote velocity, pressure, the SGS stress tensor, time and the Reynolds number, respectively. J is the Jacobian, Um is the contravariant velocity component in the generalized coordinate system, and G mn is the mesh skewness tensor. The SGS stress tensor is given as a residual stress of Eqn.6 and modeled by the dynamic Smagorinsky model as follows,

r,j-~1

I= (2sTsTy ,2

-/

-

, ~ = j1/3 , $ 7 __

L V -u t u t -u i u g

,

O ~max j O ~m~iU "at--a- ~m~xi-~U ~ mj

M tj - A

2

1

Mi,)

_

NUMERICAL DIS CRE TIZA TION AND ALGORITHM Staggered mesh of Maliska's type (1984) is employed for finite difference approximation of the Navier-Stokes equations. All the velocity components, u, v and w, are defined at each cell surface

261 (Figure 3). Second-order central difference and second order interpolation are used. The continuity equation is discretized using the contravariant velocity components. The numerical procedures are based on the fractional step method with the Adams Bashforth scheme for the convection and the SGS stress terms, and the Crank Nicolson scheme for the viscous term, which is briefly described below.

n

u7 = u i + A t x J

(

ln_l

C'/--~C

i

1

2ReO~m

O (Umu,),, _ O~,,, ( J Oxj

(9)

Um9 = )1 ~/)~mu , * 0 (

O0n+l)

O~m Gmn O~n

(10)

I OU:

(11)

= At O~m

9 ( ~ 1

U~+I =Um - A t Gm" O~,,

u.+, = j( Ox~ i

pn+l =

0n+l

(8)

a~n

a (1 O< ,. )"

nw - m~

Ci

0 (Gin,, O(u7 +uT)) )

.-[-----

(12)

)

~C}~mUn+l

__

(13)

I i~tX~=_~(GmnOCn+l 1 2P.e

0~:;m~

(14)

"0~:;n )

At first, the intermediate Cartesian velocity component can be calculated implicitly by Eqn.8, and the intermediate contravariant velocity components can be obtained by conducting coordinate transformation according to Eqn.10. Then, the scalar potential field 0 of next step can be computed iteratively by solving the Poisson equation, Eqn.ll, with the successive over relaxation (SOR) method. Also, the contravariant velocity components at next time step can be calculated by Eqn.12. Finally, velocity component and pressure at next step can be obtained by Eqns.13 and 14.

U2~ u2 ~[ PQ u2~ ~u

Figure 3 Staggered grid of Maliska's type

B O U N D A R Y CONDITION OF M A I N COMPUTA TIONAL D O M A I N The boundary conditions for main simulation are assumed to be as follows: Bottom surface: No slip condition for velocity, Neumann condition for pressure; Top surface: Free slip condition for velocity, Neumann condition for pressure; Spanwise: Periodic condition for velocity and pressure; Outflow boundary: Convective boundary condition of the form b 0-7+ C~x x = 0 is applied for velocity and pressure. In order to ensure global mass conservation, c is taken to be the bulk velocity. Inflow condition: Spatially developed turbulent inflow data is imposed. An auxiliary simulation is carried out simultaneously to generate the inflow turbulence.

262

M O D E L I N G FOR BO UNDAR Y S U R F A C E WITH R O U G H N E S S BLOCKS To represent a rough surface, rectangular blocks are actually distributed on the surface as a roughness element without any assumption such as roughness length. A method proposed by Goldstein (1993) is used to impose the no-slip boundary condition at surfaces of a roughness element. The following forcing term (Eqn.15) is added to the governing equations. The coupling of this term and the time derivative acceleration term corresponds to the equation for a kind of oscillation system.

fi(t) = ~ ,~ui(t')dt" + flui(t )

(15)

Goldstein provided Eqn.16 for the stable condition of At and the parameters of oscillation equation, cr and/3, when the time marching of the forcing term is done with a second order accurate Adams Bashforth scheme. At < - fl - 4(fl2 - 2a/c) (16) 6g where, k is a problem dependent constant of order one. In our case, it is appropriate to set k equal to 1. o: and fl are selected so that damping ratio derived from o~and/3 equals 1, that is to say, to reach the critical damping condition. In this way, the right hand side of the Eqn.16 can be prescribed Figure 4 velocity vectors around roughness elements as one parameter. At can be decided from numerical stability of flow computation. We have to give one parameter to satisfy Eqn.16. For faster convergence of an oscillation system, we should decide one parameter to minimize the right hand side of the Eqn.16 as small as possible. Figure 4 shows an example obtained by the present method for computed flows around roughness elements.

N U M E R I C A L E X A M P L E S A N D CONDITIONS FOR L E S OF THE FLO W 0 VER A H I L L The main computational domain for a hill model is set to be 40H*15H*4H, with corresponding grid numbers of 220, 129 and 40 in the streamwise, wall-normal and spanwise directions. The grid is fitted for a hill shape and normalized near the surface of the hill. The auxiliary computational domain for turbulent boundary layer has 30H*15H*4H with 120"129"40 grid points. For a rough surface, we actually array rectangular blocks in a staggered pattern on the ground surface. Its roughness density is equal to 4.1%. A rectangular block used is 0.5H*O.25H*O.3H discretized by 2*32*3 meshes. Additive forcing term realizes stable computation even for coarse resolution in the horizontal directions to a roughness element. Mesh sizes in the horizontal directions are Ax+(wall unit)=34.5 and Az+= 13.8 for the case at Rex=790, generally fine enough for wall-turbulence structures. The Reynolds number based on the friction velocity, ur, and the boundary layer thickness, 6, of the turbulent oncoming flow is Rex=660, and is set to Re=l,930 for smooth surface if the velocity at the hill top and the height of the hill, H, are taken as the reference velocity and length, respectively. For rough surface, Rex is 790-2,000 for two cases of oncoming turbulent-boundary flows and the corresponding Re of hill flows is 1,740-3,240. The boundary layer thickness for all cases is almost 5.8H. Considering the roughness height 0.25H, the roughness Reynolds numbers (Re*) based on the roughness length can be estimated 1.07 and 4.01 for both cases.

WIND TUNNEL E X P E R I M E N T S We also provide experimental data of flow velocity. The experiments were conducted in the 1.0m wide, 0.8m high and 7.0m long test section of an open circuit wind tunnel. Rough surface condition

263 was modeled by placing a large number of small roughness elements (5mm cube) on the tunnel floor and hill surface in a staggered arrangement whose roughness density equals to 4.1%, where roughness density was defined as the total roughness front area per unit ground area. Measurements of the reversed and high turbulent flow were performed by using split-fiber probe in conjunction with a 90N10 DANTEC constant temperature anemometer system. The depth of the oncoming boundary layer was controlled to be 6.25H for both surface conditions. Rex of the oncoming boundary layers were 1488 and 1984, Reynolds number based on the velocity at the crest height and the hill height was 5000 and 4300 for smooth and rough conditions respectively. Re* for rough surface is equal to 1.17. Snyder and Castro(2002) show the surface becomes aerodynamically rough at Re*>l in the case of a sharp-edged roughness element used. N U M E R I C A L R E S UL TS

We show the numerical results concerning both of spatially developing turbulent boundary layer and hill flows. We also compare the numerical results with experimental data. Please note the Reynolds number of the experiment is a little higher than the LES case. Figure 5 shows the vertical profiles of averaged velocity and turbulence intensity for the spatially developing turbulent boundary layer on the smooth flat plate at Re (the momentum thickness Reynolds number) = 1760. It can be concluded that the generated inflow turbulent statistics are in good agreement with the experimental data by DeGraaff et al. (2000). Figure 6 shows the vertical profiles of averaged velocity and turbulence intensity for the rough-wall turbulent boundary layer. They also agree well with our experimental data. Figure 7 illustrates the instantaneous vorticity contours over a hill in the turbulent boundary layer. In the case of a steep hill with smooth surface, the separated shear layer is strongly affected by the oncoming turbulence, so shows a flapping motion and forms an unsteady small wake. In the case of a gently-sloped hill with rough surface, we cannot find a separation region clearly. It can be seen that the fine structures of turbulence are enhanced by the surface roughness, accordingly sufficient turbulent energy is generated close to the ground surface. Time-averaged streamlines around a smooth and a rough (represented by roughness rectangular blocks) hill show that the recirculation zone of the roughhill wake (7.8H) becomes a little narrower and longer than the smooth hill wake (7.3H) (Fig.8). In the case of the gently-sloped hill, we cannot find clearly the separation region behind both the smooth and the rough hills. Figures 9 and 10 display the vertical profiles of averaged velocity and turbulence intensity for a smooth and a rough hill with a gentle slope. We can recognize larger turbulence intensity in the wake of the rough hill. LES results are in very good agreement with the experimental data. Figures 11 and 12 display the vertical profiles of averaged velocity and turbulence intensity for a smooth and a rough hill with a steep slope. It is very interesting that the turbulence intensity in the separation region of rough surface becomes weaker than the one of smooth surface. Consistency with experimental data is good even in the wake region for rough case at higher Re, due to sufficiently high Re*. CONCLUSION We have carried out LES analysis to investigate the turbulent boundary-layer type of flows over a smooth or a rough 2D hill model with a steep or a relatively moderate slope. Also, we can obtain the time-sequential data of inflow turbulence with reasonable consistency with experimental data as a target, by using the method by Lund (1998) or Nozawa and Tamura (2002) for a smooth- or a roughwall turbulent boundary layer. We discuss unsteady phenomena of the wake flows over a smooth and a rough hill-shaped obstacle and clarify roughness effects on the flow patterns and the turbulence structures distorted by the hill with two kinds of slope. It is shown that the separation bubble formed

264

in the wake of a rough hill becomes larger than the one for a smooth hill. In comparison with the experimental data at a little higher Reynolds number, LES results in the case of a gently-sloped hill show sufficiently good agreement for the wake flows including turbulence intensity. However in the case of a steep smooth hill there is certainly a discrepancy between LES and experimental data for the turbulence structures in the separated region. For a steep rough hill LES and experimental data agree well at higher Re due to sufficiently high Re*. R e f e r e n ces

Allen T. and Brown A.R. (2002). Large-eddy simulation of turbulent separated flow over rough hills Bound.-layer Met. 102, 177-198. Athanassiadou M. and Castro I.P. (2001). Neutral flow over a series of rough hills: A laboratory experiment. Bound.-layer Met. 101, 1-30. Britter R.E. and Hunt J.C.R. et al. (1981). Air flow over a two-dimensional hill: studies of velocity speed-up, roughness effects and turbulence. Quart. J. R. Met. Soc. 107, 91-110. Brown A.R. and Hobson J.M. et al. (2001). Large-eddy simulation of neutral turbulent flow over rough sinusoidal ridges. Bound.-layer Met. 98, 411-441. DeGraaff D.B. and Eaton J.K. (2000). Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319-346. Gong W. and Taylor P.A. et al. (1996). Turbulent boundary-layer flow over fixed aerodynamically rough twodimensional sinusoidal waves J. Fluid Mech. 312, 1-37. Henn D.S. and Sykes R.I. (1999). Large-eddy simulation of flow over wavy surfaces. J. Fluid Mech. 383, 75-112. Jackson P.S. and Hunt J.C.R. (1975). Turbulent wind flow over a low hill. Quart. J. R. Met. Soc. 101,929-955. Lund T.S., Wu X. and Squires K.D. (1998). Generation of turbulent inflow data for spatially developing boundary layer simulations. J. Comput. Phys. 140, 233-258. Mason P.J. and King J.C. (1985). Measurements and predictions of flow and turbulence over an isolated hill of moderate slope. Quart. J. R. Met. Soc. 111,617-640. Maliska C.R. and Raithby G.D. (1984). A method for computing three-dimensional flows using non-orthogonal boundary-fitted co-ordinates. Int. J. Numerical Methods in Fluids 4, 519-537. Nozawa K. and Tamura T. (2002). Large eddy simulation of the flow around a low-rise building immersed in a rough-wall turbulent boundary layer. J. Wind Engrg. Ind. Aero. 90, 1151-1162. Ross A.N. and Arnold S. et al. (2004). A comparison of wind-tunnel experiments and numerical simulations of neutral and stratified flow over a hill. Bound.-layer Met. 113,427-459. Snyder W.H. and Castro I.P. (2002). The critical Reynolds number for rough-wall boundary layers. J. Wind Engrg. Ind. Aero. 90, 41-54. Tamura T. and Cao S. (2002). Numerical study of unsteady flows over a hill for the oncoming boundary-layer turbulence. ETMM5, 237-246 Tamura T., Cao S. and Shindou T. (2003). Turbulence structures of flow over a two-dimensional hill-shaped model. Proceedings of FEDSM'03 Symposium on Modeling and Simulation of Turbulent Flows, 4TH ASMEJSME Joint Fluids Engineering Conference 45354, 1-8 Ying R. and Canuto V.M. (1997). Numerical simulation of flow over two-dimensional hills using a second-order turbulence closure model. Bound.-layer Met. 85,447-474. 30 . . . . . . . . . . . . . . .

I~_ Present calcu. Reo =1760 DeGraafsRe o =1430 log law

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265

(a) averaged velocity (b) turbulence intensity Figure 6 Vertical profiles of averaged velocity and turbulence intensity for the spatially developing turbulent boundary layer on the rough flat plate

Figure 7 Instantaneous vorticity contours over a hill in turbulent boundary layer

Figure 8 Time-averaged streamlines around a steep hill

266

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Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

267

FLOW FEATURES IN A FULLY DEVELOPED RIBBED DUCT FLOW AS A RESULT OF LES M. M. Loh~sz, 1'2 P. Rambaud 2 and C. Benocci 2 Department of Fluid Mechanics, Budapest University of Technology and Economics, Budapest, 1111, Hungary 2 Von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, 1640, Belgium

ABSTRACT The present contribution concerns the simulation of a turbulent flow in a stationary square section ribbed duct by means of LES method. The pitch distance of the ribs was fixed to p/h=10, together with a high blockage value of 0.3 h/D, and a Reynolds number (defmed with bulk velocity (U) and hydraulic diameter (D)) of 40000. These parameters were chosen to reproduce the experiments of Casarsa (2003). An attempt of topology description is proposed, using concepts of streamsurface, vortex core detection, wall streamline and bifurcation lines. Furthermore, knowledge on flow topology derived from these concepts is qualitatively compared with isosurface of second scalar invariant of the velocity gradient tensor (Q) used on the averaged and/or instantaneous velocity fields. Differences between a 2D fibbed channel and the present flow, where sidewall effect are dominants, are highlighted. Especially, the existence of a strong secondary flow on the windward side of the rib is revealed by the visualization of streamsurfaces, wall streamlines and bifurcation lines. Vorticity dominated regions are detected and extracted using isosurfaces of the (Q). Thanks to well-positioned streamsurfaces and a vortex core detection method, the effect of the lateral walls on the wake of the rib is put in evidence. Apart of this analysis of the average velocity field, a sequence of instantaneous images displaying creation of coherent structures on the leading edge of the rib is presented. These spanwise structures are shaped in Lambda (A) type vortices by a mechanism similar than the one already described in Dubief & Delcayre (2000), showing the capabilities of the present approach to reproduce averaged and instantaneous features of complex flows. KEYWORDS Ribbed duct, LES, flow topology, Q criteria INTRODUCTION Flow in ribbed ducts can be considered representative of the intemal cooling channels used to cool turbine blades. The presence of ribs increases turbulence levels and, as a result, it enhances heat transfer. Detailed knowledge of the heat transfer and flow field is important for the designer to avoid material damage caused by overheating. In past years, this class of flow has been extensively investigated by experimental means at von Karman

268

Institute for Fluid Dynamics (Rau & al. (1998), ~akan (2000), Casarsa & Arts (2002a) and Casarsa & al. (2002b). The complex topology of the flow had already been put in evidence by ~akan (2000) and studied in depth by Casarsa & Arts (2002a), Casarsa & al. (2002b) and Casarsa (2003) by means of Particle Image Velocimetry (PIV). To better understand the complexity of this flow, a companion numerical research had been performed by means of Large-Eddy Simulation (LES). The practical interest and the fundamental challenge of the complexity of this flow have motivated earlier DNS and LES investigations. As a nonexhaustive bibliographic survey on this topic one may list the first investigations of the flow in a fibbed plane channel by Ciofalo & Collins (1992), and Yang & Ferziger (1993) as well the most recent contributions by Miyake & al. (2002), Cui & al. (2003), Leonardi & al. (2003), Leonardi & al. (2004), Tucker & Davidson (2004), Ashrafian & al. (2004) and Nagano & al. (2004). In these investigations, the flow was considered as 2D-like with the use of periodic boundary conditions in the transversal direction. Therefore, these simulations were only a partial representation of the case of practical interest, namely, flow in fibbed duct, where the presence of sidewall creates a 3D flow in the mean. In fact, the sidewall strongly affects the flow and a truly three-dimensional representation is needed. This necessity leads to focus only on such class of simulations, of which, to authors' best knowledge, only few studies had been dedicated up to now. Among them, one may cite; Watanabe & Takahashi (2002), Murata & Mochizuki (2000), Murata & Mochizuld (2001), Abdel-Wahab & Tafii (2004), Ahn & al. (2004), Sewall & Tafti (2004) and Tyagi & Acharya (2004). In most of these investigations, the main interest was focused on the reproduction and prediction of Eulerian statistics (average) and resolved turbulent field but without real topology analyses. However, as already mentioned above, experimental results of Casarsa & Arts (2002a) and Casarsa & al. (2002b) and Casarsa (2003) have put in evidence the real complexity associated to this topology and the difficulty to extract it from 2D experimental measurement (PIV). It is the reason why the present investigation focuses on the topology of the averaged flow field. For this purpose, different visualization tools have been proposed in the past (Garth & al. (2004) gives a summary), and among them the streamlines and stream surfaces proved to give the most fruitful information. For example, they have allowed putting in evidence the secondary flow in transverse plane as well the mechanics of the separation before and on the rib. Further insights on separation and reattachment regions have been gathered by using wall streamlines and bifurcation lines. Furthermore, existence and importance of coherent structures in separated wall flow have already been put in evidence (Dubief & Delcayre (2000), Lesieur & al. (2003), Ashrafian & al. (2004) and Tyagi & Acharya (2004)) applying the Q criterion from Hunt & al. (1988). In the present article, this criterion has been applied to the averaged and instantaneous velocity field, bringing complementary information leading to a better understanding of the complex character of this flow.

Description of the Flow The present authors have studied the flow in a square section duct where successive ribs of square cross section are mounted on one wall perpendicularly to the stream direction. A Reynolds number of 40000 based on the hydraulic diameter (D) and the bulk velocity (U) is defined. The dimensionless lengths characterizing the geometry are the rib size (h/D = 0.3) and the pitch distance (p/h = 10). In Casarsa & Arts (2002a) and Casarsa & al. (2002b), it has been experimentally found that for such configuration the flow starts to repeat itself in every pitch length after the fourth ribs. Therefore, the simulation is confined to one pitch length and periodic boundary conditions in the streamwise direction are used. Furthermore the incompressible hypothesis is adopted for this study. NUMERICAL SOLUTION

SGS modelling The flow field is simulated in the general framework of Large Eddy Simulation (LES). A complete survey of this technique is not the goal of this article but may be found in Sagaut (2002). For LES, the general strategy is based on the hypothesis that the turbulent field can be separated in the large, energy containing

269

eddies, to be explicitly resolved by the mean of a time-marching calculation, and in the small scales eddies to be modelled with an adequate sub grid stress model (SGS). Boris & al. (1992) proposed an alternative approach to the modelling with the monotonically integrated LES (MILES) concept. In this last approach, the implicit dissipative-diffusive effect of a high-order upwind scheme with a proper monotone integration of the fluxes is preferred to an explicit SGS model. A similar choice is adopted hereafter. The present investigation was performed applying Fluent 6.1, a general-purpose commercial code for fluid dynamics simulation produced by Fluent Inc. The relevant transport equations are discretized in space with the Finite Volume approach, implemented in a cell-centred collocated variable arrangement in an unstructured structure. The equations for the three components of the momentum are solved sequentially and the pressure-velocity coupling is performed applying the well known SIMPLE algorithm (see Ferziger & Peri6 (2002) for details). The interpolation of momentum fluxes in the advection term is made with a second order upwind scheme limited by a slope limiter, while the pressure is interpolated with a second order interpolation scheme. Time integration is done using the second order Gear's method in an implicit formulation. The solid walls are modelled using the classical linear-logarithmic law condition for velocity. As anticipated, a periodic condition is imposed for the streamwise direction and a forcing term is added to the streamwise momentum equation to maintain a constant mass flux in time. The time step was fixed at 0.005D/U, which approximately corresponds to an average CFL value of 0.3 over the whole domain with a maximum value of 3 encountered in some small, high velocity cells. Nevertheless, this average value respects the criteria proposed by Choi & Moin (1998) to ensure accuracy in time for LES. Eulerian statistics for mean and resolved turbulent quantities were satisfactorily validated against the corresponding measurements (Loh~isz& al. (2003) & (2004)). In Figure 1 profiles extracted in the symmetry plane and at half rib height are shown, except some small differences in the streamwise velocity, the agreement in averaged velocities is remarkable good. This is allowing the present authors to consider that the methodology is valuable and adequate to resolve and to reconstruct the topology of the flow. xn,-

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Figure 1" Comparison between computed (symbol) and PIV result of Casarsa (2003) (solid line) Left) In the symmetry plane (Z/h=0) a) X velocity, b) Y velocity Right) At half rib height (Y/h=0.5), a) X velocity, b) Z velocity Grid

Simulations were performed over a grid, unstructured in X-Y planes and structured in spanwise direction. The present unstructured mesh (union of quadratic cells) is presented in Figure 3. In the near wall regions, a structured layer with a thickness of 0.08D is used. The size of the first inner cell is 0.003D and the cellto-cell ratio is 1.09 when moving away from the wall. Finally, the wall layer spans respectively over 8 cells on the lateral sidewalls, 14 cells over the top of the rib and 10 cells above all other walls. The typical distance from the wall of the first cell centre is about 3,+=5 in wall units based on an instantaneous wall shear velocity.

270

Figure 2: Grid layout on lateral, bottom and rib walls for Z<0 (black lines). Streamlines in the symmetrical plane (white lines). Streamsurfaces illustrating upstream recirculation tube and recirculation tube connecting bottom wall upstream the rib and the top comer of the duct (light grey). RESULTS

Topology of the averagedflow The general phenomenology of flow above a ribbed surface characterized with high pitch distance and at high Reynolds number is illustrated in Figure 2. Interested reader should refer to Cui (2003) for complementary aspects on this configuration. In Figure 2, the streamlines in the symmetrical plane underline four important recirculation region located respectively: in the comer upstream of the rib; on the top of rib starting from the leading edge; in the leeward comer of the rib; in a large 'wake' downstream of the rib. Although the mean flow in the symmetrical plane may be found similar to the one of a 2D situation, outside the symmetrical plane, the effect of the side and upper wall is strongly felt and the flow becomes completely 3D. Figure 2 shows that the flow in the comer upstream of the rib has a high spanwise velocity component. This component is visualized by the spiralling streamlines marked on a streamsurface forming a tube trapped in the upstream comer of the rib (this streamsurface is going trough the line X/h=- 1.08, Y/h=0.05 in the upstream comer recirculation region). {~akan (2000), Casarsa & Arts (2002a) and Casarsa & al. (2002b) and Casarsa (2003) found similar phenomenon in their experiment. Figure 2 shows the wall streamlines for the entire flow field. It appears that the previously mentioned tube escapes upward in the streamwise direction and maintained in the low momentum area formed by the sidewall boundary layer (Figure 2), while its 'trace' on the sidewall is materialized by bent wall streamlines. Beside these wall streamlines showing a high upward movement in the windward proximity of the wall, Figure 3 also displays bifurcation lines (thick black/white lines), defined as borders where the streamsurfaces detached or reattached themselves from/to the walls (Homung & Perry (1984). As a matter of fact, for a general 3D flow, a wall shear stress component exists aligned in the direction of the bifurcation line, making arbitrary and useless the definition of the separation and reattachment point used for a 2D flow (zero wall shear stress). It is the reason why the extraction method ofHaimes & Kenwright (1999) is used here to remove this arbitrariness and using the cell size to mark the bifurcation line. Unfortunately, this representation is still subject to computational noise due to the always-perfectible level of convergence and may appear on the figures as white and/or black points. These series of points have to been seen (when possible) as continuous lines. As a qualitative scenario concerning the average results, one might propose that the blockage induced by the rib forces the flow in front of the rib to swirl in the sidewall tube transporting it over the rib in an

271

upward motion. This motion is compensated in the central part of the duct by a downward current, alimented by a streamwise elongated couple of counter-rotating structure in the upper part of the duct (Figure 4 (left)). In Figure 4 (fight), attention must be focused on the presence of the high spanwise velocity regions due to the previously mentioned blockage effect of the rib.

Figure 3: Wall streamlines and bifurcation lines; positive ("separation") black lines, negative ("reattachment") white lines. The sidewalls have a strong influence on the shape of the separated region located on windward side of the rib and they also have an influence, although weaker, on the big recirculation region in the 'wake' of the rib. This sidewall effect is linked to a secondary flow which is induced by a spanwise pressure gradient. The static pressure is presented in Figure 5 with respectively the symmetrical plane (left) and the sidewall (fight). Around the stagnation point of the rib leading wall, a region of high pressure is present, resulting in a spanwise pressure gradient driving the fluid toward the side wall (as explained above). On the same Figure 5, one may also notice that an opposite pressure gradient region exists in the 'wake' of the rib. This last pressure gradient induced a motion away from the sidewall towards the symmetrical plane. This swirl away from the wall was already visible through its footprint on wall streamlines of Figure 3 and is confirmed by pattern of the single streamline drawn in the wake of the rib in Figure 4 (right).

Figure 4: Left) Secondary flow patterns in time-averaged flow (black lines), wall streamlines (white lines). Right) Streamsurface starting from X=-2, Y-0.02 (light grey); Streamline reaching the bottom

272

wall at X=5, Z=-0.3 (black line); isosurfaces of regions of high (0.3 U) spanwise velocity; bifurcation lines as in Figure 3

Figure 5: Left) Pressure contours on symmetrical plane, Right) Pressure contours on sidewall. Another consequence of the upstream separation is the production of a high amount of vorticity. Experimental investigation performed at VKI by Casarsa & Arts (2002a) and Casarsa & al. (2002b) and Casarsa (2003) gave indication that a major part of the vorticity field is induced by the sidewall: the boundary layer on the wall displaced by the rib produces extra vorticity in streamwise direction. These rotation-dominated regions can be visualized in instantaneous numerical results by to isosurface of the second scalar invariant (Q) of the velocity gradient tensor (Hunt & al. 1988):

l (~u~o Q=5

- SuSo. )

Where ~ and S are the anti-symmetrical and the symmetrical part of the velocity gradient tensor. Q is nowadays a commonly accepted and applied criterion to visualize the coherent structures embedded in the instantaneous field (for example: Dubief & Delcayre (2000) and Lesieur & al. (2003)) as it will be presented in the next section. Nevertheless, such a detection tool applied on an averaged field had also been found interesting to investigate rotation dominated regions (Ooi & al. (2002)) and it will therefore be presented in this section together with traditional stream surfaces.

Figure 6: Left) left side (Z<0): streamsurface released from location X=-l.08, Y=0.05; fight side (Z>0): Q=0.2 (U2/h2) isosurfaces; streamlines bounding these isosurface (black lines) and wall streamlines (black lines); vortex cores (thick white lines). Right) Streamsurface of time averaged field released from a line at X/h=-3.8, Y/h=0.5 and X/h=-3.8, Y/h=0.25

273

For this purpose, the composite Figure 6 (left) presents in the same frame the information given by a Q visualisation, applied on our averaged flow feld, and the traditional stream surfaces. The left part of this figure displays the stream surface emitted from a line (X/h=- 1.08; Y/h=0.05; -1.6
Figure 7: Left) Streamsurface released from location X=-0.501, Y=I.01 (light grey) and bounding streamlines (deep lines); bifurcation lines thick black and white lines Right) Streamlines entering the downstream recirculation region (released only in negative direction), vortex cores shown by thick black lines This 'wake' structure is better visualized by Figure 7 (left) which shows the details of the separation region on the top of the rib. It may be noticed that this structure is weakly influenced by the sidewall. This fact is put in evidence by the streamlines forming the stream surface (see on the top of the fib), which are almost parallel to the streamwise direction Figure 7 (left). On the contrary, it appears that this wake structure (having a shape and a position controlled by the fib geometry) has a determinant impact on the wall streamlines on the sidewall, as it is illustrated in Figure 7 (fight). On this last picture, one may see a streamline flowing over the rib and entering in the 'wake' recirculation from the side to move towards the symmetrical plane. The bold line inside of the rotating structure represents the core of this recirculation region and was detected by the method described in Sujudi & Haimes (1995). Coherent structures in the instantaneous realisations

Recent studies on separated wall flows (Dubief& Delcayre (2000), Lesieur & al. (2003), Ashrafian & al. (2004) and Tyagi & Acharya (2004)) have shown that the Q criterion is a robust and trustable tool to illustrate coherent structures on multiple type of flow including 2D cavity, back-step and boundary layer. Dubief& Delcayre (2000) and Lesieur & al. (2003) have especially shown that separation on leading edge of a sharp obstacle creates a sheet of vortices due to the Kevin-Helmholtz (KH) instabilities. These vortices are deformed in Lambda (A) shaped vortices which are further convected in the bulk flow. A similar phenomenon is found out on the top of the fib in the present simulations showing the adequacy of

274

our method to reproduce known events (Figure 8).

Figure 8: Formation and destruction of Lambda structure; first row Q=60

(U2/h2)

(U2/h2), second row Q=450

On this figure, the formation of KH structures can be well followed through a sequence of 'Q images', where one image is displayed for every 5-time step. The formation of a vorticity sheet attached to the rib leading edge and its perturbation by the KH instability is shown for two different values of Q. The rib acts as a structure 'generator'; the entire periodic flow is seeded by the structures shed from the rib or the sidewalls. These structures seem to have a lifetime maybe higher than the convective time (which corresponds to the travel a distance of one pitch). Therefore it may be speculated that when the turbulent regime is established by the rib may be in an auto-excited state (it produces structures influencing the production of structures). CONCLUSION Results of Large Eddy Simulation have been used to understand the flow in a ribbed duct representative of systems designed for internal cooling of turbine blades. The flow field of this high blockage ribbed duct had been qualitatively analysed and the effect of the lateral walls have been found very important. The mean flow becomes completely three dimensional in the neighbourhood of the rib. It is observed that the rib induced a flow swirling towards the sidewalls before moving away the rib. At this level, a part of the flow continues to swirl in the direction of the upper comers when another part of the flow enters in the wake recirculation to swirl back to the symmetrical plane. This topology analyses underlines the perturbation induced by the rib. It is found out that the lateral swirls, which create the secondary flow are associated with spanwise pressure gradient. The 'communication' between the two-walls (fibbed-not fibbed) is well underlined by the stream surfaces trapped in an upward motion. This phenomenon has to be kept in mind when non-staggered face-to-face fibbed walls are chosen in the design process. The sidewalls together with the rib produce vorticity (rotating structures) analogous to the upstream comer vortex on the fibbed wall. These vorticity structures arch over the rib close to the lateral wall. This phenomenon was also highlighted using isosurfaces of second scalar invariant of the velocity gradient tensor (Q). The recirculation region on the top of the rib remains almost unchanged in spite the strong side wall effect on the windward side. The form of wake of the rib is only slightly affected by the sidewalls: the region in the immediate vicinity of the lateral wall is of course influenced, but the central region remains quasi-2D. The fluid can enter in the wake only through this perturbed side region, which might be an important result concerning heat transfer. Analysing the instantaneous flow field, it is found that the most relevant phenomenon is the creation of spanwise vortices and Lambda shape structures on the leading edge of the rib and their transport in streamwise direction.

275

The validation of the present method and its ability to reproduce known complex phenomena involving coherent structure generation makes it a promising tool for on going similar study with heat transfer. AKNOWLEDGEMENT

M. M. Loh~tsz would like to thank to the support of the Hungarian National Fund for Science and Research under contract No. OTKA T 037651. REFERENCES

Abdel-Wahab, S. and Taffi D. K. (2004). Large Eddy Simulation of Flow and Heat Transfer in a 90 ~ Ribbed Duet with Rotation- Effect of Coriolis Forces. Proceedings of ASME Turbo Expo 2004 Power for Land, Sea, and Air, June 14-17, Vienna, Austria Ahn, J. Choi, H. and Lee, J. S. (2004). Large Eddy Simulation of Flow and Heat Transfer in a Channel Roughened by Square or Semicircle Ribs. Proceedings of ASME Turbo Expo 2004 Power for Land, Sea, and Air, June 14-17, Vienna, Austria Boris, J. P. Grinstein, F. F. Oran, S. S. and Kolbe, R. L. (1992). New insight into large eddy simulation. Fluid dynamics research. 10. 199-228. (~akan M. (2000). Aero-thermal investigation of fixed fib-roughened cooling passages. Ph.D. Thesis, Universit~ Catholicque de Louvain, Von Karman Institute for Fluid Dynamics, June Casarsa, L. and Arts T. (2002a). Aerodynamic Performance of a Rib Roughened Cooling Channel Flow with High Blockage Ratio. 11thInternational Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8-11 Casarsa, L. (~akan, M. and Arts, T. (2002b). Characterization of the velocity and heat transfer fields in an internal cooling channel with high blockage ratio. Proceedings of ASME TURBO EXPO 2002 June 3-6, 2002 Amsterdam, The Netherlands Casarsa, L. (2003). Aerodynamic performance investigation of a fixed fib-roughened internal cooling passage. PhD Thesis, Universita degli Studi di Udine, Von Karman Institute for Fluid Dynamics Choi H., and Moin P. (1994). Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow, Journal of Computational Physics, 133, 1-4. Ciofalo, M. and Collins, M. W. (1992). Large-eddy simulation of turbulent flow and heat transfer in plane and fib-roughened channels. International Journalfor Numerical Methods in Fluids, 15, 453489. Cui, J. Patel, V. C. and Lin, C-L. (2003). Large-eddy simulation of turbulent flow in a channel with rib roughness. International Journal of Heat and Fluid Flow, 24, 372-388. Dubief, Y. and Delcayre, F. (2000). On coherent-vortex identification in turbulence. Journal of Turbulence, 1, 011 Ferziger, J. H. and Peri6, M. (2002). Computational Methods for Fluid Dynamics, Springer Garth, C. Tricoche, X. Salzbrunn, T. Bobach, T. and Scheuermann, G. (2004). Surface Techniques for Vortex Visualization. Joint EUROGRAPHICS- IEEE TCVG Symposium on Visualization Haimes, R. and Kenwright, D. (1999). On the velocity gradient tensor and fluid feature extraction. AIAA Paper No. 99-3288, Norfolk VA, June, 1999. Homung, H. and Perry, A. E. (1984). Some aspect of three dimensional separation Part I.: Streamsurface bifurcations. Zeitschriftj~r Flugwissensehafien und Weltraumforschung, 8, 77-87. Hunt, J. C. R. Wray, A. A. and Moin, P. (1988). Eddies, Streams, and Convergence Zones in Turbulent Flows Center for Turbulence research, Proceedings of the summer Program.

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Lesieur, M. Begou, P. Briand, E. Danet, A. Delcayre F. and Aider J. L. (2003). Coherent-vortex dynamics in large-eddy simulations of turbulence, Journal of Turbulence, 4, 016 Leonardi, S. Orlandi, P. and Antonia, R.A. (2003). Direct numerical simulations of turbulent channel flow with transverse square bars on one wall, Journal of Fluid Mechanics, 491, 229-238. Leonardi, S. Orlandi, P.Djenidi, L. and Antonia, R.A. (2004). Structure of turbulent channel flow with square bars on one wall. International Journal of Heat and Fluid Flow, 25, 384-392. Loh~isz, M. M. Rambaud, P. and Benocci, C. (2003). LES simulation of ribbed square duct flow with Fluent and comparison with PIV data. Conference on Modelling Fluid Flow CMEF'03 The 12th International Conference on Fluid Flow Technologies, Budapest, Hungary Loh~isz, M. M., Rambaud, P., and Benocci, C. (2004). MILES flow inside a square section fibbed duct. RTO Meeting, A VT-120 Workshop on "Urban Dispersion Modelling" April 1-2., Rhode Saint Genbse, Belgium Murata, A. and Mochizuki, S. (2001). Comparison between laminar and turbulent heat transfer in a stationary square duct with transverse angled rib turbulators. International Journal of Heat and Mass Transfer, 44, 1127-1141. Murata, A. and Mochizuki, S. (2000). Large eddy simulation with a dynamic subgrid-scale model of turbulent heat transfer in an orthogonally rotating rectangular duct with transverse rib turbulators. International Journal of Heat and Mass Transfer, 43, 1243-1259. Nagano, Y. Hattori, H. and Houra, T. (2004). DNS of velocity and thermal fields in turbulent channel flow with transverse-rib roughness. International Journal of Heat and Fluid Flow. 25, 393-403. Ooi, A. Petterson Reif, B.A. Iaccarino, G. and Durbin, P.A. (2002). RANS calculations of secondary flow structures in fibbed ducts. Centerfor Turbulence, Research Proceedings of the Summer Program 2002 Rau G., Moeller D., t~akan M., and Arts T. (1998). The Effect of Periodic Ribs on the Local Aerodynamic and Heat Transfer Performance of a straight Cooling Channel. ASME Journal of Turbomachinery, 120, 368-375. Sagaut P. (2002). Large Eddy Simulation for incompressible Flows. An Introduction 2 na Edition, Springer Sewall, E. A. and Tafti, D. K. (2004). Large Eddy Simulation of the Developing Region of a Stationary Ribbed Internal Turbine Blade Cooling Channel. Proceedings of ASME Turbo Expo 2004 Power for Land, Sea, and Air, June 14-17, Vienna, Austria Sujudi, D. and Haimes, R. (1995). Identification of Swirling Flow in 3 D Vector Fields. Tech. Report, Dept. of Aeronautics and Astronautics, MIT, Cambridge, MA Tsujimoto, K. and Nakaji, M. (2002). Numerical simulation of channel flow with a rib-roughened wall. Journal of Turbulence, 3, 035 Tucker, P.G. and Davidson, L. (2004). Zonal k-I based large eddy simulations. Computers andFluid. 33, 267-287. Tyagi, M. and Acharya, S. (2004). Large Eddy Simulation of Flow and Heat Transfer in Rotating Ribbed Duct Flows. Proceedings of ASME Turbo Expo 2004 Power for Land, Sea, and Air, June 1417, Vienna, Austria Watanabe, K. and Takahashi T. (2002). LES simulation and experimental measurement of fully developed fibbed channel flow and heat transfer. Proceedings of ASME TURBO EXPO 2002 June 3-6, 2002 Amsterdam, The Netherlands Yang, K-S. and Ferziger J. H. (1993). Large-eddy simulation of turbulent obstacle flow using dynamic subgrid-scale model. A/AA Journal 32:8, 1406-1413.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

277

Coherent Structures and Mass Exchange Processes in Channel Flow with Spanwise Obstructions Andrew McCoy, George Constantinescu +, and Larry Weber IIHR-Hydroscience and Engineering, Department of Civil and Environmental Engineering, The University of Iowa, Iowa City, IA 52242-1585, USA + Email: [email protected]

ABSTRACT Large Eddy Simulation (LES) is used to investigate three-dimensional flow phenomena induced by a pair of spanwise vertical obstructions (groynes) in a channel with lateral vertical walls and a flat bed. The main phenomena that govern the physics of this flow include the formation of a horseshoe vortex (HV) at the base of the upstream obstruction and of detached shear layers originating at the tips of the obstructions. In practice, this flow is similar to river flow around groynes. The structure and spectral content of coherent structures inside the HV system are investigated. It is found that the flow in the turbulent HV region is characterized by a very sharp increase in the level of turbulence fluctuations which explains the initiation of scour in the case of a loose bed. The distribution of the bed shear stress was determined and analyzed relative to the capacity of the flow to entrain sediment. The mixing processes (exchange of dissolved matter) between the main channel and the embayment were modeled using a passive scalar. The decay of scalar mass in the groynes area corresponding to the mixing between the dissolved matter present initially in the embayment and the channel was measured and global exchange coefficients corresponding to the initial & final stages of decay were determined.

KEYWORDS Turbulent Horseshoe Vortex System, Coherent Structures, Detached Shear Layers, Large Eddy Simulation, Massively Separated Flows, Mass Exchange, Contaminant Transport

INTRODUCTION Natural river channels are subject to continuous change in geometry due to the interaction between the flow and erodible boundaries, especially if hydraulic structures that disturb locally the flow are present (e.g., scour around bridge-pier and abutment foundations). In particular, the introduction of groynes or spur-dikes is one of the most effective approaches to stabilize eroding banks and to sustain navigable channels at proper depth and location. Though many studies have focused on scour-depth prediction by trying to correlate scour depth with global flow and geometrical characteristics, using data from small-scale laboratory experiments (e.g., Melville, 1997), only a modest few (e.g., Dargahi, 1990) have

278 tried to study the highly turbulent flow fields and the associated coherent structures that initiate and sustain scour. Erosion around these hydraulic structures can lead to significant expense for damage of the structures and poses a high safety hazard. Prominent macro-turbulence flow features associated with the formation of a scour hole around a spanwise obstruction/groyne are the necklace-like vortical structures, commonly known as the horseshoe vortex (HV) system, and the wake eddies shed behind the obstructions. Regardless of the exact form of the obstruction, the HV system is the main mechanism driving the formation and evolution of the scour hole in the case of a movable bed (Dargahi, 1990). To understand how scour develops and rigorously estimate scour depths, it is necessary to describe these flow structures and to quantify their effect on the fiver bed near the base of the structures (e.g., the sharp increase in the local bed shear stress and turbulence fluctuation levels). Another area of great interest is the role played by groyne fields in the exchange of dissolved matter with the main channel. The interaction via the mixing layer at the groyne field/channel interface and via the coherent structures shed from the groyne head can substantially modify the transport and distribution of a pollutant cloud in a river. For instance, flow and exchange of dissolved matter between a main channel and an array of shallower emerged groyne fields were investigated in laboratory models by Uijttewaal et al. (2002) for different aspect ratios and shapes of the groynes. Similar experiments were conducted by Weitbrecht et al. (2003) using a Lagrangian Particle Tracking Method coupled with PIV for velocity and turbulence quantities measurements at the water surface. However, most experimental models estimate exchange rates without taking accurately into account the distribution of the dye concentration over the depth. LES can clarify some important aspects related to the way in which the exchange processes take place over the depth of the groyne field/channel interface and assess the importance of 3D effects on the dominant quasi 2D structures observed in horizontal planes. Quantitative knowledge of these exchange processes will allow calculating global exchange parameters and longitudinal dispersion coefficients that can be used in forecasting models for river pollution. LES has been largely used to study turbulent flows in simple configurations at fairly low Reynolds numbers. Most predictions of engineering flows are obtained using the RANS approach in which the effect of most of the scales on the mean flow is accounted via a RANS turbulence model. In contrast to that, in LES the dynamically important scales in the flow are directly computed, and only the effect of the filtered scales on the large scales is modeled. The interest in LES is mainly motivated by its greater accuracy over RANS, in particular its ability to better predict turbulent mixing. In the present paper we are interested in using LES to investigate the fundamental flow phenomena present in a simplified geometry (rectangular channel with two spanwise vertical obstructions) that is representative of flow in natural rivers with groyne fields. Specific objectives include: 1) studying the formation and structure of the turbulent HV system as it varies in space and evolves in time; 2) studying the effect of the HV system on the bed shear stress; 3) analyzing the mean turbulence characteristics and spectral content of the HV system; 4) studying the exchange of dissolved matter between the embayment and the main channel.

NUMERICAL SOLVER AND GRID GENERATION The numerical solver utilized to perform the present simulations is a massively parallel DNS/LES code developed at CITS in Stanford University that allows for flow simulation in complex geometries. It discretely conserves energy on hybrid unstructured meshes allowing for simulations at high Reynolds numbers without numerical dissipation. All operators in the finite volume formulation are discretized using central schemes. A collocated, finite volume scheme on unstructured grids with arbitrary elements is used to solve the filtered Navier-Stokes equations with a dynamic Smagorinsky model (including scalar transport). The reader is referred to Mahesh, Constantinescu, and Moin (2004) and

279 Mahesh et al., (2002) for an in-depth description of code validation for both internal and external flows with varying degrees of complexity. The depth of the main channel, D, is used as the length scale. The mean velocity in the main channel, U, is used as velocity scale. The two spanwise obstructions extend up to the free surface. The physical domain extends 3D upstream of the first obstruction and 14D downstream of the second obstruction (Figure 1). The domain was inspired by and is representative of one of the experiments performed by Tominaga et. al (2001). The channel Reynolds number is 13,600. The width of the channel is B-3.75D. The length, 1, and width, b, of each obstruction are 0.625D, and 0.25D, respectively. The space between obstructions is S=1.25D. The obstructed area is 17% of the total channel section. The computational mesh contains approximately 3 million hexahedral elements arranged in an unstructured fashion which allows rapid variation in the characteristic size of the elements while maintaining high overall quality of the mesh cells and low stretching ratios. The first row of cells from the wall is located within the laminar sublayer (y+
Figure 1: General sketch of flow in a channel with lateral obstructions

Figure 2: Detail of mesh near the dike showing grid and domain partition on processors

RESULTS Horseshoe Vortex System Due to the adverse pressure gradients and downflow introduced by the presence of the upstream Obstruction, the incoming boundary layer separates and its vorticity reorganizes at the base of the obstruction forming the HV system. On the side with the obstructions the alignment of the HV is in the spanwise direction parallel to the groyne. As the HV approaches the tip of the groyne it is stretched and elongated as it starts turning around the groyne before being lifted from the bottom and dissipated into the downstream flow. The complex spatial structure of the HV system is shown in Figure 3 using instantaneous streamlines in vertical planes (see also Figure 5a) as the HV system stretches around the obstruction. The vertical planes progress from a plane making a 25 ~ angle with the lateral channel wall to a plane making a 65 o angle. The extent of each frame (except h) is bordered by the upstream obstruction wall and the channel bottom. Two main coherent structures corresponding to vortices A and B are observed inside the HV system. Their relative strength is varying along the HV due to the splitting (e.g., frames b & c for B and f & g for A) and remerging (e.g., frames f & g for B) of these vortices. Vortex B appears to also interact with secondary vortices C and D which form immediately downstream of the separating line of the upstream boundary layer. As vortices A and B are lifted away from the channel bottom the HV system starts loosing its coherence and its structure becomes very complex. Figure 4 helps visualizing the evolution of the HV system in time over

280

Figure 3: Instantaneous spatial structure of HV system as it wraps around upstream obstruction wall, commencing with the plane 25 ~ from the lateral channel wall and terminating at the plane 65 ~ from the lateral wall; a) 25~ b) 30~ c) 35~ d) 37.5~ e) 40~ f) 45~ g) 55~ h) 65 ~ The location of each plane in relation to the upstream obstruction is shown in Figure 5a. 0.6D/U. Attention is focused on the plane situated 45 ~ from the lateral channel wall (turbulence statistics are displayed in Figure 6 for same plane). It is obvious that the flow inside the HV system is highly turbulent, characterized by random interactions of eddies inside it. Monitoring the flow over very large time intervals (-150D/U) showed that the HV system was always present, though its overall structure and strength was found to vary significantly. In particular, in Figure 4 one can observe the subsequent merging and splitting of main vortices L and M (correspond to A and B in Figure 3) and floor-attached vortex N starting downstream of the separation line. Though in the frames shown vortex M appears to be consistently stronger than L, at other times the situation is reversed. Sometimes L is practically absent. The vortex N was found to be relatively stable over long time periods and at irregular times to shed smaller vortices that move toward the main vortex M and eventually merge with it (N1 and M2 in frames g & h). At other times N appears to lift itself from the floor. This structure is changing in time mostly due to the merging and destruction of eddies part of vortices L and M. Figure 5a shows the contours of the mean non-dimensional bed shear stress relative to the mean value ( r o = p u ~ , u ~ / U =0.052) corresponding to fully turbulent flow in a channel of identical section without spanwise obstructions. Maximum values of'~/~o are close to 16 which show the large effect of these obstructions on sediment entrainment at the start of the scouring process. As expected, the highest values are recorded beneath the main HV system around the tip of the upstream obstruction. For reference, the critical bed shear stress "Ccrcorresponding to entrainment conditions was calculated using Shield's diagram for a typical sediment size of ds0-0.45mm (D=0.08m, U=0.17m/s). The critical shear velocity was estimated at 0.016m/s and the particle Reynolds number at 6.13 giving a nondimensional stress of 0.035. It corresponds to a ~cr/Xo value of 3.3. Areas of entrainment for mean flow are shown in frame c. Figure 5b illustrates the complexity of the instantaneous distribution of the non-dimensional bed shear stress. Its largest instantaneous values can be as high as 22. Though most

281 of the very large values are again recorded beneath the HV system, high values are also present beneath the detached shear layers corresponding to the shedding of vertical eddies that form due to

Figure 4: Instantaneous streamlines in plane 450 from lateral channel wall showing evolution of HV system in time a) 1.08 D/U; b) 1.20 D/U; c) 1.28 D/U; d) 1.32 D/U; e) 1.36 D/U; f) 1.44 D/U; g) 1.52 D/U; h) 1.68 D/U. The location of the plane is marked with bold in Figures 5a and 5b.

Figure 5: Non-dimensional bed shear stress relative to the mean value corresponding to fully turbulent flow in a channel without spanwise obstructions a) Mean distribution; b) Instantaneous distribution; c) Mean distribution. Entrainment region as predicted by Shield's theory; d) Instantaneous distribution. Entrainment region as predicted by Shield's theory.

282

Figure 6" Mean turbulence statistics in a plane 45 ~ from lateral channel wall a) velocity streamlines; b) u '2 ;c) v'2 ;d) w'2 ; e) p,2 ;f) u'v' amplification of Kelvin-Helmholtz (K-H) instabilities. A comparison between the mean and the instantaneous bed shear stress illustrates a profound point with regard to scour prediction for loose bed cases. The instantaneous distribution of the bed shear stress suggests that sediment transport models which predict entrainment based only on the condition that the mean bed shear stress is larger than a critical stress would not be very accurate when applied to complex junction flows. Successful prediction of sediment transport and scour requires use of models that can account for the turbulence characteristics near the bed including the effect of coherent structures and for the temporal variability in the distribution of the bed shear stress at least in a statistical way (e.g., variance along with mean value should be part of the model). The point is made clear by comparing frames a & b and c & d, respectively. Use of eddy-resolving techniques which have the ability to capture these phenomena (e.g., macro-turbulence events) as part of the simulation will reduce the amount of modeling needed in the equations that describe the sediment transport and bed evolution. Figure 6 displays the mean flow and turbulence statistics in the plane oriented at 45 ~ relative to lateral channel wall. The streamlines in Figure 6a show the presence of two vortices marked M and N. When compared to the instantaneous eddies depicted in Figure 4, it is seen that by time averaging the primary vortices L and M results in only one vertical structure, M, in Figure 6a. Vortex N corresponds to the floor-attached vortices in Figure 4. The resolved turbulent fluctuations in Figures 6b-6d show that the turbulent kinetic energy is the highest inside of the HV system, as expected. The intensity of pressure fluctuations also peaks inside the HV system confirming its role in initiating and sustaining the scouring process. Figure 7 shows the time history and power spectrum of the streamwise velocity u for point p l situated inside the main vortex M (see Figure 4e). The pressure power spectrum is also shown at points p l and p2 (situated inside the floor attached vortex N). It is evident that the time series in Figure 7a contain a wide range of velocity oscillations consistent with the chaotic nature of the motions inside the HV system. The pressure and u spectra at p l display a range of energetic frequencies between St=0.1 and 10 corresponding to a broad spectrum. Clear peaks are observed in the u spectrum for pl at St-0.11 and its first two sub-harmonics (0.22 and 0.33) due to strong nonlinear interactions. The main low frequency component (St-0.11 for p 1 and St-0.28 for p2) is quite

283 strong and may be due to the presence of aperiodic oscillations inside of the HV similar to those observed experimentally by Simpson (2001) for turbulent flow past surface mounted cylinders. A secondary peak at St=0.53 is observed in the pressure spectra at both points. a) P1

Time Series- u velocity

60-

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Power Spectrum

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um

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.

.

.

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d)

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20

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Figure 7" Velocity time series and velocity & pressure power spectra within the HV system a) uvelocity time series at p 1; b) u-power spectrum at p 1; c) Pressure power spectrum at p 1 (inset shows pressure power spectrum as a log-log plot); d) Pressure power spectrum at p2. The position of p 1 and p2 are shown in Figure 4e.

Detached Shear Layer The presence of spanwise obstructions induces massive separation upstream and downstream the embayment with formation of tornado-like vortices and shedding of eddies from the groyne tip that are transported and grow along the detached shear layer and interact with the eddies inside the embayment. Figure 8 shows contours of magnitude of out of plane vorticity component Iw=[ in the free surface plane along with the variation of the pressure and its power spectrum at point p3 within the detached shear layer emanating from the groyne tip. The formation and shedding of small vortex tubes in the detached shear layers due to the growth of K-H instabilities is clearly captured. One can also observe a wide range of eddies which populate the flow inside the embayment, in particular upstream of the tip of the second groyne, and inside the wake downstream the embayment. The vortex tubes inside the detached shear layer play an important role in the mass exchange between the embayment and channel. The pressure spectrum at point p3 has a two peak shape centered around St=3.8 and 7.5 associated with the shedding of vortex tubes. No energetic frequencies are observed at low Strouhal numbers. The energy associated with both ranges is comparable. We suspect that the reason for the second peak that is normally not present in the case in which there is only one obstruction is due to the interaction of the detached shear layer with the flow in the embayment and with the downstream obstruction. This creates a feedback effect at the tip of the upstream obstruction similar to 'resonance' phenomena observed in flow over cavities.

Passive Scalar Transport To study mass exchange processes between the main channel and the embayment a passive scalar was

284

Figure 8: Flow dynamics in the detached shear layer at the free surface a) Instantaneous contours of out-of-plane vorticity magnitude; b) Pressure time series at p3; c) Pressure power spectrum at p3. introduced in the embayment area (initial concentration C0=l) once the hydrodynamic solution was statistically steady (t=0). Both the momentum and scalar transport equations were then solved for an additional 90D/U. Mass exchange primarily occurs as eddies shed from the groyne tip (Figure 8a) are transported along the detached shear layer and interact with eddies within the embayment through the mixing layer. The process is highly three-dimensional. Except near the free surface, the scalar does not appear to exit the embayment by moving in quasi-parallel planes to the channel bottom, rather there seems to be considerable mass exchange between the different levels within the embayment before the scalar moves out of it. Details of the mixing process in the free surface plane can be inferred from Figure 9 which shows contours of the instantaneous scalar concentration in the flow domain at different times. At t=83D/U (frame f) 99% of the scalar mass left the embayment. The concentration scale is different in the frames such that flow structures and scalar fields are easily seen. In practice, predictive one-dimensional transport models that rely on empirical dispersion coefficients are utilized to globally characterize the mass exchange between channel and embayment. These coefficients are based on dead zone theory. An equation is presented to describe the 1D mass exchange of a conservative pollutant flowing in a channel with embayments (Uijttewaal et al., 2001):

aMe - - T (Mr -Me)

(1)

at

Me - Mr (Me -Mr)t=o

T= am1 aiurk

=

exp(- 1

At)

(2)

(3)

where Me is the pollutant mass fraction in the embayment (Me ( t " - 0 ) ' - M 0 - initial pollutant mass), M r is initial pollutant mass fraction in main channel (=0), k is the non-dimensional exchange coefficient related to the characteristic decay time T of the 1D exchange process, a m is mean water depth (=D), a i is water depth at the interface (=D), l is embayment width (0.625D), u r is main channel velocity (=U).

285

Figure 9: Instantaneous contours of passive scalar concentration in embayment and channel. Mixing starts at t=0D/U when the concentration in the embayment is set to C/C0=I.0; a) t = 0.25D/U; b) t = 7.75D/U; c) t = 15.25D/U; d) t = 22.75D/U; e) t = 55.75D/U; f) t = 83.25D/U. The exchange coefficient k is found by plotting the mass decay in the embayment, shown in Figure 10, and using (2) and (3). Two distinct values of the exchange coefficient are seen to characterize the mass exchange. For the first 12 nondimensional time units, corresponding to the initial phase of decay, the exchange coefficient is 0.061. About 68% of the initial pollutant mass leaves the embayment during the initial phase. Then k drops abruptly to 0.032 corresponding to the final phase of decay when most of the scalar originally situated close to the lateral walls and especially near the bottom is first entrained away from the solid boundaries inside the embayment before being transported into the main channel. This process is expected to result in a less intense mass exchange.

CONCLUSIONS The present simulations demonstrate that an accurate LES model can effectively complement the information obtained from experimental studies toward understanding the complex flow phenomena present in channel flow with spanwise obstructions including the formation of the horseshoe vortex system at the base of the obstruction, the interaction of the detached shear layer with the flow within the embayment and the wake-boundary layer interactions. In particular, the instantaneous and averaged structure, spectral content, spatial extent and time evolution of the coherent structures within

286 the HV system were analyzed. The instantaneous and time-averaged distribution of bed shear stress on the channel bottom were found to differ substantially and the scour region corresponding to the 100

~,

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30

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~-.-4---~

,

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Ill

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i 40 ,

t(om)

,

,

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60

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.--

70

t --

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80

Figure 10: Variation of total scalar mass in the embayment. Also shown are two straight lines corresponding to the two dimensionless mass exchange coefficients (initial and final phase of decay). initiation of scour (flat bed) was estimated based on the bed shear stress distribution. Though as expected the sediment entrainment is highest beneath the HV system, high values of the bed stress were recorded beneath the detached shear layer. Additionally, exchange of dissolved matter between the embayment and the main channel was computed. The decay of pollutant within the embayment was quantified enabling calculation of a global 1D exchange coefficient. It was found that the exchange process in not uniform. In the initial phase of decay over which about 68% of the total mass leaves the embayment, the exchange coefficient is about twice the value recorded for the final phase.

REFERENCES

Dargahi, B. (1990). "Controlling Mechanism of Local Scouting." Journal of Hydraulic Engineering 116:10, 1197-1214. Mahesh, K., Constantinescu, S.G., and Moin, P. (2004). "A Numerical Method for Large Eddy Simulation in Complex Geometries," Journal of Computational Physics 197:1, 215-240. Mahesh, K., Constantinescu, S.G., Apte, S., Iaccarino, G., F. Ham, and Moin, P. (2002). "Progress Toward Large Eddy Simulation of Turbulent Reacting and Non-Reacting Flows in Complex Feometries," Annual Research Briefs 2002, Center for Turbulence Research, Stanford University, CA. Melville, B. W. (1997). "Pier and Abutment Scour: Integrated Approach." Journal of Hydraulic

Engineering 123:2, 125-136 Tominaga, A., Ijima, K., and Nakano, Y. (2001), "Flow Structures Around Submerged Spur Dikes With Various Rlative Height." Proceedings of the 29thIAHR Congress, Bejing, China. Uijttewaal, W., Lehmann, D., van Mazijk, A. (2001). "Exchange ProcessesBetween a River and Its Groyne Fields" Model Experiments." Journal of Hydraulic Engineering 127:11,928-936. Simpson, R.L. (2001). "Junction flows." Annual Review of Fluid Mechanics, 33, 415-443. Weitbrecht, V., Uijttewaal, W. and Jirka, G.H. (2003). "2D Particle Tracking to Determine Transport Characteristics in Rivers with Ddad Zones." Proceedings Int. Symp. Shallow Flows, Delft, The Netherlands. 2,103-110.

Engineering TurbulenceModellingand Experiments 6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

287

LARGE EDDY SIMULATION OF NATURAL CONVECTION BOUNDARY LAYER ON A VERTICAL CYLINDER

D. G. Barhaghi ~ and L. Davidson ~ and R. Karlsson a,b ~Division of Thermo and Fluid Dynamics, Department of Mechanical Engineering, Chalmers University of Technology, SE-412 96 G5teborg, Sweden bVattenfall Utveckling AB, SE-814 26 A1vkarleby, Sweden

ABSTRACT Large eddy simulation of natural convection boundary layer along a constant temperature vertical cylinder is studied and the results are compared with the existing experimental data. The highest local Grashof number is Grz = 5 x 1011. It is shown that although there are some discrepancies between the results in the region close to the wall, there is qualitative agreement between them. In order to verify the credibility of the simulations, cross correlation of Reynolds stresses is studied and interpreted.

KEYWORDS Large eddy simulation, LES, Natural convection, Free convection, Cylindrical coordinate system, Shell and tube, Boundary layer

INTRODUCTION Natural convection is an interesting scientific subject for which many aspects need further research. The applications include not only industrial fields like power generators, reactors, turbines, heat exchangers and other power conversion devices, but also natural phenomena like atmospheric and oceanic currents, bio-heat transfer, green house effects and heat transfer in stellar atmospheres. Compared to the number of experimental and numerical researches which have been carried out for other methods of heat transfer, few belong to the natural convection field. Among the experimental researches, the latest is that reported by Tsuji and Nagano (1988b). In that work, characteristics of natural convection boundary layer are compared with the previous investigations. It is shown

288

that natural convection boundary layer has a unique turbulent structure which is rarely seen in other turbulent boundary layers. The results suggest that for values of y+ between 20 to 100, u'v' is not correlated with the mean velocity gradient, O~/Oy. However, comparison of Reynolds shear stress with previous experiments showed a different behavior near the wall. Although no negative region was observed by Tsuji and Nagano (1988b), almost all previous works had shown a negative region close to the wall. It is also shown that the cross-correlation coefficient for shear stress (Ru-~r analysis.

u'v'/?-~ • v '2) becomes zero near the wall which is in contradiction to analytical

Turbulent natural convection around a heated vertical slender cylinder was studied by Persson and Karlsson (1996) and new turbulent structures were presented for the near wall region. Again it was shown that there exists a negative shear stress region close to the wall. Comparing normal stresses and temperature fluctuations with those proposed by Tsuji and Nagano (1988b), different behavior can be observed between these two experiments. Although the temperature fluctuations presented by Tsuji and Nagano (1988b) are larger than the stream-wise normal stress, it is vice versa in Persson and Karlsson (1996) work. However the location of the peaks remains in close agreement. Natural convecting boundary layers have also been studied using DNS or LES. However, most of the studies include flows either in cavities or differentially heated channels. Among them are researches conducted by Miki et al. (1993), Versteegh and Nieuwstadt (1998), and Peng and Davidson (2001). From previous investigations it could be found that natural convection phenomenon has still many aspects which are not clear and there are some other aspects for which there is no consensus. Another fact is that very few researches have been conducted on the natural convective boundary layer along a vertical cylinder and most of the studies are performed on flat plates, channels or horizontal cylinders. This fact besides existence of an experimental apparatus in the Mechanical Engineering Department of Chalmers University of Technology, created the motivation for designing an LES problem in order to study the natural convection heat transfer along a vertical cylinder.

GEOMETRY

AND GRID SETUP

As it is shown in Figure 1, a sector of a cylindrical concentric vertical shell and tube is adopted as computational domain. All the dimensions are in millimeters. The motivation for using such a geometry was an experimental equipment with almost the same dimensions at the Department. The reason for choosing such a geometrical configuration is that although an idealized natural convecting boundary layer takes place in infinite surroundings, it is nearly impossible to achieve such an ideal condition neither in experiments nor in numerical calculations. Both experiments and calculations are very sensitive to the location of the infinite boundaries and any disturbances there. An inlet and outlet are designed to prevent stratification in the stagnant region which would affect the boundary layer growth near the hot tube. The angular extent of the geometry was chosen by comparing the results of different geometries. An ideal angle should cover all turbulent structures and give averaged two-dimensional angleindependent results. In this work, different angles, 0 = 18~ 36 ~ 54 ~ 72 ~ and 90 ~ have been applied to the numerical domain.

289

/'~

% \

g-~,

Figure 1: Computational geometry

Figure 2: Grid configuration

Having studied the flow by different grid sizes, a final 98 x 402 x 162 grid (r, z and 0 directions) with 0 = 90 ~ was chosen. Different grid densities were applied at the inlet in the z-direction compared to the rest of the domain. The reason was that at the inlet the flow is in the radial direction and a turbulent three dimensional flow is introduced at the inlet. Therefore, in order to resolve these small structures a finer mesh was applied. A stretching of 9 percent is used in the transition between the inlet region and the main part of the domain, see Figure 2. The spatial resolution along the hot cylinder is A0 + < 3.5 in the span-wise direction and Az + < 55 in the stream-wise direction. The highest value for perpendicular non-dimensional distance for the wall adjacent node is ( r - Ri) + = 0.32. It is found that the maximum SGS viscosity (ut) in the boundary layer is less than 50% of fluid viscosity and in the region ( r - Ri) + < 100 it is less than 10% of fluid viscosity indicating that the boundary layer is very well resolved (see Barhaghi (2004)).

NUMERICAL

METHOD

Governing Equations The continuity, Navier-Stokes and energy equations in cylindrical coordinate system are solved. The Smagorinsky model is chosen in order to model sub-grid scales. In the momentum equations, the turbulent diffusive cross terms arising from ~ ueZ oxi are neglected.

Finite Volume Approach A conventional finite volume method (Versteegh and Malalasekera, 1995) is used to solve the governing equations. As it is very important not to dissipate the turbulence by conventional numerical schemes, it is customary to discretize the governing equations by central difference scheme. However, this approach causes a so called unphysical fluctuation or wiggle problem which

290

is related to the unboundedness of the central difference scheme. Especially, this problem can be encountered in regions where turbulence intensity is not high enough or in laminar regions and in regions where the grid is not fine enough. Both of these problems occur in the present LES simulations near the inlet. Using pure central difference scheme, these unphysical fluctuations were generated near the inlet and propagated throughout the computational domain. To remedy this problem, a blend of central difference scheme with deferred correction (Ferziger and Peric, 1996) and Van-Leer scheme were used (DahlstrSm and Davidson, 2003). The second-order Crank-Nicolson scheme is used in order to discretize the equations in time. The numerical procedure is based on an implicit, fractional step technique with a multi-grid pressure Poisson solver (Emvin, 1997) and a non-staggered grid arrangement (Davidson and Peng, 2003).

B o u n d a r y Conditions and Fluid Data Simulation of laminar regions with LES method may provoke the problem of unphysical fluctuations. In order to create turbulence at the inlet, the velocity field of the DNS of a channel flow for which the bulk velocity was 0.6re~s, is scaled and implemented as inlet boundary condition for velocities. No slip boundary condition is used for all solid boundaries. For the temperature, Dirichlet boundary condition is applied at the hot tube and the inlet. The temperature is set to 80~ and 25~ on the hot tube and at the inlet respectively. Except for the outlet, homogeneous Neumann boundary condition is used over all remaining boundaries. At the outlet, for both temperature and velocities, a convective boundary condition (Sohankar et al., 1998) is applied. The dynamic viscosity and the Prandtl number of the fluid are p = 18.9 x 10 .6 and Pr = 0.7, respectively. In order to accelerate the convergence of the numerical computations to a fully developed condition, either the results of a 2D-RANS simulation or whenever existed, the results of previous simulations for different grid configuration, were interpolated and applied as initial boundary condition.

RESULTS

Once fully developed condition is achieved, sampling can be started. Sampling should last sufficiently long to ensure independent results regarding the number of samples. This is checked by comparing the results of two different averaged result with different number of samples.

Fully Developed Flow A s s e s s m e n t In order to assess whether or not the flow has reached to the state of fully developed condition so that sampling can be started, the instantaneous local Nusselt number at different heights has been calculated. The instantaneous, filtered and averaged Nusselt number are shown in Figure 3 for two vertical levels, one near the inlet and one near the outlet. It can be observed that almost for the last 15000 time steps, Nusselt number fluctuates uniformly. The filtered value, which is an average of 2000 instantaneous neighboring Nusselt numbers, shows it more clearly. These figures verify that the flow has reached the state of fully developed condition and statistical averaging can be started.

291

300

'

' Instantanelous ~.~ Fi~te;:e;d| ~

18001 1600II

'

'Instantaneous ---Filtered I ~

250

jl[[t~ | 11111[~It~tlt~i Nu

I ti lIll

200

150

108. Time step

.5

ltl

1400~~~]~1t[

[It [l[;lve~~rlatgel I[[~~

ooo n

11

.

~ in"

Time step

(a) Nusselt no. at z/H = 0.2

15

2

~ in'

(b) Nusselt no. at z / H = 0.9

Figure 3: Assessment of fully developed flow regarding local Nusselt number Figure 4 shows velocity and Reynolds shear stress obtained from the three last 5000 time steps of simulation. As can be seen, very little difference exists between the results. For the shear stress, the big difference lies in the outer region near the outer shell far away from the boundary layer. Again from these two figures, it can be seen that the simulation has reached to the state of fully developed condition. Hereafter, the statistics and averaged values are calculated using the last 15000 time steps.

Grid I n d e p e n d e n t Result In numerical simulations, grid independent result is always desirable. This can be checked by comparing the results of different grids. However, as LES is inherently dependent on the grid resolution, no final grid-independent result exists. In this computations, a final 98 x 402 mesh in r and z directions, respectively, is chosen. However it is found that the angular width of the domain has a crucial effect on the final results. This is shown in Figure 5 where velocity and shear stress for different sizes of the computational domain are depicted. Although the grid spacing in the 0-direction is the same for all grids it can be seen that the difference between the curves for

(~- P~)/(Ro- P~)

10-a

10.2

10-1

(~- ~)/(Ro- ~ )

0~

10.3

10-2

5000timestep 5000timestep --3 rd 5000timestep 1.5

10-1

1 st

-

2 nd

12.2

5000timestep --- 3rd 5000timestep .....

i'"-, ,.?

(v'v')

Vz+ 4

-2

-

',.i

0.5

2 nd

10~

102

( r - Ri) + (a) Dimensionless velocity

-0.5

1'0~

1'02

( r - Ri) + (b) Reynolds shear stress

Figure 4: Velocity and shear stress at z / H = 0.8, G r = 2.9 x 1011 for contiguous averaged data

292

(~- P~)/(Ro 10 -3 --

P~)

-

10 -2

10 -~

( ~ - P~)/(Ro 0~

10 -3

1.4

0 = 54 ~

-_-_ o - 720

1.2

-

P~)

10 -2

10 -~

..... ~ =

72 ~ = 90 ~

1

,'

0.8

o6

Vz+

U .2

0.4 0.2 0

10 ~

-0.2

102

(a) Dimensionless velocity

10 ~

(b) Reynolds

10 2

shear stress

Figure 5" Velocity and shear stress at z / H - 0 . 8 , G r z - 2 . 9 x 1011 for different domains 0 = 72 ~ and 0 = 90 ~ is less than that for 0 = 54 ~ and 0 = 72 ~ for both velocity and shear stress. Although it is not shown, the difference between the results of other grids with 0 = 18 ~ 0 = 36 ~ was even larger. However, as can be seen in Figure 6, the difference between Nusselt numbers for grids with 0 = 72 ~ and 0 = 90 ~ is negligible compared to that between 0 = 54 ~ and 0 = 72 ~

Mean Fluid Flow and Heat Transfer Parameters Figures 7(a) and 7(b) compare the results of the present study and previous experiments. As can be seen, although the results of present study are in good agreement with the results of Tsuji and Nagano (1988a), the results of Persson and Karlsson (1996) are rather different.

As a common behavior in all graphs, however, except for a very small distance from the wall , ( r - Ri) + < 1.2, velocity does not follow the law of the wall, v + - ( r - Ri) +, the same way it does for forced convecting boundary layers. However, Figure 7(b) shows that temperature follows the law of the wall in the region ( r - R i ) + < 5, alike forced convecting flows. Although no self similar behavior regarding dimensionless velocity v z+ could be observed from the 103

y

102

Nu z 101

10 o

.... --105

8 = 54 ~ 8 = 72 ~ 8 -- 90 ~ 101 o

Raz

Figure 6: Nusselt number at z / H - 0 . 9 , G r z - 4 . 2 x 1011 for different domains

293

(~- ~)I(Ro- R~)

( ~ - ~)/(Ro - ~ ) 10 -3

180

10 -2

10 -3

10 0

10-

/~ ......."..~"'~§

o

[]

10 -2

s

6

10 0

/- ununUunnu'-

T+1O

Vz + 4

10 -1

Present s t u d y (Or= = 1.8 x 1011) Tsuji and Nagano ( O r = l . 8 x 1011) P~_rsS2;ra(d K a ; : s ; o n ( G r = l x 1 0 1 ~

2 0 * o

-2

Tsuji and Nagano ( G r = l . 8 x 1011) Persson and Karlsson (Gr = 1 x 1010)

---

v +=(r-Ri)

-4

+

i

'

10 ~

10 ~

1'0e

10 2

( ~ - P~)+

(~ - P~)+

(a) Dimensionless velocity

(b) Dimensionless temperature

Figure 7: Comparison of velocity and temperature with experimental data results of Tsuji and Nagano (1988a), graphs of the v+ vs. (r other, showing a self similar behavior, see Figure 8.

-

Ri) + in this study collapse on each

Finally, Figure 9 shows variation of Nusselt number as a function of Rayleigh number. Compared to the results of Tsuji and Nagano (1988a) and proposed correlation for the laminar region, there is a very good agreement for both the laminar and the turbulent region. However, the major difference is in the transition region. The transition for the studied flow seems to be starting from Raz = 106 compared to the flow along the flat plate for which transition starts from Ra = 8 x 10s. The difference is probably due to first the turbulent inlet, second, inherent difference between characteristics of boundary layer along a vertical flat plat and a vertical slender cylinder and third, existence of the outer shell which embraces the whole cylinder and provokes instabilities by creating some recirculating flows in the proximity of boundary layer. Another difference in Figure 9 is the overshoot of Nusselt number in the commencement of turbulent boundary layer which was not observed by Tsuji and Nagano (1988a) but had been mentioned in previous researches (e.g. Cheesewright, 1968).

..... 8

"'"

Grz

= 1.5 x 101~ -- 3.6 • 101~ l~r = 8.4 • 1 0 m r

Grz

-

Grz Grz

i 10 3

"~ ~x

o [] ---

1

Present study Tsuji and Nagano ( T w = 100~ Tsuji and Nagano (Tw -- 60~ Nu = 0.387(Gr Pr) ~ /

~r"

10 2

Nuz

Vz+

101

0 1 . . . . . . . . . . . . . .0. . . . . . . . . . . . . . . . . .1. . 1010 10

10

2

T

10

3

( ~ - R~)+ Figure 8: Dimensionless velocity profiles for different Grashof numbers

10 0

~

eo o"

10 5

101~

Raz Figure 9: Nusselt number as a function of Rayleigh number

294

(~-

R~)/(Ro- J~+)

I 0 -3

3.5

I 0 -m s t

s'~

I 0 -~

,,

3.:[

(~-~)/(Ro-~) 1 0 -3

1 0 -2 s s

i

i

i

2.5

2.5

1.5

1.5

10 -1

9

tv/--~Tt, ,' t* ]

i

i

', ~ o & ~oo~+-

o

s

[]

-+~

. []

+:o/

0.5

[] []

.' 0.5-'''~Y~'moooaoOn~.-,'"'"~

10 ~

10 2

( v - R+)+ (a) Experimental data from Tsuji and Nagano (1988a), Gr = 8.99 x 101~

o " ~ ' - ~ " + ~ 10 ~

,.,

"+

,s

;'6. ,,,'"+ +

__

~

Ii m

"r" 0_"o~o "

00 +'''0 ,

10 2

o %, ~,

X_.~

"., "".

~+ ~'"

( r - R+)+ (b) Experimental data from Persson and Karlsson (1996), Gr = 1 x 101~

Figure 10" Normal stresses and temperature fluctuations. Lines: simulation; markers: experiments

R e y n o l d s Stresses and Turbulent Heat Fluxes Figure 10 compares normal stresses and temperature fluctuations with the experimental data. In Figure 10(a), both the predicted ~ / t * and the stream-wise normal stress have higher values compared to the experimental data for the flat plate, wall-normal stress, however, have smaller values for ( r - Ri) + < 92.5. No constant value region such as that observed in the experiment for ~ / u * in the 7.5 < ( r - Ri) + < 40 can be observed in the LES-simulations. Nevertheless, the graphs are qualitatively in a good agreement and locations of maxima match each other. Discrepancies between experiments and computations are even larger in Figure 10(b). Contrary to computations and the other experimental data, normalized temperature fluctuations are smaller than stream-wise velocity fluctuations. Also, locations of maxima are under predicted. Experimental wall-normal stress in Figure 10(b), shows the same behavior as computations contrary to that presented by Tsuji and Nagano (1988a). However, none of the experiments show a tendency for this parameter to become zero sufficiently close to the wall, reflecting the unreliability of measurements close to the wall. Reynolds shear stress together with stream-wise and wall-normal turbulent heat fluxes are shown in Figure 11. Again, although the values in Figure 11(a) are in close agreement, this is not the case in Figure 11(b). While wall normal turbulent heat flux remains positive close to the wall for both computations and experiments in Figure 11(a), it gets negative values for experiment shown in Figure 11(b). Also contrary to the computation and experiment shown in Figure 11(b), no negative shear stress can be observed in the experimental values shown in Figure ll(a). However, this negative region shown in Figure 11(b) is shifted toward the wall for experiment. In the experiment of Persson and Karlsson (1996) no negative region close to the wall for streamwise turbulent heat flux can be observed, albeit the existence of this region is confirmed by Tsuji and Nagano (1988b). Finally, Figure 12 compares the cross correlation coefficient between the shear stress and the

295

(~- ~ ) / ( n o - ~ ) 10

-3

10-2

2.5

1O-1

-0.5

...,"

1'0o

~,o~176 ..-3-:/-.a , o= o E.z_.~ I

~

(?~-

v,~t, ok

i ,,;

10 -1

' ' '% .

,""'"'-

u't*

.....

10 .2

/.~.-

,,

1.5

10 -3

,

i -

1.5

"~ "~

t / V V

0.5

%'1

102

i

"

0

]

-0.5

10~

10

2

( T - Ri) §

Ri) +

(a) Experimental data from Tsuji and Nagano (1988a), Gr = 8.99 x 10l~

(b) Experimental data from Persson and Karlsson (1996), Gr = 2 x 1010

Figure 11" Turbulent shear stress and heat fluxes. Lines: simulation; markers: experiments normal stresses. From the Taylor expansion, V~rV'z = O(r3), v~2 = O ( r 4) and Vtz2 = O(r2). So Ru~ should have a constant value very close to the wall when r approaches zero. This can be found from Figure 12 for the present simulation and Persson and Karlsson (1996), for which the constant values are Ru~ ,~ - 0 . 3 and R ~ ~ -0.09, respectively. As this coefficient becomes zero in the experiment of Tsuji and Nagano (1988a), again it can be deduced that the measured stresses for this experiment are unreliable in the region close to the wall.

CONCLUSIONS Natural convection boundary layer along a constant temperature vertical cylinder is studied. It is shown that linear law of the wall for the velocity is valid just for a very small region close to the wall, ( r - Ri) + < 1.2. The profiles of the velocity collapse on each other in the turbulent region showing a self similar behavior. Nusselt number is compared with the experimental values and it is deduced that the transition

(~- P ~ ) / ( n o - n~) 0.6 0.4

-3 10-2 ................................ Present

*

Tsuji

n

p ........

study

and

§ ~ ~*~ § ~

Nagano

*

d Karl ....

.,.::;','

0.2

* *

§* ~ ~ ' ~ - % j "

:..t"

=.,=,.

RUU

o

nnno

-0.2

-0.4 ~

10-1

10

10 ~

R

(~-/~)+

102 !

Figure 12: Cross correlation coefficient comparison, Ruv

!

x/~v~v'2r

296

for this simulation has started earlier compared to experiments on the vertical wall. The existence of an overshoot in the commencement of turbulent region is also verified. Comparison of the calculated Reynolds stresses with those obtained from experiments shows that the measured values are not reliable close to the wall. This is also verified by considering cross correlation coefficient. Existence of a region with negative values for both shear stress and stream-wise turbulent heat flux is shown and it is verified t h a t the wall normal heat flux remains positive even close to the wall.

ACKNOWLEDGMENTS Financial support from the Swedish Research Council and part of the computer resources provided by the Center for Parallel C o m p u t i n g (PDC), K T H are greatly acknowledged. Authors are also indebted to Dr. Shia-Hui Peng for his support and sharing his ideas and knowledge.

REFERENCES Barhaghi, D. G., 2004. Dns and les of turbulent natural convection boundary layer. Thesis for Licentiate of Engineering 04/05, Dept. of TherIno and Fluid Dynamics, Chalmers University of Technology, G6teborg, Sweden. Cheesewright, R., 1968. Turbulent natural convection from a plane vertical surface. Journal of Heat Transfer 90, 1-8. Dahlstr6m, S., Davidson, L., 2003. Large eddy simulation applied to a high-reynolds flow around an airfoil close to stall. AIAA paper 2003-0776. Davidson, L., Peng, S.-H., 2003. Hybrid LES-RANS: A one-equation SGS model combined with a k - w model for predicting recirculating flows. International Journal for Numerical Methods in Fluids 43, 1003-1018. Emvin, P., 1997. The full multigrid method applied to turbulent flow in ventilated enclosures using structured and unstructured grids. Ph.D. thesis, Dept. of Thermo and Fluid Dynamics, Chalmers University of Technology, G6teborg. Ferziger, J. H., Peric, M., 1996. Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin. Miki, Y., Fukuda, K., Taniguchi, N., 1993. Large eddy simulation of turbulent natural convection in concentric horizontal annuli. Int. J. Heat and Fluid Flow 14, 210-216. Peng, S.-H., Davidson, L., 2001. Large eddy simulation for turbulent buoyant flow in a confined cavity. International Journal of Heat and Fluid Flow 22, 323-331. Persson, N. J., Karlsson, R. I., 1996. Turbulent natural convection around a heated vertical slender cylinder. In: 8th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics. Lisbon. Sohankar, A., Norberg, C., Davidson, L., 1998. Low-Reynolds number flow around a square cylinder at incidence: Study of blockage, onset of vortex shedding and outlet boundary condition. International Journal for Numerical Methods in Fluids 26, 39-56. Tsuji, T., Nagano, Y., 1988a. Characteristics of a turbulent natural convection boundary layer along a vertical flat plate. International Journal of Heat and Mass Transfer 31 (8), 1723-1734. Tsuji, T., Nagano, Y., 1988b. Turbulence measurements in a natural convection boundary layer along a vertical flat plate. International Journal of Heat and Mass Transfer 31 (10), 2101-2111. Versteegh, H. K., Malalasekera, W., 1995. An Introduction to Computational Fluid Dynamics - The Finite Volume Method. Longman Scientific & Technical, Harlow, England. Versteegh, T. A. M., Nieuwstadt, F. T. M., 1998. Turbulent budgets of natural convection in an infinite, differentially heated, vertical channel. International Journal of Heat and Fluid Flow 19, 135-149.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

297

DEVELOPMENT OF THE SUBGRID-SCALE MODELS IN LARGE EDDY SIMULATION FOR THE FINITE DIFFERENCE SCHEMES

M. Tsubokura (1), T. Kobayashi (2) and N. Taniguchi (3) (1)Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan (2)Japan Automobile Research Institute and University of Tokyo 1-1-30 Shiba-Daimon, Minato-ku, Tokyo 105-0012, Japan (a)Institute of Industrial Science, University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

ABSTRACT A novel method is applied to Subgrid-scale (SGS) stress and heat flux modelling and eddyviscosity/diffusivity-type models are derived, which are specially designed for the dynamic procedure using the finite difference (FD) scheme. The models are validated in turbulent channel flows at various grid resolutions, and show better agreement with DNS results than the classical Smagorinsky's model. The most notable feature of these new models is their insensitivity to the discretized test filtering operation, contrary to the Smagorinsky's model which shows strong dependence on the adopted filter parameter. Effects of the FD errors on the accuracy of SGS models are intensively studied in a turbulent pipe flow. The mixed high and second-order FD scheme is adopted and dependence of the results on the order of accuracy is investigated. At the second-order accuracy, the numerical error seriously affects the obtained results, and the mean velocity profile predicted by LES without SGS model (coarse DNS) unexpectedly shows good agreement with DNS results. At the higher-order accuracy, the coarse DNS underestimates the mean velocity, while the proposed eddy-viscosity model shows good agreement with the empirical law at the logarithmic layer and mitigates the serious overestimation of Smagorinsky's model.

KEYWORDS Large eddy simulation, Subgrid-scale model, Eddy-viscosity model, Smagorinsky's model, Finite difference scheme, Numerical error, Turbulent channel flow, Turbulent pipe flow

298 INTRODUCTION The difficulty of Large Eddy Simulation (LES) lies in the fact that not only the subgrid-scale (SGS) turbulence models but also the discretization methods strongly affect the reproduced flow field. Especially in lower-order Finite Difference (FD) schemes commonly used in engineering problems, an obtained flow field is contaminated severely by numerical errors typically at a higher wavenumber region close to a minimum grid width. The contamination of numerical errors near the cutoff wavenumber becomes more serious when we use the dynamic procedure of Germano et al. (1991). Because in the procedure, the grid-scale (GS) velocity component near the cutoff region is utilized to dynamically estimate an unknown model coefficient by explicitly filtering the GS turbulence (which is called test filtering). In fact, we found in the previous work (Tsubokura et al. in 2001) that the original dynamic Smagorinsky's model, which had successfully been demonstrated in a turbulent channel flow using the spectral method, seriously overestimates the mean velocity when we apply the lower-order FD scheme at the same grid resolutions and numerical box as the spectral method. We will see in this study that results of LES/DNS are dependent strongly on a FD order of accuracy. The second difficulty of LES using the FD scheme arises when we suppose to use explicit filtering operations required such as in the dynamic procedure or in the scale similarity concept of Bardina (1983). The important thing to be noted as regards the disrecetized filtering operations is that the ratio of the filter and the grid width given as A / h or [X/h is the numerical parameter to be determined prior to the numerical simulation. In fact our previous study shows (Tsubokura et al. in 2001) that dependence of the simple dynamic Smagorinsky's model on the discretized test filtering cannot be negligible. Because utilization of the lower-order FD scheme is always the case in engineering LES, it follows from what have been said that any newly developed SGS models always should be validated under the conditions of various numerical errors, and even more hopeful is that SGS models themselves should be derived taking the supposed numerical errors into consideration. Accordingly, the objective of the present work is to derive such SGS stress and heat-flux models which are supposed to be used in the FD scheme. Then the proposed models are intensively studied and assessed in turbulent channels and pipe flows at various grid resolutions as well as FD orders of accuracy to investigate how the FD errors affect the LES results.

SGS MODELLING

In a previous study, we proposed a straightforward way to develop SGS stress (Tij = u~u 3 - uiftj) and heat-flux (q3 = 0u 3. - 0 ~ j ) models based on the simple economic idea that wealth is proportional to earnings times the t i m e for earning (e.g. Launder in 1988, Yoshizawa et al. in 1996), in which they are modelled as q-ij ~ PijT,

(1)

qj ~ P i T ,

where the production terms of SGS stress and heat-flux are given as

Pij

O~tj = -~

O~ti

o~--2~ - ~5~-522~,

O~tj P~ - - q ~ ~

O0 - ~

o~

(2)

Applying this well-known procedure in RANS modelling to SGS modelling brings two merits in the context of numerical procedures. First, by supposing to adopt the dynamic procedure of Germano et al. (1991) to determine the model coefficient in the model, the time scale T in eq. (1) can be modelled not in a physical

299 sense but in an arbitrary or a numerical manner. In the present work, the time scale is estimated directly from the strain-rate tensor as T o( 1/S (where o is a magnitude of the strain rate tensor Sij = (O~i/Oxj + Of~j/Oxi)/2 and is given as S = v/2Sijc3ij) from the simple dimensional analysis to avoid the explicit length scale A usually included in SGS model. In fact, when A is explicitly included such as the Smagorinsky's model, the corresponding ratio of the length scale A / A becomes the parameter to be determined a priori ; which seriously degrades the attraction of the dynamic procedure. Secondly, by applying the scale similarity model of Bardina in 1983, the production terms in eq. (2) are rather easily modelled and various extension can be achieved. In our procedure, the SGS stress and heat-flux included in the production terms of eq. (2) are modelled as: Tij( ~ ) --" Ui~tj -- UiUj,

~(~) ---- 0~j -(lj

g~j

(3)

9

Thus we can obtain the following anisotropic eddy viscosity/diffusivity models: 9(ani) __

~J

-

qj(ani+v.g.) __

-

C / (sire) O~tj

sl ~

~

_(sire) O~ti )

+ 'j~

Co / (sire) O0

Isl ~j~

o~

2 C T(sim) O~t l

+~

~

Old'

(4)

~(sim) O~tj

~x~ + ~

~x~ )'

(5)

where superscript * indicates the trace-free component of stress tensor. Supposing that SGS stress and heat-flux included in the right-hand side of eqs. (4) and (5) are isotropic, following isotropic representation can be obtained: (~im)

,(~o) = - 2 C ~kk 3

~-ij

(i~o)

q~

s

(6)

(~im) ~-ik 00

=-Co ~[

(7)

o~j

In particular, we can derive a more advanced heat-flux model by coupling the effect of GS velocity gradient included in eq. (5) with the isotropic eddy diffusivity model (eq. (7)), (~im)

_(iso+v.g.) __

Co

~J

gl ( a

-

Ts

O0

_(sire) O~tj

Ox~ + ~

-~x~)

(8)

In the dynamic procedure to determine model coefficients C and Co, so-called the subtest-scale (STS) stress T~y = u~uj - u~uj (in which overtilde denotes the test filtering operation) and corresponding heat-flux also must be modelled. Here the STS stress is modelled on the analogy of the adopted SGS models. For example, the STS stress model corresponding to the isotropic model of eq. (6) is given as •(sim)

T/~) = --2C~kk= ~ii,

(9)

3Is _(~im) (~im) by just replacing the .lij and Sij with Tij and ~iy. We have many candidates for the expression of T~(f.ira) included in the modelling of STS stress, but finally we have adopted following models by considering the consistency of the numerical error in the dynamic procedure (Tsubokura in 2001)" ri(f.ira) : u i u~-~- u i u5 .

(10)

As noted below, owing to these procedures, the proposed models are found to be insensitive to the discretized test filtering operation, which is valuable for engineering LES using the FD scheme.

300

DISCRETIZATION Numerical Methods

Contrary to the Reynolds-averaged numerical simulation, it is more appropriate for direct numerical simulation (DNS) and LES to adopt a higher-order FD scheme in order to properly capture the fine scale turbulence near the minimum grid width. One of stumbling blocks toward the engineering DNS/LES has been that higher-order FD usually violates the simultaneous conservativity of mass, momentum and kinetic energy, which causes unreliable or unstable results. In fact, until recently, the scheme which conserves both momentum and kinetic energy provided that mass is conserved is only the standard second-order accurate staggered grid FD scheme on a uniform mesh developed by Harlow & Welch (1965). Therefore the solution of the engineering DNS/LES has been only to adopt sufficiently fine grid resolutions in the second-accurate FD scheme compared with the spectral method. The recently developed fully conservative high-order FD scheme for uniform Cartesian staggered grids by Morinishi et al. (1998) provides great progress in this matter. In the present work, we have adopted the fully conservative method and the extension of the method to the non-uniform grids in cylindrical coordinates (Morinishi et al. in 2004) to investigate the effect of FD errors on the results of LES by changing the order of FD accuracy. Thus the governing equations are discretized on the staggered grid system. The second-order splitting method proposed by Dukowicz & Dvinsky (1992) is adopted as a coupling algorithm of the momentum and continuity equations. The corresponding Poisson equation for the pressure is solved using FFT algorithm in a periodic or homogeneous direction while tri-diagonal matrix algorithm is adopted in a normal-wall or a radial direction. For the time marching method, the third-order Runge-Kutta scheme and the second-order Crank-Nicolson scheme are combined to avoid an excessively fine time increment caused by possible extremely fine grids in a vicinity of the solid wall (as well as near the pole in the cylindrical coordinate).

Explicit Filtering In the present work, an explicit filtering operation with different filter width must be expressed in a discretized form; one is for the grid filtering and the other is for the test filtering operations. The one-dimensional grid filtering operation discretized by second-order FD is given as, /~2 u(X/_l) - 2u(X/) ~- u(X/+l) h2 '

ft(Xr) = u(Xi) -~ 24

(11)

where XI indicates the location of the discretized velocity on the numerical grid and I is its index, z~ is the grid-filter width and h is the grid width given as XI - XI-1. The discretized test filtering operation is obtained by just replacing t h e / ~ with/~. The explicit spatial filtering is conducted only in the homogeneous directions. The important thing to be noted as regards disrecetized filtering operations is that the ratio of the filter and the grid width given as A / h or [k/h is a numerical parameter. Consequently in the dynamic procedure discretized by FD scheme, two parameters of ~/zX and A / h (when adopting Smagorinsky's model), or A / h and A / h (when adopting eq. (6)) are to be determined a priori. Dependence of the results on these parameters will be mentioned in the next section.

301

Table 1: Filter parameters for discretized filtering operations.

(5/h) ~

a 1

b 4/3

c 2

(h/h) ~

A

B

C

4

6

8

Table 2" Grid resolutions and computational domains for a plane channel flow. R e , - = 180 Lx x L~ x Lz NxxNyxNz h~+ h+

47r(~ x 4/37r~ (24~96) x65x(24~96) 23.6 ~ 94.2 7.9 ~ 31.4

RESULTS IN TURBULENT Dependence

PLANE

on the Grid Resolution

[ Re,. = 590 27r(~ x/~ x 7r(~ 48x65x48 64x65x64 51.7 57.9 25.8 29.0 Re~- = 395

CHANNELS and Discretized Filtering

First dependence of the proposed isotropic (eq. (6)) and Smagorinsky's models on the discretized grid and test filtering operations as well as the grid resolutions is intensively studied (Tsubokura in 2001) in a plane turbulent channel flow. The accuracy of FD is the fourth order in horizontal directions while the second-order is adopted for normal-wall direction. The parameters for both grid and test filtering adopted in this study are summarized in Table 1. The grid resolutions at three different Reynolds numbers (based on the friction velocity u,. and channel-half width) are summarized in Table 2. Five different grid resolutions are tested at Re,. = 180 by changing the grid width for both streamwise and spanwise directions (24 x 24, 32 x 32, 48 x 48, 64 x 64, 96 x 96). Figure 1 indicates the mean velocity and streamwise GS turbulent intensity profiles at Re,. = 395,590 obtained by the proposed isotropic model, as well as the dynamic Smagorinsky's model for reference. For the proposed model, filtering parameters of (Ab) in Table 1 which corresponds to ( A / h ) 2 = 4 and (/X/h) 2 = 4/3 are used. Concerning the filter ratio of ~ / / ~ required for the dynamic Smagorinsky's model, we have adopted 22/a. The DNS results shown in the figure was obtained by Moser et al. (1999). The well-known overestimation of the dynamic Smagorinsky's is obvious, while they are mitigated by the proposed model. The mean velocity profiles of DNS and our isotropic model are almost identical. The most important feature of the proposed model is their insensitivity to the discretized test filtering operation, which is indicated in Figure 2. In the figure, the mean velocity profiles at Re,. = 180 and corresponding bulk mean velocities obtained by LES against the streamwise width of the grid in wall coordinate are plotted. The grid dependence of each model is evident, and the dynamic Smagorinsky overestimates the bulk velocity at all resolutions tested here (see Figure 2(b)). Another important feature of the dynamic Smagorinsky is the strong dependence on the discretized test filtering operation, and when larger parameter is adopted the overestimation is more enhanced. The proposed model shows better performance than Smagorinsky's model at all grid resolutions. It is true that the proposed model also indicates the dependence on the discretized filtering parameter, but when we look more carefully into that, it is found to be insensitive to the test filtering when the grid filtering parameter is fixed. This promising feature of the proposed model is easily identified by comparing the results of (Ab) and (Cb), or (de) and (Be) in the

302 Figure 2(c). In other words, the proposed model is dependent on only the grid filtering parameter

s

20 a>

: ropo,e ,i,o, o ic

o

:Smagorinsky _o~176 ~ :DNS "~

0"~"15" ~

,~10-

S""

,,~;;;~ i - ]

A '~

y ~. ~

5.

3t b>....

I I20 ~+ ~" "~1 ,_

I2

5~

i::::~

-~..... /~

~a ~ ~

-4

o o^ =~ 'm-g~ 22o , 0o ~ -3 =-

~

f~

~ ~ h.~ ] o ~,, -~ I

../

.,,,7

-,%%~

-21,

% 1

-5 ....[ . . . . . . . . . . . . . . . . . . . . . . 10 100 Y+

0

-~l

.................. 10 y+

100

Figure 1" (a)Mean velocity and (b) r.m.s, of streamwise velocity profiles in wall coordinate at Re~=395, 590 in plane channels

A s s e s s m e n t o f S G S Heat Flux Modelling The flow field we have adopted in order to assess the SGS heat-flux models given as eqs. (7) and (8) is a stably stratified fully developed open channel flow. For a SGS stress model, the same isotropic eddy viscosity model given in eq. (6) is adopted. No-slip and free-slip velocity boundary conditions are imposed on the lower and upper walls respectively. As regards the temperature, constant temperature-flux is imposed on the lower wall, while it is supposed to be zero on the upper wall. Because the constant flux on the lower wall continuously drains heat away from the flow field, constant heat source is supposed in the entire domain. These conditions produce the statistically homogeneous state in the horizontal direction. We set Re~ = 640, Ri~ = / 3 g a Q / u a = 3.05, and Pr = 0.7, where u~, 6 and v are the friction velocity on the lower-wall, the width between lower- and upper- walls, and the kinetic viscosity,

(a)

0

:proposed isotropic

o

:Smagonsnky

X

:no model

oo. ~

o O y j ~ ~,~~ , x ~

o~

"~

xxx~

20 ..................................................................................

2OI

[ ...... 9..... Iq ~ < ~ - - -

19

[. . . . .

k-o-

1~

&-

s........,

x

Sma'tB) Proposed(Ab) Proposedl Bc). ;Io model m I ) \ S(K.i~)

~ ....... Aa

(c)

Ab

.I ...o . 'Ju

,\c

--181- -"

"~//

}

("b :Be

........

~g ,~lcl;

/

.l~

O"/

to

16 ck-,~... -la x, \ .....

i

1

, ...............

10

y+

i

14

,~.,."

..-' A.,....

,)r /x / 14~

~'"'X/'/"

"X""x'///

I00 25

50

~,5

100

0

25

50 hv- 75

'.'00

Figure 2: (a)Mean velocity profiles at Re, = 180 (filter parameter, (Ab); grid resolution, 32 x 32) and (b)(c)Bulk velocity at various grid resolutions at Re~ = 180 in plane channels. (b) :Comparison between Smagorinsky and the proposed model, (c): Dependence of the proposed model on the filtering parameter.See Table 1 for an abbreviation.

303 respectively, while/3, g, and Q are the volumetric expansion coefficient, gravitational acceleration and the constant heat source, respectively. It should be noted that the Ri,- = 3.05 corresponds to approximately 0.1 in the bulk Richardson number defined by the temperature difference between the upper- and lower-walls and the mean streamwise velocity on the upper-wall. The size of the numerical box is Lx x L v x Lz = 27c5 x 5 x 7r5, which is divided by 144 x 64 x 144 numerical grids. Figure 3 shows the (a)mean and (b)GS turbulent intensity, and (c)GS temperature-flux predicted by the proposed SGS stress and heat-flux models. The GS component of r.m.s, obtained from DNS is also displayed (Tsubokura in 2003). Generally speaking, our LES shows good agreement with DNS results. The improvement of GS streamwise heat-flux obtained by considering the GS velocity gradient is found in Figure 3(c), in which the is 9 heat-flux model given as eq. (7) overestimates the GS streamwise heat-flux while it is mitigated by coupling the effect of the GS velocity gradient (eq. (8)) and excellent agreement is achieved. 30 -

A

20-

V 10-

I0

/-

-%%,

100

y+

j

6-1

:~-

.....

9 .------- 9 1 4 9~ - -

.TL,~"e,,'..U

1

10

y+

l(X)

'

lO

y+

"-

ISO

/

1-1.5

/

A

100

Figure 3: (a)Mean vel., (b)r. m. s. of vel., and (c)GS Streamwise. and normal-wall heat flux at R e , = 640 and Ri~- = 3.05 in a stably stratified turbulence: thick line, DNS(GS, filtered); thin line, DNS(full scale); open symbol, LES of eqs. (6) and (8) (iso+v.g.); closed symbol, LES of eqs. (6) and (7) (is 9 filter parameter, (Ac).

RESULTS IN TURBULENT Dependence

PIPE FLOW

on the discretization

error

Secondly, we focus on the effect of FD errors on the accuracy of LES, and dependence of the proposed is 9 model given in eq.(6) on the numerical error is intensively studied in a turbulent pipe flow by changing the FD accuracy from second up to twelfth order (which is almost comparable to the spectral method). Mixed high and second order FD scheme is adopted for spatial discretization, in which axial and azimuthal direction maintain higher order accuracy while radial direction is second-order accurate. The computational domain is Lx = 27cR, L~ = R and Lo = 27r

304 for the streamwise, radial and azimuthal directions respectively where R is the radius of the pipe. In all cases tested here, the Reynolds number based on the friction velocity, u~, and the radius of the pipe is 590. In the first place, we focus on only the FD errors by excluding the effect of the modelling errors from the obtained results. For this purpose we have conducted LES without SGS model (see Figure 4) at two typical grid resolutions of Nx x Nr x N0=64 x 32 x 64 and 64 x 32 x 128. In the figures, solid lines show the linear law in the inner layer and the empirical logarithmic law (Zagarola & Smits in 1998). The overestimation of the mean velocity at the logarithmic layer is clearly observed at coarser grid resolution of No = 64. Even though it is slightly modified by using the higher order FD, this overestimation is indeed critical because, as we will see below, one of the important feature of the eddy-viscosity SGS model is to increase the mean velocity at the logarithmic layer. It means that as long as the eddy-viscosity SGS model such as Smagorinsky's model or the proposed model of eq. (6) is adopted, the overestimation of the mean velocity indicated in Figure 4(a) cannot be improved. On the other hand, when we adopt the finer grid resolution of No = 128, LES without SGS model generally underestimates the mean velocity at the logarithmic layer at fourth or higher-order accuracy. This underestimation may be caused by the lack of the dissipation which should be ideally compensated by the SGS model. The notable feature appears when we adopt second-order accuracy, in which the underestimation by the higher-order accuracy is compensated with the second-order numerical errors and predicted mean velocity unexpectedly shows good agreement with the empirical law without SGS model. This feature clearly indicates that FD error at the lower-order accuracy has a similar effect to the SGS models. It also should be noted that the difference between second order and higher order is remarkable. Considering these results obtained by LES without SGS model, the grid resolution of Nx x Nr x No= 64 x 32 x 128 is adopted hereafter. The corresponding grid widths in wall coordinate are + Ax + = 57.9 for the streamwise direction, Armi ~ = 1.57 and Ar+ax = 44.9 for the radial direction, and (rA0)+~n = 2.2(near the axis) and (rAO)+ax = 29.0(near the wall) for the azimuthal direction. Figure 5 shows the comparison of the two SGS models indicated as"Smagorinsky" and "Proposed isotropic" at two different FD accuracy of the second and the eighth order. At the second-order accuracy, both SGS models overestimate the mean velocity at the logarithmic layer. But this overestimation does not necessarily suggest the fundamental drawbacks of the tested eddy-viscosity models. As shown in Figure 4(b), owing to the large FD errors at the second-order accuracy, coarse DNS (LES without the SGS model) shows good agreement with the empirical law, accordingly the SGS model is not able to improve the mean velocity at all. In other words, the discretization errors and the SGS model affect the predicted mean velocity almost comparably at the second-order FD which degrades the expected feature of the SGS model. In fact, coarse DNS at less FD errors in Figure 5(b) underpredicts the mean velocity, while this underestimation is properly improved by the SGS models. Between the "Smagorinsky" and "Present" isotropic eddy-viscosity models, the latter shows better correlation with the empirical law at the eighth order while the former slightly overpredicts it.

CONCLUSIONS The new SGS eddy viscosity/diffusivity models are proposed which are supposed to be used in the FD scheme, and are validated in turbulent channel and pipe flows at various grid resolutions and FD orders of accuracy. Generally speaking, the proposed isotropic model shows better agreement with DNS results than Smagorinky's model when the same grid resolution or FD accuracy is adopted, and at the lower-order accuracy Smagorinsky's model seriously overestimates the mean velocity at the logarithmic layer. The findings of the present study can be summarised as follows:

305

_(a): N0=64

..7"

_(b): N0=128 :2nd

. . . . . . . . . . . . . . . . . . .

........... .....

:4th

~.

:8th

/1

......

; ;;k/

..... ..........

.

'~

:8th ~ ~ / / ~ .- ~ ~ J :12th~.,.g'*

10

,,,~

0

........

!

1

........

10 y+

i

....

.

0

100

.

.

.

.

1

.

.

.

10

! y+

......

100

Figure 4: Mean velocity profiles at Re,- = 590 in a turbulent pipe without SGS model: (a), No = 64; (b), No = 128. (b): 8th order

(a): 2nd order

-

-

~ /!

+

. ,.-""

15 ~ ~a%

- "~"~; -

"~e

/ ..

~

1

........ ~

-

10

,+

DNS :Smagorinsky iPre~ nt ...

100

" -

f .,,7

......".........

.........

:coarse D N S

........

:Smagorinsky

:~.'.~e~on, 1

10

y+

.

.

.

100

Y

Figure 5: Mean velocity profiles at ReT = 590 in a turbulent pipe, filter parameters are (~/h) 2 = 4/3 and (Zx/h) 2 = 4: (a), 2nd order; (b), 8th order. 1. Dependence of Smagorinsky's model on the discretized test filtering parameter 2x/h is crucial and its larger value enhances the overestimation of the mean velocity. Contrary to the Smagorinsky's model, the proposed isotropic model has the independent property of the test filtering parameter and is dependent only on the grid filtering parameter. Considering the fact that the actual grid filtering operation is dependent on the FD scheme or the order of the accuracy, this dependence is acceptable and should be optimized. According to the grid-resolution study, 2x/h ~ 1 is proposed. However, considering the recent suggestion of Guerts and FrShlich in 2002 that the contributions of modelling and spatial discretization errors varies depending on the grid filtering parameter, more careful optimization might be required in a near future. 2. The effect of GS velocity gradient on the SGS heat flux estimation is validated and the validity of the SGS heat-flux model with GS velocity gradient given as eq. (8) compared with the simple eddy diffusivity model of eq. (7) is demonstrated. 3. The effect of the numerical error by the second-order accuracy is rather big, and the mean velocity predicted by LES without SGS model unexpectedly shows good agreement with DNS results. Accordingly fourth or higher-order FD accuracy is required to properly distinguish the effect of SGS models from the FD errors.

306

ACKNOWLEDGEMENTS

This work has been conducted in part under the project of "Frontier Simulation Software for Industrial Science" at Institute of Industrial Science (IIS), the University of Tokyo, as a proposal to IT-program organized by Ministry of Education, Culture, Sport, Science and Technology of Japan. We are also indebted to Dr. Y. Morinishi for his providing the simulation code used here.

REFERENCES

Bardina, J. (1983). Improved turbulence models based on large eddy simulation of homogeneous incompressible turbulent flows. Ph. D. dissertation, Stanford University Dukowicz, J. K., and Dvinsky, A. S. (1992). Approximation as a higher order splitting for the implicit incompressible flow equations. J. Comput. Phys. 102, :334-336. Guerts, B., and FrShlich, J. (2002). A framework for predicting accuracy limitations in large-eddy simulation. Phys Fluids 14 , L41-L44. Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991). A dynamic subgridscale eddy viscosity model. Phys Fluids A3 , 1760-1765. Harlow, F. H. and Welch, J. E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182-2189. Launder, B. E.(1988). On the computation of convective heat transfer in complex turbulent flows, J. Heat Transfer 110, 1112-1128 Morinishi, Y., Lurid, T. S., Vasilycv, O. V., and Moin, P. (1998). Fully conservative higher order finite difference schemes for incompressible flow. J. Cornp. Phys. 142, 1-35 Morinishi, Y., Vasilyev, O. V., and Ogi, T. (2004). Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations. J. Comp. Phys. in press. Moser, R. D., Kim, J., Mansour, N. N. (1999). Direct numerical simulation of turbulent channel flow up to Re~ = 590, Phys. Fluids 11,943-945 Tsubokura, M.. (2001). Proper representation of the subgrid-scale eddy viscosity for the dynamic procedure in large eddy simulation using finite difference method. Phys. Fluids 13, 500-504 Tsubokura, M., Kobayashi, T. and Taniguchi, N. (2001). Development of the isotropic eddy viscosity type SGS models for the dynamic procedure using finite difference method and its assessment on a plane turbulent channel flow. JSME Int. J. Set. B 44,487-496 Tsubokura, M. (2003). Subgrid scale modeling of turbulence for the dynamic proccdure using FDM and its assessment on the thermally stratified turbulent channel flow, ASME Journal of Applied Mechanics, submitted Yoshizawa, A., Tsubokura, M., Kobayashi, T. and Taniguchi, N. (1996). Modeling of the dynamic subgrid-scale viscosity in large eddy simulation, Phys. Fluids 8, 22542256 Zagarola, M. V. and Smits, A. J. (1998). Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33-79.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

307

A S S E S S M E N T OF THE DIGITAL FILTER APPROACH FOR GENERATING LARGE EDDY S I M U L A T I O N INLET C O N D I T I O N S I. Veloudis, Z. Yang, J.J. McGuirk, G. J. Page Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, LE11 3TU, UK.

ABSTRACT

The effect of inputting either uniform or spatially varying turbulence scales into an inlet condition generator for LES prediction has been investigated. LES calculations of a channel flow with a periodically repeating constriction were used as a benchmark test case. Simulations using a streamwise periodic boundary condition (PBC) were first compared with a published highly resolved prediction (Temmerman et al (2002)) to establish the accuracy of the present LES results. Post-processed statistics from this simulation were then input into the Digital Filter Generator (DFG) approach of Klein et al (2003). Two time series were created using the DFG. In the first, as well as specifying the first and second moments of the velocity field over the inlet plane from the PBC simulation, the turbulence scales input into the DFG were spatially uniform at values appropriate to a single point in the channel centre. In the second, the turbulence scales were allowed to vary in the wall normal direction, their variation again being deduced from the PBC simulation. LES calculations were repeated using these time series at channel inlet. Analysis of the results and comparison to the PBC predictions showed that the use of spatially varying turbulence scales increased the accuracy of the simulation in some important areas, although at some significant cost. For practical use of the DFG technique in LES calculations of general flows, a spatially varying scales approach is likely to be necessary and worthwhile, but efforts to make this more general and cost effective are needed.

KEYWORDS LES inflow conditions, digital filter generator, turbulence scales, periodic hill channel flow

INTRODUCTION

One of the important factors for both Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) of general flows is the proper specification of time varying inlet boundary conditions, since in many cases these can significantly affect the development of the downstream flowfield. It has been shown by Klein et al (2001a, 2001b), Stanley et al (2000), and Lund et al (1998) that DNS and LES results for a plane jet, a spatially developing boundary layer and the breakup of a liquid sheet are very sensitive to the inlet conditions. Le et al (1997) demonstrated in a backward facing step flow

308 simulation that, if the inflow information did not contain the phase information of realistic turbulent eddies, the turbulence level in the separated shear layer decayed rapidly and a relatively long inlet section was needed to develop physically acceptable inlet profiles. Maruyama (1999) performed Large Eddy Simulation of a high Reynolds number turbulent boundary layer and found that the inflow turbulence characteristics had a significant impact on the downstream flowfield. In flows that are fully developed, or take place in streamwise repeating geometries, the question of generating inlet conditions is circumvented by the use of periodic boundary conditions, where outflow and inflow conditions are numerically linked. However, this method is of course limited to cases where periodic conditions are applicable. Spalart (1988) extended the idea of outlet/inlet linkage to a turbulent boundary layer simulation by using a coordinate transformation technique; however, the method is complex and restricted to flows where mean streamwise variations are small compared with transverse variations. Lund et al (1998) presented a similar method for generating turbulent inflow data for simulation of spatially developing boundary layers. Their approach was to "recycle" the turbulent velocity field to the inlet from predictions at a station near the domain exit after appropriate scaling, and is essentially a variant of the Spalart method. The so-called 'precursor' method, where a separate LES calculation is performed to generate unsteady inlet conditions, is perhaps the most accurate current technique. It produces realistic turbulent structures at the inlet plane and can be the only way of allowing the LES method to capture the correct flow physics in the domain of interest (see for example Akselvoll and Moin (1996), or Tang et al (2004)). This approach is, however, expensive since it effectively requires two LES predictions to be performed. Hence, the search for alternative and improved synthetic inflow condition generators is still an important objective. A few simple suggestions have been made, based on superimposing disturbances onto a specified time-averaged velocity profile. The disturbance can be a 'white noise' (random number) signal but this is known to produce poor results since the non-correlated nature means the input disturbance decays rapidly. Le et al. (1997) extended this idea by superimposing fluctuations with given second moments and spectral shape onto a mean velocity profile and applied this method to a backward facing step flow simulation. They found that their method was not very satisfactory as the turbulence level still decayed rapidly. Recently, Batten et al (2004) proposed a new method to generate synthetic inflow turbulence based on superposition of sinusoidal modes with random frequencies and wave numbers, constructing a field with given moments and spectra. Keating and Piomelli (2004) applied this method to a spatially developing duct flow and a boundary layer on a flat plate. Their study showed that the method proposed by Batten et al does display advantages over other methods based on random number sequences. However, a relatively long transition distance was still required for the turbulence structure in the inlet profile to become self-consistent. A promising method has been developed by Klein et al (2003). This uses a digital filtering procedure to generate velocity time series at each point in the inlet plane that reproduce specified first and second moment single point statistics as well as a locally prescribed autocorrelation function. The nondimensional assumed shape of the autocorrelation function is made dimensional at each point using input turbulence length (or time) scales. Klein et al have demonstrated the procedure in two test cases: DNS of a plane turbulent jet and 2D DNS of the primary break up of a liquid jet. The method seems flexible, more accurate, and more cost effective than most other existing methods. While first and second moments are often available (e.g. from experimental data), or relatively easy to estimate, turbulence length and time scales are more problematic and rarely available from measurements. In the validation and test cases presented in Klein et al (2003), it seems that only turbulence scales that were spatially uniform over the entire inlet plane were used to generate the time series for all grid points in the inflow plane. The question of the sensitivity of resulting LES calculations to the choice of turbulence scales used as input to the DFG has not so far been investigated. Klein et al emphasise that their method can deal with spatially varying scales, but warn that if the variations are 'strong', distorted correlation functions can result. No estimate is provided, however, to quantify how large a variation is permissible, nor what level of benefits can accompany use of spatially varying scales. The

309 main focus of the present paper is therefore to investigate this. The effect of inputting spatially variable turbulence scales into the DFG technique and performing flowfield simulations will be examined by comparing two LES calculations of the same flow problem. The first DFG-driven simulation will use inflow data generated with spatially uniform turbulence scales, following the practice of Klein et al (2003): we call this a 'Level 1' simulation. The second LES prediction will use inflow data generated with spatially varying turbulence scales in the wall-normal direction; we refer to this as a 'Level 2' simulation. The two simulations will be compared with a 'baseline simulation' from which the information needed for DFG input (first and second moments and turbulence scales) is extracted. For convenience, a flow problem allowing periodic boundary conditions in the streamwise direction has been chosen for the baseline simulation. It is of prime interest to see the level of fidelity achieved by the two DFG-driven simulations in reproducing the predictions of the baseline simulation.

NUMERICAL METHOD For the purposes of this study, an in-house CFD code, DELTA, was employed. This code was originally written as a RANS code for aerospace applications such as Hot Gas Ingestion (Page et al (1991)). It is based on a multi-block, structured mesh, finite volume formulation. It uses a collocated variable arrangement on a curvilinear non-orthogonal mesh, in combination with Rhie and Chow smoothing to avoid pressure-velocity decoupling. The code adopts an appropriately modified pressurecorrection algorithm so that it can predict problems with a wide range of flow speeds from incompressible to strongly compressible. It was converted to run in LES mode essentially via four modifications: (i) (ii) (iii)

(iv)

an explicit time stepping method was introduced employing a 3rd order accurate low storage Runge-Kutta method, 2 nd order central differencing was used for convection term discretisation rather than the MUSCL technique used previously, correct scaling of the Rhie and Chow smoothing terms was adopted to take account of the very small time steps needed in LES calculations, combined with very significant variations in cell volume size across the solution domain, a sub-grid-scale (SGS) model was introduced; for the present calculations a simple Smagorinsky (1963) model was selected, including the usual Van Driest near wall damping modification.

For the generation of LES inflow data, the DFG algorithm of Klein et al (2003) was used. The DFG technique generates artificial inflow data that are characterised via input statistical properties. These are the profiles in the inlet plane of first and second moments (mean values and Reynolds stresses) of the three velocity components and length (or time) scales of the turbulence in the 3 co-ordinate directions. These statistical properties are obtained here from a separate LES prediction simulation. Using this information, the DFG method produces time series of the inflow data in two steps. In the first step, a provisional time series for a velocity fluctuating about a zero mean is generated (Um). A random data series r m is defined with r m = O, rmrm = 1, rmr. = 0 ( m :/: n ) . Um is then constructed from a convolution of the random data series and a digital linear non-recursive filter: N

Um = Zbnrm+n n=-N

(1)

where the b, are filter coefficients and N is the extent of the filter support. Due to the statistical properties of a random data series, it follows that:

310 N

~_~bjb j-k UmUm+ k m j---N+k ~ m

(2)

N

UmU m

j=-N

Eqn. 2 provides a relation between the autocorrelation of U m ( R u u - the left hand side) and the filter coefficients. If a functional form of the autocorrelation is assumed, the values of bn that will give rise to a time series Um possessing this autocorrelation may be obtained by inversion of Eqn. 2. Klein et al suggested that a non-dimensional form of Run (x, r), where r is the distance vector and r = could be

I,I,

prescribed via just one parameter, a turbulence length scale L. Batchelor (1953) has shown that in the case of homogeneous turbulence, Ruu takes the form

7g'r2 / (3)

Ruu (r, O, O) =exp -~--~) If we specify L in terms of the local grid spacing, say Ax, and write L = nAx, Eqn. 3 becomes:

UmUm+k=Ruu(kAx)=exp( UmUm

~r(kAx)21 4("Ax) 2

( erk 2"] =

expt--~n2)

(4)

Klein et al (2003) show that this results in explicit relations for the filter coefficients:

bk ~

bk

with/~k = exp --~-n2 )

(5)

Hence, by specifying the desired length scale through an appropriate choice of n, and assuming the Ruu functional form of Eqn. 4, a filter kernel can be generated, such that when convoluted with rm, a time series for Um is produced which possesses the desired length scale. In order to extend the procedure to three dimensions, three 1D filter coefficients are multiplied to produce the final 3D filter coefficient. This requires the definition of 3 length scales (Lx, Ly, Lz), or what is equivalent, 3 values for n (nx, ny, nz). Eqn. 1 is then used to produce the zero mean fluctuating velocity field. The second step of the DFG procedure is used to modify the time series generated in the first step so that it possesses desired mean values and correlations between different velocity components. This is achieved by employing Lund's method (1998). The time series from the first step, ui, satisfies: ui - O, u~uj = 6,j. The following transformation gives the final result:

U, = Ui + a~juj m

(6)

with the a o. given by:

o ),,2

o

t,,R3ilail (R32-a21a3,)la22 (R33-a21-a322~,2

/

311 B

Where (.) denotes an appropriate averaging procedure, R o. is the second order correlation tensor and Ui is the velocity time series that has the desired first and second moments and the input turbulence scales (and, it should be noted, also the input autocorrelation function). The above description has been cast in terms of turbulence length scales. It is, however, convenient to modify the implementation such that for the streamwise flow direction a time scale Tx is used instead of a length scale Lx, since this allows direct connection between the streamwise direction and the timemarching nature of the LES solution. This implies that temporal correlations are modelled via the same exponential form as given in Eqn. 3 transformed into the time domain. The method is otherwise identical, with the integral time scale linked to the simulation time step At via Tx -ntAt. If the time and length scale parameters nt, ny, nz are the same at every grid node in the inlet plane, then just one 3D filter coefficient suffices. If the scales vary spatially over the inlet plane, a number of filters are required. In principle, one 3D filter coefficient may be created for each grid point. However, the risk of the method producing distorted correlation functions increases as the spatial variation of scales grows, and in addition, the computational cost of the DFG process itself increases significantly. Hence, there is a cost-effective trade-off that has to be found, particularly since, if the DFG method becomes as expensive as a precursor LES, its usefulness disappears. TEST CASE AND RESULTS OF PERIODIC BOUNDARY CONDITION SIMULATION In order to carry out an assessment of the DFG algorithm, the streamwise repeating constricted channel flow problem studied by Temmerman et al (2002) was selected as a test case. The flow geometry can be seen in Figure 1. The channel height is Ly = 3.035h, where h is the constriction height, the length in the streamwise direction is Lx = 9h, and in the spanwise direction is Lz = 4.5h. The Reynolds number based on h and the bulk velocity, UB, at the constriction crest is 10,595. For the calculations performed h was 0.028m and UB 5.53m/s, identical to the values used in Temmerman et al (2002). A periodic boundary condition (PBC) simulation was performed to obtain the information required by the DFG method. The PBC simulation was processed to deduce statistical data from a simulation record length (-0.64 secs) corresponding to around 40,000 time steps or 14 flow through times (note TFT = LJUB 45 msec), providing statistically stationary first and second moments of the velocity field. The time step used At was 3.51 e-4 TFT (At -~ 16 psec) corresponding to a maximum CFL number of around 0.2. The simulation was performed on a grid with 160x64x62 (x, y, z) cells, where the grid spacing was constant in the streamwise, x, and spanwise, z, directions while in the wall normal direction, y, it was refined close to the lower wall. Note that this grid (-650,000 cells) is substantially coarser than the 196x128x186-5 million cell grid used in the well-resolved simulations of Temmerman et al (2002).

Figure 1: Contours of U/UB from PBC LES Figure 1 presents contours of the mean streamwise velocity component U divided by UB. The flow separates from the curved surface of the constriction and reattaches on the lower plane channel wall to form a separation bubble of length 4.41 x/h. Predicted values for separation and reattachment locations

312 are compared with reference data taken from Temmerman et al in Table 1.

U/U,, u'u'/U~ and

u'v'/U~ profiles in the y direction at x/h = 0.05, 2.0 and 5.0 can be seen in Figures 2, 3 and 4, again compared with the reference LES data. The position of each station is identified by the dotted lines in Figure 1. TABLE 1 SEPARATION

BUBBLE

DATA

FROM

REFERENCE

AND PBC SIMULATIONS

Reattaehment Point ! Separation Bubble (x/h) ! Length (x/h)

Simulation

Separation Point (x/h)

Reference Data

0.22

4.72

Periodic Boundary Conditions

0.09

4.50

......... II....

4.5 4.41

,

Figures 2 (a) (b) and (c) show that the current PBC simulation is close to the results of the reference LES (Temmerman et al (2002)) in terms of the mean axial velocity development. At the inlet plane (x/h - 0.05) the profiles are similar, although the present LES shows higher velocities close to the lower wall, and a mass balancing underprediction in the region near the upper wall. The high velocity near the lower wall is probably responsible for the earlier separation shown in Table 1. At x/h = 2.0, the separation bubble features are in close agreement with the reference data. The overprediction of the axial velocity in the central region and underprediction close to the upper wall are presumably related to the inlet profile differences. Finally, Figure 2 (c) shows that after reattachment, the flow develops in a similar manner in the two calculations, except near the lower wall. The differences noted are small, but presumably related to the lack of resolution (compared to Temmerman et al's grid) in the current LES calculation near the lower wall, and the use of a wall function as opposed to a well-resolved LES grid. The comparison between predictions of axial normal stress and shear stress show reasonably close agreement between the two calculations as can be seen in Figures 3 and 4. Although there are differences, it is felt that for the present purposes, the benchmarking of the current PBC data against the reference LES data of Temmerman et al (2002) is sufficient to justify the use of the PBC simulation as valid test data in the DFG study to follow.

3y/h

3

|

2

fyY~ f

U/UB

(a) I

02

[

04

I

06

I

08

I

I

)

(b) -02

0

I 02

I 04

I 06

I 08

I

02

04

ll6

08

Figure 2" Axial velocity profiles at (a) x/h = 0.05, (b) 2.0, (c) 5.0. Squares: Ref. LES data, line" PBC

313 3

3

3

2

2

1

1

|

u'u'/U 2,

(a) c

.

I Q(lZ

I QO~

I Q~

I Q(13

9

Q(~

e04

e08

068

C

Q~

004

Figure 3" Axial stress profiles at (a) x/h = 0.05, (b) 2.0, (c) 5.0. Squares: Ref. LES data, line: PBC /h n ~ u ii

u'~n

|

n

.

II

.

'

uv/U~ I -ool

l 2

2

l(c)

(a) 1o~

.~

-Q(13

-~

-~

0

-OC2

4301

0

Figure 4: Shear stress profiles at x/h - (a) 0.05, (b) 2.0, (c) 5.0. Squares: Ref. LES data, line: PBC EXTRACTION OF LENGTH AND TIME SCALES FROM PBC LES SOLUTION

Using the PBC simulation, time series were extracted at selected points in the (y, z) space covering the inlet plane. From these time series, a longitudinal integral time scale could be calculated via the autocorrelation of the axial velocity in the usual way:

(8)

Tx(y,z ) = f u(x,,,y,z,t)u(x,,,y,z,t +dr)dr

Similarly, lateral integral length scales Ly and Lz were calculated from spatial correlations in the relevant directions. For input to the DFG calculation at Level 1, the turbulence scales were calculated using PBC data from a single point at mid-span and mid-channel height. The values obtained can be seen in Table 2 in terms of time step (nt = Tx/At) and mesh spacing (ny = Ly/Ay, nz = Lz/Az): TABLE 2 n VALUES

Simulation Periodic Boundary Conditions

I! II

FOR INLET

PLANE

MID POINT

nt

ny

nz

342

5

3

314

For input to a DFG calculation at Level 2, PBC time series were processed at a series of points in the wall normal (y) direction along the mid-span line. The resulting distributions are shown as dotted lines in Figure 5. To reduce the cost involved in using the DFG approach to generate inlet plane data with spatially varying scales, these profiles were digitised into 8 zones. The digitisation can be seen as the solid line in Figure 5; this resolution was considered sufficient for the current study. 900 800 700 600 500 nt 400

.....t

300

25

6

20

5

4

15

..,.

nz 3

10

200 100 0

2

5

...... -

0 ' 0

0.01

0.02

0,03

0.04

0.05

0.06

0.00

i 0.01

-

" "".

1

,L-.-

i 0.02

y (m)

0.03

0.04

0.05

0.06

0.00

y (m)

i

"',

0 0.01

0.02

0.03

0.04

0.05

0,06

y (m)

Figure 5" Profiles (dotted line) of nt=Tx/At, ny =L/Ay, nz=Lz/Az, solid line-digitised profiles It should be noted at this point that the computational cost of the DFG algorithm is significantly affected by two parameters. The first is the maximum value of nt, ny and n~. The larger the maximum value of n, the larger the working arrays generated by the algorithm. This results in increasing computational time required to produce each step of the time series. The second constraint is the number of filters used. The cost of running the DFG method to produce the inlet plane time series increases dramatically the larger the number of different turbulence scales are specified in the inlet plane. It is for this reason only a coarse digitisation has been chosen in this initial study. LEVEL 1 & LEVEL 2 LES RESULTS AND COMPARISON WITH PBC SIMULATION Before proceeding to the presentation of the Level 1 and 2 results, some comments may be made about 1,2 DFG

DFG

Assumption Assumption .........

0.8

,,

(Point A) ( P o ~ t

B)

f

Ruu 0.4

-0.2 -0.4

Time (ms) Figure 6" Autocorrelations in streamwise direction the improved realism achievable via a spatially varying scales input into the DFG algorithm. Figure 6 shows autocorrelations of the axial velocity deduced from the PBC simulation at two points. The first. point (A) is the mid-span mid-height point selected to provide the turbulence scales input to the DFG Level 1 simulation. Point B is a point close to the lower wall on the mid-span line. Two observations may be made. The first is to note the shape of the autocorrelation function predicted by the PBC simulation. The predicted shape at neither point follows the Gaussian shape assumed in the DFG

315 method. The record length used to obtain these autocorrelations is at least 60 integral timescales long, so the PBC predicted shapes are numerically accurate and departures from a Gaussian shape are determined by the local flow physics. Hence, the Klein et al (2003) DFG approach will never be able to reproduce the inlet turbulence structure precisely, since it generates time series that are forced to fit Gaussian correlation functions. The second point is that the integral time scale at point B is around twice that of point A. Hence a Level 1 approximation, which would assume the same turbulence scales at both points, will necessarily introduce errors (the correlation function labelled DFG (point A) would be used at both points). A Level 2 approximation, although it will not remove the Gaussian function restriction, will at least allow this to be scaled appropriately at different points, as shown in Figure 6 by the two curves labelled 'DFG' Profiles of U/U,, at two streamwise locations and of u'u'/U 2 and u'v'/U~ at x/h = 2.0 predicted by the PBC and Level 1 and Level 2 DFG simulations are shown in Figures 7 and 8. Although the mean velocity profile within the separated zone is close to the PBC simulation in both DFG calculations, the Level 2 data are closer and, further downstream, only the Level 2 simulation reproduces the recovery region after reattachment as in the PBC case. A similar observation may be made for the turbulence statistics where, at x/h=2.0, the Level 1 data show significant underprediction of the fluctuating field in 3 _

2 _

1 -

- f

-0 2

0

012

014

016

018

0 2

016

014

018

Figure 7: U/UB profiles at x/h = 2.0 (left) and 5.0 (fight). Solid line" PBC, dashed line: Level 1, dash dot line: Level 2 3 "31

o

0 02

Figure 8:

0 04

u'u'/U~

0 06

0 08

-0 04

-0 03

-0 02

-0 01

0

profiles (left) and u'v'/U~ profiles (fight) at x/h : 2.0. Solid line: PBC, dashed line: Level 1, dash dot line" Level 2

both normal and shear stress profiles. These profiles are fairly typical and indicate the closer fidelity to the target PBC solution achieved by a spatially varying scales Level 2 approach to synthetic inlet profile generation. It should also be commented that, although the spatial variation of scales in the wall normal direction was quite significant (Ly for example varying by a factor of 5, see Figure 5), this did not seem strong enough to cause distortion errors as noted by Klein et al.

CONCLUSIONS The comparisons between a target PBC LES calculation and two DFG-driven LES calculations of the same flow presented in the present paper have highlighted the following main points:

316

(i) the Gaussian autocorrelation shape assumed by the DFG will necessarily differ in general from that determined by the flow physics, and this limits the ability of the DFG method to reproduce turbulence structures at an inlet plane with complete fidelity compared to, say, a precursor calculation, (ii) the use of spatially varying turbulence scales in the DFG input cannot remove this restriction, but does allow the integral scale of the Gaussian correlation to match the local turbulence structure better, (iii) a DFG-driven LES calculation using spatially varying scales at inlet predicted the test problem better than a constant scale simulation on the evidence of both mean and turbulence profiles, (iv) although the strength of spatial variation in the turbulence scales was relatively strong (a factor of 5), this did not prove problematic for the DFG method, (v) there is clearly a compromise between the computational cost of the DFG and the accuracy of the simulation, however, on the evidence presented here it is believed that, for general flows, a spatially varying scale approach will probably be necessary, and this suggests further work on the DFG approach to reduce its computational cost for variable scale input would be worthwhile.

REFERENCES

Akselvoll K. and Moin P. Large Eddy Simulation of turbulent confined co-annular jets. (1996). J. of Fluid Mech., 315, 387-411. Batchelor G. (1953). The Theory of Homogeneous Turbulence, Camb. Univ. Press, Cambridge, UK. Batten P., Goldberg U., and Chakravarthy S. (2004). Interfacing statistical turbulence closures with Large Eddy Simulation, AIAA Journal, 42, 485-492. Keating A. and Piomelli U. (2004). Synthetic generation of inflow velocities for Large Eddy Simulation, AIAA 2004-2547, 34thAIAA Fluid Dynamics Conference. Klein M., Sadiki A., and Janicka J. (2001 a). Influence of boundary conditions on the Direct Numerical Simulation of a plane turbulent jet. Proc. of TSFP2 1, 401-406. Klein M., Sadiki A., and Janicka J. (2001b). Influence of the inflow conditions on the Direct Numerical Simulation of primary breakup of liquid jets. Proc. oflLASS-17-Europe 1, 475-480. Klein M., Sadiki A., and Janicka J. (2003). A digital filter based generation of inflow data for spatially developing Direct Numerical or Large Eddy Simulations. J. Comp. Phys. 186, 652-665. Le H., Moin P., and Kim J. (1997). Direct Numerical Simulation of turbulent flow over a backwardfacing step. J. Fluid Mech., 330, 349-374. Lund T. S., Wu X., and Squires K. D. (1998). Generation of turbulent inflow data for spatially developing boundary layer simulations. J. Comp. Phys. 140, 233-258. Maruyama T. (1999). On the influence of turbulence characteristics at an inlet boundary for Large Eddy Simulation of a turbulent boundary layer. Proc of ETMM4 (Corsica). Page G. J., Zhao, H., and McGuirk J.J. (1991). A parallel multi-block Reynolds-Averaged NavierStokes method for propulsion installation applications. Proc. 12th Int. Symp. on Air Breathing Engines (Melbourne) 1,864-876. Smagorinsky J. (1963). General circulation experiments with the primitive equations, I: The basic experiment. (1963). Monthly Weather Review, 91, 99-164 Spalart P. (1988). Direct Numerical Simulation of a turbulent boundary layer up to Re0 = 1410. J. Fluid Mech. 187, 61-98. Stanley S. and Sarkar S. (2000). Influence of nozzle conditions and discrete forcing on turbulent planar jets, AIAA Journal, 38, 1615-1623. Tang G., Yang Z., and McGuirk J.J. (2004). Large Eddy Simulations of dump diffuser aerodynamics. Paper in preparation. Temmerman L., Leschziner M.A., Mellen C., and Froehlich J. (2002). Investigation of wall-function approximations and sub grid scale models in Large Eddy Simulation of separated flow in a channel with streamwise periodic constrictions. Int. Jnl. of Heat and Fluid Flow 24, 157-180.

4. Hybrid LES/RANS Simulations

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

319

H Y B R I D L E S - R A N S : C O M P U T A T I O N OF T H E F L O W AROUND A THREE-DIMENSIONAL HILL L. Davidson ~ and S. DahlstrSm ~ ~Department of Thermo and Fluid Dynamics Chalmers University of Technology, SE-412 96 G6teborg, Sweden http: //www. tfd. chalmers, se / -lada

ABSTRACT The main bottle neck for using Large Eddy Simulations at high Reynolds number is the requirement of very fine meshes near walls. Hybrid LES-RANS was invented to get rid of this limitation. In this method unsteady RANS (URANS) is used near walls and at a certain distance from the wall, where the LES resolution is sufficiently good, a switch to LES is made. A one-equation RANS turbulence model is used in the URANS region and a one-equation SGS model is employed in the LES region. The matching between URANS and LES takes place in the inner log-region. In the present paper an improved LES-RANS method is evaluated for computing the flow over a three-dimensional axi-symmetric hill. The improvement consists of adding instantaneous turbulent fluctuations (forcing conditions) at the matching plane between the LES and URANS regions in order to provide the equations in the LES region with relevant turbulent structures. The turbulent fluctuations are taken from a channel DNS. Three different approaches are compared in the present paper: hybrid LES-RANS with forcing conditions, hybrid LES-RANS without forcing conditions, and LES employing the WALE SGS model

KEYWORDS

LES, hybrid LES-RANS, DES, forcing conditions, 3D hill, URANS

INTRODUCTION When simulating bluff body flows, LES (Large Eddy Simulations) is the ideal method. Bluff body flows are dominated by large turbulent scales which can be resolved by LES without too fine resolution and accurate results can be obtained at affordable cost (Rodi et al., 1997; Krajnovid and Davidson, 2003, 2004). On the other hand, doing accurate predictions of wall-bounded flows with LES is a challenging task. The near-wall grid spacing should be about one wall unit in the wall-normal direction. This is similar to the requirement in RANS (Reynolds-Averaged NavierStokes) using low-Re number models. The resolution requirement in wall-parallel planes for a wellresolved LES in the near-wall region expressed in wall units is approximately 100 (streamwise) and

320

wall

URANS region LES region URANS region

Iy+l

wall x

Figure 1: The LES and URANS region (left) and the computational domain (right). 20 (spanwise). This enables resolution of the near-wall turbulent structures in the viscous sublayer and the buffer layer consisting of high-speed in-rushes and low-speed ejections (often called the streak process). At low to medium Reynolds numbers the streak process is responsible for the major part of the turbulence production. These structures must be resolved in an LES in order to get accurate results. Thus, for wall-bounded flows at high Reynolds numbers of engineering interest, the computational resource requirement of accurate LES is prohibitively large. Indeed, the requirement of near-wall grid resolution is the main reason why LES is too expensive for engineering flows, which was one of the lessons learnt in the LESFOIL project (Davidson et al., 2003; Mellen et al., 2003). The object of hybrid LES-RANS (Xiao et al., 2003; Davidson and Peng, 2003; Temmermann et al., 2002; Tucker and Davidson, 2004; Tucker, 2003) is to get rid of the requirement of high near-wall resolution in wall-parallel planes. In the near-wall region (the URANS region), a low-Re number RANS turbulence model (usually an eddy-viscosity model) is used. In the outer region (the LES region), the usual LES is used, see Fig. 1. The idea is that the effect of the near-wall turbulent structures should be predicted by the RANS turbulence model rather than being resolved. The matching between the URANS region and the LES region usually takes place in the inner part of the logarithmic region. In the LES region, coarser grid spacing in wall-parallel planes can be used. In this region the grid resolution is presumably dictated by the requirement of resolving the largest turbulent scales in the flow (which are related to the outer length scales, e.g. the boundary layer thickness), rather than the near-wall turbulent processes. The unsteady momentum equations are solved throughout the computational domain. The turbulent RANS viscosity is used in the URANS region, and the turbulent SGS viscosity is used in the LES region. Although good results have been presented with hybrid LES-RANS, it has been found that the treatment of the interface between the URANS region and the LES region is crucial for the success of the method. The resolved turbulence supplied by the URANS region to the LES region does not have any reasonable turbulent characteristics and is not representative of turbulence at all. This results in too poorly resolved stresses on the LES side of the interface and this gives a hack - also referred to as a shift - in the velocity profile approximately at the location of the matching plane. In DahlstrSin (2003); Davidson and Dahlstr6m (2004) and Davidson and Billson (2004) an improved hybrid LES-RANS was presented in which fluctuations were a d d e d - taken either from channel DNS or synthesized - at the LES side of the interface. The method was applied to flow

321

in a channel and in an asymmetric diffuser. The method was shown to give good agreement with experimental results. An interesting- and rather similar approach - was recently presented by Batten et al. (2004) in which synthetic turbulent fluctuations was used to trig the resolved turbulence when going from an URANS region to an LES region. In the present paper three different methods are evaluated for computing the flow over a threedimensional axi-symmetric hill, see Fig. 1: hybrid LES-RANS with forcing conditions, hybrid LES-RANS without forcing conditions and LES using the WALE SGS model. The predictions are compared with experiments of Simpson et al. (2002) and Byun et al. (2003, 2004). It can be mentioned that a wide range of turbulence models were applied to this flow including many different two-equation eddy-viscosity models, the explicit algebraic Reynolds stress model and a full Reynolds stress model. All RANS models fail completely in capturing this flow (Haase et al., 2005).

EQUATIONS The Navier-Stokes equations with an added turbulent/SGS viscosity read

Ogi

0

lop

O---t -~- ~X j ( (t i (t j ) = ~ l i

0 [

fl O X i -~- ~

Ofti] i)fti

( l/ -~- l/T) O X j

J

' ~x~ = 0

(1)

where uT = ut (ut denotes the turbulent RANS viscosity) for y < ym~ (see Fig. 1), otherwise uT = usg~. The turbulent viscosity UT is computed from an algebraic turbulent length scale and for kT a transport equation is solved, see below. The density is set to one in all simulations.

HYBRID LES-RANS A one-equation model is employed in both the URANS region and LES region which reads Ok, T ot

0 +

0 -

[ [ (" +

Ok T

] j +

/.3/2 -

"T

PkT --

Sij

'

=

-- 212T Sij

(2)

where l]T : ckl/2g. In the inner region (y < Yml) kT corresponds to RANS turbulent kinetic energy k; in the outer region (y > Ymt) it corresponds to subgrid-scale kinetic turbulent energy (ksgs). No special treatment is used in the equations at the matching plane except that the form of the turbulent viscosity and the turbulent length scale are different in the two regions. At walls kT -- O. More details in Davidson and Dahlstr6m (2004).

Forcing conditions The DNS fluctuations are added as momentum sources in the cells in the LES region adjacent to the matching plane. The sources for the three momentum equations read

SU----3'PU'DNSV'DNsAn,

SV----'~P@NsV'DNsAn,

SW----')'PW~DN6.V~DNsAn

(3)

where An is the area of the control volume, and ~ = C.~kT(X,Yrnl, Z)/kfluct, where kfluct is the turbulent kinetic energy of the DNS fluctuations and c~ = 0.4. The time history of the channel DNS database along a line yml is used. The streamwise variation is obtained by employing Taylor's hypothesis. These DNS fluctuations are consider to be generic, i.e. to be used for all types of flow. The object of the DNS fluctuations is to act as a forcing term, and as such the exact form of the fluctuations is not critical: the forcing term should force the momentum equations to start

322

700

600

o o~ oo

'<•400

+

500

oo oo

~

oo oo

15

~"~'- 300 +~ <1 200 100

o4

-

o'"

; xlH

~'o

i

1'5

1'0

1;

z/H (b) - - N~ = (~%/AX)z=O;--- Nz = (~%/ZXz)~=0.

(a) Az + at z = 0; Az + z / H = 1.65; o. Ax+/5 at z = 0.

Figure 2: Grid spacings.

>

Forcing

> ,

No forcing ,

0.6

0.6

:c 0.4

0.4

0.2

0.2

,

0 1

1.5

~IH

2

1

1.5

2

~IH

Figure 3: Velocity vectors in symmetry plane. Every 2nd vector in x direction and every 3rd vector in y direction are shown. The vectors above the figures have the magnitude of 0.1Uin.

to resolve large-scale turbulence. The forcing term should be physical, i.e. in some sense mimic turbulent fluctuations, but it is probably also important that its time scale and length scale are related to the that of grid cells. There is no point in using a forcing term whose length scale is much larger or smaller than the computational cell, because such a term cannot trigger the equations into resolving turbulence. The location of the interface is set to y+ ~ 40 at the inlet, and in the domain it is defined by the time averaged streamline starting at this y-location. More details can be found in Davidson and DahlstrSm (2004) and Davidson and Binson (2OO4).

The Numerical Method An incompressible, finite volume code is used (Davidson and Peng, 2003). For space discretization, central differencing is used for all terms. The Crank-Nicolson scheme (with c~ = 0.6) is used for time discretization of all equations. The numerical procedure is based on an implicit, fractional step technique with a multigrid pressure Poisson solver Emvin (1997) and a non-staggered grid arrangement.

323

~

z=O

o

c

0.8

~-~

o

~

~

!

~

o i

!

"~

o

,,

o/,,

0.6

y/H

~

0.4

~

L-

0.2

O0

J

~ " "

J

1

0.5

:1!

,,

-

,,

'

o~

'!

Ol, !

0

,,

0.5

oil

1

0

o,

.

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

Figure 4: Streamwise velocity component (g)/Ui~ at x - 3.69H. hybrid LES-RANS with f o r c i n g ; - - - hybrid LES-RANS without forcing;-.- LES; o experiments (Simpson et al., 2002)

it

0.8

:

0.6

y/H

-D--

I

0.4 II

,,

4; t,,

fli

/4'

Ii

-II

i, II

02 /,~ /t

Oo'.2

-o.~

o

-0.2 -0.1

0

-0:2-0~

0

-0:2&-7 d

-0.2 -0.1

0

t

-012 -0.1

o

hybrid LES-RANS with forcing; Figure 5: Spanwise velocity component (@/Ui,~ at x - 3.69H. - - - hybrid LES-RANS without forcing;--- LES; o experiments (Simpson et al., 2002) RESULTS A 162 x 82 x 130 (z, y, z) mesh is used (1.7 million cells). It is nearly orthogonal in the near-wall region of the hill. The inlet is located at x = - 4 . 1 H and the outlet at x = 15.7H, see Fig. 1. The mean inlet {g} profile is set according to experimental data (measured at z = 0 without hill). Instantaneous turbulent fluctuations are taken from a channel DNS of Re~ = u~5/~ = 500 (Davidson and DahlstrSln, 2004). The flow in the lower half of the channel flow (half width 5) is then used to superimpose turbulent fluctuations in the boundary layers at the lower and upper wall. It turned out that RMS of the instantaneous fluctuations from the channel DNS matched the experimental RMS profiles reasonably well and no re-scaling of was applied. Slip conditions are used at the side walls and homogeneous Neumann conditions are employed for all variables at the outlet. A time step of AtUi~/H = 0.026 is used which gives a maximum instantaneous CFL of approximately 2 (the CFL number is larger than 1 in approximately 100 cells). The computations are run 8 000 time steps to make sure that the flow is fully developed which corresponds to 10 through flows. The flow is then time averaged during 5000 time steps. An estimate if this is sufficient is

324

experiments

>

with forcing

>

LES

.

:a

,

0.5

'

'

/

I

l

I

'

;

.

.

.

.

.

- - x \ \

. . . . . . . .

>

~ ~

without forcing

.

; ; -2

. . . . .

-1.5

z/H

.

.

.

.

.

.

.

.

.

.

" \ \ 1

<,

-1

-0.5

0-2.5

-2

-1.5

z/H

-1

-0.5

0

Figure 6" @}-{@} velocity vectors, x / H - 3.69. Every 3~e and 2ne vector in z and y direction, respectively, are shown. The vectors above the figures have the magnitude of 0.1U in. . . . . . .

x/H-

~'~

>

1.5

"", "x'"x

x/H-

-

2.5

_2

\ \

0.5

-s

-2

-1.5

-1

z/H

-0.5

I

0-2.5

-2

-1.5

z/H

-1

-0.5

Figure 7: {9}-{@} velocity vectors at x / H - 1.5 and x / H = 2.5. Hybrid LES-RANS with forcing. Every 3~d and 2nd vector in z and y direction, respectively, are shown. The vectors above the figures have the magnitude of 0.1Uin. to check how symmetric the flow is. A qualitative answer is obtained by looking at the spanwise velocity component at plane x = 3.69H, see Fig. 5. For example, after sampling during 5 000 time steps the maximum asymmetry in the (~} profile is O.027Uin and occurs near the symmetry plane at Izl/H = 0.16, where the flow is highly unsteady. Two global iterations are required for each time step and the CPU time for one time step is 25 seconds on a single AMD Opteron 244 processor. Some details of the grid are shown in Fig. 2. The streamwise and spanwise grid spacing in wall units are, as can be seen, fairly large, with the latter varying between 200 and 300, except at the foot and the crest of the hill. The Ax + values start at around 500 at the inlet, and attains values of 40 in the recirculating region. When we look at the grid spacings expressed as the ratio of the boundary layer thickness and the grid spacing we find values between 5 and 20 in the hill region for

325

with forcing

without forcing

LES _..._,. ......_..

1.5

--.--_.,-.~.....,,...,,....~

. . . . .

""-~"~'.'~"~"~'~

0.5

...4.....

",~ "N " ~ ~

[2::2h~.7~ZCZZ ~ I . . t ..

["2'"

__.._.....

-0.5 ::2 : 7.:R22 : ~ C'<"~. . . . . -1

. . . . . . .

/~'It

t//

I

. . . . .

..-/l\

. . . . .

.....-I

\ i

\ I

_.._.....,....,,tliii.... - . ~

~-.\\ .

.

/

\ l

'~

\~

I

f

I

l i l t . . - . -

\\\'~

. . . . .

"

I

. . . . . . .

i

. . . . . .

-

I..-It%\\\\

f i__.._

-"

" ~ ' ~

........ -.....,...

.

.

-%

.

~ .I . . . . . . I

._,._.............,,/fl///~_

-1.5

.

-

. . . . . .

- - .

.

.

.

3

.

0

. 1

2

3

V((/<~-----

;

0

2

1

z/H z/H Figure 8" Unit velocity vectors ((~2> - (@}) along the lower wall. Every direction are shown.

-.-. 3

z/H 3~d vector in x and z

o o 0

0.8 0.6 y/H

o

o

.

i

o

o

o

I

-1.79H

-1.14H

-0.81H

,<\

D O

I I

~~

', !

0.4

Ii ~

0.2

.

00

._.Z) 0.02

0

0.2

0

0.02

0

0.02

0.01

0

O.Ol

resolved Figure 9: Turbulent kinetic energy. Hybrid LES-RANS with forcing, x/H = 3.69. energy O. , ~ 2. - - - modelled energy kT/U~n; o experiments (Simpson et al., 2002). both Nx and Nz, except at the separation region where d95 goes to zero (Fig. 2b). The object with hybrid LES-RANS and DES is to free the grid spacing of the Reynolds number dependence, and make it dependent only of outer scalings. Spalart et al. (1997) suggest that the value of Nx and Nz for a well-resolved LES should be 20, assuming that the viscous-influenced region is accurately modelled with (U)RANS. N~ ~_ Nz ~- 8 was used in the airfoil simulations by DahlstrSm (2003). Vector fields in the symmetry plane z = 0 are shown for the hybrid LES-RANS with and without forcing conditions in Fig. 3. The separation region for the LES simulations takes place earlier and is much shorter (not shown). The flow separates at x / H ~_ 1 and re-attaches at z / H _~ 2 both with and without forcing. This is in reasonable good agreement with LDV measurement of Byun et al. (2003). Figure 4 presents the streamwise velocities at x / H = 3.69. Fairly good agreement is obtained with the hybrid LES-RANS, both with and without forcing. The LES simulations give a slightly worse

326

agreement with experiments. The predicted velocity profiles in all simulations are too full. This indicates either too a slow predicted recovery rate or too a small predicted recirculation bubble. It can also be seen that all predicted velocity profiles in Fig. 4 are larger than the experimental velocities. The reason for this difference is not clear. The secondary velocity fields at x = 3.69H are presented in Figs. 5 and 6. The overall agreement between predictions and experiments is rather good. The magnitude of the predicted (~) by LES are too large at z / H >_ -0.49 compared with experiments. The velocity fields in Fig. 6 for both hybrid LES-RANS predictions and experiments show a large clock-wise vortex whose centre is located at y / H ~ 0.2, z / H ~ -1.2. The centre for the LES simulations, however, is located too close to the centre, compared with experiments. The main difference between the hybrid LESRANS simulations and the experiments is found in the central region at y / H ~ 0.7, z / H "~ -0.5. Here the large clock-wise vortex in the experiments is broken, and it looks like a trace of a decaying counter clock-wise vortex (it is also clearly seen in the experimental spanwise velocity profile at z,/H - -0.49 in Fig. 5). No such vortex is visible in the predictions. In Fig. 7 the secondary velocity fields are presented for x / H = 1.5 and x / H = 2.5. Here it can be seen that at x / H = 2.5 a streamwise vortex has been created induced by the reattaching flow in the symmetry plane. It has its centre at z / H ~_ -1.3, and it gets larger further downstream (cf. Fig. 6). No vortex is present at x / H = 1.5. The streamwise vortex predicted with hybrid LES-RANS without forcing (not shown) is very similar to that in Fig. 7, whereas the streamwise vortex predicted with LES (also not shown) is virtually non-existent. The direction of the flow at the wall is visualized in Fig. 8. Here the flow patterns observed in the symmetry plane and the cross planes are recognized. The flow approaches the hill and is forced to diverge in the lateral direction (+z direction). For the two hybrid LES-RANS simulations a recirculating region is formed at 1 < x / H < 2, Izl/H < 0.5 (slightly wider without forcing), whereas that predicted with LES is much larger. Figure 9 shows the turbulent kinetic energies. As can be seen, the agreement between predictions and experiments is good. The largest discrepancies are seen in the central region for -0.16 _> z / H >_ -0.49, y / H > 0.5, in which the experimental values are considerable larger than the predicted ones. This is probably related to the trace of a experimental streamwise counter clock-wise vortex observed in Fig. 6 which was not seen in the predictions. This vortex could be responsible for generation of turbulent kinetic energy which is advected downstream. It could also be that the position of the vortex is unsteady, which would show up as high turbulent kinetic energy. As can be seen from Fig. 9 the modelled turbulence is much smaller than the resolved one, except close to the wall at z / H < -1.14.

CONCLUSIONS Two hybrid LES-RANS m e t h o d s - one standard and one in which DNS fluctuations are added as forcing conditions- and one LES with the WALE model have been used to predict the flow around a three-dimensional axi-symmetric hill. A mesh of 1.7 million cells is used. The two hybrid methods give both results which are in fairly good agreement with experiments. The agreement of the LES results with experiments slightly worse but still acceptable. As mentioned in the Introduction, steady RANS simulations fail completely.

327

,.~" .,~ o,e

or e

~15

15 A

Jv ~

vlO

10

0

1 00

.......

101'

1 02

,,,

0 ~

. . . . .1 '00

y+ (a) Forcing. Figure 10: Channel flow. R e ~ - - u ~ - 5 / u - 500. x/5 - 7; DNS (Davidson and DahlstrSm, 2004); +" 2.5 ln(y +) + 5.2.

........

1 01'

.........

102

y+ (b) No forcing.

z / 5 - 15;_ _ z / 6 - 23;o

It has earlier been found that standard hybrid LES-RANS (i.e. without forcing) gives rather poor results for channel flow, whereas when forcing conditions are employed the agreement is excellent (Davidson and DahlstrSm, 2004; Davidson and Billson, 2004). It is thus somewhat surprising that for the 3D hill flow hybrid LES-RANS without forcing gives as good results as with forcing conditions. This was also found for the flow in the plane, asymmetric diffuser (Davidson and DahlstrSm, 2004). Standard hybrid LES-RANS performs poorly in fully developed channel flow (periodic boundary conditions), because the only boundary that the LES region sees is the interface to the URANS region, and the turbulence that is transported across this boundary represents a poor turbulent boundary condition. On the contrary, both in the diffuser flow and the 3D hill flow realistic turbulence is imposed as inlet boundary conditions. Of course, if the inlet is situated very far upstream, the flow will forget the inlet boundary conditions, but in both the diffuser flow and the 3D hill flow, the inlet is located rather close to the expansion and the hill, respectively. In Fig. 10, the two hybrid LES-RANS methods are used to compute developing flow in a channel. Instead of using periodic boundary conditions in the streamwise direction, inlet and outlet conditions are used. Instantaneous inlet boundary conditions are prescribed from channel DNS data (Davidson and DahlstrSm, 2004). Velocity profiles are shown for three streamwise locations downstream of the inlet, namely x/5 = 7, 15 and z/5 = 23. As can be seen, the agreement for the hybrid LES-RANS with forcing conditions is perfect at all three locations thanks to the added DNS fluctuations at the interface, whereas in the hybrid LES-RANS simulations without forcing conditions the resolved turbulence is gradually dissipated. The distance from the inlet to the hill foot in the present 3D hill simulations is approximately 25. In the diffuser simulations the distance between the inlet to the start of the expansion (diffuser region) is 165. Acknowledgments.

This work was financed by the FLOMANIA project (Flow Physics M o d e l l i n g - An Integrated Approach) and is a collaboration between Alenia, AEA, Bombardier, Dassault, EADS-CASA, FADS-Military Aircraft, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Iraperial College, TU Berlin, UMIST and St. Petersburg State University. The project is funded by the European Union and administrated by the CEC, Research Directorate-General, Growth Programme, under Contract No. G4RD-CT2001-00613.

328

REFERENCES Batten, P., Goldberg, U., Chakravarthy, S., 2004. Interfacing statistical turbulence closures with largeeddy simulation. AIAA Journal 42 (3), 485-492. Byun, G., Simpson, R., Long, C. H., 2003. A study of vortical separation from three-dimensional symmetric bumps. AIAA paper 2003-0641, Reno, N.V. Byun, G., Simpson, R., Long, C. H., 2004. A study of vortical separation from three-dimensional symmetric bumps. AIAA Journal 42 (4), 754-765. Dahlstr6m, S., 2003. Large eddy simulation of the flow around a high-lift airfoil. Ph.D. thesis, Dept. of Thermo and Fluid Dynamics, Chalmers University of Technology, G6teborg, Sweden. 1 Davidson, L., Billson, M., 2004. Hybrid LES/RANS using synthesized turbulence for forcing at the interface. In: Neittaanm/iki, P., Rossi, T., Korotov, S., Ofiate, E., P~riaux, J., KnSrzer, D. (Eds.), ECCOMAS 2004. July 24-28, Finland. 1 Davidson, L., Cokljat, D., Fr6hlich, J., Leschziner, M., Mellen, C., Rodi, W. (Eds.), 2003. LESFOIL: Large Eddy Simulation of Flow Around a High Lift Airfoil. Vol. 83 of Notes on Numerical Fluid Mechanics. Springer Verlag. Davidson, L., DahlstrSm, S., 2004. Hybrid LES-RANS: An approach to make LES applicable at high Reynolds number (keynote lecture). In: de Vahl Davis, G., Leonardi, E. (Eds.), CHT-04: Advances in Computational Heat Transfer III. April 19-24, Norway. 1 Davidson, L., Peng, S.-H., 2003. Hybrid LES-RANS: A one-equation SGS model combined with a k - w model for predicting recirculating flows. International Journal for Numerical Methods in Fluids 43, 1003-1018. Emvin, P., 1997. The full multigrid method applied to turbulent flow in ventilated enclosures using structured and unstructured grids. Ph.D. thesis, Dept. of Thermo and Fluid Dynamics, Chalmers University of Technology, G6teborg. 1 Haase, W., Aupoix, B., Bunge, U., Schwamborn, D. (Eds.), 2005. FLOMANIA: Flow-Physics ModellingAn Integrated Approach. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Springer. Krajnovid, S., Davidson, L., 2003. Numerical study of the flow around the bus-shaped body. ASME: Journal of Fluids Engineering 125, 500-509. Krajnovid, S., Davidson, L., 2004. Large eddy simulation of the flow around an Ahmed body. In: 2004 ASME Heat Transfer/Fluids Engineering Summer Conference. Charlotte, USA. 1 Mellen, C., Fr6hlich, J., Rodi, W., 2003. Lessons from LESFOIL project on large eddy simulation of flow around an airfoil. AIAA Journal 41 (4), 573-581. Rodi, W., Ferziger, J., Breuer, M., Pourqui4, M., 1997. Status of large-eddy simulations: Results of a workshop. J. Fluids Engineering, 248-262. Simpson, R., Long, C. H., Byun, G., 2002. Study of vortical separation from an axisymmetric hill. International Journal of Heat and Fluid Flow 23 (5), 582-591. Spalart, P., Jou, W.-H., Strelets, M., Allmaras, S., 1997. Comments on the feasability of LES for wings and on a hybrid RANS/LES approach. In: Liu, C., Liu, Z. (Eds.), Advances in LES/DNS, First Int. conf. on DNS/LES. Greyden Press, Louisiana Tech University. Temmermann, L., Leschziner, M., Hanjalid, K., 2002. A-priori studies of near-wall RANS model within a hybrid LES/RANS scheme. In: Rodi, W., Fueyo, N. (Eds.), Engineering Turbulence Modelling and Experiments 5. Elsevier, pp. 317-326. Tucker, P., 2003. Differential equation based length scales to improve DES and RANS simulations. AIAA paper 2003-3968, 16th AIAA CFD Conference. Tucker, P., Davidson, L., 2004. Zonal k-1 based large eddy simulation. Computers & Fluids 33 (2), 267287. Xiao, X., Edwards, J., Hassan, H., 2003. Inflow boundary conditions for LES/RANS simulations with applications to shock wave boundary layer interactions. AIAA paper 2003-0079, Reno, NV.

1 can be downloaded from www.tfd.chalmers.se/-lada

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

329

A P P L I C A T I O N S OF A R E N O R M A L I Z A T I O N G R O U P BASED H Y B R I D RANS/LES M O D E L C. De Langhe, B. Merci and E. Dick Department of Flow, Heat and Combustion Mechanics, Ghent University Gent, 9000, Belgium

ABSTRACT A hybrid RANS-LES model has been constructed using a Renormalization Group approach. The resulting model has explicit filter width dependence in the effective viscosity and in the time scale of the transport equation of the mean dissipation rate. A two-equation RANS limit of the model exists for filter widths that are large compared to the integral length scale. In this paper, after a general overview of the model, its performance is illustrated in low-Reynolds mode for flow over periodic hills and in high-Reynolds mode for flow in a sudden pipe expansion.

KEYWORDS

Hybrid RANS/LES, subgrid modelling, renormalization group, turbulence.

INTRODUCTION In a good large-eddy simulation, most of the turbulent kinetic energy should be resolved, meaning that the filter width should lie in the inertial range. In complex geometries, one is often not sure whether this is reached, and generally many grid points are needed to fulfil this requirement in the whole flow domain, especially near walls. Hybrid RANS/LES models, also called very-large-eddy simulation (VLES) models, can be seen as a general class of models which do not require the filter width to lie in the inertial range. A class of models that are able to model integral length scales are the RANS models, and therefore the usual approach to VLES modelling has been to modify an existing RANS model by making it filter width dependent. The general comments, however, on these models are that they lack physical basis and often contain empirically calibrated terms, coming from comparing RANS simulations with experiments and DNS. These calibrations are not always valid for LES subgrid models. This work is an attempt to overcome some of these shortcomings. To that end, transport equations for subgrid quantities are derived with the renormalization group. The main advantage of the approach is that, within one framework, LES subgrid models as well as RANS models can be constructed.

330 G E N E R A L C O N S I D E R A T I O N S F O R VLES M O D E L L I N G It is clear that transport equations for mean turbulent quantities are necessary in a model that removes large, anisotropic, integral range structures from the turbulent field. Further it is well known that principally two of these quantities are necessary to define the integral turbulent length and time scales required by the turbulent viscosity. On the other hand, when the filter width gets smaller, these subgrid quantities should change accordingly, as less is being modelled. A general approach to VLES modelling has been to start from a conventional RANS model by making the length scale in these models filter width dependent. The DES models belong to this class (Spalart e.a. (1997)). For a twoequation model, this results in an increase of the dissipation term in the k-equation. In another approach, proposed by Speziale (1998), the turbulent viscosity, as calculated with a RANS model, is multiplied by a filter width dependent function. As we feel that the comments from the introduction apply to these approaches, we investigated the construction of subgrid viscosity and transport equations directly form the Navier-Stokes equation by means of a renormalization group procedure.

R E N O R M A L I Z A T I O N GROUP AND VLES M O D E L L I N G Applications of renormalization group (RG) to the turbulence problem have a wide history. The first attempt to tackle turbulence with RG techniques was by Yakhot and Orszag (1986), who applied the dynamical RG, mixed with EDQNM renormalized perturbation theory, to calculate a variety of turbulence constants and some standard type turbulence models, like a Smagorinsky type subgrid model and a k - c type RANS model. Many subsequent RG work in the turbulence literature consisted of remarks on, and improvements of, the original work of Yakhot and Orszag. The work of Giles (1994a,1994b) follows a different RG procedure, analogous to the original RG method used in statistical mechanics. This led to some different constants than the YO method, and was less flawed by uncontrolled approximations. Moreover, for our purpose, Giles' approach to the derivation of turbulence transport equations is more amenable for VLES. Unlike the approach of Yakhot and Smith, the approach of Giles does not rely on heuristic arguments for the construction of the production and destruction terms in the c-equation. When one wants to adapt the YO RANS equation to their LESform, extra terms arise with no clear physical interpretation (De Langhe (2003)).

THE M O D E L

Model Equations The investigated model is the RG k - c model derived by Giles (1994a,1994b). But, where in the construction of the RG RANS model, the iterative RG procedure stops when the integral wave number A e is reached, leading to an effective viscosity and model coefficients that depend only on k and c, now the wave number A c at which the RG procedure is stopped is kept as a variable in the equations and coupled to the filter width. For the technical details of the RG calculation we refer to the original papers of Giles and De Langhe (2003). In De Langhe (2003), we explained that, for high Reynolds numbers and for the filter width wavenumber A c in the inertial range, the RG procedure leads to an effective viscosity v ( A c ) - ac-'J3A~.4/3, with a = 0.46, and with the mean dissipation rate c determined by the transport equation

(1)

331

D___~g:Dt v ( A ')A2cc (Ce,PK

- Ce2 E) + ~

(ggV(Ac)

i )"

(2)

The model constants are 4 C~, = 3

a'= 1.39

C~2-2,

and PK =-roSa the production of turbulent kinetic energy, with z"0 the Reynolds stress tensor and

So -(OjU

+0,Uj)/2 the strain rate tensor (U denotes the resolved velocity field). The subgrid kinetic

energy equation used for determining the RANS limit and for post-processing (De Langhe (2003)) is

>x

-

D--t-: PK- v ( A c ) A ~ K

a 0x) + ~ x (C~V(Ac) Ox; "

(3)

The above subgrid model has two special features. First, the effective viscosity depends only on the mean dissipation rate. The transport equation for the turbulent kinetic energy is only necessary to determine the RANS limit of the model (see later). The second feature concerns the explicit filter width dependence of the time scale in the g-equation, which gets smaller for decreasing filter width. The physical interpretation of this time scale was given by Lumley (1992) for RANS models, as the time it takes for information to travel from integral length scales to dissipative length scales. As the filter width plays an analogous role in our LES model as the integral length scale in RANS models, the obtained dependence of the time scale on the filter width is also expected purely on physical grounds. The g-based subgrid model thus has a clear physical interpretation, with a varying filterwidth changing the dynamical behaviour of the model through the time-scale. In contrast, to the more usual transport-equation subgrid models, which are based on the K-equation with a filter width dependent destruction term, do not have this dynamical behaviour, as they generally no longer depend on an equation for the dissipation rate. We should also mention that the destruction term in the K-equation was not obtained by the RG procedure. To obtain explicit filter width dependence in this destruction term, it was modelled, in a similar way as done for K-equation based subgrid models, by using the time scale from the __ g-equation,

i.e.

1 _ v A 2c' a s __ g -

K

VA2c~

Limiting Behaviour of the Model The RANS limit follows if A c < A ..... where Ae, is a wave number constructed from the subgrid K and c as

Aes =

2

CK

] 3/2 -E

~ K

E ~ ,Tz"--3/2' K

where C~ is the Kolmogorov constant (calculated by RG (Giles (1994a)) as C K - 1 . 4 4 ) . The expression for Ae' is analogous to that of the wavenumber corresponding to the integral length scale, but now with subgrid values for K and c instead of ensemble averaged values. Substitution of A c with Ae, in Eqns. (1), (2) and (3) leads to the RANS model

332

--2 v - C/~ K g

DF De

~

e

L,, - T(

-

(4)

a~) a

ae

(av Z--),

(6)

with Cv - 0 . 1 and the other constants as above. In practice, the above behaviour is obtained by taking A c = nCL, where L - rain (A, @ )

and A is the

filter width. Near-wall Behaviour

When the model is to be integrated up to solid walls, extra modifications are necessary, which are slightly different for the LES and the RANS mode of the model. A wall boundary condition for the dissipation rate equation, which is independent of K , is also used. The complete explanation and motivation for these modifications can be found in De Langhe (2003). L E S mode

In LES-mode, the near-wall behaviour is partly regularized through a ramp function in the eddyviscosity formulation (Yakhot & Orszag (1986)): near the wall the effective viscosity reduces to the molecular viscosity when the filter width gets small enough. The ramp function also determines the DNS limit of the model in well-resolved regions of the flow away from walls. Further, an additional production term in the e-equation is also included. This term corresponds to a production term in the exact (unaveraged) e-equation, that gets important near walls where the second order derivatives of the velocity get large. It vanishes, however, under averaging with the (low order) RG calculation, and evaluation of this term with the RG method would require a (difficult) higher order analysis. Instead, in the present work, we adopt the model for this term as used in the Yang-Shih model (Yang & Shih (1993)). The resulting low-Reynolds subgrid model is

a EA_~4 - C v ( A . ) - v o I+R(v---~

De

2

-v- ( a = c)aDt

a

(7)

ae)

~(Ce'Pk-CeiT)+E+-'~xi(O~V(Ac)()x

where v 0 is the molecular viscosity, R ( x ) - max(0,x) is the ramp function and A c - ~ A . The model constants are a - 0.46

C-IO0

and the extra production term E is

a ' - 1.39

C~-1.33

Cc2 - 2 . 0

333

a2Uii2.

(8)

E = v0V(Ac) ~xj~xk

RANS mode In RANS mode, the low-Reynolds modifications essentially result from a smoothening of the ramp function. The complete low-Reynolds RANS model is --2

V=Vo+

8 DK ~ ~)K Dt = PK - g + ~x, (aq"-~x/)

a

ae

with Cu = 0.1 and the other constants the same as in the LES case. The E term is again defined by Eqn. (8). The transition function f is given by

f(y+)=l[l+tanh(n(y+-C'))l with n =0.065 and C ' = 37 (these constants were determined by comparison with DNS data for channel flow (De Langhe (2003)). The time scale ~" is the sum of the turbulent and the Kolmogorov time scales

K

T= _-=-+ 8 u

Wall boundary condition for e As boundary condition for c at a solid wall we use ew = 0.22u4 / Vo-

RESULTS

Periodic hill The first application is the calculation of the flow over the periodic hill, that was analysed by Temmerman and Leschziner (2001) with highly resolved LES. The Reynolds number, based on the hill height h and bulk velocity on top of the hill, is 10595. The domain dimensions are 9h in the streamwise direction and 3.036h in the direction normal to the upper wall. A spanwise dimension of 4.5h was used. The grid is refined in the streamwise direction on the top of the hill. The total number of grid points used was about 2.3-105 . The code used a 4-th order accurate central scheme in space and a second order accurate scheme in time. In this simulation, the anisotropic grid measure of Scotti

334

et al. (1993) was used to determine A c, and the maximum viscosity ratio was around 40. Due to the low Reynolds number, most of the flow was computed with the model in LES form. Only the first few near-wall cells in the center of the channel (y+ < 3 ) and a region of y+ < 10 near the top of the hill were computed in RANS mode. The separation point lies completely in the RANS zone, while the length of the recirculation bubble is mainly determined by the LES part of the flow. In Figure 1 we show the turbulent structures for this flow using the second invariant of the strain-rate tensor, Q.

Figure 1. Isosurfaces of Q = 10 show the coherent flow structures for the periodic hill. Shown in Figure 2 are the stream traces for the mean (time- and spanwise-averaged) velocity. The reattachment point lies at x / h = 5.1, which is in reasonable agreement with the value 4.7 obtained from highly resolved LES, in which about 20 times more grid points were used (Temmerman et al. (2003)). The separation point lies at x / h = 0.22, in good agreement with the reference LES.

Figure 2: Mean flow stream traces for the periodic hill. Also shown are mean streamwise velocity profiles and resolved shear stress profiles at different streamwise locations (Figure 3). The locations shown are in the recirculation zone (x/h = 2), just behind reattachment (x/h = 6 ), and halfway the ascending side of the hill (x/h = 8). The results obtained for this flow are generally in much better agreement with the benchmark data than RANS models, some of which give seriously erroneous predictions of the stresses and the reattachment length. In Temmerman et al. (2003) the same case was studied with LES, on different grids and with different subgrid models and near-wall treatments. It is instructive to compare our results with the ones obtained for the coarsest grid used in Temmerman e.a. (2003), which still has about three times more grid points. Especially for the mean velocity field, our results compare favourably (we should note that this is probably not entirely due to our model, but also due to the finer grid we used near the hill crests, which, as found in Temmerman e.a. (2003), is an important factor to obtain an accurate separation prediction).

335

1.2 1

0.9

08:

0.9 08

0.8

07;

0.7

0.7

06L

0.6 .=0.5 0.4

~

'F

0.9-

1.1

0.6

0.5 :

0.3 O.2 0.1 0, -0.1 -O.2 -0.3 -

-o 03

0,2 ;

o

o.2r

o.1

o

o.1

1

Ol

0.3

o o

o , , ~ I

3

{

. . . .

. . . .

Wh

i

0015 -

l

001 0.005

-O.005

~

,,,.,

~-O015

o -o 005

.ool -O 02

o

i

i

i

o

I

I

9 o

o -

Ooo ,

2

y/h

~:~'= -001

.

0

O.4 ~ o

r

!

0.5

03 L

0.005

-O.Ol

5

0,4 ;

y/h

o,

o

oo o

~o

-0.02 !i

OooooOo~ i

i

=

y/h

i

I

2

,

,

I

i , l

3

-o o15i

~176176176

_

-0 030 y/h

y/h

Figure 3. Mean streamwise velocity and resolved shear stress profiles for the periodic hill flow at x/h = 2 (left), x/h = 6 (middle) and x/h = 8 (right). Lines" benchmark LES, 9 our computation.

Sudden pipe expansion Secondly, the model was tested on a typical hybrid RANS/LES test case, i.e. separated and confined flow at high Reynolds numbers. The geometry was taken in accordance with the experimental setup by Szczepura (1985). The expansion ratio is 1.9455, and the Reynolds number, based on the largest diameter D and the bulk velocity at the outlet, is Re = 2-105 . The flow was computed with a second order central scheme in space and second order accurate time discretization. The inlet conditions were constructed from a hybrid RANS/LES pipe flow simulation, taking into account the specific grid resolution at the inlet. A white noise fluctuation, with a magnitude corresponding to a turbulence level of about 5 %, was superposed on that inlet velocity field. The total length of the inlet pipe was about 5 D , and the length of the section after the expansion about 50D. The total number of grid points was about 7-105 . The resolution is finest near the expansion, and quite coarse otherwise. Twice the cube root of the cell-volume was used as filter width. The flow field is not integrated up to the wall, but wall-functions were used instead. The low-Reynolds modifications, as presented earlier in this paper, were therefore discarded, and the high-Reynolds model, i.e. Eqns. (2), (3) and Eqn. (7) (to assure a DNS limit) for the LES part and Eqns. (4)-(6) for the RANS part, is used, with the following modifications in the wall-adjacent cells. First, the e-equation is not solved in the first cell, but instead, when P denotes a wall-adjacent cell, the dissipation is computed as -

,,~314--312

~

ep = ~

Kp

/gyp

(9)

336 with ic= 0.39 the von Kfirm~n constant, Ke the turbulent kinetic energy at point P and yp the distance from point P to the wall. Further, in the wall-adjacent cells, the production term in the turbulent kinetic energy equation is computed as ~,.2 __fd/4~l/2

K--

A/)t~a Kp Yp

(10)

"

The velocity at the point P is computed from

u. K" pr

U* -

r

rd/4~l/2

tJ pt-'u Kp L

C~/4 --1/2

and

y=

Kp Yp

v0

with E = 9.793 and Up the mean velocity at point P . When P lies in the RANS zone, these are the standard wall functions, based on equilibrium conditions that are valid in the logarithmic layer. When m

P lies in the LES zone, the effective viscosity is given by the usual expression, Eqn. (7), with e computed from Eqn. (9), and ~ e therein computed from the RANS K-equation with the production term, Eqn. (10). This is a valid assumption, when an equilibrium spectrum (i.e. a constant e in the inertial range) is assumed in the log-layer. Figure 4 is an instantaneous plot of the axial velocity field. In Figure 5 we show the turbulent viscosity ratio and Figure 6 is a contour plot which depicts the RANS (black) and LES (white) zones. One sees that there is RANS activity near the wall (typically in a zone of two to three near-wall cells) and in some spots in the mixing layer. The occasional switch to RANS in the mixing layer is an artefact of comparing the filter width with the subgrid length scale (instead of the integral length scale). But as the viscosity varies continuously when switching from LES to RANS, and because these switches only occur in a few neighbouring cells at a time, these occasional switches to RANS do not influence the results.

Figure 4. Instantaneous axial velocity field in a mid-plane for the sudden pipe expansion.

Figure 5. Instantaneous turbulent viscosity contours for the sudden pipe expansion.

337

o

s

I Figure 6. RANS (black) and LES (white) zones for the sudden pipe expansion. A quantitative comparison with the experiments is provided by measurements of the axial velocity at R = 0.4771 diameters in Figure 7. It should be mentioned that the experiments were only done for the region x/D < 3, and the data for x/D > 3 result from a linear extrapolation to the cut-off value =0.1757 (which is the axial velocity at R - 0 . 4 7 7 1 D for fully developed pipe flow at the same Reynolds number). Figure 7 shows that our computation is in very good agreement with the experimental results. Only the secondary recirculation bubble is not well represented, which is likely due to the adopted wall-function approach.

-,/

0.2

0

0

0.15 0.1

=0.05 ~

0

o

o

o

o

o

o

0, -0.05 -0.1 -0.15 -o.2

' ' '~

....

~ ....

~ D 3~ '

'

, , l l , e ~ l n l 4 5

Figure 7 Axial velocity at a distance R - 0.4771 diameters. The line denotes the experiment, and symbols denote the result from our computation.

CONCLUSION

Two applications of a recently developed hybrid RANS/LES model were presented. The results for the periodic hill show good agreement with reference data from a highly resolved LES of the same flow, in which about 20 times more grid points were used. The simulation results of the sudden pipe expansion show very good agreement with the available experimental measurements.

338 ACKNOWLEDGEMENTS The authors thank F. Magagnato and colleagues at the University of Karlsruhe for making available their CFD-code SPARC. The first author was funded by the research project BOF/GOA 12050299 from the research council of Ghent University, during the execution of this work. The second author works as Postdoctoral Fellow of the Fund for Scientific Research- Flanders (Belgium) (F.W.O.Vlaanderen).

REFERENCES De Langhe C. (2003). Renormalization Group Approach to Hybrid RANS/LES Modelling. PhD thesis, Ghent University, http://sfxserv.rug.ac.be/execl/fulltxt/thesis/801001345024.pdf Giles M.J. (1994). Turbulence renormalization group calculations using statistical mechanics methods. Phys. Fluids, 6:2, 595-604. Giles M.J. (1994). Statistical mechanics renormalization group calculations for inhomogeneous turbulence. Phys. Fluids, 6:11, 3750-3764. Lumley J. (1992). Some comments on turbulence. Phys. Fluids A, 4, 203-211. Scotti A., Meneveau C. and D.K. Lilly (1993). Generalized Smagorinsky model for anisotropic grids. Phys. Fluids A, 5:9, 2306-2308. Spalart P.R., Jou W.-H., Strelets M. and Allmaras S.R. (1997). Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. Advances in DNS/LES, 1st AFOSR Int. Conf. on DNS/LES (1997), Columbus Oh.. Speziale, C.G. (1998). Turbulence modelling for time-dependent RANS and VLES: A review, AIAA J, 36:2, 173-184. Szczepura, R.T. (1985). Flow characteristics of an axisymmetric sudden pipe expansion - results obtained from the turbulence studies rig. Part 1 mean and turbulence velocity results. CEGB Berkeley TPRD/B/0702/N85. Temmerman, L. and Leschziner, M.A. (2001). Large-eddy simulation of separated flow in a streamwise periodic channel constriction. In Turbulence and Shear Flow Phenomena, Second International Symposium. KTH, Stockholm, June 2001. Temmerman L., Leschziner M.A., Mellen C.P. and Fr6hlich J. (2003). Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flows in a channel with streamwise periodic constrictions. Int. J. of Heat and Fluid Flow, 24, 157-180. Yakhot, V. and Orszag, S.A. (1986). Renormalization group analysis of turbulence, J. Sci. Comput., 1, 3-51. Yang, Z. and Shih, T. (1993). A new Time Scale Based k - g Model for Near-Wall Turbulence, AIAA J, 31, 1191-1198.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

339

APPLICATION OF ZONAL LES/ILES APPROACHES TO AN UNSTEADY COMPLEX GEOMETRY FLOW P. G. Tucker and Y. Liu Civil and Computational Engineering Centre, School of Engineering, University of Wales Singleton Park, Swansea, SA2 8PP, UK

ABSTRACT Flow and heat transfer inside an idealized electronics system is simulated using Large-eddy Simulation (LES) related approaches. These include: Yoshizawa LES (Large Eddy Simulation), DES (Detached Eddy Simulation), LNS (Limited Numerical Scales) and other hybrid LES-RANS (Reynolds Averaged NavierStokes) approaches including a new ILES (Implicit LES)-RANS method. Modelling this unsteady complex geometry flow is found challenging. Performances of the LES related methods are compared with zonal EASM (Explicit Algebraic Stress Model) unsteady RANS (URANS) results and measurements. For mean velocities, the (I)LES-RANS methods have similar accuracies to the zonal EASM and LES. Velocity results are in a reasonable agreement with measurements. However, as far as heat transfer is concerned, none of the models investigated performs well. Significant heat transfer discrepancies exist. The LNS performs poorly for both the flow field and heat transfer and DES proves impossible to converge. This is partly attributed to the irregular interface arising from the DES interface being grid controlled.

KEYWORDS LES, zonal LES, ILES, URANS, heat transfer, electronics systems

INTRODUCTION

With ever increasing power densities the reliable prediction of fluid flow and heat transfer in electronics is becoming especially important. In reality, flow inside most electronic systems is turbulent and due to the geometrical complexity also can exhibit large scale more coherent unsteadiness. Present-day useable turbulence modelling techniques for industrial applications involve Reynolds Averaged Navier-Stokes (RANS) or Unsteady RANS (URANS). To a much lesser extent pure Large-eddy Simulation (LES) and blends of LES and (U)RANS can be used. In (U)RANS, the Navier-Stokes equations (NSE) are essentially time averaged removing high frequency temporal components and smaller spatial scales associated with turbulence. Hence flow solutions have a more regular smoother appearance. In contrast to (U)RANS, in LES the NSE are spatially filtered. Eddies smaller than a characteristic resolution A (typically defined by

340

grid spacing), known as subgrid scales (SGS), are filtered out and hence need to be modelled. Eddies larger than A are resolved. Since most turbulence is resolved and not subjected to the vagaries of modelling, LES is capable of capturing unsteady flow feature better than (U)RANS and can give more accurate solutions. Despite LES being superior to URANS, it still has some theoretical and practical drawbacks (see Boris et al., 1992; Grinstein and Fureby, 2002). An alternative to LES is Monotonically Integrated LES (MILES) or ILES introduced by Boris et al. (1992). In ILES, monotone higher order convection algorithms are used to discretize the unfiltered NSE. The monotone algorithms can through numerical dissipation produce a builtin filter and a corresponding implicit SGS model. Grinstein and Fureby (2002) demonstrate that ILES can be successfully used to simulate a wide range of flows and owing to the absence of subgrid stress term calculations, can also lead to substantial savings in computational effort. Although ILES has some theoretical justification, from a pragmatic view point it is likely that in most industrially related LES type simulations the numerical dissipation takes on a greater role than that implied by the SGS model. Under these circumstances it appears most sensible to improve solution accuracy by switching the SGS model off. It is in this spirit the ILES is explored here. Of course the temporal rather than spatial discretization can be used to supply filtering/dissipation. Novel use of the former is tried here. In addition to ILES, along different lines other approaches aimed at reducing LES computational costs are hybrid LES-RANS methods. The first of these - Detached Eddy Simulation (DES) - was proposed by Spalart et al. (1997). With this, to avoid the need for such fine grids, intricate, fine, anisotropic boundary layer features requiring especially high resolution are (U)RANS modelled. However, the accuracy advantages of LES are utilised away from walls where structures are larger and less intricate. Broadly following this strategy, various combinations of RANS and LES models have been investigated (see Davidson and Peng, 2001; Temmerman et al., 2003 and Tucker and Davidson, 2004). Notably, Tucker (2004) produces hybrid ILES-RANS solutions. With this, the very different turbulence length scales implied by the RANS and ILES models are blended using a Hamilton-Jacobi (HJ) equation. Essentially another hybrid LES-RANS method is the Limited Numerical Scales (LNS) approach proposed by Batten et al. (2002). The idea of LNS is to derive the SGS stress from the underlying Reynolds stress via a latency parameter, ct, based on the ratio of products of turbulence length- and velocity-scales. When (z = 1, non-linear (cubic) RANS modelling is applied and when 0{< 1, LES modelling is used. With LNS, unlike other hybrid LES-RANS methods, in which RANS modelling is used near walls and LES away from them, depending on mesh resolution RANS modelling or LES may be used anywhere. If the grid is LES sufficient LES will automatically be used. This study considers the configuration shown in Figure 1, which represents an idealized central processor unit. Two, by electronics standards, powerful (just making it sensible to explore hybrid approaches) fans drive the air flow. To investigate heat transfer, a heater is mounted on one horizontal surface. Since the ratio of Gr/Re2 (Gr and Re are the Grashof and Reynolds numbers, respectively) is low (< 0.01), buoyancy forces can be neglected. The Figure 1 geometry flow is complicated exhibiting transition from laminar to turbulent flow, unsteady separation, reattachment, strong streamline curvature and regions with impingement. Furthermore, it is expected (see Chung and Tucker, 2004) that large coherent vortex structures will convect over the heated surface. Also, there is a possibility of non-unique solutions (see Henkes, 1990; Shyy, 1985). This paper extends previous (U)RANS (Tucker and Pan, 2000) studies on this unsteady complex geometry flow. The use/applicability of above-mentioned hybrid (I)LES-RANS, LNS and DES approaches is considered. To set the performances of these new models into context results are compared with zonal k//EASM (Explicit Algebraic Stress Model, see Gatski and Speziale, 1993) URANS and Yoshizawa (1993) LES results. Also, all results are compared with LDA and heat transfer measurements.

341

Region1 Heater~ Baffle .L_.I plate-'---I Fan2~ ~ Region_.2~

i

Grill1 / / ,,~_ Cut-out ~ C_rrill2

g

Crrill3 ~~ ~.Grill 4

outlet ~ •

Region3 ! Flowinlet/outlet

Figure 1" Schematic of an idealized system studied. NUMERICAL METHOD The conservation of mass and momentum equations are solved in a standard weak conservation form and hence for brevity are not shown here. Instead attention is focused on the turbulence modelling and numerical solution procedures. Where appropriate tildes are used to indicate that variables can have dual meanings in the sense that for URANS modelling they are temporal averages and for LES spatial.

LES/ILES In conventional LES, for a SGS model based on the eddy viscosity, the SGS stress tensor, z-0 , is defined as

r~ = 2/,ts~sS~ - z'k----~-~6~

(1)

where/Xsa s is the SGS eddy viscosity, S,j = (Off, / ~xj + ~9~'j//gx, )/2 is the strain rate tensor based on the resolved velocities and 6 U the Kronecker delta. Here for LES computations ,Usas is evaluated from the Yoshizawa model which solves a transport equation for k. In an ILES context, although #sos = 0, an effective eddy viscosity can be produced by numerical diffusion.

Hybrid (I)LES-RANS Broadly four hybrid (I)LES-RANS approaches are tried. These are DES, Tucker and Davidson's (2004)

k-l based zonal LES (ZLES), zonal ILES (see Tucker, 2004) and LNS. In Z(I)LES, the Wolfshtein (1969) k.4 RANS model is used near walls. Either the Yoshizawa LES or ILES are used for the core region. The interface between the RANS and (I)LES models is set at the dimensionless wall distance Yi~,= 30. Results are compared when the wall shear stress used in y~, is temporally and spatially averaged ( yi~..... = 30) and also based on instantaneous local values ( y,* = 30). For k-I/ILES just yi~.... = 30 is usedand the wall distance is calculated using a HJ equation (see Eq. (4)) (Tucker, 2004). The latter smoothly blends the dramatically different modelled turbulence length scales implied bythe RANS and ILES modelling. For ZLES multigrid based smoothing operators are used. For DES, near walls the Spalart-Allmaras (1994) (S-A) (U)RANS model is used. Away from them essentially Smagofinsky LES is applied. The interface between these zone is set at Yint = 0.65max(Ax, Ay, Az) where the A terms are grid spacings. Clearly the interface is grid controlled and this can create irregularities.

342

Limited Numerical Scales (LNS) In Batten et al.'s (2002) LNS method, the eddy viscosity takes the following form 2

k ~ = ac;L p6"

(2)

where o~is a so called the latency parameter. It is defined as

(3)

a ' : min [(LV)LES, (L V )~vs ]/(L V )RaNs

where (LV)LeS = Cs(U)~ISI is the product of the LES length and velocity scales and (LV)RANS:

6 + C~k2/e for RANS, in which the constant Cs =0.05, L" = 2max[~c, Ay, Az],

Is I-

~/25,~s,~and 6 =

10 2~ The transport equations for k (turbulent kinetic energy) and e (dissipation rate) and other parameters can be found in Batten et al. (2002). As mentioned before, to large extent, a mesh used decides the model switch from RANS to LES or from LES to RANS.

Wall distance Function Wall normal 'distances' d (the tilde identifies that these can be modified distances and again have dual turbulence modeling meanings) are either evaluated using Poisson (Tucker and Pan, 2000), Eikonal or HJ equations. The HJ equation (Tucker, 2004) is expressed as

Iv l

= 1 + f(d)V2~r +

(4)

g(d)

Heref(d) =eod and g(d) = e, (d / L)". The length scale L is the distance from walls to the ILES region and n is a positive integer. When e 0 = e 1 = 0, Eq. (4) reduces to the hyperbolic natured Eikonal equation. Weak viscosity solutions of this give exact nearest wall distances d = d . The Eikonal equation can be solved by propagating fronts from solid surfaces. Here, the equation is propagated towards the RANS/ILES interface. Then, for zonal ILES solutions, at the interface, the condition d = 0 is set and Eq. (4) solved using a Newton approach with e0, el > 0. The Laplacian enables a smooth transition between the modelled RANS length scale (that needs an accurate d) and the ILES zone (needing d = 0). The functionf(d ) forces the Laplacian to tend to zero near walls. This ensures near wall distances are accurate. The function g(d) controls the RANS length scale in the vicinity of the ILES zone. Typical Eq. (4) d distributions for various e0 and el combinations can be found in Tucker (2004). For DES d is initialized as 0.65max(Ax, Ay, Az). The front propagation naturally terminates at the RANS-LES interface creating a potentially economical ready to use DES distance scale field.

Heat Flux Modelling The simple eddy diffusivity model is used i.e. h~ = r T/ PrT OT / Ox. In URANS, r and the turbulent Prandtl number Prr = 0.9. For LES, r

=r

= 1,it

(eddy viscosity)

s and Prr = 0.4. For ILES r

= 0.

Calculation of Turbulence Intensity Conventionally, the turbulence intensity is defined as T, = u / U ( U - time mean velocity in the x direction;

343

u'- fluctuating component). For LES related zonal methods, modelled turbulence contributions are neglected here. u is obtained only from the resolved field. Therefore, T~ is given by T, = ~/{fffi')- {if) ~//if/, where {o} denotes a time-averaging operation.

Boundary Conditions and Numerical Details Detailed boundary information for the Figure 1 geometry can be found in Tucker and Pan (2000). Here only brief description is given. At inlets and outlets, the total pressure is fixed and the normal velocity is set to conserve mass flow rate. No-slip and impermeability boundary conditions are applied for velocities at solid walls. Fans 1 and 2 are modelled as a momentum source. The grills are modelled as energy sinks. The Figure 1 domain size is L = 0.75 m, H = 0.64 m and W = 0.2 m in the x, y and z directions respectively. A 105 (x) x 99 (y) x 51 (z) non-uniform grid is used for all computations. At first off-wall nodes, the average y+ -- 2. Except for k-I/ILES and DES computations, wall distances are obtained from the Poisson equation. For DES the Eikonal is tried. The k-I/ILES distances are generated by solving Eq. (4). The flow equations are solved using a standard LES suitable staggered grid finite volume code (Tucker and Pan, 2000). Second-order central differences are used to discretize the convective and diffusion terms. At walls second order backwards differences are used. The Crank-Nicholson scheme is employed to discretize the temporal terms. Since non-dissipative spatial differences are preferred, for ILES the temporal discretization is used to dissipate turbulence. This is achieved by leaving momentum source terms and treating one component of the non-linear convective terms in a dissipative 1st order fully implicit manner.

RESULTS AND DISCUSSION For mean velocity profiles and turbulence intensities comparisons are made with the Laser Doppler Anemometry (LDA) measurements of Tucker and Pan (2000). These are available at the six profiles shown in Figure 2. These measurements have an estimated uncertainty of + 5%. Velocities are normalized by the average axial velocities of the two fans (U0 = 4.5 m/s). A thermistor based measurement of Tucker and Pan (2000) for the average coherent flow unsteadiness amplitude at around the centre lines of the six profiles is also compared with. This measurement has an uncertainty of + 10%. Heat transfer comparisons are made along the heated surface centreline in the x direction. The measured Nusselt number error is + 5 % (Liu, 2004). Like the k-I/(I)LES, for zonal URANS EASM predictions, near walls Wolfshtein's k-l _ C,,k,,~ model is used. The interface between the k-l and the EASM is set at y~, = 60 ( y" =/_5, ~ //z).

LES/URANS Zones and Resulting Flow Fields Figure 3 gives contours identifying RANS and LES regions at around the mid x - y plane for the LNS, k//ILES and the DES. For the k-I/ILES and DES for clearer observation the interface distance has been exaggerated. For DES this is achieved by taking CDES = 1.3. For the k-I/ILES Yi*.,= 100. The general interface nature of the k-I/ILES is similar to that for k-I/LES (with the averaged wall shear stress) but with ILES, the length scale drop is much more severe. Except for LNS, dark areas represent the (I)LES region and the light dark colour gives the URANS zone. In the LNS, the dark area is mostly URANS and the light zones (~ < 1) are LES. As can be seen, for the LNS and DES the interfaces are quite irregular (if an instantaneous local wall shear

344

Figure 2: Positions of six profiles investigated.

Figure 3: Contours of RANS and LES regions at mid x -y plane: (1) for LNS; (2) for k-I/ILES and (3) for DES. stress value is used, it is also irregular for the k-l/LES - hence this approach is not recommended). This is to be expected. Via Eq. (3) instantaneous values decide the LNS interface. For DES it is controlled bythe irregular grid. Compared with the other hybrid methods used here, LNS and DES are both found difficult to converge. This is mostly attributed to the irregular interfaces. Also, for LNS solving one more transport equation for e detracts from convergence. Although the S-A URANS model proved stable, DES was impossible to converge for the current complex geometry case. Therefore, DES results cannot be presented here. The lower modelled viscosity for DES, relative to LNS, is likely to be the key aspect preventing convergence. Notably, the k-gILES gives a smoother interface and hence sensible LES boundary conditions. Figure 4 shows instantaneous streamlines at the mid x - y plane from the k-I/ILES, k-I/LES (at Yi*,,= 30), LES, LNS and k-//EASM approaches. Massive separation, numerous vortex structures and strong streamline curvature can be seen. Comparison of the plots suggests that the k-//(I)LES and LES capture more unsteadiness activity than the URANS k-//EASM and LNS. In the channel like region, containing profiles (1) and (2) all models, except for the LNS, give a significant backwards mean U velocity (U/Uo .~ 2). However, for LNS the flow in this channel is considerably lower (U/Uo ,~ 0.1) constituting a significant qualitative solution difference. The relatively high k-I/(I)LES and LES flow activity can clearly be seen in Figure 5, which compares the

345

Figure 4: M i d x - y plane instantaneous streamlines from the k-1/ILES (1), k-1/LES (2), LES (3), LNS (4) and k-//EASM (5).

Figure 5: Time histories of u-velocity at the central point of profile 5 for the k-I/ILES (1), LES (2) and LNS and k-//EASM (3). temporal u-velocity variations at the central point of Profile 5 for the k-I/ILES, LES, LNS and k-//EASM. Examination of other temporal u-velocity variations at the central points of other five profiles (Profiles 1 4 and 6) shows similar results. Cleary, the lower dissipation k-I/ILES, which has zero subgrid scale eddy viscosity, shows most small scale/high frequency unsteadiness activity. This is to be expected. Table 1 presents the average system unsteadiness amplitude (A) against average turbulent viscosity (/zt). The error in A is also given. Encouragingly, the LES related methods predict unsteadiness amplitudes in reasonable accord with the measurement. The k-I/ILES which just has dissipation produced from numerical scheme, as would be expected, predicts more unsteadiness than the other methods. The LNS predicts the unsteadiness well. However, Figure 5 shows that, unlike the other LES related methods, the frequency is relatively low. The URANS k-//EASM gives the lowest amplitude. This is typical (see Tucker et al. 2003) and to be expected of a URANS method. To more immediately compare model performance quantitatively, average percentage errors based on six

346

TABLE 1 PERCENTAGE ERRORS IN A, U AND TI

Model

/~....

Measurement

A

Error (%) in A

Error (%) in U

Error (%) in Ti

-15

+27

0.245

k-t/ILES

7.7e 7

0.262

+7

LNS

1.3e -3

0.235

-4

k-I/LES ( Yi~,=30)

1.4e 4

0.199

-19

-15

-25

k-I/LES ( y,~ =30)

1.3e4

0.218

- 11

-16

-21

Yoshizawa LES

7.1 e 4

0.221

- 10

-15

-21

k-//EASM

1.8e 3

0.057

-78

-15

-29

.....

profiles (involving around 100 data points) for each model are considered using 21B~xp - B,um I/ 21 nexp l, i=l

i=l

where n is the total number of experimental points, the subscripts, 'exp' and 'num', representing experimental data and numerical values, respectively. Results are summarized in the right hand columns of Table 1. The '+' and '-' symbols represent under- and over-predictions, respectively. The errors in u show that except for the LNS, all other models generally produce similar velocity accuracies. The table also shows that the Yoshizawa LES gives the lowest average Ti error. However; if modelled fluctuations are also considered, it would be expected that the difference between the zonal k-I/LES and LES would be smaller. This is because with the k-I/LES, the k-l RANS is used in the near-wall regions and some modelled turbulence is transported out of the near wall region. Hence, the modelled parts have more influence on the flow in the k-l/LES than in the LES. The k-I/ILES gives a larger average error than the k//LES but significantly, unlike all other methods, the Ti error is now positive. Hence it would seem the LES type solutions have too much dissipation of resolved energy and the zonal ILES too little. Table 1 shows that the LES related approaches improve intensity predictions compared to the k-//EASM result. It should be mentioned that the LNS performs poorly and in many places gives more than 50 % average errors. Therefore, we do not present the LNS results. Even for the case of a simple empty two-dimensional box with a heated sidewall (see Henkes, 1990), and marginally more complex geometries (see Quere et al., 2004) the question of solution uniqueness can be fairly vexing. Shyy (1985) also grapples with this problem for simulations of essentially a simple bifurcating duct, finding two distinct solutions. Similar uniqueness issues occur with multi element airfoil configurations. For these a controlling factor is where peaks in turbulence energy, produced by upstream elements, strike the faces of the downstream elements (see Tucker et al., 2004). For the current system, strangely experimental evidence suggests the flow character can perhaps depend on the external environment i.e. the location of the unit in a room. The poor LNS results could in part be attributed to the LNS model triggering a solution of a slightly different character to that given by the other models. Further possible reasons for discrepancies of both the LNS and other models are given later.

Heat Transfer Results Figure 6 shows time-mean local Nusselt number Nux along the heated surface centreline in the x direction for all the models examined, where x,, represents the starting point of the heater. The local Nusselt number

347 250! o Measurement ........ Y It =30 ..... y ~,v,=30 .... k-I/ILES LES § k-I/EASM

200! [ F

~5o ! 4

100

(

50~

O

[ i '

+ ~-

o [~:~ 0.00

o 0

f--/2s163

f _ ~ ...... .~..:1-~ >~-f~-~- , o , ~ ~-~

,

.

"~ ~

~ ................................................................................. 0.05

0.10

0.15

0.20

X-Xo

Figure 6: Local Nux distributions at the heater surface centreline. is defined as Nux = (x-Xo)q/(k(T ~ -Tr,s)), where q is the measured convective heat flux, Ts and Trey surface and reference (the location of the reference point is located just upstream of the heater) temperatures and k the thermal conductivity of air. It is found that the LES related approaches underpredict Nux especially the LES. The k-I/ILES and k-I/LES approaches predict similar Nux distributions.. The k-//EASM over-predicts Nux. However, it should be noted that when used in a high Reynolds number form on a coarser grid the EASM gives impressive agreement (Liu, 2004). Nevertheless, since the onedimensional stationary flow wall functions are totally inappropriate for the heated flow regions this result must be regarded as fortuitous and hence is not shown. It might be expected that the added resolved turbulence activity for the less dissipative k-l/ILES approach would yield the highest heat transfer levels for the LES related methods. However, this is not so. Instead it is k-I/LES with the dynamic interface. It seems possible that the interaction of the complex irregular strongly time varying interface for this approach excites the near wall flow hence increasing heat transfer. However, if this is the case the improved heat transfer is occurring for non-ideal reasons. Unlike for the flow field, for Nux all models show extremely large errors. However, perhaps this is not too surprising. The high resolution simulations of Chung and Tucker (2004), just focusing on a sharply turned flow region, show a 500% change in Nux can be induced by small upstream flow perturbations. For the complex Figure 1 geometry there are, in addition to the substantial turbulence modelling errors especially for heat flux modelling, also significant problem definition issues, i.e. questions on the impact of how well losses introduced by grills and also the energy input from fans are modelled. The latter, in practice, introduce significant flow unsteadiness and swirl. Also, the lower fan shown in Figure 1 is significantly obstructed by a casing component and consequently a characteristic curve had to be specially produced by the fan manufacturer. The problem definition questions, combined with experimental, grid dependence and turbulence model errors could easily account for the complex geometry Nux errors and also the velocity and turbulence intensity errors. Another key question, as noted earlier, is solution uniqueness. Electronic systems flows are especially complex with many flow inlets and outlets and hence the potential for multiple solutions is perhaps significant. This is another aspect perhaps worthy of future exploration. However, local grid refinement studies were not successful in improving Nux prediction. This needs to be further explored.

CONCLUDING REMARKS The k-I/(I)LES, DES, LNS and LES approaches have been applied to a complex non-isothermal electronics flow. Comparisons were made with measurements and EASM URANS simulations. Except for the LNS, overall all models investigated gave similar time-mean velocity predictions with the LES

348

related methods giving improved turbulence intensity predictions. The LNS method performed poorly. This might be attributed to solution non-uniqueness issues. DES solutions could not be converged. This is partly attributed to the grid controlled irregular interface. For all methods heat transfer results were relatively poor. The EASM over-predicted heat transfer and the others under-predicted it, especially the pure LES.

References Batten P., Goldberg U. and Chakravarthy S. (2002). LNS-an Approach towards Embedded LES. Paper Number AIAA-2002-0427. Boris J.P., Grinstein F.F., Oran E.S. and Kolbe R.L. (1992). New Insight into Large Eddy Simulation. Fluid Dynamics Research 10, 199-228. Chung Y.M. and Tucker P.G. (2004). Numerical Studies of Heat Transfer Enhancement in Laminar Separated Flows. Int. J. Heat and Fluid Flow 25, 22-31. Davidson L. and Peng S.H. (2001). A Hybrid LES-RANS Model Based on a One-equation SGS Model and a Two-equation k-g2Model. Proc. 2rd Int. Symp. on Turbu. and Shear Flow Phenomena, 175-180. Gatski T.B. and Speziale C.G. (1993). On Explicit Algebraic Stress Models for Complex Turbulent Flows. J. Fluid Mech. 254, 59-78. Grinstein F.F. and Fureby C. (2002). Recent Progress on MILES for High Reynolds Number Flows.ASME J. Fluids Engrg. 124, 848-861. Henkes R.A.W. (1990). Natural-convection boundary layers, PhD Thesis, Technical University Delft. Liu Y. (2004). Numerical Simulations of Unsteady Complex Geometry Flows, PhD Thesis, University of Warwick, UK. Quere P.L., Xin S.H. Gadpin E. Daube O. and Tuckerman L. (2004). Recent Progress in the Determination of Hydrodynamic Instabilities of Natural Convection Flows. Proc. CHT-04 ICHMT Int. Syrup. On Advances in Computational Heat Transfer, Norway, CHT-04-K4. Shyy W. (1985). A numerical study of annular dump diffuser flows. Computer Methods in Applied Mechanics and Engineering 53, 47-65. Spalart P.R. and Allmaras S.R. (1994). A One-equation Turbulence Model for Aerodynamic Flow. La Recherche AerospatiaIe 1, 5-21. Spalart P.R., Jou W.H., Strelets M. and Allmaras S.R. (1997). Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach. First AFOSR Int. Conf. on DNS/LES in Advances in DNS/LES, 137-147. Temmerman L., Leschziner M.A. and Hanjalic K. (2002). A-priori Studies of a Near-wall RANS Model within a Hybrid LES/RANS Scheme. Engrg Turbu. Modelling and Experiment 5, 317-326. Tucker P.G.and Pan Z. (2000). URANS Computation for a Complex Internal Isothermal Flow. Comput. Methods Appl. Mech. Engrg. 190, 1-15. Tucker P. G., Liu Y., Chung M. and Jouvray A. (2003). Computation of an unsteady complex geometry flow using novel non-linear turbulence models. Int. J. for Numerical Methods in Fluids 43: 9, 979- 1001. Tucker P.G. and Davidson L. (2004). Zonal k-l Based Large Eddy Simulations. Computers and Fluids 33, 267-287. Tucker P.G. (2004). Novel MILES Computation for Jet Flows and Noise. Int. J. Heat and Fluid Flow 25:4, 625-635. Tucker P. G., Rumsey C. L., Spalart P. R., Bartels R. E. and Biedron R. T. (2004). Computations of all distances based on differential equations. 34 th AIAA Fluid Dynamics Conference and Exhibit, Portland, Oregon, Paper Number AIAA-2004-2232. Wolfshtein M. (1969). The Velocity and Temperature Distribution in One-dimensional Flow with Turbulence Augmentation and Pressure Gradient. Int. J. Heat Mass Transfer 12, 301-318. Yoshizawa A. (1993). Bridging Between Eddy-viscosity-type and Second-order Models Using a TwoScaie DIA. Proc. 9th Int. Symp. on Turbulent Shear Flow, Kyoto, 23.1.1-6.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

349

INTERFACE C O N D I T I O N S FOR HYBRID RANS/LES C A L C U L A T I O N S Anthony Keating, Giuseppe De Prisco, Ugo Piomelli and Elias Balaras Department of Mechanical Engineering, University of Maryland, College Park, MD 20742

ABSTRACT Hybrid RANS/LES methods, in which the attached boundary layers are simulated by RANS while the non-equilibrium regions of the flow are computed by LES, have received considerable attention over recent years. One issue that may affect (sometimes significantly) the accuracy of the results in hybrid methods is the generation of turbulent eddies capable of supporting the Reynolds stresses in the LES region from a smooth RANS field (in which the Reynolds stress is due entirely to the model). We present results obtained in spatially developing turbulent channel flow in which synthetic turbulence is coupled with a controlled forcing technique that generates physically realistic turbulent eddies. This technique substantially reduces the distance required for the development of realistic turbulence. We discuss the effect of the free parameters used in this technique, and their optimization. Simulations are also performed that evaluate the effect of the approximations required when the full Reynolds-stress tensor is not know, and only statistics typically available from RANS simulations (the mean velocity and the Reynolds shear stress) are available. The effect of the quality of the synthetic turbulence at the inflow plane and the overall computational cost is also discussed. Finally, results from a simulation of an accelerating boundary layer are given. KEYWORDS

Large-eddy simulation, Inflow boundary conditions, Hybrid RANS/LES, Synthetic turbulence, Turbulent channel flow, Accelerating boundary layer INTRODUCTION

The generation of realistic instantaneous turbulent fields from statistical data is an issue becoming increasingly important, given the attention that hybrid RANS/LES methods have received in recent years. In these methods, a RANS calculation is carried out in regions where standard turbulence models are expected to be accurate (attached thin shear layers, for instance), while LES is used in non-equilibrium flow regions. One issue that arises in these methods is the behavior of the flow in the transition zone between the RANS and LES regions. In the RANS zone the flow solution is either steady, or only contains information on the largest scales of motion if unsteadiness is present; most or all of the Reynolds shear stress is provided by the turbulence model. In the LES region, on the other hand, the resolved scales must

350 supply most of the Reynolds shear stress and small scale structures must be present to provide it. Typically, a transition zone exists in which the resolved, energy-containing eddies are gradually generated and grow. The generation of realistic turbulent fluctuations at the RANS/LES interface is a critical factor in determining the length of this unrealistic transition region. There are at least two ways to provide an interface condition in a hybrid calculation: a single grid can be used that spans the RANS and LES regions, and the eddies may be generated in the LES region either naturally, by the instabilities present in the flow [the approach followed in simulations of massively separated flows performed using DES (Spalart et aL, 1996; Squires 2004)], or by some form of stochastic forcing. The other technique involves the use of two separate computational domains, in which the RANS results are used to supply the statistics of the turbulence to be used as inflow for the LES. The latter approach is related to the specification of inflow conditions for LES, which generally use one of three types of inflow boundary-conditions: random fluctuations superimposed on a mean profile; instantaneous velocity fields saved from a periodic, precursor simulation; or recycling and rescaling a plane of velocity downstream of the inflow plane. Within the framework of hybrid methods, the use of velocity fields obtained from a separate calculation is not feasible, since the flow in the RANS region cannot be computed by LES at a reasonable cost (otherwise there would be no reason for the use of hybrid methods). However, Schltiter, Pitsch and Moin (2003) proposed an inflow boundary-condition for the LES region based on the rescaling of an instantaneous flow database saved from a separate calculation, based on the results of a RANS calculation in the upstream region. Simulations of confined swirling and non-swirling jets using this method yielded results in good agreement with experiments. It is unclear how the method would behave if the flow database were generated in conditions significantly different from those at the LES inflow. The use of random fluctuations superimposed on a mean profile is also a feasible solution to the interface problem. The fluctuations (whose moments would be obtained from the RANS) could be either generated by localized forcing, or computed separately and superimposed on the random inflow profile, again obtained from the RANS. This approach, however, was found to require long transition distances for realistic turbulence to be generated (Le et al., 1997). More recent applications based on the synthetic, homogeneous turbulence generation method by Kraichnan (1970), which has been recently extended to inhomogeneous flows by Smirnov et aL (2001) and Batten et al. (2004), suffer from the same limitation (Keating et al., 2004). One inflow-generation technique that could be applied both to a single-grid calculation such as DES and to a method that uses the RANS data from a separate computation for the generation of the LES inflow was proposed by Spille-Kohoff and Kaltenbach (2001) and has been recently investigated by Keating et al. (2004). These studies showed that, by using synthetic turbulence at the inflow plane coupled with controlled forcing, the development length of the turbulent eddies could be substantially reduced. The forcing method enhances wall-normal velocity fluctuations at several control planes downstream of the inflow plane and is modulated so that a "target" Reynolds shear-stress profile is achieved. In those simulations, statistics calculated from an LES were used, and the full Reynolds-stress tensor was available, as well as the dissipation rate; some of these statistics may not be available in a hybrid RANS/LES simulation. In this paper, we discuss results obtained using only statistical data that is typically available from a RANS simulation: the mean velocity profile and the Reynolds shear stress, (u~v'). We also present a number of improvements in the implementation of the forcing, as well as the effect of the quality of the synthetic turbulence at the inflow. Finally, we discuss the additional computational cost of the forcing method, and apply the method to a more challenging flow configuration.

351 G O V E R N I N G EQUATIONS AND N U M E R I C A L M E T H O D The governing equations for the LES of an incompressible flow are obtained by applying a filtering operation to the continuity and Navier-Stokes equations. The filtered equations were solved on a Cartesian staggered grid. Conservative second-order finite differences were used for spatial discretization while a fractional-step method (Kim and Moin, 1985) coupled with a second-order implicit Crank-Nicolson method (for the wall-normal diffusion term) and a third-order explicit Runge-Kutta method (for the remaining terms) was used for time integration. The subgrid-scale stresses were parameterized using the dynamic eddy-viscosity subgrid-scale model (Germano et al., 1991; Lilly, 1992) with the eddy-viscosity coefficient averaged over Lagrangian flow pathlines (Meneveau et al., 1996). Synthetic turbulence generation The synthetic turbulence generation method of Batten et al. (2004) is used to create a three-dimensional, unsteady velocity field at the inflow plane. This method requires as input the mean velocity field, the Reynolds-stress tensor and the specification of a time-scale of turbulence, rb, calculated as the ratio of the turbulent kinetic energy, k, to the turbulent dissipation rate, ~ (Batten et al., 2004). The method involves the summation of sines and cosines with random amplitudes and phases that yield a velocity field having specified length- and time-scales, and energy spectrum. In all simulations presented here we used 200 random modes; this number was required to ensure that the resulting statistics were independent of the number of modes used. Further details of the method can be found in Batten et al. (2004). We will also present results obtained generating synthetic turbulence by the simpler method of Lund et al. (1998) which gives a random field with length and time-scales equal to the grid size and time-step. While this method produces less realistic structures, it is substantially cheaper than Batten's method, since it does not require the summation of a large number of modes at each grid point. Controlled f o r c i n g The controlled forcing methods proposed by Spille-Kohoff and Kaltenbach (2001) adds a forcing term to the wall-normal momentum equation that amplifies the velocity fluctuations in that direction, thus enhancing the production term in the shear-stress budget. A controller is used to determine the forcing amplitude based on the error in the Reynolds shear-stress: e(y, t) = (u'v')*(Xo, y ) - (u'v') z't (Xo, y , t)

(1)

where (u'v')*(Xo,y) is the target Reynolds shear stress at the control plane x = Xo, and (ulv') z,t (Xo,y, t) is the current Reynolds shear stress, averaged over the spanwise direction and time using an exponential window. The forcing aims at the enhancement (or damping) of local flow 'events' that contribute to the Reynolds shear stress. This is achieved by setting the force magnitude to f(xo,y,z,t)

=r(y,t)

[u(xo,y,z,t) -

(u)Z't(xo,y)]

(2)

where f is related to the error by r ( y , t) = de(y, t) + ~

f0 ~

e(y t d t'

, ')

(3)

In our previous simulations (Keating et al., 2004) ot = 1 and/3 = 1 were used; we subsequently found that the transient period for the controllers could be significantly reduced (without adverse effects on other parameters) by increasing/3 to 30. All the simulations presented in this paper, therefore, were carried out with ot = 1 and/3 = 30. Further details on the forcing method may be found in Spille-Kohoff and Kaltenbach (2001) and Keating et al. (2004).

352

SPATIALLY-DEVELOPING C H A N N E L F L O W RESULTS The spatially-developing turbulent flow was simulated in a channel with a length of 1On:3 and width Jr 3, using 240 x 65 x 64 grid points in the streamwise, wall-normal and spanwise directions (Ax + = 51.7, Az + = 19.4, AYmi + n = 0.6, Ay+max = 33.7). The Reynolds number Rer (based on friction velocity u r and channel half-height 3) was 400. No-slip boundary conditions were used at the upper and lower walls, while a convective boundary condition (Orlanski, 1976) was used at the exit plane. The simulations using synthetic inflow conditions are compared to a baseline simulation, which used planes of velocity fields from a periodic simulation of turbulent channel flow at the same Reynolds number. To evaluate the effect of the inflow boundary conditions, we use three indicators: the coefficient of friction, Z"w Cf

=

1

2

'

(4)

the errors in the turbulent kinetic energy (TKE) and Reynolds shear stress, f+* Iq,e(y)- q2(x, y)[dy

e q2 (X ) =

f +_~[q2 (y ) [dY

,

1 f+* ] ( u ' v ' ) , ( y ) - (u'v')(x,y)[dy euv(X) = ~-~ max(l(u'v'),[)

(5)

where q20, ) and (u'v'),(y) are calculated from the periodic simulation and q2(x,y) and (u'v')(x,y) are obtained from the spatially developing simulation. These indicators allow us to compare easily the spatial development of the first and second moments of the velocity field for the different inflow boundary conditions. A calculation in which the inflow was supplied from a separate, precursor, simulation will be used as a reference.

Effect of a continuous force In earlier work with the controlled forcing in a similar configuration (Keating et al., 2004), we applied the force at four control planes (located at x/3 = 1.3, 2.6, 3.9 and 5.2). The forcing was very effective in generating realistic turbulence in a short distance, as shown in Fig. 1, in which the three measures of effectiveness are plotted. It can be observed that by the last control plane (within 53 of the inflow plane) the correct Reynolds-stress profile had been established, although it took 10 additional channel halfheights for the mean profile and the TKE to adjust and reach their expected values. This is a significantly shorter transition distance than was found when the synthetic turbulence was used on its own (Keating et al., 2004). This localized forcing, however, resulted in large fluctuations in the shear-stress upstream of each control plane (Fig. lc). While these fluctuations do not propagate downstream of the control region in the staggered code used here, when implemented in a co-located Navier-Stokes solver they were found to cause large instabilities. In order to reduce them, we now gradually increase the force, f ( x o , y , z , t), over a number of planes upstream of each control plane. A number of envelopes were tried; we had the most rapid decay of the errors using a hyperbolic tangent envelope that reduced f to approximately half of its maximum value over a distance of 1.33. Note also that when the force was increased gradually in this manner, two control planes (at x/3 = 1.3 and 2.6) were sufficient to reduce the error in the Reynolds shear stress to acceptable levels. Figure 1 shows the effect that distributing the force has on C f, o n eq2 and euv. The C f and the error in TKE are essentially unchanged from the case in which the force was discretely applied at four planes. Both CU and TKE recover to fully developed values around 15 channel half-heights downstream of the inflow plane. Making the force distribution continuous reduces substantially the fluctuations in the shear

353 8

(a)

7 %

6

X

"~,,,"

5 4

3 1

0

5

10

5

10

0.8 0.6 0.4

ks-

9%

-- ..

,.

0.2 0 0.6

0

15

~~,,,I, V.~;....

(c)

1111 I

f ',:: 1::

0.4

,

i~, Ii ~1 lilt I I i I I i~

0.2

'

I

|l~t400

i 0

",k

0

'I:

ii i I I

,~

;"r ~'"",.e','-..-.--..,_. _ __ i

. . . . . .

..

.

5

. ~ . _

..-

.~.~

10

9

15

x/5

Figure 1" Downstream development of (a) the coefficient of friction, and the integrated errors in (b) the turbulent kinetic energy, and (c) the Reynolds shear stress. , precursor simulation; , discrete controlled forcing, 4 p l a n e s ; - - - , controlled forcing gradually increased, 2 planes.

Figure 2: Contours of streamwise velocity fluctuations at y/8 = 0.01. (a) Precursor simulation 9(b) discrete controlled forcing, 4 planes; and (c) continuous controlled forcing, 2 planes. Contour lines are evenly spaced at +0.3, +0.24, +0.18, +0.12 and +0.06. Dashed lines indicate negative contours.

354 stress upstream of the control planes, and the error is reduced to an acceptable level in a shorter distance than when discrete forcing was applied. Instantaneous streamwise velocity fluctuations are shown in figure 2 for a plane parallel to the wall, at y / 6 = 0.01 (y+ _~ 4). The effectiveness of the forcing method in enhancing the strong turbulent eddies rapidly, thereby re-generating the streaky structure of the wall-layer, is apparent. The stronger force applied when only two planes are used results in streaks that are initially unphysically long; they break down rapidly, however, and the correct eddy structure is rapidly established. Effect o f reduced available information In this section, we discuss results obtained using the reduced information typically available from a RANS simulation, rather than the complete statistical data (full Reynolds-stress tensor and dissipation rate) that were used in the calculations described above. In order to use Batten et al.'s (2004) method to generate a synthetic velocity field at the inflow, the four non-zero components of the Reynolds-stress tensor and the time-scale, rb, are required. An eddy-viscosity turbulence model only gives k and e, or (in the case of one-equation models) the eddy viscosity yr. We examine here the more restrictive case in which only vt and the mean velocity profile are obtained from the model. To relate the TKE, k, to the Reynolds shear stress, (u'vl), we use the experimental result (Bradshaw et al., 1967; Townsend, 1962): 0u l - ( u ' v ' ) l = vt -~y = a l k where a l = ~ stresses:

(6)

and c u is typically 0.09. We then distribute k equally among the normal Reynolds 2 (u'u') = (v'v') = (w'w') = - k . 3

(7)

To approximate the time-scale, we use the definition of eddy-viscosity from the k - ~ turbulence model to express ~ in terms of k and vt (Menter, 1997): -- c#kZ/vt

.

(8)

The time-scale in Batten et aL's method is the ratio of k over ~, which, using equations (6) and (8) can be estimated as k 1 rb = - = (9)

Both of these approximations were found to agree reasonable well with the values of k and rb calculated from a periodic LES. Figure 3 shows the development of the skin friction coefficient and the integrated errors in the TKE and Reynolds shear stress when (7-9) are used to generate the synthetic turbulence at the inflow plane. The error behavior is very similar to the case in which the full statistical information is used, with C f and the error in TKE reduced to acceptable levels within 15~ and the error in Reynolds shear stress reduced to acceptable levels before the second controller. The only observable difference is the slightly higher error in TKE near the inflow plane, which is caused by the underprediction of k by equation (6) at the inflow. Effect o f inflow quality Batten et aL's (2004) method involves the summation of a large number of random modes, and its computational cost is relatively high (see the next section). Lund et al. (1998) proposed a randomfluctuation inflow-generation method that created a random field with assigned second moments; the

355

8 7

%

x

6 5 4 3 1

0

'5

10

15

,

"'.

(b)

0.8 0.6 f - \ 0.4

; .............

r .,.~ i ,

.................................................... ,,,,

0.2 0 0.6

0

5

10

(c)

\i'.

0.4

. . . . . ..~ 0.2

15

k:..

-

~.

~,,.,

0

,o-'~

....

.. . . . . .

~

o,"~

5

10 x/6

Figure 3: Downstream development of (a) the coefficient of friction, and the integrated errors in (b) the turbulent kinetic energy, and (c) the Reynolds shear stress. ~ , precursor simulation;---, continuous controlled forcing, 2 planes, Batten et al.'s synthetic turbulence at inflow; - . - , continuous controlled forcing, 2 planes, reduced information at inflow, Batten et aL's synthetic turbulence at inflow; ...... , continuous controlled forcing, Lund et al.'s random fluctuations at the inflow. length- and time-scales of this random field are the grid size and time-step. This method was used to generate synthetic turbulence at the inflow and compared with the data obtained using Batten et al.'s (2004) method. Figure 3c shows that the controllers are able to reduce the error in the Reynolds shear stress; downstream of the control planes, however, the flow undergoes a second transition and does not recover before the end of the domain. We believe that the lack of length- and time-scale information at the inflow is the cause of this behavior: the turbulent eddies decay too rapidly, and the forcing required to achieve the desired target is too high, increasing the amplitude of the fluctuations without affecting the correlation between u' and v' that is a result of the establishment of the proper eddy structure. Computational Cost

In the simulations presented here, we found that the cost of the controlled forcing was negligible, the only additional CPU time being due by the additional transient that is required for the controllers to adjust to the flow, and by the generation of the synthetic turbulence. As mentioned previously, this transient has been reduced by increasing fl in the controller to 30, and, in the simulations presented here, was approximately 1208/Ub, or 4 flow-through-times. A similar startup period is required by recycling methods such as that of Lund et al. (1998) to obtain the required running averages. Since the method of Batten et al. (2004) involves the summation of a large number of modes for each grid point on the inflow plane, it is relatively expensive ( ~ 15% of the total CPU time).

356 0.006 0.005 0.004

[/J

0.003 0.002 0.001

0

;0

~0

;0

4o

;0

;0

~0

80

x/6

Figure 4: Downstream development of the coefficient of friction in the accelerating boundary layer. , baseline simulation using recycling/rescaling;---, truncated domain, synthetic turbulence at inflow; - - - , truncated domain, controlled forcing. A C C E L E R A T I N G BOUNDARY LAYER RESULTS To investigate the performance of the synthetic turbulence and forcing technique in a non-equilibrium flow, simulations of an accelerating boundary layer, closely matching the experiments of Warnack and Fernholz (1998) have been performed. A baseline simulation was first performed using the rescaling/recyclind method of Lund et al. (1998) at an inflow Reynolds number, Res. = 1200. For this simulation the domain length was 6008~, the width was 258~) and the height was 258~). The freestream acceleration (from U ~ = 1 to Uo~ = 3.5) began at approximately 1008~ and was finished at 4508~). A relatively coarse grid (256 x 96 x 64) was used as comparisons were primarily being made between this simulation and simulations of a truncated domain that used synthetic inflow boundary conditions. The truncated domain used for the simulations using synthetic inflow boundary conditions started at x/8~ = 150 and therefore had a length of 4508~ and 192 grid points in the streamwise direction (the resolution was the same as the baseline simulation in all three directions). Inflow statistics were extracted from the baseline simulation to provide the required mean velocity profiles, Reynolds stress tensor and dissipation rate for the synthetic turbulence method of Batten et al. (2004) and the controlled forcing technique. Two truncated domain simulations were performed: one with synthetic turbulence at the inflow only, and another with synthetic turbulence at the inflow plus controlled shear stress forcing at four downstream planes. Figure 4 shows the downstream development of the coefficient of friction, C f for the three cases. The baseline simulation (using the rescaling/recycling at the inflow) shows good qualitative agreement with the experiments of Warnack and Fernholz (1998) although the C f in the recovery region is underpredicted (simulations using finer grids show better agreement). The truncated domain simulation using synthetic turbulent alone shows a sharp reduction in C f just downstream of the inflow, as the flow begins to laminarize. Because of this laminarization the flow transitions early (around x / 8 -- 50 instead of x / 8 ~ 53). When controlled shear stress forcing is added, the inflow generates realistic turbulence very quickly, and shows good agreement with the baseline simulation. The slow recovery of turbulence when only synthetic turbulence is used is also evident in contours of the wall-normal Reynolds stress shown in figure 5b. When compared to the full domain simulation results shown in figure 5a (that uses rescaling/recycling at the inflow), there are excessively low fluctuations in the boundary layer from the inflow until x/8* ~ 350. Figure 5c shows results from the simulation that uses controlled forcing. The large fluctuations near the inflow are caused by the forcing method. It is clear that the boundary layer is turbulent during the first stages of the acceleration. It is also interesting to note the increase in (v'v') at x/8* ~ 380 is also reproduced well when the controlled forcing method is used.

357

Figure 5: Contours of the wall-normal Reynolds stress, (vlv'). (a) Full domain, recyclind/rescaling; (b) truncated domain, synthetic turbulence at inflow; and (c) truncated domain, controlled forcing. CONCLUSIONS The use of a controlled forcing technique coupled with synthetic turbulence has been shown to be effective in generating fully developed, realistic turbulence within a short distance. This method shows promise for use in hybrid LES/RANS methods, where the inflow plane of the LES region is defined by statistical quantities obtained from RANS. We eliminated the strong oscillations in the Reynolds shear-stress upstream of the control planes observed in earlier work by using a continuous force instead of a single control plane. We investigated the effect of replacing the normal Reynolds stresses and dissipation with semi-empirical estimates that only use the eddy viscosity and velocity gradient (information supplied by any RANS turbulence model). We found that these approximations were used did not result in significant differences in the spatial development of the flow downstream of the control planes. We determined that simple synthetic turbulencegeneration methods based on random numbers with no imposed length- and time-scales are not as effective as the Batten et aL (2004) synthetic method. Despite its cost, the latter is preferred. Simulations of an accelerated boundary layer, where the inflow plane was placed at the very beginning of the acceleration showed good agreement with results obtained using the rescaling/recycling method (which required a longer inlet region where the flow was in equilibrium). Overall, the controlled forcing appears to be relatively inexpensive and robust. We plan to continue testing this method, extending our simulations to adverse pressure-gradient separated boundary-layer cases, and implementing it in actual hybrid RANS/LES.

Acknowledgments This work is supported by the Air Force Office of Scientific Research under Grant No. F496200310112, monitored by Dr. T. Beutner. The authors thank Drs. Paul Batten and Hans Kaltenbach for their helpful discussions regarding the implementation of their synthetic flow generation and controlled forcing methods.

358

References Batten R, Goldberg, U. and Chakravarthy, S. (2004). Interfacing statistical turbulence closures with largeeddy simulation, AIAA J. 42:3,485-492. Bradshaw, R, Ferriss, D.H. and Atwell, N.R (1967). Calculation of boundary layer development using the turbulent energy equation, J. Fluid Mech. 23, 31-64 Germano, M., Piomelli, U., Moin, E and Cabot, W. (1991). A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3:7, 1760-1765. Keating, A., Piomelli, U., Balaras, E., and Kaltenbach, H.-J. (2004) A priori and a posteriori tests of inflow conditions for large-eddy simulation, Phys. Fluids 16:12, 4696-4712. Kim, J. and Moin, E (1984). Application of a fractional step method to incompressible Navier-Stokes equations, J. Comput. Phys. 59:2, 308-323. Kraichnan, R.H. (1970). Diffusion by a random velocity field, Phys. Fluids 13:1, 22-31 Le, H., Moin, R and Kim, J. (1997) Direct numerical simulation of turbulent flow over a backward-facing step, J. Fluid Mech., 330, 349-374. Lilly, D.K. (1992). A proposed modification of the Germano subgrid-scale closure model, Phys. Fluids A 4:3, 633-635. Lund, T.S., Wu, X. and Squires, K.D. (1998). Generation of inflow data for spatially-developing boundary layer simulations, J. Comput. Phys. 140, 233-258. Meneveau, C., Lund, T.S. and Cabot, W.H. (1996). A Lagrangian dynamic subgrid-scale model of turbulence, J. Fluid Mech. 319, 353-385. Menter, ER. (1997). Eddy viscosity transport equations and their relation to the k-e model, J. Fluids Eng. 119, 876-884. Orlanski, I. (1976). A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phys., 21, 251-269. Schltiter, J.U., Pitsch, H. and Moin, E (2003). LES inflow conditions for coupling with Reynoldsaveraged flow solvers, AIAA J. 42:3,478-484 Smirnov, A., Shi, S. and Celik I. (2001). Random flow generation technique for large eddy simulations and particle-dynamics modeling, J. Fluids Eng. 123, 359-371. Spalart, R R., Jou, W. H., Strelets, M., & Allmaras, S. R. (1997) Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach, In Advances in DNS/LES, edited by C. Liu and Z. Liu, (Greyden Press, Columbus), 137-148. Spille-Kohoff, A. and Kaltenbach, H.-J. (2001). Generation of Turbulent Inflow Data with a Prescribed Shear-Stress Profile, n DNS/LES Progress and challenges, edited by C. Liu, L. Sakell, T. Beutner (Greyden, Columbus, OH), 319-326. Squires, K. D. (2004). Detached-eddy simulation: current status and perspectives, In Direct and largeeddy simulation V, edited by R. Friedrich, B. J. Geurts, and O. M6tais (Kluwer, Dordrecht), 465-480. Townsend, A.A. (1962). Equilibrium layers and wall turbulence, J. Fluid Mech., 11, 97-120. Warnack, D. and Fernholz, H.H. (1998). The effects of a favourable pressure gradient and of Reynolds number on an incompressible axisymmetric turbulent boundary layer. Part 2. The boundary layer with relaminarization, J. Fluid Mech., 359, 357-381.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

359

APPROXIMATE NEAR-WALL TREATMENTS BASED ON ZONAL AND HYBRID RANS-LES METHODS FOR LES A T H I G H R E Y N O L D S N U M B E R S F. Tessicini, L. Temmerman t and M. A. Leschziner Department of Aeronautics, Imperial College London, London SW7 2AZ, UK t Current address: Numeca International s.a. Av. F. Roosevelt 5 B-1050 Brussels, Belgium

ABSTRACT Two strategies, combining a LES scheme with different near-wall RANS approximations, are investigated by reference to simulations for plane channel flow and two separated flows at moderate and high Reynolds numbers, respectively. One strategy is a hybrid modelling scheme, wherein the subgrid-scale model in the outer LES domain is interfaced with a RANS model in a predefined nearwall layer. The other is a zonal method in which a thin-shear-flow RANS solution in the near-wall layer, embedded within the LES domain which covers the entire flow, is used to provide boundary conditions for the LES computation, the two thus being loosely coupled. Both methods allow the thickness of the near-wall RANS layer to be chosen freely. In the hybrid LES-RANS scheme, the near-wall layer is interfaced to the outer LES region, subject to compatibility conditions for velocity and turbulent viscosity imposed across the interface. These conditions are extracted dynamically as the simulation progresses. In the zonal approach, a mixing-layer model provides the eddy-viscosity field in the near-wall layer, while in the hybrid approach, a two-equations ( k - e ) model is used. KEYWORDS

Turbulent flow, Hybrid LES-RANS, Zonal model, Two-layer model, Near-wall simulation 1

INTRODUCTION

Large Eddy Simulation is now almost routinely used to investigate fundamental aspects of turbulence mechanics, to help validate statistical closures and to obtain predictions for flows in which unsteady events associated with turbulence are of major interest or influence. Although LES continues to be an expensive approach at practically relevant Reynolds numbers, the expense is tolerable when the flow being simulated is remote from walls. However, flows which are substantially affected by near-wall shear and turbulence pose serious resource challenges as a consequence of the need to increase the near-wall grid resolution in line with N ,.~ O(Re~ to restrict the distance between the wall and the nodes closest to the wall to around y+ = 2 (where y+ is the distance in wall units based on the the local instantaneous friction velocity) and to maintain a

360

cell-aspect ratio of order Ay + = O(1), As + = 0(50) , where s is the mean direction close and parallel to the wall. Thus, at high Reynolds numbers, the utility of LES in a practical context depends greatly on the availability of acceptably accurate near-wall approximations that allow the resolution requirements to be reduced to economically tenable levels. Over the past few years, a whole range of approaches to this problem have been proposed. These include log-law-based wall-functions, various zonal and seamless RANS-LES hybrid schemes, notably the DES method, and the immersed boundary method. So far, no one particular method has been demonstrated to be definitively superior to others, and all involve restrictions and limitations which adversely affect the resulting solution in some circumstances. Even in a simple fully-developed channel flow at high Reynolds number, no method is able to give a solution that is without defect in the vicinity of the edge of the near-wall layer within which the approximate model is applied. In earlier work by Temmerman et al. (2002), Hadziabdid et al. (2003) and Temmerman et al. (2004), a RANS-LES hybrid method has been investigated, in which a conventional low-Re model is applied within a near-wall layer, the thickness of which can be chosen freely. Coupling to the LES domain is effected via compatibility constraints, including a dynamic process which adjusts the turbulent viscosity at the RANS side of the interface by reference to the subgrid-scale viscosity in the LES region. In the present paper, this methodology is implemented in combination with a two-equation low-Re k - e model in the near-wall layer and a one-equation subgrid-scale model in the LES region. To permit this, an implicit solution method of the turbulence equations within the layer had to be devised to procure stability and boundedness. In parallel to the above work, a second two-layer approach has been pursued. In this, the near-wall layer is numerically separated from the LES domain. Simplified (parabolized) versions of the momentum equations are solved in the layer, together with the turbulence-model equations (at this stage, a mixing-length model) with 'boundary conditions' taken from the outer LES domain at the interface. The solution is then used to extract the wall-shear stress, which is then used as a boundary condition for the LES domain. This is, essentially, an application of the method of Balaras and Benocci (1994), Cabot (1995) and Wang and Moin (2002), here within a body-fitted finite-volume framework. Both methods are contrasted against each other, as well as against a log-law-based wall-function approach. Their performance is examined for fully-developed channel flow at R e -- 42200, to a spanwise homogeneous, separated flow in a channel constricted by hill-shaped protrusions on one wall at R e = 21500, and to a spanwise homogeneous flow separating from rear upper side of a hydrofoil at R e -- 2.2 • 106.

2 2.1

O U T L I N E OF L E S - R A N S S T R A T E G I E S The

hybrid

strategy

The principles of the hybrid scheme are conveyed in Fig. l(a). The following provides a summary of a more detailed description provided by Temmerman et al. (2004). The thickness of the nearwall layer may be chosen freely, although in applications to follow, the layer is simply bounded by a particular wall-parallel grid surface. The LES and RANS regions are bridged at the interface by interchanging velocities, modelled turbulence energy and turbulent viscosity, the last subject to the continuity constraint across the interface, V"Lmod Turbulence in the RANS layer E S "-- t. Zrood RANS. is here represented by means of the two-equation model of Lien and Leschziner (1994). With the turbulent viscosity given by ut -- C t, k 2 / c , matching the subgrid-scale viscosity to the RANS viscosity at the interface implies: . rood

< VLES > C~.~.. = < f~. k 2 / c >

(1)

361

where < 9> indicates averaging over any homogeneous directions. Boundary conditions for solving the k-equation in the RANS layer are provided by the subgrid-scale energy in the LES domain, while the dissipation rate is evaluated from the subgrid-scale energy as kl"5/(Const x A) where A represents the cell size (AxAyAz) 1/3. The turbulence equations in the sublayer are solved by a coupled, implicit strategy, replacing an earlier sequential, explicit solution applied to oneequation models, which was found to cause stability problems with two-equation models. The smooth transition from the RANS value C~, = 0.09 to the interface value C~,,i,~tis effected by the empirical exponential function (see Temmerman et al. (2004)): 1 - ~;(-y/A) c , = 0.09 + (c~,~nt - 0.09) 1 _ exp(-yint/Aint)

(2)

Figure 1: Schematics of (a) the hybrid LES/RANS scheme; and (b) the two-layer method.

2.2

The zonal two-layer strategy

The objective of a zonal two-layer strategy is to provide the simulation with the wall-shear stress, using information from the outer flow near the wall. The wall-shear stress can be determined from an algebraic law-of-the-wall model or from differential equations solved on a near-wall-layer grid refined in the wall-normal direction- an approach referred to as "two-layer wall modelling". In geometrically simple flows, such as that in a channel, the wall-parallel velocity can be fitted to the log law to predict a wall-shear stress, the simplest form being:

Um= u~-/n 9ln(ymu~./u) + B

(3)

The two-layer wall method, shown schematically in Fig. l(b), was originally proposed by Balaras and Benocci (1994) and tested by Balaras et al. (1996) and by Wang and Moin (2002) to calculate the flow over the trailing edge of an hydrofoil. At solid boundaries, the LES equations are solved up to a near-wall node which is located, typically, at y+ = 50. From this node to the wall, a refined mesh is embedded into the main flow, and the following simplified turbulent boundarylayer equations are solved: o~

(~ + ~'~)o ~ j

= F~,

362

o< Fi = - - ~ +

Ou-~ Ox ,

@ (4)

+ Ox----~'

where n denotes the direction normal to the wall and i identify the wall-parallel directions Balaras et al. (1996). In the present study, only the pressure-gradient term has been included in Fi. The effects of including the remaining terms are being investigated and will be reported in a future paper. The eddy viscosity ut is here obtained from a mixing-length model with near-wall damping, as done by Wang and Moin (2002): ~-At= ~y+(1 - e-A~-~:)2,

(5)

/2

where y+ is the distance in wall units based on the local, instantaneous friction velocity. The boundary conditions for equation (4) are given by the unsteady outer-layer solution at the first grid node outside the wall layer and the no-slip condition at y = 0. Since the friction velocity u~ is required in equation (5) to determine y+ (which depends, in turn, on the wall-shear stress given by equation (4)), an iterative procedure had to be implemented wherein ~t is calculated from equation (5), followed by an integration of equation (4).

3

FULLY-DEVELOPED

CHANNEL FLOW

The performance of the present near-wall practices is first assessed By reference to a turbulent channel flow at a Reynolds number of 42200, based on half-channel width and bulk velocity. The computations are summarised in Table 1, and related solutions are presented in Fig. 2. The table also lists the modelling practices adopted and location of the interface. Thus: 'Ref LES' identifies a highly-resolved reference solution ; 'IEQ' and '2EQ' denote, respectively, that a one-equation ( k - l ) and the two-equation k - c model have been used in the RANS layer; 'j17' relates to the grid line at which the interface has been placed, and this corresponds to the wall distance identified in the last column; and 'WALE' and 'k-eq' signify the subgrid-scale model used ( Nicoud and Ducros (1999) and Yoshizawa and Horiuti (1985), respectively). Both the hybrid and two-layer-model TABLE 1 G R I D S , MODELLING P R A C T I C E S AND I N T E R F A C E LOCATIONS USED FOR C H A N N E L - F L O W SIMULATIONS

Case

Cells number

SGS model

interface location (y+)

Ref LES

512 x 128 x 128

WALE

2EQ-j17

64 x 64 x 32

k-eq

113

1EQ-j17

64 x 64 x 32

k-eq

113

Two-layer Fi = 0.

64 x 64 x 32

WALE

60

log-law WF

64 x 64 x 32

WALE

60

simulations were performed on a domain of 21rh x 2h x lrh. For the hybrid RANS-LES method, the near-wall cell dimensions were Ax + = Az + = 196 and Ay+ = 0.8, while the wall-normal cell height for the two-layer approach was Ay+ = 60, the other two dimensions remaining unchanged. Thus, Ax + the cells have an aspect ratio h-~z+ = 1, in accordance with a recommendation by Shur et al. (1999). For the highly-resolved LES computation (Ref LES), the dimensions of the near-wall cells were Ax + = Az + = 24.5 and Ay+ -- 1.5. The velocity profile given in Fig. 2 show fairly close agreement

363

between the two-layer and hybrid-scheme solutions, on the one hand, and the highly-resolved LES solution on the other. All profiles feature inflections, which are almost always observed with this type of approximations, and these signify insufficient turbulence activity (resolved + modelled) in the interface region. Of the solutions, those arising from the wall-function and the two-layer implementations are virtually identical. This is not surprising, as the latter is 'sublayer-resolving' version of the former, both operating within layers of identical thicknesses. The solution obtained with the one-equation hybrid scheme is furthest from the highly-resolved solution, but here the interface is at y+ = 113, relative to 'only' 60 for the two-layer scheme. However, the two-equation form gives an improved solution, which is quite close to the reference profile. The shear-stress plot shows, for the two-equation hybrid method, that the modelled proportion of the total shear stress rises rapidly towards the wall to a peak of almost 80% of the maximum. This is consonant with the objective of such a method, namely to delegate an increasing proportion of the turbulence activity to the statistical model as the wall is apnroachod. 2EQJ17 30

..... I

.

....... i

........

f

. ....... i

......

~,. 2EQJ17 20

[

ref. LES

,,~..~,.''

0.8

i ,t~-.,xr

two-lay. 1/~r'~" wall-f u n c ~ ~ , ~ . ~ hyb.d~NS~ES

U+I5

~0.6

~

0.4

0.2

5 0

'":: resolved! - - modelled I..... total i I - - ref. LES

A ->

I Interflw.eTwo.-Lay~and Wall

10

~'~-. .9~"~.,. .." ....... :.~:~.,, .:" '...'.. :~.

1

10

+ 100

1000

10000

Y

0'0

0.2

0.4

y/h

0.6

0.8

Figure 2: Velocity and shear-stress profiles for channel flow at R e - 42200; Ref. LES with 512 • 128 x 128 nodes, relative to 64 • 64 x 32 for the wall approximations 4

CONSTRICTED-CHANNEL

FLOW

The near-wall approximations are next applied to the case of a channel with periodically arranged hill-shaped constrictions along one wall, which cause massive separation at their leeward side. The computational domain is 9h • 3.036h • 4.5h, with h being the hill height, and extends from one hill crest to the next (see Fig. 3), with periodic conditions applied at both ends. The Reynolds number, based on the flow rate, is 21560. This value is a rather low, making this flow a less than ideal test case. However, there are very few alternative flow configurations at higher Reynolds numbers for which there are benchmark solutions. An extensive study of this flow, with particular emphasis on the sensitivity to wall functions and SGS models, is reported in Temmerman et al. (2003). The availability of extensive data from two highly-resolved simulations over a grid of about 5 million cells allows the accuracy of the present approximate methods to be assessed. This computation forms the first entry in Table 2. Five simulations with approximate near-wall methods are reported: three with the hybrid scheme, with different interface locations, one with the two-layer scheme and the fifth with the log-law wall function. In all cases, the WALE SGS model was used. All were performed with substantially coarser grids than that of the reference solution, but there are significant differences in respect of the wall-normal grid between the meshes for the hybrid and two-layer methods, because the former requires the grid to be wall-resolving in the wall-normal direction. The resolution of the grid in the streamwise direction is equivalent to that of the coarsest grid used in Temmerman et al. (2003) for the study of wall-functions. The spanwise resolution was chosen to keep the ratio A x + / A z + as close to unity as possible, and this thus gives a much coarser grid than the one used in the reference computation. The

364

configuration sketch in Fig. 3 contains dashed lines that indicate the physical location of the interfaces corresponding to j18, the 4th entry in Table 2. Fig. 3 also gives the distribution of the universal wall distance between the wall and the wall-nearest grid surface. As seen, typical y+ values are 10, 30, 50 and 100 for the interface locations j5, j9, j13 and j18, respectively. Fig. 3 demonstrates that the last (j18) extends almost to the centre of the recirculation zone. Table 2 gives results for the mean separation and reattachment locations, while Fig. 4 shows velocity and turbulence/SGS-viscosity profiles, both at x/h = 2 (roughly in the middle of the recirculation zone). The hybrid method gives the correct separation point, but predicts an excessively long recirculation zone, a defect also observed in earlier applications of the method in combination with a one-equation model in the near-wall layer ( Hadziabdi~ et al. (2003), Temmerman et al. (2004)). The log-law-based wall function returns a seriously delayed separation and premature reattachment, a result also reported in Temmerman et al. (2002). Finally, the two-layer method, here operating with Fi = ~Oxi ' also predicts late separation, but the reattachment location agrees fairly well with that of the fully-resolved simulation. Reference to the velocity profiles reveals that all calculations, but that with the wall function, agree reasonably well with the reference solution. Two especially encouraging feature are, first, that the two-layer model, which is an especially simple and economical implementation, gives a significantly superior representation to that of the wall-function, and second, that the results of the hybrid model are, essentially, insensitive to the location of the interface. As seen from the profiles of turbulence/SGS viscosity, increasing the thickness of the near-wall layer leads, as expected, to a steep rise in the turbulence activity represented by the statistical model; yet, the quality of the result obtained from the simulation does not deteriorate. TABLE 2 GRIDS, SEPARATION AND REATTACHMENT POINTS AND INTERFACE LOCATIONS FOR CONSTRICTED-CHANNEL SIMULATIONS

Case Ref.

5

Cells number

(x/h)s~p (x/h)r~at 4.72

interface location(j)

196 x 128 x 186

0.22

-

2EQ-j5

112 x 64 x 56

0.25

5.43

5

2EQ-j13

112 x 64 x 56

0.23

5.76

13

2EQ-j18

112 x 64 x 56

0.23

5.69

18

Two-layer Fi - o3_ Oxi

112 x 64 x 56

0.42

5.12

5

log-law WF

112 x 64 x 56

0.58

3.05

5

HYDROFOIL FLOW

This separated flow evolves along an asymmetric trailing edge of a model hydrofoil. The Reynolds number, based on free stream velocity U~ and the hydrofoil chord, is 2.15 x 106. The corresponding Reynolds number, based on hydrofoil thickness, is 101,000. Simulations were performed over the rear-most 38% of the hydrofoil chord. The flow had previously been investigated experimentally by Blake and numerically by Wang and Moin (2000). The computational domain is 0.5H x 41H x 16.5H, where H denotes the hydrofoil thickness. Table 3 lists the simulations performed. The present 'coarse-grid' results are compared to those obtained by Wang and Moin (2002) on a C-grid of 1536 x 96 x 48 nodes, claimed to be well-resolved. The distance, in wall units, between the wall-nearest LES-grid point and the wall, in the straight portion of the hydrofoil, is A, while

365

Figure 3: Left: Interface location for the constricted channel flow. Right: Distribution of y+ of the interface j5-j18. (see Table 2) .

Figure 4: Velocity and turbulent/SGS-viscosity profiles for periodically constricted channel flow at Re = 21560; Ref. LES with 196 x 128 x 186 nodes, relative to 112 x 64 x 56 for the RANS/LES hybrid and 112 x 64 x 56 for the two-layer approximations. the corresponding interface in the hybrid scheme is at y+ - 60. The inter-nodal distance in both streamwise and spanwise directions is Ax + = 120. The two-layer grid contains only one quarter of the number of nodes of that used for the reference simulation. Inflow boundary conditions were taken from Wang & Moin. These had been generated in two parts: first, an auxiliary RANS calculation was performed over the full hydrofoil, using the k - v 2 f turbulence model by Durbin (1995); the unsteady inflow data were then generated from two separate LES computations for flat-plate boundary layers at zero pressure gradient. Discrete time series of the three velocity components at an appropriate spanwise ( y - z) plane were saved. These data, appropriately interpolated, were fed into the inflow boundaries of the present simulations. The upper and lower boundaries are located at 41 hydrofoil thicknesses away from the wall, to minimize numerical blocking effects. At the downstream boundary, convective outflow conditions are applied. Results given in Figs. 5 and 6 relate mainly to the two-layer method. For the hybrid (2-equation) scheme, only skin-friction data are included because the results are not yet definitive. Fig. 5 gives wall-normal profiles of the mean-velocity magnitude, U = (U 2 + V2) 89 and of streamwise rms velocity along the hydrofoil (Ue is the boundary-layer-edge velocity). The streamwise locations are identified by 'B'-'F' in the inset in Figure 5. Agreement is good for all the simulations in the first two sections, which are located in the straight part of the hydrofoil, along which the flow is attached. Starting from section D, at x / H = -1.625, the wall-function and the twolayer method with Fi = 0. present velocity profiles that differ from the reference simulation and from the implementation with Fi = ~ In sections D E and F, the flow field is affected Oxi "

366

TABLE 3 G R I D S , MODELLING P R A C T I C E S AND INTERFACE LOCATIONS FOR HYDROFOIL SIMULATIONS

Case

Cells number

Ref LES.

SGS model

interface locat. (y+)

1536 • 48 • 96 (C grid)

Dynamic

-

Two-layer Fi = o02_ Oxi Two-layer Fi = 0.

512 • 128 • 24

Dynamic

40

512 • 128 • 24

Dynamic

40

2EQ

512 • 128 • 24

k-eq

60

log-law WF

512 • 128 • 24

Dynamic

40

by a strong pressure gradient that is not accounted for in the simplest version of the two-layer approach (Fi = 0.). As regards the rms value of the streamwise-velocity fluctuations, the wallfunction approach again gives essentially the same result as the the simplified wall model. The implementation with Fi = o_a Oxi predicts well the location of rsm peak in all sections, but the actual value is too high, perhaps reflecting the weakness of the quasi-steady assumption upon which the simplified form of Fi is based. Figure 6 gives profiles of the normalized mean-streamwise velocity in the wake of the hydrofoil. The peaky feature predicted in the first profile at the hydrofoil trailing edge ( x / H = 0.) is caused by the rather coarse mesh resolution in the wall-normal direction, a consequence of the use of the H-topology in which the single-cell near-wall layers merge in the wake region and extend to the exit plane. This could be circumvented with a C-topology grid, but a resolution penalty would then arise further downstream. Finally, Figure 6 gives the skin-friction coefficient, Cf, both on the upper and lower surfaces. All models but the hybrid scheme predict similar skin-friction levels on the lower surface (lower curves) along which the flow is attached. The tendency of the hybrid scheme to return too low values for the skin friction has also been observed in other applications. While Cf on the upper surface is also fairly well returned over most of the surface, the wall-function and two-layer methods fail to capture the sharp decline in the skin friction associated with the onset of the deceleration of the boundary layer prior to separation. In contrast, the hybrid scheme resolves this feature, albeit only qualitatively. Apart from modelling defects associated with the approximations, this difference reflects the fact that the substantially coarser grid used with the wall-function and two-layer methods at the upper curved surface does not allow the precise geometric variation of the surface, especially its curvature, to be resolved accurately by the coarse near-wall LES mesh. Nevertheless, all wall models may be claimed to give a credible representation of the skin friction, including that in the post-separation region. 6

CONCLUDING

REMARKS

No approximation of the near-wall region within a LES scheme that sacrifices full resolution will give an entirely satisfactory representation of this region, and this statement also applies to the methodologies investigated in the present paper. The overall objective can only be to achieve an acceptable compromise between economy and predictive realism. Of the two methods examined, the hybrid approach is more expensive, but provides a numerically consistent framework and gives a better resolution of the turbulent near-wall structure. Most encouraging is the fact that it yields results which are only weakly dependent on the location of the interface - within reasonable limits, of course. Relative to the earlier use of one-equation modelling in the near-wall layer, twoequation modelling does not seem to offer decisive benefits. Whatever model is adopted, small-scale (high-frequency) information is progressively lost as the near-wall is increased, and this needs to be compensated for by some form of spectral enrichment (see, for example, Hadziabdid et al.

367

,

..~

,fl

0.5

..~

~:~ 0.4 ~

/

0.3

] 0

Figure

0

5:

,

,

o.~.

o.,

:

,

0.5 0.4 t

'

~

o.3

J

. . . . 0.5 1

Mean

/

' ."~ 2 2.5

1.5

3

u/u~

3.5

magnitude

4

4.5

5

velocity

0

5.5

0

" 0.1

0.,

o.~

0.,

rms(u )/U~

0.,

o.~

(Left),

rms streamwise velocity (Right) at present results wall model Fi = o_z Oxi full L E S , - - - - - wall function,---- wall model Fi = O.

x/H = - 3 . 1 2 5 , - 2 . 1 2 5 - 1 . 6 2 5 , - 1 . 1 2 5 , - 0 . 6 2 5 ;

0.007 0.006 0.5

~

0.005 0.004

~

0.003

k

~'~

:

o

(

@

'

0.002 0.001 0

-0.5

,0.001 .0.002 ,0.003 0

1

2

3

U/U~

4

5

.0.004

-8

-7

-6

-5

-4

x/H

-3

-2

-1

0

Figure 6: (Left) Profiles of the normalized mean streamwise velocity in the wake at

x/H = 0, 0.5, 1.0, 2.0 and 4.0 and mean skin friction coefficient Cf (Right) present results wall model Fi = O_p_ . . . . full LES,----wall function,--wall model Fi = 0, = two equations Oxi ' Rans-LES, Cf only preliminary results.

(2003)). The principal attraction of the zonal two-layer strategy lies in its economy and simplicity. It is, essentially, a method for generating, numerically, a solution in the near-wall layer, which is an improvement on the analytically prescribed log-law-based wall function. In fact, the two are virtually equivalent for Fi=0 in equation (4). Inclusion of the pressure gradient in Fi is found to yield improvements, and the results presented for all three flows are clearly encouraging. On the other hand, the method does not allow thick near-wall layers to be prescribed without significant errors being provoked due to a serious deterioration in resolution. This also applies to wall-lawbased approaches. Finally, it is arguable that the quasi-steady implementation, in which Fi in equation (4) only contains the pressure gradient, is too simple. This may be appreciated upon noting that fluctuations in pressure gradient are balanced, at least to a significant extent, by inertial perturbations, even close to the wall. It thus follows that the transport terms need to be included in Fi, which brings about a significant complication of the algorithm, with attendant cost implications. This is the subject of ongoing efforts.

368 7

ACKNOWLEDGEMENT

This work was undertaken, in part, within the DESider project (Detached Eddy Simulation for Industrial Aerodynamics ), a collaboration between ALA, CFX, DASSAV, EADS- M, ECD,LML, NLR, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and NTS. The project is funded by the European Union and administrated by the CEC, Research Directorate-General, Growth Programme, under Contract No. AST3-CT-2003502842. REFERENCES

Balaras, E., Benocci, C., 1994. Subgrid-scale models in finite difference simulations of complex wall bounded flows. In: Applications of Direct and Large Eddy Simulation. AGARD CP551, 2-1 - 2-6. Balaras, E., Benocci, C., Piomelli, U., 1996. Two-layer approximate boundary conditions for largeeddy simulations. AIAA J. 34 (6), 1111-1119. Cabot, W., 1995. Large-eddy simulations with wall-models. Tech. Rep. Annual Research Briefs, Center for Turbulence Research, Stanford, USA. Durbin, P., 1995. Separated flow computations with the k-epsilon-v2 model. AIAA J. 33 (4), 659-664. Hadziabdid, M., Hanjalid, K., Temmerman, L., 2003. Merging LES and RANS strategies: zonal or seamless coupling? In: Friedrich, R., Geurts, J., M~tais, O. (Eds.), Direct and Large Eddy Simulations V. Kluwer Academic Press, 451-464. Lien, F., Leschziner, M., 1994. A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure. Part 1: Computational implementation. Comput. Methods Appl. Mech. Engrg 114, 123-148. Nicoud, F., Ducros, F., 1999. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turb. and Combust. 62 (3), 183-200. Shur, M., Spalart, P., Strelets, M., Travin, A., 1999. Detached-eddy simulation of an airfoil at high angle of attack. In: Rodi, W., Laurence, D. (Eds.), Engineering Turbulence Modelling and Experiments 4. Elsevier Science, 669-678. Temmerman, L., Hadziabdid, M., Leschziner, M., Hanjalid, K., 2004. A hybrid two-layer URANSLES approach for large eddy simulation at high Reynolds numbers. Int. J. Heat and Fluid Flow(in press). Temmerman, L., Leschziner, M., Hanjalid, K., 2002. A-priori studies of a near-wall RANS model within a hybrid LES/RANS scheme. In: Rodi, W., Fueyo, N. (Eds.), Engineering Turbulence Modelling and Experiments V. Elsevier, 317-326. Temmerman, L., Leschziner, M., Mellen, C. P., FrShlich, J., 2003. Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions. Int. J. Heat and Fluid Flow 24 (2), 157-180. Wang, M., Moin, P., 2000. Computation of trailing-edge flow and noise using large-eddy simulation. AIAA J. 38 (12), 2201-2209. Wang, M., Moin, P., 2002. Dynamic wall modelling for large-eddy simulation of complex turbulent flows. Phys. Fluids 14 (7), 2043-2051. Yoshizawa, W., Horiuti, K., 1985. A statistically derived subgrid scale kinetic energy model for the large eddy simulation of turbulent flows. J. Phys. Soc. Japan (54), 2834-2839.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

369

LES, T-RANS AND HYBRID SIMULATIONS OF THERMAL CONVECTION AT HIGH RA NUMBERS

S. Kenjere~ 1 and K. Hanjalid Department of Multi Scale Physics, Faculty of Applied Sciences, Deft University of Technology Lorentzweg 1, 2628 CJ Delft, The Netherlands

ABSTRACT The paper reports on application of different approaches to the simulations of thermal convection at high Rayleigh (Ra) numbers. Based on new well-resolved LES in 107_ 1012). For such applications, the direct and large-eddy simulation techniques, DNS and LES, become prohibitively expensive, with the upper limit for DNS being at present at Ra < 10 s and for LES at Ra < 10 l~ Complex flow domains pose additional challenge for DNS and LES, often requiting sophisticated grid design, clustering and local refinement. But most of all, the orientation of the imposed temperature gradient with respect to the gravitational vector can have a dramatic influence on the character of the 1Corresponding author. Tel." +31-15-278-3649; Fax" +31-15-278-1204. E-mail: [email protected]

370

underlying physics, posing different demands on simulations. A case in point is the thermal convection between two parallel infinite plates heated differentially: the horizontal orientation produces very different flow and turbulent structure from the vertical one even at the same Ra number. These two configurations have, therefore, been considered as "a must" for testing the generality of any statistical turbulence models. The horizontal configuration, corresponding to the classic Rayleigh-Bdnard convection, is especially challenging for simulation because of high demand for numerical resolution of the the wall boundary layers, which becomes progressively thinner with an increase in the Ra number. The application of the standard wall-functions with RANS or LES is known to produce very erroneous heat transfer coefficient (Nusselt numbers), leaving the integration up to the wall as the only choice for accurate predictions of wall phenomena. In addition to accurate predictions, industrial applications require a fast and numerically robust approach which will make parameterization studies possible. In this paper we report on development and validation of different modelling and simulation approaches for turbulent thermal convection. Because of grid resolution problem with conventional LES for high Ra and Re numbers and still large uncertainties in RANS for thermal convection, we focus on combining the best features of RANS (proper near-wall behaviour) and LES (proper spatial and time resolution of turbulence structures away from a solid wall). Because the avaliable DNS or LES results for thermal convection are scarce in the literature and limited (at present) to low values of Ra (_<2x 107 for DNS and <_10s for LES) we first performed a series of well-resolved LES in 107<_Ra_<10 ~ range. After that we investigated possibilities for merging RANS and LES methods by employing hybrid "seamless" approach. Comparative assessment of LES (both well-resolved and coarse), T-RANS and hybrid method is performed focusing on the wall-heat transfer and different subscale/subgrid contributions to second-moments.

EQUATIONS AND SUBSCALE/SUBGRID MODELS The conservation equations for momentum and energy for buoyancy-driven incompressible fluid flow (with Boussinesq approximation)can be written as :

D(U~)Dt- oxjO(O(Ui}u Oxj - 7-ij)

IO((P)--PREF)+/3gi((T)--TREF)p Oxi

D{T) 0 (O{T)) (q) D t = Ox j c~-~z j - 7-~J + --p%

(1)

(2)

where () stands for Reynolds (time or ensemble) averaged quantities in RANS and spatially filtered quantities in LES. The turbulent stress (~-~j) and heat flux (~-0~)represent the unresolved turbulence contributions - on subgrid or subscale levels for LES and T-RANS approach, respectively. In order to close the system of equations, these contributions must be modelled. In the present work, we adopted the well tested and tuned three-equations (k) - (e) - (02 ) ASM/AFM model for the T-RANS approach, Kenjereg and Hanjalid (1999,2002). For the subgrid contributions for LES we adopted the buoyancy-extended Smagorinsky (1983) model of Eidson (1985). The hybrid approach is based on the seamless coupling between the T-RANS (in the near-wall region) and LES (in the rest of the flow) by applying an approach based on the proposal of Dejoan and Schiestel (2001). A short overview of all models used is given next.

Subscale Model for T-RANS In order to properly capture the near-wall behaviour of turbulence quantities, the low-Reynolds (integration up to the wall) three-equation model is used, Kenjereg and Hanjalid (1999,2002). The turbulent

371

stress and heat flux are expressed via a "reduced-algebraic" truncation of their parent differential equations, while retaining all source terms:

2 (k>~j - ~ 7-~j= 5 (

+

Oxj

O(T)

+ Coil (g~-oj + gj~-o~)7-

Oz~ O(U~)

)

Toi -- --CO Tij-~x j + ~Toj OXj -[- f]flgi(02> -+- (s

(3)

T

(4)

The system is fully closed with three-additional equations for remaining turbulence quantities:

O(V,) D(k> = 79k - T i j ~ -- fl9irOi - (r Dt OXj D(e)Dt = D~ _ -T1 Cr

'

D(O e) (OT) = 790 -- 2Toj Dt ~xj

2(r

'

Ut = Cufu(k)T

Tij O
-Jr-S l

(5)

where T=(k)/(e) is the characteristic time-scale while fu=exp [ - 3 . 4 / ( 1 + Ret/50) 2] and L=I 0.3 exp (-Re2t) are low-Re damping functions. These damping functions are based on the local turbulence parameters instead of the local wall-distances which makes them more appropriate for complex geometries, and Ret is turbulent Reynolds number defined as Ret=(k)2/u(e>. Instead of solving an additional equation for the dissipation of the temperature variance (e0), the constant thermal-to-mechanical turbulence time scale ratio is assumed, i.e. R=T/To=(O 2) (e)/(k> (e0)=const. All coefficients of the subscale T-RANS model are given in Tab. 1. Table 1: SUBSCALE T-RANS MODEL COEFFICIENTS

Co

G~

G2

G

~

.

R

0.2

1.44

1.92

0.09

0.6

0.6

0.5

Subgrid-scale Model for LES The LES subgrid model is based on the simple eddy-viscosity Smagorinsky formulation with an extension that includes buoyancy production effects - as proposed by Eidson (1985):

Prt Ox----f ' (Skj

'

-Pr--~t

Prt Ox~5 (~kj

(6)

where C=0.01 and Prt=0.4 are the subgrid model coefficients, [S]=(2SijSij) U2 is the modulus of the strain rate (Sij= 1/2 (O(Ui) / Oxj + O(Uj )/Ox~) ), and A = ( A x A y A z ) 1/3=(A Vol) 1/3 is characteristic filter length. It is noted that this formulation implies the representation of the subgrid heat flux via the simple-gradient-diffusion-hypothesis (SGDH), i.e. Ta=--u SCs/PrtO(T)/Oxi.

Hybrid RANS/LES: "Seamless" Approach In the present work we adopted the "seamless" hybrid approach based on the work of Dejoan and Schiestel (2001). This method uses a single RANS model to provide the eddy viscosity for the unresolved/subscale turbulence in the complete flow domain. Here the RANS model is the above defined three-equations model and the only difference is in the redefinition of the coefficient C~2 in the equation for the dissipation of turbulence kinetic energy (c). Two variants are considered: Dejoan and Schiestel(2001) : C~2 = C~I +

O.48

LRANS ) n 1 +/3 LLES

New : G2 = G1 +

0.48

(7)

372

where LRANS:nZw and LLEs=(AVol)1/3 are the RANS and LES length scales - respectively, and/3 and n are empirical constants that need to be determined, Dejoan and Schiestel (2001). In order to further reduce the empirical input into the seamless approach and to make the method more suitable for the complex geometries, the parameter c~=max(1, LRANS/LLEs) is introduced with LRANS---f~(]r Eq. 7. NUMERICAL METHOD The discretized equation set is solved using a finite-volume Navier-Stokes solver for three-dimensional flows in structured non-orthogonal geometries. The Cartesian vectors and tensors components in the collocated grid arrangements are applied for all variables. The SIMPLE algorithm is used for coupling between the velocity and pressure fields. The second-order central difference scheme (CDS) is applied for the discretization of the diffusive terms in all equations and of convective terms in (U, V, W, T) equations (T-RANS, LES and HYBRID approach). The second-order linear upwind scheme (LUDS) is applied for convective terms in subscale turbulence equations, i.e. (k, c, 02). The time integration is performed by fully implicit second-order three-time-levels method which allows larger time steps to be used as compared to explicit time marching integration. The solver can be run in serial (single processor) or parallel mode utilizing the domain decomposition MPI directives. For the latter, the number of CPUs varied from 8 (for T-RANS, HYBRID and coarse-LES studies) to 64 (for fine-LES simulations) always resulting in an ideal speed-up on the SGI Origin 3800 platform. 2 RESULTS AND DISCUSSION As mentioned in the introduction, we performed first a series of well-resolved LES of thermal convection between two flat horizontal walls with aspect ratio (4:4:1) heated from below and cooled from above in the range of high Ra numbers (up to Ra=109) - all for Pr--0.71 - an extension in two orders of magnitude in Ra compared with (at the present) the highest Ra reached by the DNS, Kerr (1996), Kerr and Herring (2000). The buoyancy extended version of the Smagorinsky formulation is used for this purpose, Eidson (1985). The grid resolution was 2562 • 128 control volumes - with the thickness of the first control volume next to wall of AZw--2.5• 10 -4 and 10 -4, which should ensure proper resolving of the wall thermal boundary layers. The typical values of the non-dimensional time step was At=t V//39AT/H=5 • 10 -3 and 10 -3 for Ra-- 107 and 109, respectively. The simulations produced good agreement with the available experimental, DNS (for Ra=107) and T-RANS results - in terms of the integral heat transfer (Nusselt number), Fig. 1-above. This is not surprising since the numerical grid was fine enough to fully resolve all scales (both spatially and in time) at Ra--107 - producing practically a DNS. For Ra=109 we made additional grid clustering and we claim that we have a well-resolved LES up to the wall. The sufficiency of the spatial and temporal numerical resolution is illustrated in Fig. 2left showing the time evolution of the maximum value of the ratio between the turbulent and molecular viscosity. Obviously, the application of a conventional LES to higher values of Ra is seriously questioned. A simple analysis based on the ratio between the dissipation length scale and the distance between the walls shows that an immense grid resolution is needed for fully resolved simulations: e.g. for Ra=1014 and 1017 - DNS will require 3.5• 104 and 2.5• 105 grid points in the vertical direction, Kerr (1996). Adding to this the limits associated with the time step and the need to perform the time integration over at least a few convective time scales - one can conclude that at present (even when using massively parallel systems) the upper affordable limit of DNS is Ra~101~ In order to overcome this problem, a possibility of merging RANS and LES approaches is investigated next. The goal of this merging is to find a proper 2SARA Computing and Networking Services, Amsterdam,The Netherlands; www.sara.nl

373

Nu 102.

....

DNS.......

'

'

~ Kerr:(1996) O Groetzbach(1983) [ x W. . . . . . (1994)

I

........

'

[ I [

LES:

........

'

I

i O- -O Eidson (1985) I [~" "O Kimmei and D. . . . . dzki (2000) I I V Wong and Lilly (1994) I i 2D T-RANS: I [ * Kenjeres and Hanjalic (2000) I .. lO1 -

.,, y "*~

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''1

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_~1 .~.s.s~ ~,~'-"[ .-..~" 4

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[] T-RANS o present

.....

Chavanne et al. (1997)

i O LES - present

" i,,,I

. . . . . . .

'

Su

,I

,

,

,

,,,,,I

106

l0 s ' ......

I

........

I

. . . . . . . .

107 ........

I

........

I

. . . . . . . .

I

........

I

........

- - " Expt. Fleischer & Goldstein (2002) I ] ExpL Niemela et al. (2000) [ I .... Expt. Ch . . . . . . . t a1.(1997) I I . . . . Expt. Chavanne et al. (2001)-ultimate [

.

109 I

........

I --

104

I

l0 s I

......

,

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/ ~ / ~/ /

/

~,~j.] / " / ~,." ,_ -~

103

102'",

109

,, ..... I

, .,,,,,,I

10 TM

1011

, , .... .I

1012

........

I

1013

........

I

1014

........

I

........

10 TM

I

......

1016

,,q

1017

Ra

Figure 1" Comparison of the computed Nu(Ra) results with several DNS, LES and experimental correlations for a turbulent thermal convection of air over flat horizontal walls: Above- in intermediate range of Ra (6.5• 105 <_Ra<_2x109); Bellow- for high Ra range including ultra-turbulent regime (2x 10 l~<_Ra<_2• 1016). method which will make possible to provide accurate heat transfer prediction in complex geometries while employing affordable - moderately dense (RANS-type) - numerical grids. In our previous works, Kenjere~ and Hanjali6 (1999,2002), Hanjali6 and Kenjere~ (2001), the potential of the transient-RANS (T-RANS) approach for accurate prediction of heat transfer for similar configuration has been reported. Here, the large eddy structures are fully resolved in time and space while the single-point RANS three-equation (k) - (e) - (02) model is used for the parameterization of unresolved (subscale) turbulence. Excellent agreement with experimental and DNS simulations of other authors have been obtained. In addition to accurate predictions of integral and local heat transfer, the first and second moments of turbulence quantities, very realistic representations of the unsteady behaviour of large convective structures have been obtained. In the present study, we extended the upper limit of Ra even further- up to 2• 1016 observing clearly a tendency towards the Kraichnan (1962) ultimate regime for R a > 1013, Fig. 1-below. This regime is characterized by a change in slope of integral heat transfer, i.e. Nuc~Ra 1/2 as experimentally observed by Chavanne et al. (2001). It should be mentioned that even for the largest simulated values of Ra, a relatively modest grid was used with 1283 CVs, though with a strong clustering in the near-wall region. For these large values of Ra, larger geometrical domains (8:8:1) are simulated. The key of success for accurate predictions lies in the anisotropic subscale representation of the second-moments which are dominant contributions in the near-wall region.

374

6

'

I

'

I

'

I

'

I

'

100

L

Z i

'

I

'

i

'

i

'

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I

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'-i

Expt. Niemela et al. (2000)

l

Ra=109

80

:=

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.........................

"

|

3

60 f

so 1

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Ra=lO 7

..... -----.... -----

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I

~

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610

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!

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65

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70

/

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i

,

130

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,

140

,

150

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T i

i

160

~

170

i

180

;

~9iO '

Figure 2: Left- Ratio between turbulent and molecular viscosity for different values of/~a (107,109) - the wellresolved 2569. x 128 LES" Right- Time evolution of overall Nusselt number at horizontal walls - comparison between experimental, well-resolved and coarse LES, T-RANS and HYBRID approach (with different values of and n as well as with the a parameter). 2 "x" ~ . ' ~ . \

"%, ", l

0.975

'

t

I---

"

i

I 0.1

a

I

0.2

~

I

0.3

I

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0.4

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~'.

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~

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1.6

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~ 0.925

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HYBRID: 822 x 72 TRANS." 822x721

I

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376

Figure 7: Trajectories of massless particles of an instantaneous velocity field portraying the resolved flow structures in the central vertical plane: Above- well-resolved LES; Middle- HYBRID; Below: T-RANS approach; Ra=109; please note that the T-RANS and HYBRID results are shown as a zoom-in of the 8:8:1 aspect ratio geometry.

Figure 8: Trajectories of massless particles of an instantaneous velocity field portraying the resolved flow structures in the central horizontal plane (z/H=0.5 -above) and inside thermal boundary layer (z/H=lO -3 -below) Left- well-resolved LES; Middle- HYBRID; Right- T-RANS approach; Ra= 10 9. hybrid "seamless" approach of Dejoan and Schiestel (2001) seems attractive since the above presented three-equations (k)-(e)-(O 2) ASM/AFM model can be used to provide the eddy-viscosity for subscale turbulence over the complete flow domain. The major advantage of this method in comparison with the "zonal" hybrid strategy lies in its simplicity - it requires redefinition of only one empirical coefficient

377

(C~2). This coefficient is now expressed as a function of the ratio of the characteristic RANS and LES length-scales. For LRANS/LLEs _< 1 range the RANS subscale model dominates and conventional TRANS numerical method is active. For LRANS/LLEs > 1, a reduction in C~2 (towards C~1) causes an increase in (c) which in turn reduces the turbulent viscosity. Finally, for C~2 - C~1, the DNS limit is reached. Additional appealing features are that this method does not require 'a priori' specification of the interface location and it does not exhibit discontinuities, as reported in hybrid methods with a demarcation (interface) between the RANS and LES regions. In order to perform comparative assessment between different simulation approaches, the Ra = 109 case is selected. This value of Ra is large enough to be representative of highly turbulent thermal convection. The well-resolved LES results (on 2562 • 128 grid) are used as a reference for comparison. Simulations employing T-RANS, Hybrid and coarse-LES approach are run in parallel - but on a significantly coarser mesh with 822• CV, which represents just 5% of the total grid cells employed for fine LES. As a starting point in our analysis we recall that LES on this coarse mesh resulted in an underprediction of the integral heat transfer of 50%, Fig. 2-right. Our first goal is to reduce this enormous discrepancy to levels acceptable by industrial standards. The T-RANS approach resulted in excellent agreement with both experimental data of Niemela et al. (2000) and the fine-LES results. The long-term averaged vertical temperature profiles are shown in Fig. 3-1eft. It can be seen that the coarse LES shows huge discrepancy in slope compared to fine-LES and T-RANS profiles. Different sets of parameters/3 and n for HYBRID approach have been tested in order to bring heat transfer prediction in good agreement with the fine-LES, Fig. 3-right. The strong damping of C~2 coefficient in the RANS (LRANS/LLES<_I -/3=4, n = 2 / 3 ) or buffer region (I
378

CONCLUSIONS A comparative assessment of different strategies in performing simulations of thermal convection in turbulent regime is presented. In order to improve the potential of the conventional LES for accurate predictions of the near-wall heat transfer on moderately dense (RANS-type) grids, different variants of hybrid RANS/LES merging have been tested. It is demonstrated that applications of T-RANS approach resulted in good agreement with the available DNS and fine-LES (for low R a ' s ) and experimental results over a range of Ra. The key of success lies in modelling of the subscale turbulence contributions by lowR e three-equation (k) - (e) - (02) model which provides correct behaviour of turbulence variables in the near-wall region. In order to make T-RANS approach capable of capturing the low-intensity flow instabilities - such as found in side-heated configurations - a simple redefinition of the modelled coefficient C~2 (as a function of the RANS and LES turbulence length-scales ratio) is proposed. This "seamless" hybrid approach was then successfully applied to simulations of thermal convection at Ra=109. Results show significant improvements in capturing of the smaller flow structures and slight improvements regarding the subscale turbulence variables in the buffer region. Especially encouraging are the results obtained with the wall-distance-free variant of the model based on the oz parameter which has a potential especially for simulating flows in complex geometries. ACKNOWLEDGEMENTS The research of S. K. has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW).

REFERENCES Chavanne X., Chilla E, Chabaud B., Castaing B. and Hebral B. (2001). Turbulent Rayleigh-B6nard convection in gaseous and liquid He. Phys. Fluids, 13:5, 1300-1320. Dejoan A. and Schiestel R. (2001). Large-eddy simulations of non-equilibrium pulsed turbulent flow using transport equations subgrid scale model, in. E. Lindborg et al. (eds.) Turbulence and Shear Flow Phenomena 2 (Proc. Int. Symp., Stockholm, Sweden, 27-29 June 2001) 2, 341-346. Eidson T. M. (1985). Numerical simulation of the turbulent Rayleigh-B6nard problem using subgrid model. J. Fluid Mech. 158, 245-268. Hanjali6 K., Had~iabdi6 M., Temmerman L. and Leschziner M. (2004). Merging LES and RANS strategies: zonal or seamless coupling? in. R. Friedrich et al. (eds.) Direct and Large Eddy Simulations V, Kluwer Academic Publishers, 451-464. Hanjali6 K. and Kenjere~ S. (2001). T-RANS Simulation of Deterministic Eddy Structure in Flows Driven by Thermal Buoyancy and Lorentz Force. Flow, Turbulence and Combustion, 66, 427-451. Kenjere~ S. and Hanjali6 K. (1999). Transient analysis of Rayleigh-B6nard convection with a RANS model. Int. J. Heat and Fluid Flow 20:3, 329-340. Kenjereg S. and Hanjali6 K. (2002). Numerical insight into flow structure in ultraturbulent thermal convection. Phys. Rev. E 66:3, Art.No.036307, Part 2B, 1-5. Kerr R. M. (1996). Rayleigh number scaling in numerical convection. J. Fluid Mech., 310, 139-179. Kerr R. M. and Herring J. R. (2000). Prandtl number dependence of Nusselt number in direct numerical simulations. J. Fluid Mech., 419, 325-344. Kraichnan R. H. (1962). Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids, 5, 13741389. Niemela J. J., Skrbek L., Sreenivasan K.R. and Donnelly R.J. (2000) Turbulent convection at very high Rayleigh numbers. Nature, 404:6780, 837-840. Smagorinsky J. (1983). The beginnings of numerical weather prediction and general circulation modeling: early recollections. Adv. Geophys. 25, 3-23. Speziale C. G. (1998). Turbulence Modelling for Time-Dependent RANS and VLES: A review". AIAA J. 36:2, 173-184.

5. Application of Turbulence Models

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Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

381

INDUSTRIAL PRACTICE IN TURBULENCE MODELLING AN EVALUATION OF QNET-CFD A. G. Hutton QinetiQ Ltd., Farnborough, GU14 0LX, UK

ABSTRACT From an examination of the challenges encountered in the industrial practice of CFD, it is argued that RANS-based turbulence modelling, offering any degree of reliability across the spectrum of applications, is inevitably a knowledge-based rather than predictive discipline. A large body of relevant knowledge has been generated across industrial sectors, largely as the outcome of numerous validation exercises. The collaborative project, QNET-CFD set out to pull together this somewhat dispersed resource, and process and structure it into a form from which well founded best practice advice can be derived. Following a brief explanation of the methods and constructs developed within the project, the value and potential of the approach is explored and the key lessons of importance to its future evolution are discussed.

KEYWORDS QNET-CFD, computational fluid dynamics, turbulence models, best practice, quality, validation, knowledge base

THE CHALLENGES FOR INDUSTRIAL PRACTICE Computational Fluid Dynamics (CFD) is now widely used across all sectors of industry and its uptake is growing rapidly. It is emerging as an important and for, many applications, the primary tool for carrying out design and assessment studies. However, experience reveals that, in the industrial context at least, it is a difficult technology to apply successfully. The practitioner must be highly skilled and knowledgeable in order to set up and/or select model inputs appropriately, and then to be in a position to interpret the results reliably against design or assessment requirements. A degree of uncertainty tends to prevail even in the hands of an expert. With few notable exceptions, industrial flow regimes are turbulent and it is the selection (if not the construction) and deployment of a suitably competent, practicable model of turbulence which perhaps poses the greatest challenge for industrial practice. It is important to emphasise that time and costs are always key considerations. A set of design studies must take no longer than one or two days (for example, in the field of aeronautics the aim is calculate a complete wing lift-drag polar overnight) and the computer resources available to most design or assessment engineers are

382 relatively modest. Consequently, with one or two exceptions, closure at the level of large eddy simulation (LES) or even hybrid RANS/LES methods is not considered practicable at the present time. RANS-based turbulence modelling will remain the industrial workhorse for the foreseeable future and this is the source of a major conundrum. Despite many years of research, a generic model which is competent across a wide class of industrial applications has yet to emerge. Indeed over one hundred RANS turbulence models or model variants have been published in the literature. Fresh ones appear on a regular basis which try out new ideas or which fix inadequacies discovered in predecessors and these are then validated over a relatively narrow range of flows. Confusion tends to reign in industry. Which of these models is best for a given application in terms of competency, robustness and affordability? How far can they be pushed outside their range of validation? How should other modelling choices be made (e.g. mesh resolution, boundary condition set-up etc.) so as to ensure that competence, which may well have been demonstrated under entirely separate conditions, is not undermined by the impact other sources of error on model performance. In the face of these uncertainties, k-e with wall functions is often selected on the basis that, historically, it is the most used and thus best understood (the safety in numbers option).

QNET-CFD; A WAY F O R W A R D At the Expert Programme on Turbulence organised by the Isaac Newton Institute, Cambridge UK (Hewitt and Vassilicos (2004)) it was concluded that there is no universal state of turbulence and therefore no universal RANS closure model. The implications of this statement for industrial practice are indeed considerable. It would seem that a predictive RANS-based genetic theory based on fundamental principles is not possible. Instead, turbulence models must be customised and calibrated to capture the essential flow physics governing the important design or assessment parameters associated with a given or possibly narrow class of application. Knowledge and understanding of such performance is not, in general, transferable to other applications. In short, trustworthy turbulence modelling in industrial practice is inevitably a knowledge-based discipline relying on two key elements, a) the assembly of a body of knowledge of sufficient scope and quality and b) an effective mechanism for structuring and collating this body of knowledge into guidelines and procedures for the effective deployment and interpretation of turbulence models. The first of these elements relies strongly on a well founded validation methodology for underpinning quality. This is a far from trivial task. In recent years, many validation exercises have been organised in which a number of computations, using a range of turbulence models have been compared against industrial test data. These frequently prove inconclusive and often generate more questions than answers (e.g. different sets of calculations using the same turbulence model and often the same code seldom agree with one another let alone the measurements, Hutton and Casey (2001)). Clearly, validation is an important and topical issue (Oberkampf et al. (2002)). The AIAA Committee for Standards in CFD have invested and continue to invest much thought and effort into building a rigorous validation methodology founded on fundamental notions of verification and validation (AIAA (1998)). This is a very demanding exercise which can be expected to exert a strong impact on validation practice as it matures in the future. However, in order to address what is clearly a pressing need, the QNET-CFD consortium adopted a more pragmatic approach which sought to put both the above elements in place based on the current state of knowledge and validation practice, but in a flexible way which could accommodate developments as this state evolves.

383

The QNET-CFD Project (www.qnet-cfd.net)

Over the past decade or so, an enormous body of knowledge has been generated across all industrial sectors, largely as an outcome of numerous validation comparison exercises. However, this body is somewhat dispersed, is relatively underused and tends not to migrate across sectors. QNET-CFD set out to pull together this important resource, critically examine and filter it, and then structure it into a form from which well founded best practice advice could be derived. The project ran for four years from 2000 to 2004 and was fully funded by the EU as a Thematic Network. It addressed all aspects of quality and trust in the industrial application of CFD (i.e. numerics, meshing, boundary conditions etc.), since these are intrinsically intertwined with perceived turbulence model performance (a primary cause of the difficulties in validation practice). The network comprised 43 members drawn from industry, government research establishments, academia and the principle code vendors. Activities were organised across six so-called thematic areas (TAs) each broadly aligned with one or more industrial sector. These were External Aerodynamics (TA1); Combustion and Heat Transfer (TA2); Chemical and Process, Thermal Hydraulics and Nuclear Safety (TA3); Civil Construction and HVAC (TA4); Environment (TA5); and Turbomachinery Internal Flows (TA6). Each TA pursued the same programme in parallel which was hierarchically structured around the notions of an application challenge (AC) and an underlying flow regime (UFR). An application challenge is a test case of industrial complexity by which the competency of CFD is judged within a given sector (e.g. wind tunnel flow over a civil aircraft, dispersion around a building, mixing in the lower plenum of a PWR etc.). As a minimum, the engineering parameters which are of key importance to the design or assessment engineer (e.g. lift and drag, concentrations, entrainment coefficients etc.) must have been measured. CFD computations of sufficient quality must be available. An underlying flow regime is a generic flow configuration or process which captures a key element of the fluid physics associated with an AC (e.g. shock-boundary layer interaction, bluff-body wake, jet impingement, buoyancy influenced jet mixing etc.). The set of UFRs connected to a given AC represent the important mechanisms which control the fluid dynamic behaviour of the AC. It is to be expected that detailed and comprehensive measurements of both the velocity field and turbulence variables are available for UFRs thus facilitating insight and understanding of turbulence model behaviour. The basic idea is that UFRs are well studied under controlled conditions and this is sufficient to derive firmly founded advice which can then be lifted into best practice for the associated AC. Each AC and UFR was documented according to a specified format and then tested and filtered against rigorous project quality procedures to ensure fitness-for-purpose. From this basis, best practice advice was formulated for each UFR and AC, taking care that this was fully supported by the documented evidence. Where feasible, the UFR best practice advice was integrated into that set out for the associated ACs. The outcome of this process is illustrated by the following example which is the best practice advice for the shock-boundary layer interaction UFR, based upon the axisymmetric bump test data of Bachalo and Johnson (1986).

Numerical Modelling Issues

9 Discretisation Method USE a high order accurate scheme, at least second order accurate in space, with as little numerical dissipation as possible 9 Grids and grid resolution

384 For low Reynolds number turbulence models, USE a mesh that has wall adjacent cell heights of y§ < 1. For a wall function approach, USE a mesh with wall adjacent cell heights in the range 50
Physical modeling Turbulence modelling Do NOT USE standard linear models such as k-e or k-t0. These typically result in a delayed shock position with weak or non-existent flow separation. If a linear model is to be used, USE the Menter SST (Shear Stress Transport) model. However, note that this model tends to predict a slightly earlier separation. If a better prediction of shock location, pressure plateau and separation location is required, USE a cubic non-linear model such as the cubic k-e model of Suga or that of Apsley and Leschziner. The Speziale variant performs least well in capturing the separation location and pressure plateau and should not be used. Transition modelling No transition modelling has been applied as part of this UFR.

Application Uncertainties The advice given here is not applicable to laminar boundary layers undergoing transition due to shock wave interaction. 9 The advice is limited to transonic flows. None of the models considered perform particularly well in predicting velocity profiles in the flow recovery region. The complete output of QNET-CFD comprises 44 application challenges and 42 underlying flow regimes each of which has been fully documented and quality reviewed. This considerable body of knowledge and advice has been assembled into a Knowledge Base (KB) which can be accessed and navigated by means of an internet browser. The hierarchical structure of the KB is set out in Figure 1. Whilst it is possible to go to directly to any part at any level, it is envisaged that most users will enter through that TA which is identified with his/her sector and marshal relevant advice and guidance via the application challenge which is most closely associated with the problem under

385

Ir

Figure 1: Structure of the QNET-CFD Knowledge Base analysis. The top level content of the KB together with its look and feel can be explored by locating http://eddie.mech.surrey.ac.uk on the web.

AN EVALUATION OF QNET-CFD QNET-CFD has pioneered a new, highly pragmatic approach to turbulence modelling in industrial practice, one which is based upon the assembly and collation of knowledge and experience into a form which guides model selection and the interpretation of performance against design and assessment requirements. The value and practicality of this paradigm must be judged in the light of operational experience. Early use amongst the network membership is very encouraging in this regard. A key metric in the longer term, however, will be whether or not the KB is deemed worthy of investment on the part of the CFD community to consolidate and further expand its content (the KB will be launched shortly into the public domain under the stewardship of ERCOFTAC). Like all pioneering steps, the concept will take root and flourish if the underlying principles are sound, but it can be expected evolve as it is improved and refined along the way. At a minimum level, QNET-CFD has succeeded in pulling together a dispersed and fragmented resource, generated by many man years of CFD testing and evaluation across a variety of application areas. In this way, the knowledge and insight of a relatively small, disparate collection of experts has been transferred to a large population of users. This in itself can be expected to exert a significant influence on the quality with which industrial CFD is practised in the future. As the project unfolded, a number of valuable lessons were learned regarding the validation of CFD methods against existing test data and the extraction of well-founded advice. Many of these are recorded in comprehensive state-of-the-art reviews developed for each Thematic Area, which can be found in the KB. Space limitations dictate that only the more important genetic issues can be discussed here.

386 A great deal of thought and effort was invested in the development of procedures for checking the quality of experimental test data and CFD simulations in the context of both ACs and UFRs. The aim was to ensure fitness for purpose without being over-restrictive. In the event, for many of the studies examined it was not possible achieve compliance with all quality criteria. This was often because of insufficient documentation (e.g. details of the numerical discretisation scheme or experimental error estimates unrecorded) but also in some cases due to poor practice. (It should be stated that many of the data and calculations in question were published several years previously and did not always conform to modem good practice or indeed were not generated specifically for validation purposes). Thus, rather than acting as an accept/reject gateway, the quality review procedures were designed to assign a 'quality status' with recommendations on how this status could be improved. The AC or UFR was rejected if this status fell below a certain threshold. Researchers and practitioners who are embarking upon an experimental test or a CFD calculation for validation purposes should be encouraged to consult these quality checklists during the planning and reporting stages in order to strive for an outturn with high quality status. Knowledge and capability in the field of industrial CFD is changing rapidly. For example the TA6 Coordinator reported that, during the course of the project, the typical grid size for a single blade row simulation for the design of a new turbomachinery component in his company increased by nearly an order of magnitude. From past experience, greater resolution is likely to effect the interpretation of turbulence model performance. Thus rather than providing a single snapshot, the KB must become a living resource which is constantly re-generated as the state of the art evolves. It transpires that for many application challenges, important design or assessment parameters (e.g. wing or blade loading, pressure recovery coefficient etc.) are fairly insensitive to turbulence model details, provided other modelling aspects are sensibly addressed. These so-called first order parameters are often well predicted even when the details of the turbulence are not. Of course the situation can be expected to change if the AC is operating off-design and model sensitive effects such as separation and re- attachment then become important. This observation may be viewed as common knowledge by some of the more experienced practitioners, but it is nonetheless important to provide advice and guidance on situations which are comparatively robust and straightforward. Application uncertainties arise from the fact that an AC (or less frequently, an UFR) cannot be fully specified for CFD purposes. For example, the fine details of the geometry such as notches or leakage gaps etc.; or aspects of inflow conditions, such as flow angle or swirl etc. may not be known with sufficient precision. In many cases these are of only secondary importance. However, occasionally the key design or assessment parameters prove particularly sensitive to such uncertainties to the extent that their influence on reliability is on a par with that of model choice. Under such circumstances, there is good reason to reject the AC (or UFR) on quality grounds. However, such situations may well arise in practice, and so cannot be ignored within the general framework of guidelines and advice.

CONCLUDING REMARKS The QNET-CFD project has conceived and pioneered an entirely new approach to the assembly of fluid' dynamics knowledge and its delivery in a structured form, directly underpinning CFD practice across a range of application areas. In particular, it provides an effective mechanism for matching turbulence model selection to application demands as well as the interpretation of performance against design and assessment requirements. In the absence of a genetic, predictive theory for closing the RANS equations, a knowledge-based approach such as this would seem to be necessary. The potential of the approach as the basis of a valuable industrial tool must be judged in the light of critical experience. As it was developed and refined within the project, the fundamental importance

387 of the validation process to the quality of the emergent best practice advice became apparent. In order to strengthen this aspect it is suggested that practitioners embarking on a test or calculation for validation purposes should be guided by the QNET-CFD quality review checklists during the planning and reporting stages. Clearly, the knowledge base is currently in the early stages of its life cycle. It must aspire to become a living resource which grows strengthens and adapts with the evolution in knowledge, capability and validation practice, a goal to which ERCOFTAC is fully committed.

REFERENCES AIAA (1998). Guide for the Verification and validation of Computational Fluid Dynamics Simulations. AIAA Report AIAA-G_077-1998 Bachalo W.D. and Johnson D.A. (1986). Transonic Turbulent Boundary Layer Separation Generated on an Axi-symmetric Flow Model. AIAA Journal 24, 437 Hewitt G.F. and Vassilicos J.C. (2004). Prediction of Turbulent Flows, Cambridge University Press, UK Hutton A.G. and Casey M.V. (2001). Quality and Trust in Industrial C F D - A European Initiative. AIAA Paper 2001-0656 Oberkampf W.L., Trucano T.G. and Hirsch C. (2002). Verification, Validation and Predictive Capability in Computational Engineering and Physics. Proc. of Foundations '02 - A Workshop on Model and Simulation Verification and Validation for the 21 st Century, ed. Dale K. Pace, John Hopkins University, Maryland, USA

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

389

THREE-DIMENSIONAL FLOW COMPUTATION WITH REYNOLDS STRESS AND ALGEBRAIC STRESS MODELS G.B. Deng 1, P. Queutey & M. Visonneau Laboratoire de M~canique des Fluides, UMR 6598, Ecole Centrale de Nantes 1 Rue de la Noi~, B.P. 92101, 44321 Nantes Cedex 3, France

ABSTRACT A quadratic explicit algebraic stress model (EASM) that takes into account the variation of production-to-dissipation rate ratio is compared with an implicit algebraic stress model (ASM) and with their parent Reynolds stress model (RSM) in this paper. A new implementation of ASM model where the turbulent eddy viscosity provided by the explicit solution is employed is found to be robust. Computations for ship flows at model and full scale are performed to assess the accuracy of different models. Explicit and implicit algebraic stress models give similar prediction for the flow investigated. The RSM model provides better prediction in the region dominated by convex curvature. However, no much improvement is observed near the concave surface. KEYWORDS

Reynolds stress model, implicit and explicit algebraic stress model, bilge vortex, curvature. INTRODUCTION Due to the Boussinesq assumption, linear eddy viscosity model often fails to give satisfactory prediction for complex three-dimensional flow. On the other hand, because of its numerical instability, Reynolds stress model is not frequently used in industrial applications. Although it is capable to overcome the inherent shortcoming of a linear eddy viscosity model, its superior accuracy is not yet demonstrated for a wide range of industrial applications, especially those involving flow separation. Industrial applications appeal for turbulence models going beyond the Boussinesq assumption while maintaining numerical robustness in term of CPU time and stability as with a linear eddy viscosity model. To meet this end, alternatives have been proposed. Implicit and explicit algebraic stress models deduced from Reynolds stress model are among the most promising. Algebraic stress model (ASM) was first proposed by Rodi[15]. It is deduced from the Reynolds stress transport model using a weak equilibrium assumption. The ASM model is not widely 1Correspondence author ([email protected])

390

used because of insufficient robustness and singular behavior that occasionally occurs. Pope [14] has proposed an explicit solution to the algebraic stress equation for two-dimensional flow. This methodology got a renewed attention almost 20 years later due to the works of Gatski & Speziale [4] who extended this approach to three-dimensional flow. They employed the term explicit algebraic stress model (EASM) for this methodology. Since then, the EASM model has been applied to a wide range of flows. Progresses have been made continuously which improve the predictability of the model. To differ from the EASM model, the ASM model proposed by Rodi is sometimes referred as implicit algebraic stress model. The first important improvement in the development of the EASM model concerns the treatment of a nonlinear term due to the production-to-dissipation rate ratio appearing in the algebraic stress equation. Gatski & Speziale [4] assumed a constant value for this ratio determined from a weak equilibrium assumption. Resulting formulation becomes singular when flow is far from equilibrium. A regularization is applied to avoid the singularity. Consequently, the EASM model thus obtained may differ considerably from the original ASM model when flow is far from equilibrium and gives an erroneous asymptotic behaviors for large strain rates as noticed by Speziale and Xu [19]. Girimaji [5] and Johansson & Wallin [8] discovered independently that the variation of the production-to-dissipation rate ratio can be taken into account in an EASM model by solving a nonlinear equation for a scalar quantity. For two-dimensional flow, this nonlinear equation is of cubic order. As the roots of a cubic equation are multiple, care must be made to avoid unphysical root. Based on an asymptotic analysis, Jongen and Gatski [10] have shown that the correct choice is the root with the lowest real part. For three-dimensional flow, the order of the nonlinear equation is of sixth order [21], which is too complex to be applied in real applications. The second progress concerns the tensor bases employed to represent the explicit solution. The most widely used EASM model is based on a three-term tensor bases proposed by Gatski and Speziale [4]. This quadratic model is the exact solution of the algebraic stress equation for two-dimensional flow, but it is only an approximation for three-dimensional flow. For complex three-dimensional flow, such an approximation may become insufficient. Exact explicit solution for the algebraic stress equation for three-dimensional flow has also been provided by Gatski and Speziale [4]. But it relies on a ten-term tensor bases which is too complex to be applied in numerical computation due to high computational cost. Wallin and Johansson [21] provide an exact solution to the algebraic stress equation for three-dimensional flow that contains only five tensor terms for a particular version of the LRR pressure-strain rate model. The approach employed by Gatski et al. and Wallin et al. relies on the exact solution of the algebraic stress equation. Jongen and Gatski [9] proposed a projection method that allowed to obtain an approximated solution based on a five-term tensor bases to the algebraic stress equation for three-dimensional flow. In spite of those achievements, EASM models based on higher order tensor bases are not widely used, possibly because of the solution of the associated nonlinear scalar equation used to determine the production-to-dissipation rate ratio is too complex. Unlike EASM model, implicit ASM model does not suffer from the above mentioned shortcomings. Exact solution to the algebraic stress equation can be computed numerically. The variation of production-to-dissipation rate ratio can be easily taken into account in an iterative procedure. However, early implementation of the ASM model suffered usually from numerical instabilities. During the present study, we found that implicit ASM model can be implemented as robust as EASM model when the turbulent eddy viscosity provided by the explicit solution is employed in the implementation. The last but probably the most important improvement made in the development of EASM model which can be applied to implicit ASM model as well concerns the weak-equilibrium assumption. Girimaji [6] showed that instead of applying the weak-equilibrium assumption in the coordinate system where computation is performed, the curvature effect represented by the convection of the Reynolds stress tensor can be taken into account in an EASM model when the

391

above mentioned assumption is applied in a flow oriented orthogonal coordinate system. This is the so called curvature correction approach in algebraic stress model. Although the formulation is well established, one open question remains concerning the choice of the orthogonal coordinate system. Solutions proposed so far either apply only to 2D flow [16] or suffer from numerical stability problem [20][22]. To author's knowledge, no viable solution exists at the present time for 3D flow. As the main interest of the present paper concerns three-dimensional flow, curvature correction will not be considered further in the following. The objective of the present paper is to compare the performance of explicit and implicit algebraic stress model with their parent Reynolds stress transport model for three-dimensional flow computation. The first model selected for the present study is a quadratic explicit algebraic stress model detailed in Rumsey et al. [16] that takes into account the production-to-dissipation rate ratio. To evaluate the reliability of this model which is only an approximation for twodimensional flow, numerical resolution to the original algebraic stress equation is performed. This is the so-called ASM model. Both models are compared with their parent Reynolds stress transport model refereed as RSM model. All of them are based on the same pressure-strain rate correlation model, namely the quasi-linear SSG model [18] and adopt a near wall low Reynolds number formulation. In order to provide a useful assessment for industrial relevant applications, it is desirable to choose three-dimensional test cases for which detailed experimental data are available and Reynolds stress transport model has been proven to give much better prediction than linear eddy viscosity model. Flow around ship is one of the few test cases that fulfillthese requirements. Detailed experimental studies and extensive numerical investigations at model scale reveal that the prediction of ship after-body flow is a challenging task due to the interaction of the bilge vortex with the boundary layer that is strongly influenced by the effect of convex and concave curvature. Linear eddy-viscosity model fails to give satisfactory prediction, while better result can be obtained with Reynolds stress model [1] [II]. Computation with different turbulence models will be performed at model scale for an oil carrier called the KVLCC2. Results obtained will be compared with detailed experimental data available [12]. Numerical simulation will also be performed for a research vessel called the Nawigator XXI at full scale, one of the configurations chosen in an European cooperative project known as EFFORT (European Full-scale Flow Research and Technology) carried out during the period from 2002 to 2005 in which the authors participate.

Turbulence Models The Reynolds stress transport equations can be written as:

DT~j = p~j + r

Dt Where "riy- uiuj is the Reynolds stress tensor, P,j = -

P/j

_ c~j + D~j

(1)

is the production terms given by:

OUi OUj~ 4 Tyk--~Xk + "r,k--~Xk] = - 2 k (bikSkj + S, kbky) + 2k (bikWkj - Wikbkj) -- -~kSij

2 An isotropic model is employed for the dissipation rate tensor Cij --- 5(~ijC. The term Dij combining the effect of turbulent transport and viscous diffusion is modelized with the Daly and Harlow model:

O ( k OTij

Dij -- ~x k

Cs-7-kl

+

OTij)

where Cs-0.22. We adopt the quasi-linear SSG model for the pressure-strain rate correlation r

[lS]:

392

where C O = 3.4, C 1 = 1.8, (72 = 0.36, Ca = 1.25 and C4 = 0.4. The quadratic term in the original SSG model is omitted in the present study, since it introduces much more complexities when deriving the explicit solution to the algebraic stress equation. Other commonly used pressurestrain rate correlation models such as the IP and the LRR model can also be represented by the above formulation. But only the quasi-linear SSG model is employed in the present study. Sij and W~j appearing above are the strain rate and rotation rate tensors defined respectively as:

s j=

+ Ox ] , w j=

and bij is the Reynolds stress anisotropy tensor defined as bij - r~j 1 9 The RSM model is 2k - 5(~iJ implemented with a low Reynolds number formulation combined with a resolution of a transport equation for the turbulence frequency w [1]. To deduce the transport equation for the Reynolds stress anisotropy tensor bij, the turbulent kinetic energy equation k=~-~/2 is required. It can be easily obtained from the contraction of the Reynolds stress transport equation (1) using the fact that the pressure-strain rate tensor is traceless (r = 0). The result reads, Dk = PDt

e+ D

(2)

where P is the turbulent production P = --~-ikOui/Oxk, and D = D~i/2. Prom equation (1) and (2), the following transport equation for the Reynolds stress anisotropy tensor can be deduced: Dbij Dt

1

2k

(

-~ )D c [( _~)P CO ] 1(4 ) D~j - ~ = 1+ -- + - 1 bij - 62 Sij -s e --2 - -2

1

( 2 ~ + -~1 (2 2 (2 - C3) \bikSJk + bjkSik - -~bmnSmnSij/

-- C4)(bikWkj

-- Wikbkj)

(3)

To obtain an algebraic equation for bij, approximation must be made on the left land side of equation (3). In the present study, the first term D b i j / D t is assumed to be zero using the socalled weak-equilibrium assumption. Better approximation can be obtained with the curvature correction approach as proposed by Girimaji [6]. But this approach will not be considered in the present study for the reason mentioned in the above section. The following approximation described by Gatski and Jongen in [2] is applied to the second term,

1

:

-~25ij) D] = 1~-~(Dij - ~ S i j D ) - D C-7-s

2 The term Dij - 5(SijD is assumed to be zero. D / e is evaluated by using equation (2),

D c

1Dk c Dt

(P-l)

= C c 2 - C ~ 1 - (TP - l )

Here, equilibrium assumption D(k/e)/Dt=0 is applied to obtain the approximation for Dk/Dt by using the transport equation for k and e for homogeneous flow [18]. With the above approximation, the following algebraic equation for the Reynolds stress anisotropy tensor bij can be obtained: c (~_P

--+

_~

+

Cc2-Cel)

b~j -

1(4

-C2

)

S~j

2 ) + -~ 1 (2 -- C4)(bikWkj - Wikbkj) = 0 1 (2 -- C3) ( bikSjk + bjkSik - 5bmnSrnnSij 2

(4)

In the ASM approach, equation (4) is solved numerically. The nonlinear term P / e is treated explicitly in an iterative procedure using the value obtained at the previous iteration. The symmetry of the Reynolds stress anisotropy tensor is taken into account, but not the property of

393

traceless. Thus a 6 x 6 matrix is inverted at each point to obtain the solution for the Reynolds stress anisotropy tensor. In the EASM approach, an explicit analytic solution of equation (4) is derived. Mathematical background and detailed derivation of the EASM model can be found in [2][3][4][9]. Final result of the explicit solution of equation (4) for two-dimensional flow is detailed in [16] which is briefly repeated here for completeness. The Reynolds stress tensor is given by: IS The turbulent eddy viscosity is determined from:

( 0000 ) where OL1 is obtained from the solution to the following cubic equation: (OL1/T) 3 -[- p(Ctl/T) 2 -[- q(OZl/W) + r = 0

(6)

where z = k / e is the turbulence time scale, and p =

')'1 1 ( 222 2 ) ')'lal ~2~-270' q = (2r12r270)2 .y2_ 2rl2T2,y0aI _ 5r ] r a 3 + 2R2r12"r2a 2 , r = (2r/2r2"Y~ 2

The root of equation (6) may be real or complex. The correct root is the root with the lowest real part [10]. Rumsey and Gatski have presented a very convenient way for numerical computation to obtain the correct root for equation (6) using only real number calculation. Details can be found in [16]. Other parameters are given by r/2 = SijSij , { W 2} -- - W i j W i j , R 2 = --{W2}/T] 2, al = 1 (4 __ C 2 ) ,

a2 = [1 (2 -- C 4 ) ,

a3 = [1 (2 - C 3 ) ,

a4 -- [')/1 - 2")/o(O~1/T)~2T2] -1 T, ")/0 = C1/2,

")/1 =

C~ - C~I) / (C~2 - 1), C~1 = 1.44 and C~2 = 1.83. Coefficients 6'1 to 6'4 are the coefficients for the quasi-linear SSG model for the pressure-strain rate correlation already given previously. To implement the EASM model and ASM model, two transport equations must be solved to provide the turbulent velocity and length scales. Any existing two-equation model can be used for this purpose. However, model recalibration may be necessary in order to obtain optimum results, especially when the model is integrated down to the wall without using wall function. An investigation on this issue can be found in [7]. In the present study, no attempt to model recalibration is made. An existing model is employed, namely the k-a~ BSL model proposed by Menter [13]. Compared with the original model, only two natural modifications are introduced. Firstly, the turbulent eddy viscosity is replaced by the value given by (5). Secondly, the contribution of the nonlinear part of the Reynolds stress to turbulence production is taken into account in threedimensional flow computation. This modification is not needed in 2D because the contribution is zero. The EASM model thus implemented can be integrated down to the wall and gives correct log law behaviors without any recalibration. The present implementation of the ASM model differs from the previous one by the fact that equation (6) is also solved to obtain a turbulent eddy viscosity given by (5). The turbulent eddy thus obtained is employed in the scale-determining two-equation model. In addition, it is used to decompose the Reynolds stress into a linear part and a residual part according to _asm _ 5 2 kt~ij + 2 u t S i j where -~j 7"ij = ( 2 k(Sij - 2UtSij) -+- 7-[j with T~ = ~ij _~sm is obtained from the numerical solution to the algebraic stress equation (4). The linear part is treated implicitly in the momentum equations as for a linear eddy viscosity model, while the contribution of residue term ~-~ is added explicitly in the source term in the momentum equation as it is done for the higher order nonlinear terms for the EASM model. Occasionally, the solution of the algebraic Reynolds

394

Figure 1: U velocity contours at propeller plane. stress equation (4) becomes singular. In this case, it is replaced by the result provided by the EASM model. With this implementation, the ASM model gives identical results within numerical discretization accuracy for two-dimensional flow as the EASM model. It has been found that computation with the ASM model is as stable as the EASM model, even for three-dimensional flow where the two models no longer provide identical result. NUMERICAL

RESULTS

All turbulence models are implemented in an in-house code called ISIS using the same numerical method. The Reynolds averaged Navier-Stokes equations are solved on an unstructured grid using a finite volume approach. The convection flux is evaluated with a second order upwind blended interpolation, while a second order central difference scheme is applied for diffusion terms. Computations are performed for the KVLCC2 oil carrier at model scale with a 121 x 81 x 41 structured grid with a double model. Results obtained with a 159 x 103 x 61 finer grid show a nearly grid independent solution. Streamwise mean velocity contours predicted by the RSM model are compared with the measurement data in figure l(a). The numerical result obtained with the SST model is also displayed on the same figure to illustrate the improvement of the RSM model compared with a linear eddy viscosity model. Results obtained with the EASM model are compared with those obtained with the RSM model and the ASM model in figure l(b). The EASM model gives a better prediction than the SST model. But all the distinguished features presented in the result predicted with the RSM model are not inherited by the EASM model. Discrepancies observed are unlikely due to the approximation of explicit solution expressed in a three-term tensor bases, since the ASM model provides similar solution. They can only be attributed to the weak-equilibrium assumption employed to obtained the algebraic stress equation (4). A detailed comparison between numerical prediction and measurement data reveals that globally, the RSM model consistently provides a more accurate simulation of the attached boundary layer around the ship except near the concave wall. This behavior is best illustrated by the comparison between the numerical predictions and the measurement data at the measurement station (X/L=0.45) upstream the propeller plane shown in figures 2 and 3. Profiles for U velocity, turbulent kinetic energy and two shear stress components ~ and ~ as function of y for several depths (Z/L=-0.01,-0.02,-0.0a,-0.04,-0.05 and-0.06) are compared. Normal stresses are not shown, since they behave similarly as the turbulence kinetic energy. The first three profiles from the top represent mainly the attached boundary layer along the stern. Flow in this region is dominated by

395

Figure 2: U velocity and turbulent kinetic energy at station 1 as function of y for several depths. the development of a turbulent boundary layer on a convex wall. The characteristic of a turbulent flow on a convex wall as well as turbulence model prediction capability for this kind of flow are well understood. The convex curvature strongly attenuates the turbulent kinetic energy and the shear stress. Linear and nonlinear eddy viscosity models (including explicit algebraic stress model) without curvature correction are unable to predict correctly this attenuation. On the other hand, due to the built-in mechanism represented by the exact convection and production terms, Reynolds stress model is capable to predict correctly this phenomenon [17]. W h a t we observed in this region confirms the observations just mentioned. The RSM model is capable to predict a correct low level of turbulent kinetic energy and of the main shear stress uw. It also provides better prediction on the mean velocity field. On the contrary, both quantities are over predicted by the EASM model and by the ASM model. Below this region, we are in the core of the bilge vortex which interacts with the boundary layer developed on the hull. The hull surface in this region exhibits a very complex convex-to-concave transition. The effect of a concave wall is opposite to that of a convex wall. Turbulent kinetic energy and shear stress are amplified. Turbulence modelization for a flow on a concave wall is much more challenging due to different physical phenomena. A convex wall mainly attenuates pre-existing eddy, whereas a concave wall may involve the reorganization of the pre-existing eddy structures and the creation of new large-scale eddies. It is argued that Taylor-Gortler vortices exist on a concave wall, which may introduce additional complexities for turbulence modelization. Built-in mechanism presented in the Reynolds stress model is no longer sufficient to ensure a good prediction. The RSM model does not provide so much improvement compared with other turbulence models. The three different models are also applied to a numerical simulation for a research vessel called Nawigator X X I at full scale with a constant advance speed of 13 knots, giving a Reynolds number about 4 x 10 s based on the ship length. Numerical computation is performed with double model on a 120 x 100 x 59 structured grid without using wall function. Propeller effect is not taken into account in the computation. Measurement data are not yet available. Only numerical results are presented in this paper. Numerical prediction of streamwise velocity contours and turbulent kinetic energy contours with different turbulence models at the propeller plane are presented in figure 4. Some differences are observed between the EASM model and the Reynolds stress model. The intensity of the bilge vortex is lower than with the EASM model. It also predicts a higher

396

Figure 3: ~-~ and ~

shear stress at station 1 as function of y for several depths.

Figure 4: U velocity and turbulent kinetic energy contours at propeller plane for XXI

theNawigator

.

turbulent kinetic energy near the hull. The ASM model gives similar prediction as the EASM model. These results are similar to what we observe at model scale although less differences are observed. CONCLUSIONS Ship flows at model and full scale are computed to assess the accuracy of a quadratic explicit algebraic stress model by comparing to the exact numerical solution of the algebraic stress equation. Both models are found similar for the flow investigated. Result of the explicit solution is employed in the implementation of the implicit algebraic stress model. The ASM model thus implemented is found to be as stable as the EASM model. Results obtained with the implicit ASM model suggest that it is unlikely possible to improve the accuracy of the quadratic EASM model for the flow investigated in the present study by using higher order tensor bases. In addition to the effect of adverse pressure gradient as well as the interaction between the 3D boundary layer and the bilge vortex, stern flow is found to be dominated by the effect of curvature. Globally, the RSM is found

397

to perform better than the EASM model and the ASM model, especially in the region dominated by convex curvature effect.

Acknowledgements This work is supported by the European EFFORT project (Projet n ~ GRD2-2001-50117). Part of this work is carried out on the basis of CPU allocations on the two national computer centers of France CINES and IDRIS, and on the CCIPL regional computer center as well.

References [1] G.B. Deng and M. Visonneau. Comparison of explicit algebraic stress models and secondorder turbulence closures for steady flows around ships. In Proc. 7th Int. Conf. on Numerical Ship Hydrodynamics, Nantes, France, 1999. [2] T.B. Gatski and T. Jongen. Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Progress in Aerospace Sciences, 36:655-682, 2000. [3] T.B. Gatski and C.L. Rumsey. Linear and Nonlinear Edy Viscosity Modelling. in Closure Strategies for Turbulent and Transitional Flows, B. Launder and N. Sandham eds., Camgridge University Press, 2002. [4] T.B. Gatski and C.G. Speziale. On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech., 254:59-78, 1993. [5] S.S. Girimaji. Fully explicit and self-consistent algebraic reynolds stress model. Theoretical and Computational Fluid Dynamics, 8:387-402, 1996. [6] S.S. Girimaji. A galilean invariant explicit algebraic reynolds stress model. Phys. Fluids, 9(4):1067-1077, 1997. [7] A. Hellsten. New Two-Equation Turbulence Model for Aerodynamics Applications. thesis, Helsinki University of Technology, 2004.

PhD

[8] A.V. Johansson and S. Wallin. A new explicit algebraic stress model. In Proc. Sixth European Turbulence Conference, Laussane, Suiss, 1996. [9] T. Jongen and T.B. Gatski. General explicit algebraic stress relations and best approximation for three-dimensional flows. Int. Journal of Engineering Science, 36:739-763, 1998.

[10]

T. Jongen and T.B. Gatski. A unified analysis of planar homogeneous turbulence using single-point closure equations. Journal of Fluid Mechanics, 399:117-150, 1999.

[11]

L. Larsson, F. Stern, and V. Bertram. Benchmarking of computational fluid dynamics for ship flows: The gothenburg 2000 workshop. Journal of Ship Research, 47(1):63-81, 2003.

[12]

S-J. Lee, H-R Kim, W-J Kim, and S-H Van. Wind tunnel tests on flow characteristics of the kriso 3,600 teu containership and 300k vlcc double-desk ship models. Journal of Ship Research, 47(1):24-38, 2003.

[la]

F.R. Menter. Zonal two-equations k - w Paper, 93-2906, 1993.

[14]

S.B. Pope. A more general effective viscosity hypothesis. J. Fluid Mech., 72:331-340, 1975.

turbulence models for aerodynamic flows. AIAA

398

[15] W. Rodi. A new algebraic relation for calculating the Reynolds stresses. ZAMM, 56:219-221, 1976. [16] C.L. Rumsey and T.B. Gatski. Recent turbulence model advances applied to multielement airfoil computations. Journal of Aircraft, 38(5):904-910, 2001. [17] C.L. Rumsey, T.B. Gatski, and J.H. Morrison. Turbulence model predictions of etrat-strain rate effects in strongly-curved flows. AIAA Paper, 99-0157, 1999. [18] C.G. Speziale, S. Sarkar, and T.B. Gatski. Modeling the pressure-strain correlation of turbulence: An invariant dynamical systems approach. Journal of Fluid Mechanics, 227:245-272, 1991.

[19]

C.G. Speziale and X.H. Xu. Towards the development of second-order closure models for nonequilibrium turbulent flows. Int. J. Heat and Fluid Flow, 17:238-244, 1996.

[2o]

S. Wallin, A. Hellsten, M. Schatz, T. Rung, and D. Peshkin. Streamline curvature correction algebraic reynolds stress turbulence modelling. In Third Int. Symposium on Turbulence and Shear Flow Phenomena, Sendai, Japan, 2003.

[21] S. Wallin and A.V. Johansson. An explicit algebraic reynolds stress model for incompressible and compressible turbulent flows. Journal of Fluid Mechanics, 403:89-132, 2000. [22] S. Wallin and A.V. Johansson. Modelling streamline curvature effects in explicit algebraic reynolds stress turbulence models. Int. J. Heat and Fluid Flow, 23:721-730, 2002.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

399

COMPARISON OF TURBULENCE MODELS IN CASE OF JET IN CROSSFLOW USING C O M M E R C I A L CFD CODE A. Karvinen and H. Ahlstedt Institute of Energy and Process Engineering Tampere University of Technology P.O. Box 589, FI-33101 TAMPERE, HNLAND

ABSTRACT A circular jet in a crossflow is simulated numerically using a commercial CFD code. A grid resolution study is performed to find a grid that yields a grid-independent solution. The three high-Reynolds-number k-e models, the two k-co models, the six low-Reynolds-number k-e models as well as the Reynolds stress model have been used. In all calculations, the near-wall regions are resolved all the way down to the wall and no wall functions are used. Jet to crossflow velocity ratio is 3.3. The Reynolds numbers of the crossflow and the jet are 317,000 and 8,130, respectively. Results show that a grid should be very fine to obtain a grid-independent solution. There are considerable differences between the turbulence models. Results of standard k-e model are closest to the measured values. Results of the SST k-o3 model are also quite satisfactory. The RNG k-e model and the standard k-0~ model gave nonphysical velocity profiles. Contrary to expectations, Reynolds stress model did not predict velocity profiles well.

KEYWORDS CFD, jet in crossflow, k-e model, RNG k-e model, realizable k-e model, k-o3 model, SST k-o~ model, lowReynolds-number k-e model, Reynolds stress model.

1

INTRODUCTION

Jet in crossflow occurs in many applications, e.g. in vertical and/or short take-off and landing (V/STOL) aircraft, in combustor chambers, in oil or gas flow into a container, in chemical industry, and in waste discharge into water bodies and into the atmosphere. Jet in crossflow has been under intensive study for over fifty years. The first studies focused on determining the place of the core of the jet (Jordinson 1956, Margason 1968). In the articles Ramsey & Goldstein (1971), Andreopoulos & Rodi (1984), Kim et al. (2000), and Meyer et al. (2002) the shape of the jet has been studied experimentally. Numerical studies have been done by e.g. Patankar et al. (1977), Alvarez et al. (1993), Yuan & Street (1998), and Yuan et al. (1999). A more extensive review can be found in Schetz (1980) and Margason (1993).

400 2

GOVERNING EQUATIONS AND THEIR NUMERICAL SOLUTION

In this work, incompressible steady-state flow is assumed. The buoyancy has not been taken into account. On this basis, the Reynolds-averaged Navier-Stokes equations (RANS) are

~ui --0, ~Xi Uj .OU. i . . . 1 Op q--V ~2Ui + ~ (_UiU..--TZ-T..t'~,j)

bxj

p bxi

bxj~xj

(1) (2)

k

where ui is the mean velocity component in x i direction, P is the density, p is the mean pressure, v is the kinematic viscosity, the overbar denotes time averaging, and the prime denotes fluctuating part.

2.1

Turbulence Models !

!

In the k-e models and the k-co models, the Boussinesq hypothesis is used. The Reynolds stresses, -uiuj, are approximated with

2 k ~)ij, --U~U~ = 2 v t S i j - -~

~uj)

l {Oui

Sij -- -~ ~OXj -[- ~Xi

'

(3)

where vt is the turbulent (or eddy) viscosity.

In the s t a n d a r d k-e model (Launder & Spalding 1972, Launder & Sharma 1974), the turbulence kinetic energy, k, and its dissipation rate, e, are obtained from the modeled transport equations:

Uj~ ~E

-~) [ (

UJ~bxj "-- bXj

V --I--

(4)

-}-Vt$2 -- E,

Vt) ~~X~] E 2 E2 V nt- ~Oe -}-f i e ~vtS - C2e--k-,

(5)

where S -- 42SijSij. The turbulent viscosity is computed from vt = C~k2/E, where C~, is a constant. The model constants have the following values: Cl, -- 0.09, Cle = 1.44, C2E = 1.92, •k - 1.0, and ~e = 1.3. The transport equations for k and e in the RNG

~k _

~9

uJ-~xj - ~xj t)E

~ (

u j ~ -" ~Xj ~Xj

~eVt

~E)

~

k-e model

(

(XkVt~xj

(Yakhot & Orszag 1986) are

(6)

q- VtS2 -- E,

E 2 E2 C/tTI3(1--TI/TI0) E2 q- Cle~VtS -- C2e-- --,

k

1 .qt_~113

(7)

k

where 1"1= Sk/e, 110 = 4.38, ~l = 0.012, and vt is obtained in a similar way as in the standard k-e model. The quantities ctk and o~e are inverse effective Prandtl numbers (not constants as in Yakhot & Orszag (1986)), see Fluent Inc. (2003) for more details. The model constants are: Cu = 0.0845, Cle - 1.42, and C2E -- 1.68. The transport equations for k and e in the realizable k-e model (Shih

uj~

0xj

=

V -+-

et al. 1995, Fluent Inc. 2003)

-I- Vt$2 -- E,

are (8)

401 Oe

O

uJN=N

s

("-'- ~

k + q'V~'

(9)

where Cle is not a constant. The turbulent viscosity is obtained in a similar way as in the standard k-e model, but now C# is not a constant. The model constants are: C2e = 1.9, ok = 1.0, and 6E = 1.2. In the standard k-e0 model (Wilcox 1998), the turbulence kinetic energy, k, and the specific dissipation rate, m, are obtained from the following transport equations:

Uj~jXj = ~Xj V+ ~kk

+ VtS2-- ~*f~*k0)'

(10) (11)

ujN = N

+

N

+

-

and the turbulent viscosity is obtained from vt : k/o3. The functions f~*** and f~i are auxiliary functions, see Fluent Inc. (2003) for more details. The model constants are: ~i : 0.072, [3* = 0.09, ok = 2.0, and aco = 2.0. In this work, the low-Reynolds-number version of the k-m model is used (Wilcox 1998).

The SST k-m model (Menter 1994) has a form similar to that of the standard k-e0 model:

Ok ~ ( v t ) UJ~xj --" Oxj v -+-~kk Oxk] + VtS2-- ~*k(o,

(12)

~g --- ~Xj CJ [( V "Jr---(~eo Vt) a~X~l-Jr-S2 -- ~iO)2 "Jr-Dr UJ-~xj

(13)

where ~i = Fl~i,1 + ( 1 - F1)~i,2 and the turbulent viscosity is obtained from Vt "--k/mmax[1,f~F2/alo3]. The quantity f~ is the mean rate-of-rotation, F1 and F2 are the blending functions, and Ok and ~co are the auxiliary functions (not constants). The term Do is the cross-diffusion term. See Fluent Inc. (2003) for more details. The model constants are: al = 0.31, [3* = 0.09, 13i,1 = 0.075, 13i,2 -- 0.0828, 6k,1 = 1.176, ~ , 1 = 2.0, Ok,2 -- 1.0, and ~co,2 = 1.168. See Fluent Inc. (2003) for more details of the low-Reynoldsnumber correction to the turbulent viscosity which is enabled in this work. In the Reynolds stress model (RSM), there are the exact transport equations for the transport of Reynolds stresses, -- U iU j~ (Gibson & Launder 1978, Launder et al. 1975):

~k

~ ~Xk UiUjUk+ 7 (~kjui' "Jr-~ikUjI],J-- ~uiuk~Xk q- UjUtk:~xk) + ob

OXk ~Xk ( ~ )

+ y ~OXj +~Xi --2V~xk~~___~--2~k( u~utmEikm+u~utm~jkm)' (14) (~ij

Eij

where D~j is the turbulent diffusive transport, (~ij is the pressure-strain term, and Eij is the dissipation tensor. See Fluent Inc. (2003) for more details on how these terms are modeled. The model constants are: C1 = 1.8, C2 = 0.60, CIE = 1.44, C2e = 1.92, C# = 0.9, ok = 1.0, and 6e = 1.3. In the the low-Reynolds-number k-e models, the standard k-e model is modified to take into account the low-Reynolds-number effects. In this work, the following low-Reynolds-number k-e models are used: Abid

402 (Abid 1991), Lam-Bremhorst (Lam & Bremhorst 1981), Launder-Sharma (Launder & Sharma 1974), YangShih (Yang & Shih 1993), Abe-Kondoh-Nagano (Abe et al. 1994), and Chang-Hsieh-Chen (Chang et al. 1995). In these models the transport equations for k and e are

UjOXj "-- ~Xj uj ax---7.-- ~xj

V + ~kk

+ VtS2 -- E,

(15)

~xj + C l e f l ~vtS - C 2 e f 2 ~ + E,

V+ ~

(16)

where e = eo + g. The eddy viscosity is obtained from vt = Cl~f~k2/E. The damping functions (Table 1) depend upon one or more of the following parameters: Ret = k2/~v, Rey = x/--ky/v, y* = v-3/4gl/4y, and y+ = uxy/v. The quantity y is the distance from the wall. The model constants are in all models the same as in the standard k-e model. TABLE 1 LOW-REYNOLDS-NUMBER k-E MODELS Model Abid Lam-Bremhorst Launder-Sharma Yang-Shih Abe-Kondoh-Nagano

(1 + 3.4/x/R~t)tanh (y+/80) [1 - exp(-0.0165Rey)]2 ( 1 + 20.5/Ret) exp [-3.4/(1 + Ret/50) 2] [1 -- exp(-- 1.5 • 10-4Rey - 5 • 10-7Re 3__ 10_lORe5)_] 1/2/(1_-t- 1/~/~t) [1- exp(-y*/14)] 2 [1 + 5 exp(-Ret/200)2/Re~/4]

Model Abid Lam-Bremhorst Launder-Sharma

2.2

1

[1- exp(-0.0Zl5Rey)] 2 (1 + 31.66/Rr

Chang-Hsieh-Chen

Yang-Shih Abe-Kondoh-Nagano Chang-Hsieh-Chen

1

1 + (0.05/f~) 3 1 ~-~t/(1 + x/l~t)

.f2 1

1 - exp(Re2) 1 - 0.3 exp(-Re 2)

2v (oqV~/o3y) 2 o 2v (~x/~/~y) 2

o o 2VVt(O2u/Oy2)2

0

vv~ (a~u/ay~) ~

~/(1 +~) exp(-y*/3.1)]2 [1 - 0.3 exp(- (Ret/6.5)2] [1-0.01 exp(-Ret2)] [1 - exp(-0.0631Rey)]

[1 -

o o

Calculation of Near- Wall Area

When using some of the high-Reynolds-number k-e models or the RSM, the model should be modified in the vicinity of the no-slip walls. In this work, the method is used where the whole domain is subdivided into two regions, based on Rey. In the fully turbulent region (Rey > 200) the k-e model or the RSM is used. In the area where Rey < 200, basic model equations are retained, but (Wolfshtein 1969, Chen& Patel 1988): Vt --- C/~v/-k~/~,

k3/2

~.lz= Cey (1-- e-Rey/&') , gE = Cey (1-- e-Rey/Ae) ,

where A~ = 70 and AE = 2Ce (Chen & Patel 1988). Between the inner (Rey < 200) and outer (Rey > 200) region, vt and e are blended to ensure a smooth transition (Fluent Inc. 2003).

2.3

Calculation Procedure

All the calculations have been done using FLUENT 6.1.22 (Fluent Inc. 2003), a commercial software based on the finite volume method. All terms in all equations are discretised in space using second-order central

403 differencing, apart from the convection term, which is discretised using a second-order upwind scheme. Pressure-velocity coupling has been done using the traditional SIMPLE algorithm. Default values of all constants are used. All options which are enabled by default are used. In addition, a differential formula for effective viscosity/teft =/Z -t-/zt Q.t is a molecular viscosity) is used in the RNG k-e. model to take into account low-Reynolds-number effects. Also a low-Reynolds-number correction to the turbulent viscosity is enabled when using either of the k-o) models. All calculations are iterated until they approach the machine number, therefore it can be assumed that iteration error is negligible. Relaxation parameters are reduced when needed.

3

C O M P U T A T I O N A L D O M A I N AND M E S H

The schematic of the problem is shown in Figure 1. Two Reynolds numbers based on the crossflow free stream velocity (uoo) and the distance from the leading edge of the flat plate to the jet axis, and based on the jet average velocity (wj) and the jet diameter are Rex ..~ 317,000 and ReD ~ 8,130, respectively. The working fluid is air with density p = 1.225 k g / m 3 and dynamic viscosity/Z = 1.7894 x 10 -5 kg/ms (v =/z/p). v r

velocity = 1.435m/s turbulent ~-intensity = 2.5% = turbulent ~ viscosity ratio = 5

= crossflow(u**~ 1.5 m/s), with boundary layer similar to that in experiments =

~L

supply blocks

x

,,.--

t

i

_ 3.75D _!_

fully developedpipe flow (wj = 4.95m/s)

I

Z

.;i i 125D

C

25D main blocks

wall = symmetry plane

=

velocity = 4.95 m/s L turbulent intensity -- 5.2% hydraulic diameter = 0.024 m

Figure 1" Computational domain, boundary conditions and schematic of velocity profiles in block interfaces. Domain width is 25D (modeled half 12.5D) and sidewall boundary condition is symmetry plane. Pipe diameter D = 24 mm.

In the experiments ((~zcan & Larsen 2001), special consideration was given to establishing the fully developed incoming flow conditions on the flat plate and in the pipe. The boundary layer of the flat plate was turbulent. In the simulation, this is achieved using a long enough development distance of the boundary layer (128.75D) so that the boundary layer velocity and Reynolds stress profiles close to the jet axis (x = - 3 . 7 5 D ) are similar to measurements (Figure 2(a)). In the measurements, turbulent boundary layer is produced using a vortex generator and a shorter development distance. In the experiments, pipe flow was fully developed. Also in the simulation, a long enough pipe (75D) is used to produce a fully developed pipe flow (Figure 2(b)). The computational grid consists of four blocks (Figure 1 and Figure 2(c)). Only one half of the domain is modeled because of symmetry. The pipe consists of a so-called supply block and of a so-called main block, where 11,800 and 11,040 control volumes have been used, respectively. Also the crossflow consists

404

of two blocks, i.e. the supply block (117,000 control volumes) and the main block where six different grid resolutions have been used such that the total number of control volumes used is 267,640, 387,340, 549,340, 910,840, 1,405,240 or 2,302,990. Grids in all blocks have been constructed so that the dimensionless wall unit, y+, is close to unity at the wall-adjacent cells of all no-slip walls. All grids are constructed so that grid is finer in the vicinity of the jet than elsewhere. 1.4

!

RSM 9

.

.

,

.

..

1.3

Exp

1.2

f

/

/

Y/

~

RSM

1.1 - - Std. k-m

0 .... 0

0.5

1

1

3'0

4'0

5'0

z/O

u/u~

(a) Plate boundary layer

(b) Longitudinalcenterlinepipe velocity

(c) Grid 1,405,240

Figure 2: Grid and incoming velocity considerations.

4

4.1

RESULTS

Grid Resolution Study

Results show that a grid should be very fine m obtain a grid-independent solution (Fig. 3). This is in close agreement with the observations made in Karvinen & Ahlstedt (2003). Velocity profiles of the standard k-e model do not change significantly when grid has been made finer than 549,340 control volumes. However, we can see slight changes in the turbulent kinetic energy. RSM velocity profiles change significally even when grid is changed from 910,840 to 1,405,240 control volumes. Unfortunately, we were not able to carry

8

7 6

8

!

9- . - -.-. - -9

267,640 | 387,340 | 549,340 ] 910,840 ] 1 405 240 '1 2~302~990 J Exp. ,]

9- . 267,640 - - 387,340 - - 549,340 9- . 910,840 - - 1,405,240 - - 2,302,990

7 6

8

9- . - -9- . -9

8f

| | 7~

267,640 387,340 549,340 910,840 1,405,240 Exp.

9- . --.-. --

[

iI

5

5

2'

3

~.~

2 21 "

~

~

0

0.5

1

15

0;

x

0.2

k-e, u/uo.

1

0.4

k/uL

U/U.. (a) Std.

~*"'~'~ "~

1

-~

267,640 387,340 549,340 910,840 1,405,240

(b) Std.

k-e, k/u~

0:6

-%

o

0.5

u/u~

(c) RSM,

i

u/uo.

115

ok 0

•

0.2

k/u~

(d) RSM,

-0.4

0.6

k/u~

Figure 3: Horizontal velocity and turbulence kinetic energy in plane y = 0 in location x/h = 1.5. Measurements (Ozcan & Larsen 2001).

405 out calculation using RSM and finest grid (2,302,990 control volumes), but because velocity profile and turbulent kinetic energy profile change only slightly when grid is changed from 910,840 to 1,405,240 control volumes, it can be assumed that grid of 1,405,240 control volumes produces a quite grid-independent solution. Therefore, grid consisting of 1,405,240 control volumes is used exclusively from here onwards. 4.2

Turbulence Model Comparison

In this section, all turbulence models introduced above are compared. The standard k-e model reproduced the horizontal velocity profiles of experiments best, as can be seen from Figure 4. Also the SST k-m model gave quite satisfactory results conceming horizontal velocity profiles only. There are some problems with the Abid model, the Abe-Kondoh-Nakano model, and the Chang-Hsieh-Chen model. They show that there is negative vertical velocity of a value as large as 0.7 m/s in the location x/D -- 1.5 (Figure 5(b)), which is not seen in the results of Ozcan & Larsen (2001). Also both k-o3 models predicted this, but the value was slightly lower. It can be seen from Figure 6 that all models predicted turbulent kinetic energy levels that are lower than measured. Note that the profilies are given in slightly different locations than other quantities. The value of the realizable k-e model and the shape of profile of the RSM are closest to the experiments, but the value of the RSM is much lower than measured. Also turbulent kinetic energy dissipation is difficult to predict correctly. These observations are consistent with the common knowledge that the primary variables are easier to predict than the turbulent variables.

8

9- -.-. - --

7 6

Std. k-• RNG k-e Real. k-e Std. k-03 SST k-co RSM

9 Exp.

8

I

[ | | [ i

8

9- -

1

A -

7

--

6

YS -9- .- AKN - - CHC

5

LS

7 t 6

-.-. - --

5

| ]

9- -

LB LS YS AKN CHC Exp. 5

5

,-

A

Real. k-e | Std. k-o~ | SST k-c0 RSM Exp. ~,~,,,

9

9 E x p . /

8

I 9Std. k-E - - RNG k-e

LB

.~. "~-~4

3

""*

~

/

'~

ii

i 3

I.\

2" . i ~ :4

\

1>

0'

-0.5

"

0

1

'

0.5 1 U/U**

(a) x/D = 1.5

'

--00.5

1.5 U/U~

(b) x/D = 1.5

~'~'jt

0

,

0.5 1 U/U~,

(c) x/D = 3.0

,

1.5

o -0.5

0

0.5 1 U/U~

1.5

(d) x/D = 3.0

Figure 4: Horizontal velocity, u/u,, in plane y = 0. Measurements (Ozcan & Larsen 2001).

5

CONCLUSIONS

The grid resolution study showed that accurate calculation of the jet in a crossflow requires a very fine grid. However, results change very little when a grid is refined from 1.4 million to 2.3 million control volumes. A grid with about 900,000 cells, or even slightly less, gives good enough results in practical applications. There are considerable differences between the turbulence models. Results of the standard k-e model are closest to the measured values. Results of the SST k-o3 model are also quite satisfactory. The RNG k-e model

406

81 7 l" | | 6

i L

/

Real. k-e Std. k-to SST k - ~ RSM

-,-. - m

,

r''.

A

8r

7

- - - LS . - . YS - - AKN CHC

7I | / 6t

6

.~x~

~,4

81 [ i

/

,~

~'~,4

.~x~

, ~ |. g.

/

""

Stdk

8I

--.-. - --

7

~,,:~x~

Real. k-e Std. k-to SST k-to RSM

~4

at 0

0.5

1

w/u~

1.5

--0.5

(a) x / D = 1.5

0

0.5

w/u~

1

1.5

~ ~.

,

O-LO.5

(b) x / D = 1.5

0

-~0.5

w/u.

~1

AKN -.~x~

.

31 -~1.5

. - . YS - CHC

x~,. ,\

~4

!f

--0.5

/ _[

O[ /

~

~

0'5 i w/u~

(c) x / D = 3.0

~'5

(d) x/D = 3.0

Figure 5: Vertical velocity, w/u=, in plane y = 0. Measurements (0zcan & Larsen 2001).

8

~- - ~ , 4 . G k k e _ e

7~

--9- . - i

l 6[

Real. k-e Std. k-to SST k-to RSM

. 7 i

.

YS AKN CHC

9 ~x~

[.

.

".

--~4

"

3

/'.~

2

.

6

~xp

5 "~. . ,. ' .: . . . . . ~ ' ~ 4 ~ - ~ "

.

.

".,.

- - Real. k-e . - . Std. k-to - - SST k-to

..\,.

- 9 ~SM Exp.

. - . YS - - - CHC 9

AKN Exp.

"

[[M_'. - ~

5

5

~--~4

--~4

3 ":l

Std. k-e .... RNG k-e

~

3

2

:.

2 t.

1 0

0.2

0.4

0.6

0- 0

:,' - "" 012

k/u 2 (a)

x/D =

10 0'.4

0'.6

10

0

0.2

k/u 2 1.83

(b)

x/D =

~4

O.6

0

0.2

k/u 2 1.83

(c)

x/D =

0.4

0.6

k/u 2 3.67

(d)

x/D =

3.67

Figure 6: Turbulence kinetic energy, k/u 2, in plane y - 0. and the standard k-co model gave nonphysical velocity profiles. Some models predicted quite large negative vertical velocities in the vicinity of the jet, which are not seen in the experiments. Contrary to expectations, the Reynolds stress model did not predict velocity profiles well. All models gave lower turbulent kinetic energy levels than experiments. Also turbulent kinetic energy dissipation is difficult to predict correctly.

ACKNOWLEDGMENT The authors gratefully acknowledge the support from Tekes - The National Technology Agency of Finland, Andritz Oy, Enprima Oy, Fortum Nuclear Services Oy, Kvaerner Power Oy, Metso Paper Oy, Nokia Oy, Numerola Oy, Patria Aviation Oy, and Process Flow Ltd Oy.

407

8

8

- - Std. k-e RNG k-e - - Real. k-e RSM

6

.-. A - - LB LS , - , YS - - AKN CHC

7

6 5

-m -m

Std. k-e RNG k-e Real. k-e RSM

81f

-. - m ,-, - --

6

A LB LS YS AKN CHC

5

4r-

k.':2.-.2 `

3 I ~

3

it

21

~

8

7

// ~ ]

el(u31O)

~ ~8 ~

(a) x/D = 1.5

0.2

q'-~4 t ~ / /

q'-~4 "

0.4

El(u31D) (b) x/D = 1.5

0.6

El(u310) (c) x/D = 3.0

0.8

00

0.2

0.4

0.6

0.8

El(u31D) (d) x/D = 3.0

Figure 7" Turbulence kinetic energy dissipation, e/(u3/D), in plane y = 0. REFERENCES Abe K., Kondoh T., and Nagano Y. (1994). A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flow - I. Flow Field Calculations. International Journal of Heat and Mass Transfer 37, 139-151. Abid R. ( 1991). A two-equation turbulence model for compressible flows. Technical Report AIAA-91-1781, AIAA 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hawaii. Alvarez J., Jones W.R, and Seoud R. (1993). Predictions of momentum and scalar fields in a jet in crossflow using first and second order turbulence closures. In Computational and Experimental Assessment of Jets in Cross Flow, AGARD-CP-534, pages 24-1...24-10, 1993. Andreopoulos J. and Rodi W. (1984). Experimental Investigation of Jets in a Crossflow. Journal of Fluid Mechanics 138, 93-127. Chang K.C., Hsieh W.D., and Chen C.S. (1995). A Modified Low-Reynolds-Number Turbulence Model Applicable to Recirculating Flow in Pipe Expansion. Journal of Fluids Engineering 117, 417-423. Chen H.C. and Patel V.C. (1988). Near-wall turbulence models for complex flows including separation. AIAA Journal 26, 641-648. Fluent Inc. (2003). FLUENT 6.1 User's Guide, Lebanon. Gibson M.M. and Launder B.E. (1978). Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 86, 491-511. Jordinson R. (1956). Flow in a jet directed normal to the wind. R & M No. 3074, British A.R.C. Karvinen A. and Ahlstedt H. (2003). A comparison of turbulence models and the calculation of the nearwall area in the case of a jet in a crossflow. In Peter R~back, Kari Santaoja, and Rolf Stenberg, editors, VIII Finnish Mechanics Days, pages 99-110. Helsinki University of Technology.

408 Kim K.C., Kim S.K., and Yoon S.Y. (2000). PIV Measurements of the Flow and Turbulent Characteristics of a Round Jet in Crossflow. Journal of Visualization 3, 157-164. Lam C.K.G. and Bremhorst K.A. (1981). A Modified Form of the k-e Model for Predicting Wall Turbulence. Journal of Fluids Engineering 103, 456-460. Launder B.E., Reece G.J., and Rodi W. (1975). Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics 68, 537-566. Launder B.E. and Sharma B.I. (1974). Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc. Letters in Heat and Mass Transfer 1, 131-138. Launder B.E. and Spalding D.B. (1972). Lectures in Mathematical Models of Turbulence, Academic Press, London. Margason R.J. (1968). The path of a jet directed at large angles to a subsonic free stream. Technical Report NASA TN D-4919. Margason R.J. (1993). Fifty years of jet in cross flow research. In Computational and Experimental Assessment of Jets in Cross Flow, AGARD-CP-534, pages 1-1...1-33. Menter F.R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32, 1598-1605. Meyer K.E., Ozcan O., and Westergaard C.H. (2002). Flow Mapping of a Jet in Crossflow with Stereoscopic PIV. Journal of Visualization 5, 225-231. Patankar S.V., Basu D.K., and Alpay S.A. (1977). Prediction of the Three-Dimensional Velocity Field of a Deflected Turbulent Jet. Journal of Fluids Engineering 99, 758-762. Ramsey J.W. and Goldstein R.J. (1971). Interaction of a Heated Jet With a Deflecting Stream. Journal of Heat Transfer 94, 365-372. Schetz J.A. (1980). Injection and Mixing in Turbulent Flow, volume 68 of Progress in Astronautics and Aeronautics, AIAA, New York. Shih T.-H., Liou W.W., Shabbir A., Yang Z., and Zhu J. (1995). A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Flows. Computers Fluids 24, 227-238. Wilcox D.C. (1998). Turbulence Modeling for CFD, DCW Industries, Inc., La Canada, California. Wolfshtein M. (1969). The velocity and temperature distribution of one-dimensional flow with turbulence augmentation and pressure gradient. International Journal of Heat and Mass Transfer 12, 301-318. Yakhot V. and Orszag S.A. (1986). Renormalization Group Analysis of Turbulence: 1. Basic Theory. Journal of Scientific Computing 1, 3-51. Yang Z. and Shih T.H. (1993). A k-e model for turbulent and transitional boundary layers. In R. M. C. So, C. G. Speziale, and B. E. Launder, editors, Near-Wall Turbulent Flows, Elsevier Publisher. Yuan L.L. and Street R.L. (1998). Trajectory and Entrainment of a Round Jet in Crossflow. Physics of Fluids 10, 2323-2335. Yuan L.L., Street R.L., and Ferziger J.H. (1999). Large-Eddy Simulations of a Round Jet in Crossflow. Journal of Fluids Engineering 379, 71-104. Ozcan O. and Larsen P.S. (2001). An experimental study of a turbulent jet in cross-flow by using LDA. Technical Report MEK-FM 2001-02, Technical University of Denmark.

6. Experimental Techniques and Studies

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

411

TIME R E S O L V E D PIV M E A S U R E M E N T S FOR V A L I D A T I N G LES OF THE T U R B U L E N T F L O W W I T H I N A PCB E N C L O S U R E MODEL G. Usera 1'2 , A. Vernet l, J. A. Ferr61 1Dep. Enginyeria Mecb.nica, Universitat Rovira i Virgili, Av. Pa'fsos Catalans 26, 43007 Tarragona, Spain 2Instituo de Mecfinica de los Fluidos, Universidad de la Repfiblica, J. H. Reissig 565, 11300, Montevideo, Uruguay

ABSTRACT In this paper a novel conditional sampling method based on a fuzzy clustering technique is presented. The technique is applied to the analysis of the large-scale structures of the turbulent flow in an experimental model of a printed circuit board (PCB) enclosing. To overcome the limitations of the conventional (unconditional) time averaging, a set of conditionally sampled averages, or cluster averages, are obtained. Further, the time history of the flow is analysed on the base of this set of cluster averages, yielding time sequence relations among them that simplify their interpretation in terms of flow dynamics. In this way, events of alternate vortex shedding have been identified in the selected portion of the flow. KEYWORDS PIV, LIF, LES, turbulence, fuzzy clustering

INTRODUCTION

The overall aim of the ongoing research is to investigate the velocity field in complex domains and its influence on the heat transport by natural and/or forced convection. In the present stage of the investigation only the velocity field is under study, while plans for simultaneous measurement of velocity and temperature, through two-colour Laser Induced Fluorescence (LIF) are being implemented. The increasing availability of full field measurement techniques with good time resolution (e. g. time resolved PIV) and of higher order numerical simulation methods for turbulent flows, (e.g. LES or DNS) provides the fluid dynamic researcher with databases of increased level of flow detail in the form of extended time series. To efficiently use these types of data, specialised methods need to be developed. Moreover, the validation of numerical simulation methods against time resolved PIV also

412

requires methods that allow to check the transient dynamics of the simulation rather than just the mean statistical properties. Some of the widely used techniques that can be found in the literature are, among others, the Proper Orthogonal Decomposition (POD), the Pattern recognition (PR), and, to a lesser extent, the Singular Spectral Analysis (SSA) and Fuzzy Clustering (FC). Proper orthogonal decomposition (POD) involves the computation of a subset of the eigenvectors and eigenvalues of the correlation tensor, in decreasing order of the eigenvalues magnitude. The eigenvectors with the largest eigenvalues can give an insight into the underlying large-scale structure of the flow. The use of these leading eigenvectors as starting templates for pattern recognition in the search of coherent structures has been considered by Kopp et al. (1997). In addition, the projection of the original data set onto the subset of eigenvectors, allows for reducing the dimensionality of the data set with a loss of information limited to the finer, and thus less energetic, scales of the flow. Is in this last manner in which POD is applied within this work. A summary of the applications of the POD technique has been given by Berkooz et al. (1993). While POD is typically applied in the spatial domain, a class of techniques exist, named generically Singular Spectral Analysis, which extend the application of the POD concept to the time domain, Ghil et al (2001). Multivariate SSA allows for simultaneous decompositions of the spatial and temporal domains into data adapted basis functions. These techniques will not be considered in this work, but will be in the following stages of this research program. Pattern recognition comprises another set of techniques used to identify flow structures governing the dynamics of the flow (Ferr6 and Giralt 1989 and Ferr6 et al. 1990). However, usually only a moderate percentage of the instantaneous frames contained in the time series is involved in the pattern classification. On the other hand, fuzzy clustering is a technique that allows the classification of all the instantaneous frames into a specified number of conditionally averaged or coherent subsets (Vernet and Kopp 2002). This technique can identify flow structures whose existence remains hidden within the usual unconditional averaging procedure. In this paper a novel conditional sampling method based on a fuzzy clustering technique is presented. The technique is applied to the analysis of the large-scale structures of the turbulent flow in an experimental model of a printed circuit board (PCB) enclosing. To overcome the limitations of the conventional (unconditional) time averaging, a set of conditionally sampled averages, or cluster averages, are obtained with an improved signal-to-noise ratio. Further, the time history of the flow is analysed on the base of this set of cluster averages, yielding the time sequence relations among them that simplify and complement their interpretation in terms of flow dynamics.

EXPERIMENTAL SETUP AND T E C H N I Q U E Time resolved PIV recordings were obtained using an 420 by 480 pixels Motion Scope PCI 1000 S digital camera capable running at 125 and 250Hz sampling rates for 512 image strips. Illumination was provided by a Monocrom DPSSL 532nm Pulsed Laser source, with a cylindrical lens. Licopode spores were used as seeding material. A sketch of the experimental model, which was run with water, is showed in figure 1. The entrance section is a square of side h=.024m. Two sets of measurements at different locations were obtained, at Reynolds numbers ofRe=l.16xl03 and 5.79x103 relative to the entrance section, which correspond to mean velocities at this section of Uo=4.8xl 0 .2 m/s and 2.4xl 0 -1 m/s respectively. In this paper only the

413

results at the lower Reynolds number are presented, with an emphasis on the separation region near the lower channel entrance, as pictured by region (D) in figure 1. The PIV processing algorithm is of the iterative pattem deformation type, following roughly the ideas of Nogueira et al. (2001), and is described in detail in Usera et al. (2004). Special care has been taken to improve the performance of the PIV interrogation near the wall boundaries.

Figure 1. Sketch of experimental model with main paths of flow. (a) Side view with separation region shown at the lower channel entrance. (b) Front view. In order to have an overview of the characteristics of the system analysed a numerical simulation was performed. Figure 2 show the vortical structures visualised by means of L2 iso-surfaces (Jeong and Hussain, 1995) obtained in the numerical simulation. It can be seen that the flow in the upper channel is dominated by two counter rotating vortical structures. The non-stationary characteristic of the flow is also apparent from figure 2.

Figure 2. Vortical structures educed by L2 surfaces from related numerical simulation.

ANALYSIS OF MEASUREMENTS In the following analysis we will resctrict ourselves only to a selected portion of the flow, due to space constrains. The selected portion is the entrance to the lower channel in the vecinity of region (D) in figure 1. Figure 3 shows the streamlines for the mean velocity field obtained from the ensemble averaged velocity field of the complete 2550 frames PIV data set. It is seen that the m e a n velocity field is characterized in the region analised by a strong recirculation down stream of the leading edge of the plate that separates the upper channel from the lower one, with the centre of the re-circulation vortex at

414

x/h=l.7 and y/h=0.8. Thus, the main flow entering the lower channel is deflected downwards, passing through a section of haight about 2/3 the total height of the channel.

Figure 3. Mean velocity field streamlines in lower channel entrance region, showing re-circulation. This type of representation of the mean flow should not be pushed to far in drawing conclusions about the dynamics of the flow. For instance, if the instantaneous flow was to be permanently as pictured in figure 3, the heat transfer from the plate near the re-circulation region could be adversely affected. Thus, it is important to assess the transient evolution of the flow in this region, since transport of heat away from the plate might be enhanced by transient structures. The reminder of this analysis will be based onto the vorticity field, since it is expected to be well suited to signal the underlying structures in the re-circulation region, which are expected to be mainly vortical structures. In figure 4, the mean dimensionless normal vorticity field is pictured, as defined in Eqn. 1. 9

W z

.

--

2U0

m

.

2U0

where Uo is the main velocity relative to the entrance section and h/2 is the half height of the channel.

Figure 4. Normalised mean vorticity contours (Wz/(2Uo/h) ). Further plots will be restricted to the dashed region. Figure 4 shows that the relative maximum of vorticity are roughly aligned with the shear layer defined between the main flow entering the lower channel and the re-circulation area. This layer weakens towards x/h=2.5 marking the downstream extension of the re-circulation area. A region of interest around the re-circulation area has been signalled with a dashed line in figure 4. Further plots will be restricted only to this regionof interest. We proceed now with the analysis of the transient structures arising in this region. The first step is to compute the auto-correlation functions for the vorticity time series at three selected locations (figure

415

5). These locations are indicated in figure 4 by small squares and their co-ordinates are y/h=0.8 and x/h=l.0, 1.5, 2.0. The first one falls right outside the re-circulation region in the main flow while the second and third lay within this region. Dimensionless integral time scales for vorticity fluctuations at these locations are T/dt=l.5xl0 l, 2.4x10 l, 2.4x10 ~, respectively, while the dimensionless sampling time for each PIV series was Ts/dt=510. It is worth noting that, while the auto-correlation functions at locations 1 and 3 oscillate around zero after a few integral time scales, in the case of location 2 the auto-correlation function exhibits a deeper oscillation of larger amplitude and duration. This might signal the existence of cyclic vortex formation near this location, which in fact will be further suggested by the clustering analysis below.

0.8 0.6 .o O o

0.4 0.2 0 -0.2 -0.4 0

~.~.~~~ 50

100 150 Time steps

200

250

Figure 5. Auto-correlation ofvorticity time series at y/h=0.8 and x/h=l.0 (-o-), x/h=l.5 (-V-) and

x/h=2.0 (-~-) Next, proper orthogonal decomposition was applied to the time series of instantaneous vorticity fields within the dashed region of figure 4, which consisted of 10x48=480 grid points and 2550 time steps assembled in 5 series of 510 time steps each. The first 40 eigenvectors and eigenvalues were computed, accounting cumulatively for up to 70% of the total variance and with the last eigenvalue accounting for only 6% of it. The eigenvector fields for eigenvalues 1,2,3,4,5,10 and 20 are given in figure 6 that shows an increasingly finer hierarchy of vortical structures, as expected from a spectral like decomposition, with alternate vorticity sign within the re-circulation region. However, the analysis of the eigenvectors alone gives no indication of the temporal evolution of these structures. It should be noted that the results from POD analysis would be the same regardless of the temporal organisation of the instantaneous time steps. The information regarding the temporal organisation of the flow is kept then in the projections of the time series onto these eigenvalues and an analysis of these time series of projections is in order to recover this information. This will be the objective of the following paragraphs where a fuzzy clustering algorithm will be applied to determine a set of ensemble averages or clusters, which picture the most representative states of the flow and the transitions between them.

416

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x/h Figure 6. POD eigenvectors for vorticity field. From top to bottom: # 1, #2, #3, #4, #5, # 10, #20. The fuzzy clustering algorithm applied here is the c-Means clustering algorithm coupled with a validity criteria proposed by Xie and Beni, both of which are described in a general context in Xie & Beni (1991). Application of this method to time series of fields in fluid dynamics has been proposed in

417

Usera et al (2003). Here a brief description of the method will be given following the work of Xie and Beni.. The c-Means Clustering Algorithm seeks the minimisation, for a prescribed number of clusters c, and fuzziness index m (m> 1), of an objective function Jm, defined as:

Jm= ~ ~'~(~l'ij)md2(X j, Vi )

(2)

i=l j=l

where Xj (j=l,2 ..... n) is the data set to be clustered and d2(Xj,Vi) is a measure of the distance between the vectors Xj and the cluster centre Vi, usually taken as the Euclidean distance, and btij is the fuzzy membership function that verifies the condition: ~-"g~j =1

(3)

i=l

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(4)

j=l

j=l

Also, the fuzzy membership function can be computed from a specified set of centroids Vi from

/1

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1

(5)

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The minimisation of Jm is attained by iterating equations (4) and (5) from a starting guess of either gij or Vi until no further improvement of Jm (equation 2) is observed. Still, the c-Means Clustering Algorithm by itself gives no the clue about the number of clusters to be used in partitioning the set. In order to determine the optimum number of clusters to be used for each data set, as well as the value of the fuzziness index m, the validity criterion for fuzzy clustering, proposed by Xie & Beni (1991), was applied. For any given partitioning, regardless of the algorithm used to determine it, a compactness and separation validity function S, can be defined as:

S = i=l j=l

n.min d2(Xi,Xj) ij

(6)

The validity function S can be regarded as the ratio of the measure of the compactness of the clusters over the minimum separation among clusters. Optimum values of c and m are those for which S reaches an absolute minimum value. In some cases however one might be content to find a relative minimum value.

418

For the vorticity field data considered here an optimum value for S was found at c=13 and m=l.16. Due to space constrains not all cluster ensemble averages will be reproduced here, but only six of them selected due to frequent transitions that the flow exhibits among them. In figure 7 a first set of three clusters is presented, comprising clusters #8, #6 and #4, while in figure 8 the second set contains clusters #1, #2 and #7. The cluster index is assigned arbitrarily during the iteration process. The ordering of the clusters among each set was decided according to the frequency of the transitions of the flow between states corresponding to each cluster. 1

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x/h Figure 7. Cluster sequence 8-6-4 Figure 7 shows that an array of vorticity spots with alternate sign evolve downstream. In particular positive vorticity spot forms near the leading edge of the plate (clusters #8 and #6) and drifts then downstream (clusters #6 and #4). A similar picture emerges in figure 8, although there the vortex forming at the leading edge has negative vorticity, while a vortex with positive vorticity is evolving downstream and away from the plate. The downstream drift, and downwards form the plate, of these structures is expected to be associated with a net transport of heat away from the plate that would enhance the limited transport of heat associated solely to the re-circulation pattern of the mean velocity field. The probability of occurrence of each cluster is given in figure 9. This figure can also be interpreted as the percentage of cumulative time spent by the flow in configurations classified as belonging to each cluster. From this point of view, the series presented in figures 7 and 8 account respectively for 22% and 28% of the time, so that together they represent the evolution of the flow for about 50% of the total time.

419

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CONCLUSIONS A Fuzzy Clustering Algorithm has been presented and applied to the analysis of time resolved PIV measurements in a turbulent flow inside a PCB enclosing. The mean statistical properties and the transient dynamics of the flow have been obtained and analysed with the purpose of future validation of LES simulations of this same flow.

420

The selected portion of the flow presented in this communication shows a re-circulation region beneath the plate at the entrance of the lower channel. Events of altemate vortex shedding have been identified through the fuzzy clustering analysis. These events, which were hinted from the auto-correlation function, may enhance the transport of heat beneath the plate, with respect to that which would be produced by the mean flow. Future work will consider an evaluation of the temperature field and its relationship with the velocity through simultaneous measurements with through two-colour Laser Induced Fluorescence (LIF) in order to quantify the effect of these structures on the transport of heat.

ACKNOWLEDGEMENTS

This work was financially supported by grants 2002 SGR00189 from AGAUR Generalitat de Catalunya, DPI2003-06725-C02-01 from DGI, Ministerio de Ciencia y Tecnologia y Fondos FEDER, and a pre doctoral grant from Fundaci6n URV assigned to Gabriel Usera.

REFERENCES

Berkooz, G., Holmes, P., Lumeley, J. L., 1993, "The proper orthogonal decomposition in the analysis of turbulent flows", Annuel Review Fluid Mechanics, Vol. 25, pp. 539-575 Ferr6, J. A., Giralt, F., 1989, "Pattem-recognition analysis of the velocity field in plane turbulent wakes", Journal of Fluid Mechanics, Vol. 198, pp. 27-64 Ferrd, J. A., Mumford, J.C., Savill, A.M., Giralt, F., 1990, "Three-dimensional large eddy motions and fine-scale activity in a plane turbulent wake", Journal of Fluid Mechanics, Vol. 210, pp. 371-414 Ghil M., Allen M. R., Dettinger M. D., Ide K., Kondrashov D., Mann M. E., Robertson A. W., Saunders A., Tian Y., Varadi F., Yiou P., (2001) "Advanced spectral methods for climatic time series", Reviews of Geophysics, 40, 1, 2002 Jeong J., Hussain F., (1995) On the identification of a vortex. Journal of Fluid Mechanics, 285 69-94 Kopp, G. A., Ferr6, J. A., Giralt, F., 1997, "The use of pattern recognition and proper orthogonal decomposition in identifying the structure of fully-developed free turbulence", Journal of Fluids Enginnering, Vol. 119, pp 289-296 Nogueira J., Lecuona A., Rodriguez P. A., (2001) Local field correction PIV, implemented by means of simple algorithms and multigrid versions. Measurement and Science Technology, 12 1911-1921 Usera G., Vernet A., Ferre J.A., (2004) Considerations and improvements on analysing algorithms for time resolved PIV of turbulent wall bounded flows. 12th International Symposium. Applications of Laser Techniques to Fluid Mechanics. Usera G., Vemet A., Pallares J., Ferre J.A., (2003) On the organization of the cross-stream flow field in a square duct. 5th Euromech Fluid Mechanics Conference, Toulouse, France Vemet, A., Kopp, G. A., Ferr6, J. A., Giralt, F., 1999, "Three-dimensional structure and momentum transfer in a turbulent cylinder wake", Journal of Fluid Mechanics, Vol. 394, pp. 303-337 Xie, X. L., Beni, G., (1991) A Validity Measure for Fuzzy Clustering, IEEE Trans. Pattern Anal. Machine Intell., Vol. PAMI-13, No 8, pp 841-847.

421

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

SKIN FRICTION MEASUREMENTS IN COMPLEX TURBULENT FLOWS USING DIRECT METHODS J.A. Schetz Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute & State University Blacksburg, VA, 24061-0203, USA

ABSTRACT Skin friction (or wall shear) measurement is an important topic for both scientific and practical reasons, so there is a long history of work in the area. There are two broad classes of methods indirect and direct measurements. The direct methods employ a movable element of the surface surrounded by a very small gap and connected to some type of flexure. One then measures the displacement of the movable element or the strain in the flexure to obtain the skin friction force acting on the movable element directly. This is the preferred approach for complex flow situations, and such devices are the subject of this paper. First, an overview of prior instruments is presented to provide some of the history and characteristics of these units. This is followed by separate sections considering: 1) sensitivity and range, 2) calibration, 3) time response, 4) mitigation of vibration effects, 5) interfering inputs, and 6) uncertainty. The paper concludes with a brief discussion of recent developments and directions for future research.

KEYWORDS Skin friction measurements, wall shear measurements, boundary layer measurements

INTRODUCTION The accurate measurement of skin friction, or wall shear, in complex flows is a matter of great interest for scientific and practical reasons. Here, "complex flows" is meant to include flows that are some combination of 3D, unsteady, perhaps transitional, mixing and possibly combustion, and/or high-speed with shocks over surfaces that may be contoured, rough, porous with injection or suction and even ablating or subliming. Measurement is critical, because current computational methods do not provide sufficiently accurate skin friction results for such complex flows. Indeed, a major role for skin friction measurements is to provide a challenging standard for CFD validation studies. Also from the scientific viewpoint, local skin friction values are central to all correlating techniques for turbulent flows through the friction velocity u, = (1;w/p)1/2. These correlating techniques form the basis for the development of

422 all turbulence models. From the practical viewpoint, skin friction is a key item in assessing the performance of any fluids engineering device, and the skin friction distribution can play a very important role in identifying and correcting problem areas in such devices. An emerging area of application for skin friction measurements is as critical and sensitive inputs for flow control systems. Since skin friction (or wall shear) measurement is an important topic, there is a long history of work in the area starting with Froude (1872). There are two broad classes of methods - indirect and direct measurements. The direct methods generally employ a movable element of the actual surface surrounded by a very small gap and connected to some type of flexure. One then measures the displacement of the movable element or the strain in the flexure to obtain the skin friction force acting on the movable element directly. The author and his students have attempted to develop a compilation of the main prior works, and a Table and 50 references can be found at: http ://www.aoe.vt.edu/-j schetz/Skin Friction/C f_Gage_Table&Refs.pdf Reviews of direct skin friction measurement techniques can be found in Winter (1977) and Schetz (1997). The direct measurement approach has many advantages in principle, e.g. no prior knowledge of the flow, even whether the flow is laminar, turbulent or transitional, is needed. Also, direct methods as defined here do not require optical access to the surface. This is not to imply, however, that the direct methods are easy to apply. Almost all case of practical interest involve very small viscous forces in comparison to other forces or effects (i.e. high Reynolds numbers), so great care and otten considerable engineering creativity is required to develop an accurate and reliable device of this type. The indirect methods rely upon some analogy or data correlation to use another measurement to imply skin friction, so they all presume a great deal of prior knowledge about the flow. A good discussion can be found in Nitsche, et al (1984). These methods have their place, with ease of use being their primary advantage, but they are prone to great uncertainty for any complex flow situation where a complete, prior database does not exist. Here, we will only consider direct measurements of skin friction with a gage employing a movable surface element. An idealized sketch of a direct-measuring skin friction gage is shown in Fig. 1. The movable wall element, or floating head, surrounded by a small gap (order 0.1 mm) is attached to some sort of flexure (illustrated here as a cantilever beam). The interior of the gage and the gap may be filled with a viscous liquid that serves the purposes of: 1) providing a continuous surface for the flow, 2) minimizing the effects of pressure gradients, 3) providing damping, 4) stabilizing the temperature of internal sensors, and 5) preventing foreign matter from lodging in the gap. A key feature of a gage design is a flexure that is weak in the direction of the small wall shear force, but strong in the direction of the much larger pressure forces normal to the head. A cantilever beam has the desired behavior, but other flexure configurations are also possible.

Figure 1" Idealized Schematic of a direct-measuring skin friction gage.

423

The design and implementation of a direct-measuring skin friction gage for a particular application involves many of the same issues as for any instrument, namely: 1) sensitivity and range including damage limits, 2) calibration, 3) time response, 4) interfering inputs, 5) uncertainty, etc. These issues will be discussed in the separate sections that follow.

SENSITIVITY AND RANGE The levels of shear are generally small, ranging from a few Pascals for low-speed gas flows to several hundred Pascals for high-speed flows. The size of the movable wall element (or floating head) directly affects the force to be measured, but one also wishes to have a small head relative to some characteristic length of the flow to obtain a local measurement. All of this is connected with the design of the flexure and the sensing system. The Table on the referenced website shows that floating heads with diameters on the order of 100 mm (4 in.) are not uncommon, and few truly "small" sizes have been successfully employed. About 10 mm is about the best that can be used reliably at this time. The efforts to develop M E M S gages are excluded here, because they have yet to realize their full potential, Sheplak et al (2004). If the response of the sensor is small, a relatively large movement of the floating head and flexure is required. This is undesirable from a number of viewpoints. First, the gap around the head must be enlarged to allow for greater movement, leading to flow disturbances and errors. Second, large movements of the head imply either tilt or vertical displacement, with attendant disturbances and errors. One method of compensating for these effects is to employ a so-called "nulling" gage arrangement; the shear force displaces the head and a balancing force is applied sufficient to move the head back to its original, or "null," position. Successful gages of this type have been used, e.g. Bruno et al (1969). If, on the other hand, the combination of the head size, flexure design and sensor can be arranged to limit the motion of the head and flexure to very small values, it is possible to develop successful skin friction gages with a "non-nulling" configuration. Clearly, a non-nulling configuration is to be preferred from the point-of-view of simplicity, small overall gage size and frequency response. At Virginia Tech, we have used semi-conductor strain gages to produce successful gages for a wide variety of applications, including contoured heads, roughness on the head and even porous surfaces with injection through the head. One example in Fig. 2 (a) displays some of the other issues involved in gage design. This gage was designed and used for very hot, supersonic flow. Since the piezoresistive strain gages are quite sensitive to temperature, great care was required in the gage design. The beam extension is hollow to reduce axial heat flow with fins to increase heat dissipation into the liquid fill in the housing. The floating, sensing head is the same thickness and material as the surrounding wall to match surface temperatures. The example in Fig. 2 (b) had a quartz tube beam which permitted water cooling of the backside of the head under extreme heat flux conditions. Other sensors have been investigated for skin friction gages. Using capacitance or magnetic proximity sensors to measure head or flexure displacement has not proven fruitful. One promising sensor utilizes fiber optics. Broadband light travels down the fiber to the polished end, where part is reflected and part is transmitted across the gap (order 100 ttm) and reflected from a polished surface on the far side. See Fig. 3 (a). An interference pattern is formed, and the output can be processed to measure small changes in the gap. An early implementation of this sensor into a skin friction gage is shown in Fig. 3(b). A fortunate result of the need for a small gap (0.1 mm) from fluid mechanics aspects is the creation of a "stop" so that motion of the head is constrained and damage to the sensing units is prevented.

424

Figure 2" Non-nulling skin friction gages for hot-flow tests.

Figure 3" Fiber-optic displacement sensor and skin friction gage.

425 CALIBRATION The simplest and best way to calibrate the type of gage in Fig. 1 is to apply a known force to the head by hanging a weight from a fine thread attached to the face, first turning the gage so that the face is in a vertical plane With a given surface area of the head, these point loads can be transformed into a corresponding shear level on the face. For a gage with two-component capability for 3D flows, each component is calibrated separately. Clearly, that this calibration method becomes difficult as the floating head size decreases, especially to the range of MEMS devices. The next best method is a constant-head tank filled with liquid that empties into a narrow, wide channel of sufficient length for a fully-developed, planar, laminar flow. One can then use the exact solution along with a measured pressure distribution to find the shear on the gage head in the channel.

TIME RESPONSE When the flow is nominally steady, the time response of a gage is not critical. There are unsteady flow situations where the time response is important. Also, some high-speed flow test facilities are of the impulse type, so time response is critical, even if the intent is to study flows that are nominally steady over the short test time (about 0.5-10 msec.) Making the reasonable assumption that the sensors are not the limiting factor, the time response is governed by the stiffness of the flexure and the mass of the head. It would seem a simple matter to design for a given target time response, however these quantities also determine sensitivity to vibrations and/or acceleration loads, which is important in impulse test facilities. Computerized Finite Element analysis (FEM) is very useful in the design process, Orr et al (2004) There are two general approaches to the problem. Some utilize piezo-ceramic elements as the sensor and include compensation for acceleration loads (Fig. 4(a)), but they are limited to 2D flow. Another concept is to use a head with a very low mass and a flexure that is stiff and light (Fig. 4(b)). The material was high- temperature plastic, and the sensors were semi-conductor strain gages.

Figure 4: Skin friction gages for short duration tests.

426 MITIGATION OF VIBRATIONS All test facilities have some level of background vibration, ranging from benign to severe. Avoiding vibration effects by gage design is possible, but it is not always feasible. Adding a constraint to have the natural frequency far removed from the background can make the design problem intractable. Filling the housing with a viscous liquid to provide damping works very well, but leakage and refill are always problems. We made a successful implementation of damping with a thin rubber sheet on the head, gap and housing as in Fig. 5(a). Even a thin sheet bears a part of the load from the wall shear, so one must increase the gap size and use FEM structural analysis for design (Fig. 5(b)). Wall shear from flow over the sheet covering the gap produces a load on the sheet that is partially transmitted to the main flexure.

Figure 5: Skin friction gage with rubber sheet for damping from Magill et al (2002). Remington and Schetz (2000) considered permanent and electric magnets. The gage proved useful for test environments with moderate vibrations, but it was unable to dampen very severe vibrations. Cantilever beam flexures are unbalanced from a dynamics viewpoint and prone to vibration problems. This has led us to a wheel flexure concept shown in Fig. 6(a). The hub of the lower wheel is fixed to the housing. The thin spokes (blades) of the wheel are instrumented with metal foil strain gages, and fiber optic sensors view the polished sides of the counter-weights. Careful static balancing resulted in a gage that was quite insensitive to vibrations, Orr, Schetz and Fielder (2004).

INTERFERING INPUTS The most common interfering inputs are pressure, acceleration, and temperature. A uniform change in static pressure should have a minimal effect. Acceleration effects were discussed above. The effects of temperature variations can enter in a number of ways. First, some sensors are very sensitive to temperature changes. Second, a temperature variation can introduce thermal distortions that may be of the order of the displacements one is trying to measure. A temperature variation may also change the structural properties of the flexure. It is possible to include temperature changes in calibration, but we have found that unsteady and/or spatially non-uniform temperature changes cause more problems, and that is hard to include by calibrating at different, uniform "soak" temperatures. A

427 less obvious effect occurs where the surface heating environment produces a temperature difference between the surface of the floating head and the surrounding wall. A rapidly changing wall temperature affects local heat flux, and Reynolds analogy suggests that this will lead to a significant change in local wall shear. This has been confirmed by experiment, but there is no agreement on either the magnitude of the effect or how to calculate it. The safest course is to design the gage to minimize any temperature difference between the surface of the floating head and the surrounding surface.

Figure 6: Wheel flexure 3D skin friction gage from Orr, Schetz and Fielder (2004) UNCERTAINTY We will be concerned only with uncertainties related to the gage design itself, not with the complete instrumentation system. An obvious area of uncertainty is the design of the floating head and gap. Closely related are uncertainties from any misalignment of the head with the adjacent wall. Measurements for a few sets of conditions have been published, which provide the gage designer with useful information on how to minimize errors. The limitation to the available experimental results is that they are for a few particular cases that may or may not be applicable to a new gage design situation. For example, the helpful results of Allen (1980) were obtained for a nulling gage with a large floating head (127 mm diam.) in an unheated, supersonic flow (M=2.2). How should they be applied to a low-speed case or a hot, hypersonic flow with a much smaller head size compared to the local boundary layer thickness? The only study of these issues with modem CFD tools is MaeLean and Schetz (2003). CFD is attractive, because calculations can be made for any projected set of conditions and geometry. Typical results are in Fig. 7(a). Putting this all together, the designer can use experimental information and/or CFD tools to design skin friction gages where the errors due to head design, gap size and reasonable misalignment can be limited to about +/- 2%. The effects of an axial pressure gradient on a skin friction gage are a concern, and experiments suggest that a small lip thickness and small gaps help to minimize such influences. Also, one of the intended benefits of a liquid fill in the housing is to minimize pressure gradient effects. The idea is that low intemal velocities of an incompressible fluid cannot produce large static pressure variations. Supersonic flows can have very large pressure gradients, especially across shocks, and there has not been a definitive experimental study for the supersonic flow regime. Again, CFD would seem to be a good approach to pressure gradient issues for all regimes. Some CFD results are in Fig. 7(b) where the pressure gradients were created by imposing a virtual diverging or converging channel above the gage.

428 For hot-flow conditions, the biggest uncertainty is surely the result of unsteady and/or spatially nonuniform temperature variations. With careful design, we have conducted tests under very hot-flow conditions for a few seconds where estimated errors from temperature were kept to about +/-10%. A very crude summary of the uncertainty for direct skin friction measurements including gage design, calibration, pressure and temperature might be: 1) unheated subsonic and supersonic flows without shocks impinging on the gage about +/- 5-10%, 2) hot subsonic and supersonic flows without shocks impinging on the gage about +/- 10-20% and 3) supersonic flows With shocks impinging on the gage have uncertainties too large to be useful. To put all of this in perspective, a comparison of skin friction measurements and predictions for a flow of moderate complexity is given in Fig. 8.

Figure 7: CFD predictions of flow in skin friction gage from MacLean and Schetz (2002)

Figure 8: Measurements and prediction for a Mach 2.4 flow from Orr, Schetz and Fielder (2004)

429 FUTURE DEVELOPMENTS Looking forward, it is clear that MEMS gages or truly miniature gages fabricated by other techniques are about to revolutionize skin friction measurements. Issues such as calibration, 3D capability, robustness, high-temperature capability and sensitivity to particle contamination remain to be fully addressed, but progress is being made. Fiber-optic sensors also hold great promise, whether in miniature or simply small skin friction gages. Many of the potential benefits have already been demonstrated in the work reported in Orr, Schetz and Fielder (2004). Finally, the routine application of accurate skin friction gages for active flow control is almost upon us. We have experimented with active control for vibration mitigation, but we have not yet produced a gage ready for routine use. We and our partners at Luna Innovations, Inc. have also experimented with nanomaterials for skin friction gages, and some success with the gage in Fig. 9 has been achieved. The bond holding the nanotubes to the substrate was weak, and it failed during testing and sometimes during handling. Further work is needed in that and other areas, but good agreement with predictions was achieved in a Math 2.4 flow for a few tests.

(a) Enlarged photo ofnanotube gage head, flexure and substrate

-~ :Fiber~l~-~iliconchip ~ GN~nWthotubb~adrray ~ B Bond l~ ~NA~~ Colader SiliconSubstrate (b) Schematic of the nanotube gage Figure 9: Nanotube skin friction gage from Henderson (2004) Important issues in uncertainty quantification for all types of skin friction gages, such as the effects of pressure gradients, still need much work. Modem computational tools for statics, dynamics, heat transfer and fluid mechanics can, and have, play an important role in those matters as well as in the entire gage design process.

430 ACKNOWLEDGEMENTS

The author is indebted to a large number of bright and hard-working graduate students who played key roles at every stage in the development of our skin friction gages at Virginia Tech. In addition, Kent Murphy and his colleagues at Luna Innovations, Inc. have been excellent collaborators. Finally, the financial support of NSF, NASA and the USAF is gratefully acknowledged.

REFERENCES

Allen, J. M. (1980) "Improved Sensing Element for Skin-Friction Balance Measurements," AIAA d., 18:11, 1342-1345. Bowersox, R., Schetz, J. A., Chadwick, K., and Diewert S. (1995) "Technique for Direct Measurements of Skin Friction in High Enthalpy Impulsive Scramjet Experiments." AIAA d., 33:7, 1286-1291. Bruno, J. R., Yanta, W. J., and Risher, D. B. (1969) "Balance for Measuring Skin Friction in the Presence of Heat Transfer," Final Report. USA Naval Ordnance Lab., NOLTR-69-56. Chadwick, K., DeTurris, D.J. and Schetz, J.A. (1993) "Direct Measurements of Skin Friction in Supersonic Combustion Flowfields," d. Eng. Gas Turbines and Power, 115:3, 507-514. DeTurris, D., Schetz, J.A., and Hellbaum, R. F. (1990) "Direct Measurements of Skin Friction in a SCRAMjet Combustor." AIAA Paper 90-2342. Froude, W. (1872) "Experiments on the Surface-friction Experienced by a Plane Moving through Water." 42naBritish Association Report, 42, 118-125. Henderson, Bancrot~, IV (2004), "An Exploratory Study of the Application of Carbon Nanotubes to Skin Friction Measurements," MS Thesis, Virginia Tech, Blacksburg, VA. MacArthur, R.C. (1963) "Transducer for Direct Measurement of Skin Friction in the Hypersonic Shock Tunnel." CAL Report 129, Comell Aero. Lab., Buffalo, NY. MacLean, M. and Schetz, J.A. (2003) "Numerical Study of Detailed Flow Affecting a Direct Measuring Skin-Friction Gauge," AIAA d., 41:7, 1271-1281. Magill, S., MacLean, M., Schetz, J., Kapania, R., Sang, A. and Pulliam, W. (2002) "Study of DirectMeasuring Skin-Friction Gage with Rubber Sheet for damping," AIAA d., 40:1, 50-57. Nitsche, W., Haberland, C. and Thunker, R., (1984 )"Comparative Investigations of the Friction Drag Measuring Techniques in Experimental Aerodynamics, ,, ICAS 84-2.4.1, 14th ICAS Congress. Orr, M.W., Schetz, J.A. and Fielder, R.S. (2004) "Design, Analysis and Initial Tests of a Fiber-Optic Shear Gage for 3D, High-Temperature Flows," AIAA 2004-0545. Pulliam, W.J. and. Schetz, J.A. (2001) "Development of Fiber Optic Sensors for High Reynolds Number Supersonic Flows," AIAA 2001-0245. Remington, A.. and Schetz, J.A. (2000) "A Study of Magnetically Damped Skin Friction Measurements for High Vibration Environments," AIAA 2000-2522. Schetz, J.A. (1997) "Direct Measurements of Skin Friction in Complex Flows," Appl. Mech. Rev., 50,:11,2, 198-203. Sheplak, M., Cattafesta, L., and Nishida, T. (2004), "MEMS Shear Stress Sensors: Promise and Progress," AIAA 2004-2606. Winter, K.G. (1977) "An Outline of the Techniques Available for the Measurement of Skin Friction in Turbulent Boundary Layers." Progress in Aerospace Sci:, 18, 1-57.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

431

REYNOLDS NUMBER DEPENDENCE OF ELEMENTARY VORTICES IN TURBULENCE K. Sassa i and H. Makita 2 1 Department of Natural Environmental Science, Kochi University, Kochi, 780-8520, Japan 2 Department of Mechanical Engineering, Toyohashi University of Technology Toyohashi, 441-8580, Japan

ABSTRACT The present experiment aims to investigate the Reynolds number dependence of f'me-scale coherent eddies called elementary vortices universally existing in various turbulence fields. The elementary vortices were detected in stable-stratified grid turbulence, homogeneous shear turbulence and a nocturnal surface layer of which turbulence Reynolds number, Rx, ranges from 19 to 2391. The conditional sampling was conducted referring to the steep change in transverse component of velocity fluctuation peculiar to the Burgers vortex. The resultant wave traces given by conditional averaging assured the existence of the elementary vortex similar to the Burgers vortex in all of the turbulence fields. The diameter of the elementary vortex was estimated to be about 10 times of the Kolmogorov scale being regardless of Rx. The maximum azimuthal velocity normalized by the Kolmogorov velocity increased in proportion to Rx~ except for the data obtained in shear flow fields. Such dependence accords with the scaling law for the elementary vortex introduced by DNS studies.

KEYWORDS Turbulence, Vortex, Coherent structure, Reynolds number, Conditional sampling, Measurements, Hotwire, Universal structure.

INTRODUCTION Fine-scale coherent vortices in turbulent flow fields were firstly found in various DNS fields (e.g. Hosokawa & Yamamoto 1989, Jimenez et al. 1993 and Tanahashi et al. 2001). Now, such vortices, named the elementary vortices by Kida et al. (2002), is known to exist universally in various types of turbulence fields. It is expected to make more detailed inspection on their fluid dynamical features, which will give us new aspects of understanding on the nature of turbulence. Recently, Ishihara et al. (2003) realized the DNS turbulence field reaching Rx=1200 using one of the most powerful computer system in the world, the Earth simulator. Its resolution was, however, not enough to examine the universality in the structure of the elementary vortices in a large-scale turbulence such as an

432 TABLE 1 CHARACTERISTICSOF TURBULENCEFIELDS

ATM ST1 ST2 LSE " LSG

Turbulence field i.. . . . . noeturn'alsurface layer homogeneousshear turbulence..... homogeneousshear turbulence stable-stratifiedand excited turbui~ee stable-stratifiedgrid turbulence

U (m/s) 1.9 8.0 5.0 5.0 5.0

u'/U 0.334 0.153 0.149 0.248 0'030

,,,w'H./ 0.154 0.092 0.097 0.136 0.020 ,,

%l(mm) 0.71 ' 0.23 0.32 0.14 0.31

R~, 2.391 820 586 300 19

atmospheric turbulence field. Their high-resolution DNS has achieved only Rx<700. Then, the existence of the elementary vortex and the Reynolds number dependence of its characteristics have not been examined yet for high-Reynolds-number turbulence fields of Rx>1000 accompanied by satisfactorily wide inertial subrange in their energy spectra. As for experimental works, Belin et al. (1996), Sassa (2000) and Mouri et al. (2003, 2004) detected the elementary vortices through hotwire measurements. Especially, Belin et al. achieved Rx--5000 in a helium gas flow, however, they conducted measurements by using their own unique hotwire system. We think it preferable to get more reliable data to examine Reynolds number dependence of the characteristics of the elementary vortices through conventional experimental methods. The present experimental study investigates R~ dependence of the structure of the elementary vortex in various turbulence fields. We try to detect the vortices by single X-probe measurements and a conditional sampling method in grid turbulence fields, uniform shear turbulence fields and an atmospheric surface layer.

EXPERIMENTAL METHODS

Realization of Turbulence Fields We examined three kinds of typical turbulence fields, i.e., nocturnal surface layers denoted by ATM, stable-stratified grid turbulence by LSG and LSE and homogeneous shear turbulence by ST1,2 as shown in Table 1. Their turbulence Reynolds number, Rx, ranges from 19 to 2391. The field observation was made at the North end of Toyohashi University of Technology (Sassa et al. 2003). The campus locates on a relatively flat terrain about 4 km inland from the Pacific Ocean. A well-devdoped quasi-steady surface layer without flow separation caused by buildings and hills was formed, when a land breeze blew from North at night. Measuring equipments were settled at 8 m high from the ground. A cup anemometer and a wind direction vane were used to monitor the mean wind speed and its direction. Three thermocouples were also settled at three different heights to check the stability of the air. Velocity and temperature fluctuations were simultaneously measured by an I-X probe composed of an upstream constant current coldwire (I-probe) and downstream constant temperature hotwires (X-probe) and a high-precision thermo-anemometer originally designed by Makita et al. (1994). The frequency response is completely flat up to 3.5 kHz and the signal to noise ratio of thermo-anemometer is as good as about 70 dB. The I-X probe having a spatial resolution of 0.7 x 0.7 • 0.4 mm 3 was always directed to the mean wind direction and gave the streamwise and lateral components of the velocity fluctuation in addition to the temperature fluctuation. The data were stored in a 16bit digital recorder at 24 kHz sampling rate. In each analysis, calculation was made by selecting 20 minutes data from the all recorded data during which the wind characteristics were confirmed to be satisfactorily steady.

433

In Table 1, ST1 and ST2 are uniform shear turbulence fields generated in a laboratory wind tunnel by installing a shear generator and an active turbulence generator (M=35mm) composed of many randomly flapping agitation wings (Makita & Sassa 1991) upstream of a test section of 70 cm • 70 cm in cross section. Mean velocity at the center of the cross section was U = 8 m/see, and 5 m/see, and the Corrsin shear parameter, x/u(~gu/i)z), was 5.3 and 4.6 for STI and ST2, respectively. The turbulence energy was kept almost constant downstream for both cases. Measurements were made at the center of the cross section at X/M=100. The spatial resolution of the X-probe was 0.8 • 0.8 • 0.4 mm 3. LSE and LSG are homogeneous stably-stratified turbulence fields without shear realized by setting the active turbulence generator and a thermal stratification generator in a smaller wind tunnel (Sassa et al. 2000). The size of the test section is 35cm high and 45cm wide. For LSE, the active turbulence generator violently agitated air flow and induced strong turbulence, whereas the generator was stopped and acted as a static grid with a mesh size of M = 25 mm for LSG. For both of the cases, the mean velocity, U, and the mean temperature gradient, dO/dz, were 5 rn/sec, and 30 K/m, respectively. Measurements were made at X/M = 20 and 80 using the same I-X probe and thermo-anemometer as the field measurements. All of the data were recorded for 5 minutes and analyzed by a computer system. For all turbulence fields, the coordinates x, y and z were set to the streamwise, horizontal and vertical directions. Conditional Measurements

Generally, the elementary vortex is considered to have a configuration similar to the Burgers vortex. Then, as schematically shown in right hand side of Figure 1, a quick velocity change defined by the following equation must be observed in the wave trace of w-component when the elementary vortex parallel to the y-axis passes through the X probe. w = 2~:r

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434

We performed conditional sampling to detect the coherent vortex by referring to the large velocity gradient, dw/dt, as follows. aw/at (2) Here, C denotes the detection threshold in order to pick up a large velocity gradient. A similar detection scheme was already proposed by Mouri et al. (2003, 2004). The present scheme of the measurement, however, does not give us instantaneous 3-D velocity fields, being different from those obtained in DNS studies based on strict detection schemes (Kida and Miura 1998, Tanahashi et al. 2000, 2001). If the holographic PIV was employed as reported by Bos et al. (2002), we can get 3-D velocity fields experimentally. But, such systems are not available in high-Reynolds-number turbulence fields such as atmospheric turbulence. In the present experimental situation, the magnitude of the detection threshold affects the determination of the characteristics of the elementary vortex as shown in Figure 2. Actually, we selected a constant value of C = 2.5, in order that the diameter of the coherent vortex in our detection scheme is about 101] being almost equal to the value estimated by Kida et al. (2002). Each 1000 points data were ensemble-averaged around the detection point.

We also checked availability of the present single-probe detection scheme by comparing the present data with those obtained by an array of 5 X-probes. Now, we use the maximal value of the 2nd invariant of the velocity gradient tensor, Q (Tanahashi et al. 2000, 2001), for the 2-D velocity field obtained from the results of the measurements by the X-probes array.

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435

where Sij and W o are the symmetrical and anti-symmetrical parts of the velocity gradient tensor. The probability density distribution of the elementary vortex diameter is shown in Figure 3. Though the detected events did not agree for the two detection conditions of the large transverse velocity gradient and the maximal value of Q, the results of the distributions were almost the same each other. As shown in Figure 4, the ensemble-averaged velocity vector map clearly shows a typical vortical structure similar to the Burgers vortex even when dw/dt was employed as the detecting condition. We consider that the present single-probe detection scheme is fairly available, when the discussion is restricted on the mean characteristics of the elementary vortex.

RESULTS AND DISCUSSION

Characteristics o f Turbulence fields Table. 1 summarizes the statistical features of the turbulence fields. The turbulence intensities for ATM, ST1, ST2 and LSE exceeded 10 % and were about one order of magnitude larger than the value for LSG. The spatial resolutions of the probes employed in the present measurements are about same as the Kolmogorov dissipation scale, rl, in each case. The turbulence Reynolds number is more than 103 for the atmospheric surface layer (ATM). For the excited turbulence field of LSE, Rx reaches about 300. The value is one order of magnitude larger than that for the grid turbulence (LSG), though it does not reach the value of the atmospheric surface layer. With uniform shear, ST1 and ST2 can obtain larger Reynolds number reaching about 500. One-dimensional energy spectra of streamwise velocity fluctuation in these turbulence fields are compared in Figure 5. For LSG, the low wavenumber range of the spectral distribution is almost flat and the clear inertial subrange is scarcely observed in it. For LSE, ST1 and ATM, energy level is evidently elevated in the low wavenumber range and the inertial subrange satisfying the Kolmogorovis -5/3 power law becomes clearly observed in the spectra. Apparently, the width of the inertial subrange increases with the Reynolds number. 10

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436

Conditional Averaged Wave Traces Figure 6 shows the ensemble-averaged wave traces of the vertical velocity component. These traces have wave forms quite similar to that of the Burgers vortex as shown by dashed lines in all of the turbulence fields, clearly demonstrating the existence of the elementary vortices. The decay of the velocity outside the vortex core becomes slower with increasing Rx. This, we guess, is because the number of the vortex having the axis not strictly-directed to the y-direction increases with the turbulence intensity. Namely, the present detection method inevitably catches many unwished vortices having various irregular angles of incidence. But, such irregularity seems to scarcely affect the results of the estimation around the vortex core, when the detection threshold was properly selected. Similar tendency was also confirmed by Tanahashi et al. (2002). The maximum azimuthal velocity and the diameter of the elementary vortex are defined by the appearance of sharp plus and minus peaks and the interval of the peaks in these wave traces. The intensity of the maximum azimuthal velocity

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437 normalized by Kolmogorov velocity, Wmax/Uk,is observed to increase with Reynolds number except for ST 1 shown in Figure 6(c). Figure 7 shows the ensemble-averaged wave traces of the streamwise velocity component. As stated above, the elementary vortex can be approximated by the Burgers vortex. Then, the radial velocity component represented solely by the streamwise velocity component in these cases, must appear as a converging flow toward the vortex axis. The radial flow pattern converging to the vortex axis is slightly observed for LSG in Figure 7(a), but it cannot be observed in the other cases. These wave traces may be contaminated by the irremovable effect of azimuthal velocity component of the vortices not parallel to y-axis. In addition to the above mentioned effects, the value of streamwise velocity obtained by an X probe is inevitably contaminated by the v-component. The turbulence intensity increases as Reynolds number becomes larger. Therefore, the influence of the cooling effect of the vcomponent on each hotwire sensor must be taken into account, when the aspect of the change in streamwise velocity traces is analyzed for Rx>300. The bump observed around the detection point in the wave traces of Figures 7(b)-(d) may be introduced by the axial velocity component when the elementary vortex passes through the probe position. Similar effects were also pointed out by Mouri et al. (2003). The amplitude of the bump for ST1 is smaller than that for LSE though the Reynolds number of ST1 is larger than that of LSE. Therefore, it may be related to the turbulence intensity rather than Reynolds number (see Table 1). The present results seem to suggest that the employment of a triple-wire probe is more desirable.

Reynolds Number Dependence of Characteristics of Elementary Vortex From the wave traces, we picked up the characteristics of the elementary vortex: diameter, maximum azimuthal velocity and vortex Reynolds number, F / v , defined by the ratio of circulation of these vortex and kinematic viscosity. Figure 8 shows their dependence on the Reynolds number. In the figures, the results by Kida et al. (2002), Tanahashi et al. (2001), Mouri et al. (2003, 2004) and Belin et al. (1996) are also plotted. The diameter of the elementary vortex is kept to be about 10 times of Kolmogorov dissipation scale, being regardless of R~ in spite of the ambiguity of our detection criterion as shown in Figure 8(a). Kida et al. (2002) and Tanahashi et al. (2000) also showed the diameter independent of Reynolds number by DNS. As for the maximum azimuthal velocity, the Reynolds number dependence differs by the scaling velocity. When it is normalized by the Kolmogorov velocity, Wm~/Uk, it increases with R except for ST1 and ST2. Then, its Reynolds number dependence is about Wm~/Uk ~ R ~ . Such tendency is roughly corresponds to the scaling law introduced by Kida et al. (2002), though our power index, 0.28, is about twice as large as their value. The maximum azimuthal velocities in turbulent boundary layers measured by Mouri et al. (2004) are slightly larger than the present data. This is because they employed the detection threshold larger than the present experiment. The data of ST1 and ST2 are quite small and out of the regression line. One of the reasons may be the essential difference in characteristics of the flow fields. ST I and ST2 are the shear turbulence fields, whereas LSE and LSG are homogeneous turbulence without shear. Of course, the atmosphric boundary layer, ATM, and the turbulent boundary layer of Mouri et al. (2004) are also shear flows. But, their mean velocity gradients are so small that the small-scale turbulence fields are assumed to be almost isotropic. In the shear turbulence fields, the elementary vortices are apt to be aligned parallel to the principal axis of strain tensor (Tanahashi et al. 2001) and the number of vortex normal to the principal axis of the strain may decreases in the cases of STI and ST2. As shown in Figure 8(c), the maximum azimuthal velocity normalized by the rms value of the fluctuating velocity, Wm~/W', decreases with R~ contrary to the case normalized by the Kolmogorov velocity. Its Reynolds number dependence was given as w,,,,,/w'~ Rx*~1. The grid turbulence, LSG, has no inertial subrange in its spectrum and, generally

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,

1000 R~

Figure 8: Reynolds number dependence of (a) diameter, (b) maximum azimuthal velocity normalized by Kolmogorov velocity, (c) maximum azimuthal velocity normalized by the rms value of fluctuating velocity and (d) vortex Reynolds number of the elementary vortex. 9 :ATM, 9 :LSE, LSG, ~ :ST1, 2, o :DNS(Kida et al. 2002), O :Helium tank (Belin et al. 1996) V :Grid turbulence (Mouri et al. 2003), /x :Boundary layer (Mouri et al. 2004) -9- :Regression line for the present data ........:Re-dependence(Kida et al. 2002),----:Re-dependence(Tanahashi et al. 2002) I :critical Reynolds number(Belin et a1.1996)

439 speaking, the consistency in nature of a turbulence field is strongly affected by the magnitude of energy source. Therefore, the data of LSG may be better to be removed from the present regression analysis. In this case, Wmax/W'becomes almost independent of the Reynolds number. The vortex Reynolds number also increases with Rx as shown in Figure 8(d). But, the data are slightly scattered. So, we must conduct further experiments in order to obtain more reliable Reynolds number dependence of F / v . Belin et al. (1996) reported that the coherent vortex structure for Rx >700 is completely different from that at the smaller Reynolds numbers. We could not observe such tendency, though we have conducted the measurements in various turbulence fields of different natures. We guess that Belin et al. has some problems of their experimental system.

CONCLUSIONS The elementary vortex exists in various kinds of turbulence fields independent of their turbulence Reynolds number. Its profile is well approximated by the Burgers vortex and has almost constant diameter of d = 1011. The maximum azimuthal velocity was scaled by both Kolmogorov velocity and the rms value of the fluctuating velocity. The maximum azimuthal velocity normalized by Kolmogorov velocity increases with R ~ whereas it decreases with R~~ when normalized by the rms value.

References Belin F., Maurer J., Tabeling P. and Willaime H. (1996) Observaton of Intense Filaments in Fully Developed Turbulence. J. Phys.II France 6, 573-583. Bos F., Tao B., Maneveau C. and Katz J. (2002) Effects of small-scale turbulent motions on the filtered velocity gradient tensor as deduced from holographic particle image velocimetry measurements. Phys. Fluids 14:7, 2456-2474. Hosokawa I. and Yamamoto K. (1989) Fine Structure of a Directly Simulated Isotropic Turbulence. J. Phys. Soc. Japan 58, 20-23. Ishihara T., Kaneda Y., Yokokawa M., Itakura K. and Uno A. (2003) Spectra of Energy Dissipation, Enstrophy and Pressure by High-Resolution Direct Numerical Simulations of Turbulence in a Periodic Box. J. Phys. Soc. Japan 72:5, 983-986. Jimenez J., Wray A. A., Saffman P. G. and Rogallo R. S. (1993) The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65-90. Kida S., Goto S. and Makihara T. (2002) Elementary Vortex in turbulence fi Structure and dynamical roles. Proc. of 2002 Meeting of Japan Society of Fluid Mechanics, Sendai, Japan, 21-25, (in Japanese). Kida S. and Miura H. (1998) Swirl condition in low-pressure vortices. J. Phys. Soc. Japan 67:7, 2166-2169. Makita H., Moil S. and Yahagi A. (1994) Spontaneous generation of internal gravity waves in a wind tunnel. Stably stratifiedflows: Flow and dispersion over topography (ed.; Castro, I. P. and Rockliff, N. J.), 81-91. Makita H. and Sassa K. (1991) Active turbulence generation in a laboratory wind tunnel Adavances in Turbulence 3 Springer-Verlag. 497-505. Mouri H., Hori A. and Kawashima Y. (2003) Vortex tubes in velocity fields of laboratory isotropic turbulence: Dependence on the Reynolds number. Phys. Rev. E 67, 016305. Mouri H., Hori A. and Kawashima Y. (2004) Vortex tubes in turbulence velocity fields at

440 Reynolds numbers Re0 ~300-1300. Phys. Rev. E (to be appeared). Sassa K. (2000) Detection of Fine-Scale Coherent Structures in a High-Reynolds-Number Turbulent Flow. Special publication of NAL SP-48, 61-64. Sassa K., Matsunaga S., and Makita H. (2000) Buoyancy Effect and universal structure of a stably stratified turbulence field. Advances in Turbulence VIII (ed.; Dopazo), 291-294. Sassa K. Makita H. and Sekishita N. (2003) Detection of fine-scale coherent vortices in atmospheric turbulence. Proc. Conference on Modelling Fluid Flow '03, Budapest I, 487-494. Tanahashi M., Iwase S. and Miyauchi T. (2001) Appearance and alignment with strain rate of coherent fine scale eddies in turbulent mixing layer. J. Turbulence 2, 006. Tanahashi M., Ootsu M., Fukushima M. and Miyauchi T. (2002) Measurement of coherent fine scale eddies in turbulent mixing layer by DPIV. Engineering Turbulence Modelling and Experiments 5, Pergamon 525-534. Tanahashi M., lwase S. and Miyauchi T. (2000) Appearance and Alignment with Strain Rate of Coherent Fine Scale Eddies in Turbulent Mixing Layer. Advances in Turbulence VIII, Kluwer Academic Publishers 655-658.

Engineering TurbulenceModellingand Experiments6 W. Rodi (Editor) 9 ElsevierLtd. All rights reserved.

441

Near-wake turbulence properties in the high Reynolds incompressible flow around a circular cylinder by 2C and 3C PIV R. Perrin, M. Braza, E. Cid, S. Cazin, F.Moradei, A. Barthet, A. Sevrain, Y. Hoarau

Institut de Mdcanique des Fluides de Toulouse, CNRS/INPT UMR n~

Toulouse, France

ABSTRACT The main objective of the present experimental study is to analyse the turbulence properties in unsteady flows around bluff body wakes and to provide a database for improvement and validation of turbulence models, concerning the present class of nonequilibrium flows. The flow around a circular cylinder with a low aspect ratio (L/D=4.8) and a high blockage coefficient (D/H=0.208) is investigated. This confined environment is used in order to allow direct comparisons with realisable 3D Navier-Stokes computations avoiding 'infinite' conditions. The flow is investigated in the critical regime at Reynolds number 140,000. A cartography of the velocity fields in the near wake of the cylinder is obtained by PIV and Stereoscopic PIV techniques. Statistical means and phase-averaged quantities are determined. Furthermore, POD analysis is performed on the data set in order to extract coherent structures of the flow and to compare the results with those obtained by the conditional sampling technique. The Reynolds stresses, the strain-rate and vorticity fields as well as the turbulence production terms are determined.

KEYWORDS Detached Turbulent flow, PIV, POD

INTRODUCTION The accurate quantification of the turbulence properties in the near wake region is of a priority interest concerning the physical analysis and the turbulence modelling of unsteady separated flows past bluff bodies. This comprehension is a prerequisite for elaborating adapted and efficient turbulent modelling techniques for this category of flows characterised by a double physical nature, organised and chaotic. In this context, the main objectives of the present study are to provide a detailed cartography of the coherent structures in the near wake, subjected to the effects of the fine-scale turbulence. On this purpose, the 2D-PIV and 3C-PIV techniques have been used. The results have been processed, beyond the Reynolds averaging, by using the phase-averaging and the Proper Orthogonal Decomposition. A discussion on the coherent, organised part of the flow as well as on the incoherent, turbulent part is addressed.

442

EXPERIMENTAL SET-UP

Configuration The ex2periment has been carried out in the wind tunnel S 1 of IMFT. The channel has a 670x670 mm square cross section. The cylinder spans the width of the channel without endplates and has a diameter of 140 mm, giving an aspect ratio L/D=4.8 and a blockage coefficient D/H=0.208. The upstream velocity Uinf at the centre of the channel is 15m/s, so the Reynolds number based on the upstream velocity and the cylinder diameter D is 140,000. The free stream turbulence intensity, measured by hot wire technique in the inlet was found 1.5%. All the quantities have been dimensionless by the Uinf and D.

Measurements Wall pressure measurements have been carried out with a pressure transducer Validyne DP 15-20. The frequency response is fiat up to about 500 Hz. The acquisitions have been achieved during 2 to 5mn at a sampling frequency of 1 kHz. The uncertainty on the Cp coefficient is estimated to 3.5%. For the PIV measurements, a double-pulsed Nd-YAG laser Quantel (2x200mJ) and PCO-sensicam cameras (1280x1024 pixels) have been used. The seeding particles used have been DEHS. A typical size of the particles is l gm. The system, both camera and laser, has operated at a frequency of 4Hz. The measurements have been carried out in the near wake of the cylinder in the (x,y) plane located at the middle span position z=0 (Figure 1a)

2C-PIV: The camera was equipped with a 35mm objective lens at a diaphragm aperture of 11. The size of the measurement area has been 238 x 188 mm (1.34D x 1.7D). The software used to analyse images is a product of IMFT ("service signaux-images"). The algorithm is based on a 2D FFT cross-correlation function implemented in an iterative scheme with a sub-pixel image deformation, according to Lecordier & al. (2003). The flow has been analysed by cross-correlating 50% overlapping windows of 32x32 pixels. This yielded fields of 77 x 61 vectors with a spatial resolution of 3.13mm (0.0224D). This resolution is proven to be sufficient for the evaluation for the evaluation of the major part of the turbulent stresses, according to tests that have been carried out by using smaller PIV planes and a calculus of vector with an interrogation window of 16x16 pixels. Nevertheless, the smallest-scale turbulence beyond the above resolution cannot be provided in the energy spectrum, as is generally the case for any PIV experiment.

Figure 1 a)flow configuration b) Scheimpflug configuration 3C-PIV: three-component measurements were carried out by using stereoscopic PIV, both to check the influence of the w component normal to the plane on the results and to quantify the third normal Reynolds stresses w 2 . The angular configuration of Scheimpflug is employed (Figure lb). Similarly to Willert (1997), the two cameras were placed on either side of the

443

light sheet, thus both images are stretched identically and it is possible to view the same area by the two cameras. The procedure employed to calculate the 3 components is the same as in Cid & al (2002). The cameras were equipped with a Scheimpflug adaptor designed by the LML (Laboratoire de M6canique de Lille) and with a 35mm objective lens at a diaphragm aperture of 11. The mean spatial resolution is similar to the 2C-component one. The domain of measurement is 0.6<x/D<2.28 and -0.7
About 3000 pairs of images were analysed to generate converged turbulence statistics with the 2C-PIV. The uncertainties are estimated, using a 95% confidence interval. They are 0.02 for U, 0.03 for V mean components, and 0.015 for u 2 , 0.02 for v 2 and 0.01 for uv correlations. For the 3C-PIV, 2570 instantaneous flow fields were acquired. The differences between the 2C-PIV and 3C-PIV results are lower than 0.03 for the mean components and lower than 0.04 for the correlations. Phase averaging

The nearly periodic nature of the flow, due to the von KArmAn vortices, allows the definition of a phase and the calculation of phase averaged quantities. The flow is classically decom__posed into a mean component, a periodic fluctuation and a random fluctuation as U, = U, + U, + u,' (Reynolds & Hussain (1972)). The phase average quantity is then(U,) =U~ +U, . The phase averaged quantities have been measured with the 2D-PIV technique. The trigger signal used as an indicator of the vortex shedding is the pressure on the cylinder at an angle 0=70 ~ with the forward stagnation point. This location is near the separation and upstream the transition. Therefore, the signal has a strong quasi periodic component at the Strouhal frequency, and it is not very affected by turbulent fluctuations. A typical pressure signal and its power spectrum are shown on Figure. 2. 10' 2 to'

lg

.os

1r

.1

1o'

lo'

.2

....

b)

'~'

'~

Figure 2. a) wall pressure signal at 0 = 70 ~ b)power spectrum

Both, PIV images and pressure signal are acquired and stored to obtained phase averaged quantities. By post-processing the pressure signal, the phase is determined at each instant of acquisition of PIV. The flow fields are then ranged in 16 classes corresponding to phase angles dividing a period ; each class is in fact a window of width 2n/128. Statistics are then performed in these classes. From the trigger signal, the phase of the flow is determined by using the Hilbert transform technique, as in Wlezien & al (1979). The Hilbert transform allows calculation of the instantaneous envelope and phase from a band limited signal. Before applying the Hilbert techniques, a bandpass filter is applied to the signal. The instants where the phase cannot be determined because of the presence signal irregularities are detected and removed by a sort based on thresholds on amplitude and periods. The independence of the results from these parameters has been checked.

444

Finally, about 170 images are collected per classes and averaged. The estimated uncertainties are 0.07 for , 0.1 for , 0.05 for , 0.08 for , 0.04 for . FLOW REGIME Measurements of mean wall pressure coefficient have been carried out in the median section z/D=O, around the cylinder, every 10 ~ for different Re numbers from 65,333 to 191,333 (Figure 3a). The mean pressure drag coefficient is evaluated by integration of the pressure (Figure 3b). The base-pressure coefficient, (-Cpb) is found higher than in nonconfined flow conditions, because of the blockage ratio. This yields a drag coefficient higher than in a non-confined case. The drag decrease shows that the flow is at the beginning of the critical regime. This regime occurs at lower Reynolds number than reported in Roshko (1961 ), because of the free-stream turbulence intensity (1.5%). 1

\

o

3o

6o

~ti

12o

.a)

1~

1~

b)

lo'

.......

'1

lOI Re

........

lO'

Figure 3. a) Mean wall pressure coefficient around the cylinder. b) Mean pressure drag coefficient versus Re REYNOLDS AVERAGED FIELDS The topology of the mean flow at Re=140,000 is studied in this section according to Reynolds averaging decomposition. Streamlines, iso-U and iso-V contours measured by 2C-PIV are shown in Figure 4. Capital letters indicate Reynolds averaged quantities. As expected, a two symmetric eddies pattern is obtained, due to the averaging of the passage of the alternating vortices, resulting in a symmetric pattern for U and in an antisymetric one for V. As expected, the mean spanwise component W, measured by 3C-PIV is found to be null. The dimensionless recirculation length lc is found 1.28 +0.03 with 2C-PIV and 1.23 +0.03 with 3C-PIV. Values between 1.1 and 1.4 are found by several authors (Cantwell & Coles (1983), Norberg (1998) .... ) in the same Re number range. Given the difference in the boundary conditions (blockage, aspect ratio and inlet turbulence intensity) and experimental details having an important influence especially in the critical regime, where the global parameters vary rapidly with Re number, the present results appear to be reasonable.

Figure 4. Mean velocityfield, a) streamlines b) iso contours of U c) iso contours of V

445

The mean velocity gradients, then the strain rate tensor S and the rotation rate tensor o9 are calculated with a central difference scheme. Figure 5 shows the iso-contours of $12

=l(Ou

OV]and

2~,/)y+~xx

l(Ov ~ ] As expected, a two lobes antisymetric configuration a121=2 ~xx"

is obtained. The mean strain rate and the mean rotation rate are of the same order of magnitude. The maximum of vorticity is found at the location x/D=0.6 and y/D=0.55.

Figure 5. a) mean strain rate $12 b) mean rotation rate o912 Figure 6 a, b, c, d. represents the iso-contours of the Reynolds stress tensor components. The u 2 component has a two-lobe structure with maximum values located near x/D=l and y/D=+_0.5. On the rear axis, the maximum value is found to be at x/D-1.3, that is close to the recirculation length, v 2 has a one- lobe structure with the maximum value 0.6 on the rear axis

at x/D=l.4, w 2 presents also a one-lobe structure with lower value than the other normal stress. Its maximum value is found 0.18 at x/D =1, that is comparable with LDV measurement of Norberg (1998). Concerning the shear stress field uv, the maximum values (+0.2) are located on either side of the wake centre line at x/D=l.4 and y/D=+0.3. As a classical result, uw and vw have been found null by 3-C PIV, confirming the two dimensional character of the mean flow in the centre of the channel. As observed by many authors, it is noticeable that all the components have their maximum value near the vortex formation region. Then the turbulent kinetic energy, evaluated from the normal stresses, exhibits a one-lobe structure with the maximum located at x/D=l.25 (Figure 6d). The turbulent production term ~ / ) U ~ is evaluated and shown on Figure 6e. Significant values coincide with the P =-u~uj Oxj turbulent kinetic energy, but the maximum values are located near the shear layer at x/D=0.6

and y/D=+_0.55.

Figure 6.a) b) c) mean Reynolds stresses : normal components

446

Figure 6. d) mean shear stresses e)mean turbulent kinetic energy J) production PHASE AVERAGED FIELDS

The power spectrum of the wall pressure signal (Figure 2) exhibits a peak at the frequency 22.5 Hz, corresponding to the vortex shedding at a dimensionless frequency (Strouhal number) St=f D/U-0.21. Figure 7 shows the streamlines of the phase averaged flow at phase angles O, ~/2 and re. The periodic vortex shedding is clearly shown. The dimensionless vorticity is represented on Figure 8 at four phase angles in the mean period. It is shown that the vorticity peak at the centre of a vortex decrease from 3 to 1 (in absolute value) when the vortex moves downstream from x/D=0.6 to x/D=2. Furthermore, the region of significant vorticity (taken arbitrary to o9>0.5) increases from a width of about 0.8D to 1D. The vortices centres (identified with the Q criterion, Jeon & Hussain, (1995)) have been marked for each phase angle and the mean trajectory is shown on Figure 9. At x/D=2, the trajectories seem to be nearly parallel to the rear axis at a distance y/D = 0.25, as found by Cantwell & Coles too. The longitudinal mean celerity of the vortices is evaluated by taking the derivative of the trajectories, to reach a value of 0.7 U~nfatx/D-2. The Reynolds stresses at constant phase are evaluated. Figure 10 shows the normal components (u2), (v 2) and the shear stress (uv) at the phase angle (o-=7c The general topology is found comparable to the result of Cantwell & Coles. This indicates that the normal stresses have high values near the centre of the vortices, while the maximum of the shear stress are located around the vortices. This is observed downstream of the formation region. In the formation region, significant values of (u:) and (uv) are found in the shear layers. This topology is also found comparable to the results of Leder (1991) who measured phase averaged quantities in the near wake of a flat plate by LDA technique.

Figure 7 streamlines of the mean velocity fields at constant phase angles.

447

Figure 8.1so-contours of the mean vorticity at constant phase.

%. 5

O8ot 0

....

oo

~8 ~' ~ ~'

2

o

0

,,-

~

130000

.

.

,

:

1

. . . .

i

X

15

. . . .

t

2

,

,

Figure 9 a) Trajectories of the alternating vortices. Circles indicates the centres of the vortices and number indicates the phase (1: r .. 16:30rd16)

Figure 10. Reynolds stresses at constant phase

448

The global Reynolds stress (in statistical sense) can be decomposed in two contributions: the periodic motion and the random motion. The definition of this decomposition ensuring that the two contributions are uncorrelated, the Reynolds stresses can be expressed as:

UiUj =Uiuj +(Uitujt) Figure 11 shows the two contributions of the shear stress. The topology and the level of them are comparable, the maximum values of the random motion being located quite nearer the cylinder than the maximum value of the periodic motion. This behaviour was also indicated by Cantwell & Coles conceming the non-confined cylinder experiment.

Figure 11. Shear stress. Contributions of the periodic motion and of the random motion COHERENT STRUCTURE IDENTIFICATION BY MEANS OF POD From a data set U(X), the Proper Orthogonal Decomposition analysis consists in searching the function0(x) that is most similar to the members of U(X) on average (Berkooz & al. (1993)). This is done by solving an eigenvalue problem where the kemel is the two point correlation tensor R(x,x') = u(x)u(x'). In the present study, the snapshots method has been used with the 3000 instantaneous flow fields. The POD has been performed on the fluctuations away from the mean field. The average operator used is the statistical mean. Figure 12 shows the energy of the 100 first modes. Beyond the 10th mode the energy drops under 1% and then diminishes quite slowly. This is due to the highly turbulent nature of the flows. The reconstruction of an instantaneous field with 5, 10 and 20 modes is shown on Figure 13. It seems that 10 modes are Figurel 2. Energy of the sufficient to obtain the essential of the von K~krm~nvortices.

modes (in % of the total)

Figure 13. Reconstruction of an instantaneous field with N modes

449

If we consider the task of extracting the von K~rmfin vortices from the full fields, the POD technique allows, by the reconstruction with a sufficient number of modes, identification of the position of the vortices for each instantaneous field. There is a dispersion of the large scale motion in a population of fields at a fixed phase. Hence, the phase averaged motion shows the vortices in some average position. On Figure 14, the instantaneous velocity fields (taken at a phase angle gr-=a:) is represented. On Figures 14b,c the fluctuation by subtracting the phase averaged field and the fluctuation by subtracting the first 10 modes reconstruction are shown. Large scale fluctuations are observed near the vortices when the fluctuations are calculated away from the phase average fields. This illustrates the dispersion of the vortices at a constant phase. Therefore, the contribution (u i 'uj') evaluated in the previous section is due both to the random small scale fluctuation and to the dispersion of the large scale vortices.

Figure 14. a) Instantaneous field: U. b) fluctuations away from the phase averaged ~(k) ---._ field.'U (U) c)fluctuation away from the 10 mode reconstruction: ~ a k (x) k=l 1

According to the diagonal decomposition of the correlation tensor in the POD, the contribution of the N first modes and the rest can be evaluated as: k

N

~

k=l

k=N+l

Figure 15 shows the two contributions of the shear stress, by taking N =10 modes.

Figure 15. Shear stress a) Contribution of the l Othfirst POD modes b) contribution of the other modes It appears that the essential of the shear stress is produced by the large scale vortices. The random contribution to the shear stress is essentially located, as expected, in the shear layers (15b). By comparing 15b and 11b, it can be seen that the phase-averaged chaotic contribution to the shear stress includes the effect of the large-scale irregularities due to large-scale chaotic

450

motion, whereas the POD decomposition allows extracting better the finer-scale chaotic contribution. CONCLUSION The present study is an analysis of the coherent flow pattem and of the impact of the random turbulence in the flow past a circular cylinder at the beginning of the critical regime. The confined character of the flow allows further 3D numerical simulations of the exact flow configuration, by using reasonable grid sizes comparing to non-confined experiments. The present study identifies the topology of the near-wake coherent structures and of the turbulent stresses according to the Reynolds-averaged, phase-averaged and POD decompositions. Acknowledgement : Part of the present study was carried out in the context of the European Commission Research Programme FLOMANIA G4RD-CT2001-00613.

REFERENCES Berkooz G., Holmes P. & Lumley J. L. (1993). The proper orthogonal decomposition in the analysis of turbulent flows, Annual Rev. Flui Mech. 25, 539-75 Cantwell B. & Coles D. (1983), An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J. Fluid Mech., 136, 321-374 Cid E., Cazin S. & Drouin V. (2002), Validation de PIV st6r6oscopique et application a un 6coulement a6rodynamique de sillage 3D. 8e Congres Francophone de Fdlocimdtrie Laser. Jeong J. & Hussain F. (1995), On the identification of a vortex, J. Fluid Mech, 285, 69-94. Lecordier B. & Trinite M. (2003), Advanced PIV Algorithms with Image Distortion Validation and Comparison from Synthetic Images of Turbulent Flow, PIV03 Symposium Busan, Korea. Leder A. (1991), Dynamics of fluid mixing in separated flows, Physics of Fluids A 3, 17411748 Norberg C. (1998), LDV-measurements in the near wake circular cylinder, ln: Conference on Bluff Body Wakes and Vortex lnduced Vibrations Presented at ASME Fluids Engineering Division (Annual Summer Meeting), Washington DC (FEDSM98-5202) Reynolds W. C. & Hussain, A. K. M. F (1972), The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments, J. Fluid Mech, 54, 263-288. Roshko A. (1961), Experiment on the flow past a circular cylinder at very high Reynolds number, J. Fluid Mech, 10, 345-356. Willert C. (1997), Stereoscopic digital particle image velocimetry for application in wind tunnel flows, Meas. Sci. Tech. 8, 1465-149. Wlezien R. W. & Way J. L. (1979), Tecniques for the experimental investigation of the near wake of a circular cylinder, AIAA Journal, 17:6, 563-570.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

451

SINGLE- AND TWO-POINT LDA M E A S U R E M E N T S IN THE TURBULENT NEAR WAKE OF A CIRCULAR CYLINDER A. Ducci, E. Konstantinidis, S. Balabani and M. Yianneskis Experimental and Computational Laboratory for the Analysis of Turbulence, Department of Mechanical Engineering, King's College London, Strand, WC2R 2LS, UK

ABSTRACT Turbulence characteristics were determined in the near wake of a circular cylinder at R d = 6700 and 7200 by means of LDA measurements. Conditional averaging was employed to separate the contributions from large-scale organized motions associated with the passage of vortices shed from the cylinder and two-point measurements provided information on the spatial coherency of the fine-scale turbulence during the early stages. Between five and ten diameters downstream of the cylinder, the contribution from coherent motions to the total stresses shows a significant drop associated with the diminishing strength of the shed vortices. The corresponding contribution from random turbulence shows less of a variation but the conditionally averaged distributions tend to become more homogeneous and isotropic. This trend is reflected in the measurement of the Taylor length microscales. The dissipation rate of turbulence kinetic energy was determined by direct measurement of the mean squared spatial velocity gradients using various assumptions. Using a band-pass filtering technique to remove the coherent part of the signal, it is shown that 85-90% of the total dissipation is due to random fluctuations.

KEYWORDS Laser-Doppler anemometry, turbulence, near wake, coherent structures, two-point measurements.

INTRODUCTION

The measurement of turbulence in real engineering flows is often complicated by the presence of coherent structures. As an example, one might consider the trailing vortices that form behind the impeller blades in an agitated vessel (Lee and Yianneskis, 1996). These structures play an important role in the transport of momentum, heat and mass and their understanding is of fundamental and practical significance in turbulence research. It is not surprising that a large body of literature has been devoted to their study and appropriate eduction schemes have been developed. The degree and type of flow organization due to coherent structures varies from one configuration to another or even within different regions of a particular flow. The wake behind two-dimensional bluff cylinders in cross-flow

452

is an archetypal example of organization in coherent structures. The region up to ten diameters downstream of the cylinder, termed the near wake, is dominated by large-scale structures and a high degree of organization due to the shedding of coherent vortices which form an alternating vortex street. In the intermediate region between ten and fifty diameters downstream of the cylinder, there is a gradual decay of the peak vorticity but the overall organized motion does not vary appreciably with distance due to a balance between vortex diffusion and stretching. Further downstream in the far wake, the effect of coherent structures is weak and the decaying turbulence approaches local isotropy. As a result, the intermittency properties have been found to agree with a Gaussian distribution. There are numerous experimental studies of the turbulence characteristics in wakes, e.g. Townsend (1949), Cantwell and Coles (1983), Hussain and Hayakawa (1987), Zhou and Antonia (1993), Matsumura and Antonia (1993), to mention but a few. Many different approaches have been employed using mostly hot wires, including single, X-arrays and rakes. Furthermore, various conditional averaging techniques have been developed that enable the organized motions due to vortex shedding to be separated from random fluctuations due to overriding turbulence. Previous studies have concentrated on the intermediate and the far wake whereas the near wake region has received much less attention. This partly stems from difficulties in obtaining reliable measurements with hot wires in highly turbulent and directionally unsteady flows as is the near wake (Ong and Wallace, 1996). LaserDoppler anemometry (LDA) offers a suitable technique to study the near wake region in great accuracy and in a non-obtrusive manner, e.g. see Lyn et al (1995). The description of turbulence is incomplete without information on the length scales of the flow. In the context of large eddy simulations (LES) information on the spatially coherency of the flow is required in order to make important decisions on grid resolution and sub-grid scale modeling. The success of more traditional approaches to turbulence modeling also depends on this information which may be used to validate closure assumptions or determine empirical constants involved in the models. An experimental route to obtain this kind of information is the two-point correlation function which requires the simultaneous measurement of the instantaneous velocity fluctuations at two points. Such measurements may be used to estimate micro and integral turbulence length scales, and the dissipation rate of turbulence kinetic energy. Since the subject of this paper is not on experimental techniques, a discussion on the limitations and difficulties using this approach and particularly in relation to LDA, is omitted. Relevant measurements have been attempted in the far wake by few investigators (Browne et al, 1987; Schenck and Jovanovich, 2002) using elaborate and intrusive hot wire assemblies. However, two-point measurements in the turbulent near wake behind cylinders are scarcely found in the published literature. This paper describes an experimental study of the turbulent near wake of a circular cylinder by means of single- and two-point LDA measurements. The study aims at improving understanding of the relation between large-scale coherent structures and fine-scale turbulence in vortex shedding flows and expanding the availability of data to other properties such as length microscales and dissipation. To the knowledge of the authors, the present study is one of the very few in which two-point LDA measurements are attempted in the near wake of a cylinder. EXPERIMENTAL DETAILS AND ANALYSIS TECHNIQUES

Experimental setup Experimental data has been obtained in a recirculation type water tunnel with a test section of 72 x 72 mm 2. The test section was preceded by a contraction section (contraction ratio 0.11) fired with hexagonal honeycomb and screens to condition the inflow. The latter was uniform except the thin boundary layers on the walls with a turbulence intensity of 0.033. The test section was made of transparent acrylic plastic to facilitate optical access for LDA measurements. A circular cylinder with a

453

diameter d = 7.2 mm was placed centrally and spanned the entire test section. Mean inflow velocities of U = 0.93 and 1 m s-1 were used in the experiments so that the Reynolds number, based on U and d, was Rd- 6700 and 7200, respectively. In the coordinate system used, x is aligned to the main flow direction, y and z are perpendicular and parallel to the cylinder axis, respectively. The corresponding velocity components are denoted u, v and w (not measured), respectively. All measurements were obtained at x/d = 5 and 10 and in a x-y plane midway along the cylinder span assuming a uniform flow on the spanwise direction on the average. However, it should be pointed out that the instantaneous flow structure contains both coherent and random fluctuations associated with three-dimensional structures and turbulence, respectively. A three-probe four-component LDA system (DANTEC TM)was used for the velocity measurements in the near wake of the cylinder. It consisted of a 5 W argon-ion laser, transmition and collection optics built-in with the probe-heads (backscatter) and four burst spectrum analysers for processing of the Doppler signals. The system could be configured in various ways to obtain multi-component and multi-point measurements as required. The flow was seeded with spherical particles with an average diameter of 10 ~m and specific gravity of 1.1 in order to obtain high signal-to-noise ratios. All individual Doppler bursts (velocity samples) from the analysers were registered with reference to a common clock source and stored on hard disk for subsequent processing.

Conditional averaging for triple decomposition The triple decomposition (Reynolds and Hussein 1972) has been applied to the instantaneous velocity signal, i.e.,

q=Q+'q+qr

(1)

m

where q (- u, v) is the instantaneous quantity, Q is the global mean (time-averaged) quantity, ~" is the periodic (or coherent) fluctuation and qr is the random (or turbulent) fluctuation. The triple decomposition is used here as it enables the contributions from the coherent and random components to the global Reynolds stresses to be determined. Furthermore, it allows estimation of the turbulent velocity scales and of the transfer of kinetic energy between organized motion and turbulence. However, it should be pointed out that by virtue of the methodology, the random component computed by the triple decomposition incorporates indistinguishable contributions from both small and large scales due to the inherent 'jitter' of the periodic flow field. A conditional averaging method was employed similar to that used by Matsumura and Antonia (1993). The phase on which ensemble averaging is based is provided by a reference oscillatory signal (ufluctuation) from one measurement probe located just outside the wake. The reference signal was band-pass filtered around the main frequency of vortex shedding in the wake, )Co. This frequency was detected from the main peak in the frequency spectrum and corresponded to a Strouhal number fod/U of about 0.21. The phase was calculated from the filtered signal as

(2)

where t,., and t2, ~ correspond to the peaks of the cyclic velocity fluctuations. Therefore, the interval t2.~-t,.~ defines the instantaneous period. It was verified that the average value of the instantaneous periods corresponds to To - f o ~. A total number of approximately 105 samples were acquired for each

454

velocity component at every measurement location and the shedding period was divided into twenty phase-intervals. Estimation o f spatial velocity gradients f r o m two-point measurements The mean squared spatial gradients of the instantaneous velocity fluctuations

((~///(~x) 2 , (t~///O,~) 2 ,

(Ov/Ox)2 and (~/Oy) 2 are estimated by calculating the finite difference functions f, e.g. for the first gradient,

(3)

f (Ax) = [u(x, y,z,t) - u(x + Ax, y,z,t)] 2

using simultaneous LDA measurements at two points separated by Ax. Similar difference functions are used for the other gradients. The condition of simultaneity is imposed by finding the particles that satisfy the following criterion: /~1)

- tj

(2)

(4)

< r w

where the superscripts refer to the two probe volumes and rw is a time coincidence window. This parameter has been carefully selected a posteriori (30 its) in order to avoid space-time correlation measurements (rwUr/-1 ~ 0.75). Once the function f is known, the spatial gradients are estimated by finding the slope of straight line fits to (Ax 2, f(Ax 2) data. Figure 1 shows the lines fitted to the data obtained at a measurement location in the wake and the variation of the gradients across the wake. The fitting range is between 0.05 mm (2.4r/) and 0.3 mm (6r/). Here r/=(V3/~) TM is the Kolmogorov length scale, g is the dissipation rate of turbulence kinetic energy determined experimentally from two-point measurements and v is the kinematic viscosity of water. The resolution of the present two-point measurements is better than that achievable by conventional PIV due to limitations associated with the sizes of particles and interrogation area. The procedure described has been assessed in grid-generated turbulence where the dissipation rate can be readily obtained from the decay of turbulent kinetic energy and a systematic error analysis has been carried out to determine and quantify the different sources of error affecting the measurement of spatial gradients. The error in the estimation of the spatial gradients is in the range of 16-22%. A more detailed description of the LDA configurations employed to measure the different spatial gradients and the sources of error can be found in Ducci and Yianneskis (2005). 150 125

I

(a)

I

O

I

|

> O

5-

o

100 ~

7 5 f 50

~7

V

~o

~

I

4r 3 1.

v

O

~

9 ~

9 o

(0u/0x)~

@

(OvlOy)2

9 0

(aulOy)~ (Ovlox)2

9 o 0 ~I

2

[]

o

0

o"

<>

zx (Au (ax))~ v (Au (Ay))2

25 0

0.0

1 ~t

0.1

n

i 02

(Av (ay)) 2 0

'. 03

AX2, Ay2 (mm 2)

'. 04

(Av (Ax) 2

' 05

00.0

(b)

tll

,. 05

1.'0 y/d

Figure 1" (a) The finite difference functions fused in the determination of the mean squared spatial gradients. (b) The calculated gradients across the wake.

2.0

455

MEAN VELOCITY AND REYNOLDS STRESSES Figure 2 shows the mean velocity deficit and the Reynolds stresses in the near wake. Values are appropriately normalized with the mean velocity deficit in the wake centreline U o - U - U and the wake half-width h, i.e. the lateral distance from the centreline to the point where the mean velocity is 0.5 Uo, which are summarized in Table 1. In what follows, an asterisk denotes normalization by Uo and h. The mean velocity distribution at x / d = 10 agrees well with that in the far wake but some deviation is observed at x / d = 5 as might be expected due to the proximity to the formation region. The total Reynolds stresses u'u* and u'v* do not change appreciably between the two axial stations but v'v* drops markedly. It should be noted that the absolute values of all three stresses actually decrease downstream but this effect is concealed by the normalization used. The maximum values of u'u" and u'v ~ across x / d = l0 compare favourably with the results of Matsumura and Antonia (1993) obtained for R d = 5830. However, v'v" is significantly lower in the present experiment. This is due to the dominance of the coherent motions to the transverse v fluctuations as is shown below, which are more pronounced in their experiment. The coherent part of the velocity fluctuations is deduced by conditional averaging, and their average over the shedding period is included in Figure 2. The coherent flow motions make a more pronounced contribution v ' v * than to either u'u ~ or u*v ~ . Note that the coherent stress ~*~* is nearly zero in the wake centreline, with maxima on either side in contrast to u*u ~ which displays a single maximum in the centreline. The percentage contribution of coherent stresses to u*u ~ , v~ and u'v * averaged across the wake drops from 16, 57 and 32 to 5, 26 and 9, respectively, between x / d = 5 and 10. This marked drop in coherent stresses may be attributed to the diminishing strength of the vortices shed by the cylinder with downstream distance. This contrasts the evolution of the wake further downstream of x / d = 10 which is characterized by a relevant constancy of the normalized coherent stresses (Matsumura and Antonia 1993). 1.00

!

x/d =

x/d =

,

i

x / d = 10

x/d = 5

10

0.75 ,.,O O

~

o

00

0.0 2.0

0.25

~

[300 no

2

1

i

i

1

0.00

2

0 O0

o

[]

oo

0000

o0

O0

0~ o

I

I

1

O

DoDoDD

/

[]

--

[]

~

~176 o

~

O

8 2

I

2

I

1

I

0 y*

10

[]

[]

oooOoooOOo

=

DDDD [] D [] DODD

..

[] O

~176176176

~00000000_0 I ~ 000000 0 1 y*

1 xld

O

~DDOOOO~ ~

00

2

o

~ooOqOOOOOOO

xM=5

x / d = 10

0

[] []

~oon

I

I

,

[] 1.0 -

0000

2

DD

-

oOO O

i

i [] []

[]

D oD 000 0

xld = 5

0.0

~ o

oo

0.2

0.5

Do O

~ Do

0.50

0.4

1.5

D o n " I'D r'l D D

oOO D~

0.6

DD o~

!

1

2

Figure 2: Mean velocity deficit and Reynolds stresses in the near wake. ~ mean velocity, [2 total stresses, 9 coherent stresses from conditional averaging. The solid line is the mean velocity deficit in the far wake, x / d = 420 (Browne et al, 1987).

456

TABLE 1 CHARACTERISTIC SCALES IN CYLINDER WAKES

Study Present Present Matsumura and Antonia (1993) Antonia and Mi (1998)

Ra

6700 5830 3000

x/d

Uo/U

h/d

10 10 10

0.29 0.23 0.19 0.22

0.70 0.94 0.81

0.68

ORGANIZED MOTION AND TURBULENCE STRUCTURE Figure 3 shows the results of ensemble averaging the velocity fluctuations at constant phase at x/d = 5 and 10. The streamlines display characteristic foci and saddle points associated with the periodic passage of turbulent vortices as seen by an observer moving with the convection velocity of the vortices, Uc ~ 0.9Uo. The streamwise coordinate is obtained using Taylor hypothesis, i.e. x* = - U c t / d . The actual shape of the sectional streamlines strongly depends on the convection velocity. Therefore, the choice of convection velocity of 0.9Uo was made such that the streamline pattern matches the vorticity contours. In this figure and in the rest of the paper the flow is from left to right. The contours of ensemble-averaged spanwise vorticity ~ deviate substantially from a circular shape at both axial stations. The circulation around the vortices and the peak vorticity at their centres decrease with downstream distance from the cylinder but the vortex structure does not change appreciably between the two stations. Comparison of peak vorticity and vortex convection velocity with other published data is shown in Table 2. There is satisfactory agreement with other reported data but those of Matsumura and Antonia (1993); their measurements gave significantly higher peak vorticity which indicates more pronounced organized motion than in the present case and may account for the difference noted in the v'v" profiles.

Figure 3" Streamline patterns in a reference plane which translates with the convection velocity of the vortices and corresponding contours of ensemble-averaged vorticity at constant phase.

457

TABLE 2 VORTEXCHARACTERISTICSIN CYLINDERWAKES Study

Rd

Present

6700

Armstrong et al (1986) Matsumura and Antonia (1993) Zhou and Antonia (1993)

21500 5830 5600

x/d lO 10 10

4.04 2.54 3.83 4.63 2.5

0.87 0.87 0.91 0.91 0.85

Figure 4: Ensemble averaged contours of random Reynolds stresses at x/d - 5 (left) and 10 (right); (a) (u*~u~),(b) (v~v~) and (c) (u*~v~).Contour levels increase by 0.1.

458

The large-scale coherent motions give rise to periodic velocity fluctuations which are not discussed further here for economy of presentation. The associated turbulence fields are shown in Figure 4 in terms of the random stresses (u',,u~.), (v',v',) and (u'~v',). The notation (-) stands for ensemble averaging at constant phase. High values of (u',u',) and (v',v',) occur near the vortex centre at both x/d = 5 and 10. However, the lobes of high (u',.u'~) become elongated at the downstream station and the absolute peak value of (v',v',) decreases by 25%. At x/d = 10, (u',u',) and (v~v',,) have a similar magnitude indicating some degree of isotropy of the fine-scale turbulence. The general appearance of (u',v',) contours coincide with the results obtained by Matsumura and Antonia (1993) at x/d = 10, however, there are some notable differences. High (u',.v',,) regions are found in the flow separatrices between successive vortices (saddle points) with absolute values of 0.4. In comparison the results obtained by Matsumura and Antonia show high (u',,v',,) regions close to the wake centreline with peak values of 0.21. At x/d = 5, there is a secondary peak near the vortex centre which does not exist at the downstream station.

TURBULENCE SCALES AND DISSIPATION Two-point measurements were employed to determine the length scales of turbulence and the viscous dissipation rate of turbulence kinetic energy. The Taylor length microscales, 20, were determined from the mean squared spatial gradients via equations of the following form (Hinze 1975),

G

2U2 =

(5)

~

The four different Taylor length scales determined are shown in Figure 5(a). The length scales are quite uniform across the wake because the measurements do not extent beyond the turbulent core. The Reynolds number based on turbulence intensity and Taylor micro-scale 2~ in the wake centerline is Rx = 306. With the exception of 2)~ that has a slightly higher value, the other three microscales have a similar value (2/d ~ 88 This indicates that the smallest scales of turbulence possess some degree of isotropy. This allows estimation of the dissipation rate of turbulence kinetic energy, 6, based on the assumption of local isotropy. The simplest method for the estimation of c is: 2

c=l

k, OxJ

(6)

A less restrictive assumption is that of homogeneity. Using the four directly measured spatial gradients and substituting the remaining unknown terms via isotropic relations, the equation for c may be reduced to

E

e = v -(8u~2 +8

tax;

+ 2 ( 0 u / 2 +2

ray)

(7)

Wygnanski and Fielder (1969) proposed the following 'semi-isotropic' expression:

(8)

459

with k - - ( o ~ Y / ~ x ) 2 / ( ~ / / / ~ X ) 2 . The estimates of the dissipation rate e using the equations above are shown in Figure 5(b). Equations 7 and 8 give similar results across the wake. Equation 6 yields higher values than Eqns 7 and 8 particularly in the wake centerline. This difference may be attributed to the fact that the flow is not completely isotropic so close to the cylinder. It should also be noted that the error in the estimation of the mean squared spatial gradients amounts to 16-22%. Despite this uncertainty the data do provide an estimate of the dissipation rate in the turbulent near wake of a circular cylinder, a parameter of prominent importance in turbulence modeling, for which data is scarcely found in the published literature. The dissipation rate determined above contains contributions from both coherent and random fluctuations. In order to separate these two contributions a band-pass filtering technique was employed instead of conditional averaging due to the lack of a reference signal in the two-point measurements. The periodic fluctuations due to the coherent vortices were subtracted from the coincident data which is subsequently used in the determination of the mean squared spatial gradients. Although this methodology is known to have some limitations, these are not expected to affect significantly the present results as the vortex shedding frequency is low compared to that of the turbulence fluctuations. The random part of the dissipation rate determined in this way is also shown as filled symbols in Figure 6. The results indicate that 85-90% dissipation occurs due to the random fluctuations. 3

.

,

,

9

0.6

!

A

0.5 []

,

!

! 9 9 9

DD

~Dn

n

0.4

[] ~

g

[] O ~

Eq. (6) Eq. (7). Eq. (8)

[]

0.3

[]

0.2

oo.o

,

o15

'

1 io

o

2 xx

0

3,ry

A

2yy

~7

2

'

y/d

11.5

'

0.1

o.oo.o

2.o

(a)

'

o15

'

11o

'

1 15

'

20

y/d

(b) Figure 5" Taylor microscale (a) and dissipation rate of kinetic energy (b) across the wake at x / d = 10. Open symbols denote total dissipation and solid symbols the dissipation due to random fluctuations.

CONCLUDING REMARKS The motivation for this work is to provide a description of turbulence in the near wake of a circular cylinder which is dominated by highly organized coherent motions associated with vortex shedding. To this end, single- and two-point LDA measurements were carried out five and ten diameters behind the cylinder at Rd = 6700 and 7200. Conditional averaging was employed in order to separate the contributions from coherent and random motions to the total Reynolds stresses. The results show the early stages of wake evolution and indicate that there is a substantial decrease in the coherent Reynolds stresses between x / d = 5 and 10. The random stresses show less of a variation with downstream distance but the corresponding conditionally-averaged distributions indicate a tendency to become more homogeneous and isotropic. Turbulence length scales and the dissipation rate of kinetic energy were determined from the mean square spatial velocity gradients. The four different Taylor microscales determined have similar values at x / d = 10 and are quite uniform across the wake which further supports the tendency of the smallest scales towards isotropy (which is certainly not completed in the near wake). The assumption of local isotropy gives higher values, particularly in the wake centreline,

460

for the dissipation rate of turbulence kinetic energy than those obtained based on the assumptions of either 'homogeneity' or 'semi-isotropy' which give similar values. Viscous dissipation occurs mainly by random fluctuations which account for 85-90% of the total dissipation. The study is one of the few experimental attempts to study the near wake of a cylinder in a nonobtrusive manner and also to measure directly the spatial velocity gradients, which is known to be a formidable task. Despite the uncertainty associated with the measurements reported here and the indicative nature of the dissipation estimates due to the hypotheses employed the present study makes a contribution to the understanding of turbulence in bluff-body wakes and provides valuable data for quantities difficult to measure directly.

REFERENCES

Antonia, R.A. and Mi J. (1998) Approach towards self-preservation of turbulent cylinder and screen wakes. Experimental Thermal and Fluid Science, 17, 277-284. Browne, L.W.B., Antonia, R. and Shah, D.A. (1987). Turbulent energy dissipation in a wake. Journal of Fluid Mechanics, 179, 307-326. Cantwell, B. and Coles, D. (1983). An experimental-study of entrainment and transport in the turbulent near wake of a circular-cylinder. Journal of Fluid Mechanics, 136, 321-374. Ducci, A. and Yianneskis, M. (2005). Analysis of errors in the measurement of energy dissipation with two-point LDA. Experiments in Fluids (Accepted). Hinze, J.O. (1975) Turbulence, 5th ed., McGraw-Hill. Hussain, A.K.M.F., and Hayakawa, M. (1987). Eduction of large-scale organized structures in a turbulent plane wake. Journal of Fluid Mechanics, 180, 193-229. Lee, K.C. and Yianneskis M. (1998). Turbulence properties of the impeller stream of a Rushton turbine. American Institution of Chemical Engineers Journal, 44, 13-26. Lyn, D.A., Einav, S., Rodi, W. and Park J.H. (1995) A laser-Doppler velocimetry study of ensembleaveraged characteristics of the turbulent near wake of a square cylinder. Journal of Fluid Mechanics, 304, 285-320. Matsumura, M. and Antonia, R.A. (1993). Momentum and heat transport in the turbulent intermediate wake of a circular cylinder. Journal of Fluid Mechanics, 250, 651-668. Ong, L. and Wallace, J. (1996). The velocity field in the turbulent near wake of a circular cylinder. Experiments in Fluids, 20, 441-453. Reynolds, W.C. and Hussain, A. (1972). The mechanics of the organized wave in turbulent shear flow. Part 3. Theoretical models and comparison with experiments. Journal of Fluid Mechanics, 54, 263-288. Schenck, T. and Jovanovi6, J. (2002). Measurement of the instantaneous velocity gradients in plane and axisymmetric turbulent wake flows. Journal of Fluids Engineering, 124, 143-153. Townsend, A.A. (1949). Momentum and energy diffusion in the turbulent wake of a cylinder. Proceedings of the Royal Society of London. Series A, 197, No 1048, 124-140. Zhou, Y. and Antonia, R.A. (1993). A study of turbulent vortices in the near wake of a cylinder. Journal of Fluid Mechanics, 253, 643-661. Wygnanski, J.C. and Fiedler, H. (1969). Some measurements in the self preserving jet. Journal of Fluid Mechanics, 38, 577-612.

461

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

AERODYNAMICS

OF A HALF-CYLINDER

IN GROUND

EFFECT

X. Zhang*, S. Mahon, M. Van Den-Berg and C. Williams Aerospace Engineering, School of Engineering Sciences University of Southampton, Southampton SO17 1BJ, U.K. ABSTRACT The flow around a half-cylinder placed in proximity of a moving ground is studied in a wind tunnel model experiment, with the aim of providing physical insight and a database. Reynolds number based on the base height of the model varies between 6.8 • 104 and 1.7 x 105. Measurements include forces, pressures, oil flow visualization, hot-wire and particle image velocimetry (PIV). Two distinct flow regimes exist with the critical gap to diameter ratio (h/d) between 0.525 and 0.55. When the half-cylinder is placed away from the ground (h/d > 0.55), the flow is characterised by strong periodic, alternate vortex shedding behind the model base, and high base pressure with associated high drag coefficient. The Strouhal number of the vortex shedding does not vary with the height and stays consistent at 0.22. A sudden jump in the force behaviour is observed between h/d - 0.525 and 0.55. Below h/d < 0.525, periodic vortex shedding disappears with a corresponding reduction in the base pressure, and the drag coefficient experiences a sudden drop. KEYWORDS

Half-cylinder, ground effect, moving belt, vortex, unsteady aerodynamics. NOMENCLATURE d

eL c~ D E(f) h L P qcr St

u~

x, y, z ~, V, W

base height drag coefficient lift coefficient pressure coefficient; (p - poo)/qoo drag spectrum frequency Height/gap of the model above the ground lift pressure dynamic pressure; pU~/2 Strouhal number; fd/Uoo free stream velocity cartesian coordinates: x+ve downstream, y+ve up, z+ve to starboard velocities in the x, y, z directions respectively

*[email protected] (Xin Zhang)

462

a

spanwise vorticity, (Ou/Oy - Ov/Ox)d/U~.

1. I N T R O D U C T I O N This study investigates the flow around a half-cylinder placed near a moving ground. The method is wind tunnel model tests. We wish (a) to provide physical insight into changes in force behaviours of a simple generic geometry in close proximity of ground, with major, representative physical features, and (b) to generate a well defined database to validate predictive, numerical methods, such as detached eddy simulation. An example of engineering relevance of this study is the prediction of force (and its change) behaviours of an inverted wing in ground effect (Zerihan and Zhang [2000], Zhang and Zerihan [2003]). The industry application can be found with high performance and race car front wings. A turbulent wake is generated off the finite trailing edge of the wing, containing discrete, alternate vortices away from the ground. In the close proximity of the ground, there is a change in the fundamental characters of the turbulent wake. A large flapping motion is coupled with the vortex shedding in the wake. A wall jet type of flow appears on the ground, beneath the turbulent wake. With these changes in physics there is a significant variation in the force behaviour. Recent attempts in trying to simulate the flow using computational fluid dynamics, e.g. (Mahon and Zhang [2005]), highlight modelling difficulties and the need for a benchmark case. Another example is the unsteady flows associated with cylindrical bodies on a landing gear. The vortex containing turbulent wake is responsible for far field noise generation. A highly relevant research area is the flow around a circular cylinder placed near a flat plate (Bearman and Zdravkovich [1978], Buresti and Lanciotti [1979], Angrilli et al. [1982], Taniguchi and Miyakoshi [1990], Price et al. [2002], Zdravkovich [2003]). Most of these model test studies employed a fixed ground, e.g. a flat plate. The studies produced some interesting but nonconclusive observations. Bearman and Zdravkovich [1978] conducted wind tunnel tests at Re = 2.5 and 4.5 x 104. A thick ground boundary layer (0.8d) was present on the fiat plate. They noticed a critical gap ratio, h/d, of 0.3 below which vortex shedding disappeared. Buresti and Lanciotti [1979] performed hot-wire measurements at Re = 0 . 8 6 - 3 x 105. The tests were conducted at a fairly high turbulence level (0.9%) and gave a critical gap ratio between 0.3 and 0.4, depending on the surface finish. The load cell measurement of Taniguchi and Miyakoshi [1990], mainly concerned with the effect of boundary layer, showed a critical gap ratio between 0.3 and 0.9. These studies are useful in that they reveal the existence of two flow regimes. The wall condition obviously inhibits normal motion of the fluids and exerts a larger influence at small gap ratios. The flow is seen to accelerate in the gap, leading to asymmetric near-wall pressure distributions. Many questions though remain unanswered. The flow is obviously governed by parameters such as Reynolds number, boundary layer thickness, turbulence level, surface finish, etc. We have seen a largely diverse set of critical gap ratios being reported, range from 0.3 to 0.9. It is interesting to note for a normal flat plate a critical gap ratio of 0.5 was given by Everitt [1982] and 0.35 by Kamemoto et al. [1984] for a triangular body. Unlike other studies, Angrilli et al. [1982] reported an increase in the shedding frequency as the gap became smaller. The effect of Reynolds number remains to be quantified. The flow around a half-cylinder contains less degree of freedom compared to a circular cylinder. However it does contain most of the major physics discussed earlier. There are few detailed studies of flows around a half-cylinder (Kumarasamy and Barlow [1995], Kumarasamy [1995]). In Kumarasamy's wind tunnel tests, a fixed ground was used and both surface pressure and hot-wire measurements were employed. The tests were done at Re = 4.67 x 105. A critical gap ratio of 0.33 was identified. However there was no sudden jump in the force behaviour. It does not appear that

463

the lift trend at large gap ratios are correct. A relevant study of a half-ellipse was conducted by Kim and Geropp [1997, 1998] which highlighted the effect of the moving ground condition, as with Zdravkovich [2003] using a circular cylinder. Kim and Geropp used laser doppler anemometry. It is somewhat surprising that similar vortex structures were observed at h/d = 0.2 and 1.0. The few isolated studies are either severely limited in parameter range or restricted by the use of physically incorrect, e.g. fixed ground plane. As a result a number of questions remain to be answered, one of which is the identification of critical flow regimes. The present flow would allow a well defined flow separation, thus facilitate the set-up of a numerical model. The use of a moving ground plane in the experiment ensures correct boundary conditions being observed.

2. D E S C R I P T I O N

OF EXPERIMENTS

2.1. H a l f - C y l i n d e r M o d e l a n d W i n d T u n n e l The half-cylinder model spans the wind tunnel and is aligned with the ground. Its axis lies parallel to the ground and the base normal to the freestream. A side-view of the model is shown in Fig. 1 and model installation in Fig. 2. The model is equipped with end-plates and the size of which is shown in Fig. 1. The model is mounted on a strut connected to an overhead force balance. In Fig. 1, the coordinate system used in the study is defined. The experiment was performed in the University of Southampton 2.1 m by 1.7m wind tunnel. The tunnel is of a conventional closed jet, closed circuit design. For correct modelling of the ground plane the tunnel is equipped with a large, moving belt rig. A system is located upstream of the belt for removal of the boundary layer that grows along the floor of the wind tunnel. The boundary layer is sucked away through a slot and a perforated plate. With the boundary layer suction applied, the velocity reaches the freestream value less than 2mm from the ground, corresponding to h/d < 0.02. The freestream turbulence level is 0.3%. The half cylinder model has a diameter of I00 mm and a span of 500 mm and is manufactured using a stereolithography technique. This ensures a high quality test specimen with pressure tappings. There are a total of 47 pressure tappins around the circumference of the model and 26 tappings in the spanwise direction. Measurements include surface oil flow, pressure tappings, forces, PIV and hot-wire anemometry. The range of model height (gap), h, varies from 0.15d to 1.75d. The surface of the model is polished by 1500 grit wet and dry sand paper. Reynolds number based on the maximum freestream speed and surface roughness is about 5. Reynolds number based on the model base height varies between 6.8 x 104 and 1.7 x 105 . 2.2. M e a s u r e m e n t s

Point turbulence measurements were obtained using a Constant Temperature Anemometry (CTA) system. Two single-wire probes were used simultaneously, both positioned at a centre span location. The streamwise location of the probes was set at x/d - 1.0 with vertical locations of y/d -- 0.5 and y/d = -0.5. Each probe consisted of a Platinum (10% Rhodium) wire, 2.5 microns in diameter orientated in a vertical sense. The ratio of the length of the active element within each probe, to the wire diameter was 200. A Newcastle anemometer system (Anemometer modular rack system - Instruction manual [1996]) was used incorporating a Whetstone bridge and signal conditioner with the bypass frequency set at 9.2 kHz. The analogue signals were recorded and processed using custom designed LabVIEW based software (Heist [1997]). Velocity measurements (u) for a single model configuration were obtained over a period of 50 seconds at a sampling rate of 10 kHz.

464

Figure 1. A sideview of the test model.

Figure 2. Model installtion in the wind tunnel.

Two-dimensional Particle Image Velocimetry (PIV) measurements were performed using a Dantec PowerFlow system. The region of the flow field mapped extends x/d = 1.8 downstream including the base flow region from the ground plane to above the model. To facilitate optical access a Perspex endplate was used. The Gemini PIV 15 laser was mounted directly downstream of the model configured to produce a vertical, streamwise laser sheet. The Dantec Hisense digital camera (type 13, gain 4) was equipped with a 105mm focal length lens and mounted perpendicular to the laser sheet. This setup generated u - v velocity contour maps in the x - y plane. In order to illustrate the flow field phenomena of interest, results were processed on a very fine grid with the spacing between grid points corresponding to 1.4mm (0.014d). This led to some noise within the results especially where measurements were taken through the endplate. The raw PIV images were processed using a three stage strategy. Initially the images were cross-correlated with an interrogation area of 32 by 32 pixels and a vertical and horizontal overlap of 75%. The resulting vector maps of dimension 157 by 125 then underwent consecutive velocity-range validation and moving-average validation schemes. No filtering was used as this was found to 'blur' the results significantly in regions of high velocity gradients. Other tests conducted included force measurements, surface pressure measurements and oil flow visualization. The forces were measured using a two-component load cell (vertical and horizontal forces) rigidly attached to the base of the main supporting strut. The surface pressures were measured using a 64 channel Zero, Operate, Calibrate (ZOC) pressure transducer. Surface streaklines were obtained using an oil flow visualisation technique, performed using a mixture of titanium dioxide and light oil. 2.3. E r r o r s a n d U n c e r t a i n t i e s The height of the model above the ground was set using a metal shim which was slid between the base of the model and the ground plane. The height of the model was set with an accuracy of +0.05mm. The angle of the back face with respect to the ground plane was set using a digital inclinometer with an accuracy of • ~ The freestream velocity within the wind tunnel test section was set using a value of dynamic head with an accuracy of =t=0.05mm of water. The worst case uncertainties in the CL and Co measurements were :i=0.0103 and -+-0.0014 respectively. These uncertainties incorporate the quoted accuracy of the equipment and are arrived at using the root-sum-square method proposed by Moffat (Moffat [1982, 1998]). The surface pressures were measured to within an accuracy of •

of water. The worst case

465

Again this uncertainty incorporates uncertainty within the Cp measurements was • the quoted accuracy of the equipment and is arrived at using the root-sum-square method proposed by Moffat. 3. R E S U L T S A N D D I S C U S S I O N 3.1. S u r f a c e Flow It appears that the basic flow type does not change within the Reynolds number range tested. However, changes are observed when the gap between the model and the ground is narrowed. Figs. 3 and 4 give examples of the changes. These pictures only show the central portion of the model viewed from above. Nevertheless the basic features do not vary across the span, indicating a fairly two-dimensional flow field. This is more so away from the ground at larger gap ratios, e.g. Fig. 3. At h / d - 1, flow separation/transition is observed at around x/d = -0.09 (Fig. 3). It is not clear from the surface flow whether the flow re-attaches to the surface to form a bubble. Further studies are needed. On the surface nearest to the ground, flow separation is also observed, the extent of which is smaller. At h/d = 0.2, transition is observed at around x / d - -0.13 (Fig. 4). On the surface nearest to the ground, no transition and separation is present and the flow separates at the edge.

Figure 3. A top view of the surface streaklines at hid - 1, Re = 1.36 x 105. Flow from left.

Figure 4. A top view of the surface streaklines at h/d = 0.2, Re = 1.36 x 105. Flow from left.

3.2. Force B e h a v i o u r The change in force behaviour is illustrated in Fig. 5. Only one set of data are show in Fig. 5 as measurements indicate only small differences within the tested Reynolds number range. Clearly there exist two flow regimes. At small gap ratios, h/d <_ 0.525, CD is nearly constant at 1 and CL shows a reduction as the gap ratio increases. Co experiences a sudden jump to a higher value

466

at h/d = 0.525. Above h/d = 0.55, CD is about 50% higher, suggesting the introduction of new physics. The CL curve on the other hand shows only a small kink between h/d = 0.525 - 0.55. If we take h/d = 0.525 - 0.55 as the critical gap ratio, it is much closer to the value (0.5) reported for a normal flat plate (Everitt [1982]) than some of the values reported for circular cylinders.

0.8 2.5

0.4

2

0 o

o~

-0.4

1.5

-0.8 1

~

o ....

a

o'5 ....

~ .... h/d

CD

1:5 ....

2 -1"2

Figure 5. Force behaviour at Re = 1.7 x 105.

3.3.

Surface Pressures

Surface pressures reflect the changes in the flow characters. Examples of Cp distributions in the two flow regimes are given in Figs. 6 and 7. At large gap ratios, h/d > 0.55, the surface pressures are nearly symmetric. The stagnation point at 0 = 180 deg is followed by flow acceleration and suction towards the edges of the half-cylinder. Although the distribution of pressure tappings do not allow for an accurate identification of the suction peaks, it does seems that the suction peaks appear before the edges. This feature agrees with the surface flow visualization. The base suction is of higher value than that at smaller gap ratios and is nearly constant. At small gap ratios, h/d < 0.525, the surface pressures show an asymmetric distribution. The stagnation point moves towards the ground, by as much as 20 deg from large gap ratios to the smallest gap ratio, h/d = 0.15. Suction appears later than that of large gap ratios. The suction peaks are of smaller value as well. The base suction is still nearly constant but is reduced significantly. This accounts for the drop in Co. 3.4. F l o w S t r u c t u r e s b e h i n d t h e B a s e The flow field behind the base is surveyed using PIV. Only limited cases are covered. However, the flow features should not vary across the Reynolds number range. Results suggest that periodic shedding of alternate vortices exists at h/d >_ 0.55. The vortex shedding disappears at smaller gap ratios (h/d <_ 0.55). The discrete distribution of sensors within the PIV camera can cause a phenomena known as peak-locking within the measurements (Gui and Wereley [2002]). To quantify any peak-locking the statistics of the particle motion within the data field were analysed. It was noted that peak-locking was not present within the data. At large gap ratios, h/d >_ 0.55, the period shedding of alternate vortices is supported by two contra-rotating mean vortices behind the base. This feature is visible in Fig. 8, which shows the lower limit of the gap ratio of this flow regime. The two contra-rotating vortices define a recirculating flow region just behind the base. At this Reynolds number and away from the ground, the mean flow is fairly symmetric. The gap is narrowed, the length of the re-circulating region, as

467

Figure 6. Surface pressure distribution h / d = 1, Re = 1.7 • 105. Flow from left.

Figure 7. Surface pressure distribution at h / d = 0.2, Re - 1.7 • 105. Flow from left.

defined by streamwise position of the bifurcation point, experiences small variations and stays at x / d = 1.3 until h / d = 0.6. Below h / d = 0.6 the mean flow becomes asymmetric and the length of the re-circulating region increases, to x / d = 1.4 at h / d = 0.575 and x / d - 1.58 at h / d - 0.55. When the gap ratio is reduced to h / d = 0.525, a sudden change in the flow structure occurs. The periodic shedding of vortices disappears below this gap ratio. The mean velocity field does not show two contra-rotating vortices in a re-circulating region (see Fig. 9. There is evidence of an elongated re-circulation region, but the area covered by PIV does not allow for a complete description. The mean velocity in the region behind the base is generally of lower value than that in the vortex shedding region. The wake expands downstream. This expansion movement is more pronounced in the shear layer emanated from the edge away from the ground. The movement of the shear layer emanated from the edge nearest to the ground is constrained by the presence of the ground. It does, however, moves towards the ground and induces a wall-jet like near ground flow. The wall-jet type of near ground flow is also noted in wing in ground effect aerodynamics (Zhang and Zerihan [2003]). The change in the vortex shedding pattern is illustrated in Fig. 10 where the spectra of the streamwise velocity fluctuation at two heights are shown. The spectra were calculated from the velocities measurements obtained using a hot-wire technique. The time step in the physical domain was 1 • 10 -3 seconds corresponding to a spectral resolution of 1Hz in the spectral domain. In the vortex shedding regime of the flow, the velocity spectrum is dominated by a narrow band peak at St = 0.22. This number does not change in the vortex shedding regime. Measurements at the two probes locations ( x / d = 1, y / d = -0.5 and x / d = 1, y / d = 0.5) indicate that the signals are dominated by a periodic feature and have a 180 deg phase difference. At the small gap ratios tested (h/d < 0.525), the velocity spectra are of a broad band nature.

468

Figure 8. Mean velocity vectors at h/d = 0.55, Re = 6.8 x 104. Flow from left. Every fifth vector shown.

10'

10 0

I I I 1

Ii Ii Ii

Figure 9. Mean velocity vectors at h/d = 0.525, Re = 6.8 x 104. Flow from left. Every fifth vector shown.

h/d=0.2 h/d=1.0

E

~10" I,LI 1 0 .2

100

200 300 f(Hz)

+YL 400

500

Figure 10. Spectra of the streamwise velocity fluctuations at x/d = 1.0, y/d = 0.5; Re = 1.36 x 105. 4. S u m m a r y Measurements were conducted of flow around a half-cylinder placed in proximity of a moving ground, at Re = 6.8 x 1 0 4 - 1.7 x 102. Two distinct flow regimes were identified: one with alternate vortex shedding behind the base and another without vortex shedding. The critical gap ratio lies between 0.525 and 0.5. At large gap ratios (h/d >__0.55), periodic vortex shedding was observed behind the base, leading to a high base suction. The vortex shedding was supported by a mean base flow of two contrarotating vortices with a re-circulating region length of z/d = 1.3. The shedding frequency is nearly constant at St = 2.2. The pressure distribution was nearly symmetric. Separation was observed on the surface leading to the edge of the model. The lift and drag coefficients experienced only small variation as the gap ratio varied. When the gap ratio was reduced to below h/d >_ 0.55, a sudden drop in the drag coefficient was observed with a 50% reduction in value. At small gap

469

ratios (h/d <_ 0.525), no periodic vortex shedding behind the base was present. The base suction, though still nearly constant, reduced. The surface pressures showed asymmetric distribution. The stagnation point experienced a downward shift toward the ground. Transition was observed on the surface leading to the edge away from the ground. No flow separation was recorded on the surface nearest to the ground. A wall-jet type of flow existed near the ground. The lift coefficient experienced notable increase as the gap ratio is reduced in this regime. The distinct flow features and force behaviour could potentially serve as a calibration case for moving ground experiments. Acknowledgements The authors would like to thank Airbus UK and BARfl for providing the test models and Sammie Chan for their help in the wind tunnel tests. References Anemometer modular rack system - Instruction manual. Department of mechanical engineering, university of newcastle, 1996. F. Angrilli, S. Bergamaschi, and V. Cossalter. Investigation of wall induced modifications to vortex shedding from a circular cylinder. J. Fluids Engineering, Transcations of the ASME, 104:518-522, 1982. P. Bearman and M.M. Zdravkovich. Flow around a circular cylinder near a plane boundary. J. Fluid Mech., 89(1):33-47, 1978. G. Buresti and A. Lanciotti. Vortex shedding from smooth and roughened cylinders in cross-flow near a plane surface. Aeronautical Quarterly, 30:305-321, February 1979. K.W. Everitt. A normal flat plate close to a large plane surface. Aeronautical Quarterly, 33: 90-103, 1982. L. Gui and S.T. Wereley. A correlation-based continous window-shift technique to reduce the peak-locking effect in digital PIV image evaluation. Experiments in Fluids, 32:506-517, 2002. D. Heist. User manual for labview software : "analogue instruments", 1997. University of Surrey Report No. ME-FD/95.37. K. Kamemoto, Y. Oda, and M. Aizawa. Characteristics of the flow around a bluff body near a plane surface. Bull. JSME, 27(230):1637, 1984. M. Kim and D. Geropp. Ldv measurements in the near wake behind a half ellipse close to a stationary and moving wall. In 7th Int. Conf. on Laser Anemometry-Advances ~ Applications, Karlsruhe, Germany, 8-11 September 1997. M. Kim and D. Geropp. Experimental investigation of the ground effect of the flow around some two-dimensional bluff bodies with moving-belt techniques. Journal of Wind Engineering and Industry Aerodynamics, 74-76:511-519, 1998. S. Kumarasamy. Incompressible flow simualtion over a half cylinder with results used to compute associated acoustic radiation. PhD thesis, University of Maryland, Maryland, USA, 1995.

470

S. Kumarasamy and J. Barlow. Interference of plane wall on periodic shedding behind a half cylinder, 1995. AIAA Paper 95-2285. S. Mahon and X. Zhang. Computational analysis of pressure and wake characteristics of an aerofoil in ground effect, 2005. to appear. R.J. Moffat. Contibutions to the theory of single-sample uncertainty analysis. Journal of Fluids Engineering, 104:250-260, 1982. R.J. Moffat. Describing the uncertainties in experimental results. Fluid Science, 1:3-17, 1998.

Experimental Thermal and

S.J. Price, D. Summer, J.G. Smith, K. Leong, and M. Paidoussis. Flow visualization around a circular cylinder near to a plane wall. J. of Fludis and Structures, 16(2):175-191, 2002. S. Taniguchi and K. Miyakoshi. Fluctuating fluid forces acting on a circular cylinder and ineterference with a plane wall. Ezperiments in Fluids, 9:197-204, 1990. M.M. Zdravkovich. Flow Around Circular @linders Vol 2: Applications. Oxford University Press, Oxford, UK., 2003. J.D.C. Zerihan and X. Zhang. Aerodynamics of a single element wing in ground effect. Journal of Aircraft, 37(6):1058-1064, 2000. X. Zhang and J.D.C. Zerihan. Off-surface aerodynamic measurements of a wing in ground effect. Journal of Aircraft, 40(4):716-725, 2003.

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

471

TURBULENT WALL JET INTERACTION WITH A BACKWARD FACING STEP N. Nait Bouda 1, C. Rey

2, J.M. Rosant 3 and T. Benabid i

1 Laboratoire de Mdcanique des Fluides Th6orique et Appliqu6e, Facult6 de Physique U.S.T.H.B. BP 32, El Alia Bab Ezzouar 16111, Alger ALGERIE 2 Laboratoire de Mod61isation et Simulation Num6rique en M6canique et G6nie des Procddds UMR 6181 CNRS-Universit6 Paul Cdzanne Aix-Marseille III, IMT, Technopole de Chateau Gombert, F-13451 Marseille Cedex20. 3 Laboratoire de M6canique des Fluides, UMR 6598 CNRS-ECN, Ecole Centrale de Nantes, F-44321 Nantes Cedex 3

ABSTRACT : Laser Doppler measurements were conducted in a turbulent wall jet which involves a recirculation zone due to the presence of a backward facing step. The initial development of the wall jet was controlled and results were compared to earlier measurements. Great attention was paid to the separation and reattachment processes and the influence of the inflow configuration on these. Reynolds stress u'v' and intermittency factor (derived from flatness factor) were determined in order to get some idea of the flow structure and interaction process downstream of the step. It was found that by diffusion the turbulence in the external region definitely interacts with the near-wall turbulent structures in separating and reattaching flows.

KEYWORDS: Wall jet, turbulence, intermittence, separation, reattachment, backward facing step.

INTRODUCTION: The turbulent wall jet is a basic flow of fundamental interest for turbulence researches because of its double characteristics. The inner layer of the plane wall jet is similar to a turbulent boundary layer, while the outer layer resembles a free jet. Apart from this fact, a sudden change in the surface geometry downstream creates a separated shear layer. A widely known separated flow is the backward-facing-step flow. This is of general interest, and experimental facilities are available to study accurately the separation and reattachment process. The wall-bounded recirculation region past the step is a generic flow configuration, representative of many engineering applications.

472 The situation of a wall-jet flow with a backward facing step is encountered in many industrial processes involving fluid separation. For instance, it occurs at the entrance of a semipermeable membrane for filtering devices or in cyclone separation systems. In environmental applications, the winds acceleration over hills generates wall-jet type flows. The situation is also found in waterbodies with widening zones where one has an effluent discharge. Further we find these aspects in the formation of snow-drifts, the displacement of dunes and the silting of rivers. The study in a laboratory model under more academic configurations is thus of special interest for the understanding and the control of these phenomena. Moreover such studies can be used for testing numerical flow simulation methods. So far, much attention has been given to the statistical properties of the flow field (de Brederode and Bradshaw 1972, Eaton and Johnston 1981). The flow configuration with a sudden channel expansion has been extensively studied (Etheridge and Kemp (1978), Driver and Seegmiller (1985), Isomoto and Honami (1989)). This has led to schematic representation of separated flows with a large recirculation zone located past the step corner, that reattach after a certain distance downstream of which the boundary layer redevelops. So far, few experimental studies have considered the description of the backward facing step flow in terms of eddy structure. Multidirectional quantitative measurement techniques such as particle image velocimetry are well suited for the study of instantaneous flow structure and evolving dynamics of turbulent flows. Actually PIV is now commonly used and has become a very efficient tool to investigate the complex dynamics and structure of flows ( see Scarano and Riethmuller (1999), Kostas et al. (2002) and C. Schram et al. (2004) for recent reviews). Vortical structures in the flow and their interactions are believed to play an important role in the generation of turbulent stresses and entrainment by shear. The aim of the present experimental investigation is to get new information on the influence of the external flow on the structure of the recirculation region and its development. A sketch of the flow structure is given in figure 2. Zone 1 is an ordinary boundary layer of thickness 8 that determines the initial conditions. The sudden change in surface geometry causes the boundary layer to separate at the step edge; the flow then behaves as a free shear layer (zone 2). This zone gradually transforms into a mixing layer that impinges on the wall and borders a closed recirculation zone (zone 3). Previous studies (Chandrasuda and Bradshaw (1981), Eaton and Johnston (1981)) have shown that the recirculation zone is not a stagnant fluid zone. The large eddies are affected by the turbulence of the external flow and the wall conditions. The reattachment zone 4 is defined as the location where the large eddies impinge on the wall (Badri Kusuma et al. 1992). The flow is unsteady and highly threedimensional with large-scale structures (Kostas et al. 2002). Downstream of the reattachment region, the boundary layer develops further and undergoes relaxation (zone 5). In the case where the incoming flow is a wall jet, zones 6 and 7 remain free shear layers. The simultaneous presence of two different turbulent zones in the wall jet suggests that intermittency plays an important role in the region of maximum velocity. The tests conducted by Alcaraz (1977) have confirmed this observation. Experimental results presented here are obtained by laser Doppler anemometry and we focus our attention on the evolution of the internal turbulence structure in the recirculation zone through measurement of the flatness factor and the Reynolds stress -u'v'.

EXPERIMENTAL APPARATUS: In this section, the test facility and instrumentation are described.

Flow configuration

473

A subsonic open circuit wind tunnel (located in the Fluid Dynamics Laboratory of Ecole Centrale de Nantes FRANCE) was modified to obtain the basic flow field (wall jet and step, Badri Kusuma 1993). A schematic diagram of the test section is shown in figure 1. The three parts are: I" a convergent section ended by a nozzle of 40mm height (H) and 700mm width (/). II: an inlet section, upstream of the step, of about 1100mm length (L) and 700mm width (/). III: a backward facing step, with a step height h of 20mm. The aspect ratio AR = / =35 (> 10), is largely sufficient to ensure two-dimensional flow according to h criteria of de Braderode and Bradshaw (1972). The reference velocity, measured at a point 340mm upstream of the step (at y = Ymax)' is Ue= 6m/s; the resulting Reynolds number based on the step height is Reh= 7600. Therefore, the experimental configuration corresponds to a turbulent <<separation and reattachment>> phenomenon following Adams et al (1984). At the same station, x = -340mm (x = 0 is at the step location), the turbulence intensity in the main flow is 13%. The inner layer thickness at the edge of the step is about 6 --- 20mm. Under this condition where 6 --=__ 1, the free shear layer is affected by the upstream flow but not governed by it (Bradshaw and h Wong (1972)).

Figure. 1. Schematic diagram of test section (dimensions in mm) Dashed line defines the position of the longitudinal section of measurements (z =0 plane)

700

h=20 II00 700

nozzle .._~I!

I

.....

Xr

'

I

Figure: 2. Schematic picture of flow structure

474

Photograph showing the recirculation flow structures, using laser tomography (Badri Kusuma 1993); Zone 3 and 4. A complete description of the flow structure was already given by Badri Kusuma (1993), using visualisation and hot-wire measurements. The description was only qualitative in the recirculation zone where hot-wire measurements are of limited validity. Nevertheless, it was proven that the reattachment zone was shorter in the case of incoming wall-jet flow, compared to the case of incoming boundary-layer or channel flow.

Laser Doppler Anemometer system: A commercial DANTEC 2D Laser Doppler Anemometer was used to measure two orthogonal components of velocity. The system consists of a continuous laser source (Argon-ion type with emitting power of 2W), transmitting optics (beam splitter and focusing lens) and receiving optics (photodetectors PM55X08 and PM57X08). In addition, a Bragg cell was used in order to distinguish between negative and positive flow directions. The beam splitter generates three laser beams (blue, green, blue + green) with two different wavelengths (blue = 488 nm, green -- 514.5 nm) that permit simultaneous acquisition of two velocity components. Depending on the working area, two different lenses were mounted. For the measurement in the transverse plane (xOz), the optical probe head is installed inside the wind tunnel and a lens with a 160 mm focal length is used. Then, the size of the measuring volume is 0.078x0.93 mm 3 for the streamwise velocity component, and 0.074x0.88 mm 3 for the spanwise velocity component. For the measurement in the normal plane (xOy), the optical probe head is outside the wind tunnel and with a lens of 399 mm focal length with which it is possible to measure until the centerline of the test section. The sizes of the measuring volume are now larger, 0.19x5.8 mm 3 (streamwise velocity component) and 0.18x5.5 mm 3. The data were acquired via AT interface board which is controlled with Floware software (Dantec products).

RESULTS:

Control of the flow upstream. Preliminary measurements were made at different test sections upstream of the step in order to control the wall jet flow development along the centreline. In particular, measurements of streamwise and spanwise velocity components were performed and are presented for the station x = -17h. In figure 3

475

u'w' we present normalized shear stress ,--75-- and spanwise mean velocity ~

Y . We Yl/2 can essentially confirm the two-dimensionality of the incoming flow upstream of the step considering the low values found for these quantities. Umax

plotted versus

max

b/

In figure 4, the normalized mean streamwise velocity U~max is plotted versus

Y , and shows a good Yl/2 collapse with the results obtained by Eriksson and al. (1998) in a two-dimensional developed turbulent wall jet. On the same figure, the mean normal velocity profile

v

is given, as obtained from the

U max

continuity equation by integrating the interpolated streamwise mean velocity data.

2,0

_

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,

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0,2

,

I

,

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0,4 0,6 u/Ureax

,

I

0,8

,

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Figure 4: Mean velocity profiles m

Umax

u'w' = f(__~y ) ' U 2max Yt/2

u

v

Umax ' Umax

= f(Y) Yl/2

upstream the step (x=-17h)

The flow downstream and the recirculation zone

In the second stage of the experiment, measurements were performed in the region after the step. The longitudinal mean velocity profiles h- normalized by Urea x a r e given in figure 5. The development of the recirculation zone can be identified in this representation; it corresponds to the negative velocity values. The reattachment length is then evaluated to be Xr = 4h.

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m~

u

= f(y / h ) .

The normal stress - u ' v ' , normalized by Um2a,is presented in figure 6; in the recirculation zone it shows a fairly constant behaviour all over the height of the flow field. This observation is contrary to the experimental study performed in a turbulent-boundary-layer configuration by Schram et al (2004), where an inner peak at y=h is reported. Beyond these sections (x> 4h), the Reynolds stress profile tends to approach such a behaviour. In a wall-jet flow, where the maximum velocity is greater than the external velocity, turbulence produced in the external region diffuses towards the inner one, contrary to the situation in boundary-layer and internal flows. The influence of the external zone with its large energetic structures on the inner region under the influence of a step changes the turbulent diffusion which undergoes amplification towards the wall. So, it produces a larger interaction between the different zones of the flow, which induces flapping phenomena in the recirculation zone and impingement of the large structures so that the reattachment length X is reduced to X = 4h. These mechanisms can be well identified from the curves presenting the Reynolds stress evolution (figure 6). The different location of the centre of turbulent production due the shear (u'v'> 0 corresponding to ~ < 0 )

0y

and of the eddy (vortex) centres (u'v'> 0

corresponds to a positive eddy rotation) develops a competition between these mechanisms. Then we observe a tendency to a redevelopment when the step effect vanishes at x/h = 8.

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Figure 6: Evolution of Reynolds stress profiles downstream of the step. Statistical assumption of independence between a signal u' and its intermittency properties make it possible to establish that F-3/7u. 7u is the intermittency factor and 3 is the value of the flatness factor F of the signal u' when it is assumed to be Gaussian:

(au'/a,) ~ Fu' : [(a.,/at)~ l The results in figure 7 correspond to the study of the flatness factor for u' and v' signals, respectively the longitudinal and normal fluctuation components of the velocity, presented in dimension of internal intermittency factor as: 3 3 7u = ~ ; 7 / ~ and

7~=_V-;7/~~

It is noted that the intermittencies of the structures behave differently for the 2 fluctuating components of turbulent velocity, with close values in the recirculation zones, and very different values in the external zones of the jet.

478 Figure 7 shows the evolution of the intermittency factors relative to the longitudinal and normal velocity components for various x stations: In the recirculation zone, 0 < x < +4h, and for the region corresponding to zone 2 defined above, (confined between y= lh and y= 3h), some disturbances are noted in the streamwise intermittency factor. These are likely to be due to a flapping phenomenon in the longitudinal direction. On the other side, approaching the reattachment zone, 7v' tends to 1; this can be attributed to the mixing character of this zone. Downstream of the reattachment zone, Tu' tends to 1 in the whole flow field.

16i

16

- e - - Yu' - - o - %, x= +lh

14 12

16[

- e - - ~u,

1

1

--0-- yv' = 2h 1 c~~%x

- - ~ - Y.' - - o - Yv, x : 3h

~olJl

1 10

"10 >.8

8

o

I ,,

8. 2 i

i

1.0

1.5

2.0

16[ --e-- Yu, o Tv, 14[ x = +4h

|

>: ~0~S

;t.

0.5

1.5

1.0

0.0

0.5

1.0

,

1.5

0.5

1.0

, , 16[--e-- Yu - o - - Yv 14~ x = +5h

16[ --e-- Tu,

to', 10[{~ I.

10tr,r~,,

,

e_

2.0

0.5

1.0

o

14f

.. ,

2.0

1.5

i

I

2.0

Tv,

x = +6h

,~" I.

:I!} i9 .

. 1.5

.

. 2.0

. 0.5

.

. 1.0

1.5

2.0

Figure 7 9Evolution of streamwise and normal intermittency factors (derived from flatness factor measurements) downstream the step. Using statistical analysis of the velocity field, Schram et al. (2004) have identified two recirculation bubbles in the configuration corresponding to an expansion ratio value about 1,25 (defined as the ratio of the test section height downstream of the step to the height upstream). The small bubble is located close to the lower step comer - counter clockwise - and the large one, downstream - clockwise -. It is noted that the dimensions of the small recirculation zone are approximately l h in the streamwise

479

direction and 0.7h in the normal wall direction. So, if we admit the same process in our configuration, the point of their collapse (x=+h) could be a seat of a high activity in the inner layer as y << h. An often-quoted parameter

y~/2U2max(product

of the jet half width to the maximum velocity) is

presented in figure 8. Generally this product is constant (Alcaraz 1977). We can notice the constancy of the values upstream and a sudden change to a higher value in the recirculation zone. 4

Ill 2

9

Upstream the step

-i6'-i4'-i2'-io'

-~ ' -~ ' -~ ' -i ' ~

9

9

Downstream the step

i }'i'~'g'~

-~ fi

g

lb

x/h

Figure 8 9Evolution of U2max.yl/2 in the test section

CONCLUSIONS: Our contribution to the knowledge on backward-facing step flows concems the special flow configuration studied. We consider as incoming flow a wall-jet type flow that presents a particular structure with two centres of turbulence production. One is due to an inner shear with small-scale eddies linked to the inner layer and on the other side, a free-shear jet flow with entrainment of fluid mass characterised by large eddies. This new inflow configuration produces some changes in the flow downstream. The present results complement the measurements performed by Badri Kusuma et al. (1993) and contribute to underline the role of the external flow large eddies on the wall region and on the recirculation zone. These structures affect the separation zone and tend to increase the flapping phenomena up to the reattachment point. As a consequence, the reattachment length decreases. The flatness factors, given as internal intermittency factors, confirm the description and quantify the governing mechanisms in the recirculation zone. REFERENCES"

Adams E.W, Johnston J.P and Eaton J.K. (1984). Experiments on structure of turbulent reattaching flow. Report MD-431 thermosciences division. Dept of Mech. Eng. Stanford university. California. Alcaraz E. (1977). Contribution a l'6tude d'un jet plan turbulent 6voluant le long d'une paroi convexe/l faible courbure. These d'Etat, Universit6 Claude Bernard Lyon. Badri Kusuma M.S., Rey C. and Mestayer P. (1992). The effects of wall roughness and the external flow structure on backward facing step flows. 11 th Australian fluid mechanics conference University of Tasmania, Hobart, Australia. 14-18 December, II, 795-798. Badri Kusuma M.S., (1993). Etude exp6rimentale d'un 6coulement turbulent en aval d'une marche descendante: cas du jet pari6tal et de la couche limite. Th6se de Doctorat, Universit6 de Nantes et Ecole Centrale de Nantes, n ~ ED 82-38.

480 Bradshaw P. and Wong F.Y.F. (1972). The reattachment and relaxation of a turbulent shear layer.

Journal of Fluid Mechanics, 52,113-135. Chandrasuda C. and Bradshaw P. (1981). Turbulence structures of a reattaching mixing layer.

Journal of Fluid Mechanics, 110, 171-194. De Brederode V. and Bradshaw P. (1972). Three-dimensional flow in nominally two-dimensional separation bubbles. I. Flow behind a rearward-facing step. Aero Report 72-19. Imperial College of Science and Technology, London. Driver D.M. and Seegmiller H.L. (1985). Features of a reattaching turbulent shear layer in divergent channel flow. AIAA Journal, 23:2, 163-171. Eaton J.K. and Johnston J.P. (1981). A review of research on subsonic turbulent flow reattachment. AIAA Journal, 19, 1093-1100. Eriksson J.G. Karlsson R.I. and Persson J. (1998). An experimental study of a two-dimensional plane turbulent wall jet. Experiments in Fluids, 25, 50-60. Etheridge D.W. and Kemp P.H. (1978). Measurements of turbulent flow downstream of a rearward facing step. Journal of Fluid Mechanics, 86:2, 545-566. Isomoto K. and Homani S. (1989). The effect of inlet turbulence intensity on the reattachment process over a backward facing step. Journal of Fluid Engineering, 111, 87-92. Kostas J., Soria J. and Chong M.S. (2002). Particle image velocimetry measurements of a backward facing step flow. Experiments influids, 33, 838-853. Scarano F. Benocci C. and Riethmuller ML. (1999). Pattern recognition analysis of the turbulent flow past a backward facing step. Physics in Fluids, 11, 3808-3818. Schram C. Rambaud P. Riethmuller ML. (2004). Wavelet based eddy structure eduction from a backward facing step flow investigation using a particle image velocimetry. Experiments in Fluids 36, 233-245.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

481

THE ROLE OF PRESSURE-VELOCITY CORRELATION IN OSCILLATORY FLOW B E T W E E N A PAIR OF BLUFF BODIES

Shinnosuke Obi, Norihiko Tokai, and Keita Sakai Department of Mechanical Engineering Keio University, Yokohama 223-8522, Japan

ABSTRACT Turbulent flow measurements were conducted between two bluff bodies set in uniform flow in tandem arrangement. The velocity obtained with PIV were averaged with respect to either time or phase of periodic pressure oscillation induced by vortex shedding from the bluff body, i.e., Reynolds decomposition or threelevel decomposition. The Reynolds stress caused by periodic fluid motion was found excessively large compared with those related to turbulent fluctuation in entire flow field. The PIV data were used to solve discrete Poisson equation of instantaneous pressure. The effect of organized vortex motion was recognized as the strong correlation between velocity and pressure gradient, which could explain the poor performance of RANS turbulence models in predicting this kind of flows.

KEYWORDS

Turbulent Wake, PIV, Reynolds Stress, Pressure-Velocity Correlation, URANS, Three-level Decomposition

INTRODUCTION

Turbulence models based on the Reynolds-averaged Navier-Stokes (RANS) approach often fail to predict flows associated with massive separation, in contrast to LES that correctly captures large-scale turbulent fluid motion typically found in such flows. It is generally recognized that the poor performance of the RANS models is due to the shortcomings of the statistical approach itself in representing the coherent structure in turbulence. On the other hand, the development of conventional RANS models have relied upon the knowledge of simple shear flows in nearly equilibrium state, hence it is likely that the turbulent transport process, which is remarkable in non-equilibrium flows, is not correctly incorporated in the existing models. In particular, the treatment of pressure diffusion transport, which is remarkable in free shear flows, is still far below satisfactory level, partly because of the lack of experimental information on the pressurevelocity correlation. The knowledge about such processes in flows out of equilibrium state is desired for the development of RANS-based turbulence models.

482 iT,;

,~ Flow E~

2H

~_ 1

Measured by t.DV ~ ~ i

3m

Upstreamcolum'n4N~'Y >x ~.~

4H

kJ2H Measuredby PIV i T ...... I._y

Downstreamcolumn)

Figure 1" Measurement area The turbulent wake of a bluff body put in uniform flow is a typical example of the problems where RANSbased models exhibit their disadvantage. The length of the wake calculated by RANS models is usually too long compared to experiment and/or LES (e.g., Bosch and Rodi, 1998). This is usually attributed to the vortices shed from the body; hence the application of unsteady RANS (URANS) has become attractive alternative to LES (Iaccario et al., 2003). The theoretical basis for URANS is, however, not firm enough to make this approach a standard tool for engineering design. Detailed discussions based on experimental data, and preferably on DNS, are desired. Turbulent flows associated with periodic motion have often served as attractive topics for experiments in the past. The three-level decomposition technique (Hussain and Reynolds, 1970) is often applied to analyze the interaction between periodic motion and turbulence. The flows under external periodic forcing offer the possibility of flow separation control, hence there are a number of studies to clarify the interaction of oscillatory motion and turbulence (e.g., Yoshioka et al., 2001a, 2001b). From an experimental point of view, flows associated with oscillation at known frequency are convenient for phase-averaging. On the other hand, flows comprising natural vortex shedding as represented by the wake of a bluff body require certain treatment for acquisition of meaningful data (Lyn and Rodi, 1994). The oscillatory motion in turbulent flow is extracted in the present study and its significance is discussed. The present study considers the flows associated with natural vortex shedding at nearly constant frequency. Such problems have been considered by Lyn and Rodi (1994) as well as Lyn et al. (1995). The interest in the present study is to analyze the suitability of URANS approach to this kind of flow. The three-level decomposition is applied to separate organized, oscillatory fluid motion from turbulent fluctuation, so that the influence of the shed vortex becomes evident. Besides, the instantaneous pressure field is estimated from the velocity data obtained with PIV, so that the pressure-related statistics are evaluated. Discussions are made on possible requirements for the RANS approach to handle this kind of flow.

EXPERIMENT Test Section

The flow around a pair of bluff bodies as shown in Fig. 1, representing recent multiple bridge construction (e.g., Vezza and Taylor, 2003), was considered. The issue here was the wind load exerted on the body which was located in the wake of another, i.e., the vortex shed from the body in upstream hits the other and causes fluctuating force with significant amplitude. To understand the flow dynamics in such a configuration, the present study focused on the region between two bodies as shown by a box marked with broken lines. The bodies were set in a test section with rectangular cross section (0.13mz0.3m) which was connected to a closed loop circuit of water at room temperature. The dimension of the bluff body H was 30mm, yielding

483

Figure 2: Instrumentation. the blockage ratio of 0.1 and aspect ratio 4.3. The velocity of the oncoming uniform flow Uc was varied through the range 0.1 m/s < Uc < 0.3m/s. The Reynolds number based on H and U~ was 3000 < Re < 9000. The free stream turbulence was measured to be below 1%. The mean flow was assumed to be homogeneous in the spanwise direction; hence the velocity measurements were undertaken at the center of the span.

P I V Measurements

An in-house PIV system comprising an 8-bit digital camera with 1008 x 1018 pixels (REDLAKE, ES 1.0) and a ND:YAG Laser with 30mJ output power (New Wave, Solo PIV) was used. The thickness of the light sheet was approximately lmm. White Nylon 12 particles, 90#m in mean diameter and specific gravity of 1.02, were used as a tracer. A pair of successive images were acquired with a frame grabber (National Instruments, PCI-1422) and sent to a 32bit PC at a interval of At -- 2.5ms controlled by an external trigger (National Instruments, PCI-6602). The resulting conversion factor to yield physical length was 0.055mm/pixel. The velocity vectors were evaluated by direct correlation method using an interrogation area of 32 x 32 pixels with 50% overlap in both directions, and erroneous vectors were eliminated by applying a spatial filter. In reference to the vortex shedding frequency, the data acquisition rate was determined to be 1Hz.

Phase Detection

The phase-averaging of the velocity field was conducted with reference to pressure fluctuations on the surface of the upstream body. The instrumentation illustrated in Fig. 2 was used to synchronize the pressure and PIV measurements. The reference pressure was measured at the round surface of the body, cf. Fig. 3(a); this location was selected because strong velocity fluctuations were observed in preliminary measurements using LDV. The static pressure was sensed at a round hole of 1.0mm in diameter and evaluated with a precision transducer (Validyne, DP103) in differential mode with respect to the pressure measured on the wall of the water channel at a position away from the obstacle, (z/H, y/H) - (-8.5, -2.8). The analogue electronic signal was amplified and digitized with a 16bit A/D converter (National Instruments, PCI-6040E)

484 0.15[

i i i

0.1

i

cO

0.05

(3

~-.

0 |

Or)

i

-0.05

i i i

I..

13_

i

|

-0.1

i

| '

o

0.5

tA tP

t~

i

1'.5

2

Time

(b) Definition of tA, tB to detect phase.

Figure 3: Principle of the phase-averaging with respect to the pressure on the surface of the body.

10 -6

>' 0.10 E~10 -8

-8 ~- o.05 10 -1~ 0

0.1

0.2 St = f H I U c

0.3

0 0.5

0.4

(a) Spectrum of filtered pressure fluctuation

1 (t B-tA ) fs

1.5

(b) PDF of characteristic period

Figure 4: Characteristics of pressure fluctuation on the surface of the body. at a rate of 400Hz. The pulse signal for controlling the illumination of ND:YAG Laser was recorded simultaneously with the pressure signal for detection of phase as described later. Since the frequency of vortex shedding was evaluated to be fs=2.1Hz by preliminary experiments, a low-pass filter at 4Hz was applied to the pressure signal to eliminate noise at higher frequency. The phase with respect to the dominant frequency in pressure fluctuation was evaluated according to the practice proposed by Lyn and Rodi (1994). As illustrated in Fig.3(b), the phase angle q~ was detected as follows: A pair of successive peaks were sought in the low-pass filtered fluctuating pressure signal and the instants at which the pressure reaches maxima were specified as tA and tB, respectively. The instant at which the velocity data was stored, tp, was used to determine the phase according to

4, =

t p -- tA 2 ~ - ~ .

tB--tA

(])

Note that the time t p was the midpoint of the instant at which two successive images were acquired. This procedure was necessary to allow a slight change in the dominant frequency in pressure fluctuation. The spectra of the filtered fluctuating pressure, shown in Fig. 4(a), indicated the dominant frequency of St = f s H / U ~ 0.21, though, at the same time, a significant amount of the fluctuation exists at a lower frequency

485 ,

o

o

1.5 XR

1

O

O

0.5

30()0 40()0 5000 6000 7000 8000 9000 10000 Re Figure 5: Length of recirculation zone behind the body in upstream side. than fs. These slow fluctuations resulted in the modulation of f~, as illustrated by the probability density function of the characteristic period (tB -- tA), in Fig. 4(b). The velocity data were separated into 20 groups according to r so that each group corresponded to a discrete phase angle of A r -- 0.17r, the average over each group providing a discrete variation over 0 _< r < 27r. When the characteristic period (tB -- tA) did not satisfy the condition 0.5 < fs(tB -- tA) < 1.5, the pressure fluctuation was considered to be out of phase, and the corresponding velocity data were rejected. As a consequence, the valid data were approximately 82.2% of whole sample of 8000 points, resulting in about 330 samples for every phase angle band. It should be noted that the all data, i.e., 8000 points, were used for time-averaging.

RESULTS Effect o f Reynolds N u m b e r

The length of the time-averaged recirculation zone is first examined because it is known to reflect the gross influence of the variation of Reynolds number on flow around a bluff body (Zdravkovich, 1997). Fig. 5 presents the length of the recirculation zone XR, determined as the streamwise coordinate on the centerline at which the streamwise mean velocity changes its sign, as a function of Reynolds number. It is indicated that the recirculation zone reaches the second body at lower Reynolds number, Re <_ 4000, while xn stays nearly constant for 4000 <_ Re <_ 9000, yielding x R=0.8. This is in good accordance with a separate experiment conducted in a wind tunnel (Kuroda et el., 2003). The results shown in the subsequent sections are obtained at Re = 9000 so that the influence of Reynolds number is minimized.

Streamlines

The phase-averaging procedure described in the preceding section enables the instantaneous velocity to be decomposed into the periodically varying part and fluctuation around it:

~ = (u~) + u'~,

(2)

where the velocity in bracket, (ui), is phase-averaged velocity that varies with time or with phase r Fig. ~6(a) shows the streamline pattern obtained by smoothly connecting the time-averaged velocity vectors. A pair of recirculation zone is observed behind the body in upstream side. On the other hand, the

486

Figure 6: Comparison of streamlines

Figure 7" Streamwise normal component of Reynolds stress. phase-averaged streamlines obtained from (ui), shown in Figs. 6(b) and 6(c), demonstrate the evolution of wavy pattern as a consequence of the shed vortex. The phase r -- 0 corresponds to the instant at which the fluctuating pressure detected on the body surface reaches the maximum. In reference to the location of the pressure measurement, cf. Fig. 3(a), it is recognized that a vortex is about to leave the opposite side of the body. Reynolds stresses

The instantaneous velocity ~i may also be expressed by a sum of mean velocity Ui, periodic component ~2i, ! and turbulent component u~ as izi = Ui + ~zi + u'i. (3) Subtraction of the mean velocity Ui from (ui) introduced in Eq. (2) provides the periodic velocity fluctuation ~ , hence the separation of periodic and turbulent velocity fluctuation is possible. The distribution of streamwise normal component of Reynolds stress u 2 is shown in Fig. 7, comparing the Reynolds stresses calculated from the average of whole sample (total), those calculated from the periodic velocity fluctuation, and turbulent components. The u2-component shows two separate regions of high intensity, located along the shear layer separated from the first body. It reaches the maximum at x / H ~ 0.5, and gradually decreases downstream. Comparing the turbulent and periodic parts, one can see that the turbulent part is concentrated to the near-wake region where a pair of steep peeks is observed, while the

487

Figure 8: Reynolds shear stress.

Figure 9: Transverse normal component of Reynolds stress. periodic part develops in the middle of the observed field, reflecting the fluctuation due to shed vortices. m

The shear stress component ~-~ shown in Fig. 8 resembles to that of the u 2 component in the sense that there are two separate peaks, positive and negative, along the separated shear layer. The evolution of the periodic part, Fig. 8(c), occurs in the middle of the two bodies, in accordance with the observation of fi2 in Fig. 7(c). On the other hand, the v2-component shown in Fig. 9 indicates qualitatively different tendency; a broad peak is located in the center of the wake and a remarkable peak is observed right in front of the body located downstream. The decomposition of this quantity into turbulent and periodic parts reveals the fact that this apparently strong turbulent fluctuation is mostly due to the periodic motion, i.e., the periodic vortex shedding from the body in upstream. It is interesting to note that the distribution of the turbulent part shows a pair of peaks at x/H=O,the same location as those of the other components, implying that the fluctuation there is a consequence of energy redistribution as it would occur in canonical shear flows. In contrast, the periodic part shows nothing common to the distribution of turbulence; the remarkable peak in front of the second body is considered to be a consequence of flip-flop motion of the impinging streamline there, as inferred from Figs. 6(b) and (c).

488 DISCUSSION From the results presented in the preceding section, it is expected that the URANS approach predicts the present flow with reasonable accuracy because the oscillatory vortex motion may well be captured. At the same time, it is unlikely that the existing RANS provides the remarkable peak of v 2 component in front of the body in downstream. A straightforward conclusion might be: Use the URANS instead of RANS. However, what to do when the organized vortex motion is not that evident, such that the spectrum of velocity fluctuation has no significant peak but only a broad distribution? In the present section, a possible modification to the models within the framework of RANS approach is discussed. The unsatisfactory performance of the RANS models is often attributed to the existence of coherent structure in massively separated shear layer. However, once averaged over sufficiently long time, there is no trace of coherence in the governing equation. For the improvements of RANS models, the discussions should be restricted to the terms appearing in the exact transport equation of Reynolds stresses. The most probable element that may reflect the large-scale vortex motion would be the terms containing velocitypressure correlation, i.e., pressure diffusion and re-distribution. Here, these two terms are treated together as velocity-pressure gradient correlation. Taking advantage of the PIV measurement, the distribution of instantaneous pressure gradient is inferred by solving discrete Poisson equation of pressure (Sakai et al., 2002, Tokai, 2003). Provided that the considered flow is fairly two-dimensional at any instant, the Poisson equation of pressure may read

02p O2p ( O~ O~ Ox--5 -t- ~ ~ 2p Ox Oy

Oit O~b) Oy Ox "

(4)

The terms on the right-hand-side are available from PIV measurement, hence Eq. (4) can be solved under the suitable boundary condition. In the present study, the boundary conditions are specified using pressure gradient normal to the boundary which is estimated based on the convection terms in momentum equations, i.e.,

On=-

itn On + it~ Os - '

(5)

where n and s denote the coordinates normal and parallel to the boundary of the control volume, respectively. The contribution of unsteady term is not negligible, though it is omitted due to the lack of temporal resolution of the present instrumentation. Using the available PIV data, numerical solutions of Eq. (4) are obtained on an array of equidistantly distributed 56 • 60 grids in the x- and y-directions. It should be noted that the instantaneous pressure gradient can be estimated by this method while the pressure itself is not available unless a reference pressure is measured at least at a point in the control volume. Nevertheless, the correlation between the velocity and pressure gradient is readily estimated as a field value. The results are presented in Fig. 10 for the three components of Reynolds stress under consideration,

Op

0t9

(6)

1-Iij --- ?_ti~xj -+- Uj OX i .

The role of this term may be interpreted as the composite of pressure diffusion and re-distribution of Reynolds stresses. Since the pressure diffusion does not appear in divergence form in the individual stress equation, this term does not reflect the transport of stress. Nevertheless, the comparison with the corresponding production term,

ouj

Pij = -puiuk ~

ou~

- puyuk -~Xk '

(7)

well explains the function of these processes. The production term of individual Reynolds stress shown in Fig. 11 points to the fact that the t e r m 1-Iij functions as a sink, for example, it is readily seen that Pal and I-IlX show analogue distribution with opposite sign. The same is true for P12 and 1-I~2. However, the situation is

489

Figure 10: Correlation of velocity and pressure gradient

Figure 11" Production term of total Reynolds stresses. different for the transverse normal stress component v 2, namely, the production term P22 becomes negative in front of the downstream body where v 2 shows an extreme peak. Interestingly, this peak is provided by the II22 term shown in Fig. 10(c). It is therefore inferred that the excessively large Reynolds stress related to oscillatory or organized turbulent motion is a consequence of the production due to velocity pressure gradient correlation. The above observation points to the fact that the oscillatory fluid motion is represented by the velocitypressure gradient term from the RANS point of view, which has not been considered within the conventional framework. In addition, the strong anisotropy of the normal stress components is another unusual factor, because the common algebraic relationship between Reynolds stresses cannot express such a strong anisotropy. Although any concrete modeling strategy is not known at the present, the modification of the existing RANS model for this process may improve the prediction in flows with massive separation, and if it is possible, the differential Reynolds stress model that expresses the transport of individual stress component is probably the most potential candidate.

490 CONCLUSIONS The flow around a pair of bluff bodies set in tandem in uniform flow has been experimentally investigated. The three-level decomposition of the velocity data obtained by PIV has revealed that the strong turbulence intensity found between the bodies is caused by the oscillatory motion due to vortex shedding. The effect of vortex motion is recognized as a strong correlation between velocity and pressure gradient. The generally recognized poor performance of RANS models in predicting this kind of flows is attributable to the absence of such terms that represent this process.

ACKNOWLEDGEMENTS The authors are grateful to Prof. S. Masuda, Keio University, and Dr. S. Kuroda, IHI, for invaluable discussions. The financial support for the present work has been provided by the Ministry of Education, Science, Sports and Culture, through Grant-in-Aid for Scientific Research (B), 15360100, 2004.

REFERENCES

Bosch, G. and Rodi, W. (1998). Simulation of vortex shedding past a square cylinder with different turbulence models, Int. J. Numer. Meth. Fluids 28, 601-616. Fuchs, W., Nobach, H. and Tropea, C. (1994). Laser Doppler anemometry data simulation: application to investigate the accuracy of statistical estimate. AIAA Journal, 32:9, 1883-1889. Hussain, A. K. M. E and Reynolds, W. C. (1970). The mechanics of an organized wave in turbulent shear flow, J. Fluid Mech. 41, 241-258. Iaccarino, G., Ooi, A., Durbin, P. A. and Behnia, M. (2003). Reynolds averaged simulation of unsteady separated flow, Int. J. Heat and Fluid Flow 24, 147-156. Kuroda, S., Sugimoto, T., Shito, M. (2003). Private communication. Lyn, D. A. and Rodi, W. (1994). The flapping shear layer formed by flow separation from the forward comer of a square cylinder, J. Fluid Mech. 267, 353-376. Lyn, D. A., Einav, S., Rodi, W. and Park, J.-H. (1995). A laser-Doppler velocimetry study of ensembleaveraged characteristics of the turbulent near wake of a square cylinder, J. Fluid Mech., 304, 285-319. Sakai, K., Maeda, T., and Obi, S. (2002). Evaluation of the pressure-velocity correlation in turbulent wake of a rectangular cylinder, Proc. 5th JSME-KSME Fluids Engineering Conference, Nagoya, OS 14-4-2. Tokai, N. (2003). Visualization of pressure field in turbulent wake between two bluff bodies, Proc. ASME FEDSM'03, 4th ASME-JSME Joint Fluids Engineering Conference, Honolulu, FEDSM2003-45758. Vezza, M. and Taylor, I. (2003). An overview of numerical bridge deck aerodynamics, The QNET-CFD Network Newsletter, 2:2, 21-26. Yoshioka, S., Obi, S. and Masuda, S. (2001a). Organized vortex motion in periodically perturbed turbulent separated flow over a backward-facing step, Int. J. Heat and Fluid Flow 22:3, 302-307. Yoshioka, S., Obi, S. and Masuda, S. (2001b). Yoshioka, S., Obi, S. and Masuda, S. (2001). Organized vortex motion in periodically perturbed turbulent separated flow over a backward-facing step, Int. J. Heat and Fluid Flow 22:4, 393-401. Zdravkovich, M. M. (1997). Flow around circular cylinders, Oxford University Press, Oxford, UK.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

491

TURBULENT STRUCTURES IN A SUPERSONIC JET-MIXING LAYER INTERACTION E. Collin, P. Braud and J. Delville Laboratoire d'Etudes A~rodynamiques UMR 6609 CNRS Universit~ de Poitiers ENSMA F-86036 Poitiers Cedex, France

ABSTRACT In the context of hypermixing, the interaction between a supersonic mixing layer and a small control jet is experimentally studied. This interaction results in a flapping motion of the control jet, a distortion of the mixing layer, and an increase of the Reynolds stresses. The increase in the turbulence quantities is mainly contained in large-scale, well-organized coherent structures that are generated by the interaction. Characteristics of these structures are analysed through linear stochastic estimation strategies. KEYWORDS

supersonic jet, flow control, turbulent flows, coherent structures, linear stochastic estimation

1

INTRODUCTION

Radial pneumatic injection has been proved an efficient means of mixing enhancement in turbulent supersonic jets (Davis, 1982; Parekh et al., 1996; Freund and Moin, 2000; Delville et al., 2000, 2003). Such control techniques are suitable for aeronautic applications, since aircraft engines can provide pneumatic sources for flow control. Radial pneumatic injection into a supersonic jet is however poorly documented, compared with impinging jet studies for example. This paper outlines an experimental study of the interaction between the mixing layer of a primary round supersonic jet and the flow issuing from a small transverse rectangular injector. The work presented here is the continuation of a parametric study of the mixing enhancement capabilities of this interaction (Collin et al., 2000), an analysis of the CJ penetration behavior (Lardeau et al., 2003) and a study focused on the plane of symmetry of the manipulated primary jet (Collin et al., 2004). The snapshot shown in figure 1 illustrates the nature of the C J-mixing layer interaction. Some important observations arising from these studies are:

492

Figure 1: Snapshot of the manipulated mixing layer (Lardeau et al., 2003). Main jet is in the left hand side, and CJ is in the lower left part of the picture. Note the difference of behaviour between the bottom (manipulated) and upper (non-manipulated) mixing layers. Main jet boundary layers seeded with alcohol droplets 9 The mixing enhancement is obtained only just downstream of the C J, the mixing layer spreading rate very rapidly recovers a non manipulated value (Collin et al., 2000). 9 The CJ penetration is strongly intermittent and exhibits a flapping character. The corresponding flapping frequency is of the order of the natural Kelvin-Helmholtz frequency just upstream the CJ interaction (Lardeau et al., 2003; Collin et al., 2004). 9 The flapping CJ motion induces highly organized, azimuthally-oriented, coherent structures. This plays an important part in the mixing enhancement process (Collin et al., 2004). 9 The intermittent CJ penetration has been reproduced in incompressible flows, at lower Reynolds number as well (Lardeau et al., 2003), indicating that the flapping motion can be encountered over a wide range of flow regimes, with or without shock wave effects. 9 The flapping is a result of absolute instability conditions just upstream of the C J, and a coupling between the main mixing layer local Kelvin-Helmholtz and the CJ jet column instabilities (Collin et al., 2004). The present paper focuses on the three-dimensionnal effects of this interaction, both on the turbulent flow field (section 3) and on the organisation of the coherent structures (section 4). Section 2 deals with the experimental set-up and specific methods used in this work.

2

EXPERIMENTAL

SETUP

The experimental study is performed on a primary, ideally expanded, supersonic jet surrounded by a subsonic coflow. Characteristics of both the jet and the coflow are given in table 2. The convergent-divergent nozzle diameter is D = 50mm, and the test section is 500x500mm 2. The main jet is supplied by high pressure air tanks, while the coflow is entrained. The CJ consists in a rectangular jet blowing radially into the main jet. The C J, whose section is 1 1 x l m m 2, is oriented so that its span is normal to the primary jet axis. The CJ is located at the streamwise position X / D = 0.1 and at the radial position r / D = 0.6, as shown in figure 2. At this downstream location, the natural (i.e. the CJ being inactive) mixing layer thickness is ~0 = 4ram. The velocity ratio between the CJ and the main jet is r - Ucj/U1 = 0.84, which corresponds to a nearly sonic CJ Mach number. The CJ nozzle is a simple convergent.

493

TABLE 1 C H A R A C T E R I S T I C S OF THE MAIN ROUND J E T C O N F I G U R A T I O N

FLOW

main jet

entrained coflow

Velocity

U1 =385ms-1

U2 =45ms-1

Mach number

M1 - U1/al=l.38

M2 =- U2/a2=0.12

Stagnation temperature

260K

290K

Velocity difference Mixing layer convective velocity Convective Mach number

AU - U1

U2=340ms -1

Uc -- (a2U1 + alU2)/(al + a2)=230ms -1 Mc - (U1

U2)/(al + ae)=0.47

Figure 2: Schematic view of the flow configuration The (x, y, z) coordinates correspond respectively to the longitudinal (downstream-oriented) axis, the vertical ascending axis and the transversal axis. The origin is chosen at the center of the main jet nozzle. The velocity componnents on the (x, y, z) axes are (u, v, w). A part of this paper deals with a cynindrical system of coordinates (x, r, 0). By convention, 0 = 0 corresponds to the center of the CJ. In the cylindrical system, the velocity componnents are written: (ux, ur, ue). The large windows of the wind tunnel allow measurements by optical means. Non intrusive measurements are performed using a 2 component Laser-Doppler Anemometry (LDA) system. For exploration of the mean and turbulent field (section 3), two different optical configurations are used in order to obtain the (u, v) or (u, w) velocity components separately. Traverse systems permit these velocity measurements to be performed over a 3D mesh. Coherent structure analysis, as discussed in section 4, is based only on (u, v) components. In this experimental configuration, a sub-miniature fluctuating pressure sensor is added just downstream of the CJ nozzle. This sensor gives information about the CJ penetration status. The pressure signal p(t) is sampled by the external analogic input of the LDA system. The measurements are performed at high average sampling rates (greater than 20kHz), and using long runs (more than 30,000 samples per position). Auto and cross-correlation functions for the randomly sampled data are calculated using the slotting technique (van Maanen and Tummers , 1996; Nobach et al., 1998). The pressure sensor configuration is shown in figure 3a, and the autocorrelation function of pressure fluctuations Rpp is plotted on figure 3b. This function is plotted against AtUc/5~o, where Uc is the convective velocity of the main mixing layer, and 5~0 the main jet mixing layer thickness just upstream of the CJ impact. The shape of Rpp is approximately sinusoidal. The

494

Figure 3: a) Detailed picture of the test section and location of the reference pressure sensor (units are millimeters); b) autocorrelation coefficient of the pressure signal Rvp(solid line: CJ activated; dashed line: CJ inactive)

period 1/fo of the oscillations corresponds to a Strouhal number St = foS~o/Uc = 0.2. This is coherent with previous results (Collin et al., 2004), where it has been shown that the frequency of the CJ flapping is of the order of the main mixing layer Kelvin-Helmholtz frequency, just upstream the CJ nozzle.

3

MEAN VELOCITY AND TURBULENT

QUANTITIES

Figure 4a shows the mean velocity field with an activated CJ. Positions of the main jet and the CJ are indicated as solid volumes. The flow is characterized by a strong distortion of the main jet. Regarding the mean velocity field only, the CJ seems to act just like a tab (Foss and Zaman, 1999). We notice also that disturbances are detected deep inside the main jet, just downstream the CJ. These may be generated by shock waves as the CJ penetrates into the supersonic region of the main jet. Nevertheless, the entire mean velocity field is not similar to what is obtained with tabs. Indeed, the ratio beween the axial and the azimuthal vorticity components, wx/Wo, is nowhere more than 5% whereas it should reach values up to 60% in the case of a jet with tabs, for this downstream location. Figure 4b illustrates the low levels of a~. The turbulent kinetic energy distribution, k = 1 @-ff+ ~ + ~-~), is plotted in figure 5a. From this point of view, the CJ again acts like a tab: the turbulent kinetic energy is almost doubled inside the distorted region of the main manipulated jet. One notice also that very strong values of k are encountered in the impact region. However, the turbulent shear stresses u~ur, plotted in figure 5b, have a character unlike that obtained with tabs. Indeed, the CJ generates much larger values of uxur than would be expected with tabs (Foss and Zaman, 1999). To summarize, from several points of view, the CJ effects on the main jet look like what is obtained with tabs. Yet the mean and turbulent flow properties are different in the case of the C J: no average longitudinal vorticity is detected, and high values of U~Ur arise. The mixing enhancement process is then different, compared to tabs. According to figure 1, the mixing enhancement seems to be related to the penetration instability rather than to the longitudinal vorticity.

495

Figure 4: a) Mean velocity field at sections z = 0, x/D = 0.08, 0.46 and 0.84; b) streamwise vorticity distribution at x/D = 0.84 (dashed lines represent mean longitudinal velocity countours

(~-u~)/Au)

Figure 5: a) Turbulent kinetic energy k and b) shear stress UxUr at sections z = 0, x/D = 0.08, 0.46 and 0.84 4

COHERENT

STRUCTURES

As the mixing enhancement mechanism is mainly based on the flapping of the CJ, the largescale coherent structures that characterize the flow behind the CJ are here studied. Since these structures are mainly azimuthally-oriented, this part of the study only concerns the (u, v) velocity components. As a consequence, results can not be given in terms of cylindrical coordinates in this section. The auto-correlation coefficient function Rvv for z = 0 and x/D = 0.6 is plotted in figure 6a. Strong oscillations are detected in v for this region of the flow. The associated frequency f0 corresponds to a Strouhal number St = foS~o/Uc = 0.2. Since 5~ = 1.55~0 at x/D = 0.6, f0 does not scale with the theoretical Kelvin-Helmohltz instability for this cross section. Strong oscillations are also detected in the cross-correlation function Rpv, as shown in figure 6b. Similar observations can be made for Ruu and Rpu. Large values of Rp~ are encoutered even further downstream, without any frequency shift, as shown in figure 7. This corresponds to velocity measurements near the high speed boundary of the mixing layer, for z = 0. Significant correlation levels are recorded even at x/D ~_ 3, which is unusual considering the primary jet Reynolds number. In fact, it seems that the entire distorted region of the mixing layer oscillates at a high frequency. Figure 7 does not give

496

Figure 6: a) Correlation functions Rvv and b) Rpv at x / D = 0.6 and z = 0

Figure 7: Cross-correlation function Rpv for several x / D locations correct information concerning the convective velocity of the coherent structures. This velocity is biased by the local velocity in the mixing layer: figure 7 gives a convective velocity of 294ms -1, while the measurements in Collin et al. (2004) lead to the main mixing layer convective velocity Uc=230ms -1 Provided the cross-correlation functions Rpu and Rpu reach significant values in the distrorted region of the flow, a linear stochastic estimation (LSE) of the (u, v) velocity components is possible. The reference parameter of the LSE is the pressure p(t). Details of the LSE can be found in Adrian (1979); Adrian and Moin (1988). Since only one reference variable is available in our case, we compute a multi-time-delay estimation. The estimated velocity field (g, ~) can be written as:

N (t + k dr) g(y, z , t) = }2k=oAk(Y,z).P ~(y, z, t) = ~k=o Y Bk(y, z).p (t + k dt)

(1)

where Ak(y, z) and Bk(y, z) are a priori unknown real coefficients, which are computed by a least mean square method. Values of p(t) are randomly sampled (see section 2), but given the shape of Rpp (figure 3), the pressure signal is approximated by a simple sine function. The data of the LSE written in 1 are N = 128 and dt = 8 10-%. Hence, N dt covers approximately a flapping period of

497

Figure 8: LSE results at x/D = 0.6: velocity vectors and Wz vorticity contours for z = 0, 3D isosurface wzD/AU = - 120

Figure 9: Q criterion isosurfaces, from DNS results of Lardeau et al. (2003) the CJ. The estimated velocity field for x/D = 0.6 is given in figure 8. The spanwise vorticity Wz is computed using Taylor's hypothesis: O~ 10~ (2) = N + ot where Uc is the convective velocity of the main mixing layer. Coherent structures are clearly visible in the distorted region of the flow. These large horseshoe-like vortices have a leading edge near the high-speed side of the mixing layer, at z/D = 0. They are similar to those identified by the direct numerical simulations of Lardeau et al. (2003) (see figure 9), although these simulations were performed for an incompressible plane mixing layer and a round CJ, at very low Reynolds number. The nature of the interaction between a jet and a mixing layer does not seem to vary with Re or Mc. A comparison of the turbulent quantities x/~u2, x/~v2 and ~ between the estimated and the original velocity fields is given in figure 10. Roughly 50% of the original energy is reproduced with the LSE. In other words, in the centre of the distorted region of the flow, half of the energy for (u, v) velocity componnents is contained in the large scale vortices. The increase in the Reynolds stresses discussed in section 3 may thus be considered as mainly be due to the coherent structures, instead of fine-scale turbulence production.

498

0.200 -

~J'- -

0.175 -

a

iI

0.150 -

iI

0.125 -

i11 ~

,,~

,,' / %0 ' s ~'~

/,'~/%x~, ',

0.100 0.075

\*

o"

0.050 0.025 0.000 0.0250. 8

I

0.7

I

0.6

l

0.5

I

0.4

I

0.3

o'.2

Y/D

Figure 10: Comparison between the original (dashed lines) and the estimated (solid lines) turbulent field for x / D = 0.3 and z = 0. Symbols refer to "D = x/=~u2/AU, ~ = x/~v2/AU, ~, = ~-~/AU 2

5

CONCLUSION

The interaction between a control jet and a supersonic mixing layer has been experimentally studied with non intrusive methods, and special attention paid to the turbulent properties, the 3D effects and the coherent structures. The present study confirms previous results (Collin et al., 2000; Lardeau et al., 2003; Collin et al., 2004): the pneumatic radial injection into a main supersonic jet generates a mixing enhancement and a distortion of the main jet geometry. Such feature is similar to what is obtained when a jet is manipulated by tabs. However, no further comparison can be made between the pneumatic injection and the tabs. It is shown here that the hypermixing process in the case of radial pneumatic injection is directly linked with an unsteady behavior at the interaction between the annular mixing layer of the main jet and the radial control jet (C J). The CJ flaps at a frequency that corresponds to the local Kelvin-Helmohltz instability of the main annular mixing layer in front of the CJ nozzle. The flapping CJ induces large scale coherent structures, which are azimuthaly oriented, and whose dynamics is very deterministic. These vortices play a very important part in the increase of Reynolds stresses in the manipulated flow. However, the large-scale structures that are generated downstream the pneumatic injection can not be assimilated to the well-known Kelvin-Helmohltz vortices. Indeed, in our case the characteristic frequency of the vortices remains constant in the whole distorted region of the flow, even very far downstream the injection. The fact that the flapping CJ generates oscillations in a large region of the flow at a frequency f0 that matches the main jet mixing layer properties could explain that most of the mixing enhancement is performed very near the CJ nozzle, and that the spreading rate downstream of X / D = 1 recovers a value close to the one obtained in the non-manipulated case. Indeed, the Cd acts like a pulsed jet at frequency f0, for which the flow is receptive only if 6a~ _~ 6~0. For future works, the nature of interaction between a pneumatic injector and a supersonic mixing layer could be used as a trigger. Hypermixing devices could contain two rows of CJ. Given the fact that the CY-mixing layer interaction results in very deterministic vortices, in a tandem CJ configuration, the second row of CJ could control with closed loop strategies the large-scale, well-organized coherent structures generated by the first row.

499

REFERENCES

R.J. Adrian, Conditional eddies in isotropic turbulence, Physics of Fluids 22 11 (1979) 2065-2070 R.J. Adrian and P. Moin, Stochastic estimation of organized turbulent structure: homogeneous shear flow, J. Fluid Mech. 190 (1988) 531-559 E. Collin, S. Barre and J.P. Bonnet, Analysis of jet interaction for supersonic flow control, Proceedings A VT/RTO Meeting 51 (Braunschweig, Germany, 2000) E. Collin, S. Barre and J.P. Bonnet, Experimental study of a supersonic jet-mixing layer interaction, Physics of Fluids 16 3 (2004) 765-778 M.R. Davis, Variable control of jet decay AIAA Journal 20 5 (1982) 606-609 J. Delville, E. Collin, S. Lardeau, S. Denis, E. Lamballais, S. Barre and J.P. Bonnet, Control of jets by radial fluid injection, ERCOFTAC Bulletin 44 (2000) 57-67 J. Delville, E. Collin, S. Lardeau, S. Denis, E. Lamballais, S. Barre and J.P. Bonnet, Control of jets by radial fluid injection, in: J. Priaux, M. Champion, J.J. Gagnepain, O. Pironneau, B. Stoufltet and Ph. Thomas, eds., Fluid Dynamics and Aeronautics New Challenges, (CIMNE, Barcelona, Spain, 2003) J. Foss and K.B. Zaman, Large and small-scale vortical motions in a shear layer perturbated by tabs, J. Fluid Mech. 382 (1999) 307-329 J.B. Freund and P. Moin, Jet mixing enhancement by high-amplitude fluidic actuation, AIAA Journal 38 10 (2000) 1863-1870 S. Lardeau, E. Collin, E. Lamballais and J.P. Bonnet Analysis of a jet-mixing layer interaction, Int. J. Heat and Fluid Flow 24 (2003) 520-528 (ETMM5 special issue) H.R.E van Maanen and M.J. Tummers, Estimation of the autocorrelation function of turbulent velocity fluctuations using the slotting technique with local normalisation, Proceeding 8th Int. Syrup. on Application of Laser Technology to Fluid Mechanics 36 4 (1996) 1-2 H. Nobach, E. Muller and C. Tropea, Correlation estimator for two channel non coincidence laserdoppler anemometer, Proceeding 9th Int. Syrup. on Application of Laser Technology to Fluid Mechanics 32 1 (1998) 411-415 D.E. Parekh, V. Kibens, A. Glezer, J.M. Wiltse and D.M. Smith, Innovative jet flow control : Mixing enhancement experiments, AIAA Paper 96-0308 (1996)

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

501

TURBULENT PROPERTIES OF TWIN CIRCULAR FREE JETS WITH VARIOUS NOZZLE SPACING T. Harima 1, S.Fujita 1 and H. Osaka 2 1Department of Mechanical and Electrical Engineering, Tokuyama College of Technology, Takajo 3538, Kume, Shunan, 745-8585, Japan 2Department of Mechanical Engineering, Yamaguchi University, Tokiwadai 2-16-1, Ube, 755-8611, Japan

ABSTRACT The turbulent flowfield of three-dimensional turbulent jets issuing from two circular nozzles has been investigated experimentally. The aim of this report is to supply a series of turbulent data of twin circular jets for the cases of three different nozzle spacing (S/d=2, 4 and 8) in the region where the interference between two circular jets exists. Turbulence quantities which include the turbulence intensity and the Reynolds shear stress, were measured using an X-type hot wire probe (5 ~tm in diameter, lmm effective length) operated by the linearized constant temperature anemometers (DANTEC). The exit plane Reynolds number based on nozzle exit velocity Ue (~40m/s) and nozzle diameter d, was kept constant at 25000 through out this experiment. From this experiment, it was found that switching tendency between the major and minor axes in the contour plot of urmdUe was observed in nozzle spacing S/d=2 and 4. Also, the development of the turbulence intensity in the inner region between two circular jets was suppressed by the interference of each circular jet.

KEYWORDS Twin Circular Jets, Interference, Control, Turbulent flowfield, Turbulence Intensities, Reynolds Shear Stress, Contour Plot

INTRODUCTION

The turbulent flowfield of three-dimensional turbulent jets issuing from two circular nozzles has been investigated experimentally. In our previous report [Harima et al. (2001)], the properties of the mean velocity field was clarified experimentally. From the results, for the case of nozzle spacing S/d=2, the contour plot of the streamwise mean velocity U/Ue showed the switching of the major axis between the y and z axes. Furthermore, the streamwise variation of the length scales on the y and z directions for each circular jet showed the switching tendency similar to the results of the elliptic [Quinn (1989)] and

502 the rectangular [Marsters & Fotheringham (1980)] jets. However, details of the turbulent properties were not investigated. Multiple circular free jets are very useful in many engineering fields, for example, the thrust augmenting ejectors, the pollutant disposal devices and the air conditioners. So far, only a few studies on the multiple circular jets have been reported. For example, Knystautas (1964), Pani et al. (1983) and Okamoto et al. (1985) clarified the mean velocity field. However, the turbulent flowfield of the multiple circular jets were not investigated in detail. On the other hand, the mean velocity field of two parallel plane jets [Tanaka (1970), Tanaka (1974), Marsters (1983) and Elbanna & Gahin (1983)] related to the present jet were reported, but the turbulent properties have not been investigated. The aim of this report is to supply a series of turbulent data of twin circular jets for the cases of three different nozzle spacing (S/d=2, 4 and 8) in the region where the interference between two circular jets exists.

NOMENCLATURE d Re U Ue Uox u w uw x, y, z Zpm

: : : : : : : : : :

Nozzle diameter Reynolds number (=Ue d/v) Streamwise mean velocity Streamwise mean velocity at the circular nozzle center of the exit plane (x/d=O) Streamwise mean velocity on the x axis for the single circular jet Longitudinal fuctuating velocity Lateral fluctuating velocity Reynolds shear stress Cartesian coordinate system with origin at the middle point between two nozzle centers Distance from the x axis to a position where the mean velocity profile takes maximum value on the z axis for each nozzle pitch

Greek Symbol v : Kinetic viscosity of air Subscripts e : Value at the circular nozzle center of the exit plane ox : Value on the x axis for the single circular jet p : Nozzle pitch (=S/d) pox : Value on the x axis for each nozzle pitch pmz : Value at the location where the streamwise mean velocity profile U/Ue takes the maximum value on the z axis for each nozzle pitch rms : Root mean square

E X P E R I M E N T A L SETUP AND P R O C E D U R E The configuration of the flowfield and the coordinate system are presented in Figure 1. The twin circular nozzles are installed horizontally with the nozzle center spacing S/d (2, 4 and 8), and the nozzle diameter d is 10 mm. Longitudinal and lateral fluctuating velocities and Reynolds shear stress were measured using an X-type hot wire probe (5~tm in diameter, l m m effective length) operated by the linearized constant temperature anemometers (DANTEC). The signals from the anemometers were passed through the low-pass filters and were sampled using A.D. converter at 10 kHz. The processing of the signals was made using a personal computer. Acquisition time of the signals was

503 usually 80 seconds. The exit plane Reynolds number based on the nozzle exit velocity Ue and the nozzle diameter d, was kept constant at 25000 (Ue ~.~40m/s) through out this experiment. In order to indicate the nozzle exit conditions, the profiles of both the streamwise mean velocity and the longitudinal turbulence intensity measured using a single hot wire probe at the sections of both x/d=O and 1 on the slice plane of z/d=4 for the case of S/d=8, are shown in the left hand side and the right hand side of Figure 2, respectively. At the nozzle exit plane, the streamwise mean velocity profile has a flat shape (x/d=O), and the magnitude of the exit longitudinal turbulence intensity urms is 0.8% of the exit streamwise mean velocity Ue. Here, the uncertainty of a single hot wire associated with u,'ms is estimated at +3% which contains a calibration error. The reason of this low uncertainty is that the flow direction is almost parallel to the x axis and the turbulence intensity is very small at the nozzle exit region. Furthermore, there seemed to be some errors in the results for the longitudinal and lateral turbulence

s S'

sss

"~,

Honeycombs & 2 Screens

~

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Figure 1" The configuration of the flowfield and the coordinate system 1.0

0.8

10.`o

// 0.4 0.2 1

J

y/d

9 = .... 0.5

5.

T....... 00 -

~o.o5 I

0.5

,

,

y/d 1

0

Figure 2: The profiles of both the streamwise mean velocity and the longitudinal turbulence intensity at the sections of both x/d=O and 1 on the slice plane of z/d=4 for the case of S/d=8

504 intensities and the Reynolds shear stress measured using an X-type hot wire probe and the linearized constant-temperature anemometers, so efforts were made to estimate uncertainty in the following. In accordance with Vagt (1979), thermal diffusion response equations were expanded with Taylor series and calculated up to the second order of fluctuating quantities. Here, the higher order terms were neglected, provided they were very small compared to the first order terms. The estimated total error of turbulence intensities is +5.6% for Urmsand +5.3% for Wrmsrespectively, and that of Reynolds shear stress is + 11% in the jet center region. The total error contains a calibration error, an error caused by neglecting higher order terms of fluctuating velocities and the cross-flow error [Hussein et al. (1994)]. Furthermore, it should be noted that the turbulence intensity in the lateral region away from the jet center contains larger error compared to that in the jet center region.

E X P E R I M E N T A L RESULT AND DISCUSSION Figure 3 shows the streamwise variation of the positions of Zpmld. Here, Zpmld is the distance from the x axis to the position where the streamwise mean velocity profile on the z axis takes a maximum value for the cases of S/d=2, 4 and 8. From this result, it should be noted that the positions of Zpm/d for all the cases of S/d move toward the twin jets center (y/d=z/d=O) with increasing streamwise distance owing to the entrainment between two circular jets. This result was also reported in the two plane parallel jets by Marsters (1983). 9

'

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~

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,

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I

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Figure 3" The streamwise variation of the positions of Zpmld for the cases of S/d=2, 4 and 8

x/d=7.6

0.2 G)

. . . . .

8.3

\'IZ"

9.1 . . . . . . . .

0.1

J

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i v

0.01

0.005

,

1

,

,

(urms)pmz/Ue 9Twin Jets (S/d=2) ................. /x Twin Jets (S/d=4) [] Twin Jets (S/d=8)

I , . . . .

5

10

,

x/d

,

I[

JJ J---J J1

I

I , , ,,i

50

Figure 4: The streamwise variation of the longitudinal turbulence intensities

100

(Hrras)pox/Ueand (Urms)pmz/Ue

505 Figure 4 shows the streamwise variation of the longitudinal turbulence intensities of (Urms)pox/Ueo n the x axis and (lgrms)pmz/Ue a t the location where the streamwise mean velocity profile U/Ue takes the maximum value on the z axis. The result of the single circular jet is also shown in this figure for comparison. The streamwise locations where (U~ms)pmz/Uetakes the maximum value are x/d~7.6 (S/d=8), 8.3 (S/d=4) and 9.1 (S/d=2) respectively, here the streamwise position taking the maximum value was determined using third order regression curves. And it should be noted that the streamwise distance of the potential core [Harima et al. (2001)] of Upmz/Ue showed almost the same value for all cases of S/d. The location of the maximum value of (blrms)pmz/Uefor the case of S/d=8 is almost identical to that of the single circular jet, because the effect of interference between two circular jets does not exist in the upstream region. These results mean that each streamwise location of the maximum value of (u~ms)pmz/Uefor the case of S/d=4 and 2 are moved toward the further downstream section compared to the result of single circular jet as S/d decreases. Therefore, the development of the longitudinal turbulence intensity at each circular jet center (z=Zpm) is suppressed by the interference between two circular jets. On the other hand, the (~lrms)pox/Uefor each case (S/d=2, 4 and 8) takes maximum value at x/d~7.7, 15.8 and 27.4, respectively, as increasing S/d. These streamwise positions taking the maximum value locate in more upstream region than those of streamwise mean velocity Upox/Ue [x/d~12 (S/d=2), 22 (S/d=-4) and 40 (S/d=8)] [Harima et al. (2001)] on the x axis. From this result, it can be clarified that the effect of the interference to the longitudinal turbulence intensity in the inner region of two circular jets, appears more faster than that to the streamwise mean velocity. In order to realize the results mentioned above in addition, the relationship between S/d and the streamwise position x/d taking the maximum value for (Urms)pm~/Ueand (Urms)pox/Ue,is shown in Fig. 5. The streamwise position x/d taking the maximum value for (lgrms)pmz/Ue decreases slightly with increasing S/d, and the value of x/d approaches to the result of the single circular jet. On the other hand, the streamwise position x/d taking the maximum value for (urms)pox/Ue increases monotonically with increasing S/d. Figures 6, 7 and 8 show the longitudinal turbulence intensity contour plots for the cases of S/d=2, 4 and 8, respectively. Each upside contour plot shows the result of the present jet, and the downside contour plot shows the result of the single circular jet, for comparison. The difference in the streamwise variation of the contour plot for each S/d will be investigated. For the case of S/d=2, all contour lines at the section of x/d=lO are transformed, and the contour line Urms/Ue=O.10 is dented near the x axis (y/d=z/d=O) without linking with the other side contour line. A 10

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Figure 5" The relationship between S/d and the streamwise position x/d taking the maximum value for ( Urms)pmz/Ue and ( Urms)pox/Ue

506

switching tendency of the major axis on the y and z axes is observed in the streamwise region from the section of x/d=-20 to 40 as found in the rectangular jet by Marsters (1980). Then, for the case of S/d=4, the contour lines (Urms/Ue
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Figure 6: The longitudinal turbulence intensity contour plots for the case of S/d=2

, , , i , , , , i i !

. . . . . . . . . . . ,|,

-2

1

z

S,?~e,

2

?,,r?ular

z/d

8

6

l e ~ , 1.

8

10

Jets

12

14

(u.../Ue)

"

(S/d=8) x 10

'.

321

0 . . . . . .

i . . . . . . . . .

i

0

,

-2

-2

-4

-4

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Single

Circular

.,,,..,,.,,=,,,t,..,.,, -2 0 2 z/d 6

7J/

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1

-8

-10

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Twin

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14

(u../Ue) x 10 =1

0 2 4 6 z/d 10 10 . . . , . . . , . . . , . . . , . . . , . . . , . . .

0

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12

(b) x/d=20

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...................

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(u~./Ue) X 10 2~ Sing.le. C i r , c u l a r , l e t , t

8

-4

, 0

0

4

9 9i ,

x 10 2

Single Circular Jet .. , . . . , . . . , , , ,/ 2 z/d 6 8 10

..............

.

0

"o

0

' ' "( 'u'. ~' '. /"U"et)

2

..........

-6 -2

2

9, i ,

2

0

6

(b) x/d=20

8

' , , '

, -2

-4

12

,~1~,,1~=

(b) x/d=20

0

10

"o

(u~./Ue) x 10 Single Circular Jet~l

2

-4

.......... ~w~n'~;-'&~;-~i 1

0 0

9

, i , , , i , . , i , , , i , i

-10

z/d

(a) x/d = 10

" " " " " "'l")'e~4/]]" " " " ' " "

= i-

6

- Io. . . ~. . . . .0. . .

2

t

O

O

4

(u"/Ue) x 10 ' 1

, , i , , , ! , , i I , , ,/

Cu~,/ue) x 10 '_1

I I I ' I I I '

0

2

(a) x/d = 10

Twin Jets (S/d=2) t

O

-6

0

t

...,...,...,.

~ ...... 0

,

6 Twin

(a) x/d = 10

2

4

' ' ' , ' ' ' i ' , , i , , , i , , , i , ,,

' t

"o

0 "o

z/d

4 ......

="

Jet

8

(c) x/d=40 Figure 7" The longitudinal turbulence intensity contour plots for the case of S/d=4

-8

-10

:"~

(u~o/Ue)x 10 2-

Single C i r c u l a r J e t . . i . , , i . , , i . , , i . . . i = = i [ | | ,

-2

0

2

z/d

6

8

10

(c) x/d=40 Figure 8" The longitudinal turbulence intensity contour plots for the case of S/d-8

507

0.20 cx/d e 40 0.15

QIO " 50 o 20 a 75 0 3 0 v 100

J 0.05

0.00

5

z/d

10

(a) S/d=2 0.20

i

0.15

x/d ~5 e40 ~ 1 0 " 50 o 20 ~ 75

I,

0.05: 0.00

'

5

0

z/d

10

15

(b) S/d=4 0.20

.

.

.

.

i

ex/5d elO o 20 o30

0.15

e 40 a 50 cr 75 v 100

0.05

0.00

0

5

z/d

10

15

(c) S/d=8 Figure 9: The streamwise variation of the profiles of longitudinal turbulence intensity tlrms/Ue o n the z axis for the case of S/d=2, 4 and 8

0.20

|

x/d e5 e40 e l O ~, 50 o 20 o~ 75

0.15

0.05

0.00 -5

0

z/d

5

10

Figure 10: The streamwise variation of the profiles of longitudinal turbulence intensity ~lrms/Ueon the z axis for the case of single circular jet

508 0.010

!

,

,

x/d o5 e40 , 1 0 .', 50 o 20 o' 75

'~ 0.000

-0.005

-0.010

. . . .

I 5

. . . . z/d

I 10

(a) S/d=2 0.010

9

,

,

!

,~. ~'~ d ,~

x/d r ~10 o 20 o 30

.....

e40 ', 50 o 75 vlO0

0.000

-0.005

-0.010

'

0

'

' 5

z/d

'

~ 10

. . . .

(b) S/d=4 0.010 ox/5d e 40 e l O " 50 o 20 ~ 75 0 3 0 v 100 0 . 0 0 0 .,-

-

-

--

-0.005

-0.010 = 0

. . . .

~ 5

. . . . z/d

, 10

(c) S/d=8 Figure 11" The streamwise variation of the profiles of the Reynolds shear stress uw/Ue 2 on the z axis for the cases of S/d=2, 4 and 8 0.010

. . . .

i

x/d ~5 e 40 ~ 1 0 " 50 o20 cr 75 o 30

p~ ~ ..~ ~ ~ 0.000

-0.005

-0.010

-5

. . . .

i 0

,

z/d

, 5

.

,

,

t 10

Figure 12: The streamwise variation of the profiles of the Reynolds shear stress ~ l U e 2 on the z axis for the case of single circular jet

509 the shape of the contour plot at the section of x/d = 10 is almost the same as that of the single circular jet. The contour plots of x/d=20 and 40 have each major axis in the z direction. From the results of S/d=2 and 4, it is easily suspected that the contour plot of S/d=8 in the downstream section will have a major axis in the y direction. From the mentioned above, it was found that the switching tendency between the major and the minor axes of the contour plot Urms/Uewas observed for the cases of S/d=2 and 4. Figures 9(a), (b) and (c) show the streamwise variation of the profiles of longitudinal turbulence intensity u,,,dUe on the z axis for the cases of S/d=2, 4 and 8 respectively. The profiles of Urms/Ue for the single circular jet on the z axis, are shown in Fig. 10 for comparison. For the case of S/d=8, the profiles at the sections of x/d=lO, 20 and 30 are almost the same as those of the single circular jet, because there is no affect of interference on the longitudinal turbulence intensity between two circular jets in this region. In the region of x/d>40, the profiles are transformed and show smaller values in the inner region compared to those of outer region from the circular nozzle jet center (z/d=4) owing to the interference between two circular jets. Here, comparing with the results of two-dimensional parallel jets [Tanaka (1970)] (S/d=12), the profiles in the inner region of the two-dimensional parallel jets showed the same tendency with those of the present jets. The profiles of x/d>20 (S/d=4) and x/d>lO (S/d=-2) show the similar distortion with the case of S/d=8 in the inner region from each circular nozzle jet center (z/d=2 and 1), respectively. From the results mentioned above, the profiles of the longitudinal turbulence intensity Ur,ndUe on the z axis for the cases of all S/d are transformed and show the smaller values in the inner region from the circular nozzle jet center owing to the interference between two circular jets. Furthermore, the streamwise location where the effect of the interference can be found, increases with increasing S/d. Figures 11(a), (b) and (c) show the streamwise variation of the profiles of the Reynolds shear stress ~-~/Ue 2 on the z axis for the case of S/d=2, 4 and 8, respectively. For comparison, the profiles of u---~/Ue2 for the single circular jet are also shown in Fig. 12. For the case of S/d=8, the profiles at the section of x/d =10, 20 and 30 are almost the same as those of the single circular jet, as shown in the profiles of UrmdUe. For the case of S/d=4, the magnitude of the peak value in the profile of I~-~/Ue2l in the inner region (z/d<2) is smaller than that in the outer region from the circular nozzle center (z/d=2) at the section of x/d =10, 20 and 30. In the region of x/d>40, the profiles show the normal shape of circular jet. For the case of S/d=-2, the peak value of [~-~/ee2] in the inner region (z/d< 1) at x/d = 10 is smaller than that in the outer region (z/d> 1). From the result mentioned above, the magnitude of the Reynolds shear stress is also suppressed by the interference between two circular jets.

CONCLUSIONS (1) The development of the turbulence intensity at each circular jet center is suppressed by the interference between two circular jets. The effect of the interference to the longitudinal turbulence intensity in the inner region of two circular jets appears more faster than that to the streamwise mean velocity. (2) The switching tendency between the major and the minor axes of the contour plot Urms/Ue was observed for the cases of S/d=-2 and 4. (3) The profiles of the longitudinal turbulence intensity UrmdUe on the z axis for the cases of all S/d are transformed and show the smaller values in the inner region from the circular nozzle jet center owing to the interference between two circular jets. Furthermore, the streamwise location where the effect of the interference can be found, increases with increasing S/d.

51o

(4) The magnitude of the Reynolds shear stress is also suppressed by the interference between two circular jets.

References Elbanna H. and Gahin S. (1983). Investigation of two plane parallel jets. AIAA Journal 21:7, 986991. Harima T., Fujita S. and Osaka H. (2001). Mixing and Diffusion Processes of Twin Circular Free Jets with Various Nozzle Spacing. Proc. of Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics 2001, 2, 1017-1022. Hussein J. H., Steven P. C. and William K. G. (1994). Velocity measurements in a high-Reynoldsnumber, momentum-conserving, axisymmetric, turbulent jet. Journal of Fluid Mech., 258, 31-75. Knystautas R. (1964). The turbulent jet from a series of holes in line. Aeronautical Quarterly 15, 128. Marsters G.F. and Fotheringham J. (1980). The influence of aspect ratio on incompressible turbulent flows from rectangular slots. Aeronautical Quarterly 31:4, 285-305. Marsters G.F. (1983). Interaction of two plane parallel jets. AIAA Journal 15:12, 1756-1762. Okamoto T., Yagita M., Watanabe A. and Kawamura K. (1985). Interaction of twin circular jets. Bulletin of the JSME 28:238, 617-622. Pani B. and Dash R. (1983). Three-dimensional single and multiple free jets. ASCE Journal of Hydraulic Engineering 109:2, 254-269. Quinn W.R. (1989). On mixing in an elliptic free jet. Phys. Fluids, A 1(10), 1716-1722. Tanaka E. (1970). The interference of two-dimensional parallel jets (lst Rep). Bulletin of JSME 13:56, 272-280. Tanaka E. (1974). The interference of two-dimensional parallel jets (2nd Rep). Bulletin of JSME 17:109, 920-927. Vagt J.D. (1979). Hot wire probes in low speed flow. Prog. Aerosp. Sci. 18, 271-323.

511

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

LDA-MEASUREMENTS OF T H E T U R B U L E N C E AROUND A VENTURI

IN A N D

R.F. Mudde 1, L. Deutz 1, V.A. Nievaart 2, and H.R.E. van Maanen 3 t Kramers Laboratorium voor Fysische Technology, Applied Sciences, TUDelft 2 Interfaculty Reactor Institute, TUDelft 3 Shell E&P, Rijswijk, The Netherlands

ABSTRACT The flow in and around a Venturi (100 mm diameter pipe, 60 mm diameter throat, water) is experimentally investigated using LDA. The Reynolds number of the flow upstream of the Venturi is 1.9.105. The experiments provide radial profiles of the mean velocity and turbulence intensities upstream, in the throat, the diffusor part and at various locations downstream of the Venturi. Furthermore, autocorrelations and spectra are presented. Special attention is given to the LDA set up, allowing measuring as close as 120 #m to the wall. The experiments show that the flow in the Venturi and at least 5 diameters downstream of the end of the diffusor is not in equilibrium. The turbulence levels in the throat are much lower than in well developed flow; the opposite is true downstream of the diffusor. A low turbulence intensity jet flows out of the throat into the diffusor, where the turbulence intensity in the wall region is high. The dissipation in the throat is higher than the production. The spectra show that the power of the velocity fluctuations in the throat is low and in the downstream part after the diffusor it is high as compared to the well developed pipe flow.

KEYWORDS Venturi, LDA, turbulence, Spectra, Velocity Profiles

INTRODUCTION For the optimum production and reservoir management of oil and gas fields, continuous measurement of the flow rates of individual wells is required. However, all these flows are multi-phase. Despite many years of research, multiphase flow metering is still under development. The traditional technique is to use so-called test-separators which basically are big settling tanks, in which gravity separates the various phases. After this separation, single-phase flow rates are determined by ordinary flow meters. Although the test-separator is a simple device, it has several disadvantages: costs, size and a limited operation range. Because of its costs, a test-separator is a "shared" meter, which means that each well is only monitored during a short period, after which another well will be measured. This means that during extended periods of time no metering is available for the well, hampering optimal operation of the connected wells. Thus, there is a need in oil and gas production for a robust device that is able to provide

512

continuous readings for many years in off-shore applications without any need for maintenance. Worldwide, the production of natural gas is increasing. Natural gas is regarded as a clean fuel and large resources are still available. However, the number of wells producing only gas is negligible. Practically all reservoirs produce gas in combination with condensate (liquid consisting of hydrocarbons) and water. Below a certain temperature and pressure, these two products will condense in the flow line, requiring multi-phase metering. A Venturi flow meter is being considered as a good, robust candidate for measuring such multi-phase flows with a high (> 90%) Gas Volume Fraction, so-called wet gas. However, the geometry of a Venturi will influence the turbulence of the flowing medium in a complicated way. CFD calculations for single phase flows have been shown to be inaccurate and very sensitive to the closure relations used. In order to understand the behaviour of the Venturi in multiphase flows, a deep understanding of properties of single phase flow in the Venturi will be helpful. Therefore an experimental program has been initiated to get a better understanding of the turbulence in and around a Venturi in a single phase flow. This will form a basis for assessing the operation of the Venturi in the case of a wet gas. The Venturi is a constriction in the flow line that consists of a contraction followed by a smooth diffusor. Detailed information about the flow properties, including the turbulence close to the wall are scarce in literature. This prompted us to perform LDA experiments in and around a Venturi. As the flow in wetgas flow lines is highly turbulent, we have concentrated on high Reynolds numbers. Our Venturi (throat diameter 60 mm) is placed in a flow line with a diameter of 100 ram. The experiments are performed with water at a Reynolds number of 1.9.105, well into the turbulent flow regime. The objective is to provide reliable LDA data in and around the Venturi. Special attention is given to the flow properties close to the wall. Therefore, LDA with a high spatial and temporal resolution is needed. This study deals with optimizing the LDA set up and measuring and analysing the development of the flow from 2 diameters upstream to 5 diameters downstream of the Venturi.

HYDRODYNAMICAL

BACKGROUND

Contractions A contraction in a flow line leads to a rapid increase in the mean axial velocity with a considerable flattening of the radial velocity profile. Furthermore, the turbulence intensity will be appreciably reduced when passing through the contraction. In (Spencer et al. 1995) measurements on a contraction with water flow are described and compared with computer simulations at Re = 105. The contraction has a total angle of 40 ~ the upstream pipe diameter is 100 mm, the downstream one 54 mm. The authors

re••rtt•era•ia•pr•fi•es•ft•emeanve•••it•an••ft•etur•u•entkineti•energy(- 89 2 diameters upstream, 1 and 10 diameters downstream of the contraction. At the upstream location, the agreement between experiment and simulation is excellent. The same holds for the mean velocity profiles just after the contraction and l0 diameters downstream. However, the simulations overpredict the turbulent kinetic energy just after the contraction by an order of magnitude. At the 10 diameter downstream location, the predictions of the energy are still much higher than the measured ones.

Diffusers achieved much more attention, see e.g. (Azad 1996). Usually, flow through diffusers does not possess the simplified features of a fully developed flow. It might be expected that any mathematical model of turbulence with reference to the existing experimental data for symmetric equilibrium flows will be inadequate when used to predict diffuser flows in moderate to strong adverse pressure gradients (APG) (Okwuobi and Azad 1973). Azad and co-workers therefore performed an extensive experimental program to acquire more quantitative data.

513

Flow through a conical diffuser develops towards a jet-type of flow with a considerable increase in turbulence levels. The flow is subject to an APG which is strong enough to cause separation of the mean flow. Efficient conical diffusers should have a total divergence angle of 6 - 8 ~ In such a diffuser, the APG is strong enough to cause appreciable instantaneous flow reversals (instantaneous reverse flow up to 30% of the time), but the time-average flow is non-separated. (Okwuobi and Azad 1973) provide velocity profiles in a diffusor (upstream diameter 101.6 mm, divergence angle 8 ~ diffusor length 720 mm). They are, for Re = 1.52.105 and 2.93-105, presented in wall units in fig.(1). The numbers at the velocity profiles refer to the measuring location. The position 'ref' is 304 mm upstream of the diffusor, the positions 1 through 11 are distributed downstream of the start of the diffusor with 60 mm spacing. Obviously, the standard logarithmic law of the wall does not hold in the diffusor. The profiles at positions 6 - 10 show an inflection point indicating an adverse pressure gradient. Note that the deviation from the law of the wall keeps increasing with the downstream position. The same authors also show, that the turbulence intensity increases considerably in a diffusor and that the degree of anisotropy decreases from the wall to the diffusor axis.

_ Re= 152000

r

10

J.

2

3O

u+

u+= 5.75 log

20

r'

r

0

101

L

102

1'03

y+

104

Figure 1: Velocity profiles in wall units at two Reynolds numbers in a diffuser, from Okwuobi and Azad (1973) Venturi

(Deshpande and Giddens 1980) performed extensive measurements of the axial and tangential velocity in a 50.8 mm pipe with constriction of 25.4 mm at Re = 1.5.104. Their constriction had a cosine shape. They found a flat axial velocity profile in the throat with a very steep gradient at the wall; the maximum velocity was not found at the axis, but close to the wall. Downstream of the constriction, flow reversal in the wall region was found. It took at least 10 diameters for the mean axial flow to recover. Furthermore, the axial centreline velocity was seen, after an initial increase due to the constriction, to drop below the value of a well developed flow at the same flow rate, before recovering to this value. As to the turbulent velocity fluctuations it was reported that the axial component, Urms, was slightly attenuated I in the converging part, whereas the azimuthal one, Wrms, was slightly amplified. In the diverging part, a sharp increase in these fluctuations was found. The flow became highly anisotropic with U r m s more than 50% larger than Wrm s ~ . (March 1998) reported experiments in a Venturi at Re = 4.0 9105. Where the

514

turbulence intensity upstream was 7%, in the throat it was only 1%.

EXPERIMENTAL

SET-UP

The flow line with the Venturi is schematically shown in fig.(2). The water flow is driven by gravity from the buffer vessel (V = 12m 3) through a magnetic flow meter (Krone Altoflux Altometer K300) and the Venturi into a smaller storage vessel. From this vessel, the water is p u m p e d back into the storage vessel. The flow rate of the p u m p is regulated such that the level in the buffer tank is constant within a few millimetres, generating a very constant, oscillation free water flow. The flow line that contains the Venturi has an inner diameter of 100 ram. The length of the straight pipe upstream of the Venturi is 7 m, the downstream length 3.7 m.

Figure 2: Flow line with Venturi The Venturi is sketched in fig.(3), its throat diameter is 60 mm. In order to be able to measure close to the wall (with LDA), the Venturi is placed in a square box with fiat surfaces, also filled with water, to reduce cylinder lens effects at the pipe wall. To improve the optical properties further, a part of the wall of the throat is replaced by a thin transparent sheet (thickness 250 #m). In this way, the optical distortion of the laser beams is kept to a minimum and the LDA measuring volume can be positioned parallel to the wall, significantly improving the spatial resolution in the radial direction (Van Maanen and Fortuin 1983). This configuration also enabled detection of the scattered light at an angle of 90 ~ which significantly enhances the quality of the Doppler signals as will be discussed below. As the pressure in the throat is low, the pressure in the square box can be set at a desired (low) pressure to ensure that the thin sheet is not sucked inwards. Similar measures are taken in the diffusor part of the Venturi. Measuring points up and downstream of the Venturi are equipped with special plastic screws with fiat faces on the air side to minimize distortion of the laser beams. The diameter of these screws is large enough to be able to measure further than the axis of the flow line. The LDA equipment consists of a 5W Argon laser (Spectra-Physics) and TSI ColorBurst, ColorLink and IFA750 signal processor. Lenses with focal lengths of 122.2 m m and 250 m m are used. The raw data are analyzed using in-house developed software. The set up can be operated both in back scatter mode and in side scatter mode. We employed the latter (at 90 ~) in order to arrive at an as small as possible measuring volume (size ~ 150#m). The LDA probes are mounted on traversing tables with an accuracy of 10 pro. The closest measuring location which could be achieved was 120 # m from the wall. The upstream water centerline velocity is set at 2.0 m/s, resulting in a Reynolds number based on the flow in the pipe of 1.9.105 . Radial traverses of the axial and tangential velocity components are made at various positions: 1 diameter (1D) upstream of the start of the contraction, in the middle of the throat,

515

Figure 3: The venturi and its dimensions (in mm). Note that the flow is from right to left half way the diffusor and 0D, 2D, 3D, 4D and 5D downstream of the end of the diffusor part. The exact positions are given in fig.(4). 136

109

110

152

100

I..... I i ...... i

'5D down'

'4D down'

'3D down' '2D down'

"

'0D down'

106

191

"~i~ diffuser

throat

'ID up'

713

Figure 4: Exact location of the eight measurement locations, dimensions in rnrn

SIGNAL ANALYSIS Signal to Noise Ratio The LDA set up is optimized by using diagnostic tools (Van Maanen 1999), i.e. time-between-data distribution, pdf of the velocities and the auto correlation function of the velocity fluctuations. A significant improvement has been obtained as can be seen from the auto correlation functions of fig.(5), where the effect of changing from back scatter to side scatter on the SNR is shown. This improvement is caused by:

9 Strong reduction of the amount of light, scattered by the wall which reaches the detector. 9 The spatial filtering eliminates those parts of the measurement volume which produce low quality Doppler signals. 9 The small dimensions of the measurement volume (in three dimensions!) reduce the velocity differences, caused by gradients, across the measurement volume, which cause a noise contribution (Van Maanen 1999). A low noise contribution to the velocity fluctuations is important as the value of the auto correlation function at time shift zero, is, in the ideal case, directly related to the stress vtv I.

516

The velocity measured, v(t), can be written as the sum of the true velocity, U + u'(t) and a noise contribution, n(t): v(t) = U + u'(t) + n(t). Assuming the noise to be uncorrelated, the auto correlation function of the measured velocity can be written as (subtracting the mean U for convenience, hence using the fluctuating part v' = u' + n):

ACFr v' (r)

=

'

lim 1 T - ~ o o -T

f_/n

T v'(t)v'(t + 7)dt

--

lim T1 fo T n(t)n(t + r)dt lim Tl f o r u'(t)u'(t + r)dt + T-+oo T-+oo

=

ACF~,,u,(r) + a n2zX(~)

-

-

(1)

with A ( r ) = 1 for r = 0 and A ( r ) = 0 otherwise.

7

0.2 o

'%,\, %L

'

",r

' ' time shift (ms)

l o o

0.2 0

time shift (ms)

100

Figure 5: left" backscatter LDA (SNR = 2.7), right: side-scatter with reduced optical distortion (SNR = 13.3). Hence, the noise, being uncorrelated, only contributes at time shift zero. For time shifts larger than zero the noise is uncorrelated and therefore the ACF drops. Based on this, (Van Maanen 1999) defines the following formula for the signal-to-noise ratio (SNR)

SNR =

~/ ~

(1 - at2)

(2)

with at the contribution of the turbulent velocity fluctuations to the A C F at r = O.

Auto Correlation Function gJ Spectrum Using the slotting technique with local normalization (see (Van Maanen 1999)), the ACF is derived directly from the data. The algorithm has been further improved by assigning products of two individual velocity estimates to 'slots' of finite width, depending on the time difference of the two velocity estimates. This product is divided across two adjacent slots, dependent on the actual time difference and the slot time differences. The sums, accumulated in the slots are normalised by the "local" products (see e.g. (Van Maanen 1999) and (Van Maanen et al. 1999)). Due to statistics: the more data, the less erratic the ACF will be. The Auto Power Spectral Density (spectrum for short) can be obtained from the A C F via the Fourier transform. In practice, this results in spectra (APSDd) with high variances at the higher frequencies. This problem is solved by fitting a smooth, parametrized, curve through the A C F . The parameters directly describe the power spectrum without the need for a Fourier transformation to yield the A P S D f (see (Van Maanen 1999)). Because the spectrum for lower frequencies is more accurately described by the Fourier Transform of the ACF, the best spectrum is obtained by combining the two spectra: A P S D ( k ) = w(k) 9APSDd(k) + (1 - w(k)) 9A P S D I ( k ), with k = ~ the wave number (~ is the mean axial velocity). The weighing function w(k) has the property that it is 1 for small k's and drops to zero for larger values. The k-value of the drop depends on where the noise in A P S D d becomes larger.

517

RESULTS Mean Velocity Profiles The mean axial velocity profiles are shown in fig.(6). As can be seen, the velocity profile in the throat is very flat, almost like an ideal plug flow. Five diameters downstream of the Venturi, the mean velocity profiles are still flatter than that at 1D upstream, showing that the flow has not recovered completely. Note that the profiles directly behind the Venturi are not axi-symmetric. This was persistent in the experiments. No swirl is present in the flow as the average tangential velocities are zero. No explanation could be found. Inspection of the velocity time series, measured close to the wall in and downstream of the diffusor, shows temporal flow reversal similar to that reported by (Okwuobi and Azad 1973). However, the period of 'negative velocities' is much smaller than that of positive ones, resulting in a positive mean axial velocity at every measured position.

~.

~,~ 5m/s

" '5D down' '4D down' '3D down' '2D down'

'0D down' diffuser

throat

'ID up'

pipe

Figure 6" Mean axial velocity profiles in and around the Venturi The development of the flow can be seen better by plotting the velocity in wall units (u+ = ~U,) as a U. see fig.(7). The flow upstream of the function from the distance to the wall (in wall units y+ -- Y-V), Venturi agrees well with the theoretical prediction of the "law of the wall". The velocity in the throat of the Venturi shows, of course, a very flat profile. In the diffusor, the deviation is further increased and the flow is seen to relax slowly to the theoretical curve when going downstream. From this figure it is obvious that at 5D downstream the equilibrium turbulent pipe flow has not been reached.

0Ddown 2Ddown 30 u+

(-)0

30

o,,O,io~

oo.OO "~ 9 3Ddown

U+

(-)

~" 4Ddown

20

l up /

5Ddown

o.oooooOO'~176176 theory

theory.... ~ f f u s e r 0 100

101

102

103

104

Y+

105

(-)

0

100

101

l02

103

104

y+

(-)

Figure 7: Mean axial velocity profiles as a function of the distance to the wall (in wall units)

105

518

Turbulence Intensities The axial and tangential turbulence intensities, Iax and Itan, have been calculated via the definition: I - x/'u'u' with g the local mean velocity. The intensities have been corrected for noise contributions by inspecting the ACF. This correction is better for signals with a high SNR, stressing the importance of high quality data. As the definition of the intensities contains the local mean velocity, the values of the intensities go up when approaching the wall. The axial turbulence intensities are shown in fig.(8). 80,

80

Ddown

Iax

60 Iax

(%)

(%)

60

2Ddown

40

40

!

........ ,',~~176

-..."...........

20

2Ddown ...

own . . . . . . . . .

............... -ii;i

........

?ffu'~'" . . . . . . . . .

0

0

0:2

0:4

0:6

0:8 y/R

i (-)

1:2

1.4

0

0

0.2

0.4

~ ........

0.6

0.8 y/R

1 (-)

1.2

1.4

Figure 8: Turbulence intensity profiles of the axial velocity The turbulence intensity in the throat is much lower than 1D upstream of the Venturi (where the turbulence intensity is equal to that of the pipe further upstream), in agreement with the literature. In the diffusor, the intensity in the central part is still low: the flow that comes out of the throat moves as a jet of low turbulence intensity into the diffusor section. The intensity in the wall region has increased to values much higher than that of a well developed turbulent pipe flow. The increase continues further downstream. Just outside the diffusor the levels in the center are around those of the turbulent flow in the pipe; in the wall region they have increased further. Further downstream, the profile becomes flatter but is higher than that of the turbulent flow. From the turbulence intensity it is obvious that the flow 5D downstream is not in equilibrium: the turbulence intensity profile is flatter than for turbulent flow in a fully developed pipe flow and its magnitude is roughly twice as high, indicating that the dissipation is not in equilibrium with the production. The tangential velocity fluctuations show a similar turbulence intensity development along the Venturi.

ACF ~ A P S D Auto correlation functions have been calculated from the velocity time series using the slotting technique as described in (Van Maanen et al. 1999), with a slot width of 70ps, for three different radial positions (y/R = 0.01, 0.1 and 1.0). In the throat of the Venturi, the ACF drops very quickly, resulting in a low Taylor time scale. This indicates a high dissipation and a low production rate. In the diffusor, the centreline dissipation is less than in the throat, but higher than 1D upstream of the Venturi. Close to the wall, in the diffusor, the production of turbulent kinetic energy increases, supposedly as a result of the flow reversal that is observed in the time series of the velocity at that position. The macro scale, T~n ~ u also provides an indication for the dissipation rate: the lower Tm the higher the dissipation rate. In the throat and diffusor, the value of Tm are 22~ and 51% of that 1D upstream, in agreement with the higher dissipation found from the ACFs. From the ACFs, the Taylor time scale has been estimated by fitting a parabola to the small time shift part. This time scale is converted to the Taylor length scale: Lx = Tx 9v--~. The results have been collected in fig.(9). In the throat, the length scale, especially in

519

the center increases by a factor of 2. The decrease of the corresponding Taylor time scale is insufficient to counterbalance the increase in the mean velocity due to the contraction. As can also been seen from the graph, the Taylor length scale at 5D downstream is in the central region back at its original value. However, in the wall region this is not the case. This, again, illustrates that the flow has not returned to equilibrium. It should be noted, that the Taylor time scale follows from a fit through the data which introduces an uncertainty that has not been assessed here. 10 9

"~ E .c

....,

E~ E

m >.,

,/ !

7

/

6

5

---~--

throat 4~

8

p~e__

4

--

_1Dupe'

13- - . . . . . . .

<4;0

/"

'~f" i ,/' 0Ddown ....~ ,,~dif ',,

..

-2;o

' t'-4- - E l - -

"(3

;

0.01 0.1 1.0

',

.," J

!

,,,, Q

y/R y/R

2 D d o w ~ - " ~ , , 3Dd~

~

43""

y/R =

-.~-

200

X 4Ddown , ..la'-"fq 5gdown

Azr" .o.

_ -lET"" " ..'" ,.o" .~

400

" -.

600

"o

800

1000

location relative to center of throat (mm)

Figure 9: Development of the Taylor length scale for three radial positions. The power spectra, calculated from the A C F s are plotted in fig.(10). The spectrum in the throat is much lower than the others. Furthermore, its shape is distorted: the f a m o u s - 5 / 3 slope is no longer present. This indicates that the turbulence is indeed far from equilibrium: the dissipation outweighs the production. When moving out of the diffusor, the power increases up to 2D downstream, after which it slowly decreases towards the equilibrium value. From 2D down to 5D down, the inertial range of k-values stays roughly the same. The power of the high length scales (low k) decreases slowly and the power in the small scales increases monotonically when moving downstream.

CONCLUSIONS LDA proved to be a valuable tool to resolve the turbulence properties of a single phase flow through a Venturi. The diagnostic tools proved to be helpful in optimising the LDA set-up. Measurements of average velocities, turbulence intensities, auto correlation functions of the turbulent velocity fluctuations and power spectra of these showed to be feasible. The steep velocity gradient in the throat of the Venturi, caused by its plug-flow like behaviour, could be measured with a high resolution by the use of the thin-foil construction in combination with the sidescatter technique, which gave high quality Doppler signals, combined by a small measurement volume, which approached a spherical shape. The turbulence level in the throat proved to be very low and the dissipation showed to be larger than the production of turbulence. This is in agreement with literature results, but more detailed information is now available on the properties of the turbulence. The flow reversal in the divergent section of the Venturi gives rise to high turbulence levels (mainly production) which gradually dissipate downstream, but at 5D down, the flow is still not in equilibrium.

520

Figure 10: Auto Power Density functions for y / R = 1.0 at varies locations along the flow line. A comprehensive data-set is now available on the properties of the turbulent flow in and around a Venturi, which is essential in understanding the properties of a Venturi as a flow meter and which can also be used to verify CFD calculations. The next step will be the measurement of the properties of the turbulence of the liquid film at the wall in two-phase flow.

References Azad, R. (1996). Turbulent flow in a conical diffuser: A review. Exp. Therm. and Fluid Sci. 13, 318-337. Deshpande, M. D. and D. P. Giddens (1980). Turbulence measurements in a constricted tube. Journal of Fl'uid Mech. 97, 65-89. March, J. F. (1998). The calibration of a turbine meter as a secondary flow-rate standard by a one-point relative method using laser doppler velocimetry. Meas. Sci. ~ Techn. 9, 129-132. Okwuobi, P. A. C. and R. S. Azad (1973). Turbulence in a conical diffuser with fully developed flow at entry. Journal of Fluid Mech. 57(3), 603-622. Spencer, E. A., M. V. Heitor, and I. P. Castro (1995). Intercomparison of measurements and computations of flow through a contraction and a diffuser. Flow Meas. Instr. 6(1), 3-14. Van Maanen, H. R. E. (1999). Retrieval of Turbulence and Turbulence Properties from Randomly Sampled Laser-Doppler Anemometry Data with Noise. Ph.D. thesis, Delft University of Technology, The Netherlands. Vail Maanen, H. R. E. and J. M. H. Fortuin (1983). Experimental determination of the random lumpage distribution in the boundary layer of turbulent pipe flow using laser-doppler anemometry. Chem. Eng. Sci. 38, 399-423. Van Maanen, H. R. E., H. Nobach, and L. H. Bendict (1999). Improved estimator for the slotted autocorrelation function of randomly sampled lda data. Meas. Sci. ~ Techn. 10, 4-7.

7. Transition

This Page Intentionally Left Blank

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

523

M O D E L L I N G OF U N S T E A D Y T R A N S I T I O N WITH A D Y N A M I C INTERMITTENCY EQUATION

K. Lodefier ! and E. Dick I 1 Department of Flow, Heat and Combustion Mechanics, Ghent University, Ghent, B-9000, Belgium

ABSTRACT A transition model for describing wake-induced transition is presented based on the SST-model by Menter, with the k- co part in low-Reynolds form according to Wilcox, and two dynamic equations for intermittency: one for near-wall-intermittency and one for free-stream-intermittency. The total intermittency factor, which is the sum of the two, multiplies the turbulent viscosity computed by the turbulence model. The quality of the transition model is illustrated on the T106a test cascade using experimental results by Stieger and Hodson for transition in separated state and on the T106d test cascade using experimental results from Hilgenfeld, Stadtmtiller and Fottner for transition in attached state. The test cases differ in pitch to chord ratio, Reynolds number and inlet free-stream turbulence intensity. The unsteady results are presented in S - T diagrams of the shape factor and wall shear stress on the suction side. Results show the capability of the model to capture the basics of unsteady transition.

KEYWORDS Unsteady Transition, Intermittency, LP turbine, RANS, T106.

INTRODUCTION

In order to further increase the loading of the LP blades in a gas turbine, periodic impact of wakes coming from previous blade rows is used to force transition. Without wake-induced transition, the boundary layer at the suction side would massively separate in high load conditions. Under the impact of the wakes, the boundary layer turns turbulent. In between two wakes, the boundary layer state relaxes to a laminar and even a separated state. This is the becalming effect. Without the periodic impact of the wakes, the flow would be separated, resulting in high losses. Due to the transition to turbulence, the separation disappears periodically, and this makes the time averaged losses decrease.

524 In the experiments, the upstream blade row has been replaced by a moving bar system. The diameter of the cylindrical bars has been chosen to produce a wake with a velocity deficit and a turbulent kinetic energy that is comparable to that of a blade wake. In 3D flow, the unsteady vortex shedding behind a cylinder results in a highly unsteady wake. The vortices break up in the span-wise dimension relatively short after the cylinder. When performing 2D unsteady calculations, it is possible to resolve the shedding, but resolving the break-up of the shedding in the third dimension is not possible. As a consequence, the shedding vortices extend much further downstream of the cylinder, compared to experiments. It was therefore decided not to include the cylinders in the computation. Instead, a sliding wake profile has been applied at the inlet. The approach adopted in this paper is an unsteady extension of the steady model developed by the present authors (2003). The model is a combination of a dynamic equation of intermittency with a two-equation k-co turbulence model. The time dependent behaviour of the intermittency equation was modified. Experimental observations show that the leading edge of a turbulent spot moves with 0.88 times the local free-stream velocity, whereas the trailing edge moves at a lower speed of 0.5 times the local free-stream velocity. These velocities are clearly visible in a space vs. time diagram (S- T, see Stieger and Hodson). These features of turbulent spots are in steady flow cases averaged into the transitional shape for intermittency described by Narasimha. In unsteady flow cases, however, the Narasimha-law, which is a time average, is no longer justified. The transition has an evolution in time and space. These are connected via the leading edge velocity of the turbulent spots. We observe that the spontaneous time evolution of the intermittency equation ofLodefier & Dick (2003) is much too slow. To speed up the equation, we modify the time scale in the production term. The result is now a fast time response of the intermittency equation.

D E S C R I P T I O N OF THE T106a T E S T C A S E

Geometry The computations are performed on the T106 LP turbine blade, measured by Stieger & Hodson (2003). The flow is unsteady due to incoming wakes from a moving bar system located 0.33Cax upstream of the cascade leading edge. The basic parameters are listed in Table 1. The computational grid consists of a T106 profile with pitchwise periodic boundary conditions. In Figure 1, a zoom on the leading edge of the grid shows the boundary layer refinement on the profile. The moving bar system is not included in the computation. Instead, a moving wake profile is used as inlet condition. The inlet of the computational domain is located between the bars and the leading edge at 0.15c upstream of the leading edge. The grid consists of 147000 cells, while the y+ value in the first grid point in wall vicinity is below 0.2. Per pitchwise traverse 800 time steps are used. This high accuracy is necessary to properly simulate the unsteady movement of the wake. Precursor calculation

The inlet profile is determined from a precursor calculation. A 2D RANS computation of the wake of a cylinder at R e = 2600 is compared with experimental results of Stieger and Hodson. The experimentally measured wake spreading is computed by steady RANS between x / d = 5 and x / d = 62 on a rectangular grid of 150ram x 4 0 m m . Only half of the wake is computed. In the wake center, a symmetry condition is applied. The y coordinate is the distance form the symmetry axis. Wake characterizing parameters at x / d = 5, such as wake width (Yl/2) and velocity deficit (Uoo-Ucenter),are extracted from the experimental results. Since the turbulent velocity fluctuations at that location are highly anisotropic, the level of turbulent kinetic energy used in a steady isotropic RANS simulation cannot be that of the experiment. The turbulent kinetic energy in the wake and the background dissipation have been used as degrees of freedom to match the experimental results at x / d = 62. The wake behind a cylinder has a self-similar profile as described by

525 Pope (2000). Therefore we impose a self-similar profile at the inlet of the 2D RANS precursor calculation, with the inlet conditions (x/d=5) of Table 2.

U--Uoo-(Uoo-Ucenter)eXp-(ln2) y-~/2

(1)

, k=koo+(kcenter-koo)exp-(ln2) y--~/2

The inlet conditions for the specific dissipation are those described by Wilcox (1993):

(1) = 0)oo "at"C 114

x/k lm,x

,

lmu = 0.18Yl/2

(2)

1.6 -

9 $1k~lfind Hocbon

-

1.4

1

1.2 1 0.8

J

0.6

0,4' 0.2 0 ~

%,

j.,,'

-0"20

....

9 9 0.251 . . . .' 0.. 5 '.1 0 . 7 5 1

....

s/smax

11

Figure 2:T106a; pressure distribution.

".:/

"..

/

Figure 1" Grid at the leading edge of the T106a profile and a sketch of the experimental setup (upside down).

TABLE 2 T106A; INLETCONDITIONSPRECURSOR RANS COMPUTATION Ucenter/Uoo

0.3

yl/2

0.018mm 6.0m2/s e

kcenter

koo f.Ooo

0.005m2/s 2 80S -1

TABLE 1 T106A CASCADE blade chord axial blade chord pitch to chord ratio Reynolds number inlet flow angle design exit angle bar diameter bar pitch/cascade pitch axial distance bars to LE flow coefficient free-stream turbulence intensity

c Ca~

g/c Re2c al d

Vx/Ub

198mm 170mm 0.799 2.6 x 105 -37.7 ~ 63.2 ~ 2.05mm 1.00 0.33Co.x 0.83 < 0.5%

526 Inlet conditions

From the RANS wake calculation, the profiles at plane x/d = 42.7 under the relative flow angle fl = 63.2 ~ are used as inlet conditions for the T106 cascade calculation. This wake profile is applied as a sliding wake profile at the inlet. The procedure ensures that the RANS computation of the wake has good agreement with the experimentally measured behaviour. The inlet velocity angle in the wake is somewhat smaller than in the free stream. A correction for inlet angle has been applied over the whole inlet profile such that the time- and area-averaged inlet angle equals 40.7 ~, which gives better agreement with the experiments, as suggested by the experimenters. The time averaged pressure distribution over the rear part of the profile is in good agreement with the experiment, as illustrated in figure 2.

DESCRIPTION OF THE T106d TEST CASE

Geometry This test case was experimentally investigated in the high-speed cascade wind tunnel of the Universit~it der Bundeswehr Mtinchen (Hilgenfeld, Stadtmtiller and Fottner 2002). The cascade has the same LPT rotor blade as T106a, but with increased pitch-to-chord ratio. Bars are moving parallel to the inlet plane 70mm upstream of the leading edge. The basic parameters are listed in Table 3. This is operating point 1 of the test case documentation. The computational grid consists of a T106 profile with pitchwise periodic boundary conditions. The moving bar system is not included in the computation. Instead, a moving wake profile is used as inlet condition. The inlet of the computational domain is located between the bars and the leading edge at 60mm axially, upstream of the leading edge. The grid consists of 107000 cells. The y+ value in the first grid point in wall vicinity is below 0.4. Per pitchwise traverse 3 x 800 time steps are used. This high accuracy is necessary to properly simulate the unsteady movement of the wake.

9

2

calcuUluon

.,,p.,.n,t

1.5

o"

1

~ ,u.---~,.-,-i-,.

0.25

-,

q

0.5 x I I,x

7,,,

i

0.75

. . . .

i

1

TABLE 3 T 106D CASCADE blade chord c 100mm axial blade chord Cax 85.967mm pitch to chord ratio g/c 1.05 Reynolds number Re2c 2.0 x 105 exit Mach number Ma2t~ 0.401 inlet flow angle a/ -37.7 ~ bar diameter D 2.05mm bar pitch/cascade pitch 40mm/105mm axial distance bars to LE 70mm Strouhal number 0.84 <1% free-stream turbulence intensity turbulence level in the bar wake 7%

Figure 3" T106d; pressure distribution. Inlet conditions

The bar pitch had to be reduced to 35mm to obtain a multiple of the blade pitch in the calculation. The inlet angle had to be increased to 41.7 ~ in order to get good comparison with the experimental pressure distribution. An important remark is that increasing the inlet angle to 42.7 ~also gives good agreement for the pressure distribution, but has an important influence on the leading edge separation bubble. This change of the leading edge separation bubble alters the boundary layer integral parameters in the

527 acceleration phase of the suction side, which influences the receptivity to the Mayle transition criterion, resulting in too early transition prediction. Due to the small height of the blade (176 mm), the flow is three-dimensional. Comer vortices are present at the end walls in the decelerating part of the suction side. The pressure side remains relatively unaffected. This results in a measured value of the axial velocity density ratio AVDR of 0.8718 (from private communication). This effect has been incorporated in the calculation by means of a source term in the x- and y- momentum equations. This source term is a constant multiplied by the velocity component, and is only active in the rear half part of the decelerating channel between two blades. Its value has been tuned such that the source term prevents the trailing edge from separating. Compared to experiments, the computed trailing edge shape factor is higher. This indicates that the applied acceleration force is less strong than what would be an equivalent force of the experimental 3D effects. It was the choice of the authors to keep the acceleration force as low as possible. In the unsteady results, there is a closed separation bubble which varies in size under the effect of wake passage, but essentially is located between x/lax = 0.6 and 0.75. This is well captured in the calculation. The transition is of bypass type, and starts just upstream of the bubble. The transition is completed over the separation bubble. This makes the test case extremely sensitive to the inlet conditions. In the calculation with inlet angle 42.7 ~ the transition starts earlier, and fully suppresses the separation bubble. Verification of the inlet conditions is achieved by means of the time averaged pressure distribution over the profile, which is shown in Figure 3.

UNSTEADY TRANSITION MODEL We have chosen to work with the turbulence weighting factor ~, which is the sum of the 'near-wall' intermittency factor 7 and the 'free-stream' factor ~. The intermittency factor 3' represents the fraction of time during which the near-wall velocity fluctuations, caused by transition, have a turbulent character. The intermittency factor tends to zero in the free stream. The free-stream factor ~ expresses the intermittent behaviour of the turbulent eddies, coming from the free stream, impacting into the underlying pseudolaminar boundary layer. Near the wall, the eddies are dampened and the free-stream factor goes to zero. The free-stream factor is unity in the free stream of a gas turbine engine where the flow is always heavily turbulent. Both components of the turbulence weighting factor have been modelled by a convectiondiffusion equation.

Equationfor free-stream factor ( Cho & Chung (1992) investigated the transition between a turbulent jet and the surrounding laminar free stream. They developed an intermittency model which describes the intermittent submission of turbulent eddies coming from the jet disturbing the laminar free stream. Based on this work, Steelant & Dick (2001) included the dissipation term Er in their conditionally averaged transition model. Pecnik et al. (2003) did a recalibration of two constants to fit them into the globally averaged model.

0 /d /us O( U OUO( O(p() Ot ~ O(pU~() Ox, =-E( +~x, [( + ~ - ( ) ~ x , ] E ( = ' C31tCU~2 On On

(3)

The factor C3 was determined by Steelant & Dick to be C3 = 2.5, but Pecnik et al. argued that for globally averaged NS, instead of conditionally averaged NS, the value has to be adapted to C3 = 15. The term E~guarantees a zero normal variation of the free-stream factor near the wall. In combination with the

528 boundary condition ( = 0, this leads to a zero free-stream factor throughout the major part of the boundary layer. The diffusion coefficient g~ has been determined by Steelant & Dick to create an inverse Klebanoff profile for the free-stream factor prior to transition. The coefficient is a function of ~, but the Steelant & Dick term was a function of ~. Since this is a free-stream term, only the free-stream factor ~ should influence this diffusion coefficient. The factor C1 has been redetermined by Pecnik et al. to be C1 = 3.5.

/z< :/zClf,,,< Tu-~ E-In (1- ()-]-,/4(,-r

, f~,< = 1- exp -265

(4)

Equation for near-wall intermittency factor a(pr)

+

+

Pr = 2fl(1-r)x/-ln(1- y)P[U~F, +(Uf~ - U ~ ) ( 2 - F , ) ]

(5) (6)

The intermittency equation is a convection-diffusion equation with a source term, as developed by Steelant & Dick (2001); with ay=1.0. The role of the diffusion term is to allow a gradual variation of y toward zero in the free stream. The boundary condition for ), at the wall is a zero normal derivative. A damping functionf accounts for distributed breakdown of the turbulent spots. The production term has some important differences compared to the original one. When going to unsteady cases, the original production term appears to be a very slow term. The time behaviour becomes slow near the wall due to the low local velocity. The local velocity has been replaced by the expression between square brackets in Eqn. 6. During unsteady transition, the starting function F~ = 2 (see next section). As a consequence, the free-stream velocity Uoo,instead of the local velocity U is a factor in the production term. Further, the reduction term for distributed breakdown fx in the original source term also slows down the time behaviour. Therefore, it is no longer used during unsteady transition. These changes make the unsteady behaviour of the intermittency factor closer to reality. The accurate time and space dependent build up of intermittency is hereby not fitted with experimental data. As explained hereafter, the starting function is chosen such that the steady behaviour of the equation is not changed.

Detection of start of transition

Tl O6a The transition criterion is specific for unsteady transition in separated state. The transition is started based on flow separation. When the flow is separated, and the free-stream turbulence level is superior to 3%, transition is started. This means that downstream of that location Fs is set to two. Once activated, the starting function remains activated until the free-stream turbulence intensity drops below 3%. Then Fs is set equal to zero, so that the production term becomes a destruction term, leading to the decrease of the intermittency factor. The steady version of the production term is recovered when Fs is equal to unity. The procedure requires the definition of free-steam turbulence intensity, which is a non-local property. This is the turbulence intensity at the boundary layer edge for different streamwise locations on the profile. A streamwise coordinate s is created by solving the following equation: O(U,s)/8x, = U in the flow field. Boundary layer edge ("free-stream") properties are linked to this streamwise coordinate. The boundary layer edge is determined at a level of rotation equal to 1% of the maximal rotation at that streamwise location.

529

The turbulence level of 3% has been chosen as a threshold value for wake identification. The exact value of that threshold is not critical, and was determined by examination of turbulence intensity in S-T plots. The wake center has Tu = 5% at about S/Sma~ss= 0.7. In between two wake centers Tu = 0.45% at that location.

TlO6d Due to the low ratio of bar pitch to chord pitch, there is not a very clear distinction between the wakes. The turbulence intensity at the suction side boundary layer edge has only moderate variation around an averaged value of 2%. Therefore the Mayle criterion has been used to detect start of bypass transition in attached state (Mayle 1991). Distributed breakdown has not been taken into account. The starting function Fs is equal to one downstream of detection of transition.

TURBULENCE MODEL The turbulence model used is the SST k - co turbulence model from Menter (1994). The low-Reynolds modifications of Wilcox (1993) have been added. In order to suppress the excessive production of turbulent kinetic energy near the stagnation point, a time scale bound has been applied according to Durbin (2002). For the NS part of the calculation, the turbulent viscosity from the turbulence model is multiplied by the turbulence weighting factor r. For both the k and co equations, the production part of the source term is multiplied by a function of v. Before transition and inside the boundary layer (v = 0), the production term is multiplied by/l//~t. Consequently, the turbulence model is started up early but excessive creation of turbulent features is avoided. As a result, at start of transition, turbulent properties have a non-zero value. Small, but non-zero starting values are essential to let grow turbulent properties sufficiently fast in the transition zone. Outside the boundary layer or after transition, the full production terms are used. These considerations are incorporated in Eqn. 7. The factor 0.1 is introduced in order to ensure sufficient production throughout the domain. The dissipation part in the transport equations for k and co is left unaltered in the complete domain.

(7)

N U M E R I C A L CODE The calculations are performed with Fluent 6.0.20. Via User Defined Transport Equations (UDF) the two turbulence equations and the two intermittency equations are added to the Fluent NS solver. This is a finite volume (cell centered) solver, with implicit dual time stepping for convection and with pressure correction.

N U M E R I C A L RESULTS TlO6a

For a typical thin LP turbine blade about 60 percent of the profile losses are created on the suction side. We therefore have particular interest in the unsteady transition on the suction side. Here, the unsteady transition model is used. For the pressure side, a bypass transition mechanism (Lodefier & Dick 2003) is applied which does not influence the suction side results. In Figure 4, the wall shear stress is plotted on an

530 S - T diagram over the entire suction surface. Zones with negative wall shear stress have been blanked. The space coordinate S is the relative suction surface length S/Smo~. The time coordinate t/Tis rescaled with the wake period T. The solid line on the S - T diagram represents a particle path with the trailing edge free-

stream velocity. The dashed line represents half the trailing edge free-stream velocity. Clearly visible are two separation bubbles, with a velocity that is lower than half the free-stream velocity. Unsteady transition is activated if the flow is separated and the free-stream turbulence intensity is superior to 3%. The S - T diagram indicates that some time after their birth, the separation bubbles are suppressed by the transition to turbulence of the boundary layer (t/T = 0.4). Later, at t/T = 0.65, a small separation bubble appears at S = 0.77, which starts up transition at this location. The transition is stopped when the free-stream turbulence intensity drops below 3%. We clearly see the transition to high wall shear stresses being convected downstream along the suction side. The angle of this transition line is close to half the local free-stream velocity. Also shown in Figure 4 are the intermittency factor on the wall, and the free-stream turbulence intensity along the suction side. It is clearly visible that the combination of flow separation and free-stream turbulence intensity Tu > 3% induces transition. The transition is ended when the turbulence intensity drops below 3%. Comparison with the experiments is done by an S-T diagram of shape factor. In Figure 5, the experimental and computed shape factor is shown. There is good agreement between simulation and experiment. High values of shape factor occur at S values of 0.7 to 0.8. These high values indicate velocity profiles that are separated, or have the tendency to separate. There is also agreement with the wall shear stress diagram in Figure 4. Attention has to be made that shape factor and wall shear stress are not identical. Due to the incoming wakes, the velocity profiles are distorted. This has a different result on shape factor and on wall shear stress. Nevertheless, the regions of low wall shear stress and high shape factor are in agreement. In between two wakes (T =0.3 5), we see computed separation extending up to the trailing edge. This is not in comparison with the experiments where natural transition takes place at S - 0.9. In the simulation, no detection mechanism of natural transition is used.

Figure 4:T106a; S - T diagram over the suction side of the wall shear stress, the free-stream turbulence intensity and the wall intermittency factor

531

Figure 5: T106a; S - T diagram of the experimental (left) and computed (right) shape factor Tl O6d

Due to the increased loading, a separation bubble appears at 0.65 Cax.In steady flow (free-stream Tu = 2.5%) a large separation bubble appears. Due to the wakes in the unsteady flow, the separation bubble is reduced, and the start of transition is shifted. The transition is always of the bypass type. The transition location is located before the separation point, in the deceleration zone. As the deceleration increases, the growth rate of the spots increases. At this point a distributed breakdown model is inappropriate. At start of the deceleration phase, there is a competition between transition due to spot growth, and flow separation. This is a very delicate situation. The results are nevertheless good. This is because the experimental results were known by the authors, and have been aimed for by careful choice of the inlet angle. If the simulation goes into massive separation, a large 'hysteresis' is preventing the flow to reattach. Due to wake impact, the transition location moves forward, but the growth rate of the spots is low in the weak adverse pressure part of the suction side. The changes in intermittency are subtle, but the effect on the wall shear stress is clear. The separation bubble reduces under wake impact. See Figure 6. Separation can also be seen in the shape factor plot. This plot also shows that at the trailing edge, the computed flow is close to separation; the values of shape factor are higher than in the experiment. See Figure 7.

ACKNOWLEDGMENT The work reported was done within the research project 'Unsteady Transitional Flows in Axial Turbomachines', funded by the European Commission under contract number G4RD-CT-2001-00628.

REFERENCES

Cho R. and Chung M. K. (1992) A k-e -7 equation turbulence model J. Fluids Engineering 237, 301-322. Hilgenfeld L., Stadtmtiller P. and Fottner L. (2002) Experimental investigation of turbulence influence of wake passing on the boundary layer development of highly loaded turbine cascade blades. Flow, Turbulence and Combustion 69, 229-247. Lodefier K. and Dick E. (2003) Transition modelling with the SST turbulence model and an intermittency transport equation. ASME GT-2003-38282, Atlanta, USA.

532

Figure 6:T106d; S - T diagram of the computed free-stream turbulence intensity, the intermittency factor at the wall and the wall shear stress

Figure 7:T106d; S - T diagram of the experimental (left) and computed (right) shape factor Medic G. and Durbin P. A. (2002) Toward improved prediction of heat transfer on turbine blades. J. Turbomachinery 124, 187-192. Menter F. R. (1994) Two-equations eddy-viscosity turbulence models for engineering applications. A/AA J. 32, 1598-1605. Pecnik R., Sanz W., Geher A., and Woisetschl~iger J. (2003) Transition modeling using two different intermittency transport equations. Flow, Turbulence and Combustion 70, 299-323. Steelant J. and Dick E. (2001) Modeling of laminar-turbulent transition for high free-stream turbulence. J. Fluids Engineering 123, 22-30. Stieger R. D. and Hodson H. P. (2003) The transition mechanism of highly-loaded LP turbine blades. ASME GT-2003-38304, Atlanta, USA. Wilcox D. C. (1993) Turbulence modelling for CFD. DCW Industries, Inc.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

533

Transition to turbulence and control in the incompressible flow around a NACA0012 wing Y. Hoarau l, M. Braza 1, y. Ventikos 2 and D. Faghani 3 1 Institut de M6canique des Fluides de Toulouse, CNRS/INPT UMR n~ Toulouse, France 2 Dept of Engineering Science, University of Oxford, UK 3 Euro-American Institute of Technology, CERAM/EAI Tech, Sophia Antipolis, France

ABSTRACT The present study analyses the successive transition steps in the flow around a high-lift wing configuration, as the Reynolds number increases in the low and moderate range (800-10,000), by the Navier-Stokes approach. A very good comparison with a water-channel experiment is achieved. This flow system is mainly governed by two kinds of organised modes appearing successively as the Reynolds number increases: the von K~rm~n and the shear layer mode. A period-doubling scenario characterises the first 2D stages of the von K~rm~n mode up to Reynolds number 2000, where the shear-layer mode becomes predominant. The successive stages of the 3D transition are also analysed in detail. The history of the three-dimensional modes development from a nominally 2D flow configuration is clearly addressed, as well as the quantification of the spanwise wavelength. In a second step, the effect of wall suction has been studied both in 2D and 3D flows around the NACA0012 airfoil at 20 ~ of incidence and a Reynolds number of 800. This study has the objective to optimise the aerodynamic coefficients and to attenuate the mentioned 3D transition effects in the near wake. The receptivity of the flow to the suction is clearly shown and the suction position on the wall has been optimised according to the improvement of the aerodynamics coefficients (minimum of drag coefficient and increase of lift). In the 3D suction case, the optimum 2D suction position is used. Three spanwise suction ways are employed taking into account the spanwise natural wavelength. The secondary instability is removed in the near wake region and a good effect on the aerodynamic coefficients is achieved.

KEYWORDS

Instabilities, transition, drag reduction

534

INTRODUCTION The transition to turbulence in the flow around airfoils has received less attention till recently, when compared to similar configurations involving bluff-body wakes. The main interest in the research for unsteady flows around airfoils has focused on the high-Reynolds number range and especially to pitching motion analysis related to the dynamic stall. This interest is justified by the importance of the pitching motion of lifting bodies in the Avionics industry and especially Rotorcraft industry. However, there is a major interest, both from a fundamental and industrial point of view, with particular regard to the inherent unsteady flow around airfoils and wings: the spontaneous appearance of unsteadiness with steady external conditions. There is indeed a high interest to examine the natural transition to turbulence governing this kind of flows, because of the development of main organised modes that persist at the high Reynolds number range and they interact non-linearly with any imposed (forced) frequency oscillation. From a practical point of view, the transition mechanisms lead to a substantial growth of the mean values and amplitudes of the global parameters, an issue that is important in industrial applications in aerodynamics and in fluid-structure interaction. There are few attempts in the literature analysing the 2D unsteady separation at moderate Reynolds numbers. Mehta & Lavan (1975) in a pioneer work had simulated the starting separation vortex in a flow at low Reynolds numbers. A comprehensive review of the separation as well as of the dynamic stall can be found in McCroskey (1982). Furthermore, the birth of the natural transition in the incompressible or subsonic flow regimes were studied by Pulliam & Vastano (1993) up to the Reynolds number 3000, Ventikos (1993) as well as in the transonic regime by Bouhadji, & Braza (2002) up to the Reynolds number 10,000. Regarding the birth of the organised modes, there is little available knowledge concerning the three-dimensional mechanisms as well as the evolution of these modes over a wide, moderate Reynolds number range in the incompressible flow regime. The present paper focuses on the early stages of the natural transition, associated with the development of organised modes as the Reynolds number increases. The geometry chosen is the NACA0012 wing at a high angle of incidence 20 ~ ensuring a massively detached flow in a wide Reynolds number range. This study is based on time-dependent Navier-Stokes simulation, in two and three dimensions. The following points will be discussed: 9 Identifying the successive changes that the flow undergoes as the Reynolds number increases, as well as the related instability modes. 9 Analysing in detail the shear-layer transition under the non-linear interaction with the von K~rmb,n mode in the higher Reynolds number range. 9 Analysing the development of the 3D transition from a nominally 2D flow configuration subjected the von K&rm~n mode, that is strongly asymmetric, due to the lifting body configuration. The second part of this study deals with the effect of wall suction on a massively detached flow around a NACA0012 airfoil at 20 ~ of incidence and a Reynolds number of 800. 2D laminar computations have been performed as well as 3D computations where the effects of the suction on the 3D structures are presented.

535

GOVERNING EQUATIONS AND NUMERICAL METHOD The governing equations are the continuity and the Navier-Stokes equations for an incompressible fluid. The numerical method is based on the pressure-velocity formulation employing a predictor-corrector pressure scheme and the staggered grids of velocity and pressure. Two algorithms of similar features, both based on this methodology have been used for the 2D study, both leading to practically the same results. The first one is an implicit form of an originally explicit scheme by Amsden & Harlow (1970), extended in the present case to an implicit scheme for unsteady flows, by Braza, Chassaing, Ha Minh (1986). The Navier-Stokes equations are solved by using a general curvilinear conformal mapping, (Thompson et al (1974)). The time discretisation in 3D is achieved by an Alternating Direction Implicit fractional-step scheme, originally studied by Douglas (1962) for diffusion type equations and extended in the present study in the whole convection-diffusion Navier-Stokes systemfor unsteady flows. This scheme employes an Alternating Direction Implicit method, leading to the solution of tri-diagonal systems by a very fast algorithm. The time accuracy is of second-order. The space discretisation is done by using central differences providing a second-order of accuracy. The wall pressure is discretised by the exact form of the Navier-Stokes equations at the wall, and not by using approximate Neumann-type conditions as usually. This ensures an accurate evaluation of the wall pressure especially in the strongly detached flow regions in 3D. These overall aspects constitute the EMT2/IMFT code ICARE, widely used in an academic research context. The details of the numerical scheme in 3D can be found in Persillon & Braza (1998). The code ICARE is fully parallelised in MPI architectures, (Hoarau, Rodes, Braza, Mango, Urbach, Falandry and Battle (2001)) and allows the use of a high number of parallel processors and therefore, significant grid sizes for the DNS approach. This is decisive for realistic 3D computations. The second algorithm used is a SIMPLE-type fully implicit algorithm, whose description can be found in Tzabiras, Dimas & Loukakis (1986). The discretisation of the space derivatives is done by using central differences. Orthogonal curvilinear staggered grids are employed. The methodology is secondorder accurate in space and time. The equations are written in orthogonal curvilinear systems. The boundary conditions are free-stream at the inlet boundary and non-reflective absorption conditions for the outlet boundary, specified by Jin & Braza (1993) to minimise very efficiently any feedback effect in the incompressible flow regime. The impermeability and adherence conditions are specified for the solid wall. C type grids are employed in both cases. The computations are carried out in the supercomputers SP3, SP4 and Origin 3000 of the national supercomputer centers CINES, IDRIS and CALMIP.

INSTABILITY AND TRANSITION TO TURBULENCE

The first steps of 2D transition Firstly, computations are carried out at the very low Reynolds number regime (around Re=5) showing a fully attached steady flow. As the Reynolds number increases, a small attached vortex is created near the trailing edge and grows with the Reynolds number. Near Reynolds number 70, where the bubble has attained a nearly 40% length relatively to the chord, unsteady separation starts and a very regular vortex shedding appears. The vortex shedding pattern is attained at Re=450. At Re=800, the vortex shedding motion is very regular. This regime is analysed in detail in the present study with respect to the 2D and 3D transition mechanisms. Beyond the Reynolds

536

number 800, the vortex shedding regularity is attenuated and other predominant frequencies appear, being fractions of the fundamental, up to Re=2000. As the Reynolds number increases further on, the transition process becomes more complex, because of the development of a shearlayer instability as an incommensurate mode. In fact, two different mechanisms can be identified as the Reynolds number increases: the period-doubling mechanism and the shear-layer instability.

Figure 1. Pressure coefficient spectra showing the period doubling mechanism and Instantaneous iso-pressure contours at Re=4000 showing the shear layer instability The first mechanism corresponds to the evolution of the von K~rmhn instability and it is closely related to a period doubling scenario as is clearly shown in the spectra of the numerically obtained signals for increasing values of the Reynolds number in the range (800-1600), (fig. la). The appearance of the first subharmonic frequency of the main vortex shedding one can be physically justified by the fact that at a higher Reynolds number value, the shed vortex close to the trailing edge is weakened at exactly 2T (T being the vortex shedding period) by the opposite vortex which starts to be shed. This illustrates the phenomenon of period doubling, that is characterised by an energy- vorticity exchange process. This mechanism becomes predominant whenever the externally supplied energy (Reynolds number) to the system becomes higher than a critical value, in the context of the non-antisymmetric vortex shedding mode, owing to the liftingbody configuration. It will be remembered that this kind of mechanism does not appear in the case of a symmetric von Khrmhn vortex shedding, as for example in flows past bluff bodies. The period doubling mechanism appears repetitively as the Reynolds number increases further on, and it yields spectra with four, eight . . . . peaks. Our results compare very favourably with the ones obtained by Pulliam & Vastano (1993). These authors have shown that the period doubling continues further on, as the Reynolds number increases. The second mechanism appears beyond Re=2,000. The separated shear layer undergoes another important transition mechanism that gives rise to an incommensurate frequency, due to the development of a Kelvin-Helmholtz instability. In the present case, this instability mode is forced by the oscillatory motion of the separation point, that obeys the von Kb.rmhn instability. Figure 1b show the clear formation of Kelvin Helmholtz vortices. The length of the shear-layer vortices is smaller than the von Khrm~n ones and it decreases as the Reynolds number increases. A detailed space-time tracking of these vortices allows the evaluation of the shear-layer instability wavelength in the present case. It is found that the wavelength decreases as the Reynolds number

537

increases, according to the law ~,sl~ Re "0"44. Furthermore the variation law of ratio fsl / fv-K is o~ Re~ This law exponent is very close to Re~ that characterises the development of the instability wave prior to separation, as reported by Bloor (1964) concerning bluff-body wakes. The shear-layer frequency is an incommensurate mode in comparison to the von K~rmhn mode. This leads, in association with the period-doubling scenario, to the non-linear filling-up of the energy spectrum by a multitude of modes, that are combinations of the von K~rmhn and of the shear-layer mode, in the same way as reported by Braza, Chassaing & Ha-Minh (1990) for bluffbody wakes.

Figure 2. Evolution of the global parameters and of the spanwise velocity structure; (a) mean drag and lift coefficients versus Re and (b) time-space evolution of the w velocity component along the span at Re=800 The variation of the global parameters versus the Reynolds number is shown in figure 2a. The mean lift coefficient undergoes a smooth decrease as the Reynolds number increases, because the flow is already stalled at 20 ~, for all the Reynolds numbers examined. Therefore, this decrease corresponds to an equivalent behaviour occurring at fixed Re and increased incidence beyond stall. For the same reasons the drag coefficient shows a plateau saturation level, attained in the intermediate Reynolds number range, to decrease further on as a function of the Reynolds number. The more abrupt decrease of the drag coefficient beyond the Reynolds number 5,000 is a consequence of the multitude of the shear-layer vortices and of the formation of a quasistagnation region below the separated shear layer, up to the wall. The Strouhal number evaluated from the lift coefficient directly obeys the shedding motion of the lower trailing edge vortex, that is delayed by the creation of the above-mentioned fully developed region beyond Re=5000. Therefore, this step is qualitatively similar to the "drag crisis' appearing in bluff-body wakes at higher Reynolds number, although the wake formation region of a circular cylinder varies significantly in this range. In the critical regime, a multitude of small-scale vortices are created upstream of the separation, because of the boundary-layer transition occurring upstream of the separation. The Strouhal number also shows a decrease as the Reynolds number increases due to the same reasons. These effects are obtained in the present study by the completely non-linear approach of the Navier-Stokes system. A comparison of the above global parameters is done with a water-channel experiment Williamson, Govardhan &

538

Prasad (1995), where pressure and forces measurements were carried out by means of pressure transducers and forces balance, as well as pressure fluctuation spectra. A good agreement is obtained at Reynolds number 10,000 between the present simulation and the experiment. Furthermore, the 3D computations in the low Re-range found that the Strouhal number is essentially the same as in the 2D case. These facts ensure the validity of the numerical study from the low to the higher Reynolds number range.

The first steps o f 3D transition In this section the way of the development of the 3D transition to turbulence from a nominally 2D flow configuration is examined. The Reynolds number 800 has been selected first, because it corresponds to a very strong and regular development of the von Khrmhn mode. The initial conditions are either those of a flow at rest triggered by a very weak spanwise w velocity fluctuation imposed as a random fluctuation, or a fully developed 2D vortex shedding pattern perturbed in the same way. The dimensionless rms values of the spanwise fluctuation are of order 10.4 U~ This technique does not privilege the appearance of any wavelength and the order of magnitude of the fluctuation is very weak and less than the physically existing upstream noise in any wind tunnel. By performing a very detailed 3D study, it has been found that the flow "forgets' its initial conditions and both ways of initiating the 3D transition lead to the same final regime: the first step is the development of the 2D von Khrm~n pattern followed by the appearance of the 3D mechanisms as described below. Figure 2b shows the time-space evolution of the w velocity component along the span. After a transient phase, the onset of the 3D transition appears as the organised pattern of the iso-w velocity contours according to coherent counter-rotating cells. This step is followed by the amplification of the longitudinal and vertical vorticity components (fig. 3a-b), that are found to form the same kind of coherent cells. This spanwise-periodic fluctuation plays the role of a perturbing factor acting on the von Khrmhn rectilinear vortex rows. Consequently, the COz vorticity is modified according to the vorticity conservation equations (fig 3c). Following the elliptic stability theory (Landman & Saffman (1987)), the expected spanwise mode of an originally 2D elliptic-shape vortex (in the present case the von K~rm~n vortex rows) is a 3Dundulated large-scale vortex row according to a regular spanwise wavelength.

Figure 3. Spanwise evolution of the vorticity

539

The dynamics of this pattem are similar to the ones of bluff-body wakes DNS studies, (Persillon & Braza (1998)), but in the present case the shearing mechanism is totally asymmetric. The shape of the undulated vortices is much more stretched, according to the lifting body configuration. By performing a space-averaging of all the 3D transverse sections at the same instant, it has been proven that the alternating vortex pattern is very similar to the corresponding 2D configuration at the same phase. Therefore, the present 3D route to transition is expected to be affected by the same kind of period-doubling cascade and of the shear-layer instability, as discussed in the 2D study. However, the present 3D study is still carried out in a low Reynolds number range where these effects are not yet fully pronounced. By carrying out a Fast Fourier Transform analysis of the spanwise evolution of the secondary instability mode, it has been feasible to evaluate the preferential spanwise wavelength developed under the present conditions, ~,z/D= 0.64. This value is found in good agreement with the results concerning bluff body wakes: although the fundamentals of the shearing mechanism are different in the present case of lifting body wakes, an analogy with the bluff body ones can be done by considering an "equivalent' bluff body configuration having a characteristic vertical distance c * s i n ( ~ in respect of the upstream velocity direction, c being the chord and cr the incidence. Therefore, the "'effective" Reynolds number in the analogy bluff-body and wing-body is R e * c * s i n ( 2 0 ~ = 273. The expected wavelength for this "equivalent' bluff-body wake would be of order 0.60-0.70 according to the DNS by Persillon & Braza (1998), Braza, Persillon & Faghani (2001). The same kind of 3D dynamics govern the flow at Re=1200 and the corresponding wavelength ~,z/D= 0.62, at an effective Re=410.5. The secondary instability development is expected to continue as Re increased by providing progressively smaller wavelengths, together with the development of the previous mentioned cascades. Because of the robustness of the 2D alternating pattern even at higher Re, it is expected that both routes to transition persist and be clearly identified as Re further increases, by involving in addition non-linear effects. This study is an immediate outlook of the present one, by our research group.

CONTROL OF INSTABILITIES In this part we study the effect of a constant suction at the wall on the aerodynamics coefficients and on the second instability of a NACA0012 airfoil at 20 ~ of incidence and for a Reynolds number of 800. We first studied the effect of the position of the suction on the aerodynamic coefficients and then used the best position to study 3D effects. Two-dimensional

case

TABLE 1. Evolution of the aerodynamics coefficients with the position of the suction.

Without suction x/c=0.016, D=9.04 103 x/c=0.063, D=l.03 10-2

Drag Coefficient Cx 0.435 0.361 0.34

Lift coefficient Cz 0.968 0.899 0.96

540

x/c=0.063, x/c=0.114, x/c=0.166, x/c=0.218, x/c=0.270, x/c=0.322, x/c=0.374, x/c=0.439,

D=2.07 D=l.03 D = 1.04 D =1.04 D = 1.04 D = 1.04 D=l.03 D=l.31

102 102 10-2 102 10"2 10~ lif e lif e

0.285 0.389 0.418 0.428 0.432 0.433 0.436 0.439

1.071 0.972 0.961 0.95 0.941 0.961 0.93 0.928

The suction is applied to an initial field already submitted to the von Khrm~n instability. The suction is a generated by modifying the non-slip condition at the wall and using a negative vertical velocity. Many positions have been tested and the results on the aerodynamics coefficients are shown on the table 1, where D is the normalised debit D = ~,~,o,, * l~uc,o,,. The best position is found to be at x / c = 0 . 0 6 3 for a suction vertical velocity v = - 0 . 4 . This position corresponds to the beginning of the detachment on the extrados. It is noticeable that the suction doesn't kill the von K~rmhn instability although it has favourable effect of the drag and lift coefficients 9 Three-dimensional

case

In this last part we have chosen the optimum position found in the 2D study and we performed 3D (DNS) computations. The initial flow field is one already submitted to the first instability (von K~rm~n) and the second (spanwise undulation). The suction has been applied in the different ways: every ~,z in the spanwise direction (~,z is the spanwise wavelength of the secondary instability), every ~.z/2 and continuously. -0.45 [-

--= - - - ,~Vithout c o n t r o l --~ - _~'z, D = 1 . 0 3 E - 2 ~Lz/2, D = 1 . 0 3 E - 2

-Z, D = 1 . 0 3 E - 2 ................... Z, D = 2 . 0 7 E - 2

1.4 -

- - ---= ~ --,- - -- ......

W i t h o u t control ~z, D = 1 . 0 3 E - 2 ~./2, D = l . 0 3 E - 2

1.3

~ ...................

Z, D = 1 . 0 3 E - 2 Z, D = 2 . 0 7 E - 2

0.4

z~3~ ~. ~:!2,~ :'.

r~

i r

t t.:"~

:

0.35

9 0.3

, :L

L/

0.9

.'t

0.25

0.2 ~

30

~

,

I

32

,

t

I

I

34

I

~

t*

I

36

a)

~

~

~

I

38

~

;

~

I

40

0"630

,

;

~

I

32

~

~

,

I

34

b)

,

, t*

, I

,

,

, 38

j

~ ' 4 1I

Figure 4. Effect of the suction on the 3D aerodynamic coefficients Figure 4 shows that each configuration has a positive impact on the coefficients but the suction applied continuously in the spanwise direction give the best results. The suction speed of-0.4

541

provides better results than the -0.2 one. In fact the suction applied periodically along the span according to the natural wavelength distance, acts as a forcing that improves the aerodynamic coefficients but has little effect on the three-dimensionality of the flow in the near wall region. The suction applied continuously along the span indeed attenuates the spanwise undulation. (fig. 5). In the case of a continuous suction, the flow becomes practically 2D in the near wall region and weakly 3D in the far region.

Figure 5. Time and space evolution of the vertical velocity in the recirculation area for a suction applied every ~z (a) and continuously (b) CONCLUSION The present study analyses the successive transition steps in the flow around a high-lift wing configuration, as the Reynolds number increases in the low and moderate range (800-10,000), by the Navier-Stokes approach. A quite good comparison is performed with a water-channel experiment. According to a 2D study, it is found that the present flow system is mainly governed by two kinds of organised modes appearing successively as the Reynolds number increases, the von K~rmhn and the shear layer mode. A period-doubling scenario characterises the first 2D stages of the von K~rmhn mode. The analysis of the shear-layer mode in the flow around an airfoil as a function of the Reynolds number is carried out. The variation law of the predominant streamwise wavelength and of the shear-layer frequency are determined versus the Reynolds number. The successive stages of the 3D transition around a lifting body beyond the first bifurcation are analysed in detail in the low Reynolds number range (800-1200). The history of the threedimensional modes development, the robustness of the alternating vortex pattern and the quantification of the spanwise predominant wavelengths are clearly shown. The effect of a wall suction around a NACA0012 airfoil at 20 ~ of incidence and a Reynolds number of 800 has been studied. The optimal position for the suction has found at the beginning area of the detachment. Improvements of the aerodynamic coefficients in the 2D case are achieved, confirmed in the 3D case. Furthermore, an attenuation on the secondary instability has been achieved by employing the suction technique along the span.

542

REFERENCES Amsden, M. A. & Harlow, F. H. (1970). The SMAC method: a numerical technique for calculating in compressible fluid flows. Los Alamos Scientific Laboratory Report. L.A. 4370. Bloor, M. (1964). Transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290-304. Bouhadji, A. & Braza, M. (2003). Organised modes and shockvortex interaction in unsteady transonic flows around an aerofoil. Part I and II. J. Computers and Fluids 32(9),1233-1281. Braza, M., Chassaing, P. & Ha-Minh, H. (1986). Numerical study and physic alanalysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79-130. Braza, M., Chassaing, P. & Ha-Minh, H. (1990). Prediction of large-scale transition features in the wake of a circular cylinder. Phys. Fluids A 2, 1461-1471. Braza, M., Persillon, H. & Faghani, D. (2001). Successive stages and the role of natural vortex dislocations in three-dimensional wake transition. J. Fluid Mech. 439, 1--41. Douglas, J. (1962). Alternating direction methods for three space variables. Numerische Mathematik 4, 41-63. Hoarau, Y. (2002). Analyse physique par simulation num6rique et mod61isation des 6coulements d6coll6s instationnaires autour de surfaces portantes. Th~se de Doctorat de I'INPT. Hoarau, Y., Rodes, P., Braza, M., Mango, A., Urbach, G., Falandry, P. & Battle, M. (2001). DNS of the 3D transition to turbulence in the incompressible flow around a wing by a parallel implicit navierstokes solver. In Proc. Parallel CFD 2000, Trondheim, Elsevier, pp. 433-440. Jin, G. & Braza, M. (1993). A non-reflecting outlet boundary condition for incompressible unsteady Navier-Stokes calculations. J. Comput. Phys. 107, 239. Landman, M. & Saffman, P. (1987) The three-dimensionnal instability of strained vortices in viscous fluid. Phys. Fluids 30, 2339-2342. McCroskey, W. J. 1(982). Unsteady airfoils. Annu. Rev. Fluid Mech. 14, 285-311. Mehta, U. B. & Lavan, Z. (1975). Starting vortex, separation bubbles and stall: A numerical study of laminar unsteady flow around an airfoil. J. Fluid Mech. 67, 227-256. Persillon, H. & Braza, M. (1998). Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional navier-stokes simulation. J. Fluid Mech. 365, 23-88. PuUiam, T. & Vastano, J. (1993). Transition to chaos in an open unforced 2D flow. J. Comput. Phys. 105. Sadeh, W. & Brauer, H. J. (1980). A visual investigation of turbulence in stagnation flow about a circular cylinder. J. Fluid Mech. 99, 53-64. Thompson, J.F., Thames, F.C., Mastin, C.W. (1974) An automatic numerical generation of body-fitted curvilinear coordinates system for flows containing any numbers of arbitrary two-dimensional bodies. J. Comput. Phys. 15, 299-319. Tzabiras, G., Dimas, A. & Loukakis, T. (1986). A numerical method for the calculation of incompressible, steady, separated flows around aerofoils. Int. J. Numer. Meth. Fluids 6, 789-809. Ventikos, Y. (1995). Numerical investigation of unsteady, cavitating and non-cavitating flows around hydrofoils. PhD Thesis, National Technical University of Athens. Ventikos, Y., Tzabiras, G. & Braza, M. (1993) The effect of viscous dissipation on the organised structures in the wake past an aerofoil in transition to turbulence. In Ninth Syrup. on Turbulent Shear Flows, Kyoto, Japan, August 16-18. Williamson, C., Govardhan, R. & Prasad, A. (1995). Experiments on low Reynolds number NACA0012 aerofoils. Tech. Rep. Cornell University.

8. Turbulence Control

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

545

SOME OBSERVATIONS OF THE COANDA EFFECT

G. Han, M.D. Zhou and I. Wygnanski Department of Aerospace & Mechanical Engineering, The University of Arizona Tucson, AZ 85721, USA

ABSTRACT A curved wall jet over a convex surface represents a complex turbulent shear flow, because it is susceptible to three types of primary instability modes: the inflectional and centrifugal instabilities in the outer region and the viscous instability near the surface. The relationship between inflectional instability and large spanwise, coherent structures in turbulent jets is well known, but the existence of large streamwise vortices in centrifugally unstable turbulent flows is still questionable. Such vortices were discovered and mapped on a "Coanda flow," and the present investigation explores their evolution in the direction of streaming. These vortices meander, coalesce, and increase in scale as they move downstream, while the width of the flow and the equivalent turbulent G6rtler Number also increase.

KEYWORDS Wall jet, vortices, instability, turbulent flow.

INTRODUCTION The deflection of a jet by a convex surface is of technological interest because of its high effectiveness in exerting a force on the surface. Since a jet reverses its direction before separating from a cylinder over which it flows, it generates a low-pressure region on the cylinder that results in a side force that is almost equal to twice the jet momentum emanating from the nozzle. It can be applied to enhance the low speed maneuverability of submersible vehicles, because it is more efficient and quiet than conventional thrusters that exert a smaller reaction force. A wall jet flowing over a circular cylinder has replaced the tail rotor on NOTAR type helicopters preventing autorotation. This application is in the presence of an external stream (the rotor downwash) whereupon the Coanda effect becomes synonymous with super-circulation. Although substantial understanding of the flow was achieved in recent years, the mechanism for flow separation even in the absence of the external stream remains inexplicable.

546 The "Coanda Effect" has been extensively investigated at the University of Arizona, where the initial purpose of the investigation was to determine the effects of streamline curvature on the mean flow and on its turbulence intensities (Neuendorf & Wygnanski, 1999). Currently the investigation focuses on the centrifugal instability and the generation of the large streamwise vortices that may eventually lead to the understanding of the separation mechanism of the wall jet from the convex surface (Neuendorfet a12004, Han et a12004). The streamwise vortices in the turbulent, curved wall jet are non-stationary, meandering in both spanwise and radial directions. They may be observed at a given instant, but these observations could not be translated into statistically meaningful quantity until the appearance of the PIV (Particle Image Velocimeter) that provided quantitative information in the plane of illumination. It was shown by Likhachev, Neuendorf and Wygnanski (2001) that the longitudinal vortices have a preferred spanwise wavelength, identifiable by an averaged negative value of a two-point, cross-correlation measurement using hot wire anemometers. However, the maximum negative value of this correlation is quite small suggesting that the longitudinal vortices are either weak or they are not stationary.

Figure 1: Instantaneous streamwise vorticity contours (shades of gray) that are superposed on velocity vectors in the cross flow plane of a curved wall jet (Neuendorf et a12004) PIV measurements, taken at several cross-sectional planes in this flow, revealed the existence of counterrotating streamwise vortex pairs (figure 1) whose location across the span differed from one case to another requiring pattem-recognition techniques in order to be described in a statistically meaningful manner (Neuendorf et a12004). Triple decomposition of the data into stationary, coherent, and random constituents enabled one to describe the flow in a frame of reference that is stationary relative to the vortex cores. Freed from the high-frequency background turbulence and their own low-frequency meander, the mapped vortices provided a new insight into the effects of curvature on otherwise highly turbulent shear flow. The results suggest that the longitudinal structures are not stationary and do not contribute to mean spanwise distortions, but they are strong enough to augment the Reynolds stresses and increase the rate of spread of the flow and its turbulent intensities. Their spanwise wavelength ;~z, might have been the dominant factor in determining the width of the jet, as it was found to scale directly with the arbitrarily defined local width y2, providing the relationship z.z = 2y2. Finally, the circulation of the individual streamwise structures rx.~ increased rapidly in the direction of streaming, yet the number of these structures per unit span concomitantly decreased, leaving the circulation per unit span rx,~ almost constant (The circulation actually increased ever so slightly with increasing distance from the nozzle.) This suggests that most of the increase in the strength of the vortices may be attributed to their growth in the direction of

547

streaming. This growth may come from amalgamation of structures having the same sign of vorticity or amplification that is associated with instability (figure 2). The use of pattern recognition and enhancement techniques may raise questions about the origin of the observed vortex pairs, thus so an attempt was made to reduce their meander along the span and prove their independence of the upstream flow conditions. The introduction of a free stream normal to the axis of the cylinder has also been considered, but the results are beyond the scope of the present paper. Nevertheless some established concepts about blowing boundary layer and circulation control were put to the test. 1.5- ~

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Figure 2" The development of the streamwise component of circulation, Fx with increasing distance from the nozzle

EXPERIMENTAL SETUPS A detailed description of the experimental apparatus used in the absence of an extemal stream is available in the published articles by Neuendorf et al. (1999, 2004). The Coanda Effect was reexamined on two additional cylinders having a smaller diameter for the purpose of expanding the present investigation to include the presence of an external stream. In this case the diameter of the cylinders is limited by the size of the wind tunnel in order to avoid large blockage effects. Although this part of the investigation will not be presented, the differences in the pressure distribution generated by the various cylinders are expressed. In one of the cylinders, the nozzle was convergent and curved in order that the jet will emerge almost tangentially to the surface, while in the other the nozzle was simply cut at an inclination of 30 ~ to the surface. The pressure distribution on the surface of each cylinder was also measured in the absence of external stream in order to assess the significance of the nozzle design on the flow. The pressure distribution on cylinder N~ indicates that the flow totally separates at an angular distance of 200 ~from the nozzle (figure 3). Cylinder N~ separates around 180 ~ and on cylinder N~ even earlier than that. Variation of slot width to diameter ratio, b/D, had no appreciable effect on the separation location, nor did significant variations in Reynolds number on a given cylinder. It is suspected that the shape of the nozzle is important and that one can not normalize the flow by (po-poo)b that is ideally equivalent to 89the jet momentum, J, provided a "top hat" velocity profile emerges from the nozzle. Pressure scanners were used for pressure measurements and hot-wire anemometers were used to calibrate the flow near the nozzle. A three component PIV system was used with a light sheet being located either in the cross-stream plane at successive distances from the jet-exit or in the streamwise plane at selected distances from the surface. The accuracy of the PIV measurements was checked by comparing them to hotwire measurements. The overall agreement between two systems of measurement was within 5% except in the extreme outer region of the jet where large fluctuation of streamwise velocity and large radial velocity component render the hot wire measurements suspect. Instantaneous counter rotating vortices inside the wall-jet were educed from the instantaneously measured velocity vectors. Simple ensemble averaging of

548

the data sufficed in the present investigation to obtain statistically representative results. An effort was made to stabilize the location of the streamwise vortices. At first, a single row of small conventionall vortex generators was placed along the entire span of the cylinder at various distances from the nozzle after verifying that the flow was turbulent without them and was insensitive to changes in Reynolds number. These vortex generators (VGs for short) interfered with the natural development ofthe flow and dominated the spanwise wavelength, ;~z, of the streamwise vortices downstream. It became obvious when the ~,z observed was always equal to the spacing between adjacent VGs and it did not change throughout the flow. In their absence, the ;~z educed increased with downstream distance and scaled with the local width of the flow.

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Figure 3: A comparison of pressures measured on three different cylinders, ReN=I 1000

Figure 4: Schematic drawing of the micro vortex generators and their placement A row of micro vortex generators (~tVGs) was placed on the outer lip of the nozzle (Figure 4) in order to localize the longitudinal vortices in the centrifugally unstable region without interfering with the basic state ofthis flow. The ~tVGs are wires that protrude into the nozzle a distance of 0.4 mm only. In order not to impede the natural growth of the streamwise vortices, the diameter of the wires and their protrusion into the flow was carefully selected. They also did not change the width of the flow or its location of separation.

D I S C U S S I O N

OF

R E S U L T S

Ensemble averaged vorticity contours showing the growth of the streamwise vortices with increasing

549 distance from the nozzle are plotted in Figure 5, where y is the radial distance measured from the surface and Y2 represents the local width of the wall jet. The window-size in this figure was not changed with 0 enabling a visual comparison of the eddy sizes at various distances from the nozzle. These experiments were carried out with the ~tVGs being spaced almost eleven slot-widths (2-,/b =10.85) apart. The PIV measurements were made at various downstream locations and only representative contours are shown. Between 0=30 ~ & 40 ~ the patterns of ~ = ~*b/Um~ are very regular and their spanwise wavelength is constant (U,,~ is the local mean maximum velocity). The strong vortices are adjacent to the surface, although the l.tVGs disturb only the exterior part of the jet. At the first cross section measured (i.e. at 0=30~ the circulation associated with the counter clockwise rotating flow (positive contours inside the solid lines) is much stronger than that of the counterclockwise rotating vorticity (negative contours inside solid lines). At 0=40 ~ the negative vorticity contours became larger and moved further from the surface. The increase in the typical dimension of the individual structures between 0=30 ~ & 40 ~ suggests an increase in circulation resulting from amplification. At both values of 0, one may discern that above every strong vortex located near the surface, there is a weak one of opposite sign. Farther downstream (at 0=60 ~ the vortex pattern looses some of its coherence, and at 0=90 ~ the spanwise wavelength seem to have doubled. Between 0= 130 ~ and 150 ~ the ~ contours become more regular and larger (recall that the length scales are normalized by y2).

Figure 5: Simply ensemble averaged streamwise vorticity contours, ~, at different streaming locations for VG spacing of ~ i / b = 10.85. Solid lines represent positive (counterclockwise rotating) vorticity

550 To establish the independence of 2z from the initial spacing between the l,tVGs, 2~, experiments were carried out in which ,;/,twas altered. Two cases are shown in Figure 6 for which 2,~was changed by a factor of 3. It is clear that the initial vortex spacing observed at 0=40 ~ is determined by the spacing ofthe ~tVGs; however, farther downstream (e.g. at 0=130~ ,~z is independent of the I,tVG spacing. In fact the larger 2/b=l 6.28 may have generated somewhat smaller coherent structures at 0=130 ~ These measurements suggest that the streamwise vortices are a product of an instability that probably determines the characteristic width of the wall jet.

Figure 6: The ~ contours triggered by different VG spacing (~,i) The streamwise location at which the instability is first noticed depends on 2,~,consequently for 2/b=5.43 the first signs of this instability appear at 0=40 ~ where the positive and negative vorticity contours are of equal strength (figure 5). At this distance from the nozzle some dislocations in the strength and vertical position of the vortices are being noticed. For 2/b = 16.28 the initial wavelength o f ~ is maintained up to 0=70 ~. The vortex merging process is shown in more detail in figure 7 for VG spacing of 2/b=5.43. The vorticity contours in the cross-flow plane shown in figure 7a suggest that a braiding process is occurring, particularly around AZ/y2=I.2 This is corroborated in figure 7b where contours of radial velocity component are shown in the R0-Z plane. There is a clear merging of vortices that occurs at 0> 100 ~ Since the data shown had been simply ensemble averaged, the process must be at least quasi stationary. The concentration of streamwise vorticity is very uniform near the nozzle. Farther downstream dislocations are observed where mostly one positive vorticity concentration is displaced farther from the surface and is braiding with another positive vorticity core while suppressing the region of negative vorticity that separates them. Such an amalgamation process is best observed when the PIV plane of illumination is tangential to the surface of the cylinder (figure 7b). The wall jet velocity profiles at various cross sections relative to these longitudinal vortices are continuously distorted due to the lifting and twisting of neighboring vortices that merge or braid wherever there is not enough space to accommodate them. The braiding process is continuously evolving with increasing downstream distance.

551

Figure 7: The merging process of ensemble averaged streamwise vortices--VG spacing ~,i/b = 5.43; (a) measurements in the cross plane Y-Z; (b) Measurement carried out in the tangential plane R0-Z In order to quantitatively assess the dominant wavelength, 2, of the streamwise vortices at each value of 0, the spanwise velocity undulations were decomposed into Fourier modes within a constant observation window of 128 mm (the size of the window remained constant at all 0 locations). This represented the longest wavelength that could be considered. The amplitude of each wave measured at 0=40 ~was used to normalize the data presented herein. Figure 8 represents the spatial amplification of each wave, starting from the largest that can be accommodated within the observation window of and ending with one whose length occupies 1/10 th of this window's span. The slope of each curve represents the rate at which these waves amplify with downstream distance. The signal used to assess this amplification was the ensembleaveraged product of the instantaneous radial and streamwise velocity components UV. By repeating the procedure for a variety of VG spacing, it was again demonstrated that the dimension of the streamwise vortices in this flow is independent of the input wavelength 2/b. It appears that the rate of amplification of the 3rd spanwise mode peaks the highest at 0 = 100~ and it is overtaken by the 2 nd and 1st modes at successively larger distances from the nozzle. The 4 th mode reaches significantly lower peak amplitude than the first three. 50-

window size

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Figure 8" The increase of circulation of individual vortices and the wave mode contents and growth for ~ i / b - 16.28 normalized by the amplitude at 0 = 40 ~ The UV product is used to perform Fourier transform and decomposition The amplification of streamwise vortices with increasing distance from the nozzle was established by calculating their circulation, F, by integrating the vorticity field over the area of each individual vortex rather than the ensemble averaged vorticity map. The growth rate data of F shown in Figure 8 is for the case Li/b = 16.28, for which no vortex amalgamations were observed prior to 0= 130 ~ The circulation generally increases until 0=120 ~ whereupon the vortices start to merge.

552

1.0-

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Figure 9: Contours of the streamwise component of velocity superimposed on velocity vectors in the cross flow plane y-z (i.e. showing the v, w components). 0 = 150 ~ VG spacing 2i/b = 10.85 The constant meander of the streamwise vortices may be understood better by considering the strong cross coupling between them and a secondary instability that they create. Contours ofthe streamwise velocity U and associated streamline traces in the cross flow plane (w, v), measured at a large distance from the nozzle are shown in figure 9. The numbers on the contours indicate the values of the streamwise component of the ensemble-averaged velocity. It is apparent that the longitudinal vortices modulate the mean flow distorting the velocity across the span. The downwash created between them brings stagnant ambient fluid from above while the up wash on their other side takes retarded fluid from near the surface and brings it to the center of the jet. Little wonder, therefore, that the wall jet on the curved surface spreads out decays so rapidly in comparison to the wall jet on a flat surface. (a)

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Figure 10: Modulation of streamwise velocity by streamwise vortices (a) Spanwise modulation, at y corresponding to the vortex core, each curve shifted by 1; the lines without symbol at 500,900 & 1300 represent the unperturbed velocity profiles (b) Modulation of velocity profiles at two locations. VG spacing ~ i P O -" 10.85 A global view of the distortion of the streamwise component of velocity may be obtained from figure 10. A typical spanwise modulation of this velocity component measured at a given distance from the surface is plotted in this figure. The ensemble averaged data shown was collected between 50~ ~, and it indicates the increase in Lz with increasing O. The unperturbed spanwise variations in u are indicated by a solid lines at 0=50 ~ 90 ~ & 130 ~ in this figure. These lines indicate that the flow is two dimensional in the mean; however, there is a large variation in the shape of the velocity profile on the opposite sides of the

553

mean streamwise vortex. The entrained ambient fluid on the inflow side of the vortex (that possesses negative radial velocity component) is generally slower than the velocity in the outflow region. The unperturbed flow represents approximately the average between these two profiles. These figures reveal that the velocity modulation in both spanwise and radial (normal to the surface) directions caused by the streamwise vortices is very strong. These distorted ensemble averaged velocity profiles reconfirm the existence of inflection points in bothy and in z directions (see also Tani 1962, Swearingen & Blackwelder 1987) that add to the inviscid instability of the flow and add to the generation of time dependent vorticity. The undulations along the span (figure 10) point out the existence of radial vorticity. When these shear layers start their "roll-up" as they do in the plane mixing layer they will interact with the streamwise vortices very robustly and it may be this process that leads to the meandering and braiding observed.

CONCLUSIONS The existence of longitudinal vortices in a turbulent wall jet flowing over a convex surface was proven experimentally. The spanwise wavelength of these vortices is selected by the flow suggesting that they are a product of a centrifugal instability. The growth of the vortices in the direction of streaming is attributed to amplification and to vortex amalgamation. The latter is enhanced by a secondary instability stemming from large spanwise distortions in the mean velocity. Even weak streamwise vortices embedded in a shear flow like the wall jet create large undulations of velocity across the span by transporting fluid from the surface to the outer jet boundary and vice versa.

ACKNOWLEDGEMENT This work is supported by the Office of Naval Research under the supervision of Dr. R. Joslin.

REFERENCES

Han, G., Zhou, M.D. & Wygnanski, I. 2004 Streamwise vortices in a turbulent wall jet flowing over a circular cylinder. AIAA paper 2004-2350 Likhachev, O., Neuendorf, R..and Wygnanski, I. 2001 On streamwise vortices in a turbulent wall jet that flows over a convex surface, Phys. Fluids 13, 1822-1825. Neuendorf, R. and Wygnanski, I. 1999 On a turbulent wall jet flowing over a circular cylinder, d. Fluid Mech., 381 1-25. Neuendorf, R., Lourenco, L. and Wygnanski, I. 2004 On large streamwise structures in a wall jet flowing over a circular cylinder, Physics of Fluids 16 (7), 2158-2169. Swearingen, J.D. and Blackwelder, R.F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall, d. Fluid Mech. 182: 255-290. Tani, I., 1962 Production of longitudinal vortices in the boundary layer along a concave wall, d. Geophys. Res. 67, 3075-3081.

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

555

ACTIVE CONTROL OF T U R B U L E N T S E P A R A T E D F L O W S B Y M E A N S OF L A R G E S C A L E V O R T E X EXCITATION A. Brunn and W. Nitsche Department of Aerodynamics, Institute of Aeronautics and Astronautics, Technical University, Berlin 10587, Germany

ABSTRACT The experimental investigations in the present paper deal with the excitation of fundamental instability mechanisms in separated free shear layers on a bluff body and downstream of a diffuser by means of periodic forcing in order to reduce the expansion of flow separation. The experiments focus on an unique approach to separation control using fundamental frequencies for local forcing in two different shear layer configurations (inner and outer diffusers). Each separation process is characterized by the periodic occurrence of large spanwise vortex structures. These vortices scale with the difference in height between the ramp ends. The excitation of these large scale vortex structures by periodic forcing intensifies the momentum transfer between the separation region and the outer flow, resulting in a substantial reduction of the reattachment length. For the inner and outer diffuser configurations, a universal value for the optimum forcing frequency was established.

KEYWORDS active separation control, bluff bodies, channel flows, vortex shedding, particle image velocimetry

INTRODUCTION Flow separation from solid surfaces occurs in a variety of technical applications, such as expanding flow channels (diffusers) or car and train tails, in turbomachinery, on airfoils at high angles of attack etc. This inevitably leads to a significant decrease in efficiency (e.g. Hucho (2002); Leder (1992)). Both active and passive methods of flow control can be applied to avoid or reduce this type of separation-induced performance loss (Lin et al. (1990); Yoshioka et al. (1999); Brunn (2002) etc.) Nevertheless, practical applications are almost too complicated for an accurate analysis of these typical phenomena. Hence, generic models with the most important boundary conditions are frequently used.

556

An overcritical diffuser is the simplest geometry for studying flow separation phenomena. The separation process is characterized by the periodic occurrence of vortex structures. Figure 1 shows the development of these structures in principle (Brunn (2003)): The initial Kelvin-HelmholtzInstability (KHI) leads to a roll-up of small spanwise vortices caused by low pressure and vorticity fluctuations. These structures grow rapidly and are finally shedded from the separation region (vortex-shedding). The pressure fluctuations resulting from the shedding process propagate upstream and are reabsorbed close to the separation point, generating vorticity fluctuations, which enhance the shear layer roll-up (Kiya et al. (1997)).

Figure 1: Feedback-Mechanism of instabilities in a separated shear layer (Brunn (2003)) Investigations on active separation control in plane and axisymmetric diffusers (e.g. Brunn (2002)) or on simple bluff body geometries (Sigurdson (1995); Kiya et al. (1997) etc.) were conducted successfully using forcing frequencies in the range of the observed shear layer instabilities. In these experiments wall embedded actuators were used to generate periodical perturbations, which significantly reduce the separation length. Nevertheless, the more complicated the configuration is, the more complicated the flow structures become. This is demonstrated, for instance, by Brunn (2003) in the transition from plane diffusors to axisymmetric configurations. Here the spanwise vortex structures, which have been shed from the separation region, lose their initial two-dimensional character in their early stages of development. It follows that the practicability of the proven control methods in complex geometries and real-flow applications has to be investigated. The present experimental study focusses on the comparison of active separation control methods at a plane half diffuser and the flow behind a generic car model- the Ahmed Car Model (ACM, Ahmed et al. (1984)). The flow control on this car model is an interesting application with respect to an increased efficiency of vehicles. The pressure drag in the wake region is the major component of the total drag of a vehicle due to the flow separation at the rear (Morel (1978); Hucho (2002)). Consequently, a reduction of separation will result in a strongly decreased total drag. The ACM combines the essential geometrical parameters determining shape, length and position of the separation and is used as a reference for numerical and experimental investigations (e.g. Krajnovic (2002)). The study in hand uses the simple half diffuser configuration to demonstrate the receptivity of actuator perturbations in a quasi-two-dimensional separated shear layer in terms of frequency spectra of velocity fluctuations measured with a hot wire probe. In the second part of the study, the results of the first attempts of plane diffuser control are applied to a second, more complicated configuration to reduce the separation length behind that ACM.

557

EXPERIMENTAL

SET UP

The experimental investigations were conducted in two different flow channels: an open wind tunnel with a plane half diffuser as a test section and a closed water channel for the generic car model (Ahmed-Body, Ahmed et al. (1984)). The half diffuser has an aspect ratio A R = 10 and a slant height of H = 40 ram, with the slant angle set at c~ = 25 ~ The measurements were carried out at a Reynolds number based on the inflow velocity of ReH = 4.104. To reach a fully developed turbulent inlet flow, a tripping wire was placed at 100. H upstream the slant edge, fixing the laminar-turbulent transition far upstream. A loudspeaker-slit-actuator, situated directly at the slant edge of the diffuser, was used to generate pressure perturbations, and it was inclined at 45 ~ to the mean flow direction based on the investigations of Lin et al. (1990); Yoshioka et al. (1999) and Brunn (2003). A single hot wire probe was traversed in the symmetrical plane of the flow field to measure the velocity fluctuations (Fig.2). The complete set-up is documented in the study by Brunn (2003).

Figure 2: Cross section of the plane half diffuser with actuator and the hot wire probe The measurements at the ACM were conducted in an optically fully accessible water test section using Particle Image Velocimetry (PIV) and digital flow visualization methods. The PIV-system consists of a frequency-doubled Nd:YAG laser, two CCD-Cross-Correlation-Cameras and a Synchronization Unit.

Figure 3: Experimental Set-Up of the ACM Investigations The ACM, which stretched across the whole width of the test section, was mounted on the channel wall. The slant angle was set at c~ = 35 ~ because observations on the fully three-dimensional model by e.g. Ahmed et al. (1984) and Lienhart et al. (2002) show that the flow field of the slant region is dominated by two-dimensional spanwise vortex structures under these conditions. The initial three-dimensional flow structures at the rear side edges of the ACM should be largely suppressed through the two-dimensional stretching. All other geometrical parameters of the model are based

558

on the original data given by Ahmed et al. (1984). The Reynolds number based on the inflow velocity and the slant height was set at 49000. The boundary layer upstream of the car model was fully turbulent. The periodic pressure perturbations were generated by a water pump connected to a rotating valve (Fig. 4, left) and injected into a cavity-slit-system, resulting in an oscillating wall jet without net mass flux. The actuator is comparable to the loudspeaker slit system used at the half diffuser. The perturbations during the forcing should preferably amplify the vortex structures in the separated shear layer to increase the growth of the vortices and intensify the entrainment process.

Figure 4" Rotating valve to provide periodic perturbations in terms of streamwise vorticity (right)

RESULTS The hot wire measurements in the plane half diffuser were carried out to obtain data of velocity fluctuations from the separated shear layer in the time and frequency domains. The RMSdistribution as well as the frequency spectra were used to describe the receptivity of the flow to periodic perturbations. Figure 5 shows the measured spectra in the upper shear layer at different streamwise positions. Two forcing cases are compared with the unforced base flow (thick dashed line). These two fundamental frequencies for the vortex-shedding (fs,1, left) and for the initial shear layer instability (fs,2, right) were observed in earlier investigations described by Brunn (2003). The corresponding Strouhal numbers, based on the slant height H, are StH = 0.1 for the vortex-shedding and StH = 0.38 for the initial shear layer instability (Sto = 0.021, based on the momentum thickness). 0.8

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Figure 5" Comparison of frequency spectra in the upper shear layer of a separation at a plane half diffuser with and without forcing: Excitation with vortex shedding frequency (left) and with the initial shear layer instability frequency (right)

559

The frequency response of forcing with the initial shear layer instability merely shows the forcing frequency itself as x/H _< 4. Further downstream at x/H = 10 there is no noticeable effect. The fluctuation intensity is decreased compared to the base flow. This is confirmed by Fig. 6, where the RMS-value distribution of the velocity fluctuations is illustrated. The fluctuation intensity is significantly increased due to forcing with the vortex-shedding frequency (Fig. 5, left). In those spectra an amplification of the forced structures is evident. Subharmonic frequencies also indicate that enlarged vortex structures emerge in the shear layer, leading to a significantly increased fluctuation intensity (Fig. 6, middle) and finally to an intensified momentum transfer from the recirculation to the flow outside of the separation bubble (e.g. Chapman et al. (1958)).

Figure 6: Distribution of the velocity fluctuations behind a plane half diffuser The forcing and amplification of large scale structures is obviously the most effective way to reduce flow separation in simple diffuser and bluff body configurations (Sigurdson (1995); Kiya et al. (1997); Brunn (2002) etc.) The next step is to apply this method to more complex and relevant engineering configurations. The ACM investigated in this study shows comparable streamwise flow structures in the near wake region, which is why the experiments focus on an unique approach to drag reduction in terms of active separation control. First of all, the unforced base flow was investigated with the time-averaged flow field in terms of streamlines shown in Fig. 7. Here, an ensemble average of 200 instantaneous PIV images is depicted. Flow structures typically occuring behind bluff bodies are visible (e.g. Leder (1992)): Two counter rotating vortex structures (timeaveraged) and a stagnation point that closes the separation region. The overall separation length in this case is larger than compared to the original ACM (e.g. Ahmed et al. (1984)), because of a suppressed spanwise momentum transfer. Unlike a diffuser flow with a reattachment of the separated shear layer at the wall, two free shear layers collide in the wake behind bluff bodies and enclose a separation bubble. While the upper shear layer separates from the slant and is consequently driven by the slant configuration, the flow at the bottom is like a backward facing step flow. The exemplary snapshot of the instantaneous velocity field in Fig. 8 shows discrete structures of spanwise vortices. Typical wavelengths can be assigned to these structures, occurring in almost every instantaneous velocity field, and they scale with the slant height H. Hence, it is the obvious solution to force the flow and amplify these struc-

560

Figure 7: Time averaged flow field behind an ACM tures with frequencies according to shear layer instabilities. For this reason, a frequency range was chosen according to the Strouhal numbers 0, 1 _< StH _< 0, 3. Here, the lowest value corresponds to the vortex-shedding frequency estimated with the method of Kiya et al. (1997) and was confirmed through digital flow visualization documented by Brunn (2003). The initial shear layer instability primarily depends on the boundary layer conditions upstream of the separation point (Michalke (1965); Leder (1992)) and was calculated to be around Sto = 0.017 (StH = 0.3). All control experiments in the present study were performed with a forcing intensity of cu = 3 . 1 0 -3, where

As

c'2s

c. = Ao " go9

(1)

with go as the average velocity at the inflow, c~ as the perturbation velocity at the slit exhaust, the cross section at the inflow A0 and the active actuator area As.

Figure 8: Instantaneous velocity field behind an Ahmed Car Body with the characteristic wavelength of spanwise vortices A significant reduction of the turbulent car wake separation length was achieved in all forcing cases, but with noticeable difference in the resulting (mean) flow field (Fig. 9). The excitation with the vortex shedding instability (StH = 0.1, Fig. 9 top, right) shows a drastic reduction of the recirculation area compared to the base flow (Fig. 9 top, left). However, with increasing forcing frequency the forcing effectivity is consistently reduced and in the case of an excitation in the range of the shear layer instability (StH = 0.3, Fig. 9 bottom, right) the reduction is comparatively low.

561

Figure 9: Time averaged flow fields behind an ACM The distribution of velocity fluctuations as seen in Fig. 6 for the half diffuser, is a reliable indicator for an enhanced momentum transfer by means of local forcing (Yoshioka et al. (1999)). In Fig. 10 this distribution is depicted for forcing cases 0,1 _< StH <_0, 3 , while the picture on the top left represents the flow without forcing. At frequencies close to the vortex-shedding the RMS-values far exceed that of the base flow. At StH = 0.3 the distribution shows no significant enhancement and is comparable to the half diffuser investigations (Fig. 6). A closer look at Fig. 10 (top, middle) does not only show increased fluctuations in the slant region, but also goes to demonstrate that the momentum transfer in the near wake directly at the blunt end of the car is much more intensified than at other frequencies. As a result, the entrainment process starts much earlier and the forcing effect is stronger in the near wake region behind the car model. The phase-averaged velocity fields given in Fig. 11 show the influence of the forcing in greater detail. The trigger information from the excitation mechanism was used for phase-averaged PIV measurements. After the postprocessing an average of 20 images for each phase was calculated.

562

Figure 10: Velocity Fluctuations behind ACM This figure depicts three exemplary phases in the range 0.4 _< T _< 0.8 . Exactly one forcing cycle is completed for T = 1. The left hand side shows the forcing with the vortex-shedding frequency, while on the right hand side the flow was excited with the initial shear layer instability. In addition to the streamlines, the value of the spanwise vorticity is marked. In both forcing cases an amplification of the vortex structure close to the slant edge obviously occurs immediately after the blowing phase (Fig. 11, top). For forcing with StH = 0.3 (right) this structure is very similar to those, observed in the instantaneous velocity field in Fig. 8. However, at the low frequency forcing this structure is enlarged up to the length of the slant. Further vortex growth at high frequency forcing is comparable to the natural flow. The enlargement of the amplified vortex structure in the left image exceeds the near wake region of the slant up to the blunt end of the car model. This is the main reason why the entrainment process in this case is much more intensified. Forcing with higher frequencies, especially with the initial shear layer instability, leads to a stabilization of the initial vortex structure. This is indicated by the high local concentration of spanwise vorticity. Here, the fundamental wavelength could be found in almost every velocity field, whereas due to the low frequency forcing, the vortex structures grow rapidly and are hardly detectable further downstream. A closer look at the global flow field shows a significant oscillation of the upper shear layer. The amplitude of this oscillation is considerably higher than at high frequency forcing. Consequently, the lower shear layer, separating from the bottom of the car model, Can only be co-excited if the vortices are large enough and both shear layers interact. Both shear layers are connected with the vortex-shedding process. This proves that the combination of vortex-shedding and the enlarged entrainment due to amplified large vortex structures with a connected frequency is obviously the most effective mechanism to control such flow configurations. Figure 12 summarizes the results achieved through investigations on active separation control for the two different flow configurations. The reduction of the time-averaged separation length for different forcing frequencies (Strouhal numbers) shown in this figure is normalized with the length of recirculation without forcing. An excitation with the frequency of vortex-shedding (StH = 0.1) leads to a dramatically shorter separation bubble. With higher forcing frequency the effect decreases because of the mechanisms explained above. The key to separation control in configurations

563

Figure 11: Phase averaged velocity fields due to local forcing: StH -- 0.1 (left side) and StH -- 0.3 (right) similar to the two investigated here is the forcing and amplification of the dominating large scale vortices. The size of these vortices depends only on the geometry which causes the separation. 11 0.9 0.8

0.7 r,o

~r

0.6 ""

0.5 0.4 0

,

,

,

i

01

.1

J

,

,

[] A ,

i

0.2

,

Half Diffuser ReH 4 0 104 Ahmed Car Model ReH= 4.9.10 4 5;tH

,

0:3

,

,

,

,

0:4

,

i

i

,

0.5

Figure 12: Comparison of the reduced separation length depending on the forcing frequency for the two investigated configurations

CONCLUSIONS The current study presents experimental investigations on active separation control by means of large scale vortex structure excitation and amplification. Actuators generating periodic perturbations to the flow were used to excite separated shear layers in two different geometrical configurations: a plane half diffuser and a two-dimensional generic car model configuration. Forcing frequencies in the range of the initial shear layer instability and the vortex-shedding were used to test the receptivity of the flow. An excitation in terms of periodical perturbations at the

564

slant edge leads to increased velocity fluctuations in the shear layers, while the impulse transfer between the recirculation region and the outer flow was significantly intensified due to forcing at vortex-shedding frequencies. The most effective frequency for flow control, both for the plane half diffuser and for the ACM, was observed for the corresponding Strouhal number based on the slant height StH = 0.1 of each. The amplified large scale vortices connected with the vortex-shedding process are the key to controlling the flow configurations investigated here.

ACKNOWLEDGEMENT This research was funded by the German Science Foundation (DFG) within the scope of the Sonderforschungsbereich Sfb 557 ,,Control of Complex Turbulent Shear Flows". This support ist thankfully acknowledged by the authors.

REFERENCES

AHMED S.R., RAMM R. and FALTIN G. (1984). Some Salient Features of the Time-averaged Ground Vehicle Wake. SAE-Techn. Paper Series 840300 BRUNN A. and NITSCHE W. (2002). Separation Control in an Axisymmetric Diffuser Flow by Periodic Excitation. In: RODI W. and FUEYO N. (Eds.) Engineering Turbulence Modelling and Experiments 5, Elsevier Science Ltd., 587-596 BRUNN A. (2003). Aktive Beeinflussung abgelSster turbulenter Scherschichten in iiberkritischen Diffusoren mit Hilfe periodischer Anregung, TU Berlin, PhD Thesis, Mensch und Buch Verlag CHAPMAN D.R., KUEHN M. and LARSON H.K. (1958). Investigations of Separated Flows with Emphasis on the Effect of Transition, / NACA-Report 1356 HUCHO W.-H (2002). Aerodynamik der stumpfen KSrper- Physikalische Grundlagen und Anwendung in der Praxis, Vieweg-Verlag KIYA M. and SHIMIZU M.and MOCHIZUKI O. (1997). Sinusoidal Forcing of a Turbulent Separation Bubble. In: J. Fluid Mech. 342, 119-139 KRAJNOVIC S. and DAVIDSON L. (2002). A Test Case for Large-Eddy Simulation in Vehicle Aerodynamics. In: RODI W. and FUEYO N. (Eds.) Engineering Turbulence Modelling and Experiments 5, Elsevier Science Ltd., 647-657 LEDEa A (1992). AbgelSste StrSmungen- Physikalische Grundlagen, Vieweg-Verlag LIENHART H., STOOTS C. und BECKEa S. (2002). Flow and Turbulence Structures in the Wake of a Simplefied Car Model (Ahmed model). In: Notes on Numerical Fluid Mechanics Bd. 77, Springer, 323-330 LIN J.C., HOWAaD F.G., BUSHNELL D.M. and SELBY B.V. (1990). Comparative Study of Control Techniques for Two-Dimensional Low-Speed Turbulent Flow Separation. In: KOZLOV A.V (Ed.): Separated Flows and Jets, Springer, Berlin, 429-474 MICHALKE A. (1965). Spatially growing Disturbances in an Inviscid Shear Layer. In: J. Fluid Mech. 23, 521-544 MOaEL T. (1978). The Effect of Base Slant Angle on the Flow Pattern and Drag of ThreeDimensional Bodies with Blunt Ends. In: Proc. of Syrup. Aerod. Drag Mechanisms of Bluff Bodies and Road Vehicles. Plenum Press, New York, 191-226 SIGUaDSON L.W. (1995). The Structure and Control of a Turbulent Reattaching Flow. In: J. Fluid Mech. 298, 139-165 YOSHIOKA S., OBI S. and MASUDA S. (1999). Momentum Transfer in the Periodically Perturbated Turbulent Separated Flow over the Backward-Facing Step. In: Proc. TSFP1, 1321-26

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

565

Large-Eddy Simulation of a Controlled Flow Cavity I. Mary and T.- H. L8 Computational Fluid Dynamics and Aeroacoustics Department, ONERA, BP 72, 92322 Ch~tillon, FRANCE

ABSTRACT Numerical results from a computational and experimental research program directed towards understanding cavity noise attenuation using a vortex generator placed within the boundary layer, just upstream the cavity, are presented. A LES computation, based on the compressible filtered Navier-Stokes equations, is conducted to show the effects of such suppression resonance device. Wall functions and a local mesh refinement technique are used in a multi-block structured solver to reduce the computational cost of the simulation. The M219 cavity has been retained for this study and the inflow Mach number is equal to 0.85. Inflow boundary conditions are prescribed using profiles obtained from a RANS boundary layer calculation. The results obtained show that the vortex generator device reduces significantly and simultaneously the four Rossiter modes which are driven by the strong interaction between shear layer dynamics and acoustic propagation, leading to a self-sustained resonance phenomenon. No similar behavior is observed with the controlled cavity, in which a flat plate of zero thickness across the cavity generates a v o n Karman street and the resulting vortices, impinging in the rear part, do not produced strong acoustic tones.

KEYWORDS

LES, Cavity, Flow control, Fence vortex generator, Dynamic-pressure loads

INTRODUCTION External weapons carriages can be responsible as 30% of the total vehicle drag and lead to prohibitive increases in radar signatures of current generation combat aircraft (see Shaw et al. 1983). Motivated by these considerations, future fighter aircraft programs have incorporated internal weapons carriage systems. Unmanned Combat Air Vehicles (UCAV) featuring internal bays are currently under consideration as well. However, an internal aircraft weapons bay, when exposed to freestream flow, experiences an intense aeroacoustic environment in and around the bay with levels as high as 160 to 180 dB (see Forestier et al. 2003 for instance). This high level of noise is due to a cavity resonance phenomenon as shown in the Figure 1: in the upstream shear layer, Kelvin-Helmoltz instabilities are generated. These vortices are subjected to a fast amplification rate, but remain almost coherent until they impinge the rear wall comer of the cavity. This interaction between the vortex and the comer generates an acoustic wave, which travels

566 upstream and leads to the resonance phenomenon by creating a new perturbation near the leading edge of the cavity. Then high acoustic loads can significantly reduce the lifetime of aero-structures in the bay and can damage sensitive electronic components. Therefore aircraft design engineers are being challenged to develop innovative devices to control the acoustic environment in the weapons bay. Numerous experimental works have been carried out to study different control systems: flap, synthetic jet, spoiler .... (see Stanek et al. 2000, Stanek et al. 2001, or Illy et al. 2004 for instances). Concerning the numerical simulation, it was shown recently by LarchevSque et al. (2003) that Large Eddy Simulation (LES) can give very accurate results for a realistic non-controlled case (in term of Reynolds number) for a reasonable computational cost thanks to the use of wall functions. Therefore the goal of the present paper is to assess the accuracy of LES for a cavity flow controlled by a passive device based on a vortex generator, consisting in a zero thickness flat plate, located at the leading edge of the cavity, in the boundary layer. Indeed experimental studies have shown that such device can be effective to reduce the noise in the cavity. Therefore LES could provide additional data in order to improve the understanding of the physical mechanisms underlying the effectiveness of such actuation device. The present numerical results are compared with those obtained from a computational simulation of the baseline cavity, carried out previously. They are shown and discussed, in terms of acoustic spectra and mean flow field. Illustrations of the instantaneous flow field are presented as well.

Figure 1 " Cavity Resonance Phenomenon MATHEMATICAL MODEL

Governing Equations A dimensionless form of the three-dimensional unsteady filtered Navier-Stokes equations is used for a viscous compressible Newtonian fluid. Any flow variable 0 can be written as 0 = ~ + 0 ' , where ~ represents the large scale part of the variable and 0' its small scale part. The filtering operator, classically defined as a convolution product on the computational domain, is assumed to commute with time and spatial derivatives. Moreover it is convenient to introduce the Favre filtering defined as r = PO/-P. In conservative form, the filtered Navier-Stokes equations can be expressed in three-dimensional Cartesian coordinates ( x l , x2, x3 ) as: D

OFj aQc + Ot

OXj

~F Re

7

OXj

-0

The Reynolds number is defined as Re=p0u0L0/r o where P0, u0, L0and ~0 denote a characteristic velocity, density, length and dynamic viscosity, respectively. The conservative flow variable vector is m

defined

as Q~. = ( ~ ,

~ u l , ~ u 2 , ,~ u3,

p/(T-1) + ,~ ui u i / 2 ) r, where p , u~, u2, u3, p and ?, are the

567 density, the velocity components, the static pressure and the specific heat ratio, respectively. The inviscid flux tensor is defined as -P-~j=uj Q--~+(o,81jp,82ip,83j p, puj) r , where 8ij is the Kronecker tensor. Using the ^

^

^

^

^

Boussinesq eddy viscosity assumption the viscous flux tensor is given by F~ = (O, cr]~,azj,a3j,a# uk + qj)r

,

with cr]j= (/l(T)+/a,)( o~ + c)uj ---~0 2 b--~--k ~)u-k ) and q^j = (/t(T)+/.lt)~)T ax j

axi

3

Pr

OXj

where T is the temperature, Pr the Prandtl number. The molecular and eddy viscosity values /~(T) and /z, are given by the Sutherland formula and by the subgrid scale model expression, respectively. This system is supplemented withthe filtered state equation, thanks to the reference Mach number M0 9 P = (p T)/(7". m g ) . The eddy viscosity is computed from the Selective Mixed Scale model (see Lenormand et al. 2000 ). It is a non-linear combination of the filtered shear stress tensor, a characteristic length scale,

the small kinetic energy scale and a selective function based on the angle between the filtered vorticity and the local averaged filtered vorticity (see David 1993). The same subgrid scale model was used by Larchevaque et al. (2003) and Larchevaque et al. (2004) for the computation of the baseline cavity. Numerical Method

The solver used in this study is developed at ONERA. It is based on a cell-centered finite volume discretization using structured multi-block meshes. For efficiency reason, an implicit time integration is employed to deal with the very small grid size encountered near the wall. Then a three-level backward differentiation formula is used to approximate the temporal derivative of Qc, leading to a second-order accuracy. An approximate Newton method is employed to solve the non-linear problem. At each iteration of this inner process, the resolution of the linear system relies on Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit method. For more details about these numerical aspects see P6chier et al. (2001). Usually LES requires a high-order centered scheme for the Euler fluxes discretization, with spectral-like resolution in order to minimize dispersive and dissipative numerical errors. As several works, for instance Wu et al. (1999), have shown that LES can be carried out with low-order centered scheme in case of sufficient mesh resolution, only second-order accurate scheme is employed in this study. The AUSM+(P) scheme (see Edwards and Liou 1998, and Mary et al. 2000), whose dissipation is proportional to the local fluid velocity, constitutes the basis of the discretization, because it is well adapted to low Mach number flow field zones, which are encountered in separation zones. However several modifications have been introduced to enhance its accuracy and computational cost. As we are not interested by the scheme shock capturing properties, simplified formula are used in this study to approximate the Euler fluxes. In addition, to enforce the pressure-velocity coupling in low Mach number zones, a pressure stabilization term is added to the interface cell fluid velocity, as for incompressible flow (see Rhie and Chow 1983). To prevent spurious numerical oscillations, an upwind correction is locally added at the locations where odd/even oscillations are detected (see Mary and Sagaut 2002). The viscous fluxes are discretised by a second-order accurate centered scheme. The Jacobian matrix is diagonalized using the Coakley's method. A wall treatment, based on a stress model, is employed so as to reach a reasonable computer resource requirement. The wall stress model relies on the classical logarithmic law profile for the velocity with the Van Driest compressibility correction. A fully dimensional formulation is obtained by writting the system in the local frame of reference associated to the friction lines. The same formulation was used with success in Larchevaque et al. (2003) for open cavity simulations. The total number of cells is also greatly reduced thanks to the use of a multi-block local mesh refinement technique. The numerical treatment of the interfaces between blocks is described in Mary (2002) and Mary and Nolin (2004).

568

COMPUTATIONAL SETUP Large eddy simulations have been conducted on the M219 cavity at M,= 0.85 (see Henshaw 2000). The Figures 2 and 3 illustrate the cavity geometry, with a rectangular plan-form of 2Ox4in. cut into a flat plate 3 1in. from the rig’s sharp leading edge and I in. off-centre from the rig centrelinc. The inclined sides of the rig are not modeled, but only the flow inside cavity and below the flat plate. It is noticed as the baseline cavity.

Figure 2 : Geometry of the M219 4in. Cavity Model (dimensions in inches)

Figure 3 : Generic Cavity Rig in DERA Wind Tunnel

In ordcr to control the cavity resonance phenomenon, a vortex generator is placed just upstream the cavity, entirely within the boundary (see Figure 4). This vortex generator consists of a rectangular zero thickness vertical flat plate, across the bay leading edge. Its spanwise length is equal to the cavity width. Its bottom edge is at lin. from the top of the cavity

Figure 4 : View of the vortex generator: symmetry plan (left). Front wall plan (right) The Reynolds number is equal to 1 . 4 lo6 ~ based on the free stream velocity and the depth (D) of the cavity, chosen as the characteristic velocity and length, respectively. Inflow boundary conditions are prescribed using profiles obtained from a RANS boundary laycr computation, with the developing length equal to the one of the forward plate in the experiments of Stanek ef al. (2000) A view of the computational domain decomposition i n six structured blocks, including the cavity block (thick black lines), is prescnted in Fig. 5 . The computational dotnain ranges from -7.75 D to 12.75 D in

569 the streamwise (x) direction, from -4 D to 4 D in the spanwise (y) direction and from -1 D to 7.75 D in the vertical (z) direction. Thanks to the use of both wall stress model and local refinement (see Figure 6), the total number of cells is equal to 3.2 millions, including 1 million cells inside the cavity. In average the size of the first wall cell is around 20-30 wall units. The vortex generator is discretised with 101x 8 grid points in the (y,z) plane and it has no thickness in the streamwise direction (the velocity is simply set to zero in one row of cells). A large number of cells has been put in the mixing layer region, leading to an a priori measured number of 10 cells across the initial vorticity thickness. The dimensionless time step is set equal to 2.72x10 -2, resulting in a maximum CFL number of about 20. Statistical data are collected after a relaxation physical time of 0.1 seconds for a duration of 0.5 seconds. As the lowest frequency of the Rossiter mode is equal to 120Hz, at least 60 periods of the phenomenon are used to compute the statistics.

Figure 5 : Structured Grid (6 blocks)

Figure 6 : Local Mesh Refinement in a (x, y) Plane near a cavity comer

RESULTS For the baseline cavity, it was established in a previous paper by Larchevaque et al. (2004) that the LES computations are accurate compared to experimental results of Henshaw (2000). Emphasis was put on the spectral analysis of both the shear layer dynamics and the pressure loads. A special attention was paid to the frequency distribution of the pressure oscillation energy on the walls, including the ceiling and the resulting flow structure inside the cavity was illustrated as well. As reported in Henshaw (2000), ten equidistant Kulite Pressure Transducers (K20 closest to the cavity front wall, consecutively numbered to K29 closest to the rear wall) were located on the cavity ceiling along the rig centreline. In addition Kulite K10 and K17 are located in the front and rear wall of the cavity, respectively. The same terminology is adopted for the simulation. The Figure 7 provides pressure power spectra at two different locations of the ceiling, near the front and rear walls of the cavity. Concerning the baseline cavity, the results of the simulation are compared to the experimental data of Henshaw (2000): The full lines correspond to the simulation, whereas the square symbols denote the experimental results. The agreement with the experiment is satisfying concerning the frequency and the amplitude of the four Rossiter modes (see Rossiter 1964). Their frequency are around 120 Hz, 340 Hz, 550 Hz, 820 Hz, respectively, and their level significantly depends on the distance between the point of measurement and the rear wall. Around 140 dB are recorded at the K20 location for the first and third modes, whereas around 150 dB are recorded at the K29 position near the rear wall for the second and third modes. In the Figure 7, the results of the controlled case are shown by the dashed lines. The pressure power spectra shown in this Figure clearly demonstrate the effectiveness of the control device. The four Rossiter modes are reduced simultaneously of about 12 dB for the second and the third mode tone. The results obtained are very similar to those reported in experiments of ARA Transonic Wind Tunnel concerning the delta spoiler effects (see Stanek et al. 2000 and 2001), except the growth of the mode tone at 650Hz, phenomenon not exhibited in the simulations.

570 -

140

130

~

Baseline {Exp.] Baseline ConlTolled

150 ~4o

~1~ I

120

ca_ -~120

.. ~.-. . . . - :~,.~~'-~

g

D CO

09 110

110

. . . .

IO 0

i

i

i

i

I 500

i

i

i

i

I

1000 HZ

|

,

|

,

l

1500

i

,

,

|

| . . . . 500

i

i . . . . 1000 Hz

| . . . . 1500

|

2000

"/000

Figure 7 9Control device efficiency: SPL at Kulite K20 (left) and Kulite K29 (right) In order to better understand the flow control mechanism, instantaneous and mean flow quantities can also be meaningful. Unfortunately such experimental data are not available. Therefore the results dealing with the mean and instantaneous flow field presented hereafter only concern the numerical simulations. The Figures 8 provides a comparison between baseline and passive control cases in terms of mean flow characteristics (u-velocity component) in the symmetry plane. The main observed feature is that the vortex shedding of the fence lifts or displaces the shear layer away from the cavity, resulting a diffusive impact of the shear layer on the rear cavity wall. It seems that this displacement of the shear layer leads to an increase of both the size of the separation zone and the intensity of the backflow velocity near the ceiling.

/

_

-.~_~~______~__

0 . 8g&_-~-o . 6 ~ ~-74;

- ______=~~8-

I

.~-----

Figure 8 9 Mean Velocity in the Symmetry Plane (top: baseline case; bottom: controlled case) Turning attention to the Root Mean Square (rms) of the pressure signal at the cavity ceiling, the Figures 9 and 10 reveal the noise attenuation effects of the vortex generator as expected, particularly in the aft part of the cavity. The rms value of the pressure is almost decreased by 50% in the controlled case. Without the control device, significant pressure fluctuations occur around x/D=2.5 in the streamwise direction, whereas these fluctuations are concentrated between x/D-4 and the rear wall at x/D=5 thanks to the vortex generator. With the control device, the highest values are obtained in the corner of the cavity, whereas maximal value are located in the middle of the rear wall for the baseline case.

571

r-_...___.~ 0.04--

Figure 9 : Prms Distribution on the ceiling (top: baseline case; bottom: controlled case)

Figure 10 : Prms Distribution on the rear wall baseline case; left: controlled case)

(right:

It is obvious that the flow field is absolutely three-dimensional (due to turbulence and geometry effects). The Figure 11 shows the overall instantaneous pressure fluctuations flow field. This quantity is defined as the difference between the mean pressure and the instantaneous pressure fields. These Figures highlight the presence of vortex structures (solid and dashed lines correspond to positive and negative values of the pressure fluctuation, respectively). There are clear differences between the baseline and controlled cases concerning the size and the intensity of the vortices. For the baseline case, one can identify three or four large vortices in the shear layer. These energetic vortices are distributed along the streamwise direction and are almost two-dimensional (following the spanwise direction). For the controlled case, coherent structures are smaller, but much more numerous. In the shear layer, the large vortices have been replaced by vortices associated with the von Karman street of the flat plate control device. These vortices, which have a more pronounced three-dimensional characteristic, scale like the height of the plate and their intensity is at least five time smaller than those on the baseline case

Figure 11 : Distribution of instantaneous pressure fluctuations: baseline cavity (left) controlled (right) CONCLUSIONS Large-Eddy Simulation of a vortex generator effects on the M219 cavity at a Mach number equal to 0.85 has been conducted. It has been observed that such device reduces successfully and simultaneously the four Rossiter modes, of about 12 dB for the second and third mode tones. In opposite to the baseline cavity which generates a shear layer at the edge of the leading edge, a flat plate across the cavity produce a von Karman street which impinges at the rear part with a lower intensity. Up to now, the mechanism of the feedback with acoustical waves, not studied here, is not identified. Future research will be required to make a more connection between the state of the turbulent wake and the state of the unsteady pressure field on the cavity ceiling, in the framework of a collaborative computational and experimental program of Illy et al. (2004).

572

ACKNOWLEDGMENTS Dr. R. Ashworth and Dr. G. Foster (QinetiQ) are gratefully acknowledged for providing baseline cavity data base. REFERENCES

David E. (1993). ModElisation des 6coulements compressibles et hypersonique, Thkse de l'Institut National Polytechnique de Grenoble. Edwards J.R. and Liou M.S. (1998). Low diffusion splitting methods for flows at all speed. AIAA J., 36, 1610-1617. Forestier N., Jacquin L. and Geffroy P. (2003). The mixing layer over a deep cavity at high subsonic speed. J. Fluid Mech. 475, 101-145. Henshaw M.J. (2000). M219 cavity case. Verification and validation data for computational unsteady aerodymamics, Tech. Rep. RTO-TR-26, AC/323(AVT)TP/19, 453-472. Illy H., Geffroy P. and Jacquin L. (2004) Etude expErimentale de l'effet d'un cylindre place en amont d'une cavit6 en rdgime subsonique. 39 e Colloque d'Adrodynamique Appliqude : ContrEle des dcoulements. Paris. Larchevaque L. (2003), Simulation des grandes 6chelles de l'Ecoulement au dessus d'une cavit6. ThOse de l'universitd Paris VI. Larchevaque L., Sagaut P. and Le T.-H. (2003). Large Eddy Simulation of flows in weapon bays. AIAA Paper 2003-0778. Larchevaque L., Sagaut P., Mary I., L a b b 6 0 . and Comte P. (2003).). Large eddy simulation of a compressible flow past in a deep cavity. Phys. Fluid, 15:1, 193-210. Larchev0.que L., Sagaut P., La T.-H. and Comte P. (2004). Large eddy simulation of a compressible flow in a three-dimensional open cavity at high Reynolds number. J. Fluid Mech. 516, 265-301. Lenormand E., Sagaut P., Ta P.L. and Comte P. (2000). Subgrid scale models for large eddy simulation of wall bounded flows. AIAA J., 38"8, 1340-1350. Mary I. (2002). Large eddy simulation of vortex breakdown behind a delta wing. Engineering Turbulence Modelling and Experiment 5, edited by W. Rodi and N. Fueyo, Elsevier, 687-696. Mary I. and Nolin G. (2004). Zonal grid refinement for large eddy simulation of turbulent boundary layers. AIAA Paper 2004-0257. Mary I. and Sagaut P. (2002). LES of a flow around an airfoil near stall. AIAA J., 40:6, 1139-1145. Mary I., Sagaut P. and Deville M.. (2000). An algorithm for unsteady viscous flow at all speed. Int. J. Numer. Meth. Fluids, 34, 371-401. Pdchier M., Guillen Ph. and Cayzac R. (2001). Magnus effect over finned projectiles J. Spacecraft

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Rockets, 38:4, 542-549. Rhie C. and Chow W. (1983). Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J., 21:11, 1525-1532. Rossiter J. (1964). Wind tunnel experiments of the flow over rectangular cavities at subsonic and transonic speeds. Reports and memoranda 3438. Aeronautical Research Council. Shaw Lo, Bartel H. and McAvoy J. (1983). Acoustic environment in large enclosures with a small opening exposed to flow. J. of Aircraft, 20:3, 250-256. Stanek M., Raman G., Kibens V., Ross J., Odedra J. and Peto J. (2000). Control of cavity resonance through very high frequency forcing. AIAA Paper 2000-1905. Stanek M., Raman G., Kibens V., Ross J., Odedra J. and Peto J. (2001). Suppression of cavity resonance using high frequency forcing- the characteristic signature of effective devices. AIAA Paper 2001-2128. Wu X., Jacobs R., Hunt J. and Durbin P. (1999). Simulation of boundary layer transition induced by periodically passing wake. J. Fluid Mech., 398, 109-153.

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

575

P A R A M E T R I C STUDY OF S U R F A C T A N T - I N D U C E D D R A G - R E D U C T I O N BY DNS Bo Yu

1,2

and Yasuo Kawaguchi 1

1 Turbomachinery Research Group, National Institute of Advanced Industrial Science and Technology, 1-2 Namiki, Tsukuba, lbaraki 305-8564, Japan 2 Center for Smart Control of Turbulence, National Maritime Research Institute, 6-38-1, Shinkawa, Mitaka-shi, Tokyo 181-004, Japan

ABSTRACT The effect of rheological parameters on the drag-reduction by surfactant additives is studied with a viscoelastic Giesekus model. It is found that the drag-reduction is closely associated with the reduction of burst events. The modifications of streamwise vorticity are given. The alteration of the energy transport process is discussed.

KEYWORDS surfactant, DNS, drag-reduction, Giesekus model, rheological parameters

INTRODUCTION In a previous DNS study (Yu et al., 2004b), a viscoelastic Giesekus model (Giesekus, 1982) was adopted to model the interaction between the network structures made up of rod-like micelles and solvent for the simulation of a 75 ppm surfactant solution at Reynolds number of around ten thousand. The numerical results qualitatively agreed with the experimental data, indicating that the Giesekus model is appropriate for surfactant solutions. Experiments showed that surfactant-induced dragreduction is closely associated the concentration of the solution (Li et al., 2001). This is because the rheological properties vary greatly with the concentration (Kawaguchi et al., 2003). Therefore, it is interesting and necessary to perform a systematic investigation to study the effect of rheological properties on the turbulence structures and drag-reduction rate to further clarify the turbulent transport mechanism in drag-reducing flow. To develop a viscoelastic model for engineering applications, we need to establish a DNS database covering a wide range of rheological properties. For these scientific and engineering reasons, we carded out a series of runs for the surfactant solution with a faithful finite difference scheme (Yu et al., 2004a) for a fully developed channel flow.

576

NUMERICAL METHOD The drag-reduction by surfactant additives is related to the elasticity of the network structures formed by the rod-like micelles in the solution (Yu et al., 2004b). We employed a viscoelastic Giesekus constitutive equation to model the interaction between the elastic network structures and solvent. The dimensionless governing equations for a fully developed turbulent channel flow can be written as:

Ou[ = 0

ax;

Ou+ + + Ou: Ot* uj Ox] Ot"

+ ~Ou;

Ox'~

j

(1)

( Ou+ i + 1- fl Oc~ +r~u _Op_ ++ fl__ff___O___O__ Ox; Re~ Ox] ~, Ox] ) We~ Ox]

Ou; + o u j+ c + Ox'~ cmj- --z--7m m'

Re~

+

(2)

+

We~ [C'j +Ct(C'+m--6'mXC*--6*)--6u]=O

(3)

where c~ is the conformation tensor associated with the deformation of the network structures. Re~ (Re~ = pUsh / rio ) is the frictional Reynolds number based on the frictional velocity, half of the channel height and zero-shear rate viscosity. Wet (We~ = p2U~ /rio ) is Weissenberg number. Mobility factor a is a key parameter in determining the extensional viscosity./3 is the ratio of solvent viscosity r/s over the zero-shear rate solution viscosity r/0 (r/0 = r/a + rL ,/~a is the contribution of surfactant additive). The various rheological parameters shown in Table 1 are investigated to identify their effects on dragreduction with a fixed Reynolds number, Re~ = 150. Dimitropoulos et al. (1998) studied the effect of the variation of rheological parameters with a Giesekus model. Their studies were for dilute solutions with p no more than 0.9 and the maximum drag-reduction rate was 44%. In order to get a larger dragreduction rate, smaller p values are adopted in this study. The numerical method used here is a fractional-step method. The Adams-Bashforth scheme is used for time-advancement to ensure secondorder accuracy in time. The second-order faithful finite difference scheme of Yu et al. (2004a) is used to enhance the numerical stability.

RESULTS AND DISCUSSION The Reynolds number based on the effective viscosity ri, = rio (,&tU+ / dY § + 0 - p)C~ / We~ )/(dU +/ dy § ), at the wall is used for data reduction (Sureshkumar et al., 1997). The drag-reduction (DR) rate is defined as the reduction of the friction factor with respect to a Newtonian fluid at an equal mean Reynolds number, where the frictional factors of Newtonian fluid are evaluated by Dean's equation (Dean, 1978). Some important results are listed in Tables 1 and 2. Table 1 shows that the DR rate increases with the increase of We,, with the decrease of {x and with the decrease of 13. Figure 1 shows the velocity profiles. All the velocity profiles collapse in the viscous sublayer, and the velocity profile upshifts further in the logarithmic layer with the increase of DR rate. Table 2 shows that generally speaking, larger DR rates are associated with larger uL, smaller ~ and smaller ~=-~. With the increase of DR rate, the peak value position 8+~ of the RMS of the streamwise velocity fluctuation shifts further to the bulk flow region. 8~' and 8 L" are mean spacing between low-speed streaks and mean diameter of streamwise vortex at y +' = 15, which are estimated by the separation of minimum spanwise two-point

577

correlations Ruu and Rw. Generally, large DR rates are related to large 6~" and 8~*" 9v~ and ( a r e the mean turbulent viscosity of the flow and mean elastic:energy of the network structures, respectively. The mean turbulent viscosities of all the viscoelastic fluids are smaller than that of Newtonian fluid and decrease with the increase of DR rate. From Table 2 it is seen that drag-reduction is related to not only the magnitude of elastic energy but also its spatial distribution. Figure 2 shows that the elastic energy of Fluid A is large only in the viscous sublayer; the elastic energy of Fluid B is larger than those of Fluid G and Fluid I in the viscous sublayer but smaller in the buffer layer, and Fluid B has a smaller DR rate than Fluid G and Fluid I; the elastic energy of Fluid F is the same as those of Fluid D and Fluid E in the viscous sublayer but larger in the buffer layer, and it has a larger DR rate. All these show that the occurrence of a large DR rate generally requires a large elastic energy in a wide buffer layer. TABLE 1 COMPUTATIONAL PARAMETERS AND SOME IMPORTANT RESULTS Fluid A B C D E F G H I N *based on

13 We~ tx rlo/riw 0.5 8 0.001 1.107 0.5 12.5 0.001 1.184 0.5 20.0 0.001 1.311 0.5 30.0 0.001 1.455 0.5 40.0 0.001 1.533 0.5 50.0 0.001 1.598 0.5 20.0 0.01 1.668 0.3 30.0 0.001 1.907 0.8 30.0 0.001 1.126 x x x 1.000 rlw. A-I denote viscoelastic fluids and N

Um+ Re ,* 15.08 166 18.14 178 22.14 197 25.92 218 30.36 230 33.32 240 22.33 250 29.15 286 22.06 169 14.78 150 denotes Newtonian

Re m" 5000 6440 8710 11300 13970 15980 11180 16670 7450 4440 fluid.

DR*% 0 25.4 46.0 57.9 67.7 72.0 43.5 63.4 46.3 x

TABLE 2 SOME IMPORTANT RESULTS DR*%

Fluid

" ' "u,~ 7

' " 7v,,~ -

+ w,,s

+" 6~,,~

0 A 1.46 0.570 0.721 11.2 25.4 B 1.65 0.528 0.679 17.1 43.5 G 1.85 0.526 0.684 20.3 46.0 C 1.98 0.474 0.634 22.2 46.3 I 2.18 0.479 0.679 22.2 57.9 D 2.48 0.433 0.588 28.6 63.4 H 2.68 0.402 0.564 32.3 67.7 E 2.75 0.349 0.515 30.2 72.0 F 3.19 0.308 0.474 36.3 x N 1.43 0.593 0.747 12.2 The overbar in Table 2 means the average value over

§ 6uu

:

.

. . vt

.

. ke

130 26 9.1 0.40 166 42 8.6 0.62 234 58 8.4 0.47 215 46 8.4 0.98 211 52 8.1 0.65 307 68 6.5 1.39 358 89 5.0 2.02 323 72 4.1 1.79 486 75 3.2 2.01 94 35 10.9 x space (x, y and z) and time.

Figure 3 shows the Reynolds shear stress. Generally, the DR rate is associated with the decrease of Reynolds shear stress but there is not a clear relationship between the two. A maximum DR rate of up to 72% of Fluid F was obtained in the present study, but we did not find a diminish of Reynolds shear stress, which was observed in other experiments (Li, et al. 1998). This indicates that the diminish of Reynolds shear stress is not essential for a large DR rate. The Reynolds shear stress of Fluid A is smaller than that of a Newtonian fluid, but no drag-reduction occurs. Fluid C, Fluid G and Fluid I have a similar DR rate but the Reynolds shear stresses differ appreciably. Fluid B has a smaller Reynolds

578

shear stress than Fluid G but a larger DR rate. All the above indicates that a large decrease of Reynolds shear stress does not necessarily mean a large DR rate. Actually, the friction factor is determined by three components: viscous contribution, turbulence contribution, and viscoelastic contribution as follows:

C/=

6;( u'v'),l y ,

12fl/Re0+

U; 2

viscous contribution

turbulence contribution

~N ....... A

4540: 35: 30:

....

20 [

~

~ ~ - . ~

..... H

15 [

~

~

.

"

".%

~o

i

,,9

+~ 6

h

........... B

!

=..... C ---- D

i i

.... ......

.k

\','~

4 ...................

tt~r,i

(4)

....... ............

,. "' - "" ~

~c

C; (1 )dy" WerU22 -Y*

viscoelastic contribution

12

,~~

........... B

+~ zs:

6f(1-,8)

dy* +

".,,.,N~x .....

10 [

F lli G i|i

"" H !illi

5:

1

i

10

+. 100 Y Figure 1 Velocity profiles.

~

0.6

0.4

....

i8o

~ N

0.8

i

y

Figure 2 Mean elastic energy.

1.0

+

ib +.

. . . . . . .

.*/~-~"/.~~~__~_~.,.'~.,,, ~ l~[//':"~~s" - " " ~ ~ . ~ . . . . ~,,"--.-..,, ~ :'/,!

......

A

..........

B

.........

C

........... D E F - .... G

..... H ...... I

0.2

0.0 0.00

0.25

0.50

0.75

1.00

Y Figure 3 Reynolds shear stress. For a Newtonian fluid, the turbulence contribution is the major component and contributes more than 80% of the friction factor (Yu et al., 2004b). Yu et al. (2004b) showed that the elastic network structures can decrease the Reynolds shear stress to reduce frictional drag and exert viscoelastic stress to increase frictional drag, and that drag-reduction occurs because the decrease effect exceeds the increase effect. The quadrant analysis of the Reynolds shear stress provides detailed information on the

579

0.5"

0.5

A

0.4

- 2

0.3

[

~ 0.2 ~, o.1

0.1 A-Q3

0.0

-0.1

N-Q

-0.2

o.oo

_

_

A-Q] o.;5

051 N2

0.0 -0.1

o.;o y

o.'}5

1.bo

-o.,-o.oo

0.3

o.,

o.,t N-Q1

-0.2 o.oo 0"51

0"0I

C-Q1 o.k~

-0.1

o.~o y

o.~5

~.bo

N- 2

-0.2 o.oo

0.4 0.3

~

~

o.oo

G-Q1 o.15

o.;5

o.;o y

o.:~5

1.~

o.~o

o.:r5

1.~

o.;o y

o.~5

1.bo

N-QI ~ o.k~

0.5

0.4 t N-C 0.3

0.01 -0.1

.

0.4 N-Q

0.3

0"0t -0.1

B-Q1

051

0.4 N-~

-0.2

J-

0.4 N-Q 0.3

o.;o y

o.~5

1~

y

I-Q2

N-Q3 0.0 -0.1 N-Q1 -0.~o.oo o.~

Figure 4 Reynolds shear stress from each quadrant. contributions from various events occurring in the flow. Here we investigate how the Reynolds shear stress changes for different fluids by quadrant analysis. Fluids A, B, C, F, G and I are compared with Newtonian fluid as shown in Fig. 4. It is seen that for all the viscoelastic fluids the four quadrant events at the bulk flow region are either almost the same as those of the Newtonian fluid or stronger than those of the Newtonian fluid. These stronger events are probably due to the effect of Re ~*, i.e. the Re ~*of the Newtonian fluid is smaller that those of the viscoelastic fluids. We can expect large values in each quadrant for the Newtonian fluid for larger corresponding Re ~*. At the near-wall region, the ejection and sweep events of Fluids B, C, F, G and I are reduced but the outward motion of high-speed fluids (Q 1) and the inward motion of low-speed fluids (Q3) are either reduced or enhanced. For Fluid A, the ejection and sweep events do not change as compared to the Newtonian fluid. The decrease of Reynolds shear stress of Fluids B, C, F and I at the near-wall region is due to the reduction of the ejection events (Q2) and sweep events (Q4). For Fluid I, the decrease of sweep events is the largest contribution to the reduction of Reynolds shear stress. However, the decrease of Reynolds shear stress

580

for Fluid G is primarily due to the increase of the outward motion of high-speed fluids (Q1), the inward motion of low-speed fluids (Q3), and decrease of sweep events (Q4). In summary, dragreduction is generally associated with suppression of the ejection and sweep events.

Figure 5 Instantaneous velocities and elastic energy at a y-z plane. (a) contour of streamwise velocity and v and w velocity vectors; (b) elastic energy.

581

Figure 6 Isosurfaces of an instantaneous streamwise vorticity cox = 0.1. The typical instantaneous streamwise velocity contours, secondary velocity vectors and elastic energy at a cross-section in the y-z plane are shown in Fig. 5. It is seen that the size of the strearnwise vortex becomes larger with the increase of drag-reduction rate. The occurrence of ejection and sweep events is greatly reduced with the increase of DR% rates. The high elastic energy is released during the ejection events. The isosurfaces of instantaneous streamwise vorticity are shown in Fig. 6, in which the change of vortical structures can be clearly seen. It is seen that the streamwise vorticity becomes much weaker and more elongated with the increase of large drag-reduction rates. The significant weakening of the strength of the streamwise vorticity leads to the drag-reduction. The integrated balance equations of mean kinetic energy, turbulent kinetic energy and elastic energy can be derived as shown below: U+

.

. . . . OU +

I, Input energy

II, turbulent production

~ ,BI dU+ 12dy 9

llI, viscous dissipation

. . . . c3U+ l -uv

+

.

cqy+dy ~

V, turbulentdissipation

~-i 1 - fl

IV, work by viscoelasticstress

l-p(, +~;+]ay"

,wo, t,,

dU + , 9

(5)

=o

VI, turbulence-elasticity interaction

(6)

582

;1-flOU +

07:

1-,B/ ,. Oui*) .

Lw;7 .

c,,

1-,8

)5

o-;7- + + :wo

VII, elasticdissipation

),--0 (7)

Clearly, the energy transport process for surfactant solutions is quite different from that of the Newtonian fluid. The input energy by the mean pressure gradient is dissipated through path 1 and path 2 as shown in Fig. 7. Path 1 is identical to the Newtonian case. In path 2, the energy is dissipated by the retraction of the network structures.

Figure 7 Energy transport process. Figure 8 shows that the turbulent production and turbulent dissipation are generally smaller than those of the Newtonian fluid except Fluid G. The increase of the turbulent production of Fluid G compared to the Newtonian fluid is because the increase of the velocity gradient is much larger than the decrease of Reynolds shear stress. Note that the increase of the turbulent production corresponds to an increase of input energy due to the increase of mean velocity. A more proper comparison is based on the relative contribution of turbulent production to the total energy. The relative contributions of the turbulent production of the Newtonian fluid and Fluid G are 41% and 34% respectively. This means that for Fluid G, turbulence has also been effectively suppressed. Turbulence-elasticity interaction acts basically as a sink term, which means that the fluctuating elastic force tends to damp the spatial variation of velocity fluctuations and thereby laminarize the flow. Figure 9 shows that the large DR rates are associated with large elastic work and elastic dissipation at a wide buffer layer. The vorticity equation can be derived as follows,

~ow; +

ot*

+~o~,: =

uj Ox;

+ Ou; ~+fl

wj Oxj§

o'~,; Ox;Ox;

We,

t

(8)

where the first term on the right-hand hand is the vortex stretching term and the last term is due to

583

viscoelasticity. Figure 10 shows the stretching mode (tgu //gx > 0 ) and squeezing mode (/gu / tgx < 0 ) of the streamwise vortex (]arx0u / ax[ is the mean absolute value of nrx~ / ax over time and x-z plane). It is seen that both the stretching and squeezing modes of the streamwise vortex have been appreciably suppressed at drag-reductions. For Fluid A, the stretching of the streamwise vortex is enhanced as compared to the Newtonian fluid, which is consistent with a smaller vortex size as shown in Table 2.

0.3-1

~ N

0.2

................ B

0tjy

-i ___

0.0

---

-0.1 -0.2

0.00

0.25

0.5.o

0.75

t.oo

Y Figure 8 Turbulent kinetic budget terms. ..........

A

................ B

F C

0.4

.~

o.2 0.0

-0.z

~y

VII

-0.4 -0.6 0.00

0.25

0.50

0.75

1.00

y Figure 9 Elastic energy budget terms.

CONCLUSION Numerical simulations showed that the drag-reduction rate increases with the increase of We~, with the decrease of tx and with the decrease of 13. Large drag-reduction rate is associated with large elastic energy in the buffer layer. The vortical structures become weaker and more elongated in the streamwise direction and the stretching of the streamwise vortex is greatly reduced for a large drag-

584

reduction rate. mN 0.004 .

.

0 003.

.

is"'',,

: .,...

.

.

.

.

.

A ........... B

0.0024-

C

0.0020-

,

9 ] ",, " ",, 0.00~-. ~ ) " ' ~ ' , ,

, -

o.oo

D E F

......::.

.

~ N - ...... A

i' f ' ' " -

/ _

........... g

"",

C

: "',, 0.0016-/:.~-~",,,

D E

......

----:

:""

,.--

- ....

.

0.0004-

o

. o.~

~ o.zs

~ ..so Y

o~oo y" 0.75

1.~

o.~

....... ..is

o.~

-'--------o.~s

1.~

Y

Figure 10 Intensity of the streamwise vortex stretching;, (a) stretching mode, (b) squeezing mode.

REFERENCES

Dean R.B. (i978). Reynolds number dependence of skin friction and other bulk flow variables in twodimensional rectangular duct flow. Trans. ASME, Journal of Fluids Engineering 100, 215-223. Dimitropoulos C.D., Sureshkumar R. and Beris A.N. (1998). Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters. Or.Non-Newtonian Fluid Mechanics 79, 433-468. Giesekus H. (1982). A simple constitutive equation for polymer fluids based on the concept of deformation dependent tensorial mobility. J. Non-Newtonian Fluid Mech. 11, 69-109. Kawaguchi Y., Wei J.J., Yu B. and Feng Z.P. (2003). Rheological characterization of drag-reducing cationic surfactant solution- shear and elongational viscosities of dilute solutions. In Proc. Fluids Engineering Division Summer Meeting, Honolulu, Hawaii. Li P.W., Kawaguchi Y., Daisaka H., Yabe A., Hishida K. and Maeda M. (2001). Heat transfer enhancement to the drag-reducing flow of surfactant solution in two-dimensional channel with meshscreen inserts at the inlet. ASME Journal of Heat Transfer. 123, 779-789. Sureshkumar R., Beds A.N. and Handler R.A. (1997). Direct numerical simulation of turbulent channel flow of a polymer solution. Phys. Fluids 9, 743-755. Yu B. and Kawaguchi Y. (2004a). Direct numerical simulation of viscoelastic drag-reducing flow: a faithful finite difference method. J. Non-Newtonian Fluid Mech. 116, 431-466. Yu B., Li F.C. and Kawaguchi Y. (2004b). Numerical and experimental investigation on turbulence structures in a drag-reducing flow with surfactant additives. Int. J. Heat and Fluid Flow 25, 961-974.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

EFFECT

OF

NON-AFFINE

TURBULENCE

585

VISCOELASTICITY

ON

GENERATION

Kiyosi Horiuti, Syouji Abe and Youhei Takagi

Department of Mechano-Aerospace Engineering, Tokyo Institute of Technology, 2-12-10-okayama, Tokyo 152-8552, Japan

Abstract We investigated on the effect of non-affinity on the process of the vortical structure formation and subsequent turbulence drag reduction in the direct numerical simulation of the viscoelastic fluid in homogeneous isotropic turbulence. The non-affine effect on the polymer stress was approximated using the Johnson-Segalman constitutive equation (Johnson and Segalman 1977, JS model). We carried out DNS using the steady state solution of the JS model and the full JS model. In both cases, the largest reduction of the turbulence generation was achieved when the non-affine effect was largest, i.e., the Oldroyd-A equation, and the smallest reduction when the non-affine effect was smallest, i.e., the Oldroyd-B equation. The effect of the addition of the polymer stress on the formation of the vortical structures was examined, and it was shown that the concentration of the polymer stress was largest in the tube core region in the Oldroyd-B equation and the growth of the vortex tube core region was attenuated, whereas the concentration was largest in the vortex sheet region in the Oldroyd-A equation, thus the transformation of the vortex sheet into the vortex tube was annihilated.

Keywords Viscoelastic fluid, drag reduction, constitutive equation, affinity, direct numerical simulation, spiral vortex model, vortex sheet.

Introduction Phenomenon of occurrence of drag reduction in the polymer-diluted fluids is well known (e.g., Sureshkumar, Beris & Handel 1997). This reduction of turbulence generation is considered to be due to its viscoelastic effect. Direct numerical simulation (DNS) has been devoted to analyze this phenomenon, in which the polymer stress was added to the molecular stress due to the solvent in the Navier-Stokes equation. It was shown that the marked reduction of drag was related to the reduction of generation of vortical structures (e.g., Sureshkumar et al. 1997).

586

Structures in turbulent flow field may be divided roughly into two groups: the vortex tube-like structure and the vortex sheet-like structure. Although these two structures are not distinctively separable because a vortex tube is often formed along a vortex sheet through a rolling up of the sheet, we consider that tube and sheet structures would primarily constitute the fundamental elements of vortical structures in turbulent flows. One of the notable vortical models which causes the intense energy cascade and subsequent energy dissipation is the stretched spiral vortex model (Lundgren 1982) which yields the energy spectrum obeying the - 5 / 3 law. We speculate that by annihilating the occurrence of formation process of these vortical structures, the generation of turbulence may be reduced. The purpose of this study is to investigate the effect of viscoelasticity on the vortex formation process. The most commonly used constitutive equation to evaluate the polymer stress is the upper convective Maxwell (Oldroyd-B) equation (Oldroyd 1950) and its variants (Bird et al. 1987). The Oldroyd-B equation was derived assuming that the Newtonian fluid which surrounds the bead-spring configuration of the polymers moves affinely with an equivalent continuum, i.e., the deformation of the polymer strand is identical to the macroscopically-imposed deformation (Bird et al. 1987; Larson 1983). Assuming that molecular motions do not precisely correspond to the macroscopic deformation, Gordon et al. (1972) proposed a continuum theory in which the strand motion does not deform affinely with the macroscopic deformation (Larson 1983). This theory yields the Johnson-Segalman (JS) model (Johnson and Segalman 1977) as

Drij Ouj Oui Ouk Ouk_ 1 u(1 - fl) 2Sij, Dt = (1 - a)(Tik-~x k + -~Xkrkj) -- a(rik-~x j + -~XiTk,) -- --~ri3 -A

(1)

with the Navier-Stokes equations

Ou~ ot

O(uiu~) ~

Oxj

Op =

02ui

Ox~ + 9.--Ox~Oxk - (~ -

Ori~

~) Oxj'

(2)

where rij denotes the polymer stress tensor, A the relaxation time, c~ the proportion of contribution of the non-aNne part to the constitutive equation, /3 the ratio of solvent viscosity contribution to total viscosity of solution, D/Dt the material derivative, and Sij the strain-rate tensor, (Oui/Oxj + Ouj/Oxi)/2. When a = 0, Eq. (1) is reduced to the Oldroyd-B equation, and when c~ = 1 it becomes the (lower convective) Oldroyd-A equation. In DNS of homogeneous isotropic polymer-diluted turbulence, Brasseur et al. (2003) showed that the magnitude of the drag reduction caused in this flow was insignificant when the upper convective Oldroyd-B equation (the FENE-P model) was used. In Choi et al. (2002), it was shown that the )~-DNA induces a drastically marked drag reduction when the DNA is double-stranded. Although this DNA result was obtained in the experiment in a rotating disk apparatus, the behaviour of drag-reduction was different from that obtained using the polyethylene oxide. Because the double-stranded DNA is stiff, we consider that the DNA strand may not deform affinely with the surrounding Newtonian fluid. The objective of the our study is to develop an accurate model for the polymer stress incurred by DNA, and carry out DNS of the rotating disk apparatus experiment of Choi et al. (2002). In the present study, as for a first step to accomplish this objective, we reveal the effect of inclusion of the non-afl:ine effect on the generation of turbulence using the JS model in homogeneous isotropic turbulence.

V o r t e x t u b e f o r m a t i o n p r o c e s s in N e w t o n i a n

fluid

We begin with the summary of a formation process of vortex tube structures in Newtonian fluid. We utilized the DNS data for incompressible decaying homogeneous isotropic turbulence, which

587

were generated with 256, 256 and 256 grid points, respectively, in the z, y and z directions. Periodic boundary conditions were imposed in the three directions. The size of the computational domain was 27r in each direction, the viscosity u = 0.00014, and the time interval, At, was set equal to 0.0005. For details of the DNS data, see Horiuti (2001). Assessment was done using the data at the instant when the Reynolds number based on the Taylor microscale, Ra ~ 88. Primary elements which constitute the turbulent flow field are the vortex sheets and the vortex tubes. The vortex tubes are generally considered to be formed by rolling-up of the vortex sheets, which is attributable to the Kelvin-Helmholtz instability, e.g., Kerr et al. (1994). In this Section, we summarize the process for formation of the vortex tube along the vortex sheet in the Newtonian fluid. Then, we explore the implication of the occurrence of this formation process for turbulence energy cascade. To identify the vortex tube structures, we utilized the secondorder invariant of the velocity gradient tensor (Hunt et al. 1988), Q = --(S~kSk~ + fl~kflk~)/2, with positive values, where ft~j is the vorticity tensor, (Ou~/Oxj - Ouj/Oxi)/2. To identify the vortex sheet structures, the eigenvalue of the tensor --(Sikf~kj + Sjkf~ki), [--(Sikf~kj + Sjkflki)]+, was used (Horiuti 2003). In the present study, the eigenvalues were reordered as follows. For the eigenvalues of the strain-rate tensor, ai (i = 1, 2, 3), they were reordered so that the eigenvalue, the eigenvector of which is maximally aligned with the vorticity vector, co, is chosen as cry, the largest remaining eigenvalue, as a+, and the smallest one, as a_. The corresponding eigenvectors for eigenvalues, a~, or+, or_, were denoted as ez, e+, e_, respectively. This reordering was carried out to eliminate the crossover of the eigenvalues (Andreotti 1997; Horiuti 2001). The vorticity vector was projected onto the three eigenvectors, ez, e+, e_, and the vorticity components in the z, + , - directions were denoted as co~,co+,c0_, respectively. The same reordering was done for other eigenvalues. A conventional scenario for formation of the vortex tube is the rolling-up of the vortex sheet through the focusing of vorticity via the Kelvin-Holmhelz instability (Neu 1984), in which the assumed direction of vorticity along the sheet was always parallel to that along the vortex tube. It, however, was found in Horiuti and Takagi (2004) that in many samples which were detected in the DNS data, the direction of the axis of the vortex tube was often perpendicular to the vorticity vectors along the vortex sheet. Appearance of this vorticity configuration is inconsistent with the stability analysis for the stagnation-point flow (Kerr et al. 1994), in which it was shown that the vorticity component which is perpendicular to the direction of the diverging flow decays, and that the parallel component can grow. In Horiuti and Takagi (2004), it was shown that in this configuration, the vortex tube was not formed by the rolling-up of a single vortex sheet, but formed through the interaction of the dual vortex sheets. One of the dual vortex sheets was placed perpendicular to another sheet forming the 7-shaped arrangement and this sheet generated the stagnation flow on another sheet. In the region surrounded by the dual sheets, weak circulation associated with low pressure was generated. This low-pressure region concentrated to form the vortex tube with lapse of time. Because the vortex sheets are similar to Burgers' vortex layer (Batchlor 1967), the vorticity vectors on the two sheets are parallel to the velocity vectors on the sheets. Therefore, the direction of the vorticity vectors in the recirculating region was perpendicular to the direction of the vorticity vectors on the sheets. Thus, to adjust the spatial variations of the vorticity directions, the direction of the vorticity vector on the vortex sheets were converted as the sheets were stretched by the recirculating flow. This conversion was attributed to an appearance of the negative Crz and negative vortexstretching term, azwz,2 along the sheets. With an occurrence of negative crz, a+ and a+w~ took a large positive value, which led to the generation of the vorticity in the transverse (+) direction which was parallel to the direction of the recirculating flow. With lapse of time, the vorticity in the recirculation region accumulated to form the vortex tube, and the dual sheets

588

were stretched and entrained by the vortex tube and the stretched dual sheets were wrapped around the tube core, forming the stretched spiral vortex whose structure was similar to that proposed in Lundgren (1982). Depending on the alignment of vorticity vectors along the dual sheets and those along the core region of the vortex tube, the stretched spiral vortex consisted of three modes of configurations. In Mode 1, the conversion of the vorticity vector direction due to crz < 0 occurred on both of the dual sheets, and thus the vorticity vectors along the both sheets were parallel to those along the vortex tube. This configuration was considered in Lundgren (1982). In Mode 2, the conversion of the vorticity vector direction due to crz < 0 occurred only on one of the dual sheets, and thus the vorticity vectors along one of the two sheets were parallel to those along the vortex tube, while the vorticity vectors along another sheet remained perpendicular to those along the vortex tube, i.e., Mode 2 took an asymmetric configuration. In Mode 3; no conversion of the vorticity vector direction due to Crz < 0 took place on either of the vortex sheets, and thus the vorticity vectors along both sheets remained perpendicular to those along the vortex tube. This configuration was considered in Pullin and Lundgren (2001) and Kawahara et al. (1997). After the dual sheets were formed, the core region of the tube was formed with a very rapid accumulation of the vorticity. This tube-core formation was initiated by the appearance of the region with compression in az, i.e., az < 0, which occurred in the concentrated region along the vortex sheets. When this compression occurred, the pressure Hessian terms reacted to relax the occurrence of compression by forming a concentrated low pressure region. In Modes 2 and 3, because (7+ > 0 and a+w~_ > 0 , the low-pressure region gradually accumulated to form the tube, the axis of which was in the direction transverse (+) to the vorticity direction along the sheet. The stretched sheets were spiralling around the tube due to a differential rotation, and intense energy cascade and subsequent dissipation took place with the stretching of the vortex sheets due to differential rotation. As the Reynolds number was increased, the frequency of occurrence of the spiral vortex formation increased, and the energy spectrum showed a profile close to t h e - 5 / 3 law. We note that the tubes with the configuration of Mode 1 persisted for a rather long period of time because the circulation around the tube was large compared with those in the configuration of Mode 2. We note that the configuration with Mode 3 was rarely found in the DNS data.

D N S using the J S s t e a d y - s t a t e solution The solution of the JS model depends on the amplitude of c~. To examine a dependence of the solution of the JS model on the value of c~ and obtain an overall estimate of the energy exchange between the kinetic energy of the solvent and the elastic energy due to the polymer, we obtained an approximate solution for the JS constitutive equation as follows with the initial condition at t = 0 of Tii(O) = O.

r~ (t) ~

- 2 "(1 - Z)

g~

u(1 - ~)

g~ r

/0'e_L~_~.2Sij(s)ds

(3)

~-~x ~ j ~ j ( ~ )

in which the material derivative terms were discarded. The approximate solution, Eq. (3), contains the time-memory effect, which is important to describe the behavior of the viscoelastic

589

fluid, but, for brevity, we look into the approximate steady state solution of Eq. (3) up to the first order with regards to ~ as

rij(t) ~ - u ( 1 - fl)2Sij + 2ku(1 - fl){-(1 - 2a)2SikSkj + (&kf~kj + Sj~f~ki)}.

(4)

This solution is analogous to the approximation for the subgrid-scale (SGS) stress tensor obtained using the subgrid-scale (SGS) nonlinear model (Clark et al. 1979; Horiuti 2003), but there are notable differences. The coefficient for the (Sikf~kj + Sjkf~m) term is equal to +1 in Eq. (4) regardless of the value of a, whereas it is equal to -1 in the SGS nonlinear model for the Newtonian fluid (Horiuti 2003). In addition, the term, - ( 1 - 2a)2S~kSkj, is replaced with the term, SikSkj -- f~ikf~kj in the SGS nonlinear model. Only the amplitude of the SikSkj term depends on a. The production term of the total kinetic energy of the solvent, uiui/2, due to the second term in the JS model, pds, is given as

pJS = (1

-

fl)'riiSji

=

-4(1

-

2a)(1

-

fl)uASikSkjSii.

(5)

Noting that the derivative skewness, SikSkjSij, is generally negative on average in the homogeneous isotropic turbulence (Batchelor 1967), when 1/2 < a < 1, the kinetic energy of the solvent is converted into the elastic energy on average. On the other hand, when 0 < a < 1/2, the elastic energy is backwardly transformed into the kinetic energy of the solvent on average. It should be noted that the (Sikf~kj § Sjkf~ki) term vanishes in the pJS term, thus this term makes no contribution to the production of the elastic energy, but this term contributes to the generation of the elastic enstrophy (Horiuti 2003). Unlike in the SGS nonlinear model, the vortex stretching term, f~ikf~kjSij, does not come into the production term.

140~

2.8

I

130[2.6

.~'k:k..

,

1~0~ . . . .

00

1oo~- .......

o.,

110~"

9oL=

o.o .........0.,, ....... o.,

.

~.~

.

.

~,~, "~,~.

o.,5

2

~:

1. 5

~'X"~',.

. . . . . . . . . . . . . .

t

I:

v

F 6OF-

3.5

." ....

/

.......... 0.25

"1 ]

/ ,~.-~.,

i

o,,

. . . .

", I

i"

.................... Nawtonian

/

r

i" g / " /

~,

,

..__..

"

/" ~ , " / . /?7/

E so~

"

,,.,';'>Y , ,,';r

,oE" ~ ,

-~,~,,

1.0 Newtonian

3

^

% 7o~-

"~,

.

~

,:~Y/

20

4

F i g u r e 1 Time developments of turbulent kinetic energy obtained using the JS steady solution with a = 0, 0.25, 0.5, 0.75,1.0 and obtained from the Newtonian case.

10

u0

0.5

1

1.5

i,,,,i,,,,i,,t,l,,,,I,l,,l|) 2 2.5 3 3.5 4

t

4.5

F i g u r e 2 Time development of turbulent enstrophy obtained using the JS steady solution with a = 0, 0.25, 0.5, 0.75,1.0 and obtained from the Newtonian case.

We carried out the DNS using the JS steady solution, Eq. (4), in the decaying homogeneous isotropic turbulence. In the present and following Sections, the DNS was done using 128, 128 and 128 grid points, respectively, in the x, y and z directions, the viscosity u = 0.004, fl = 0.8, A = 0.45, and the time interval, At = 0.001.

590

Figures 1 and 2, respectively, show the time development of turbulent kinetic energy and enstrophy obtained using the steady solution, (4). A strong dependence of the results on the amplitude of a can be discernible in Figs. 1 and 2. We note that the solution with a = 0 (Oldroyd-B) diverged at t ,,~ 2.3 due to an excessive backward cascade of the elastic energy into the energy of the solvent. As a was increased, both energy and enstrophy decreased, and when 1/2 < a < 1, marked reduction of turbulence generation was achieved in comparison to the Newtonian case.

Viscoelastic effect on the tube formation process In the previous Section, we showed that the reduction of turbulence generation can be achieved using the steady solution of the JS model. Its reduction was more considerable when 1/2 < a < 1, and the extreme case of a = 1 (the Oldroyd-A equation) exhibited the largest reduction. We consider that this reduction might have been achieved due to the termination of the occurrence of the spiral vortex tube formation process and subsequent generation of energy cascade and dissipation, which occurred in the Newtonian fluid. In this Section, we investigate the effect of viscoelasticity on the process for formation of the stretched spiral vortex by solving the full 3S model, Eq. (1), without resorting to the steady state solution. As we showed in the previous Section, the calculation using the (steady-state) JS model tended to be unstable due to an occurrence of intense backward cascade (particularly in the case with a ,-~ 0), the diffusive term, ~02rij/OxkOxk, was added to the right hand side of Eq. (1) with = 0 . 0 5 u ( 1 - fl)/A to stabilize the numerical solution (Sureshkumar et al. 1997).

3

50

/,,, /

2.5

~-"~'~'~~ " ~ "

2

Oldroyd-A Oldroyd-B

\ \ '~

\ \'

45 40

/

25

\ \

01droyd-A

i

Oldroyd-B

I

iI

~tl

11. . . . . . . .2. . . . . . . . .3. . . . t

4

30 v

/

i!

35 tO 1.5

/

I

2o 1

15 10

0.5

i

i

i

i

i

i

i

i

i

i

i

i

i

,

,

I

I

. . . .

I

. . . .

I

,

t

F i g u r e 3 Time development of turbulent kinetic energy obtained using the Oldroyd-A a n d - B equations.

O0 . . . .

5

T?'6

F i g u r e 4 Time development of turbulent enstrophy obtained using the Oldroyd-A and -B equations.

As for the two extreme cases for the JS model, we carried out the DNS using the Oldroyd-A (a = 1, Case A) and Oldroyd-B (a = 0, Case B) equations. Figures 3 and 4, respectively, show the time development of turbulent kinetic energy and enstrophy obtained from the Cases A and B. More pronounced reduction of the generation of kinetic energy and enstrophy takes place in Case A than in Case B. These results are consistent with the results obtained using the JS steady solution shown in the previous Section.

591

The analysis of the turbulent structures revealed that the effect of the polymer stress was most dominant in the sheet regions in Case A, i.e., the distribution of the polymer energy was primarily concentrated in the vortex sheet regions. In Case B, the effect of the polymer stress was most significant in the tube core regions. Figures 5 and 6, respectively, show the distributions of the vortex tube (identified using the second-order invariant, Q, shown using the heavy gray contour lines) and the vortex sheet (identified using the eigenvalue of the --(Sik~kj + Sjk~ki) term shown using the heavy white contour lines) obtained from Cases A and B. Also shown in the figures are the distributions of the pressure due to the polymer stress, p, (shown in the figure using the flood contours), where p, is the solution of the following Poisson equation as

02p, = OxkOxk

F i g u r e 5 Distributions of tube, sheet and the p, term obtained using the Oldroyd-A equation.

02vii OxiOxj "

(6)

F i g u r e 6 Distributions of tube, sheet and the p, term obtained using the Oldroyd-B equation.

It can be seen in Figs. 5 and 6 that, the p, term is distributed mostly along the sheet region in Case A, whereas in Case B, the p~ term is concentrated in the tube-core region and it takes a locally maximal value at the center of the tube. Thus, in Case B, the growth of the tube core region was prevented by increasing the (locally minimal) pressure at the center of the tube core region. In Case A, examination of the enstrophy generation term showed that occurrence of formation process for the stretched spiral vortex with Mode 2 was substantially reduced because the vorticity in the stretching (z-)direction was strengthened along the vortex sheet by the backward transfer of the enstrophy due to the polymer stress into the solvent. The occurrence of compression (az < 0) was substantially reduced, and the vortex sheets were prevented from wrapping around the vortex tube because the vortex sheets were pulled back to the original flat shape due to the enstrophy generation term. Markedly fewer vortex tubes were generated along the vortex sheet in Case A than in Case B and the Newtonian case. Subsequently, generation of turbulence cascade associated with the formation of the stretched spiral vortex which was observed in the Newtonian fluid was substantially annihilated in Case A. The occurrence of formation process for the tube with Mode 1 was less disrupted by addition of the polymer stress than in the occurrence of formation process for the tube with Mode 2 because the circulation around the vortex tube in Mode 1 was generally larger than the circulation in Mode 2. We note

592

that the behaviours of the p, term obtained using the steady-state solution of the JS model were similar to those shown in Figs. 5 and 6.

Summary We investigated on the effect of non-affinity on the turbulence generation in the viscoelastic fluid in homogeneous isotropic turbulence. The Johnson-Segalman constitutive equation was used to introduce the non-affine effect into the evaluation of the polymer stress. In DNS using both the steady state solution of the JS equation and the complete JS equation, extreme cases of c~ = 1 (the Oldroyd-A equation) and c~ = 0 (the Oldroyd-B equation), respectively, showed largest and smallest drag reduction. The results obtained when 0 < c~ < 1 were intermediate between those from the Oldroyd-A and the Oldroyd-B equations. This reduction was attributed to the reduction of the formation of the stretched spiral vortex structures and subsequent reduction of generation of the energy and enstrophy. These results may indicate that as the contribution of the non-affine component becomes large, the less generation of turbulence takes place. The dominance of non-affinity is correlated with the degree of deformation of the polymer networks, which implies that the polymers with large deformation may cause more drastic drag reduction.

References Andreotti, B. (1997) Studying Burgers' models to investigate the physical meaning of the alignments statistically observed in turbulence, Phys. Fluids Vol. 9, p. 735. Batchelor, G.K. (1967) An Introduction to Fluid Mechanics. Cambridge Univ. Press. Brasseur, J.G., Robert, A., Vaithianathan, T., and Collins, L.R. (2003) Polymer-turbulence dynamics in isotropic stationary turbulence with the FENE-P model, Bull. Ame. Phys. Soc. 47 No.10, 169. Bird, R.B., C.F. Curtiss, R.C. Armstrong, & O. Hassager (1987), Liquids, 2nd ed., Wiley.

Dynamics of Polymer

Choi, H.J., Lim, S.T., Lai, Pik-Yin, Chan, C.K. (2002) Turbulent Drag Reduction and Degradation of DNA, Phys. Rev. Lett. 89, 088302. Clark, R.A., Ferziger, J.H. and Reynolds, W.C. (1979) Evaluation of subgrid-scale models using an accurately simulated turbulent flow, J. Fluid Mech. 91, 1. Horiuti, K., (2001) A classification method for vortex sheet and tube structures in turbulent flows, Phys. Fluids 13, 3756-3774. Horiuti, K., (2003) Roles of nonaligned eigenvectors of strain-rate and subgrid-scale stress tensors in turbulence generation, J. Fluid Mech. 491, 65-100. Horiuti, K., and Takagi, Y. (2004) A process for formation of multi-mode stretched spiral vortex in homogeneous turbulence, in preparation. Hunt, J. C. R., Wray, A. A., and Moin, P. (1988) Eddies, stream, and convergence zones in turbulent flows Center for Turbulence Research Report Vol. $88, p. 193.

593

Johnson, M.W., & D. Segalman (1977) A model for viscoelastic fluid behavior which allows non-affine defromation, J. Non-Newt. Fluid Mech. 2,255-270. Kawahara, G., S. Kida, M. Tanaka, & S. Yanase (1997) Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow, J. Fluid Mech. 353, 115-162. Kerr, O.S., and J.W. Dold (1994) Periodic steady vortices in a stagnation-point flow, J. Fluid Mech. 491,307-325. Larson, R.G. (1983), Convection and diffusion of polymer network, J. Non-Newtonian Fluid Mech. 13, p.279. Lundgren, T.S. (1982) Strained spiral vortex model for turbulent structures, Phys. Fluids

25, 2193-2203. Neu, J.C. (1984) The dynamics of stretched vortices, J. Fluid Mech. Vol. 143, p. 253. Oldroyd, J.G. (1950) On the formulation of rheological equations of state, Proc. Roy. Soc. London A Vol. 200, p. 523. Sureshkumar, R., Beris, A. N. and Handel, R. A. (1997) Direct numerical simulation of the turbulent channel flow of a polymer solution, Phys. Fluids Vol. 9, p. 743.

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Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

595

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF FLOW CONTROL ON BLUFF BODIES BY PASSIVE VENTILATION M. Falchi, G. Provenzano, D. Pietrogiacomi, G.P. Romano Department of Mechanics and Aeronautics, University of Rome, "La Sapienza", Via Eudossiana n. 18, 00184 Rome, ITALY romano @dma.ing.uniroma 1.it

ABSTRACT In this work, the so-called Natural or Passive Ventilation drag reduction method is investigated experimentally and numerically. Passive Ventilation is performed by directly connecting the high pressure region at the front of a body to the lower pressure in the near wake using a venting duct; so far, it is able to establish a net mass flux into the wake. In particular, in aerodynamic applications it seems to be suitable to attain a global reduction of the drag of a body moving in a fluid and a reduction of turbulence levels by means of a global modification of the body wake. Velocity field investigations using Particle Image Velocimetry measurements and numerical Reynolds averaged code are employed at moderately high Reynolds numbers to clarify the effectiveness of drag reduction on a vented bluff body. The numerical and experimental results agree qualitatively, but the amount of reduction for the vented body (about 10%) is underestimated numerically. Direct balance measurements are used for comparisons.

KEYWORDS Passive ventilation, bluff-body wake, drag reduction, PIV, RANS.

INTRODUCTION Drag reduction, in particular in the automotive and aeronautic fields, is one of the most investigated topics for its possible benefits on fuel consumption reduction and performance increases. In the subsonic range, the total drag force exerting on a body is due to the sum of the skin friction drag and pressure or form drag. As the body shape changes, the relevance of each of these two drags also changes; for streamlined bodies, the skin friction drag is the main part (85-90%), while for bluff bodies the form drag is the most important (80-85%). In particular, a bluff body, immersed in a fluid flow, is characterized by a large separated region in the rear part of the body i.e. the wake. The wake

596 is generated by the boundary layer detachment which avoids the pressure recovery; the net result, considering the high values of the pressure in front of the body, is an increase of the form drag. The Natural Ventilation technique is a Flow Control technique which modifies the bluff body wake and reduces the total drag by acting on the form drag. In this technique, a venting duct connects the high pressure region in front of the body to the low pressure region in the near wake. In this way, the pressure gradient sets up a flow inside the duct with energy levels as high as the free stream, which energises the wake. The destructive interaction of the counter-rotating vorticities of the venting duct and of the external shear layer, generating a sort of aerodynamic streamlining, partially destroys the wake and raises the pressure values (behind the body) allowing pressure recovery. The main effect is a large reduction of the form drag together with the reduction of the turbulence intensity in the wake region (Aschenbach 1972, Roshko 1991, Meier & Suryanarayana 1995, Gad el Hak 2000). It is important to note that no external power is required to generate the improvement in comparison to many other drag reduction (active) techniques, e.g. base bleed, suction or near wake-heating (Monkewitz 1992). Up to now its effectiveness has been proved only in the case of the sphere (Suryanarayana et al. 1993, Meier & Suryanarayana 1995, Prabhu & Suryanarayana 2000). The obtained results indicate a 50-60% total drag reduction in the supercritical range (i.e. for Reynolds numbers larger than 3.5x105 with a fully turbulent boundary layer), while practically no reduction for lower Reynolds numbers. The authors claim that this is properly due to the boundary layer transition which helps in transferring momentum from the side of the body to the wake and from the outlet of the venting duct to the rest of the wake. The main open question just concerns with the possibility of successfully applying the technique on bluff bodies different from the sphere, which are particularly interesting for industrial applications (it should be considered that many investigations on this point are not at full disposal of the scientific community due to their commercial use). It must be also noticed that in the sphere the constant curvature allows the boundary layer detachment point to move freely over the sphere surface following changes in the fluid dynamic field (in particular the energy of the shear layers in relation to the wake). This is not allowed in bluff bodies which are more or less truncated in the rear part; this is why it is important to verify the effectiveness of the technique in this case. In this paper, experiments by Particle Image Velocimetry (PIV) and numerical runs by Reynolds averaged code (RANS) are coupled to investigate the velocity fields around vented and non vented bluff-body models to clarify the above mentioned phenomena. An overall measurement of the drag of different models has been performed by acquiring direct dynamometric balance data.

EXPERIMENTAL SET-UP AND NUMERICAL SIMULATIONS The experimental investigations have been performed in the low speed wind tunnel of the University of Rome "La Sapienza", Italy. The tunnel has an open round test chamber (0.9m diameter, 1,2m length), a maximum velocity of 55m/s, Re = 2.34x10 U, and a level of turbulence intensity < 1%. The bluff bodies used to study the effectiveness of the technique are given in Figure la), basic model without venting duct, and Figure lb), model with cylindrical venting (different shapes of the duct have been preliminarily considered, the cylindrical one resulting the most effective). This kind of bluff body, a half-sphere on a cylinder with a round comer in the aft region, has been chosen both to maintain some of the aerodynamic features of the sphere and also to investigate a body more similar to automotive applications. The diameter of the model is D = 14 cm. The area of the venting duct cross section is equal to 2.25% of the area of the body cross section, so it is fully feasible in practical applications of the method. Direct force measurements have been performed by using a three component dynamometric balance in order to valuate the drag component force for a Reynolds number ranging from 2.0x10 s to 7x105. Streamwise velocity component measurements have been performed using Hot Wire Anemometry (HWA) and a Pitot tube with pressure gauge at different positions in the body wakes. For HWA, the sample frequency has been set equal to 1 kHz; 20,000 champions were acquired at each point. These measurements have been performed at Re = 3.38x105.

597

Figure 1. Bluff body models tested; a) non vented or closed model, b) vented model. Using a PIV system, equipped by LaVision, (x,y) two component velocity measurements have been performed. The acquisition system is composed by a double Nd-Yag laser, 532 nm wavelength, with 100 mJ per pulse, 7 ns pulse duration. The camera is a cross-correlation one with a 1376x1040 pixels resolution. The seeding particles were olive oil particles with mean diameter of 1 micron. Each acquisition is made of 1,000 images couples; instantaneous and averaged vector fields have been computed as long as other turbulent quantities - turbulence intensities, turbulent kinetic energy and Reynolds stresses -. These measurements have been performed at Re = 1.7x105. It is important to notice that all measurements on the models have been taken both in the "clean" configuration and in the "strip" one; for the "strip" configuration, an adhesive strip is stuck to the model at the end of the spherical part, while the "clean" configuration is without the strip. This choice allows the boundary layer on the model surface to exploit the transition from laminar to turbulent to attain the supercritical range (which is a crucial point for the success of the technique); the comparison between clean and strip models allows to determine the importance of the boundary layer transition. The strip was an adhesive 1.5 cm wide paper strip; in Figure 2 a picture of the vented model in strip configuration is given (the model is painted in black to reduce reflections in PIV measurements).

Figure 2. The black painted vented bluff body in strip configuration.

598 The numerical simulations have been performed to test the robustness and suitability for advanced applications of the commercial RANS code used (StarCD) by comparing the data with the experimental ones. The simulations have been performed at a Reynolds number equal to 2.38x105, on a fully three-dimensional domain; a high Reynolds number finite volume K-e model was used with about 3• cells. The wall function approach with Monotone Advection and Reconstruction Scheme (MARS) for average fields and Upwind Differences for turbulence evaluation have been used. Inlet and outlet profiles have been given at boundaries. Preliminary grid sensitive tests have been performed and a final mesh extending 10D along all directions has been selected; the residual was 10.5 for all variables.

RESULTS

Direct Drag Measurements In Figure 3, the drag coefficient measured with the dynamometric balance as a function of the Reynolds number is depicted for all the bluff body models; the drag coefficient has been calculated, once the drag, Dr, has been obtained, by Co = 2Dr/pUJ S, where/9 is the air density, U~ is the free stream velocity and S is the model cross-sectional area. As shown, for the clean configuration models - non vented and vented -, an average drag reduction of about 7-8% is attained; this is within the measurements uncertainty (about 5%), so it cannot be interpreted as effective. For the strip configurations, a different situation arises; a 20% average drag reduction has been obtained. This depends on the fact that while vented models in the clean and strip configurations have almost the same drag, the non vented ones show different behaviours. The non vented strip model has about 10% higher drag due to the fully developed turbulent boundary layer which increases the skin friction drag. So, for the strip models, the technique seems to work better; for a deeper understanding of these data, velocity measurements in front and in the wake of the models must be performed. These measurements are required also by considering that a rather crude evaluation of the drag reduction (evaluating the gain in momentum which is obtained in the vented model) gives as a result the ratio of cross-sectional areas for vented to non vented models i.e. about 2%. This value is lower than what observed from balance measurements both for clean and strip configurations; therefore, the near wake behaviour is the key point to understand the phenomenon.

0,45 0,4 0,35 Cd 0,3

X- "-X----X~

--+- u n ~

L.+..._q.. _..."+~+.~

0,25 0,2 0,15 0,1 2,00E.-14)5

+

~stfip

-x-

un~

strip

clean

9 voatexicl~n

,/ 3,00E-'405

4,00E,+05

5,~

6,00E-'405

7,00Eu.05

Re

Figure 3. Direct drag coefficient balance measurements at different Reynolds numbers.

599 1.75

1.75

1.5

1.25

1.25

0.75

;~

i

0"751- ~ : ~ : ~ : : _ ~

0.5 -: :---~: 0.5 0.25

. . . .

i

~Z~-?---:

?--~:~ ~ ~ ~ :- :: ~

~-:~:-:~

~ :~~ ~

0.25

0

0 . ~ ~ 0.5

1

x (x/D)

1.5

2

0.5

b)

-,~ ~ ~,-,_ ~_~ -,=_.~_~=-__~~ 1

x (x/D)

1.5

2

1.75

1.5_ ~ : - = - - ~ : ~ - ~ - ~ } ) ~ - i ) ~ } ~ } i i ~ i i

, i iiii iii iiiiiiiii!ii ! i ii i i i i 0.75

0.5

0.25

0.5

c)

1

x (x/D)

1.5

2

0.5

d)

1

x (x/D)

1.5

2

Figure 4. Vector fields in the near wake from PIV; a) non vented, clean configuration b) vented, clean configuration c) non vented, strip configuration d) vented, strip configuration. Re = 1.7x105.

Velocity Measurements In Figure 4, the vector fields in the near wake of the models measured by PIV are depicted; the reference system origin is located on the back flat surface, on the cylinder axis. The free stream velocity is directed from left to right. In the non vented configuration, both for the clean and the strip configurations, the presence of a counter-flow in the wake of the models from the base to approximately x./D=l is noticed; no large differences are seen between the two configurations. The wake dimensions appear to be the same. For the vented configurations the situation is different; in fact, the venting jet eliminates the counter-rotating vortices by penetrating in the wake down to about x/D=0.8-0.9 (the upper bound is reached for the strip case and the lower bound for the clean one). The difference between strip and clean conditions is caused by the strip that, by forcing the laminarturbulent transition of the boundary layer, allows a slight downstream displacement of the detachment point thus decreasing the width and increasing the length of the wake (as can be seen from Figures 1 and 2, the models have rounded edges to allow such a slight displacement). It has to be noticed that in both strip and clean conditions the venting jet is not able to emerge form the wake; thus a minor effectiveness than in the case of the sphere must be expected. This is confirmed by the fact that for x/D>1.5 higher velocity are obtained at the middle of the wake for non vented models in comparison to the vented one. In theory the presence of the venting jets should have accelerate this portion of the wake in the vented models configurations; in reality, measurements show the opposite.

600 - X - Unvented clean

---¢-- x/D=0.25 Piv data

.

- - ~ x/D=0.25 Pit~t data --O-- x/E~.25 ntmcrical data

-~1

~"'--'-0.6

-0.4

-02

0

0.2

0.4

Vented clean

- + - Unvented strip

- - ~ x/D=0.25 Hwa data

0.6

0.8

k

--4-- ~nted strip

I-,~ 1

1.2

utU

0

0.5

1

1.5

2

x (x/D)

Figure 5. Streamwise velocity profiles; transverse profile at x/D=0.25 obtained with different techniques for the clean vented model (on the left) and longitudinal profiles at y/D=O (centreline) obtained from PIV measurements for the different models (on the right).

Velocity profiles can be derived from previous data; in Figure 5, the streamwise component of velocity along transverse and longitudinal directions in the wake of the model are shown. From the transverse profiles (Figure 5, left), the overlapping between the different techniques is poor. For HWA, this is due to the unresolved velocity direction in the wake (no negative velocities), while for the Pitot tube to the poor spatial resolution of the method (which smoothes the differences in the profile). It is interesting to compare PIV and numerical results; from numerics there is a clear underestimate of the negative velocities absolute values in the separation region (about one half). Moreover, the profiles for y/D>0.5 (external flow) are also different; the PIV data show a slight increase of velocity in the shear layer which is not displayed by the numerics. These results must be coupled with the full field results obtained with the numerical computations; in Figure 6 such a field is shown for the vented case. As it is possible to observe the general feature of the fluid dynamic field are captured by the code; however, differently from experimental data, the venting jet is strongly attenuated all over the wake. This result is the best one obtained by using different turbulence and wall models; due also to the results observed in the shear layer, the sensitivity of the numerical code to the used models seems to be high. Returning back to the longitudinal profiles of Figure 5 (fight), it can be observed that the non vented profiles are independent on the presence or not of the strip (as noticed also in figure 4). On the other hand, the strip vented profiles clearly show an increase in velocity in the near wake in comparison to the clean one (i.e. an increased length of the wake as also derived from figure 4). To compare velocity data with direct force measurements, the drag has been computed from the measured velocity profiles using the Maskell's formula, modified by Cummings et al. (1996):

D = p Js u ( v

-u)as

1

Js

+ , 2)ds

(1)

where u,v,w are the velocity components and the w-component is unknown for plane measurements. The drag coefficients have been computed using transversal profiles at different distances; this has been done because this point is not completely clear from the literature (Oertel 1990). In Table 1, the data are presented for all the models. Considering the approximation level of the previous relation, the data, for the unvented configuration, can be considered in well accordance with those from the balance. On the other hand, for the vented configurations the differences are much stronger. However, even in this case, a reduction of 5% between vented and non vented models can be derived from measured profiles.

601

ii:!!i!!~!~ii!!!!ii-:i~iii!>i~iiii~!~iiii~iiiii::i~ii' _:i: :ii!ii~ii!~iiiiiiiiii~iiiiiiii!i!~iii iiiiiiii~i!ii':'ii:iii~iiii!iii2iiiiii!

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: PROSTAR3.10 ~::::::::::::::::::::::::::::::::::~:::::::::::~

~:::::~:":::::

::::::::::i:~:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

:i~:::.: ::::~::S ::::: :::: 3>::S.:

:::: :::~::: ::~::::.--~:::.::::::::i::

~

= =

:::::::::::::::::::::::::::::::::::::::::::::::::::::

z~-~-o~

VELOCITYMAGNITUDE

!~::7:::~:::::

:-:!!ii~:::::!iii~?:::::ii:!~::::::iii =::==.===.=====:.=====~;:==::======

,x~...... LOCAL

===============================

MN.

34.90

o.oooo

~ocA~.

...........~ ..........~..~..........~: ....... :~ ...........!'~'-"~: ..........."}~..........~:~"........ ~...........!;~.'"'", "PRESENTATIONGRID"

~::~::~i:~ii!7::i!::::i-:!!ill :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::

:::~,

::::::::!1. ::::::::~ :::~,.

::::-

....

....... ~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........~ . . . . . . . . . . . . . . . . . . . . . . . _~ :: ._.,.

m,

....... ,~

::_-~

......... ~

. . . . . . . . . . ~. . . . . . . . . .

::::':.~. .. . . . . . . . .

.-.

:22

:::'-

'"

::".;

:::.,:

::::7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

z.4~

-:~

~

047SOE-0$

Figure 6 • Vector velocity fields in the wake of the clean vented model from numerical computations. Re = 2.38x105.

Distances

Non vented clean

Vented clean

Non vented strip

Vented strip

1D

0.4109

0.4403

0.4071

0.3947

1.5 D

0.4942

0.4613

0.4688

0.4592

2D

0.4040

0.3897

0.3770

0.3878

Balance 0.3306 0.2703 0.3685 0.2333 measurements Table 1. Comparison between drag coefficients obtained from PIV velocity profiles for the different models. Direct balance measurements are reported for comparisons.

14[

1.4

1.2

1,2.

1

~ ~

0.8

"-~0.8

o

.

6

~

:-

~

~

-~ -

~

:"

_--- ~

_: ~

~

_~ .

.

.

.

.

.

.

0.6

_==============~~ 0.4 ~ ~

0.4

ii;:i:ii; O - ~ . ~ i ~ . ~ z ~ i ~ ; ~ -1

--

~ -0.5

x(x/D)

. :

~

-

,j

0.2

. . . . . . . . . : : : : : : : : :

: : : : : : :

. . . . . . . . . . . . : : : :..: :

. . . . . . . . . . . . : : : : : :

. . . : :

: :

: :

off ~ ~i ~ ~,i ~ ~,~ ~i ~, ~ ~ ! ;-~.=~-j:: ::, ::i ::,:- ::

0

-1

b)

-0.5

x(x/D)

0

Figure 7 • Vector velocity fields in front of the models from PIV measurements; a) non vented strip configuration b) vented strip configuration.

602

It is possible that differences in pressure, not evaluated in the experiments, play an important role in the evaluation of the drag from velocity profiles. High pressure values could reduce the drag; for this reason, pressure measurements are required in future experiments. To better understand the whole fluid dynamic fields surrounding the models, PIV measurements in front of the bodies have also been performed (Figure 7). Differences are observed only just in front of the models; far upstream the differences in the fluid dynamic field are negligible; as already reported, the momentum differences between the two cases leads to a difference not higher than 2% (difference in cross-sections). Therefore, the measured differences between vented and non vented models indicate that the drag reduction mechanism in the vented models is only partially due to modifications in the front of the body, whereas the main effect is in the bluff-body wakes.

Turbulence and Reynolds stress measurements It is important to have information about the modification of the turbulent kinetic energy in the wake; in Figure 8, the streamwise rms velocity profiles (normalised with the free-stream velocity) are given for measurements with HWA and PIV. In the near wake (x/D=0.25, left part of the figure), the strong peak for the vented case reveals an increase of turbulence of the near wake in comparison to the non vented case. The shear layer peak is slightly reduced in the vented case, thus confirming an overall redistribution of the kinetic energy. There are quite strong differences between the results obtained with HWA and PIV; the latter underestimate the level of fluctuations of about 20% probably due to the different Reynolds numbers of the tests (about a factor 2). Absolute turbulentir~ensityI - x / D = 2

Absolute turbulent intensity I - x I D = 0.25

0.220.2

'~'

0.18 0.16

.t

0.14

I ~ ~

- + - I Piv data

I-+-IPiv~m

I~

[~I

Hwa data

p- 0.12 ~

0.1 0.08 0.06 0.04

L

00~

~-%~

o ............. o 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

1.8

y (y/b)

0.26

0.24

0.14 0.12 0.1 O.C~ 0.06 0.04

0.8

1

1.2

1.4

0.16

I+,~v~a

Z

0.14

~L

II.-=

Piv dam

0.12 0.1 ' - 0.08

o

[k

0.6

Absohte turbub~ iXem~ I - x / D = 2

o.i8 f~ • o16

0.4

y (y/D)

Absokle twb~nt hlens~, I- x/D=0.25

°o~

0.2

~

lllitl ~ I~

4

l

0.06

]~

0.04

f

0.02

0.~

,~

,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 y (y/b)

0

I

I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 y (y/D)

Figure 8. Absolute turbulence intensity profile at x/D=0.25 (on the left) and x/D=2 (on the right); non vented (at the top) and vented models (at the bottom). Comparison between PIV and HWA data.

603

1.75

~

i•'1, 3 60,41'2

#u,=.,=

..... i

3 "~" 4 I';' ~~ i gl ~$

15

i ~,',~

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I d151~

o "z'~g 16";I

,41 165567

0 6"~ 0 3'~ 15'2

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41 I , ' ~ ?

41 $~$

-I .~417 -I -3 |d $~'i -3 ~157

0.75

-I "75 -3 Sdl d? -'~ '~..k?$

!!iil ~iiiiiii!i!~ili~ i~i!i~ili!!l! i!~!ii!!iiiiiii!iiii~iiiiiiii!ii -41~ ,!!!:i!!!~ !~i ~!iii! !!!ii!i!ii! !!i i!!~i i'.i ii!!ii !i~!i i!iiiiii 135

-.k " ~ ¢ I ?

..4i~s

025

025

~,,,

.4,,,e

0

o

-5 I~4 I? -a $

~,, . . . .

•

05

~,~ I s n ~

I

[

!

:

~, I

0.75

~-- -"Z:$,4157 I •4 13:

ilii~

,,, ,,

ii~i!~, '~;~ i~ !i'~!~!!i! ~ ~,,,

. . . . . .

..................

x (x~)

...............

-a,o~,~

,

-aS •

3 5~1 I? I

~

I si~s

I

- . , ,:

,'

i!ii !iii

t

I

., ,~ -3 Id:~.~

~ ..... I

I

on: ,

-aS

15

1.75

I

d~4i ? I

•

I

x (x~)

.4 9 1 5 ~ .,k -$ 9 i~ ,.k,k -a liMl?

,

~ ,--

ii;:!~~!i:iii!iliiiiiiiiii~i~i!i~i!:ii!iiii:ii!

,,"=. -4 I'~

i' .....

-3 I ~ -3 S d l a?

I I

,'

' i'"° . -~,~,,,- ,, ::"

~

:

......

x (x~)

,_~ .......

-

-a,~.

,

-aS

Figure 9. Reynolds stress measurements by PIV; non vented clean (top left), vented clean (top fight), non vented strip (bottom left) and vented strip (bottom fight). For x/D=2 (fight part of the figure), the turbulence levels are reduced; in the vented case the maximum (y/D=0.35) is about 0.14-0.16, while for the non vented case it is about 0.13-0.15. This is observed also in the outer layer (y/D>0.7); the overall picture displays a reduction of the wake turbulence levels and amplitude thus confirming the observed reduction of turbulence in the wake for the case of the sphere. However, for a bluff-body the amount of reduction is lower than for the sphere even if still interesting for practical applications. In Figure 9, the Reynolds stress obtained from PIV for the different configurations are shown. Two main things can be noted; the vented configurations present reduced local minima in respect to the non vented configurations and a complete redistribution of the stresses over the whole wake (a general reduction of the wake amplitude). In particular, a slight displacement towards the free stream direction can be noted for the local maxima and minima of the stress in vented strip configuration in respect to the vented clean one. These findings confirm the reduction in the turbulence fluctuations and in the amplitude of the wake.

CONCLUSIONS In this work the Passive Ventilation technique, applied to a bluff body has been studied by means of different experimental and numerical techniques. For the models investigated (without and with turbulence inducing strip on the surface) a drag reduction from direct balance measurements has

604 been valuated both for the clean (7-8%) and the strip configuration (20%). The velocity fields in the wake region have been studied by means of different techniques which show a redistribution of the flow turbulent characteristics due to the presence of the venting jets. Data from the velocity profiles confirms the observed reduction although on a lower extent (about 5%); the amount is still very interesting for practical applications. Turbulence and Reynolds stress smoothing are also observed all over the wake for the vented models. Pressure measurements over the models surface are planned to be done in the future to better evaluate the drag reduction mechanism.

REFERENCES Achenbach E. (1972) Experiments on the flow past spheres at very high Reynolds numbers, Journal of Fluid Mechanics, Vol. 54 Cummings R.M., Giles M.B., Shrinivas G.N. (1996) Analisys of elements of drag in threedimensional viscous and inviscid flow, AIAA, paper n. 962482 Gad E1 Hak M. (2000) FlowControl: Passive, Active, and Reactive Flow Management, Cambridge University Press Meier G.E.A., Suryanarayana G.K. (1995), Effects of ventilation on the flowfield around a sphere, Experiments in Fluids, Vol. 19 Monkewitz P.A. (1992) Wake Control, in Bluff- body wakes; dynamic and instabilities, Iutam Symposium, Gottingen,Germany Oertel H.Jr. (1990) Wakes behind blunt bodies, Annual Review of Fluid Mechanics, Vol. 22 Prabhu A., Suryanarayana G.K. (2000) Effect of natural ventilation on the boundary layer separation and near-wake vortex shedding characteristics of a sphere, Experiments in Fluids, Vol. 29 Roshko A. (1991) Experiments on the flow past a circular cylinder at very high Reynolds number,

Journal of Fluid Mechanics, 10 Suryanarayana G.K., Meier G.E.A., Pauer H. (1993) Bluff-body drag reduction by passive ventilation, Experiments in Fluids, Vol. 16

9. Aerodynamics Flows

This Page Intentionally Left Blank

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

A P P L I C A T I O N OF R E Y N O L D S S T R E S S M O D E L S HIGH-LIFT AERODYNAMICS APPLICATIONS

607

TO

O. Grundestam 1, S. Wallin 1'2, P. Eliasson 2 and A.V. Johansson 1 1Department of Mechanics, Royal Institute of Technology (KTH) SE-100 44 Stockholm, Sweden 2Aeronautics Division, FFA, Swedish Defence Research Agency (FOI), SE-172 90 Stockholm, Sweden

ABSTRACT A recently proposed explicit algebraic Reynolds stress model (EARSM) based on a nonlinear pressure strain rate model has been implemented in an industrial CFD code for unstructured grids. The new EARSM was then used to compute the flow around typical three element high-lift devices used on transport aircraft both in 2D and 3D. For 2D mean flow, various angles of attack have been investigated. Two different grids have been used, one coarse grid with 35,000 nodes and fine grid with 340,000 nodes. Furthermore, a 3D take-off configuration including fuselage was computed using a computational grid with about three million grid points. For the 2D case and pre-stall angles of attack, the new EARSM makes fair predictions. For higher angles of attack, the new EARSM and the baseline EARSM show a large sensitivity to the transition point location. The original transition setting leads to a premature stall while an alternative transition setting gives predictions that are in good agreement with experiments. For lower angles of attack, there are indications on minor improvements. One angle of attack close to the maximum lift was computed for the 3D case and compared with previous computations. No significant differences were found with the new EARSM compared with the baseline EARSM. Also the convergence rate and computational effort by using the new EARSM are comparable with the baseline EARSM.

KEYWORDS

Explicit algebraic Reynolds stress model, high-lift aerodynamics, nonlinear modelling, stall, transition, turbulence modelling

INTRODUCTION

Cornerstone problems for turbulence models are flows affected by rotation, streamline curvature and separation. These phenomena are of high interest since they play a determining role in many

608

engineering applications such as turbomachinery and aeronautics. High-lift aerodynamics is an application of great interest where all these effects are present. Since the maximum load of an aircraft is determined by its take-off and landing performance, it is crucial, not least from an environmental and economical point of view, to be able to make accurate predictions by means of CFD when designing new high-lift configurations. To be able to capture critical features of this type of flow, standard two-equation models based on the Bousinessq Hypothesis are normally not adequate. Therefore, development and testing of more advanced models such as explicit algebraic Reynolds stress models (EARSMs) are important. This was one of the topics in the European project HiAer (High Level Modelling of High Lift Aerodynamics) in which this work has been carried out. In the present work, the explicit algebraic Reynolds stress model proposed by Grundestam et al. (2004), GWJ-EARSM, has been implemented in a general 3D CFD code for unstructured grids, the EDGE-code at FOI, Eliasson (2002). This was used to compute both a 2D configuration and a 3D full span slat and flap configuration. The flow case studied is the landing configuration of a representative civil aircraft wing. This configuration is particularly interesting since the complex geometry provides many problems of engineering interest. A critical test of the model's ability to accurately predict separation involves, in particular, the separation on the suction side of the flap. Moreover, on both the slat and the main wing, there are geometrically induced recirculation regions on the pressure side close to the trailing edges. Furthermore, streamline curvature and the effects of the upstream elements on the flow around the downstream elements are other important issues. Both test cases have been investigated earlier for the evaluation of turbulence models, see Eliasson (2003), and some of these results have been included in this paper for reference and comparison. The 2D mean flow formulation of the GWJ-EARSM is described shortly in the appendix. The 3D mean flow formulation is somewhat lengthy and will therefore not be given here. For a full description see Grundestam et al. (2004). The standard K - a~ equations by Wilcox (1988) have been used as model platform. The results are compared with experimental data and the predictions made by the WJ-EARSM, Wallin and Johansson (2000), and the corresponding differential Reynolds stress model (DRSM), see appendix.

GOVERNING

EQUATIONS

Since EDGE is a finite volume solver, the governing equations are solved in their integral form. However, here the governing equations are given in their differential form. For compressible flows the conservation of mass is governed by the relation Op

0

57 + ~ ( ~ u J ) = 0

(1)

The transport equation of momentum is given by OpUi ot

0 -~

OP

0

~ ( p u j u ~ ) . . . . o~ ~ ~

,

(2~s~j - p~j~)

(2)

where S~j is the compressible version of the mean strain rate tensor (see appendix). Furthermore, the transport equation for the total energy is written as O

O~F_:

0 (~EUj) =

0 (PUj)+

(

+~-~zj ( g + ~ T ) ~ z 3 +

'U~

#+~-KK

U~)

(3)

609 1

where E = e + 7UjUj + K , e is the internal energy and K the turbulence kinetic energy. The relations between the state variables are given by the calorically perfect gas relations (P = pRT, e = R T / ( ' y - 1)). The diffusivity parameters are given by ~ = #cp/Pr and I~T --- ]-tTcp/rrT. The K - c J

platform, which is used in all computations, is for compressible flows given by

D-=(pK)

=

V-

9*p~K +

~ + --

(4)

(7 K

D-7(n~)

: ~~-

9p~ ~ +

~ + --~ ~

(5)

where #T is the eddy-diffusivity parameter and is model dependent, see appendix. The production of the turbulence kinetic energy is defined as

(6)

og~

~9 = --pUiUj OqXj

The evaluation of puiuj is dependent on the turbulence model, see appendix for further details on this.

GENERAL

ASPECTS

OF T H E E D G E - C O D E

Edge is a flow solver for unstructured grids of arbitrary elements. Edge solves the compressible Reynolds Averaged Navier-Stokes equations in either a steady frame of reference or in a frame with system rotation. Turbulence can be modelled with eddy viscosity models, explicit algebraic Reynolds stress models or full differential Reynolds stress models. The solver is based on an edge-based formulation and uses a node-centered finite-volume technique to solve the governing equations. The control volumes are non-overlapping and are formed by a dual grid obtained from the control surfaces for each edge. The governing equations are integrated explicitly towards steady state with Runge-Kutta time integration. The convergence is accelerated with agglomeration multigrid and implicit residual smoothing. Edge contains different spatial discretizations for the mean flow as well as the turbulence, different gas models, steady state and time accurate time integration, low speed preconditioning etc. For a more thoroughly description of the EDGE-code see Eliasson (2002).

THREE-ELEMENT

LANDING

CONFIGURATION

IN 2D

The 2D flow test case is a wing section of a typical three element landing configuration. Two grids have been used, one coarse grid with 35,000 nodes and one fine grid with 340,000 nodes. Transition to turbulence is prescribed at given locations on both suction and pressure sides at all three elements. Two sets of transition points have been used. The default setting from the definition used in GARTEUR Thibert (1993) and a modified setting further upstream. With the coarse grid and both transition settings, a polar has been computed with the GWJ-EARSM, the WJ-EARSM and the standard Wilcox K - a~-model. The differences between the WJ-EARSM and the GWJ-EARSM within the same transition setting are very small. Therefore, the WJEARSM with the alternative transition and the GWJ-EARSM with the original setting have been omitted. One computation at the lowest angle using the finer grid has been performed with the GWJEARSM. Efforts have also been made with the finer grid at two angles close to maximum lift

610

CL ~ Sr

9> " ~ J

0

0

0

~ '~I

~I

\ o

y, \ \ \ \ \ \

i

J

J

oz

Figure 1: Computed lift polar. Coarse grid: GWJ-EARSM ( - o ) , standard Wilcox K - co (... o) (alternative transition) and WJ-EARSM ( - - x) (GARTEUR transition). Fine grid with GARTEUR transition: GWJ-EARSM (.), WJ-EARSM ( - - []) and DRSM ( - . x).

but convergence was not achieved. The reason for this is not yet understood. As a comparison, the predictions of earlier computations using the WJ-EARSM and DRSM and the finer grid are included in the results. Comparisons are also made with experimental data. This case has previously been numerically studied by e.g. Lindblad and de Cock (1999) and Eliasson (2003) and the experimental pressure distribution was obtained within GARTEUR, see Thibert (1993). In figure 1, the coefficient of lift, Co, is shown. Before maximum lift the different models are in fair agreement with each other, although predicting a lift that is too high compared to the experiments. Also, the comparison between the coarse and fine grids is good at the lowest angle of attack. In conjunction with the GARTEUR transition setting, both EARSMs predict a premature stall which is indicated by the drop in CL. With the new transition setting, however, the stall behaviour is in good agreement with the experiments. This is further discussed below. For the highest angles of attack, it was not possible to reach a steady-state solution with the WJ-EARSM and the DRSM using the fine grid. This point was excluded from the figure. The K - co-model shows very little sensitivity to the choice of transition setting and therefore only the alternative setting is shown in figure 1. Furthermore, the standard K - co-model fails to predict the stall behaviour observed in the experiments. The overall pressure distribution, Cp, for the lowest angle of attack using the fine grid is demonstrated in figure 2. All model predictions are very similar and give too low pressure on the suction side. This is consistent with the predictions of a too high CL discussed above. The largest discrepancies in Cp can be seen in the suction peak region close to the leading edge on the main wing, see figure 2b. Here, the GWJ-EARSM shows a slight improvement over the WJ-EARSM. The DRSM is somewhat better though. In the most critical region, the separation area of the flap, none of the models are in good agreement with the experiments, figure 2c. Velocity profiles versus wall normal distance are shown in figure 3. The model predictions agree well with experiments. The GWJ-EARSM shows a small overall improvement over the WJEARSM, but this is close to negligible. The abrupt stall behaviour that is seen with the EARSMs and the default transition settings is associated with a leading edge separation on the slat. For the higher angles of attack, this separation occurs upstream of the prescribed transition point which causes a massive laminar

61]

-Cp

i

%

x/c .

b)

-cp

-Cp

.

.

/

o

i

o

o

o

.

c)

.t-..\

/

.

o

o

o

o

o

~o

o

,:.

o

o

o

o

o ~.~ i

~/ '~

i

o.

z/c

Figure 2: Computed Cp distribution for the lowest angle of attack a), the suction peak area close to the leading edge of the main wing, b) and separation region on the flap, c). GWJ-EARSM ( - ) , WJ-EARSM ( - - ) DRSM ( . . . ) a n d experiments (o)

separation. In reality, the laminar separation is strongly unstable and would generate turbulence transition. Thus, a more realistic solution is obtained by moving the transition point closer to the laminar separation. This is done for the alternative setting of the transition point and can be seen as a more smooth stall behaviour in figure 1. The principal behaviour around maximum lift is well predicted, with a shift in level, however. The rapid drop in lift at the highest angle of attack corresponds well with the experimental results. It is interesting to note that, in comparison to Eliasson (2003), the pressure distribution at the highest, post-stall, angle of attack is well predicted by both EARSMs in conjunction with either transition setting. Due to more or less overlappping model predictions only the GWJ-EARSM with the alternative transition settings is shown, see figure 4 .

THREE-ELEMENT

T A K E OFF C O N F I G U R A T I O N

IN 3D

In 3D mean flows, the algebraic expressions for the GWJ-EARSM become significantly more complex. The computation of a 3D case using the full 3D form of the model is, thus, an important step for the validation of the implementation and the investigation of the model behaviour in complex 3D mean flows. The test case considered here is KH3Y, a wind tunnel model of a transport aircraft high-lift configuration at take off. This has been experimentally and numerically evaluated in the European project EUROLIFT, Thiede (2001). The geometry considered is with the flap

612

3

i .

! .

.

i

1

.

1 g

1

3

2

4

3

position

~,

5

3

Ca

6

Figure 3" Profiles of V/p/pooU/U~ against wall-normal distance. GWJ-EARSM (-), WJ-EARSM ( - - ) , DRSM (--) and experiments (o). The numbers indicate the position of the profiles.

-Cp o

o o Go

g o

o

o

'

'

z/c

'

'

'

Figure 4: Computed Cp distribution with alternative transition setting at the highest post-stall angle of attack for the GWJ-EARSM (-), experiments (o)

613

Figure 5: The KH3Y geometry and grid.

Figure 6: The lift polar, a) and drag, b) for GWJ-EARSM (o) compared with WJ-EARSM ( - - [ n ) and experiments (o).

and slat covering the full span without flap track fairings, see figure 5. The grid is unstructured with a prismatic near-wall region and consists of about three million nodes. One angle of attack close to maximum lift has been computed using the GWJ-EARSM and compared with previous computations by Eliasson (2003) using the WJ-EARSM and experimental data. The convergence rate and computational effort to compute this case using GWJ-EARSM is comparable with the baseline WJ-EARSM. A minor increase in lift is shown in figure 6 from the GWJ-EARSM compared to the baseline WJEARSM, but the difference is almost unsignificant. Also the computed wall pressure distribution, figure 7, is almost unaffected by the extension of the modelling in the GWJ-EARSM.

614

a)

ce

c)

b)

[ ~/~

x/~

x/~

Figure 7: The computed Cp at three spanwise locations, close to fuselage, a), at mid span, b) and close to wing tip, c). GWJ-EARSM ( - ) compared with WJ-EARSM ( - - ) and experiments (o).

CONCLUSIONS/DISCUSSION The GWJ-EARSM consists of elements from nonlinear Reynolds stress modelling and, therefore, becomes rather algebraically complex, especially in 3D mean flows. Despite this, the model has been successfully implemented in a general purpose industrial CFD code and has been tested on complex industrial flows. In general, the convergence behaviour and computational effort is comparable with the baseline WJ-EARSM. The importance of a successful implementation and testing should not be underestimated, since this should be regarded as one of the major goals of all engineering types of turbulence models. For the cases studied here, the differences between the models are relatively small. For the lower angles of attack, the GWJ-EARSM shows some minor improvements compared to the baseline WJ-EARSM. More significant is the differences in the predictions due to transition prediction/setting. In conjunction with the EARSMs, the original transition setting gives a premature separtion implying stall at a too low angle of attack. With the alternative transition setting, where the slat transition is moved upstream closer to the laminar separation, the behaviour around maximum lift is significantly improved and the experimental behaviour is, at least, qualitatively captured. This illustrates the importance of the transition prediction and the interaction with the flow solver. Transition prediction and the coupling to the flow solver are an important part of the European projects HiAer and EUROLIFT.

ACKNOWLEDGEMENT This work has been carried out within the HiAer project (High Level Modelling of High Lift Aerodynamics). The HiAer project is a collaboration between DLR, ONERA, KTH, HUT, TUB, Alenia, Airbus-D, QinetiQ and FOI. The project is managed by FOI and is partly funded by the European Union (Project Ref: G4RD-CT-2001-00448). BAE-system is acknowledged for kindly providing the 59% section data of a representative civil aircraft wing for this study and Airbus-D for kindly providing the KH3Y 3D take-off configuration and data.

APPENDIX. REYNOLDS STRESS EVALUATION For the GWJ-EARSM (and EARSMs in EDGE in general) the Reynolds stress anisotropy, aij =

615

puiuj/pK-

26~j/3, is divided into two parts, one that is proportional to Sij and one containing the rest. This means that for the 2D mean flow case, ,ouiuj is evaluated from the relation 2 K

=

~- (e~)

- 9# s j +

(7)

The effective turbulent viscosity and the extra anisotropy is given by

#T = - 89

a (ex) = f l 4 ( S f l - a S )

where fll

~

--

A1 N * N.2_2HF~

f14 - -

N .2

(8)

(9)

A,

-2IIn

where S and the rotation rate tensor, f~, are defined by

f2ij = 7 \ ox~

Sij = rS~j = 5 \oxj + o~, - 5 oxk

o~, ]

(10)

and the invariants are IIs - SijSji and IIa =-- ~ij~ji. Note that, due to compressibility, the definition of S~j is necessary in order to make it traceless. N is the solution to a third order polynomial equation and is given by N* = /

_4~ + (P1 + V/~2) 1/a + sign(Pz - x/~2)lP1 - v/-~2l1/a P2 >_ 0

(11) + 2(ff

-

cos

s

rccos

and

P1 =

\-~+

-

< 0

2) IIa

A~

(12)

P2

m~ = ma - c ( x / ~ s - V / - I I a )

(14)

The parameter values are {c = 0.56, A1 = 1.2, A~ = 0, A3 = 1.8, A4 = 2.25}. For the DRSM, the equation for the turbulence kinetic energy, (4), is replaced by transport equation for the Reynolds stresses

Dpuiuy Dt = Piy - peij + pl-Iij -F Dij

(15)

where Pij, eij, Hid and 7?ij represent the production, dissipation, pressure-strain and diffusion respectively. The pressure strain and dissipation rate anisotropy, eij = e i j / e - 26ij/3, can be lumped together and the model used in the present work reads

I-[i__.~jc e ij =

- -~I ( c ~ -F C I ~ ) a ij -F C 2S ij C3(

+-~-

2

a,k&j + &kakj -- 5akz&k&j

)C4(aikf~kj_~ikakj -- -~-

)

(16)

The model parameters corresponds to the Curvature corrected WJ-EARSM, see, Wallin and Johansson (2002), C ~ = 4.6, 611 = 1.24, C2 = 0.47, Ca = 2 and C4 = 0.56.

616

References Daly, B.J. and Harlow, F.H., (1970), "Transport equations in turbulence", Physics of Fluids, 13, 2634-2649. Durbin, P.A., (1993), "A Reynolds-Stress Model for Near-Wall Turbulence", Journal of Fluid Mechanics, 249, 465-498. Eliasson, P., (2002), "EDGE, a Navier-Stokes solver for unstructured grids", Proc. to Finite Volumes for Complex Applications III, ISBN 1 9039 9634 1,527-534. Eliasson, P., (2003), "CFD improvements for high lift flows in the European project EUROLIFT.", AIAA 2003-3795. Grundestam, O., Wallin S., and Johansson A.V., (2004),"An explicit algebraic Reynolds stress model based on a nonlinear pressure strain rate model", Submitted to International Journal of Heat and Fluid Flow. Lindblad, I.A.A. and de Cock, K.M.J., (1999), CFD Prediction of Maximum Lift of a 2D High Lift Configuration, AIAA 99-3180. SjSgren, T. and Johansson, A.V., (2000), "Development and calibration of algebraic nonlinear models for terms in the Reynolds stress transport equations", Physics of Fluids, 12:6, 15541572. Speziale, C.G., Sarkar, S. andGatski, T.B., (1991), "Modelling the pressure-strain correlation of turbulence : an invariant dynamical systems approach", J. Fluid Mech., 227, 245-272. Thibert, J. J., (1993), The GARTEUR High Lift Research Programme, AGARD CP-515, High Lift Aerodynamics, Paper 16. Thiede, P., (2001) EUROLIFT- Advanced High Lift Aerodynamics for Transport Aircraft, AIR & SPACE EUROPE, Vol. 3, No. 3, pp. 1-4. Wallin, S. and Johansson, A.V., (2000), "An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows", Journal of Fluid Mechanics, 403, 89-132. Wallin, S. and Johansson, A.V., (2002), "Modelling streamline curvature effects in explicit algebraic Reynolds stress turbulence models", International Journal of Heat and Fluid Flow, 23, 721-730. Wilcox, D.C, (1988), "Reassessment of the scale-determining equation for advanced turbulence models.", AIAA J., 26, 1299-1310.

Engineering TurbulenceModelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

617

T U R B U L E N C E M O D E L L I N G IN A P P L I C A T I O N TO T H E V O R T E X S H E D D I N G OF S T A L L E D A I R F O I L S C. Mockett, U. Bunge and F. Thiele Faculty of Mechanical Engineering and Transport Systems, Technische Universit/~t Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany

ABSTRACT A thorough investigation is conducted into some open questions remaining in the Detached-Eddy Simulation (DES) technique. Principle among these are the questions of grid fineness, and the extent of dependency on the background turbulence model inherent in DES. The investigation takes the case of the NACA0012 airfoil beyond stall, for which a large number of grid fineness and turbulence model combinations are computed. In order to investigate the influence of high resolution in the LES zone, a new highly-refined grid is constructed. The results of the investigation demonstrate a strong deterioration of the results due to the uneven grid fineness distribution, and the reasons for this in relation to a general grid sensitivity present in DES are discussed. However, the physical character of the DES calculations remains encouraging, and motivated by this, an analysis of the phenomenon of stochastic weak vortex shedding cycles is undertaken. KEYWORDS DES, grid sensitivity, bluff body, NACA0012, Vortex shedding

INTRODUCTION Although in recent years many new hybrid turbulence modelling approaches have emerged, there is still an incomplete understanding and a lack of consensus regarding the entire range of simulation types between those based on the Reynolds-Averaged Navier-Stokes (RANS) equations and direct numerical simulations, Travin et al. (2004). Among some of the methods which have nevertheless reached a high level of maturity is Detached-Eddy Simulation (DES), for which further application guidelines have been issued, Travin et al. (2002). Some areas of non-clarity remain however, not least of which is the question of grid resolution for the turbulent zones outside of the RANS boundary layer. The suggestion of Travin et al. (2002) is that the grid should be of sufficient resolution for a LES everywhere outside of the near-wall RANS zone, and this strict interpretation of the DES philosophy has since been confirmed by Strelets (2004). A second theme, which interacts with the former, is the influence of the choice of RANS background model. By

618 consideration of this strict interpretation of the DES approach, the model dependency of DES should hypothetically be very slight, however as experience contradicting this was gained, it was decided to launch a careful study of these two issues. In this paper, the isothermal and incompressible flow around a NACA0012 airfoil at 60 ~ angle of attack and Re = 100000 is computed and compared with experimental data. A new level of interpretation has been made available by later experiments including for the first time a frequency analysis of the aerodynamic force coefficients on the similar NACA0021 airfoil beyond stall at Re = 270000, Swalwell et al. (2003). This is acknowledged as an ideal test case for a DES due to the massive and mainly geometry-induced, unsteady separation, Strelets (2001). Despite this, the results corroborate the above-cited statement concerning the incomplete level of understanding. However, the improvements in results through the DES modification are obvious and interesting insight into the physics of unsteady airfoil separation is obtained using the DES technique.

DESCRIPTION

OF M E T H O D

The flow is computed numerically using an in-house finite-volume based code solving either the unsteady Reynolds-averaged or spatially filtered Navier-Stokes equations in case of a RANS or Large-eddy simulation (LES), respectively. The procedure is implicit and of second order accuracy in space and time. All scalar quantities as well as the Cartesian components of tensorial quantities are stored in the cell centers of arbitrarily curvilinear, semi-structured grids that can capture complex geometries and allow for local refinement. Diffusive terms are approximated with central schemes, whereas convective terms can be treated with central or upwind-biased limited schemes of higher order, Xue (1998). A hybrid blending of both approaches for a Detached-Eddy simulation is possible, as suggested by Travin et al. (2002). The linearized equations are solved sequentially and the pressure is iterated to convergence using a pressure-correction scheme of the SIMPLE type that assures mass conservation as the pressure equation is derived from the continuity equation, Karki & Patankar (1989). A generalized Rhie & Chow interpolation is used to avoid an odd-even decoupling of pressure, velocity- and Reynolds-stress components, Obi et al. (1991).

Turbulence Treatment

Turbulence is handled using three RANS turbulence models of different degrees of complexity. The simplest is a modification of the SA model, the one-equation Strain-Adaptive Linear SpalartAllmaras model (SALSA), Rung et al. (2003), which is used without the trip function present in the original model. The linear local realizable (LLR) k-c~ Model, Rung &: Thiele (1996), as the second model is a local linear two-parameter model derived from realizability and non-equilibrium turbulence constraints. The compact explicit algebraic stress model (CEASM) used, Liibcke et al. (2002), is based on the LL k-e model as a background model. The Reynolds stresses are computed by an expression that is derived by projection of the algebraic stress model into a 5-generator integrity basis. All three models are also used for DES, where the constant CDESis calibrated by computing the decay of isotropic turbulence as described by Bunge et al. (2003). To achieve a DES, the turbulence length scale in the model is replaced by the DES grid length scale. For the SALSA model this is the same procedure as for the standard DES based on the SA model in which the wall-distance is replaced. For the two-equation model the turbulence length scale in the dissipation term of the turbulence-energy transport equation is replaced. Of the two length scales present in the background model of the CEASM, the wall-normal distance is left unchanged. Underlying the

619 choice of turbulence models, a hybrid numerical scheme for the convective terms is employed and plays an important role. The blending function of this is implemented as described by Travin et al. (2002).

Grids All grids were generated such that y+ ~ 1 is yielded in the first wall-normal control volume center. The first three grids were provided and used by Strelets (2001), and are single block Ogrids of 215000, 286000 and 343000 cells. To provide a further level of refinement, a new grid was constructed from O, C and H blocks, with a total of 1.46 million volumes, see Figure 1. This block topology enables a focussing of the cells in the near wake "focus region" as defined by Spalart (2001). The coarse, medium and fine O-grids are designated c, m and f respectively, and the very fine grid vf.

Figure 1: The very fine wake grid; entire topology (left), and three-dimensional close-up (right).

On the c, m and f grids, a non-dimensional time-step of 0.025c/U~ was employed. To correspond to the much smaller cell volumes of the vf grid a time-step of 0.0125c/U~ was used. PRESENTATION

AND DISCUSSION OF RESULTS

Improvement of DES over U R A N S Unsteady URANS and DES calculations using the grids c, m and f were conducted using each of the background turbulence models described above. In addition, DES calculations based on the SALSA and LLR models were conducted for the vf grid. For each of these calculations, the lift and drag coefficients and Strouhal number are summarised in Table 1, from which it is clear that for the c, m and f grids the DES delivers considerable improvements in predictive accuracy on all fronts. The deterioration of the results for the vf grid will be discussed in detail later on. There is a qualitative behavioural difference which underlies the improvement of DES over URANS. Figure 2 shows experimental time traces of the integral force coefficients compared to those of LLR-based DES and URANS. Despite the different profile thickness and Reynolds number, a qualitative comparison of these is justifiable. The experimental curve exhibits a highly stochastic nature, with randomly occurring low frequency modulation and areas of weak and strong activity. Furthermore, there is no standard topology to each period. The DES reproduces these qualities perfectly, albeit with slightly higher amplitudes. The URANS on the other hand delivers a far less

620 TABLE 1 DES AND U R A N S RESULTS IN C O M P A R I S O N TO E X P E R I M E N T S F O R ALL M O D E L S ,

EXPERIMENTAL FORCE COEFFICIENT DATA AS USED BY STRELETS (2001), STROUHAL NUMBER FROM EXPERIMENTS OF SWALWELL ET nL. (2003). Model I]

II

Experiments URANS

m

f c m

f

vf

II

LLR k-w

0.20

1.11 1.08 1.09 0.97 0.98 0.92

c

DES

CEASM

1.92 1.83 1.86 1.63 1.64 1.56

-

-

0.13 0.13 0.13 0.17 0.17 0.18

SALSA

II

i0.20 1.27 1.19 1.18 0.93 0.94 0.94 1.06

2.16 1.99 1.98 1.57 1.58 1.58 1.82

0.20

0.14 0.16 0.16 0.18 0.18 0.20 0.19

1.21 1.12 1.11 0.91 0.97 0.95 1.13

2.11 1.93 1.91 1.53 1.64 1.59 1.96

0.13 0.12 0.15 0.17 0.16 0.18 0.19

chaotic signal and most periods have a similar topology not evident in the experimental traces. Less dissipative URANS models manage to reproduce the weak shedding cycles, although the occurrence of these is much too seldom. This observation has also been noted by Travin et al. (2004) for the rotation-corrected SARC variant of the SA model. URANS

DES

Experiment

o"

'0i 0

50

100

150

200

0

50 100 150 N o n - d i m e n s i o n a l time

200

0

50

100

150

200

Figure 2" Time traces of CL and Co for LLR-URANS (left), LLR-DES (mid) on the f grid, and experimental data of Swalwell et al. (2003) (right).

Influence of different background R A N S models; LES grid capability To provide a concrete measure of the extent of model variation apparent (independent of the predictive accuracy), the "normalised mean relative difference" (NMRD) between the numerical results obtained for each grid is calculated as follows:

NMRDx = (I(X~~--XLL~)I+I(X~AL~A--XcEASM)I+I(XLLR--XcE~SM)I} /

3x~p

,

where X is CL, Co, or St. The values of NMRD which emerge are shown in Table 2. From these figures it can clearly be seen that a dependency on the background model still exists in DES, albeit to a lesser extent than with URANS. Furthermore, in both cases the level of dependence decreases with increasing grid refinement for the force coefficients. Examination of Table 1 shows that the strengths and weaknesses of the URANS models are still reflected in their performance as DES models.

621 TABLE 2 MODEL DEPENDENCE FOR DIFFERENT GRID RESOLUTIONS: AVERAGE RELATIVE DIFFERENCE NORMALISED WITH CORRESPONDING EXPERIMENTAL VALUE.

I] NMRD -&, c 0.116 m 0.080

f

0.065

URANS NMRD -O. 0.097 0.065 0.048

~

NMRD st I[ NMRD-Or, 0.033 0.043 0.133 0.029 0.100 0.022

DES NMR D -8, 0.040 0.024 0.012

NMR D st 0.033 0.067 0.067

As mentioned in the introduction, there ought in principle to be very little difference between the DES results for the different background models. The reason for this is twofold; on the one hand it is specificied that only the thin wall boundary layer is calculated in RANS mode, with the LES-mode of the models being computed elsewhere. On the other hand, it is assumed in this context that the grid is sufficiently resolved for an LES in the non-boundary layer regions. The validity of each of these aspects will be examined in turn.

Figure 3: Instantaneous visualisation of extent of LES zone for three grids, LLR-DES.

The shaded region of Figure 3 shows where the ratio LRANS/LLEs> 1 for the c, f and vf grids, i.e. the zones of LES-mode operation of the DES. It is immediately apparent that the LES-mode regions occupy only a small portion of the computational domain. Even in the vf grid, pockets of RANS-mode operation appear relatively close to the profile. As it depends on the local turbulence length scale, the form of the LES-mode zone is highly three-dimensional and unsteady. In addition to the lower ut levels delivered by the LES modelling in these areas, a lower numerical dissipation also comes about due to the hybrid numerical scheme. The central-based convection areas of this follow relatively closely the LES zones in the wake region. Because of this locally-defined length scale, the two-equation model enables an easy visualisation of the LES zones. Upon initial consideration, it might appear as if the use of the wall distance, dw in one-equation implementations would cause a fundamental difference in the interface behaviour. However, this very formal analysis would neglect the fact that the one-equation background model also tends to pure RANS in regions of coarse mesh far from the wall. This is because the vt dissipation term scales with 1/A 2. The demonstrated behaviour of switching back to RANS mode is therefore inherent to DES regardless of the chosen background RANS model. It should be clarified that this behaviour does not invalidate the definition of DES as a "non-zonal" hybrid approach: In each of the LES-mode and RANS-mode zones, the same model is used, although with different length-scale definitions. It is proposed that these RANS-mode zones are the principle mechanism by which the background turbulence model can exert influence in DES.

622 Turning to the LES-capability of the grids, much consideration was given to the criterion or analysis method employed to evaluate this. There is unfortunately no clear and straightforward method for such an evaluation, however that considered to be the most reliable is the spectral analysis of the resolved unsteady velocity field. Using this method, it can be estimated whether the filter cut-off wavenumber lies within the inertial subrange.

Figure 4: v~v' spectra at four downstream points along the wake centreline, LLR-DES.

Example spectra of this analysis are shown in Figure 4, where the vertical v'v ~ correlation versus Strouhal number is obtained from a PSD of the v velocity fluctuations at four points at the x distances given in the figure. It was concluded that the LES resolution capability of the c grid was questionable. However, the physical character of the DES solution on this grid was still comparable with the finer grids.

D i s c u s s i o n o f the v e r y f i n e grid results Attention will now be turned to the poor quality of the vf grid results, namely the gross overprediction of the integral force coefficients for both LLR and SALSA models. The source of the overprediction is a much stronger suction on the upper surface of the airfoil, as shown in the left-hand side of Figure 5, which shows the chordwise surface pressure coefficient distribution for the LLR-DES calculations. Closer inspection indicates that the leading-edge separation point lies further forward on the vf grid.

Figure 5: Averaged chordwise C v distributions left, and averaged streamlines right, LLR-DES.

The cause of the higher suction as well as the effect of the earlier separation can be investigated by consideration of the averaged flow-field in the near wake. The streamline visualisations in the

623 right-hand side of Figure 5 show similar flow patterns for the first two grids, but some important differences emerge in the vf grid field. Both vortices are clearly larger, and the trailing-edge vortex is much closer to the profile. Furthermore, the entire recirculation zone is longer, as indicated by the displacement of the saddle-point further aft. All of these differences combine to increase the suction on the upper surface, and thereby the CL and Co. The detailed streamline traces in the leading-edge region illustrate the earlier separation point of the v.f grid. Notable effects of this are the displacement of the wall-bound secondary vortex further towards the leading-edge, and a broadening of the entire wake width. The earlier separation point could therefore in principle explain the higher force coefficient values, as the larger wake width is in turn associated with a higher rear-face suction, Roshko (1955). Both the earlier separation point and the change to the vortex topology are also to be seen in the vf SALSA-DES calculations. When considering the possible modelling causes of these differences, many different hypotheses are feasible. Beginning with the leading-edge separation, the curve definition in the v.f grid is very highly resolved compared to the coarser grids. Unfortunately, the actual separation point is not reported in the experimental data. It may on the one hand be the case that the vf solution predicts the separation too early because of the higher resolution, and corresponding lower modelled viscosity. On the other hand, it could be that the vf solution represents the true separation point, whereas that of the coarser grids is artificially fixed to a later position by the coarse leading-edge curve definition.

Figure 6: Cell volume distribution for the two grid topologies (distortions at the block interfaces are a visualisation artefact).

Bearing in mind that the suction trend is also observable between the c and f grids for which the separation point is identical, it appears as if the grid resolution is also an important factor. If this is the case, then the spatial distribution of this resolution is also of significance. Compared to the O-topology grids, the vf grid has a much less even fineness distribution, as can be seen in the cell-volume plots of Figure 6. Most notably, the region behind the trailing edge is highly uneven, containing an area of localised coarseness. Additionally, a sudden spring in the x-resolution occurs at a vertical line approximately 1.5c behind the airfoil. This influences the location of the LES zones as well as the local numerical dissipation. It is suggested that the anomaly in the trailing-edge region lies behind the much closer position of the corresponding vortex (while the leading-edge vortex location remains unaltered). This would be hard to explain in terms of the earlier leading-edge separation.

624

Investigation of physical phenomena Leaving aside the problems encountered with the highly-refined grid, the qualitative jump in accuracy observed in the physical character of the DES calculations over URANS (see Figure 2) is highly encouraging. This, combined with a lack of experimental wake flow visualisations motivates the use of DES as a tool for investigation of the physical phenomena occurring in the wake. The principle phenomenon of interest, and that which characterises the qualitative difference between URANS and DES solutions, is the sporadic occurrence of low frequency amplitude modulations or "weak shedding cycles". Similar events have been frequently documented in the experimental force time series of circular cylinders at high-subcritical Reynolds numbers. In such cases, this has been explained by a switch between laminar and turbulent separation of the shear layer. This cannot however be the case for such sharp-edged bodies as the considered profile, and it is unlikely that the RANS-modelled boundary layers would be able to reproduce this effect. The main hypothesis, see for example Travin et al. (2004), is that the weak shedding cycles arise due to three-dimensional shedding effects, whereby the shedding occurs in a cell-wise manner of varying phase across the span and not uniformly as at lower Reynolds numbers. ,

2 O

.

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

1,5 1

0,5

50

100

150 Non-dimensional

200 time

250

300

Figure 7: Above: Detail of C L time trace, showing two points of interest, below: pressure contours and isosurfaces for the corresponding left and right-hand points of interest. LLR-DES, grid f.

A section of the CL time series is shown in Figure 7, upon which crosses mark locations chosen due to their contrasting character. Underneath this on the left and right, visualisations of the flow are shown for the left and right-hand points, respectively. The visualisations show side views of the pressure contours at a slice parallel to the span, superimposed with isosurfaces of the pressure at three of these contour levels across the entire span. The intention of this is to display the level of three-dimensionality present in the flow. Looking first at the pressure levels in the far wake, a difference in character can clearly be seen; that of the weak cycle having much lower pressure magnitudes and a much less regular structure. Turning attention to the area immediately behind the airfoil, a striking difference is apparent; on the left-hand side, a vortex with very strong suction is seen, which the isosurfaces reveal to be almost entirely two-dimensional, whereas in the right-hand picture, a region of chaotically arranged small-scale structures and very weak suction is present. Paradoxically, visualisations of the A2 criterion show the shed shear layer from the

625 leading and trailing edges to be similarly two-dimensional. This implies that the three-dimensional character is not due to out-of-phase shedding along the span, rather a complete breakdown of the organised Von Ks163 shedding to a disorganised complex turbulent wake formation. This runs contrary to the previously-stated hypothesis.

CONCLUSIONS AND OUTLOOK In the current work, important findings affecting the practical implementation of the DEStechnique have emerged. Firstly, it has been shown that, contrary to expectation, a sensitivity to the choice of background model exists in DES, and that the level of this sensitivity is higher for coarser grid resolutions. A feasible mechanism, based on the occurrence of RANS-mode zones outside of the attached boundary layer, has been proposed as an explanation for this. The evidence so-far accumulated does not however suggest that more elaborate RANS models make better DES background models. A second finding of the presented work is that the level to which the grid outside the boundary layer satisfies the requirements of LES resolution is of a lesser importance. From consideration of the results of the initial three O-grids (c, rn and f), it might appear as if grid refinement delivers higher quality results in DES. Unfortunately, from the present study it is not possible to say whether further refinement would result in an over-prediction of the force coefficients. This is because the vf grid has an entirely different topology and fineness distribution, and as such does not constitute the next step in a systematic grid refinement study. It has been shown that the solution is highly sensitive to the fineness distribution, a finding which corresponds with experience from LES. In order to establish whether the over-prediction of the force coefficients is due to the grid refinement in itself, or rather due its uneven distribution (or a mixture of the two), a new grid has been constructed. This takes as its basis the f grid topology and distribution, whereby the number of points is doubled in each index direction. Calculations on this grid are underway. The demonstrated sensitivity of DES to the grid fineness distribution has important implications for the generation of DES grids for complex geometries of practical relevance. It may for example not be possible to ensure a similar distribution of fineness across the entire wake of a bluff body. It is therefore hoped that through the continuing investigations, the mechanism behind this sensitivity can be established. It may then prove possible to develop a method to reduce this sensitivity, and thereby improve the robustness of the DES technique in industrial applications. Nonetheless, the encouragingly high physical quality of DES results for bluff body flows has been demonstrated, and an example physical investigation using DES has been presented.

ACKNOWLEDGEMENTS The authors would like to express their particular gratitude to K. Swalwell of Monash University for the very informative discussions and the very helpful provision of additional experimental data not included in the referenced paper. This work has been partially funded by the E.C. FLOMANIA and DESider projects: The FLOMANIA project (Flow Physics M o d e l l i n g - An Integrated Approach) is a collaboration between Alenia, ANSYS-CFX, Bombardier, Dassault, EADS-CASA, EADS-Military Aircraft, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and St. Petersburg State University. The project is funded by the European Union and

626 administrated by the CEC, Research Directorate-General, Growth Programme, under Contract No. G4RD-CT2001-00613. The DESider project (Detached Eddy Simulation for Industrial Aerodynamics) is a collaboration between Alenia, ANSYS-CFX, Chalmers University, CNRS-Lille, Dassault, DLR, EADS Military Aircraft, EUROCOPTER Germany, EDF, FOI-FFA, IMFT, Imperial College London, NLR, NTS, NUMECA, ONERA, TU Berlin, and UMIST. The project is funded by the European Community represented by the CEC, Research Directorate-General, in the 6th Framework Programme, under Contract No. AST3-CT-2003-502842. Part of the computations for this project were carried out on the IBM pSeries 690 supercomputer of the North German Supercomputing Complex - HLRN (Norddeutscher Verbund fiir Hoch- und HSchstleistungsrechnen ) . REFERENCES

Bunge, U., Mockett, C., Thiele, F. (2003) Calibration of Different Models in the Context of Detached-Eddy Simulation. A G STAB Mitteilungen, DGLR, GSttingen. Karki, K.C. & Patankar, S.V. (1989) Pressure based calculation procedure for viscous flows at all speeds. AIAA Journal 27, 1167-1174. Liibcke, H., Rung, T., and Thiele, F (2002) Prediction of the Spreading Mechanism of 3D Turbulent Wall Jets with Explicit Reynolds-Stress Closures. In Engineering Turbulence Modelling and Experiments, 5, 127-145, Elesevier, Amsterdam. Obi, S., Perid, M., Scheurer, M. (i991) Second moment calculation procedure for turbulent flows with collocated variable arrangement. AIAA Journal 29, 585-590. Roshko, A. (1955) On the wake and drag of bluff bodies. Journal of the Aeronautical Sciences

61(1996), 99-112.

Rung, T., Bunge, U., Schatz, M., Thiele, F. (2003) Restatement of the Spalart-Allmaras EddyViscosity Model in Strain-Adaptive Formulation. AIAA Journal 41(7), 1396-1399. Rung, T.,Thiele, F. (1996) Computational Modelling of Complex Boundary-Layer Flows. Proc. 9th Int. Syrup. on Transport Phenomena in Thermal-Fluid Engineering, Singapore. Spalart, P.R. (2001.) Young-Person's Guide to Detached-Eddy Simulation Grids. NASA contractor report NASA/CR-2001-211032. Strelets, M. (2001) Detached Eddy Simulation of Massively Separated Flows. AIAA Paper 20010879. Strelets, M. (2004) Private communication in the context of the DESider project. Swalwell, K.E., Sheridan, J. and Melbourne W.H. (2003) Frequency Analysis of Surface Pressure on an Airfoil after Stall. Presented at the 21st AIAA Applied Aerodynamics Conference, AIAA Paper 2003-3416. Travin, A., Shur, M., Spalart P.R., Strelets, M. (2004) On URANS Solutions with LES-like Behaviour. Congress on Computational Methods in Applied Sciences and Engineering. Travin, A., Shur, M., Strelets, M., Spalart P.R. (2002) Physical and numerical upgrades in the Detached-Eddy Simulation of complex turbulent flows. Fluid Mechanics and its Applications 65, 239-254. Advances in LES of Complex Flows, R. Friederich and W. Rodi (editors). Xue, L. (1998) Entwicklung eines effizienten parallelen LSsungsalgorithrnus zur dreidimensionalen Simulation komplezer turbulenter StrSrnugen. PhD thesis, TU-Berlin.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

627

THE COMPUTATIONAL MODELLING OF WING-TIP VORTICES AND THEIR NEAR-FIELD DECAY T.J. Craft, B.E. Launder and C.M.E. Robinson School of Mechanical, Aerospace & Civil Engineering The University of Manchester, Manchester, M60 1QD, UK

ABSTRACT Computations have been made to explore the ability to calculate the roll-up of a wing-tip vortex and its near-field development. The case studied is a NACA0012 half-wing with rounded wing-tip, at angle of attack, cx = 10 ~ Experiments by Chow et al. (1997) provide a detailed record of the mean and turbulence data in the near downstream region, up to x/c = 0.678 chord lengths from the trailing edge. They show that the vortex generated at the wing-tip has a strongly accelerated core with a peak velocity of Uc/U,, = 1.77 which decays slowly downstream. Initial calculations with a linear k - e model showed that the roll-up process at the wing-tip was calculated readily but it was much more difficult to predict the correct acceleration and maintenance of the high velocity in the vortex core. Second-moment closure calculations using the two-component limit (TCL) model of Craft et al. (1996) were able to calculate the correct acceleration and decay of the vortex core velocity. The near-wall region was calculated using both a wall-function and a low-Re approach. Although one would expect the low-Re model to provide a more accurate representation of the near-wall flow, there was no significant difference between the cases when calculating the high velocity at the vortex core. This indicates that the accelerated vortex core is caused primarily by the pressure gradient generated along the vortex axis and, although vorticity and low-momentum fluid from the boundary layer is "wrapped up" in the vortex, it does not penetrate into the vortex core.

KEYWORDS CFD, turbulence, streamwise vortex, wing-tip, second-moment closure, non-linear models.

1

INTRODUCTION

The serious impact of the trailing vortices from the flap-ends and wingtips of large aircraft is well known. Many examples exist of the damage that may be caused when following aircraft are caught up in the swirling wake shed from an upstream aircraft. While, at a practical level, guidelines exist for safe distances between aircraft, the issue of satisfactorily predicting the vortex formation and decay with CFD methods is far from being completely resolved. This is particularly relevant at the present time as there is

628 considerable current research aimed at developing novel wing-tip devices to improve an aircraft's aerodynamic performance. Thus, attention needs to be given to examining the effect of these devices on the downstream decay of the trailing vortices and whether, by prudent design, one may cause the vortices to decay more rapidly. From a CFD point of view, one of the the difficulties in the above arises from the fact that the flow in the vortex is turbulent and, because the flow is swirling, conventional eddy-viscosity models give a poor account of the flow development. Moreover, in order to begin the computation of the vortex development, an accurate account is needed of the complex three dimensional flow over the wing which rolls up to provide the trailing wake vortex. In the current work, the authors have studied the roll-up and near-field development of a wing-tip vortex on a NACA0012 half-wing with rounded wing-tip. The specific case examined was the subject of experiments by Chow et al. (1997). These authors used a half-wing with half-span b/2 = 0.75c, set at an angle of attack ~ -- 10 ~ The velocity of the freestream in the wind tunnel was kept constant to give a Reynolds number based on chord length of Rec = 4.35x106. (Note that Chow et al. reported the Reynolds number to be Rec = 4.6x106 but in a personnal communication with Professor Bradshaw it was established that the actual experimental conditions did produce Rec = 4.35x106). Transition was forced on both suction and pressure surfaces at a distance of 4% of the chord length measured around the arc of the leading edge. A seven-hole pressure probe and hotwires provided mean velocity and turbulence data, and measurements were made in the wake up to a distance x/c - 0.678 from the trailing edge. In defining the experiment, Chow et al. decided to use as large a model as possible while avoiding severe viscous interference due to excessive growth or separation of the tunnel boundary layers. The wind tunnel itself had a width 1.0c and height 0.666c where the chord length was c - 48 inches (1.2192m). This provided a well defined set of boundary conditions that allowed the whole wind tunnel cross-section to be included in the CFD model. Chow et al. found that a large axial velocity excess developed in the core of the wing-tip vortex, with a peak value close to the trailing edge, Uc/U= - 1.77. There are extensive measurements of flow over aerofoils in general; however, the wide variety of designs and flow conditions make it difficult to determine general features which could be considered universal for all wings. The magnitude of the axial velocity at the vortex centre is found to depend on many factors, eg: wing profile, wing-tip shape, Reynolds number and angle of attack. Whereas the streamwise vortex generated along the leading edge of a delta-wing will always have a jet-like profile, with a velocity excess at the vortex core (Cutler and Bradshaw (1993), Menke and Gursul (1997)) this is not true for all aerofoils. Some experiments (Orloff (1974), Strineberg et al. (1991), Devenport et al. (1996)) using different wing shapes and flow conditions, display a velocity deficit at the vortex core, whilst others (McAlister and Takahashi (1991), Green and Acosta (1991)) lead to an excess. However, as noted by Chow et al, a velocity excess as large as Uc/U~ = 1.77 has not previously been reported. Bradshaw's group have made calculations of this flow which are reported in Dacles-Mariani et al. (1995). They used an implicit finite-difference scheme to calculate the RANS equations with an upwind-biased, fifth-order accurate convection scheme. Turbulence was calculated using a one-equation Baldwin-Barth model which solves a transport equation for k2/e_,and does not require an additional algebraic specification of length-scale. As Dacles-Mariani et al. note, eddy-viscosity models tend to overpredict the level of turbulent viscosity within a vortex core; Dacles-Mariani et al. overcome this problem by adjusting the production, P88 of k 2/e from its usual form in the Baldwin-Barth model, PSB = C1 (k2/e) S to a form sensitised to the magnitude of the strain ratels[ and vorticity 10~] tensors: eBB = C1 (k2/~) (It01 + 2min (0, Is t -10~1)). Furthermore, Dacles-Mariani et al. introduced the factor 2 in this production term as an arbitrary constant and adjusted it to enable them to reproduce the high axial velocity in the vortex core.

629 The work reported herein forms part of a wider, collaborative study, called M-DAW: Modelling and Design of Advanced Wing-tip devices. Under this project, novel wing-tip devices are being designed and tested to reduce the drag induced on an aircraft by the wing-tip vortex and to control the downstream development of the vortex. This aims to reduce the environmental impact of the aircraft in terms of fuel consumption and noise, and to understand better the influence of the trailing vortex on downstream aircraft. The current work represents the authors' initial contribution to the project, to establish robust methods and turbulence models, for the calculation of vortex roll-up and its near-field development. The principal thrusts of this phase of the work are to define the required levels of turbulence closure and grid refinement, which will then be used to calculate the flow over novel wing-tip devices.

2

2.1

NUMERICAL PROCEDURE

Code

UMIST's in-house code STREAM, Lien and Leschziner (1994a) was used for the calculations. This is a finite volume code employing generalised, curvi-linear coordinates to define a collocated grid and the smoothing algorithm of Rhie and Chow (1983) to solve the RANS equations. Pressure is calculated via the iterative, SIMPLE, pressure-correction algorithm of Patankar and Spalding (1972). The code was used in a 3-d, fully elliptic, multi-block form, with message-passing-interface (MPI) routines to allow distributed computing on a cluster of workstations. Convection was calculated using either the first-order UPWIND scheme or the second-order total-variation diminishing scheme, UMIST (Lien and Leschziner (1994b)). The approach taken was to obtain a partial solution using UPWIND for both the mean velocities and turbulence variables, and then to switch to the UMIST convection scheme for the mean velocities to obtain the fully converged solution. In previous work to calculate streamwise vortices in a square cross-section U-bend, Robinson (2001), found that it was necessary to use higher-order upwinding on all variables to get the correct streamwise vorticity. Tests were made to establish whether this was also the case for the wing-tip roll-up calculation herein; it was found, however, that the use of higher-order upwinding on the turbulence variables did not significantly improve the vortex calculation. UPWIND was therefore retained for the turbulence variables as this gave a more stable calculation with better convergence properties.

2.2

Grid

The block-structured grid was generated using the commercial meshing code, ICEM. The whole wind tunnel coss-section was included in the grid (width: 1.0c; height: 0.666c), the downstream outlet was placed at the location of the last downstream data plane, x/c = 0.678, and the upstream inflow boundary was placed at x/c = - 1.738 measured from the trailing edge. (This equates to one chord length upstream of the 1/4 chord position). This inflow boundary was further forward than the upstream plane measured by Chow et al, x/c = - 1.134. Initial calculations using Chow's et al upstream plane as the inlet boundary showed that the pressure field was already modified by the wing and there was a consequential, albeit slight, inviscid deflection of the approaching flow. To remove any complications arising from a deflected flow at the inlet, the inlet boundary was moved to the stated position further upstream where the approaching flow was truly undisturbed. By comparing with an analytical solution of a laminar vortex, Dacles-Mariani et al. (1995) concluded that a grid spacing in the cross-stream planes of Ay and Az needed to be 5x10-3c or finer, and that the vortex was less sensitive to the streamwise grid spacing, Ax, than the spacing in the cross-stream planes. A number of grid refinements were tested for the present calculations, which were mostly shown to be too coarse (see discussion in Section 3). After refinement, two final grids were selected which had 4.2x106

630

Figure 1: Sample of 4.2xl06cell grid showing wing surface and wind tunnel wall at the wing-root. cells for a high-Re calculation and 5.4xl06cells for a low-Re calculation (the low-Re grid has a greater concentration of cells in the near-wall region). Each of these grids required 26 blocks. For the high-Re grid, the grid spacing around the surface of the wing was typically An = l x l 0 - 3 c , and at the wind tunnel walls, An = 5 - 8x10-3c (where n indicates the direction normal to the surface). In the vicinity of the vortex near the trailing edge and wing-tip, the grid spacing was zS~r= Ay = Az = l x l 0 - 3 c although it was necessary to expand this downstream, such that the maximum grid spacing in the cross-stream plane was Ay ~ Az ~ 1.5x10-2c and in the streamwise direction was z~ ,~, 3x10-2c. The only modifications for the low-Re grid were that the first node adjacent to the wing surface was placed at An -- l x l 0 - S c and there were 17 cells inside the equivalent first cell of the high-Re grid adjacent to the wing surface. A sample of the 4.2xl06grid is shown in Figure 1. A general problem with block-structured grids is that refinements in the grid tend to be propagated along block boundaries, to regions of the flow where refinement is not required, hence causing inefficiencies in the calculation. This is partially responsible for the very large number of cells required and it also meant that the vortex-wake region could not be refined in as much detail as required. To overcome this in the final phase of the current work, an approach was taken where the computation was performed in two stages. The first stage used the 4.2x106 grid as described to compute the flow over the wing, and for the second, a supplementary grid was generated, covering only the region downstream of the wing. This second grid was Cartesian, with approximately 4.3x106 cells in the downstream region alone, mostly concentrated around the vortex. Inlet conditions for this second calculation were interpolated from the result of the first calculation immediately downstream of the wing's trailing edge (x/c = 0.001). 2.3

Turbulence Models

The baseline turbulence model used was the standard high-Re k - e model of Launder and Spalding (1974). Although used in industry to calculate a wide variety of flows, this model can only be expected to produce accurate results in simple shear flows. Beyond this limit it is likely to perform poorly; particularly relevant to the present case, is that the k - e model, like any linear eddy-viscosity scheme, calculates too much turbulent viscosity in the core of a vortex causing the vortex to dissipate too rapidly. Moreover, Chow et al noted that the measured Reynolds shear stresses in their vortex were not aligned with the

631 mean strain rate, indicating that no isotropic eddy-viscosity model, can accurately calculate turbulence in the vortex. Higher levels of closure are therefore sought, and in this work the cubic non-linear k - e EVM of Craft et al. (1997) and the two-component limit (TCL) second-moment closure model of Craft et al. (1996) have been examined. The former is a two-equation k - E model which uses up to cubic correlations of the strain and vorticity tensors to calculate the stress-strain relation. It captures stress anisotropies arising from local velocity gradients but does not account for convection of the stresses. This model has had mixed success in calculating flows with streamwise vorticity; Robinson (2001) showed that it calculated the flow in a square cross-section U-bend particularly well (at least as well as the TCL model in the mean velocities) but was not able to capture the 3-d flow over the rear section of a simplified car geometry (the Ahmed body). The TCL second-moment closure is formulated to ensure that realisability constraints on the Reynolds stresses are maintained in the calculation of turbulence even in the two-component limit. As a result, the TCL model does not require geometry specific factors such as the wall-normal direction and distance to the wall. It is non-linear in the pressure-strain term with up to cubic correlations of the Reynolds stresses. This model has been shown to perform better than a linear second-moment closure model in flows with strong streamwise vorticity, for example, by Iacovides et al. (1996) and for a range of other flows, Craft and Launder (2001).

Figure 2: Contours of static pressure and streamtraces around the NACA0012 half-wing

2.4

Near- Wall Treatment

High and low-Re methods were used to calculate the viscous-affected, near-wall regions. The high-Re calculations used the log-law type wall function of Chieng and Launder (1980); the near-wall spacing of An = l x l 0 - 3 g a v e y+ ,,~ 4 0 - 120 which is adequate for a 3-dimensional calculation. Some tests were also made with the analytical wall function of Craft et al. (2002); this wall function removes the assumption of the universal log-law of the wall, and instead, prescribes a viscosity distribution across the near-wall cell. An analytical solution to a simplified momentum equation then gives the required mean values in the near-wall cell which include the effects of pressure gradient and convection. Craft et al have shown this to provide a substantial improvement over traditional log-law wall functions in a number of forced and mixed convection cases. Tests of the analytical wall function versus the log-law wall function were carried out on an early grid, having ~ 2.2x10 6 cells. For this flow, no significant improvements due to the analytical wall function were found, perhaps because (in common with log-law models) it assumes the stress and strain fields to be co-aligned.

632 uJu=

,,',\-.. j

..,-r

1

I

I~

ii

-~.

+

10t C ~ & C.~

i

1

0.0 •

0.5

linear k-e; 2_Mg - ' - linear k-e; 4Mg . Symbols: Chow et al (1997)

Figure 3"

0.0 x/c . . . . non-lin k-t; 2.M.g . . . . non-lin k-8; 4Mg

0.5

Uc/Uooand Cp in the vortex core calculated by the linear and non-linear k - e models

Low-Re calculations were performed using a zonal technique: over the first 10 cells adjacent to the wing surface and spanning the viscosity-affected region, turbulent stresses were obtained from the oneequation k - l model of Norris and Reynolds (1975). The 5.4x106cell grid used for the low-Re calculation had the near-wall node for the wing surface placed at An = l x l 0 -5 which was equivalent to y+ 0 . 9 - 2.5 at the trailing edge. For a one-equation model, where only k has to be obtained from a transport equation, this represents an acceptable refinement of the near-wall grid (though it would be insufficient for a two equation k - 8 scheme). Low-Re calculations were only made on the wing surfaces; over the tunnel walls, wall functions were used to calculate the near-wall flow in calculations using each grid.

3

RESULTS

It is relatively easy to calculate the basic vortex roll-up process at a wing-tip, as this is essentially an inviscid process driven by the pressure gradient between the pressure and suction surfaces of the wing. Figure 2 show contours of pressure and streamtraces, and gives an appreciation of the vortex roll-up process. The positive pressure gradient from the pressure to suction sides of the wing generates secondary flow at the wing-tip. Towards the wing-tip/trailing edge location, the flow separates from the wing and rolls up into a vortex which is convected downstream. The strength of the generated vortex determines the pressure drop at its core. The pressure gradient along the core either accelerates or decelerates the flow relative to the freestream depending on the sign of the pressure gradient. Figure 2 shows the complete extent of the modelled domain. The pressure contours on the wing-root wall indicate how far in front of the wing the pressure field is modified and made evident the necessity to move the inlet boundary further upstream than the upstream plane measured by Chow et al. The drop in pressure on the wing surface due to separation of the vortex at the wing-tip can be seen towards the trailing edge and the low-pressure core of the vortex can be seen in the cross-stream planes. Vortex core U-velocity and coefficients of pressure calculated by the linear and non-linear k - e models on a 2.2x106 cell grid (denoted as 2Mg) are shown in Figure 3. All calculations with these models were high-Re calculations using the Chieng & Launder wall function. Also note that position x/c -- 0.0 is the trailing edge of the wing. Both models calculate a vortex core U-velocity which peaks at Uc/Uoo-- 1.42,

633

C~& ~

UJU= I

"'~,

",~,,

-.~,

1.5"1

--

-2 1.0-

~....~:.. -:.::L.i

t

I

I

I

I

t

t

t

I

t

-4

i

i

i

i

i

t

i

i

j

i

i

0.0 x/e 0.5 0.0 x/c 0.5 TCL; high-Re; 2Mg . . . . TCL; high-Re; 4Mg - ' - TCL; low-Re; 5Mg . . . . "rCL downstream only Symbols: Chow et al (1997) Figure 4:

Uc/Uooand Cp in the vortex core calculated by the TCL second-moment closure model.

a short distance before the trailing edge, and then decreases rapidly downstream. The measurements of Chow et al. on the same figure show a peak vortex core U-velocity, Uc/Uoo= 1.77 at the trailing edge which is maintained at a high level further downstream. The reason for the poor calculated values of Uc/Uoo is apparent from the coefficient of static pressure (Ces). The measured Cps falls rapidly towards the trailing edge and continues to fall downstream, though more slowly. In contrast, the Cps calculated by both the linear and non-linear models falls rapidly but does not reach the same low value as measured. Downstream, a rapid pressure recovery is calculated and the vortex core is strongly decelerated so that Uc is less than the freestream at x/c = 0.5. The drop in Cps at the vortex core is itself due to the swirling flow of the vortex. Where the vortex is dissipated and the swirl reduced, the pressure recovers at the vortex core. Hence, the early pressure recovery in the EVM calculations indicates that the vortex is being dissipated too rapidly by an excess of turbulent viscosity. Also shown in Figure 3 are the linear and non-linear k - ~ model calculated on the 4.2x106 cell grid (4Mg). It is readily apparent that the 2Mg results were not grid independent. With the 4Mg grid, the peak core velocity calculated by the linear model almost matches measurements, Uc/Uoo= 1.80, and the non-linear model overshoots by a small amount at the trailing edge. The better calculation of swirl on the 4Mg grid leads to a better calculation of Ces, although both models calculate too large a reduction in Ces at the trailing edge. The pressure still recovers too rapidly in the downstream region and Uc is again strongly decelerated. The non-linear model calculates the rise in Uc too early but there is a minor improvement in the downstream level of Uc in comparison with the linear model. Figure 4 shows Uc/Uoo and Cps calculated by the TCL second-moment closure model. The high-Re calculation on the 2.2x106 cell grid (2Mg) shows an immediate improvement over the equivalent linear and non-linear EVM calculations (Figure 3). The core velocity reaches a value of Uc/Uoo = 1.63 at the trailing edge and although the core velocity is decelerated too rapidly, it does not fall below the freestream value over the calculated distance. With the refinement to the 4Mg grid, the TCL model calculated the correct core velocity peak, Uc/Uoo= 1.77 at the trailing edge; downstream of the trailing edge the core velocity was still decelerated by the recovering pressure gradient but the drop-off in Uc/Uoo was much closer to the measurements. Calculations were also made with the TCL model using the lowRe wall treatment on the 5.4xl06cell grid (5Mg). Chow et al. describe how the turbulent boundary layer over the surface of the wing is wrapped-up in the vortex to give high levels of turbulence in the vortex.

634

Figure 5: Contours of U-velocity calculated by the TCL model with wall functions at x/c = 0.001, fractionally downstream of the trailing edge. Similarly, vorticity is generated in the viscosity-affected near-wall region and one would assume that the more detailed near-wall treatment would produce a better calculation of the wing-tip vortex. However, a compromise had to be struck in the grid generation due to the limited computational resource available: the low-Re (5Mg) grid does have almost 30% more cells than the high-Re (4Mg) grid but the additional cells are concentrated around the wing surfaces and furthermore, to ensure smooth transition of the grid between the near-wall region and the outer flow, some of the cells from the outer region of the 4Mg grid are moved towards the wing in the 5Mg grid. This actually results in a coarser grid in the outer flow region for the 5Mg grid than the 4Mg grid. This is particularly poor for x/c > 0 where the nature of the block structuring continues the near-wall refinement along the block boundaries. The fine grid in the downstream region does not then coincide with the location of the vortex core. This does not adversely affect the accuracy of the calculation of the vortex upstream of the trailing edge of the wing, as the vortex is located in the highly-refined region near the wing. However, downstream of the trailing edge, the vortex is convected into the outer flow region where the grid is coarser which accounts for the poor performance of the TCL model on the 5Mg grid as one proceeds downstream of the wing. To overcome the problems of the poor cell distribution in the downstream region, the vortex development downstream of the wing was calculated using a two stage approach. The second stage used the supplementary grid described in Section 2 and the TCL model; its inlet conditions at x/c = 0.001 were interpolated from the earlier high-Re TCL model results. The drop off in Uc/U= shown in Figure 4 (denoted as "TCL downstream only") now closely matches the measured values and the authors are confident that a small improvement in the downstream (supplementary) grid refinement will improve the calculation to match the measured results. It is noted that Dacles-Mariani et al. in their calculations of this case using the Baldwin-Barth turbulence model were only able to calculate Uc/U= to the same level of accuracy as the "TCL downstream only" calculation by making ad hoc adjustments to the production term in their model. Figure 5 shows contours of U-velocity calculated using the TCL model with wall functions and the 4Mg grid. This is a section of the cross-stream plane located at the inlet of the downstream only calculation, x/c = 0.001, which is fractionally downstream of the trailing edge. At the trailing edge the wing-tip is located at: y/c = 0.000; z/c = 0.690. The thickening of the boundary layer due to the adverse pressure

635

Figure 6: Contours of ~ (streamwise normal stress) calculated by the TCL model with wall functions at

x/c = 0.001, fractionally downstream of the trailing edge. gradient on the upper surface of the wing is shown by the large area of low velocity fluid, and the accelerated core of the vortex is clearly visible just above the location of the wing-tip. It is interesting to note how the vortex causes a thinning of the boundary layer on the upper surface of the wing by drawing low-momentum fluid out of the boundary layer and wrapping it up into the vortex. However, this low momentum fluid remains in the outer region of the vortex and does not penetrate the core. The equivalent contour plot for the ~ Reynolds stress is shown in Figure 6. From this it can be seen that there is a large amount of turbulence generated by the high velocity gradients associated with the flow being bent around the wing-tip. As with the low-momentum fluid drawn out of the wing boundary layer, the highest levels of turbulence generated do not penetrate the core of the vortex.

4

CONCLUSION

The calculations reported herein represent the initial phase of a wider study to calculate the roll-up and far-field development of vortices generated at novel wing-tip devices. From the current work it is apparent that at least a second-moment closure is required to calculate the anisotropic Reynolds stresses in the vortex. The TCL model of Craft et al. (1996) was found to be well suited to this purpose. Lower levels of closure are not able to calculate the correct rotation rate of the vortex, due to an excess of turbulent viscosity calculated in the vortex core. The pressure gradient in the core is then not sufficient to accelerate the U-velocity in the core to the correct levels. This work has also shown the sensitivity of the calculation to the grid refinement and the necessity for a highly refined grid, particularly in the region of the vortex downstream of the trailing edge.

ACKNOWLEDGEMENTS The authors would like to acknowledge the EU for supporting this work through the 5th Framework Project: M - DAW, Modelling and Design of Advanced Wing-tip devices (G4RD-CT-2002-00837). Professor Bradshaw of Stanford University greatly assisted our work by providing machine-readable data and several helpful responses to queries. The authors' names appear in alphabetical order.

636 REFERENCES

Chieng, C. C., Launder, B. E., 1980. On the calculation of turbulent heat transport downstream from an abrupt pipe expansion. Numerical Heat Transfer 3, 189-207. Chow, J., Zilliac, G., Bradshaw, P., 1997. Turbulence measurements in the near field of a wingtip vortex. Tech. Rep. NASA Technical Memorandum 110418, NASA. Craft, T. J., Gerasimov, A. V., Iacovides, H., E., L. B., 2002. Progress in the generalization of wallfunction treatments. Int. J. Heat and Fluid Flow 23, 148-160. Craft, T. J., Launder, B. E., 2001. Principles and performance of a two-component limit based secondmoment closure. Flow, Turbulence and Combustion 66, 355. Craft, T. J., Launder, B. E., Ince, N. I., 1996. Recent developments in second-moment closure for buoyancy-affected flows. Dynamics of Atmospheres and Oceans 23, 99. Craft, T. J., Launder, B. E., Suga, K., 1997. Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model. Int. J. Heat Fluid Flow 18, 15-28. Cutler, A. D., Bradshaw, P., 1993. Strong vortex/boundary layer interactions. Experiments in Fluids 14, 321-323. Dacles-Mariani, J., Zilliac, G. G., Chow, J. S., Bradshaw, P., 1995. Numerical/experimental study of a wingtip vortex in the near field. AIAA Journal 33(9), 1561-1568. Devenport, W. J., Rife, M. C., Liapis, S. I., Follin, G. J., 1996. The structure and development of a wingtip vortex. J. Fluid Mech. 312, 67-106. Green, S. I., Acosta, A. J., 1991. Unsteady flow in trailing vortices. J. Fluid Mech. 227, 107-134. Iacovides, H., Launder, B. E., Li, H.-Y., 1996. Application of a reflection-free DSM to turbulent flow and heat transfer in a square-sectioned U-bend. Expt. Therm. F1. Sci. 13, 419-429. Launder, B. E., Spalding, D. B., 1974. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3, 269-289. Lien, E S., Leschziner, M. A., 1994a. A general non-orthogonal finite-volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure. Comp. Meth. Appl. Mech. Eng. 114, 123-167. Lien, E S., Leschziner, M. A., 1994b. Upstream monotonic interpolation for scalar transport with application to complex turbulent flows. Int. J. Num. Meth. Fluids 19, 527-548. McAlister, K. W., Takahashi, R. K., November 1991. Naca0015 wing pressure and trailing vortex measurements. Tech. Rep. TP-3151, NASA. Menke, M., Gursul, I., 1997. Unsteady nature of leading edge vortices. Phys. Fluids 9, 2960-2966. Norris, L. H., Reynolds, W. C., 1975. Turbulent channel flow with a moving wavy boundary. Tech. Rep. FM-10, Department of Mechanical Engineering, Stanford University. Orloff, K. L., 1974. Trailing vortex wind-tunnel diagnostics with a laser velocimeter. Journal of Aircraft 11 (8), 477-482. Patankar, S. V., Spalding, D. B., 1972. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787. Rhie, C. M., Chow, W. L., 1983. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, 1525-1532. Robinson, C. M. E., 2001. Advanced CFD modelling of road-vehicle aerodynamics. Ph.D. thesis, Dept. of Mechanical Engineering, UMIST, Manchester, UK. Strineberg, D. R., Farrell, K. J., Billet, M. L., 1991. Structure of a three-dimensional tip vortex at high Reynolds number. J. Fluids Eng. 113,496-503.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

637

U R A N S C O M P U T A T I O N S OF S H O C K I N D U C E D O S C I L L A T I O N S O V E R 2D R I G I D A I R F O I L : I N F L U E N C E OF T E S T S E C T I O N G E O M E T R Y M. Thiery and E. Coustols ONERA/DMAE Centre d'l~tudes et de Recherches de Toulouse B.P. 4025, 2, Avenue l~douard Belin 31055 Toulouse CEDEX 4, France e-mail: Mylene.Thiery~onecert. fr

ABSTRACT

The present article deals with recent numerical results, from on-going research conducted at ONERA/DMAE regarding validation of turbulence models for unsteady transonic flows, for which shock wave / boundary layer interaction develops. The main goal is to predict the onset of Shock Induced Oscillation (SIO) in conditions as close as possible to the experiments. SIO appears over the suction side of a two-dimensional rigid airfoil and leads to the formation of unsteady separated areas. Computations were performed with the ONERA object-oriented software elsA, using the URANS-type approach, closure relationships being achieved by the one-transport equation Spalart-Allmaras model. Applications are provided for the OAT15A airfoil data base built up from tests conducted in the ONERA S3Ch wind tunnel (the airfoil aspect ratio being 3.5). These experiments are well documented for CFD validation with mean, phase-averaged and fluctuating data. Results emphasize the importance of modelling the test section geometry when carrying out 2D unsteady computations to capture SIO as precisely as possible, even though the adaptation of wind tunnel walls had been carefully managed. Lastly, first investigations towards 3D computations (taking into account the four side-walls of the wind tunnel) are presented.

KEYWORDS

Buffet, Shock Induced Oscillation, Unsteadiness, RANS, Turbulence Modelling.

1

INTRODUCTION

The present article is devoted to the resolution of the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations, with the aim of predicting the onset of Shock Induced Oscillation (SIO) developing over a two-dimensional (2D) rigid airfoil. For transonic aircraft wing applications, such oscillations are mainly caused by shock wave /

638

boundary layer interaction, in relation with large regions of separated flows. The response of the wing structure to these aerodynamic flow instabilities (buffet) corresponds to the well-known buffeting phenomenon. These aerodynamic excitations are mainly attributed to pressure fluctuations growing in generated separated areas (e.g. shock footprint, trailing edge, ...). Several studies were devoted to the understanding, the behaviour as well as the control of self-sustained SIO over airfoils under transonic conditions (Caruana et al., 2003; Ekaterinaris and Menter, 1994; Gillan et al., 1997; Lee, 1990, 2001). Though buffeting is not dangerous and destructive for civil aircraft concerns, it mainly affects aircraft manoeuvrability; flow instabilities have then to be clearly identified in order to define precisely enough the flight envelop. The present numerical study was devoted to aerodynamic issues only, even though fluid-structure coupling should be addressed when applying to real three-dimensional aircraft wings. From numerous previous world-wide investigations, RANS methods have revealed to be very adequate for computing transonic turbulent steady flows with small extent region of reverse flows (Spalart, 2000). In the case of buffeting aerodynamics, associated with the periodic motion of the shock over the suction side of the airfoil and of induced boundary layer separation, the time-scale of the wall-bounded turbulence is much smaller than that of SIO unsteadiness. This frequency gap allows to use the URANS-type approach, the mean flow being resolved from the unsteady RANS solution while the turbulence being modelled by standard RANS-type models. Experiments were recently conducted with a 2D OAT15A airfoil in the self-adaptive transonic $3 wind tunnel of the ONERA Centre of Chalais Meudon (Jacquin et al., 2005). The experimental data are well documented for unsteady CFD validation with averaged, fluctuating and phaseaveraged data. The objectives of the present numerical investigation were to run computations for conditions as close as possible to the experimental ones. First, the effect of the boundary layer over-thickening at the tripped transition was investigated. Then, since the adaptation of the wind tmmel walls was only based on the averaged streamlines and not on the instantaneous one, the impact of the presence of these walls on the unsteady 2D computations was evaluated. Lastly, with the aim of resolving, at least partially, the transverse instabilities developing over the separated area, the possibility to perform 3D computations was discussed. The computations were performed with only the one-transport equation turbulence model from Spalart and Allmaras (Spalart and Allmaras, 1994), since the major aim of the present study was to compare results of unsteady computations performed either in free-stream or in confined (with test section walls) conditions. It should be noted that former studies were specifically devoted to turbulence validation (Furlano et al., 2001; Coustols et al., 2003) and that research applied to the OAT15A test-case is in progress, too.

2

TEST CASE-

OAT15A AIRFOIL

Experiments were recently performed in the transonic $3 wind tunnel of ONERA Centre of Chalais Meudon in the framework of SIO scrutinization (Jacquin et al., 2005). A 2D airfoil (OAT15A cross section, chord length c=230 ram, relative thickness t/c=12.5%, blunt trailing edge e/c=0.5%) was mounted in the test cross-section (width x height: 800 x 760 mm2). The experimental set-up was defined with the aim of providing a two-dimensional flow to the best possible degree; the aspect ratio of 3.5 was relatively large to minimize the 3D effects without avoiding them, yet. The upper and lower walls are self-adaptive and thus flexible instead of being slotted: then, pressure mea-

639

surements allowed to adapt the shape of these walls to the time-averaged flow streamlines. The transition was tripped at x/c=7% on both sides of the airfoil. Tests were carried out at a value of Reynolds number based on the chord length Rec=3 106, a free-stream Mach number M~=0.73, Tioc=300 I~ and several angles of attack (c~) varying from 1.36 ~ to 3.9 ~ The experimental buffet onset appeared at c~=3.25 ~ and the greatest collection of unsteady data was obtained at 3.5~ indeed, averaged (68 wall static pressure taps and Reynolds averaged LDV measurements), fluctuating (36 Kulite transducers) and phase-averaged (conditional LDV measurements) data are available for CFD validation purpose.

3

3.1

NUMERICAL

TOOLS

Solver and Numerical methods

Computations were performed with the ONERA object-oriented software elsA, solving the threedimensional compressible Reynolds Averaged Navier-Stokes equations for multiblocks structured grids, using finite volume method with cell-centered discretization (Cambier and Gazaix, 2002). The fluxes are computed with two second order accurate schemes; the Jameson scheme is used for mean flow fluxes computation with artificial dissipation terms while the Roe scheme is applied to turbulent transport equations with an anisotropic correction. For steady computations, the time explicit second order accurate integration is done with the four step Runge-Kutta algorithm. Convergence acceleration techniques are applied such as localtime stepping and FAS (Full Approximation Storage) multi-grid method. The implicit stage is an approached linearization method with a LU (Lower-Upper) factorization associated with a relaxation technique. For unsteady computations, the implicit time integration is performed with the dual time stepping method which combines a physical time step, linked to the frequency range of the phenomenon under investigation and a fictitious dual time step, related to a steady process to increase convergence between each physical time step. 300 iterations per cycle were imposed to capture unsteadiness and about 10 cycles were necessary to obtain self-sustained SIO while 5 extra cycles were used to check the periodicity.

3.2

C o m p u t a t i o n a l conditions

Computations were performed at the experimental values of Mach and Reynolds number (M~=0.73 and Rec=3 106) and the transition was fixed at the experimental location (x/c=7%). Accounting for previous studies on the SIO phenomenon, computations were at first performed using a domain extending over 50 chord length with free-steam conditions; it is the "standard" approach for predicting SIO and will be referred to the "2D inf." approach. Inviscid / viscous weak coupling computations at a steady state concluded that no corrections on M~ and c, were necessary for undertaking Navier-Stokes computations with the "2D inf." approach. This confirms that the self-adaptive upper and lower walls associated with a relatively large value of airfoil aspect ratio minimizes the influence of wind tunnel walls. Later, in order to be closer to testing conditions, two "confined" approaches were evaluated. First, the upper and lower walls were taken into account in the numerical boundary conditions and the mesh definition. Secondly, the side-walls were figured on. These approaches will be referred respectively to the "2D conf." and "3D conf." approaches.

640

TABLE 1 GRID

C H A R A C T E R I S T I C S AND L I F T C O E F F I C I E N T

O B T A I N E D AT C t - - 2 . 5 ~ W I T H T H E

APPROACH.

Ay +

Wake

Ax +

CHo coarse

317x129

~0.4

121x305

~2

77798

0.95733

CHo interm.

349x149

~0.4

129x371

~2

99860

0.9603

CHo fine

633x257

~0.2

241x609

~1

309450

0.96614

oc

0.9683

75642

0.9587

CH4

4.1

INF."

Airfoil

Interpolate

4

"2D

. 309x133

.

. ~0.4

N (total pts)

. 105x329

~2

CL (1.1%)

(1%)

R E S U L T S W I T H T H E 2D I N F I N I T E A P P R O A C H

M e s h convergence

blesh convergence was performed for steady flow conditions, c~=2.5 ~ A particular attention was paid to the refinement of the blunt trailing edge (relative thickness 0.5% of chord length), of the longitudinal discretization of the airfoil suction side, and of the transverse discretization of the boundary layers and wake, as all these elements might interfere during SIO cycles. The first grid 'CH0 fine' was generated with a rather large number of discretizing points to be able to remove one point over two in each direction and generate the 'CH0 coarse' grid. To complete the grid refinement, the ~CH0 intermediate' grid was created by refining the 'CH0 coarse' grid in both directions. The characteristics of these grids can be found in Table 1. The Richardson interpolate value of lift coefficient (Slater, 2004) was determined from these three grids. The 'CH0 coarse' grid predicted a lift coefficient 1.1% smaller than the Richardson one, allowing to assume the grid convergence. Then, an iterative process on finer constraints than the lift coefficient (e.g. Reynolds stress slope in the wake) was performed and led to the generation of the CH4 grid with 309 points along the airfoil and 65 points in the blunt trailing edge, which was chosen for unsteady computations.

4.2

Unsteady results

First unsteady computations were performed at c~=3.5 ~ and led to slowly damp the SIO (weak fluctuations can be observed in Fig. la.). It was then mandatory to increase the angle of attack to develop unsteadiness and the computed r.m.s pressure distributions are compared for c~=3.5 ~ 4.0 ~ and 4.5 ~ (Fig. la.). An increase of one degree regarding the experimental angle of attack was not sufficient to capture the main flow features; the computed fluctuations on the pressure side (x/c<0) were nearly null compared to the experimental values (about 0.05 q0) and their evolution and level in the separated area (0.5<x/c<1) did not reproduce the experimental ones. The measured r.m.s. pressure values exhibited a pronounced "V-type" pattern (i.e. higher fluctuations level) in the separated areas such as the shock footprint and near the trailing edge. However, in that latter region, the computed values revealed a flat evolution. Then, the impact of the over-thickening of the boundary layer was investigated since, a priori, the separated area might develop easier with the increase of the momentum thickness 0. The experimental transition was fixed by carborundum grains at x/c=7% and the over-thickening of

641

0.49

exp. (x=3.9

9 . . . .

0.3

0, i

exp. (x=3.5

[]

9

dP

2D inf. 2D inf.

(x=3.5 0c=4.0

2D inf.

(x=4.5

[] ~ o oo

0.3

.ii ii

"•0.2

ii s [] .!i li

O.

9

o~

exp. (x=3.5

[]

,

exp. a=3.9 2 D i n f . 0c=4.5 2 D i n f . + ~10 c ~ - 4 . 5

. . . . .

ii 0.2

[] .lli i~i

II' 0~176 ~ [] 0

&.

~"~" - ~ " -1

9

. . . . . -0.5

pressure side

[]

[]

!,i""

i ii

!ii~[]o[]

9

[]

9 "" -'~" ""

0 0

xlc

0.5

suction side

[] o o

1

b.

9

[]

-0.5

-

"l::i ".'. : [] ii~

9

OQ

:'~": ~ l r r " ~ " " ' r ' " ' ~ '~'~" z~.,,.,.. . . . . . -1

[]

,,~-eM 0

xtc

i

" 0.5

1

Figure 1" r.m.s, pressure distributions for the "2D inf." approach (a." effect of the angle of attack with c~ varying from 3.5 ~ to 4.5 ~ b." effect of the over-thickening of the boundary layer due to tripping). 0 was estimated from empirical formula (proposed by Arnal (1984)) at 80% of 0 at the transition location. Computations on the OAT15A airfoil were performed at c~=4.5 ~ to ensure large flow fluctuations: the main effect of the over-thickening was to reduce the unsteadiness over the airfoil (Fig. lb.). For 0.5<x/c<1, the decrease reached 20% regarding the computation without overthickening. Nonetheless, the effect on the maximum of fluctuations location was rather small since the over-thickening represented only 8% of 0 at the shock wave location. These results obtained with the [SA] model and the "2D inf." approach were quite disappointing when comparing to former studies related to the SIO phenomenon (Coustols et al., 2003). It was then decided to pursue work by taking into account the upper and lower walls of the wind tunnel, in order to be closer to the test conditions.

5

R E S U L T S W I T H T H E 2D C O N F I N E D A P P R O A C H

Since no SIO was observed at a=3.5 ~ with the "2D inf." approach, geometrical testing conditions were figured on by modelling the measured deflection of the upper and lower walls of the test section and considering the exact location of the airfoil in the wind tunnel The extension of the numerical domain was larger than the experimental one to be able to reproduce the measured boundary layer thickness on the wind tunnel walls, 2 chord lengths upstream of the airfoil.

5.1

B o u n d a r y conditions validation

The validation of the boundary conditions for the "2D conf." approach was performed by plotting the time-averaged pressure distribution over a SIO period and comparing it to the mean experimental value (Fig. 2). The mean pressure on the upper and lower walls (Fig. 2a.) pointed out rather appropriate boundary conditions for the computation. However, with a 2D domain, a compromise has to be found between the Mach number and the static pressure, respectively at the entrance and the exit sections, which could account for the discrepancy between experimental and computed pressure values near the entrance section. Along the airfoil (Fig. 2b.), the computed pressure coefficient for the "2D conf." approach agreed

642

a.

9

!.

.........

....

exp. 2D inf. 2D c o n f .

p :..

0 0.5

~.o.68

0.66

0.64

7

1.5

0.72

-

exp. - upl ~ r wall exp. Io~ er wall 2D

=.a-/

conf.

Air) !ill location

. . . . . . . . . . .

-0.5

-6

-4

-2

0

2

Longitudinal abscissa (wind tunnel frame)

4

b.

0

0.25

0.5

x/c

0.75

.1

Figure 2: Boundary conditions validation for the "2D conf." approach with the time-averaged pressure distributions (c~=3.5 ~ a." on the upper and lower W / T walls, b.: along the airfoil). with the experimental one in the region upstream of the shock wave and on the pressure side, which indicates that the aerodynamic conditions for the airfoil (Moo and c~) were well adjusted. Moreover, the results were similar to those obtained with the "2D inf." approach on these zones. Thus, the adaptation of the wind tunnel walls was well managed for the averaged flow.

5.2

Unsteady results

Contrary to the "2D inf." approach, the 2D confined domain allowed to develop unsteadiness. Thus, several computations were performed to validate the unsteady flow obtained with the "2D conf.", as mentioned in the following lines: 9 The exit boundary condition is a constant static pressure, which imposes perturbations to be damped. That strong boundary condition might be threw back into question by a close relationship between the SIO frequency and the domain extent. Therefore, another grid was generated to push downwards the exit from 4.5 to 8.5 chord lengths. Unsteady results remained unchanged. 9 The time step used in the dual time stepping method (At=3.333 10-5s) was validated using results from former studies on SIO (Furlano et al., 2001; Coustols et al., 2003) but also by performing tests with At=10-Ss and 10-4s. The time consistency was found again with a global time stepping method using equivalent time steps. 9 The grid refinement was also checked: the total number of grid points was reduced by 42%, saving points in the boundary layers of the upper and lower walls and in the middle of the test section. Unsteady results were not impacted and the generated 2D confined domain (~62000 pts) was thus used to create the grid for the "3D conf." approach. The mean computed pressure coefficient on the airfoil was compared to the 'min.' and 'max.' computed instantaneous distributions, respectively corresponding to the most upstream and downstream shock location in the SIO period (Fig. 3a.). The computed shock wave displacement was over 15% of chord length from x/c=0.4 to x/c=0.55. The difference between 'rain.' and 'max.' was small on the pressure coeff• plateau and rather constant along the pressure side. The "2D conf." approach exhibited a great improvement regarding the 2D infinite results on the r.m.s pressure distribution (Fig. 3b.). The 2D confined results were in really good agreement with experimental values except for the maximum of fluctuations predicted 10% of chord length downstream

643

of the experimental one. The main discrepancy with the experiments lied on the mean shock wave location, which was particularly difficult to accurately predict since the pressure gradient upstream of the shock was nearly null. 1.5 o

i

!

,,

03

9 ......

exp.

I

2D inf. 2D conf.

%%

Io;

o

=. ~, 0.5

-0.5

a.

"~'" 9 " ~"

"'r,:,,~.mean... __ .~'~"

0.25

0.5

x/c

o.1

"" -

max.

0

~ ' /i

0.75

1

0 b.

-'

-0.5

0

x/c

0.5

Figure 3: Unsteady pressure distributions for the "2D conf." approach (a.: averaged pressure coefficient, b." r.m.s, pressure). The effect of the over-thickening of the boundary layer at the tripped transition was also investigated with the "2D conf." approach. The increase of 0 at the transition location led to a weak decrease of the fluctuations levels, mainly noticeable in the separated area. However, the impact of such an over-thickening was judged minor for the test case. Lastly, the r.m.s fluctuations of the longitudinal velocity component were compared to the experimental data provided by Reynolds averaged LDV measurements (Fig. 4); the latter contained both unsteady and turbulent fluctuating components while computational ones were only deduced from unsteady component (the [SA] model providing only the turbulent eddy viscosity, not the turbulent kinetic energy). The r.m.s, fluctuations velocity field was well reproduced by computa-

Figure 4: Field of r.m.s fluctuations of the longitudinal velocity component for the "2D conf." approach (a." numerical u '2, b." experimental u '2 from LDV). tion, particularly on the main characteristics of the flow field such as the maximum of fluctuations

644 level at the shock wave footprint due to the shock wave / boundary layer interaction, the shock wave displacement and the separated area.

5.3

Phase-averaged boundary layer profiles

The experimental phase-averaged velocity profiles were obtained by LDV measurements coupled with a conditional analysis (Jacquin et al., 2005). The SIO period was discretized into 20 phases and I0 of them were compared in Fig. 5 to the numerical unsteady profiles for the "2D conf." approach (the steady result obtained with the "2D inf." approach is also provided for comparison). At x/c-0.6, comparisons between experimental and numerical results confirmed that; (i) the mean state of the boundary layer was separated, and (ii) the boundary layer was periodically attached (7 phases from 4 to II) and separated (13 phases from II to 3). The first point was predicted Phase

1

5

exp, 2D inf. 2D conf.

9

..............

9

~. Q a d

0.02'

i:

):

~..

....

-

i~

"

_

O-

:: i

i

i

0

,

i

i

i

i

i

i

i

0.5 lU|

,

i

1

i

i

i

.:.-"

.:':" i

i

0

i

i

i

i

i

i

i

,

0.5 IU ~

I

i

1

i

i

11.i,

i

i

|

0

i

i

i

i

i

i

i

i

0.5 IU ~

i.

i

i

1

i

i

i

i

.. .

...."

i

0

i

,

i

i

i

,

i

0.5 lU|

,s 1..

.

....9

i

i

i

1

i

,

,

i

i

,

i

,

,

i

i

t

lU|

i

i

i

u

"/"

.

i

i

i

if.o:

'115

'91"

0.04. _ u

-

~,

_

0.02~

9

qp

...... . iI

.

..~.~ 9

'

'

i

0

. . . .

9

, ..,..."

,..;,'~ O-

......

~mO

. . :y.:""

u

. . . .

0.5 /U.

u

1

......

i

0

,

,

,

. .o 9

'

i

. . . .

0.5 /U~

9

9

o 9

I

1

,

'

'

,

i

0

0 9149

;...

. . . .

.......

........'""......

,," ........ u

. . . .

0.5 /U|

u ' , ' ' u , ' ' , l '

1

0

,'

0.5 /U.

~"

' I , , , , i

1

.......

9

|

......

....................... 0

. . . .

I

. . . .

0.5 /U

I

1

'

,

,

Figure 5: Instantaneous velocity profiles for 10 phases over 20 in the SIO period at x/c-0.6. by the "2D inf" approach even though the solution was steady. The second point was rather well reproduced by the unsteady solution of the "2D conf." approach. Nonetheless, the evolution of the computed boundary layer was either quicker (from phase 15 to 20) or slower (from phase 7 to 13) than the experimental one, which could induce, for a given phase, a large error on the boundary layer thickness (e.g. 35% at phase 13). The error was also partially linked to the error on the location of the mean shock wave. The badly predicted boundary layer dynamics occured when strong variations on the shock wave location and on the boundary layer slope were observed; that might point out the turbulence modelling limitations. Actually, the turbulence model using the Boussinesq hypothesis did not figure on the 'history' effects and did not correctly simulate the momentum transfers in the separated areas.

645

6

INVESTIGATIONS

TOWARDS

THE 3D CONFINED

The 3D computational domain covered one half of the test section, a symmetrical condition being applied in relation to the centre-line. It was generated by multiplying the 2D confined grid (63000 pts) in the spanwise direction. The Mach number and the static pressure were fixed at the same values as the "2D conf." approach ones. For CPU time cost, the boundary conditions validation was performed with a steady computation: the limitation was not so high since the steady and the time-averaged results obtained with the "2D conf." approach were identical on the upper and lower walls (that was not the case on the airfoil due to non-linear phenomena at the shock location).

APPROACH

.e~lk._=

0.72 e/

0.7

9

~'~

Q. "~.0.68

0.66

9 -

0.64

.9- . .

-

-

exp. 2D conf. 3D conf.- 89 plane~ 3D conf.- 63 plane~ Air.] "ill location

-6

-4

-2

0

2

Longitudinal abscissa (wind tunnel frame)

4

Figure 6: Pressure distributions at the upper and lower wind tunnel walls.

The validation of the boundary conditions could be checked by comparing the pressure distributions at the upper and lower walls obtained for either 2D or 3D confined domain (Fig. 6). The 3D resolution improved the pressure distributions from x/c=-6.5 to 0, which refers to the airfoil location. However, the pressure on the upper wall was over-estimated with the "3D conf." approach, which might be linked to a blocking effect. Actually, a 3D corner separation developed at the airfoil / side-wall junction and seemed to be over-extended in the 3D RANS computations. Two refinements were tested with either 63 or 89 sections (smaller cells near the side-wall) in the spanwise direction; the pressure on the upper and lower walls was not affected (Fig. 6), the grid refinement being sufficient. These 3D effects were not negligible on the steady results and will therefore be evaluated with unsteady computations, which requires CPU time about 15 times larger than one with the 2D conf. approach. Former steady studies conducted at ONERA by Furlano et al. (2001) had demonstrated tile importance to take into account the 3D confinement of the test section. More recently, Garbaruk et al. (2003) also investigated in that way on the RAE 2822 test case. All these studies concluded that it was mandatory to figure on all of the details of the experimental setup, particularly wind tunnel walls to objectively evaluate the capabilities of turbulence models for SIO predictions.

7

CONCLUSIONS

Without any computational domain constraints, the computations with the [SA] model led to damp oscillations at c~=3.5 ~ for aerodynamic conditions similar to those of the tests (Mach number, stagnation pressure and transition tripping). Although the aspect ratio of the airfoil was rather large, the upper and lower self-adaptive walls were taken into account and tile unsteady computations predicted self-sustained shock oscillations at a frequency and levels of pressure fluctuations very close to the experimental values. Such a result is very important regarding code and turbulence model validations, for unsteady turbulent flows in the framework of SIO scrutinization. Nonetheless, when comparing the time-averaged pressure distribution on the airfoil, the shock location was not correctly predicted (,-,10%), although the agreement was perfect everywhere else. Such a difference might be reduced by considering 3D effects due to the lateral non-adaptive walls of the wind tunnel. Moreover, other instability, in particular transverse one, might probably play

646

a role in the SIO development and could be partially captured by running a full 3D computation.

Acknowledgments The authors gratefully acknowledge the Service des Programmes Adronautiques which granted research reported in the paper and the Experimental and Fundamental Aerodynamics Department of ONERA for its support on the experimental data base use.

REFERENCES D. Arnal. Description and prediction of transition in two-dimensionnal incompressible flow. A GARD - Special Course on Stability and Transition of Laminar Flow, (709), 1984. L. Cambier and M. Gazaix. elsA: an ett~icient object-oriented solution to CFD complexity. AIAA Paper 2002-0108, 40 th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, USA, January 2002. D. Caruana, A. Mignosi, C. Robitaill~, and M. Corr~ge. Separated flow and buffeting control. Flow, Turbulence and Combustion, 71(1-4):221-245, 2003. E. Coustols, N. Schaeffer, M. Thiery, and P. Cordeiro Fernandes. Unsteady Reynolds-Averaged Navier-Stokes Computations of Shock Induced Oscillations over Two-Dimensional Rigid Airfoils. In Proc. 3~d International Symposium of Turbulence and Shear Flow Phenomena, Senda, Japan, volume 1, pages 57-62, June 2003. J. A. Ekaterinaris and F. R. Menter. Computation of separated and unsteady flows with one-and two-equation turbulence models. AIAA Journal, 32:23-59, 1994. AIAA Paper 94-190. F. Furlano, E. Coustols, O. Rouzaud, and S. Plot. Steady and unsteady computations of flows close to airfoil buffeting: Validation of turbulence models. In Proc. 2~ International Symposium on Turbulence and Shear Flow Phenomena, Stockholm, Sweden, volume 1, pages 211-216, June 27-29 2001. A. Garbaruk, M. Shur, and M. Strelets. Numerical study of wind-tunnel walls effects on transonic airfoil flow. AIAA Journal, 41(6):1046-1054. June 2003. M. A. Gillan, R. D. Mitchell, S. R. Raghunathan, and J. S. Cole. Prediction and control of periodic flows. AIAA Paper 1997-0832, 35 th AIAA Aerospace Sciences Meeting & Exhibit, Reno, Nevada, USA, January 6-9 1997. L. Jacquin, P. Molton, S. Deck, B. Maury, and D. Soulevant. Experimental study of the 2d oscillation on a transonic wing. to be presented at the 35 th AIAA Fluid Dynamics Conference, Toronto, 6-9 June 2005. B. H. K. Lee. Oscillatory shock motion caused by transonic shock boundary-layer interaction. AIAA Journal, 28(5):942-944, May 1990. B. H. K. Lee. Self-sustained shock oscillations on airfoils at transonic speeds. Progress in Aerospace Sciences, 37:147-196, 2001. J. W. Slater. http://www.grc.nasa.gov/www/wind/valid/tutorial/spatconv.html, 2004. P. R. Spalart. Strategies for turbulence modelling and simulations. International Journal of Heat and Fluid Flow, 21(3):252-263, 2000. P. R. Spalart and S. R. Allmaras. A one-equation turbulence model for aerodynamic flows. La Recherche Adrospatiale, 1:5-21, 1994.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

647

Z O N A L M U L T I - D O M A I N RANS/LES S I M U L A T I O N OF A I R F L O W O V E R THE A H M E D B O D Y

F. Mathey 1 and D. Cokljat 2 1 Fluent France SA, 1 place Charles de Gaulle, Montigny Le Bretonneux, 78180, France 2Fluent Europe Ltd, Sheffield Airport Business Park, Sheffield $9 1XU, UK ABSTRACT The main goal of the current study is to apply a multi-domain RANS-LES approach to the simulation of the flow over the Ahmed body geometry, which is a car type bluff body. Initially, the full configuration is calculated using RANS approach. The LES domain is restricted to the simulation of the flow above the rear slant and in the wake of the body where three-dimensional complex boundary layers separate. Both simulations are performed separately. The LES domain obtains its mean inflow velocity profiles from the separate RANS calculation. At the LES inlet face, the Vortex Method boundary condition of Sergent et al. (2002) is used to reconstruct the turbulent fluctuations. Mean velocity profiles and pressure coefficient distributions are compared with available experiment data, showing good agreement. The sensitivity of the simulation to the type of inlet boundary condition applied at the RANS/LES interface is also discussed.

KEYWORDS

Turbulence Simulation and Modelling, RANS-LES Hybrid Method, Flow Separation, Automotive Aerodynamics.

INTRODUCTION

Despite the fact that RANS simulations are successful in predicting many parts of the flow around the vehicles, they usually fail to predict the effects of separation and unsteady wake on the body. Large Eddy Simulation can have greater success than RANS in predicting the pressure and velocity field at the rear of the vehicles. However, the main obstacle facing the LES simulations is that the near wall resolution required to represent the structure of the flow in the boundary layer (where most of the turbulence energy is produced and dissipated) can be far too expensive. For instance, the flow field above the slant and in the wake of the Ahmed reference model at sub-critical angle is considered to be complex due to the presence of a three-dimensional turbulent boundary layer and a recirculation that consists of longitudinal contra-rotating vortices. The pressure drag in the wake is

648 the major component of total drag acting on the vehicle, and is directly dependent on the rear slant angle. A detailed study on the dependence of rear slant angle on the drag coefficient has been performed both experimentally (Ahmed et al. 1984, Lienhart and Becker 2003) and numerically (Sophn and Gillieron 2002). The critical angle (of rear slant edge) is found out to be 30 ~ above which there is a sudden drop in the drag coefficient as the flow fully separates over the rear slant. However, in the cases of slant angles between 15 ~ and 30 ~ there are strong contra-rotating vortices and the flow separates from the sloping surface and re-attaches at the bottom end of the sloping surface. The wake flow is highly unsteady and unstable and this is probably why RANS simulations perform poorly for these angles. The present study focuses on the 25 ~ Ahmed body case, which has been a subject of a different LES studies in the past. For example, Hinterberger et al. (2004) performed a LES using 18.5 x 106 computational cells whereby wall functions were used at the walls. The results in the lower part of the slant were in poor agreement with the experiments of Lienhart and Becker (2003). Furthermore, Krajnovic and Davidson (2004) performed an LES simulation at a lower Reynolds number, arguing that the flow becomes Reynolds independent. This hypothesis and the fact that separation is induced by the geometry allowed better resolution of the boundary layer. However, this might not be suitable for today's realistic car geometries that may involve rounded edges. In that case, a separation might be dependent on the Reynolds quantities (e.g. the size of the boundary layer). To overcome this difficulty, the goal of the current study is to apply a multi-domain RANS-LES approach to the simulation of the Ahmed body. An LES simulation that includes the whole Ahmed body at realistic Reynolds number would require prohibitively large computational resources. Applying the RANS-LES approach could substantially reduce the computational resources if an appropriate time-dependent inlet condition can be applied at the RANS-LES interface. The method presented in this paper is based on the vortex method proposed by Sergent (2002) for LES-RANS coupling, which was adapted to general flow situation in Mathey et al. (2003).

METHODOLOGY

In order to generate the time-dependent inlet condition, a random 2D vortex method (VM hereafter) is considered. With this approach, a perturbation is added to a specified mean velocity profile via a fluctuating two-dimensional vorticity field (two-dimensional in the plane normal to the streamwise direction). The vortex method is based on the Lagrangian form of the 2D evolution equation of the vorticity and the Biot-Savart law. A particle discretization is used in order to solve this equation. These particles or "vortex points" are convected randomly and carry information about the vorticity field. IfN is the number of vortex points and S the area of the inlet section, the amount of vorticity carried by a given particle i is represented by the circulation F.i and an assumed spatial distribution r I :

I ~rSk(x,y) C,(x,y) = 4

3N(2 ln(3)- 3In(2)) ~2

r/(Y)=

/

(1)

~2

e2O-2

1 2e2~2_1 2~O-2

(2)

Where k is turbulence kinetic energy. The parameter ~ provides control over the size of a vortex particle. The resulting discretization for the velocity field is thus given by: _.

u

=

1N EF

,=,

i (xi -x)xz

1 -2,.I

2

[

1-e

- 2o.~

2o-2

e

(3)

649 Originally in Sergent (2002), the size of the vortex was fixed by an ad-hoc value of cy. In order to make this method generally applicable, a local vortex size is specified through a turbulent mixing length hypothesis. Consequently, cy is calculated from the known profiles of mean turbulence kinetic energy and mean dissipation rate at the inlet according to: 2 0 " - c k 3/2 /~"

(4)

Where c=0.16. In order to ensure that the vortex always belongs to the resolved scales, the minimum value of ~ in Eqn. 4 is bounded by the local grid size. The sign of the circulation of each vortex is changed randomly after a characteristic turbulent time scale (x)has passed. In the present work, a simplified linear kinematic model is considered for the stream wise velocity fluctuations. This model mimics the influence of the 2D vortex on the stream-wise mean velocity field. This approach combines the advantages of preserving fully spatial coherence of the vortex and being independent of x (contrary to the stochastic model originally suggested by Sergent, 2002). If the mean stream-wise velocity U is considered as a passive scalar, the fluctuation u' resulting from the transport of U by (where ~ is the planar fluctuating 2D velocity field as computed by the VM) can be modelled by u ' = - v ' . g where g i s the unit vector aligned with the mean inlet velocity gradient VfJ. When this mean velocity gradient is equal to zero, a random perturbation can be considered instead. .....,

.....,

...,,

NUMERICAL M E T H O D

All calculations reported in this paper have been obtained using a development version of FLUENT V6.2, a general-purpose control-volume code, with RANS and LES turbulence models. The steady RANS run used a second-order upwind scheme for spatial discretization. The LES runs used a second-order implicit scheme for temporal discretization and central differencing for spatial discretization. For all runs the SIMPLEC algorithm was used for the pressure-velocity coupling. The details of the finite-volume method are described in Kim et al. (1998). The numerical aspects relevant to the implementation of the LES are described in Cokljat (1999).

Turbulence Modelling The v2-f model implemented in Fluent (Cokljat et al. 2003) based on Durbin's k-e-v z model (Durbin, 1995) is considered for the RANS simulation. The model is a four-equation model based on transport equations for the turbulence kinetic energy (k), its dissipation rate (e), a velocity variance scale (vZ), and the elliptic relaxation function (f). The model does not require any damping or wall function to capture the near wall turbulence The subgrid-scale stress model, used in this study, is the WALE model of Nicoud and Ducros (1999). The model considers the symmetric part of the square of the velocity gradient tensor. With this spatial operator, the WALE model returns the correct wall asymptotic y3 variation of vt, for wall bounded flows, and can take into account all the relevant mechanisms (local strain and rotational rates) responsible for the kinetic energy dissipation.

APPLICATION

TO THE FLOW AROUND AHMED CAR MODEL

A full configuration of a reference Ahmed car model is first resolved with a steady RANS approach. The Reynolds number of the flow, based on the incoming velocity and car height H is Re=7.68xl 05, as in the experiment of Lienhart and Becker (2003). The RANS computational domain covers half of the entire body to make use of the symmetry of the geometry. The mesh (Figure 1) is refined inside the boundary

650 layer and inside the wake. The boundary layer shown in Figure 1 has been resolved using 24 layers of prism while the rest of the grid consists of tetrahedral cells. The body is placed in an open channel with a cross section of 10 H x 7 H. The channel inlet is located at 6.3 H from the front face of the body and the channel outlet is located at 10 H from rear face of the body. The grid containing 4.5 million cells was created using Gambit and Tgrid. No slip boundary conditions were used on the surface of the body and on the channel floor. Symmetry (slip) conditions were used on the top and side faces of the channel. The LES domain is restricted to the simulation of the flow above the rear slant and in the wake of the body where three-dimensional complex boundary layers separate and reattach and where strong longitudinal unsteady contra-rotating vortices exist. A block structured grid (see Figure 2) containing 1.6 million cells was created with the pre-processor Gambit. The boundary layers are fully resolved with a resolution above the slant of y+<2. The LES domain obtains its inflow velocity profiles from the separate RANS calculation. At the LES inlet face, the Vortex Method boundary condition of Sergent (2002) is used to reconstruct the turbulent fluctuations.

Figure 1: Unstructured mesh for the V2F RANS simulation.

Figure 2: Block structured mesh for the Large Eddy Simulation

RANS Results

Figure 3 shows the mean velocity profile comparison of the RANS simulation with the experiment data of Lienhart and Becker (2003) at several locations in the symmetry plane upstream of the body and on the top of the body. The RANS simulation perfectly predicts the mean stream-wise and wall normal velocity profiles up to x---0.223 m. Thus the mean velocity profiles at x=-0.222 m upstream of the slant, the kinetic energy profile and the specific dissipation rate can be considered as inlet boundary conditions for

651 the LES simulation using the VM. Further down-stream (Figure 4 and Figure 5) the velocity profiles become less accurate. Prior to reattachment, the mean stream-wise velocity profiles are in reasonable agreement with the experimental value. However the flow remains dettached along the slant further downstream. Inside the near wake (Figure 6) the simulation fails to predict the correct structure of the flow.

j

x=-0.1442 6.00e+02

x=-1.262

9

5.00e+02

4.00e+02

x=-1.162 x=-1.062

x=-0.962

x=-0.862

x=-0.562

x=-0.362 x=-0.262

6.00e+02

5.50e+02

5.00e+02

y 4.50e+02

y 3.00e+02 2.00e+02

4.00e+02

1.00e+02

3.50e+02

3.00e+02

O.OOe+O0 0

20

40

60

80

100

120

140

160

0

. 50

. 100

.

. 150

u

. 200

. 250

. 300

350

400

u

Figure 3 9RANS simulation: mean stream-wise velocity profiles upstream (left) and on the top (right) of the Ahmed Body; experiment (symbols) and RANS (solid curve). x=-0.183

x=-0.163 x=-0.143

x=-0.103

x=-0.083 3.60e+02 -

3.80e+02

3.40e+02

3.60e+02

Z

x=-0.123

4.00e+02 -

3.40e+02

3.20e+02

J

3.00e+02

f/

x=-0.063 x=-0.043

x=-0.023

x=-0.003

3.20e+02

Z

3.00e+02

2.80e+02

2.60e+02

2.40e+02

2.80e+02

-50

0

50

100

u

150

200

250

-50

0

50

100

150

200

250

u

Figure 4" RANS simulation. Mean stream-wise velocity profiles over the slant; experiment (symbols) and RANS (solid curve).

RANS/LES Results

A time-averaged mean velocity LES profiles for the separation zone above the rear slant are given in Figures 7 and 8. A good agreement is found with the experimental data for the mean stream-wise and wall normal velocity profiles. It should be also noted that the flow reattachment is well predicted by the simulation (around x=-0.103 m). These profiles are compared with a simulation performed with a random noise at the inlet (RAND hereafter) instead of the Vortex Method.

652

x=-0.183 x=-0.163 x=-0.143 x=-0.123 x=-0.103

4.000+02- x=-0.083 x=-0.063 x=-0.043 x=-0.023 x=-0.003 3.800+02

'

3.600+02 3.40e+02

y 3.20e+02

y 3.40e+02 o~

,%

9

3.00e+02 2.80e+02

3.00e+02 2.80e+02

2.60e+02 -20

0

20

40

60

80

100

2.40e+02

120

-20

0

20

40

60

w

80

100

120

100

120

w

Figure 5 9RANS simulation. Mean wall normal velocity profiles over the slant; experiment (symbols) and RANS (solid curve). 6.00e+02

6.00e+02

4.00e+02 y

4.00e+021 /

3.00e+02

y

2.00e+02 I

2.00e+02 ~,'%

O'OOe+O0-50

3.00e+02t ~ ~ -

100

u

~

1.00e+02

15

200

250

O.OOe+O0-20

0

.

.

.

.

w

Figure 6 9RANS simulation. Mean stream-wise and wall normal velocity profiles inside the wake; experiment (symbols) and RANS (solid curve). Despite close agreement at the beginning of the slant, the random noise simulation over-predict the separation. This is particularly noteworthy for x=-0.143 m up to x=-0.043 m for the mean stream-wise velocity profiles in Figure 7 and for the mean wall normal velocity profiles in Figure 8. The mean streamwise velocity fluctuations are given in Figure 9. Prior to re-attachment, the correct levels of stream-wise and wall normal fluctuations (not shown here) are well predicted by both simulations. Further downstream, both simulation slightly under-predict the fluctuations but with a reasonable agreement. RAND slightly under-predicts the level of the fluctuations compared to VM, but this can hardly explains why the flow re-attach further downstream with RAND. However since the structure of the flow is highly three-dimensional, analysis of the results in the symmetry plane only may not be sufficient to take into account all the mechanisms present in this flow. Mean stream-wise and wall normal velocity inside the wake of the Ahmed body are given in Figure 10. An overall good agreement is found with the experiment data for all quantities. Results are more accurate in the near wake region (x< 0.238 m) but the mesh is also more and more stretched as moving downstream. A close agreement is found between two Large Eddy Simulations, regardless whether the Vortex Method or the random noise is considered at inlet. At this location, the flow is massively separated. The turbulent production mechanisms inside the Ahmed body wake play a dominant role and upstream turbulence history is not an issue anymore.

653

x=-0.183

4.00e+02

Z

x=-0.163

x=-0.143

x=-0.123

x=-0.103

3.80e+02

3.40e+02

3.60e+02

3.20e+02

3.40e+02

J

~

Z

x=-0.083

3.60e+02 -

x=-0.063

x=-0.043

x=-0.023

x=-0.003

3.00e+02

3.20e+02

2.80e+02

3.00e+02

2.60e+02

2.40e+02

2.80e+02

-50

0

50

100

150

u

200

250

-50

0

50

100

150

u

200

250

Figure 7: Mean stream-wise velocity (U) profiles above the slant; Experiment (symbols), Vortex Method (solid curve) and Random noise (dashed curve). x=-0.183

x=-0.163

x=-0.143

x=-0.123

x=-0.103

3.600+02

x=-0.083

x=-0.063

x=-0.043

x=-0.023

x=-0.003

3.40e+02

3.20e+02

Z

3.40e+02

Z

3.00e+02

2.80e+02

2.600+02

2.40e+02

2.80e+02

-20

0

20

40

60

80

100

120

. 0

-20

.

.

. 40

20

w

. 60

80

100

w

Figure 8: Mean wall normal velocity (W) profiles above the slant; Experiment (symbols), Vortex Method (solid curve) and Random noise (dashed curve). 4.00e+02 -

3.60e+02

3.800+02

3.40e+02

3.200+02

Z

3.40e+02

Z

3.20e+02

3.000+02

2.80e+02 f

2.40e+02 0

10

20

30

40

50



60

70

80

90

100

0

10

20

30

40

50

60

70

80

90



Figure 9: Mean stream-wise velocity fluctuations () profiles above the slant; Experiment (symbols), Vortex Method (solid curve) and Random noise (dashed curve).

654

x=-0.183

x=-0.163

x=-0.143

x=-0.123

x=-0.083

x=-0.103

x=-0.063

x=-0.043

x=-0.023

x=-0.003

I Z

Z

3.000+02 ," =~e

i

i

3.000+02

I

i

I 1.000+02

O.000+00 -50

0

50

I00

150

,

,

200

250

0.000+00

-20

0

20

40

60

80

100

120

140

u

w Figure 10: Mean stream-wise velocity profiles (U) and mean wall normal velocity profiles (W) in the wake region; Experiment (symbols), Vortex Method (solid curve) and Random noise (dashed curve). Finally, the mean velocity streamlines of the flow around the slant and in the wake behind the body are given in Figure 11 and Figure 12. In both figure, a good resemblance is achieved with the experimental picture of Ahmed et al. (1984). These results are also in qualitative good agreement with full LES performed by Krajnovic and Davidson (2004).

Figure 11" Mean streamlines velocity: LES (left) Exp. (fight) from Ahmed at al. (1986) Next, the LES and RANS surface pressure coefficient Cp along the slant are compared with the experimental data of Lienhart and Becker (2003). In Figure 13, a very good agreement is found with the LES for the stream-wise evolution of the pressure coefficient in the symmetry plane. The peak pressure at the top of the recirculation bubble (x < 155 mm) is a little over-predicted, which can be explained by the proximity of the inlet boundary condition and the decoupling hypothesis. The cross-flow distributions at various locations are shown in Figure 14. The pressure drop close to the edges of the slant are the footprints of the stream-wise longitudinal vortices. The location is well predicted by the simulation though peaks are under-predicted as we move downstream. The resulting pressure drag coefficients are given in Table 1. The results of the RANS simulation are also reported in this table. Despite good agreement between the LES and the experimental data of Lienhart and Becker (2003) for the Cp distribution, the overall contribution to the slant pressure drag coefficient Cs is found to be less accurate when compared to the Ahmed data.

655

Figure 12" Mean streamlines velocity: LES (left) Exp. (right) from Ahmed

at

al. (1986)

For time being we have no clear explanation and further investigation will be necessary to address this issue. It should be noted that pressure drag is sensitive to the choice of the pressure reference Pref. Due to the decoupling of the LES domain, for this study Pref was chosen at the top of the outlet. This choice is however confirmed by the reasonable Cp distribution. On the contrary the RANS simulation predict a lower value of Cs, very close to the experimental data. This finding is a surprising as the Cp distribution is not accurately predicted by the RANS simulation as shown in Figures 13 and 14. This might be the result of the errors that compensate each-others, though it is not apparent from the figures where absolute value of Cp is generally widely under-predicted in separated region. -1.00e-01 -2.00e-01 -3.00e-01

9

-4.00e-01

.. oo-"~176176

-5.00e-01

~0 ~ r

-6.00e-01

,'

-7.00e-01 -8.00e-01 -9.00e-01 -1.00e+O0 -1. lOe-,,..O0

-80

-2_00-180-160-140-120-100

-60

-40

-20

0

x (mm)

Figure 13: Cp on slant (symmetry); Experiment (symbols); LES (solid curve); RANS (dashed curve)

CONCLUSION The Vortex Method of Sergent (2002) was implemented in the general purpose CFD solver Fluent. The method was used in conjunction with a zonal RANS/LES simulation of airflow around the Ahmed Body. The vortex method is used to reconstruct the turbulent fluctuations at the RANS/LES interface. Computed statistics compare well with the available experimental data. Despite the fact that in the present study the "RANS flow" is decoupled from the "LES flow", no major influence was seen on the results. This raises hope that a zonal RANS/LES approach can be used as a method of choice for the prediction of the flows around real-life car geometries. The full coupling between the RANS and LES simulation will be the subject of a future study.

656 O.OOo+O0

-_

-2.50o+01

-

_ _ -5.00e+01 _ _ -7.50e+01

i

i

i i

i t

t I,

1 I,

t ,

' 1

-

i

--_ _ _

i

| i i t

-1.00e+02

-1.25e+02

-

-1.75e+02

-

-2.000+02 -2

,

,

.

-1

0

1

.

.

.

2

3

4

Cp Figure 14: Cp distribution on the slant; Experiment (symbols); LES (solid curve); RANS (dashed curve) TABLE 1" DRAG FORCE AND FORCE COMPONENT

Cs Cb Cf CD

RANS 0.144 0.12 0.01 0.364

RANS/LES 0.16 0.098

Exp. 0.145 0.077 0.02 0.285

REFERENCES Ahmed S. R., Ramm G., Faltin G. (1984). Some Salient Features of the Time Averaged Ground Vehicle Wake. SAE Paper 840300. Cokljat D. (1999). Large-Eddy Simulation of Flow around Surface-Mounted Cubical Obstacles, Part I Numerical Aspect. LESFOIL Project Report, Fluent Europe Ltd., Sheffield UK. Cokljat D., Kim S.E., Iaccarino G., Durbin P. A. (2003). A Comparative Assessment of the V2F Model for Recirculating Flows. 41th Aerospace Sciences Meeting & Exhibit,January 6-9, Reno. Durbin P. A. (1995). Separated Flow Computation with the k-g-v 2 Model. AIAA Journal, 33(4):659-664, 1995. Hinterberger, M. Garcia-Villalba, W. Rodi (2004). Large Eddy Simulation of flow around the Ahmed body. In "Lecture Notes in Applied and Computational Mechanics / The Aerodynamics of Heavy Vehicles: Trucks, Buses, and Trains", R. McCallen, F. Browand, J. Ross (Eds.), Springer Vedag, ISBN: 3-540-22088-7. Kim S. E., Mathur S. R., Murthy J. Y., Choudhury D. (1998). A Reynolds Averaged Navier-Stokes Solver Using Unstructured Mesh-Based Finite-Volume Scheme. AIAA-Paper 98-0231. Krajnovic S. and L. Davidson L. (2004). Large Eddy Simulation of the Flow Around an Ahmed Body, ASME Heat Transfer/Fluids Engineering Summer Conference, Charlotte, North Carolina, USA. Lienhart H. and Becker S. (2003). Flow and Turbulent Structure in the Wake of a Simplified Car Model. SAE Paper 2003-01-0656 Fluent 6 User's Guide, Fluent Inc., Lebanon NH03766, USA. Mathey F., Cokljat D., Bertoglio J.P. and Sergent E., (2003). Specification of Inlet Boundary Condition Using vortex Method, in Turbulence, Heat and Mass Transfer 4, Eds: K. Hanjalic, Y. Nagano and M. Tummers, Begell House Inc.. Nicoud F., Ducros F. (1999). Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor. Flow, Turbulence and Combustion, Vol. 62, pp. 183-200, 1999. Sergent E., PhD Thesis, L'Ecole Centrale de Lyon, 2002.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

657

NUMERICAL SIMULATION AND EXPERIMENTAL INVESTIGATION OF THE SIDE LOADING ON A HIGH SPEED TRAIN N. Paradot 1, B. Angel 2, P.-E. Gautier 1, and L.-M. C16on 1 1SNCF (National French Railways), Research and Technology Department, 45 rue de Londres, F-75379 Paris Cedex 08, France 2Renuda Engineering Computation, 24 rue des Patriarches, F-75005 Paris

ABSTRACT An assessment of train safety whilst simultaneously traveling at high speed and being subjected to strong cross winds requires the knowledge of the aerodynamic load on the train. This situation currently affects the French high speed double decker train (TGV Duplex) that travels on the "TGV Mrditerranre" line between Paris and Marseille. For safety reasons, the SNCF has investigated the cross wind sensitivity of the TGV Duplex operating on this line using both reduced scale wind tunnel and hydraulic tank tests and numerical methods. Comparisons between experimental and numerical side and lift forces and lee rail rolling moments acting on the TGV have been made. The leading vehicle experimental pressure distributions have also permitted further comparisons between the experimental and numerical results. Overall, these comparisons showed good agreement between the two sets of results for TGV / wind relative flow angles of 30 ~ or less. Beyond this, the differences between the results are more pronounced with a maximum difference occurring for a relative flow angle of approximately 60 ~.

KEYWORDS railway aerodynamics, cross wind effect, RANS simulation, wind tunnel experiments

INTRODUCTION

The assessment of railway safety under strong winds requires the knowledge of the aerodynamic load on the train. SNCF investigate the sidewind sensitivity of double deck french high speed train TGV Duplex (which currently operates on the TGV Mrditerranre line between Paris and Marseille) through both numerical computations and reduced scale tests in wind tunnels and hydraulic tanks. This paper presents the crosswind study whose objectives were the understanding of the flow around the train and the evaluation of the influence of atmospheric turbulence on wind induced forces and moments acting on the train running on flat ground.

658 Wind tunnel experiments have been carried out at CSTB (Centre Scientifique et Technique du Bfttiment) in Nantes. 1/15 th and 1/25 th scale static models were used in the working section 5 wide, 6m high of the Jules Verne climatic low turbulence wind tunnel, on flat ground (with two tracks on ballast) and on 6m and 12.5m high embankments respectively, with yaw angles with respect to the relative wind direction from 0 ~ to 90 ~ (Sanquer et al (2004)). Leading vehicles of models were made of unpolished Plexiglas and fitted with pressure taps in order to provide pressure measurements using sensitive differential piezoresistive sensors through pneumatic connectors. Those instantaneous pressure measurements yield both distribution of the load on the vehicle and global load (through integration of the pressure field), with a good agreement compared to the aerodynamic forces and moments measured with a 6 component dynamometer fitted under the turntable.

PREVIOUS W O R K

Over the past four years, in order to complement the experimental work discussed above, the SNCF has undertaken a series of numerical experiments to investigate the effects of side winds on TGV Duplex aerodynamic loading. Initial studies at 1/15 th scale, Fauchier and Gr6goire (2000), involved 3D simulations of the TGV Duplex on flat ground for varying TGV / wind relative flow angles using the commercial CFD code Star-CD. The results of these simulations in terms of lateral and lift force and turning moment agreed with the CSTB data for relative flow angles of less than 40 ~ Above this angle, the loadings were over predicted with a maximum difference occurring at approximately 60 ~ There were however several differences between the numerical and experimental geometry and differences in operating conditions. In terms of the numerical geometry, the rails and ballast and the wind shields protecting the equipment in the pantograph bay were not included which would have the effect of increasing the loading, (Angel and Paradot (2004)). In terms of the numerical operating conditions, the inlet velocity profile was assumed constant. If the wind tunnel boundary layer profile had been used, the loading would decrease, (Angel and Paradot (2004)). Several 2D studies at 1/15 th scale have also been undertaken at SNCF, again usi.ng Star-CD. These numerical experiments investigated the effects of wind barriers and merlons of various sizes, porosity and physical location on the aerodynamic loading of a TGV Duplex passenger car. The car was situated on either flat ground or on a suitably scaled 6m or 12.5m embankment and was operating under 90 ~ cross winds. Results from these studies indicated that both wind barriers and merlons can reduce the aerodynamic load seen by the passenger car with certain heights, thicknesses and physical locations being more efficient at doing so than others. Some of the simulations were also undertaken at scale 1 and involved assessing the effects of the atmospheric boundary layer velocity profile on the passenger car aerodynamic loading. In these simulations, the velocity profile was scaled such that the velocity at roof level matched that seen in the experimental studies of the CSTB. These results also indicated that a reduction in aerodynamic loading could be expected. 1/15 th scale 3D studies of the TGV Duplex situated on an appropriately scaled 6m embankment and either on the windward or leeward rails, for a range of TGV / wind relative flow angles, were undertaken by Paraiso and Paradot (2003) using Star-CD. The TGV Duplex geometry used was the same as Fauchier and Gr6goire (2000) with the addition of rails and ballast although the pantograph bay wind shields were not included. A uniform velocity profile at inlet to the domain was also assumed. The results of these simulations were compared with CSTB experimental data. Overall, the numerical aerodynamic loadings of lateral and lift force and turning moment agreed with the

659 experimental results for flow angles of 30 ~ or less. Above this angle, the numerical simulations over predicted these coefficients with a maximum difference at approximately 60 ~ CURRENT W O R K At the time of writing this publication, the SNCF is further investigating the effects of side winds on TGV Duplex aerodynamic loading in a 4 part research program. These four parts are: 1) Assessment of the current numerical methodology, implementing changes where appropriate and validating these changes against experimental data; 2) Assessment of the effects of the atmospheric boundary layer on aerodynamic loading; 3) Mesh independence studies; 4) Assessment of the effects of unsteady flows on aerodynamic loading using either DES or LES turbulence models. GEOMETRY AND MESH

The geometry used for the results presented here is at 1/15th scale and consists of the TGV Duplex situated on fiat ground with rails and ballast present. The TGV Duplex itself has 3 vehicles: the leading car, a passenger car 1.5 times normal length and a "false leading car" which is representative of the actual leading car but does not contain all the geometric details. This is the same geometry as used by Fauchier and Grrgoire (2000) except for the addition of the rails and ballast and pantograph bay wind shields. Figure 1 illustrates the leading car geometry used in these simulations:

k

Z

<2wind,

b)

LI

Figure 1: TGV Duplex leading car geometry and pressures taps distribution(a), coordinates system (b) The domain size is as follows: 9 Upstream of the leading car: 1 TGV duplex length; 9 Downstream of the false leading car: 5 TGV Duplex lengths; 9 Upstream on the RHS of the TGV Duplex (+ve y): 4 TGV Duplex heights; 9 Downstream on the LHS of the TGV Duplex (-ve y): 25 TGV Duplex heights; 9 Above the TGV Duplex: 10 TGV Duplex heights. Fluid properties used are for air at standard room temperature and pressure giving o=l.205kg/m 3 and /z=l.81 e-5 Pas. The reference pressure was set to a corresponding 1 bar. The meshes used for these simulations are of a block structured type consisting of several million, predominantly hexahedral, cells, as showed in Table 1. The normal cell size at wall around the leading car is lmm for all meshes.

660

TABLE 1 MESH DESCRIPTION

Mesh size (in millions) 6.2

Mesh 0 Mesh 1 Mesh 2

Larger tangential cell size at wall (power car) 5mm 10mm 5mm

2.5 3.3

Mesh 3

2.5 m m

OPERATING CONDITIONS The operating conditions for the simulations presented here are as follows: TGV / wind relative flow angles of 30 ~ 60 ~ and 90 ~ for the CSTB boundary layer profile at inlet with a varying velocity profile and constant turbulent intensity and mixing length. TGV / wind relative flow angles of 15 ~ 30 ~ 60 ~ and 90 ~ for the atmospheric boundary layer profile at inlet with: o a varying velocity profile and constant turbulent intensity and mixing length; o varying velocity, turbulent intensity and mixing length profiles. The atmospheric boundary layer profiles were calculated using the data of Hemon (1993) and cross referenced with the data of ESDU (1992,1993). Figure 2 illustrate the various profiles used in these simulations. 5O 4O

{ 30 ~ 20

I

mixing lentgh [m] i ',

10

0

turbulent intensity {

==

==

~

',

== ,,

i

0

1

2

3

Height (m 1115th scale)

4

0

1

2

3

4

Height (m 1/15th scale)

Figure 2: CSTB and Atmospheric velocity profiles (left) and atmospheric turbulent intensity and mixing length profiles (fight)

661 ASPECTS OF CFD M O D E L L I N G Numerical scheme

The numerical scheme used the SIMPLE algorithm for relaxation factors of 0.5 for momentum and turbulence discretisation scheme MARS, with a 0.5 blending factor, momentum and turbulence equations. Closure was achieved with default Star-CD constants.

pressure-velocity coupling with under and 0.3 for pressure. The 2 nd order was used for spatial resolution of the by using the RNG k-s turbulence model

Boundary conditions applied were of the following types: 9 9 9 9 9

Inlet for all inlet boundaries with the appropriate boundary layer profile(s); Pressure for all outlet boundaries; Symmetry for the boundary above the TGV; Wall "no slip" for all TGV surfaces and walls downstream of the TGV; Wall "slip" for all walls upstream of the TGV.

This last condition was imposed in order to preserve the boundary layer profile between the domain inlets and the TGV Duplex.

Calculation times and memory requirements All simulations were run on a 4 processor ES40 Compaq cluster (type memory-channel). Each processor is an alpha 21264 type with a 667 MHz CPU and 1 Gbyte of Ram. A typical iteration took approximately 2 minutes per processor and the simulations either converged or achieved steady state leading car aerodynamic loadings between 2000 and 3000 iterations.

Convergence Convergence was assumed when all residuals decreased to 104 or less or the integral of the aerodynamic loading on the leading car became constant. For all of the calculations presented here, residuals decreased to around 10 4 for a constant aerodynamic loading integral. RESULTS The simulations discussed in this section are numbered as follows: TABLE 2 SIMULATIONNUMBERING case [

2

3

4

5

6

15~ ~ 600,90 ~

15~ ~, 600,90 ~

15~ ~, 600,90 ~

1

2

3

Angle

30o,600,90 ~

15~ ~, 60o,90 ~

Mesh

0

0

15~ ~ 60~ ~ 0

V

C

A

A

C

C

C

I

O

0

A

U

U

U

L

U

U

A

U

U

U

Here, the letter:

662 9 V indicates the velocity : I the turbulent intensity : L the length scale; 9 C the CSTB profile : A the Atmospheric profile : U a uniform prOfile. It should be noted that cases 1 to 3 are practically identical to the experimental conditions in terms of geometrical set-up and operating conditions. The results for the 11 simulations are presented in Figure 3 below for the lateral force Cy, the lift force Cz and the rolling moment CMx (given at the lee-rail point), normalised by Srer=l 0m 2 and Lrer=l 8.4m.. 14

exp. CSTB

12 >.,

o

!

0L .

- -9o - - 1: mesh 0 / CSTB profile

10 9

2: mesh 0 / Atm profile (velocity)

8

~

6

N

4

L._

,3: mesh 0 / Atm. profile (complete) --13---4: mesh 1 / CSTB profile ---0-- 5: mesh 2 / CSTB profile 6:mesh3 / CSTB profile

0

10

20

30

40

50

60

70

80

90

y a w angle [~ + 6

exp. CSTB

--o--- 1: mesh 0 / CS'I'B profile

N 5 O

2: mesh 0 / Atm. profile (velocity)

i

o9

4

X

3

3: mesh 0 / Atm. profile (complete)

- - D - - 4 : mesh 1 / CSTB profile

0

_~2

5: mesh 2 / CSTB profile 6:mesh3 / CSTB profile

0

~ 0

10

l 20

30

J 40

50

J

~

60

70

80

90

90

y a w angle [~ 0 0

10

20

I

I

30

40

50

60

70

80

I

I

I

I

I

E 0 0| 0

9 t

X

.~ -1 o-1 "~-1 $ -2 . . -2

exp. CSTB

2: mesh 0 / Atm. profile (velocity)

~-1 E

O

.-

----o--- 1: mesh 0 / CSTB profile

3: rm_.sh 0 / Atm. profile (complete)

---D--- 4: mesh 1 / CSTB profile ---0-- 5: mesh 2 / CSTB profile

........ ! ....... ~................ i ....... ~........ ,,........ ~-....... ~........ . . . . . . . i

~

i

~

i

i

i

i

6:mesh3 / CSTB profile

y a w angle [~

Figure 3" Comparison of aerodynamic coefficients

663

Table 3 presents these results in terms of percentage difference from the experimental values measured at CSTB. TABLE 3 % DIFFERENCE BETWEEN EXPERIMENTAL AND NUMERICAL DATA

case I

1 (300,600,90 ~

[

2 (15 ~176176176

3 (15 ~176176176

I

Cy

+19

+47

+23

-52

-65

-51

-62

-32

-43

-50

-42

Cz

+32

+15

+31

-84

-62

-68

-65

-90

-71

-64

-48

CMx

+29

+54

+34

-59

-61

-48

-57

-41

-44

-47

-36

1-

5.3% Leading Car

_

05 ~

0.~0.2%

1~E

Leading Car

16.1% Leading Car

0.5

- 2 - c:~,

,\

..

o _0

0 -0.5 -1

-3.515-

<>/r

-1~..5 L) -2

-2.5

~ I

~

ndwardSide

-4

:or ....

,,oo ....

"70 -'F' ' ' 100 ~

200 ....

Distance(mm)

1 --

26.6% Motrice

1- -

200 , , 300 I . . . . 40I(; ,, Distance(mm)

50.4% Leading Car

A=l-.I,, "0

'---r

Roof

LeewardSide

,I, I, ,I .... I .... I 100 200 300 400 Distance(mm)

~,.#

1

~

_

96.3% Leading Car

0.~'~\

-1

~

-1.z

~

.

~2

/ /~O / .

"z~/ /

'~k

t P

-2

-3.5~-

,//~

-5~-5.5~-

I/ ~'

-4

~

I

~ ~'C~?S' I, RQo,I, L.e~.a.rd?i.de. 0

100

200 300 Distance(ram)

400

~

/

o'~

1 , ~

l "5': I , Win(~vard:id.e t " R~ I ?eewardzSide` -

6

.

5 ~ Distance(mm)

I

WindwardSide I Roof !

_

3.51 9

,

,

,

,

I

,

,

,

,

I

200 400 Distance(ram)

LeewardSide

, , , I t

600

Figure 4: Comparison of pressure coefficients: experimental (CSTB) and computed (mesh0 and mesh 1) results The i esults f;r case 1 overestimate the lateral and lilt forces and the turning moment when compared to the CSTB data. The reasons for this overestimation are not entirely clear. The numerical set-up, in

664 terms of flow regime, turbulence model, numerical algorithm and the use of a 2 nd order discretisation scheme renders it virtually identical to that of Fauchier and Gr6goire (2000). There are geometry differences but the addition of these geometric features, i.e. rails and ballast and pantograph bay wind shields, which increases the aerodynamic loading tends to be offset by the implementation of the wind tunnel velocity profile (Angel and Paradot (2004)). This leaves the possibility that either the domain size, which is different to that of the CSTB wind tunnel, or the geometry of the TGV itself is not the same as that used in the wind tunnel tests and that one or both of these elements are having an effect on the aerodynamic loading. Figure 4 shows the relative good agreement between measured and computed pressure coefficients on the leading car, for different sections of the mock-up. The application of the atmospheric boundary layer velocity profile significantly reduces the lateral loading seen by the leading car. This is not surprising, given that this velocity profile contains less energy than that of the CSTB profile. The effect on the lift and turning moment is the same. Figure 5 presents the differences in the velocity vector field around the leading car between cases 1 and 6 at 25% of the leading car length for a relative flow angle of 60 ~

Figure 5: Comparison of velocity vectors (left) and static pressure (fight) at 25% leading car length for cases 1 (top) and 2 (bottom) with a yaw angle of 60 ~ The most significant difference here is the lower acceleration on the corner of windward side and the roof of the leading car for case 2. This results in a lower velocity around the roof flowing and into the flow field on the leeward side. The underbody acceleration is also less for this case. This results in a more balanced static pressure distribution around the leading car, as shown by Fig. 5, which illustrates clearly the reductions of the lateral and lift forces and the turning moment seen by cases 2 to 3. The total pressure distribution at this position is shown by Figure 6. The lower velocity around the roof and underbody has resulted in a smaller boundary layer separation from the leading car hence, the development of smaller vortices with lower losses. This scenario extends the whole length of the leading car, as illustrated by Figure 6 at 90% leading car length. This figure clearly shows that the atmospheric boundary layer velocity profile has resulted in vortices emanating from the roof and the underbody with lower losses. No huge difference is observed between computations with mesh 0 (case 1) and 1 (case 4).

665

Figure 6: Comparison of total pressure at 25% (left) and 90% (fight) leading car length for cases 1 (top), 2 (middle) and 3 (bottom) with a yaw angle of 60 ~ An increase in turbulent intensity and the length scale via the implementation of the appropriate atmospheric profiles, case 3, has marginally increased the lateral forces seen by the leading car at the relative flow angles simulated. Figure 7 presents a comparison of the turbulent kinetic energy and its dissipation respectively at 25% leading car length for a relative flow angle of 60 ~

Figure 7: Comparison of turbulent kinetic energy (left) and turbulent energy dissipation (fight) at 25% leading car length for case 6 (top) and 10 (bottom) with a yaw angle of 60 ~ The effect of these variables on the velocity and total pressure fields at 25% leading car length is visible but more pronounced towards the end of the leading car. Figure 8 illustrate this effect at 90% leading car length. The vortex structure on the leeward side of the leading car has been completely modified by the higher turbulence and dissipation levels introduced at inlet. The reasons for this are not yet completely understood and are the subject of further investigation.

666

Figure 8: Comparison of velocity vectors and total pressure at 90% leading car length for case 6 (top) and 10 (bottom) with a yaw angle of 60 ~ CONCLUSIONS The previous and current numerical research presented in this paper has enabled the SNCF to establish and start to implement a numerical methodology for undertaking this type of simulation. They have also provided a useful insight into the effects of various operating conditions on TGV Duplex aerodynamic loading and the change in flow structures around the leading car.

REFERENCES

Angel B. and Paradot N. (2004). Validation du cas sans couche limite atmosphrrique. SNCF internal report ref. RT/SFC/AERO-BA/040730. ESDU issue 82026 (1992) Strong winds in the atmospheric boundary l a y e r - Part I: mean hourly-wind speeds. ESDU issue 85020 (1993) Characteristics of atmospheric turbulence near the ground- Part II ; Single point data for strong winds (neutral atmosphere). Fauchier C. and Grrgoire R. (2000). Simulation numrrique d'une maquette de TGV Duplex l'rchelle 1/156me soumise ~t l'action d'un vent traversier. SNCF report ref. RT/SFC/VENTSLN5/000503/01/A. Hemon P. (1993) Un modrle de couche limite atmosphrrique pour l'application au grnie civil. IAT Note 231/93. Paraiso G. and Paradot N. (2003). Etude numrrique de l'rcoulement autour d'un TGV Duplex. SNCF internal report. Sanquer S., Barr6 Ch., Dufresne De Virel M. and C16on L.-M. (2004) Effect of cross winds on high-speed trains: development of a new experimental methodology. Jr. Wind Eng. lnd. Aerodyn. 92, 535-545

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

667

LARGE-SCALE INSTABILITIES IN A STOVL UPWASH FOUNTAIN A. J. Saddington, P. M. Cabrita and K. Knowles Aeromechanical Systems Group, Cranfield University, RMCS, Shrivenham, Swindon, Wiltshire, SN6 8LA, UK

ABSTRACT

The fountain flow created by two underexpanded axisymmetric, turbulent jets impinging on a ground plane was studied through the use of laser-based experimental techniques. Velocity and turbulence data were acquired in the jet and fountain flow regions using laser doppler velocimetry and particle image velocimetry. Profiles of mean and rms velocities along the jet centreline are presented for nozzle pressure ratios of two, three and four. The unsteady nature of the fountain flow was examined and the presence of large-scale coherent structures identified. A spectral analysis of the fountain flow data was performed using the Welch method. The results have relevance to ongoing studies of the fountain flow using large eddy simulation techniques. KEYWORDS

Fountain, impinging, jet, LDV, PIV, STOVL, supersonic, turbulent, underexpanded, VSTOL NOMENCLATURE D h NPR Pa POc F

S x

Y Z Zt

Y

nozzle exit diameter (0.0127 m) nozzle height above ground plane nozzle pressure ratio {POc/Pa} atmospheric static pressure settling chamber total pressure radial distance from nozzle axis nozzle spacing co-ordinate parallel to the ground plane in the plane of the jet centres (see Fig. 1) co-ordinate parallel to the ground plane in the plane of the fountain axis (see Fig. 1) co-ordinate normal to the ground plane (see Fig. 1) h-z ratio of specific heats

668

INTRODUCTION The wall jets created by the impingement on the ground of the individual jet flows from a jet-lift short take-off and vertical landing (STOVL) aircraft (with two or more nozzles) meet at a stagnation line and interact to form an upwards-flowing 'fountain' that interacts with the airframe. In some cases this can provide a beneficial lift-generating ground cushion (Anderson, 1990). The fountain flow regime may also give rise to a variety of undesirable characteristics: hot gas ingestion (HGI); ground erosion; acoustic, thermal and pressure loads on the airframe (Curtis, 2002). Early work revealed that the fountain is sensitive to small imbalances between the jets and appears to be unstable under certain conditions (Skifstad, 1970). Further studies have reported turbulence intensities in the fountain upwash as high as 50% and a much greater rate of spreading in the fountain when compared to a free jet (Barata, 1993). Velocities normal to the axis of the fountain upwash have been found to be in the region of twenty to thirty percent of the jet exit velocity, at least for incompressible experiments (Barata, 1993; Behrouzi and McGuirk, 1993). Positioning of the fountain is largely dependent upon the momentum ratio of the wall jets with differences in their relative thicknesses causing the fountain to appear to lean (Siclari, Hill and Jenkins, 1981). Nozzle angle relative to the impingement plane and nozzle splay angle also play an important part in the fountain location and development (Siclari et al., 1981; Behrouzi and McGuirk, 1993). Visualisation of multijet impingement has revealed the presence of large scale coherent structures, evolving from the main jets, propagating through the wall jets and dissipating in the fountain (Wohllebe and Siclari, 1978; Saripalli, 1983; Kibens, Saripalli, Wlezien and Kegelman, 1987; Cabrita, Saddington and Knowles, 2002), with possible crossover of these structures from one wall jet to the opposite side of the fountain (Childs and Nixon, 1987). This may be responsible, in part, for the large degree of spreading associated with fountain flows. Whilst it is evident that the fountain upwash flow is unsteady, only limited data on the transient characteristics of this flow region are available. Early experiments relied on intrusive measurement techniques to provide mean pressure data (Hall and Rogers, 1969) with unsteady pressures on the ground plane being used to infer additional information (Knowles, Wilson and Bray, 1993). Techniques such as particle image image velocimetry (PIV) and laser doppler velocimetry (LDV) offer the possibility of detailed non-intrusive measurements in the fountain region. Previous investigations using these techniques have used water as the working fluid (E1-Okda and Telionis, 2002) or were limited to a single nozzle pressure ratio (Elavarasan, Venkatakrishnan, Krothapalli and Lourenqo, 2000). This paper reports on PIV and LDV measurements of the three-dimensional fountain flow-field generated by the impingement of two axisymmetric, compressible, under-expanded, turbulent jets on a ground plane. FACILITIES AND INSTRUMENTATION

Impinging jet facility The experiments were conducted in a dedicated impinging jet facility at Shrivenham. A schematic of the twin impinging jet flow field is shown in Figure 1. The test rig consisted of a small cylindrical settling chamber with an internal diameter of 230 mm and a height of 210 mm. It has two internal screens and removable nozzle mounting plates that enable the configuration of various nozzle spacing and splay angles. Air is supplied to the settling chamber by two Howden screw-type compressors capable of a maximum flow rate of 0.9 kgs-1 at pressures of up to 7 bar gauge. The pressure in the settling chamber was adjusted using a pneumatic CompAir A 119 computer-controlled valve that was capable of maintaining the nozzle pressure ratio (NPR) to within +0.2% of the desired value. The settling chamber

669

was instrumented with a thermocouple and pressure transducer that provide information on the stagnation conditions.

Figure 1: Schematic of the experimental set-up illustrating the main flow characteristics.

PIV

The PIV equipment consisted of a New Wave Gemini Nd:YAG double-pulsed laser which, through the use of a combination of spherical and cylindrical lenses, created a light sheet approximately one millimetre thick, positioned perpendicular to the impingement plane and passing through the plane defined by the nozzle axes. The PIV double-pulsed image pairs were acquired using a Kodak Megaplus ES 1.0 digital camera with a maximum resolution of 1008 x 1016 pixels at a rate of 15 Hz. The camera was fitted with a 60 mm (f2.8) Nikon lens and placed normal to the light sheet. This allowed for a maximum field of view of 60 mm x 30 mm, giving a resolution of 81/zm per pixel. The time separation between the two laser pulses was varied between 2.6/zs and 6.6/zs according to the calculated isentropic jet exit velocity for the particular NPR. The jets were seeded using JEM Hydrosonic long-lasting fluid droplets of approximately 1/zm diameter generated by a TS19306 Six-jet atomizer connected to a Clarke compressor. The ambient air was seeded with smoke particles produced by a Le Maitre Turbo Mist fog generator. LDV

LDV measurements were made using a Dantec system consiting of a Lexel Model 95 water-cooled Argonion laser, a Dantec 60 x 41 transmitter with 60 x 24 fibre optic manipulators, 57N20 burst spectrum analyser, a 2D FiberFlow probe and a 1 m focal length lens with a 2 x beam expander. The lens and beam expander combination created a measurement volume of 0.15 mm x 4.2 mm x 0.15 mm. Alignment was completed using a 35/zm pinhole with a photovoltaic cell. Time-averaged LDV data was derived from a sample size of 10000. Dantec Burstware data processing software was used to export the time series data, that was further processed using MATLAB.

METHODOLOGY The configuration used for the present study comprised two identical 63.5 mm long axisymmetric convergent nozzles with an exit diameter, D of 12.7 mm. The distance between the nozzle centres, S was 88.9 mm (7 D). The ground plane consisted of a one metre square, 10 mm thick, aluminium plate. The distance between the nozzle exits and the impingement plane was varied between 2.4D and 8.4D. Data were recorded for NPRs between 1.05 and 4.

670

Approximately 500 PIV image pairs were acquired to determine the mean flow, and first order flow statistics. The commercial software, Insight v3.3, developed by TSI was used to analyse the images. A cross-correlation algorithm was used to process the images and extract the instantaneous vector field. Interrogation windows of 32 x 32 pixels were employed in the processing. The size of the interrogation window was chosen to allow for a minimum of 10 seeding particles per interrogation area and to allow for the maximum in-plane particle displacement to be less than one quarter of the size of the interrogation window (Keane and Adrian, 1990). Inherent to PIV processing are the spurious vectors, which on average, accounted for less than 3% of the total. They were removed using a pass-band filter followed by a local median filter. The resulting empty spaces were filled with interpolated values from the surrounding area. LDV data was acquired in two regions: in the jet along a line described by the nozzle axis and in the fountain along a line joining the nozzle centres at heights of z/D = 0.5, z/D = 1 and z/D = 2. RESULTS AND DISCUSSION Results are presented for the twin impinging jet experiment described above. Data will be presented for NPR -- 3 at a range of nozzle height to diameter ratios. The discussion is divided into two parts: the jet flow and the fountain flow.

Jet flow The underexpanded impinging jet flow field can be divided into three main regions (Donaldson and Snedeker, 1971): the free jet region, where the flow is primarily inviscid and contains the series of expansion and compression waves; the impingement region, which is characterised by strong gradients that alter significantly the local flow properties; the wall jet, which consists of a radial redirection of the jet flow after impingement. Figure 2 shows the centreline mean and rms axial velocity as a function of distance from the nozzle exit plane, z' (z' = h - z, where z is the ground-normal coordinate) for NPRs of two, three and four. The plots are superimposed onto time-averaged schlieren images of the experiment to the same scale. Spatial correlation of the LDV data with the schlieren images is very good. At an NPR of two, there is some acceleration of the flow as it leaves the nozzle exit and some evidence of a shock structure, although it is very weak. The rms velocity initially falls but then rises again as the potential core starts to decay. At the higher NPRs the expansion and recompression through the shock structure is clearly evident. The mean velocity data agree well with previous free jet experiments (Saddington, Lawson and Knowles, 2004) up to z'/D ~ 5 - 6 after which point the shock cells appear to shorten as impingement approaches. In general, the rms velocity component increases with distance from the nozzle exit plane, varying through the shock structure approximately in phase with the mean velocity fluctuations. Similar fluctuations in rms velocity were observed when the jets were operated without a ground plane. Figure 3(a) shows PIV-derived velocity magnitude contours with superimposed streamlines at h / D = 2.4. The effect of the presence of the fountain is clearly visible in the entrainment process occurring on both sides of the jet. The left hand side (outer side) streamlines display a continuous entrainment of ambient air along the jet shear layer and wall jet. The right hand side (inner side) displays a different pattern. This is due to the presence of the fountain where the pressure is sub-atmospheric (Abbott and White, 1989), inducing a higher velocity in the fountain-facing wall jet (inner side) and a thickening of the inner shear layer of the jet. It is difficult to determine with absolute certainty the accuracy of the PIV measurements because detailed velocity field data are not available in the literature for this flow field. A comparison of PIV-derived Mach number data with a simple shadowgraph (Figure 3(b)) indicates, however, that the PIV has been able to capture the location of the important features of the underexpanded impinging jet.

671

Figure 2: Centreline mean and rms axial velocity for an impinging jet posed on time-averaged schlieren images to the same scale.

Figure 3: PIV-derived images of the left-hand-side jet

(hiD =

(h/D =

10, NPR = 3) superim-

2.4, NPR = 3).

672

Fountain flow Figure 4 shows two sequential instantaneous velocity magnitude contours at a non-dimensional nozzle height of h/D = 4.4. The images are separated temporally by 67 ms. The behaviour of the fountain flow is quite different from the jet flow in that the instantaneous velocity fields do not correlate well with the time-averaged one (Figure 5) - indicating that it is a highly unsteady flow. This unsteadiness results from the collision of two wall jets that contain vortical structures and are themselves highly turbulent. The instantaneous velocity fields show a high degree of asymmetry, the presence of large-scale vortical structures and a stagnation region whose location has been observed to vary randomly. Although the instantaneous fountain flow is somewhat incoherent, it is clear from the images (Figure 4) that it is inclined relative to the vertical. Through the observation of a sequence of these instantaneous velocity fields it appears that the fountain inclination is related to the strength and location of the dominant vortical structures. Unfortunately, the frame rate at which these data were acquired does not allow for the temporal resolution of these structures.

Figure 4: Instantaneous velocity magnitude contours of the fountain flow

(h/D = 4.4, NPR = 3).

Figure 5 shows the mean velocity magnitude contours for two nozzle heights, h / D -- 4.4 and h / D = 6.4 at an NPR of 3 and reveals two well-defined recirculation regions formed between the fountain and impinging jet flows. At the lower height (Figure 5(a)) there is a recirculation zone to the left of the fountain, which is inclined to that side. A second recirculation zone is partially visible in the top right hand corner of this figure, however, it is the left-hand vortex which appears to dominate the flow. At a slightly higher height (Figure 5(b)) the flow is more symmetrical. Two recirculation zones are clearly visible centred laterally almost exactly midway between the fountain axis and the jet axis. The recirculation zone on the right-hand side is positioned lower than the one on the left and as a consequence, the fountain is inclined to the right. The fountain appears to be a bi-stable flow which, with symmetrical geometry and jet conditions, in the mean, would produce a fountain with no inclination. The time-averaged velocity contours were, therefore, quite surprising. The sample size of 500 is not large but should be sufficient to give a good representation of the mean. Further investigation showed the fountain inclination to be most likely dependent upon asymmetries in the geometry of the experimental rig. This was confirmed by swapping the nozzles over which, despite there being no measureable difference in geometry, changed the fountain inclination direction. These observations confirm the sensitivity of the fountain to small

673

imbalances in system (Skifstad, 1970), which has also been reported in other recent experiments on twin-jet fountain flows (Elavarasan et al., 2000; E1-Okda and Telionis, 2002).

Figure 5: Time-averaged velocity magnitude contours of the fountain flow (NPR = 3). Figure 6(a) shows the instantaneous vertical, w velocity profiles along the geometric fountain axis (i.e. x/D -- 0) for two sequential instants in time. Also shown in Figure 6(a) is the mean profile. The instantaneous profiles clearly show the unsteady nature of the fountain. The upwash velocity shows instantaneous values of up to 160 ms-1, approximately double the peak mean value, which is attained at z/D ~ 0.7. The point at which the vertical velocity is zero at z/D = 0 indicates the location of the fountain stagnation line. Only one of the instantaneous profiles shows a vertical velocity close to zero at z/D = 0, which supports the previous observation that the fountain stagnation line moves from one time instant to another. The mean profile shows a rapid increase from approximately zero velocity to its peak at z/D ,~ 0.7. The velocity then decreases asymptotically towards zero as vertical velocity decays. It is interesting to note that for the nozzle heights tested (Figure 6(b)) the mean vertical velocity profile is independent of nozzle height. The slight difference for the h / D = 4.4 case is a consequence of the large fountain inclination (see Figure 5(a)) producing a vertical velocity lower than expected. The irregular time-spaced LDV data was resampled at two times the mean data rate using a nearest neighbour resampling technique (Broersen, 1999). Figure 7 shows LDV histograms of the x-component of velocity at NPR = 3, h/D = 4.4 and z/D = 1. Either side of the fountain centre (figures 7(a) and 7(c)) the histograms present a skewed distribution typical of flows with strong gradients of turbulence intensity. At the centre of the fountain (Figure 7(b)), the histogram displays two peaks, clearly indicating an oscillation. For the spectral analysis around 200000 samples were used. These were divided into segments of 211 samples and processed using the Welch method and a Hanning window in order to reduce the spectral leakage. The Welch method splits a set of data into smaller sets and calculates the periodogram of each set. The frequency domain coefficients arising from calculating the periodograms are averaged over the frequency components of each data set. This results in a power spectrum that is a smoothed version of

674

Figure 6: Vertical velocity profiles along the geometric fountain axis (NPR = 3). the original, with less noise. In this way, the Welch method enables low pass filtering of the data. The normalised power spectral density is shown as a function of frequency in Figure 8 for three nozzle heights and two measurement heights at an NPR of 3. In general, the frequency spectra is somewhat unremarkable with the exception of the data for h/D -- 4.4, z/D = 1 where a clear peak is seen in the normalised power spectral density at approximately 240 Hz indicating that under these conditions the fountain does appear to exhibit a characteristic frequency. Fountain flow instabilities have previously been observed to occur at similar non-dimensional nozzle heights (Elavarasan et al., 2000) although no attempt was made to quantify the frequency.

Figure 7: Distribution of sampled velocity in the fountain, (NPR = 3,

h/D = 4.4, z/D = 1).

675

CONCLUSIONS An experimental study was carried out into the fountain flow created by two underexpanded axisymmetric, turbulent jets impinging on a ground plane. The entrainment characteristics of the jets were found to be altered by the presence of the fountain, which is shown to be a highly unsteady flow. Instantaneous velocity fields of the fountain show a high degree of asymmetry, the presence of large-scale vortical structures and a stagnation region whose location was observed to vary randomly. The mean vertical velocity profile through the fountain is shown to be independent of nozzle height. Spectral analysis of the laser doppler velocimetry data identified a characteristic frequency in the fountain of approximately 240 Hz at a nozzle height of 4.4 diameters and a measurement height of 1 diameter.

10 0

"o

g ~

o

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| I" /

I-

10 .2 10 ~

,~. -.*

• -;-

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~ ~L ~ ~ ~, ~

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101

~r

=

10 2

10 3

"~ I~ ='~

10 4

Hz

Figure 8: Frequency spectra in the fountain, (NPR = 3).

ACKNOWLEDGEMENTS This work was partially funded by the Engineering and Physical Sciences Research Council under grant GR/R42894/01 and their support is gratefully acknowledged.

REFERENCES Abbott, W. A. and White, D. R. (1989), The effect of nozzle pressure ratio on the fountain formed between two impinging jets, Technical Memorandum P 1166, RAE. Anderson, S. B. (1990), Jet-powered v/stol a i r c r a f t - lessons learned, in 'International Powered Lift Conference and Exhibit', London, UK, pp. II. 1.1-II. 1.14. Barata, J. M. M. (1993), 'Fountain flows produced by multi-jet impingement on a ground plane', AIAA Journal of Aircraft 30(1), 50-56.

676

Behrouzi, P. and McGuirk, J. J. (1993), Experimental data for cfd validation of impinging jets in crossflow with application to astovl flow problems, in 'AGARD Conference Proceedings CP-534, Fluid Dynamics Panel Symposium', Winchester, UK. Broersen, E M. T. (1999), The performance of spectral quality measures, in '16th IEEE Instrumentation and Measurement Technology Conference', Venice, Italy, pp. 751-756. Cabrita, E M., Saddington, A. J. and Knowles, K. (2002), Unsteady features of twin-jet stovl ground effects, in 'International Powered Lift Conference and Exhibit', Williamsburg, VA, USA. Paper no. 2002-6014. Childs, R. E. and Nixon, D. (1987), Turbulence and fluid/acoustic interaction in impinging jets, in 'International Powered Lift Conference and Exhibit', Santa Clara, CA, USA, pp. 447-458. SAE paper 872345. Curtis, E (2002), A review of the status of ground effect/environment technologies, in 'International Powered Lift Conference and Exhibit', Williamsburg, VA, USA. Paper no. 2002-5985. Donaldson, C. D. and Snedeker, R. S. (1971), 'A study of free jet impingement, part 1. mean properties of free and impinging jets', Journal of Fluid Mechanics 45(2), 281-319. E1-Okda, Y. and Telionis, D. E (2002), Experimental investigation of twin jet impinging on the ground with and without a free stream, in 'International Powered Lift Conference and Exhibit', Williamsburg, VA, USA. Paper no. 2002-5976. Elavarasan, R., Venkatakrishnan, L., Krothapalli, A. and Lourengo, L. (2000), Supersonic twin impinging jets, in '38th Aerospace Sciences Meeting and Exhibit', Reno, NV, USA. Paper no. 2000-0812. Hall, G. R. and Rogers, K. H. (1969), Recirculation effects produced by a pair of heated jets impinging on a ground plane, Contractor Report CR-1307, NASA. Keane, R. D. and Adrian, R. J. (1990), 'Optimization of particle image velocimeters', Measurement Science and Technology 1(11), 1202-1215. Kibens, V., Saripalli, K. R., Wlezien, R. W. and Kegelman, J. T. (1987), Unsteady features of jets in lift and cruise modes for vtol aircraft, in 'International Powered Lift Conference', Santa Clara, CA, USA, pp. 543-552. SAE paper 872359. Knowles, K., Wilson, M. J. and Bray, D. (1993), 'Unsteady pressures under impinging jets in cross-flows', AIAA Journal 31(12), 2374-2375. Saddington, A. J., Lawson, N. J. and Knowles, K. (2004), 'An experimental and numerical investigation of under-expanded turbulent jets', The Aeronautical Journal 108(1081), 145-152. Saripalli, K. R. (1983), 'Visualization of multijet impingement flow', AIAA Journal 21(4), 483--484. Siclari, M. J., Hill, W. G. and Jenkins, R. C. (1981), 'Stagnation line and upwash formation of two impinging jets', AIAA Journal 19(10), 1286-1293. Skifstad, J. G. (1970), 'Aerodynamics of jets pertinent to vtol aircraft', Journal of Aircraft 7(3), 193-204. Wohllebe, E A. and Siclari, M. J. (1978), 'Fountain and upwash flowfields of multijet arrangements', Journal of Aircraft 15(8), 468--473.

10. Aero-Acoustics

This Page Intentionally Left Blank

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

679

DIRECT NUMERICAL SIMULATION OF LARGE-EDDY-BREAK-UP DEVICES IN A BOUNDARY LAYER P. R. Spalart, ~M. Strelets, 2 and A. Travin 2 1Boeing Commercial Airplanes, P. O. Box 3707, Seattle, WA 98124, USA 2 Federal Scientific Center "Applied Chemistry", St. Petersburg 197198, Russia

ABSTRACT Turbulent boundary layers create aerodynamic noise inside all vehicles, and especially inside jetliners. The objective of this project is to modify the turbulence, not to reduce skin friction, but to weaken the wall pressure fluctuations, or to shift them to less damaging frequencies. In order to benefit an entire windshield, the effect would be sustained over about 25 boundary-layer thicknesses, 6 , which far exceeds the common rule that relaxation takes about 10 6. Devices are studied by Direct Numerical Simulation, which is free of modeling and provides the full details of the pressure field. The Reynolds-number limitation of DNS is tolerable, because the focus is on the larger eddies. The multi-block implicit numerical method can represent fairly complex devices at a manageable cost. Inflow turbulence is provided by a recycling procedure, derived from that of Lund, Wu and Squires but much simpler. It occupies less than 5 6. Co-rotating vortex generators are tried first. They reduce the turbulence intensity away from the wall, as hoped, but actually intensify the wall pressure fluctuations. Large-eddy-break-up devices resembling a highway bridge are tried next, and succeed in reducing the fluctuations, but only over about 6 6. Thus, the technology is not successful yet for a windshield, but might be applied to other windows, and the simulation methodology appears to be well developed and of some interest.

KEYWORDS Turbulence Control, Boundary Layer, Pressure Fluctuations, Direct Numerical Simulation, Turbulent Inflow Condition, Vortex Generators, Large-Eddy-Break-Up Device

INTRODUCTION Aerodynamic noise is a factor for pilot and passenger comfort in most vehicles. Insulation and other measures add to the cost and weight of the system, and a reduction at the source would be most welcome. Obvious sources of aerodynamic or "airframe" noise are regions of massive separation, such as behind a rear-view mirror, a windshield wiper, a spoiler, or landing gear. Separation off a smooth surface, especially caused by a shock wave, is also troublesome, and has been controlled with vortex generators.

680 Many cars have small devices on their mirrors or sun-roofs; these cannot suppress massive separation, but it appears that they alter the turbulence sufficiently to provide a noise benefit. This state of the art makes the interior noise generated by an attached flow appear as the minimum that must be accepted, but our purpose here is precisely to manipulate an attached boundary layer and lower its impact, although only over a limited area, nominally a single window on an airliner. The applicability of a successful concept could be very wide, and include ground vehicles. On the other hand, the turbulent boundary layer (TBL) is rightly considered as a very "robust" flow type, one for which modifications beyond a few % are very difficult to effect with simple devices. A case in point is the large amount of work expended on riblets, which has yet to lead to any industrial application. The same applies to LargeEddy-Break-Up (LEBU) devices. It proves very difficult to defeat the turbulence in setting the skin-friction coefficient, short of obtaining laminar flow. In addition, the TBL is robust in the sense of returning to normal quite rapidly, in a distance of the order of 10 d , where d is the full thickness of the TBL. An exception is that embedded streamwise vortices extend much farther than 10 6 ; this has made them common tools for TBL control especially when the area in need of control is not anchored. However, they act in the direction of increasing skin friction, which at first sight would increase the pressure fluctuations since the pressure rms is proportional to the skin friction, multiplied by a weak function of Reynolds number. Their effect on turbulence intensity will be considered later. Other devices will also be considered, drawing on past attempts at TBL control, but with the new situation that drag reduction is not the objective. For instance, if we imagine completely suppressing the turbulence over a region of interest, and paying a penalty only downstream, this could work. However, the study is limited to simple static external devices, with some size constraints due to visibility, so that such a deep reduction is not expected. For drag reduction, even a figure such as 3% is impressive, but noise is measured with larger ratios, and a marginal alteration of the TBL would not be sufficient. Since aerodynamic noise is not the only source and is typically comparable with engine and ventilation noise, a reduction by 3dB for this one source is a plausible minimum for what would make a modification worth implementing. This represents a 30% reduction of the pressure rms, or halving the energy, which must result from a major manipulation. Another practical aspect is that the wall pressure field is filtered by the response of the flexible structure, be it skin or window. Therefore, the pressure rms in itself is only a first indication of the eventual benefit inside the cabin. The dominant frequencies, propagation velocities, and spatial pattern (summarized the two-point, two-time correlations) also matter, and need to be provided by the simulations or experiments (unless the device is tested in situ, in flight). This new set of objectives opens a very wide design space and, also, rules out any simulation approach other than Direct Numerical Simulation (DNS) or Large-Eddy Simulation. The dimensions of the device are free within practical limits, as is the type of device, as is the mechanism postulated to obtain the desired effect on the turbulence. DNS is very restrictive in terms of Reynolds number, but it is assumed that the manipulation will primarily target the largest eddies, especially since (at least in the windshield region on an airliner) the critical region of the spectrum is for a Strouhal number, St, below 1. Here, St is defined as f 6 / U e , where f is frequency and U e the edge velocity of the BL. These largest eddies are not Reynolds-number dependent, when normalized with the shear stress and d. DNS has the great advantage of making no turbulence assumptions, which is essential when studying a new type of turbulence modification, and of providing detailed flow visualizations which would suggest why a candidate device is working or not, and in which direction to change its dimensions. A promising design found in simulations may be then tested in a wind tunnel, with a number of high-performance probes, or directly in flight, which is faster and faithfully includes all the transmission effects. However, the DNS of this flow is challenging, in that relatively complex geometries must be treated, and the simulation must be "spatial", meaning that it has good turbulent inflow conditions, covers a strong modification of the TBL, its relaxation over 25 6 or so, and an outflow. The solutions applied to this challenge are presented in the next section, while the following section presents the results, and the last one the outlook for this line of work.

681

SIMULATION APPROACH Numerical Aspects

The DNS ofa TBL with embedded devices appears to represent a step in complexity, and is possible with the NTS code (Strelets, 2001) thanks to its combination of implicit numerics and moderate numerical dissipation. The code provides the capability for time-accurate computations with structured multi-block overset grids, and employs an implicit second order in time flux-difference splitting scheme similar to that of Rogers and Kwak (1988). The flow is treated as incompressible. The inviscid fluxes in the governing equations are approximated with centred 4th -order accurate differences everywhere, except for the close vicinity of the devices' stagnation points where 3rd order upwind approximations are used. The viscous fluxes are approximated with centred 2nd order differences. At every time step, the resulting finitedifference equations are solved with the use of Gauss-Seidel relaxations by planes and sub-iterations in "pseudo-time".

Turbulent Inflow Conditions This is the more novel aspect of the method. It is desirable to have a standard TBL encountering the flowcontrol device, without a lengthy approach region. Lund, Wu and Squires (1998, LWS) show that inflow conditions based on random numbers can easily take 20 b" to recover; this would be quite wasteful when the useful region is of the order of 25 6. Their recycling method, in some sense, wasted about 8 6, which is much smaller, and could probably have been reduced further. The approach here is strongly inspired by LWS, but simpler, and a few separate improvements were made. LWS applied a different treatment to the inner and outer regions of the TBL, and blended the two formulas. This may appear optimal, but it requires rescaling the friction velocity in addition to the BL thickness, and introduces several arbitrary parameters and relatively complex formulas. The approach fails to take advantage of two favorable facts. One is that the near-wall turbulence re-generates itself much faster than the outer-region turbulence, and therefore little damage is done by applying the outer-region scaling throughout. The other is that when the recycling station is taken quite close to the inflow, which is desirable in terms of cost, the conflict between inner-region and outer-region scaling essentially vanishes. For these reasons, the present approach uses a single re-scaling. Similarly, corrections to the wall-normal velocity component v have very little effect, and are omitted. Finally, here there is no auxiliary simulation. The primary simulation provides its own recycling data. The approach consists in using the velocity field at a "recycling" station X---Xr that is part-way inside the domain to provide the velocity field at the inflow, x=0. Since the BL is slightly thicker at Xr, a rescaling in the wall-normal direction y is the minimum alteration that is needed. Let the thicknesses at x=0 and X=Xrbe 60 and 8 r . Then the inflow condition for the velocity vector U is: U(O,y,z,t) = U ( x r , y 8 r / 80,z + Az),

where the spanwise shift, Az, is introduced in order to keep turbulence at the inlet and recycling sections out of phase (in the simulations discussed below, the value of Az was set equal to half the spanwise period). This is done to disorganize any durable spanwise variations of the mean flow, which would otherwise be recycled and possibly take much time to be damped by spanwise diffusion.

682

The process is controlled by two parameters: the Reynolds number difference from inflow to recycling A R x = x r U e / v and the ratio 60 / 6 r . Assuming the approximate scaling given by Schlichting for the BL thickness, the flow should stabilize with the following value for the inflow thickness: 6 = 0.37(v1.[

Ue )

ARx

14/5

(r / r 5/4 -- 1

This gives fine control over the inflow thickness (even though the factor 0.37 is not exact and the slight difference in growth rates between the momentum thickness and the BL thickness is neglected), in that the response to small changes in A R x and 6 r / 60 can be predicted. (once the flow is well established). The initial conditions for such a simulation are of some importance. For instance, random perturbations that are too weak or to inadequate in length scales can very well "die out." If so, the simulation will become laminar (with a much thicker BL than expected with turbulence, because of the different scaling). Small values of A R x and d r / 60 are cost-effective in the steady state, but impede the maturation of the turbulence that is needed initially. Therefore, it could be helpful to start with the recycling station father downstream, and then move it closer to the inflow. The procedure used here is as follows. A RANS solution is calculated with the desired inflow thickness, and perturbations added to it. These perturbations are obtained from a code that is used to initialize simulations of homogeneous turbulence. It yields "cubes" of turbulence of size 0.063, each with random phases, which are placed side-by-side on the wall after their velocity field is multiplied by a shape function, so that they are zero at the wall and in the freestream. This works well, although nothing is done to inject the correct Reynolds shear stress. A mature solution is obtained in a few flow-through times.

RESULTS Baseline Flow

The simulation needs to be validated in terms of the standard TBL quantities, and of the wall pressure field. The unit length used below is roughly 1m, for the boundary layer near the windshield of an airliner. The unit Reynolds number is 2.2 105 (which is far below full-size values) and the inflow thickness 60 is approximately 0.03, so that the momentum-thickness Reynolds number is 666. The length of the domain is 0.9, and the spanwise period is 0.12, which is over twice the outflow BL thickness. The recycling position Xr is 0.09, giving ARx=2.104 and 6 r/60=1.07. The grid spacing in wall units is about Ax+ = 30, streamwise and Az+= 11, spanwise, which is typical in DNS, and the first wall-normal spacing Ay+ is about 1. This gives about 2.5 million grid points. The time step is 103, normalized with edge velocity and unit length. Figure 1a shows that the skin-friction coefficient C f , although it first dips by about 7%, has stabilized by x-0.12. C f is 0.0051 where R0 =740, which is within 2 or 3% ofthe accepted value. The behaviour ofthe displacement and momentum thicknesses (not shown) is very smooth. Figure l b compares the four Reynolds stresses with past DNS studies, and is again very satisfactory except for a slight deficit in u' at the higher Reynolds number. This suggests that the numerical method, grid spacing, time step, and recycling procedure are all capable of representing this TBL accurately.

683

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Figure 1: Streamwise profile of the skin friction, and wall-normal profiles of Reynolds stresses Figure 2 compares the DNS wall-pressure spectrum with experiments and empirical laws. Considering the emphasis on fairly low frequencies, an outer-layer scaling was chosen, based on 6 and U e . However, the inner-layer scaling based on wall units also gives a good comparison, in the appropriate frequency range. The scaling also uses the skin friction, instead of the freestream velocity; this is not supported by a full consensus, but represents only a mild controversy for the present purpose. In the range of S t between about 0.2 and 1, the DNS results are within 3dB of the experiments. For higher S t , they fall off due to the low Reynolds number, and do not have any of the S t l range that is expected from theory. This is consistent with the absence of any extended logarithmic layer in the velocity profile. The region below S t .~ 0.2 is more delicate. The time sample appears adequate, judging from the smoothness of the spectra, but the mechanics of the simulation system do not favour very low frequencies. Outer-layer eddies take about 0.13 time units to travel from inflow to recycling station, which corresponds to S t ~ 0.2. Thus, frequencies much below S t = 0.2 cannot be expected to have physical meaning. In experiments or in a global simulation, the spectrum cannot be expected to be unique to arbitrarily low frequencies either. At some point, the spectrmn will show a memory of the transition process, and depend on whether transition was natural and contained turbulent spots, for instance. Wind-tunnel unsteadiness will also be felt. The empirical laws are level all the way to St=0 more out of simplicity than based on conclusive data or theoretical reasons. The experiments shown, however, suggest that the universal behaviour extends at least down to S t of the order of 0.04. In summary, care is needed in the region below 0.2, but it does not contain much of the energy. The bulk of the turbulence modification is expected in the region between 0.2 and 2, in which the DNS appears reliable and the Reynolds-number effect is weak. Therefore, meaningful simulations of the TBL manipulation are possible.

,31 ............. "g;;:~:;o::~:;::;::~:;::::~.,~...:,.. ",. ~., o

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Figure 2: Spectra of wall pressure fluctuations, baseline case and experiments

684

Vortex G e n e r a t o r s ( V G 's)

The simulation shown fitted a grid around the VG's, but others were run using a body force to generate the vortices, and gave close results at a lower cost (Liu et al, 1996). The VG's are 0.025 tall and spaced by 0.12, and at 18~ incidence; their planform is typical, and the circulation is expected to be of the order of 0.02. The grid is shown in figure 3; the total number of points is now near 3 million. Figure 4 illustrates the turbulence in its baseline state and after the manipulation. The typical random bulges are first seen, and indicate that the spanwise period is adequate; two-point correlations of wall pressure were studied and although they do not firmly reach 0 within the domain, they fall sufficiently low for evaluating new devices. They also agree well with experiment, roughly following the law e x p ( - 7 z / 6 ) ; this will be essential when addressing the transmission through structure or glass. The dominant propagation velocity is near 2 / 3U e , and is also essential; it would be slightly higher at full-size Reynolds numbers. The figure then shows how, farther downstream, the single vortex per period gradually overturns the TBL, without strongly suppressing the smaller eddies. This creates a large bulge, but one that does not wander in time.

Figure 3: Grid near Vortex Generator

Figure 4: Visualization of the boundary layer upstream and downstream of the VG's.

685 The rationale behind VG' s for the present purpose was the following. Rapid rotation is known to suppress turbulence, giving hope that the wall pressures would also be calmed. Another sketchy argument is that the TBL may maintain about the same level of skin friction and "total" turbulence energy, but the Reynolds stresses are now split into two contributions, x 1and x2. xl is created by the deviation from the time average, whereas ~2 is created by the deviation from the spanwise average. Potentially, the mixing due to the vortices would take over to some extent, so that the second component would deplete the first component. Only that first component contributes to noise. This idea is supported by figure 5. The baseline case has zero x2, because it is homogeneous spanwise. The VG case has a strong x2, but xi is indeed tangibly weaker over the lower half of the boundary layer, near y - 0.01, for all x beyond about 0.4.

Figure 5: The two components of Reynolds shear stress, and the total, in the baseline and the VG case. The unfortunate finding is that the alteration of the turbulence energy does not result in any benefit in terms of wall pressure, as seen in figure 6. In fact, the spectrum is everywhere higher, by up to 2dB. The conclusion must be that this device is of no value for the purpose of interior-noise reduction. Some experimentation with VG's of smaller sizes and spacing failed to produce any promising designs. The attention then turned to completely different devices.

686

Figure 6: Spectra of wall pressure fluctuations with and without VG's.

Large-Eddy-Break-Up Device This device is closer to traditional thinking in turbulence reduction, but recall that the objective here is different, so that the failure of LEBUs to find drag-reduction applications does not rule out success here. The design, seen in Figure 7, is sized to interfere with the large outer-layer eddies, and also to create drag in the inner layer, which should depress the skin friction and consequently the pressure activity. Designs were tried with the main blade at the BL edge, and with the blade well inside the BL. Results are presented for the latter type, which is somewhat more effective. It extends roughly from x = 0.33 to x = 0.38. The entry region had to be lengthened by 0.05, because the pressure gradient due to the LEBU interfered with the recycling. Thus, moderate adjustments are needed to the recycling procedure depending on the cases.

Figure 7. Large Eddy Break Up device grid Figure 8 shows that downstream of the LEBU, the turbulence has lost its largest bulges, and generally created smaller eddies in the outer layer, which can be beneficial in terms of de-energizing larger eddies. Thus, the turbulence alteration appears consistent with the postulated mechanism, without being dramatic.

687

The pressure rms shown in Figure 9 presents a mixed result. On one hand, the target 30% reduction is achieved over the x-interval [0.4,0.65] (coupled with a similar reduction in skin friction); thus, this device is much more successful than the VG's. On the other hand, this interval of length 0.25 is only about 6 times the undisturbed BL thickness at the location of the LEBU; the recovery is quite complete after 10 5. While this behaviour is reassuring in terms of placing the DNS close to the accepted estimate, it represents a failure in terms of seeking an invention. The "robustness" of boundary-layer turbulence is in full force.

Figure 8. Visualization of the flow with LEBU. Vorticity contours. Flow from left to right. Figure 10 shows the spectra, with energy summed up for each 1/3 octave. This helps identify the frequency range which contains most of the energy, when the frequency axis is logarithmic. The decrease is quite uniform over the spectrum, and close to expected factor of 1/ 2, or 3dB (the baseline value of skin friction is used everywhere for normalisation). The new peak near St= 0.25 may be spurious, which can be tested by altering the inflow length or otherwise altering the possible feedback mechanism from x = Xr to x = 0, but the energy it contains is modest.

Figure 9: Wall pressure rms with and without LEBU.

688

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1/3-octave ,,

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Figure 10" Wall pressure spectra with LEBU (centred at x = 0.35) OUTLOOK This study has so far failed to produce a viable invention. However, the simulations have screened out two concepts that appeared to have potential, much more economically than wind-tunnel or flight tests would have. The numerical system appears quite capable of producing extensive quantitative information in this relatively complex physical situation, with proper caution towards the low end of the frequency spectrum and a sound strategy for scaling with Reynolds number. Feedback within the recycling region might be more noticeable with embedded devices than in a simple TBL, but several tools are available to detect such a feedback. Direct Numerical Simulations of a TBL, with several million points and with obstacles properly gridded, are now possible on Personal Computers. The simulations not only indicate the value of a device, but provide visualizations to suggest the next generation or at least the size of the next candidate. The search for a successful device continues.

References Rogers S. E., Kwak D. (1988). An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations, AIAA Paper 88-2583-CP. Lund T. S., Wu X. and Squires K. D. (1998). Generation of turbulent inflow data for spatiallydeveloping boundary-layer simulations. J. Comp. Phys. 140, 233-258. Liu J., Piomelli U., Spalart P. R. (1996). Interaction between a spatially-growing turbulent boundary layer and embedded streamwise vortices. J. Fluid Mech. 326, 151-180. Spalart P. R. (1988). Direct simulation of a turbulent boundary layer up to R0 = 1410. J. Fluid Mech. 187, 61-98 Strelets M. (2001). Detached Eddy Simulation of Massively Separated Flows. AIAA Paper 2001-0879.

Engineering TurbulenceModelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

689

BLADE TIP F LO W AND NOISE PREDICTION BY L A R G E - E D D Y SIMULATION IN H O R I Z O N T A L AXIS WIND TURBINES O. Fleig 1, M. Iida 2 and C. Arakawa 3 1 Department of Mechanical Engineering, The University of Tokyo 2 Department of Electrical Engineering, The University of Tokyo 3 Graduate School of Interfaculty Initiative in Information Studies, The University of Tokyo 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan

ABSTRACT The purpose of this research is to investigate the physical mechanisms associated with the tip vortex noise caused by rotating wind turbines. The flow around a NACA0012 blade section at a Reynolds number of 2.87 • 106 is calculated using compressible LES (Large-Eddy Simulation). Very small eddy scales are captured, leading to reduced SGS dependency and improved predictions of the noise spectra in the high frequency domain. The flow field and aerodynamic noise caused by a NACA0012 blade with a tip in an incident flow is simulated. Quantitative agreement is observed between simulation results and experimental measurements. The broadband aerodynamic noise emitted by a large rotating wind turbine blade of arbitrary shape is predicted, with particular emphasis on tip noise. A large-scale unsteady compressible LES combined with direct noise simulation using 300 million grid points is carried out on the Earth Simulator. Simulations for the W1NDMELIII wind turbine rotor blades are performed for two blade tip geometries. It was found that the ogee type tip shape reduces overall aerodynamic noise by 2 dB. A better understanding of the physical phenomena of tip vortex noise and a comparison of the level of tip noise emitted by different blade tip shapes will contribute towards designing new wind turbine blades and blade tip shapes with reduced noise emission. KEYWORDS LES, acoustics, wind turbines, turbo machinery, Earth Simulator.

INTRODUCTION

There is a strong need to predict blade flow and noise for various engineering applications. LES is a very promising tool for flows which are difficult to handle with statistical RANS turbulence models, especially vortex shedding and massively separated flow problems. LES can provide a more accurate turbulent frequency spectrum and time-accurate pressure fluctuations as well as acoustic phenomena and unsteady forces acting on a blade. However, LES at the Reynolds numbers occurring in

690 aeronautical applications is extremely challenging and often computationally not feasible. Research has been carried out concerning LES of airfoil and blade section flow, mostly at high angles of attack due to the importance of stall effects and massive separation in aerospace applications. LES has recently become a viable method due to increases in computer power. Mellen et al. (2002) provides an overview of the results obtained in the course of the LESFOIL project carried out by various universities and institutions across Europe. The project assessed the feasibility of LES for the computation of the flow around an airfoil. Test case is the Aerospatiale A-airfoil at a Reynolds number of 2.1x106 and an angle of attack of 13.3 degrees which corresponds to maximum lift. Various subgrid-scale models, near-wall models, grid resolutions and spanwise extents of the computational domain were investigated. Only Onera employed a grid fine enough to fully resolve the near-wall region of the flow in all three directions without the need for a wall model. It was concluded that the grid resolution has a much larger impact on the quality of the computation than the models and that it is the main factor in achieving a good solution for the flow. Although research has been carried out in airfoil flows, numerical simulations of tip vortex flow are extremely rare. To the authors' knowledge, wall resolved LES simulations of tip vortex flow of a finite blade at Reynolds numbers in the order of 106 have not been performed to date due to the excessive grid requirements in all three dimensions. The simulation of a f'mite blade with a tip requires a large number of grid points in the spanwise direction. Although there are still a lot of problems to be solved in terms of models to improve predictions of airfoil and blade section flow by LES, it is worth attempting at this stage to simulate tip vortex flow. Tip vortex flow is an important industrial problem. The aim of this work is to carry out the first simulation of tip vortex flow and related acoustics at high Reynolds number at angles of attack of up to 10 degrees. The present LES simulation is the first and largest such simulation. It can give us new insights into the physical phenomena causing tip vortex formation and tip noise. The simulation has been made possible by the Earth Simulator (Yokohama, Japan), developed by NEC in 2002. It is presently the fastest supercomputer in the world and allows us to refine the computational grid in all three dimensions. The Earth Simulator is a highly parallel vector supercomputer of distributed-memory type with 5120 arithmetic processors. It has a total peak performance of 40 Tflops and total main memory of 10 TB. The effects of the streamwise and spanwise grid resolution on the simulation results are investigated in the simulation of the NACA0012 blade section, using periodic spanwise boundary conditions. Special attention is paid to determine the grid characteristics required to capture the natural transition near the leading edge. Using the so-obtained optimum grid characteristics for the present flow solver, a finite blade with a tip with NACA0012 blade section is simulated and turbulent velocity intensity contours aft of the trailing edge associated with the tip vortex as well as acoustic noise spectra are compared with experimental measurements. Finally, numerical flow and noise LES simulation of the WINDMELIII two-bladed upwind wind turbine rotor is described. This simulation is the first ever attempt to predict the broadband noise emitted by a wind turbine blade purely by CFD. Analysis of the flow field in the blade tip region will be presented for the actual tip shape and an ogee type tip shape. The noise levels predicted in the farfield will be shown for both tip shapes. NUMERICAL METHODS

Large-Eddy Simulation The goveming equations are the filtered unsteady three dimensional compressible Navier-Stokes equations. The effects of the subgrid-scale eddies are modeled using the Smagorinsky model (1963). The Van Driest wall damping function (1954) is used to correct the excessive eddy viscosity predicted

691 by the Smagorinsky model near the wall. The solution is advanced in time using a second-order implicit approximate-factorization Beam-Warming approach with Newtonian sub-iterations and threepoint backward differencing for the time derivative. The spatial derivatives are discretized using a third order finite-difference upwind scheme. The original chord for this research was developed for the application of turbo machinery and jet engine by Matsuo (1988) in the authors group and has been devised for LES.

Acoustic modeling The basic equations governing fluid motion and sound propagation are the same. The advantage of using a compressible flow solver is that it is able to simultaneously model the acoustic field over a short distance from the blade surface as long as the computational grid is sufficiently fine. In the present work, the grid is free enough to model acoustics up to approximately 10 kHz up to 1 to 2 chord lengths away from the blade surface, taking into account refraction and wall reflection effects, thus making the acoustic solution more accurate than acoustic analogy methods. In this way the propagation of the tip vortex pressure fluctuations can be modeled, leading to the prediction of tip vortex noise. RESULTS AND DISCUSSION Before investigating the physical mechanisms associated with the tip vortex noise caused by rotating wind turbines, flow simulations of a non-rotating blade section and a finite blade with a tip are performed at Reynolds numbers and angles of attack similar to the ones arising in the blade tip region on wind turbines operating in the design condition. Near blade flow data is compared against experimental data.

Validation of flow solution- NACAO012 blade section flow at high Reynolds number The NACA0012 blade section in an incident flow at an angle of attack of 5 and 6 degrees with a Reynolds number of 2.87x106 and a Mach number of 0.205 is simulated with periodic spanwise boundary conditions. The blunt trailing edge geometry and the flow parameters are based on the experiment by Brooks et al. (1981). The chord length is 0.6096 m. Terracol et al. (2002) already simulated this flow using LES. The computational grid is an O-grid consisting of 1713 points along the airfoil surface, 65 points perpendicular to the surface and 400 grid points in the spanwise direction, with a total of 44 million grid points. The suction side consists of 1630 grid points and the pressure side has 84 points. The grid spacing normal to the blade surface, Ay+, is approximately 1.0 along the entire blade surface. In the transition region, the streamwise grid spacing, Ax+, is approximately 50.0. The spanwise grid spacing, Az+, takes a value of 50.0, resulting in a span which corresponds to 20 % of the chord length. The computational domain has a diameter of 16 chord lengths. The non-dimensional time step based on the free stream velocity is 2.0x 10.4 c/Uoo. Grid spacing effects and the effect of the Smagorinsky constant Cs can be studied to assess the effects of the numerical dissipation caused upwind scheme indirectly. This is done with respect to trailing edge wall pressure spectra.

Grid spacing effects A streamwise grid spacing greater than dx + = 50.0 in the transition region downstream of the leading edge on the suction side did not capture the instabilities associated with the transition from laminar to turbulent flow. The flow remained laminar and separated at mid-chord. The trailing edge surface

692 pressure spectra did not agree well with experimental measurements. Increasing the number of grid points in the streamwise direction along the suction side beyond 1630 did not significantly alter the numerical results. A spanwise grid spacing in the order of Az + = 100.0 was found to be too coarse to properly capture transition to turbulent flow. Early laminar separation took place. A spanwise grid spacing smaller than Az + = 50.0 leads to the occurrence of transition effects near the leading edge. The location of laminar to turbulent transition as well as the vortex shedding frequency at the blunt trailing edge became consistent as the spanwise grid spacing was further decreased towards Az + = 10.0.

Smagorinsky constant effects The Smagorinsky constant Cs was assigned values of 0.00 (no eddy viscosity), 0.10, 0.15 and 0.20, and simulation results of the trailing edge wall pressure spectra were compared with experimental surface pressure spectra by Brooks et al. (1981). The blade section has a blunt trailing edge with vortex shedding occurring at the trailing edge. The simulation crashed when applying no eddy viscosity model. This suggests that the eddy viscosity does indeed contribute to the dissipation and that not the entire dissipation is caused by the upwind scheme. Surface pressure spectra on the suction side at the trailing edge obtained by LES with Cs = 0.10, 0.15 and 0.20 as well as experimental values are shown in Figure 1. The simulation results were obtained atter calibrating the streamwise and spanwise grid spacing. They correspond to the following grid configuration: The streamwise grid spacing is set to Ax+= 50.0 in the transition region. The spanwise grid spacing is Az+ = 50.0. 400 grid points were taken in the spanwise direction, resulting in a span which corresponds to 20 % of the chord length. The LES simulation with Cs = 0.10 yields a vortex shedding frequency of 2,500 Hz, which is lower than the experimental value of 3,300 Hz. The simulation with Cs = 0.20 yields a shedding frequency of 4,400 Hz, which is much higher than the experimental value. The difference in simulation results due to varying values of the Smagorinsky constant suggests that the eddy viscosity does affect the dissipation, while the effect of the numerical dissipation caused by the upwind scheme is small. In this case the simulation results obtained with Cs = 0.15 give results that are closest to the experimental measurements. The spectra illustrate a narrowband component due to the vortex shedding emerging out of a wideband continuum which is generated by turbulent boundary layers due to the bluntness of the trailing edge. The predicted vortex shedding frequency is slightly less than the experimental shedding frequency of 3,300 Hz. It can be said that there is favorable agreement between results obtained by the present flow solver and experimental results. 100

m n '9, ,r x

95

o

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.

.

.

.

.

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.

.

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Figure 1" Surface pressure spectra on suction side at trailing edge, Experiment: Brooks et al. (1981)

693 It can be thought that the computational grid used in the present simulation is fine enough to resolve a remarkable part of the spectrum. By using a very fine grid and capturing the smallest eddies the effect of the numerical dissipation to annihilate the small scales and overwhelm the SGS contribution is reduced as well as is the dependency on the SGS contribution. Although the numerical dissipation term will not vanish even for very fine grids, using a grid as fine as the present one can partly compensate for the undesired effects of the upwind scheme. The present LES approach can be considered suitable for efficient computation of compressible LES for applications for a full wind turbine blade.

Surface pressure distribution Figure 2 compares the experimental (Gregory et al. (1970)) and computational mean surface pressure coefficient for an angle of attack of 6 degrees obtained with Cs = 0.15. Good agreement between computational and experimental results is observed. Figure 3 shows the instantaneous pressure contours for an angle of attack of 5 degrees. Instabilities associated with the laminar-turbulent transition to turbulent flow can be seen on the suction side downstream of the leading edge. At approximately x/c = 0.25, the instabilities disappear. The present configuration was found to have a grid spacing that is sufficiently fine to capture the unstable phenomena associated with the transition without applying any tripping mechanisms, without manipulating the eddy viscosity and without purposely refining the computational grid in the transition region. Cp2.0

1.0 0.0 -1.0 9 Experiment -LES

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Figure 2: Mean Surface pressure coefficient (a= 6~ Experiment: Gregory and al. (1970)

Figure 3" Instantaneous pressure contours for a= 5 ~

Figure 4: Vorticity isosurfaces COxfor a = 10.8 ~

694

NACAO012 blade with tip The flow field and aerodynamic noise caused by a NACA0012 blade in an incident flow with a tip is simulated at a Reynolds number of 4.06x105 based on the reference chord length of 0.1524m/s. Brooks et al. (1986) carried out detailed investigations about the noise emitted by a finite blade with a tip and about the tip vortex structure. A NACA0012 finite blade with a tip corresponding to the blade tip tested in the wind tunnel by Brooks et al. (1986) is simulated. The total number of grid points is 50 million. The simulation is performed for angles of attack of 0.0, 5.4 and 10.8 degrees. Figure 4 illustrates the instantaneous isosurfaces of vorticity C0x for an angle of attack of 10.8 degrees obtained by LES. It can be seen that vorticity is primarily associated with the tip vortex. The structure of the tip vortex can be seen to consist of two main vortices. The vortical structures in the vicinity of the blade tip surface generate intense fluctuations and play a major role in the contribution to aerodynamic noise. Figure 5 illustrates the turbulent velocity fluctuations u'/U associated with the tip vortex. Here, U is the velocity component corresponding to the chordline direction. The left side shows the experimental hot-wire flow measurements (1986) which were taken perpendicular to the chord line 1.3 mm behind the trailing edge. Velocity fluctuation contours obtained from the LES simulation on the same plane are shown on the right side. Brooks et al. (1986) believe that the measured u' velocity component data were accurate to within 10% for each measurement location. It is possible to identify the approximate tip vortex formation region dimensions and velocity fluctuation scaling with angle of attack and to draw some qualitative conclusions. The boundary layer fluctuations can be identified for all three angles of attack in the LES simulation results and in the experimental measurements. For an angle of attack of 0.0 degrees, a uniform boundary layer with no tip vortex formation can be identified. As the angle of attack increases to 5.4 degrees, the tip vortex starts to form. The intense pocket at mid-height which shows 5% turbulent intensity can be seen in the LES simulation and in the experiment at the same location. As the angle of attack is further increased to 10.8 degrees regions of larger intensity and size form near the tip of the blade. Two large local pockets of turbulent activity around a core at the immediate tip can be observed in the experimental measurements. In the simulation the two large pockets can also be identified at roughly the same locations. The existence of the two large pockets also agrees with the observations from Figure 4. It can be said that the locations of intensive turbulent fluctuations agree qualitatively with the experimental measurements. Experiment

Simulation

_

__

~

/

L i

Figure 5" Turbulence intensity u'/U contours on the suction side at tip. a = 0.0 ~ (top), 5.4 ~ (middle), 10.8 ~ (bottom). Experiment: Brooks et al. (1986).

695

Figure 6: Instantaneous pressure perturbation field for ~t= 5.4 ~

Figure 7: Pressure perturbations isosurfaces at blade tip for a = 5.4 ~

Figure 6 shows the instantaneous pressure perturbation field for an angle of attack of 5.4 degrees. It can be seen that the noise source lies at the immediate tip, with acoustic waves propagating away from the tip. The pressure perturbations contain contributions mainly form the fluctuations associated with the tip vortex and its interaction with the trailing edge of the blade. The propagation of the tip vortex fluctuations away from the tip vortex region is directly simulated with the present compressible LES code. The computational grid is f'me enough to allow the acoustic field to radiate two chord lengths away from the blade. Figure 7 shows the corresponding three dimensional instantaneous pressure perturbation field. The mechanisms of noise causing vortex generation and acoustic wave propagation in the blade tip region can be studied in detail with the aim of proposing noise reducing design methods for the blade tip shape.

WINDMELIII flow and acoustics A simulation of broadband aerodynamic noise emitted by a large rotating wind turbine blade is performed, with particular emphasis on tip noise. A large-scale unsteady compressible Large-Eddy Simulation combined with direct noise simulation in the near-field is carried out on the Earth Simulator to predict the far-field aerodynamic noise caused by the rotating W1NDMELIII wind turbine at the design condition. Concerning the near-field, flow and acoustics are solved with the same code, while the far-field aerodynamic noise is predicted using acoustic analogy methods. Simulations for WINDMELIII are performed for two blade tip geometries, using an extremely fine grid to resolve the smallest eddy scales.

Flow conditions and geometry The numerical simulation is carried out in accordance with an acoustic measurement experiment of a WINDMELIII test turbine performed by Nii et al. (2004). The two-bladed wind turbine of upwind type has a diameter of 15 m with a rated power output of 16.5 kW. The actual tip shape of the rotor blades has a curved leading edge and straight trailing edge, as illustrated on the top of Figure 8. Ueffis the effective flow velocity at a particular blade section. An ogee type tip shape is also simulated, as shown on the bottom of Figure 8. The ogee type tip shape has been found in outdoor noise measurements on wind turbines to be noise suppressing. The Reynolds number of these two kinds of tip shape is 1,000,000 based on the chord length of 0.23m and the Mach number is 0.16 at the tip. The computational grid is of O-grid topology consisting of 765 grid points along the airfoil surface and 193 grid points perpendicular to the airfoil surface. 2209 grid points are taken along the span direction, of which 1597 points fall on the blade. Finally the total number of grid points reaches 300 million. The

696 simulation requires 300 hours with 112 processors in the "Earth Simulator" whose maximum speed is 40 TFLOPS in the full specification of 5120 processors.

Figure 8: WINDMELIII - Blade tip shape.

Figure 9: Vorticity COxIsosurfaces (Left: actual, Bottom: ogee)

Flow field in the blade tip region The vorticity isosurfaces COxin the tip region are shown in Figure 9 for both tip shapes. Very complex three dimensional vortical flow structures associated with the tip vortex can be identified. For both tip shapes these vortical structures prevail in the immediate vicinity of the trailing edge, suggesting the importance of the tip vortex-trailing edge interaction as a noise contribution. Concerning the actual tip shape, a major part of the vortical structures can be identified at the immediate tip and in the tip vortex, in close proximity to the blades surface. Vortices that exist near the blade surface have greater contribution to the sound generation than those that exist far from the blade surface. Reduced tip vortex shedding and reduced interaction with the trailing edge suggests reduction of noise for the ogee type tip shape. There exists, however, some degree of interaction between the vortical structures and the trailing edge further inboard. For the ogee tip the trailing edge curvature leads to the formation of further but weaker circulation vortices due to local cross flows along the trailing edge.

Far-fieM noise prediction The acoustic perturbations evaluated by the LES simulation are integrated using the FW-H equation by Ffowcs (1969) and Brentner (1998), yielding the acoustic pressure level in the far-field. The simulated changes in sound pressure level for the actual and the ogee type tip shapes are shown in Figure 10 for

697 an observer point located 20 m upstream of the wind turbine. The ogee type tip shape shows a decrease in acoustic pressure level for frequencies above 3 kHz. The ogee type tip shape clearly has a noise reducing effect. A reduction of 2 dB is attained for the overall sound pressure level. The reduced acoustic emission of the ogee tip is related to the decreased interaction between the tip vortex and the trailing edge. 10 O.

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Figure 1O: Sound pressure level- Simulation results. CONCLUSION A computational model for aerodynamic noise prediction that takes into account the true shape of the wind turbine blade geometry was developed. This is an important step towards wind turbine blade design with respect to aerodynamic noise reduction. The method developed in this research allows arbitrary blade tip shapes to be modeled and their acoustic emissions to be estimated. For validation purposes, turbulent fluctuations associated with the tip vortex and the acoustic field generated by a finite blade are simulated using Large-Eddy Simulation. Complex flow features associated with the tip vortex are captured and validated with experimental measurements. It was found that the tip vortex constitutes a major noise source. The findings obtained can give us new insights into the physical mechanisms causing tip vortex flow and tip noise. Grid spacing effects and Smagorinsky constant effects were investigated for the third-order upwind difference scheme. The computed blade surface pressure distribution and surface pressure spectra at the trailing edge agreed quantitatively with experimental measurements up to high frequencies with proper choice of the grid spacing and the Smagorinsky constant. The structure of the blade tip vortex as well as turbulent velocities associated with the tip vortex agreed qualitatively with experimental measurements. The pressure spectra associated with the tip vortex and the acoustic field generated by the tip vortex agreed quantitatively with experimental measurements at frequencies that are relevant for the wind turbine flow and tip vortex noise simulation. A fine grid is required in order to simulate using LES a full blade with a tip for such high Reynolds numbers. The Earth Simulator allows us to use an extremely fine grid in all three dimensions. As a conclusion, the validation of the aerodynamic and the acoustic model used in the present numerical method has been shown to provide quantitatively accurate results for blade surface pressure spectra and acoustic wave propagation properties. The model was further applied to the blade tip of WINDMELIII. A direct noise simulation was performed with compressible Large-Eddy Simulation (LES). Acoustic wave propagation was simulated directly from the hydrodynamic field. The far-field broadband noise caused by a rotating wind turbine blade of WINDMELIII type was computed using Ffowcs Williams-Hawkings integral

698 method, with particular emphasis on tip vortex noise. The structure of the tip vortex was shown to be more complex for the actual tip shape. The degree of interaction between the tip vortex and the wind turbine blade trailing edge was reduced for the ogee type tip shape. The use of an ogee type tip shape can reduce the noise level for frequencies above 3 kHz by up to 5 dB. Overall reduction was found to be 2 dB. These numerical results will contribute to the future design of very large wind turbines such as 10MW rotors for offshore wind farms in order to suppress the noise while increasing the tip speed for cost reduction. The ogee type tip shape has been confirmed to be less noisy than the standard one in the simulation, which leads to the next idea of tip configuration for high performance of both aerodynamics and noise. ACKNOWLEDGEMENTS

The authors gratefully acknowledge the Earth Simulator Center for providing the computational resources for this research. The support of the wind turbine group at AIST (National Institute of Advanced Industrial Science and Technology, Japan) concerning WINDMELIII is gratefully acknowledged. REFERENCES

Brentner K.S. and Farassat F. (1998). Analytical Comparison of the Acoustic Analogy and KirchhoffFormulation for Moving Surfaces. AIAA Journal 36:8, 1379-1386. Brooks T.F. and Hodgson T.H. (1981). Prediction and comparison of trailing edge noise using measured surface pressure. Journal of Sound and Vibration 78:1, 69-117. Brooks T.F. and Marcolini M.A. (1986). Airfoil Tip Vortex Formation Noise. AIAA Journal 24:2, 246-252. Ffowcs Williams, J. E., Hawkings, D. L., (1969), "Sound Generation by Turbulence and Surfaces in Arbitrary Motion", Phil. Trans. of the Royal Soc. Of London, A." Mathematical and Physical Sciences, 264(1151), pp.321-342. Gregory N. and O'Reilly C.L. (1970). Low-Speed Aerodynamic Characteristics of NACA0012 Aerofoil Section, including the Effects of Upper-Surface Roughness Simulating Hoar Frost. Reports and Memoranda No.3726. Matsuo, Y., Arakawa, C., Saito, S., Kobayashi, K., (1988), "Navier-Stokes Computations for Flow field of an Advanced Turboprop", AIAA-88-3094. Mellen C.P., Fr6hlich J., Rodi W. (2002). Lessons from the European LESFOIL project on LES of flow around an airfoil. AIAA 2002-0111, 40 th AIAA Aerospace Sciences Meeting and Exhibit. Nii Y., Matsumiya H., Kogaki T. and Iida M. (2004). Broadband Noise Sources of an Experimental Wind Turbine Rotor Blade (Japanese). Transactions of the Japan Society of Mechanical Engineers Part B 70:692, 99-104. Terracol M., Manoha E., Herroro C. and Sagaut P. (2002). Airfoil Noise Prediction using LES, Euler Equations and Kirchhoff Integral. Proceedings LESfor Acoustics, GSttingen. (1994). ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics. NASA Conference Publication, Hampton, Virginia. Smagorinsky J. (1963). General Circulation Experiments with the Primitive Equations. Monthly Weather Review 91:3, 99-164. Van Driest E.R. (1956). On the turbulent flow near a wall. Journal of Aeronautical Science 23, 1007.

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

699

A ZONAL RANS/LES APPROACH FOR NOISE SOURCES PREDICTION M. Terracol ONERA, DSNA/ETRL 29 av. de la Division Leclerc, BP 72 92322 Ch~tillon Cedex, France. e-mail: terracol @onera.fr

ABSTRACT Some hybrid CFD/CAA methods have generally to be used for the numerical simulation of trailingedge noise (see Manoha et al. (2002) for instance). This study focuses on the first step of such hybrid methods, which is to get an accurate description of the unsteady aerodynamic sources by the mean of a 3D unsteady simulation of the flow. Such a simulation is however generally still away from the numerical capabilities of "usual" supercomputers. This paper proposes a zonal LES method for the numerical prediction of the aerodynamic noise sources. This method allows to perform only some zonal LES close to the main elements responsible of sound generation, while the overall configuration is only treated by a RANS approach. Attention will be paid to the tricky boundary treatment at the interface between the RANS and LES regions. The method is first assessed by the simulation of a flat plate ended by a blunted trailing-edge, and then applied to the simulation of the flow over a NACA0012 airfoil with blunted trailing-edge.

KEYWORDS RANS/LES coupling, Aeroacoustics, Airfoil, Turbulence, Trailing-edge noise.

INTRODUCTION This study deals with the numerical prediction of the aerodynamic noise generated by airfoils. More particularly, it focuses on the problem of trailing-edge noise. This complex problem remains however out of reach of DNS/LES methods. For this reason, some hybrid CFD/CAA methods have been developed in the past few years (see Manoha et al. (2002) for instance), in which an unsteady CFD simulation is used to get a prediction of the acoustic sources, which are then used as an entry data of an acoustic propagation solver. This study focuses on the first step of this hybrid method. Indeed, Large-Eddy Simulations of realistic flows remain generally still away from the numerical capabilities of "usual" supercomputers. In this paper a zonal RANS/LES method is proposed to perform some numerical predictions of aerodynamic noise sources at a moderate computational cost. The main idea is then to perform only some zonal LES close to the main elements responsible of sound generation, while the overall configuration is only treated by a much cheaper RANS approach. However, the main

700

difficulty remains the coupling between the RANS and LES regions. In this paper, a particular attention will be paid to the treatment of turbulent boundary layers at the inflow of the LES region. The proposed approach is described in the first section of the present paper. Then, a first test case is considered in the second section to assess the approach. This case deals with the flow over a fiat plate ended by a blunted trailing-edge. Finally, the third part of the paper presents some early results of the application of the zonal RANS/LES method to the numerical simulation of the noise sources generated by the flow around a NACA0012 airfoil with a blunted trailing-edge.

ZONAL RANS/LES M E T H O D

Main Equations m

The principle of the method is to decompose the filtered LES field U as a mean part U 0 and a fluctuating part U': m

U - U 0 + U'

(1)

The mean part U 0 can be computed using a classical RANS parametrization on the whole configuration, while the calculation of the fluctuating part U' is performed in a small region only thanks to a LES solver. In this region, the LES equations are formulated under a perturbation form, that is to say that it is chosen to solve the following Non-Linear Disturbance Equations (NLDE) rather than the classical LES-filtered equations (Labourasse & Sagaut (2002)):

~---U' + N ( U ' + U o ) - r L - N ( U o ) - r

bt

o

(2)

where U is the vector of the conservative variables, r 0 and r L are respectively the classical Reynolds and subgrid scale terms, and (for a sake of simplicity in the notations) N denotes the Navier-Stokes operator (including the convective and viscous terms). It is to be noted that these equations are simply obtained by subtracting the Reynolds-averaged equations from the LES-filtered equations. From the continuous point of view, they are thus strictly equivalent to the LES equations. However, as it was shown in Labourasse & Sagaut (2002), solving the perturbation equations (2) instead of the full LES equations allows to minimize the sensitivity of the solution to numerical errors and to consider some reduced domain sizes. This thus allows to perform a local LES coupled with a global RANS simulation.

Boundary Conditions for the LES region To prevent reflections at the LES zone interfaces where the mean field is imposed as a boundary condition, an extension of the characteristic theory (Thompson (1987)) to the perturbation formulation has first been retained as boundary condition in the previous works dealing with this zonal LES method (see Sagaut et al. (2004) for more details). The approach has been applied to the simulation of the unsteady flow in some realistic configurations such as a low-pressure turbine blade (Labourasse & Sagaut (2002), Sagaut et al. (2004)) and the slat cove from a high-lift wing profile (Terracol et al. (2003)). However, such a boundary treatment appears only valuable in the case in which quasi-laminar boundary layers are present at the inflow of the LES region. A particular treatment is still needed to

701

account for some fully-developed turbulent boundary layers (TBL) at inflow. In this study, two methods have been retained to account for the coherent turbulent structures present in TBL: The first one is to combine a compressible extension of the recycling strategy developed by Lund et al. (1998) with the previous characteristic boundary treatment and apply it to the perturbation variables at the LES region inflow. The principle of the method is to extract some perturbations U' in a plane downstream of the inflow plane, and inject them at the inflow location after a rescaling. In this study, the compressible extension proposed by Sagaut et al. (2004) has been retained, which consists in applying a simple recycling treatment to the pressure and temperature variables. The algorithm then reads:

u;],, (y,*,,t)= fl,U;lrec (y:c,t) rilin (y~,t) = T Irec(Y;c 't) P L, (y~,t)= P'lrec(Y;c 't)

f

i=1,2,3

(3)

where the recycling parameter fl represents the ratio of the friction velocities at the inflow

(in)

. (in)

and recycling

(rec) planes:

fl = u~

U (rreC) "

This approach has been proven to be efficient for dealing with boundary layer simulations with reduced streamwise extent in several studies, and may thus appear as a good candidate for the present zonal approach. However, it is to be noted that the main difficulty relative to this approach remains to initiate the recycling process. In general, a secondary TBL/channel flow simulation has to be used to provide some appropriate perturbations. Moreover, such a process may introduce a non-physical recycling frequency due to the artificial streamwise periodicity introduced in the simulation, which depends on the location of the recycling plane. 9 The second one is to perform a synthetic reconstruction of the typical structures present in TBL. For this purpose, an approach based on the one developed by Sandham et al. (2003) for supersonic flows has been retained. Again, the approach is combined with the characteristic boundary treatment for the perturbation variables. At the inflow, the streamwise (u'l) and wallnormal (u'2) components of the velocity fluctuations are computed as follows:

u', =U**~C~J(~y,~x) Y~l"Jexpl-(~y~X)

(4)

where U** is the reference velocity and m is the number of modes to be considered. The first mode (j=l) provides a representation of the near-wall streaks, while higher-order modes (j > 1) represent some larger structures with a growing spanwise extent through the boundary layer. The coefficients flj and toj are chosen to match the typical sizes of the coherent structures present in TBL, as in Sandham fixed to

( y~X )§ =12,

and

yj

et al.

(2003). The peak perturbation location are respectively

=(j-I)8",

j > 1, where tT is the displacement thickness at

the inflow location. The parameter ~j denotes some spanwise phase shift coefficients. The amplitude coefficients CU have been tuned to match as well as possible typical TBL RMS

702 profiles. The exponent nj has also been introduced, to modify the envelope of the first mode, with n1 = 1 - 6 ( 1 + tanh (10(y-y~a~))), and nj, l = 1. Finally, the spanwise component u'3 is derived from a divergence-free condition. In practice, a white noise is also added in the boundary layer zone, with a maximum amplitude of 4% of the reference velocity. In this study, four different modes were considered. The values of each parameter used here are detailed in table 1 (/z denotes viscosity).

TABLE 1 ANALYTICALTBL PARAMETERS

Cly U~ U (in)

U

C2j u(m)

max

Yj

Ig "1'

(

~-1

Ix ~,

ur

/

15.2

-5

12 lzlu~ in)

100

100

0

5.6 5.6 5.6

-2.8 -2.8 -2.8

8" 28" 38*

133 200 400

32 58 109

0.1 0.2 0.3

Numerical Method The spatial scheme retained in this study is the modified AUSM+P scheme developed by Mary & Sagaut (2002). This scheme takes advantage of a wiggle detector that allows to limit the numerical dissipation of the scheme to the zones where odd-even numerical wiggles are detected. Elsewhere, the scheme acts as a centered non-dissipative scheme well-suited for LES applications. For time integration, a second-order accurate implicit Gear scheme, based on an approximate Newton solver has been used. Finally, the subgrid-scale model retained in this study is the selective mixed-scale model fully described in the works by Lenormand et al. (2000)

APPLICATION TO FLAT PLATE / BASE F L O W HYBRID C O M P U T A T I O N To assess the proposed approach, the flow over a thin flat plate ended by a blunted trailing-edge has been considered. This leads to an acoustic wave emission at the trailing-edge. A reference LES on the full configuration ("FULL") taking into account the boundary layer transition process has first been carried out. In this simulation, the flat plate extends over 60h, with h the trailing-edge thickness. The spanwise extent of the domain is Lz=4h, with periodicity conditions. A laminar Blasius velocity profile with a thickness of 80 = 0.27h has been imposed at the inflow, with a small random perturbation. The Reynolds number based on the trailing-edge thickness is Reh=10,000, and the Mach number of the flow is 0.5. Figure 1 shows a schematic view of the configuration.

703

Figure 1" Flat plate / base flow configuration. In this simulation, a transition occurs at a location of 45h upstream of the trailing-edge. Then, a shorter computational domain located close to the trailing-edge has been considered. The streamwise extent of the flat plate region has been reduced to 1 lh in this case. As a mean field, we have chosen to use here the averaged LES field, to be able to compare our results with those from the full LES. At this inflow location, the boundary layer thickness i s 6 - - h . Three zonal I~ES simulations have been performed: the first one uses only the characteristic boundary treatment at inflow ("LAM"), while the second one takes advantage of an additional recycling treatment for the perturbations ("REC"), and the third one ("ANA") relies on the use of the analytic TBL model (4). Figure 2 shows the mean and RMS velocity profiles obtained in each case at a location of h upstream of the trailing-edge. It is to be noted that only the zonal simulations using a particular turbulent treatment at inflow lead to some results in good agreement with the full reference LES. These two simulations also exhibit a highly threedimensional flow behavior, with a good representation of the typical structures present in TBL (see Fig. 3).

4

/ / /

I

I

I

I

I I I1| 10 ~

I

I

I

y+

I

i

I I Ill 10 =

~=zs

I

t

I

t

1 I II|

~

10 3

Y+

Figure 2: Mean streamwise (left) and RMS (fight) velocity profiles. Symbols: FULL; dashed line: LAM; dash-dotted line: REC, solid line: ANA.

704

Figure 3: 3D view of the flow close to the trailing-edge (case ANA). Figure 4 shows the strong wave pattern emitted at the trailing-edge in the zonal simulation "ANA" (the full computational domain is shown). As it can be seen in Fig. 5, the different simulations exhibit a peak at a Strouhal number of about St=0.24 in the acoustic spectrum, with several harmonics. This figure reveals that the inflow treatment based on the recycling treatment leads to some numerical errors in the highest wavenumbers.

Figure 4: Dilatation field 0 = V.u (case ANA).

705

iilI '~176 /u 0.2

0.4

0.6

0.8

St

1

1.2

1.4

Figure 5: Pressure spectrum at 10h above the trailing-edge. Same key as fig. 2

APPLICATION TO NACA0012 SIMULATION This case deals with the application of the method to the numerical prediction of the noise generated by the flow past a NACA0012 airfoil with a blunted trailing-edge, at a 5 ~ angle of attack. The chord of the profile is c=-60.95 cm, and the Mach number of the flow is M=0.205, leading to a chord-based Reynolds number of Rec =2,860,000. The thickness of the trailing-edge is h=2.5 mm. All these parameters, except the angle of attack, match those of the N A S A experiment of Brooks & Hodgson (1981). This configuration has been extensively investigated in the works by Manoha et a/.(2002), who performed a coupling between a compressible LES performed around the full airfoil, and acoustic propagation techniques combining the use of Linearized Euler Equations (LEE) and integral methods. Despite the success of the proposed LES/CAA coupling, the LES used as a basis for the study was subject to strong limitations in terms of grid resolution. In particular, the use of a spanwise extent which was sufficient to ensure a correct development of the spanwise structures at the trailing-edge led to an under-resolution of the TBL, and thus to some too small boundary layer thicknesses at the trailing-edge. The result was a strong overestimation of the main expected frequency of the associated acoustic wave emission. It has thus be chosen to apply the zonal LES approach to the simulation of this kind of flow. A 2D steady RANS calculation (using the Spalart-Allmaras model) has first been performed over the full configuration. The grid used for this computation was composed of 321,600 meshpoints. Then, a small 3D LES region surrounding the trailing-edge has been defined, in which the proposed approach is applied (see Fig. 6).

706

Figure 6: Location of the LES region. The significant reduction of the extent of the LES region allows to consider a rather large spanwise extent of 1.67% of chord (Lz = l c m = ~ ) . The mesh considered here for the LES region, which matches the classical LES requirements in terms of resolution, is composed of roughly 5.4 millions of points (with 84 meshpoints in the spanwise direction). Finally, it has been chosen to use the analytical TBL model as inflow condition for the LES region. Figs. 7 and 8 show respectively some preliminary results of the spanwise vorticity component and of the dilatation field obtained close to the trailing-edge after a physical integration time of 26 ms.

Figure 7: Spanwise component of vorticity in the vicinity of the trailing-edge.

707

Figure 8: Dilatation field 0 = V~u A turbulent vortex shedding is clearly observed, together with an acoustic wave emission. Fig. 9 shows that this acoustic wave emission is associated to a broadband peak in the pressure spectrum, at a frequency of about 2,300 Hz, which is lower than the one reported by Brooks and Hodgson (around 3,000 Hz) in their experiment at a 0 ~ angle of attack. This difference is however not so surprising when considering the 5 ~ angle of attack which leads to a significantly thicker TBL at the trailing-edge on the suction side. Moreover, the amplitude of the main broadband peak in the pressure spectrum is well predicted (roughly 88 dB in the experiment). This result is thus much more consistent with the experiment than the one corresponding to the LES reported in the works by Manoha et al., in which a main frequency of about 5000 Hz was obtained, with a global overestimation of 3 dB of the pressure levels.

loo 95 so 85 8o 75

~" To

45 4O

10 9

KHz)

10+

Figure 9: Pressure spectrum at the trailing-edge.

708

CONCLUSION The proposed zonal LES method has been shown to allow to get a significant reduction of the cost of the simulations associated to the numerical prediction of aerodynamic acoustic sources in comparison with classical LES. The analytical model of TBL used at inflow has been shown here to be the only one which allows to reproduce properly both the TBL and acoustic properties of a trailing-edge flow. The method has been assessed on an academic configuration, and then applied to a more realistic flow (NACA0012 airfoil). In this last case, the first results obtained with the zonal approach display a quite good agreement with some reported experimental results. It is to be noted here that previous classical LES performed on the same configuration did not exhibit such a good agreement, since the grid resolution could not be fine enough on the full configuration when a significant spanwise extent was considered. In the future, a more extensive analysis and validation of the zonal NACA0012 simulation will be performed. Then, the zonal LES method will be used as the first step of a hybrid LES/CAA approach. That is to say that it will be used for a coupling with an acoustic solver (Manoha et al. (2002), Terracol et al. (2003)) to describe mid- and far-field noise radiation. REFERENCES Brooks T.F. and Hodgson T.H. (1981), "Prediction and comparison of trailing-edge noise using measured surface pressures", Journal of Sound and Vibration, 78:1, 69-117. Labourasse E. and Sagaut P. (2002), "Reconstruction of turbulent fluctuations using a hybrid RANS/LES approach", J. Comput. Phys., 182, 301-336. Lenormand E., Sagaut P. and Ta Phuoc L. (2000), "Large Eddy Simulations of Subsonic and Supersonic Channel Flow at Moderate Reynolds Number", Int. J. Numer. Meth. Fluids, 32, 369-406. Lund T. S., Wu X., and Squires K.D. (1998), "Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations", J. Comput. Phys., 140, 233-258, 1998. Manoha E., Herrero C., Sagaut P. and Redonnet S. (2002) "Numerical Prediction of Airfoil Aerodynamic Noise", A/AA Paper 2t}02-2573, 8th CEAS/AIAA Aeroacoustics Conference, Breckenridge (Co), USA. Mary I. and Sagaut P. (2002), "Large-Eddy Simulation of Flow around an airfoil near stall", A/AA J. 36-1, 1139-1145. Sandham N. D., Yao Y. F., and Lawal A. A. (2003), "Large-eddy simulation of transonic turbulent flow over a bump", Int. J. Heat and Fluid Flow, 24, 584-595 Sagaut P., Gamier E., Tromeur E., Larchev~que L., and Labourasse E. (2004), ''Turbulent inflow conditions for LES of compressible wall bounded flows", A/AA J., 42:3. Terracol M . , Labourasse E., Manoha E., and Sagaut P. (2003), "Simulation of the 3D unsteady flow in a slat cove for noise prediction", AIAA paper 2003-3110, 9 th AIAA/CEAS Aeroacoustics Conference, Hilton Head SC. Thompson K. W. (1987), "Time dependent boundary conditions for hyperbolic systems". J. Comput. Phys., 68, 1-24.

Engineering TurbulenceModelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

709

A E R O D Y N A M I C S A N D A C O U S T I C S O U R C E S OF T H E E X H A U S T J E T IN A C A R A I R - C O N D I T I O N I N G S Y S T E M A. Le Duc 1, N. PelleP, M. Manhart 1, E.-P. Wachsmann 2 1 Fachgebiet Hydromechanik, Technische Universit~t Miinchen, Arcisstr. 21, D-80290 Munich, Germany 2 Development Fluid Dynamics Simulation, AUDI AG, I/EK-41, D-85045 Ingolstadt, Germany

ABSTRACT An application of Large-Eddy Simulation to noise prediction is presented. We consider a car airconditioning jet exiting into the passenger's compartment. It produces a broadband noise. Within a hybrid approach, we compute acoustic sources using Lighthill's acoustic analogy. We solve the flow using a Cartesian solver combined with an Immersed Boundary Method, which takes into account the complex exhaust nozzle. This technique results in an efficient solver that allows for highly resolved computations of the flow field. We investigate the influence of grid resolution and boundary approximation order on time averaged velocity fields as well as on acoustic sources. The mean flow results show that the complex geometry jet does not behave like a classical turbulent jet. The numerical investigations show that a smooth representation of the surface together with an adequate grid resolution is required to represent the small vortical structures giving the main contribution to the acoustic quadrupole terms. If one requirement is not met, errors in the strength of the acoustic source terms and their dynamic response can be considerable.

KEYWORDS

Aerodynamics, Complex geometry, Acoustic analogy, Acoustic source terms, Large Eddy Simulation (LES), Immersed boundary method.

1

INTRODUCTION

With the strong development of computer power, the computation of flow and aerodynamically generated noise in complex geometries becomes a feasible task, but remains challenging. This

710 paper offers an example of such a computation. The flow considered here is the jet exiting a car air-conditioning system into the passenger's compartment. The rectangular exhaust pipe (see Fig. 1) is fitted with two sets of thin rectangular blades meant to deflect the flow. The width of the pipe is twice its height. The blades are set at high angles of attack with respect to the channel direction. The flow is characterized by a moderate Reynolds number of 25000 based on the height of the pipe, and a very small Mach number M ~ 0.01. Our goal is to predict the aerodynamic development of the turbulent jet and the aeroacoustic noise generation.

Figure 1: Front view (left) and side view (right) of the rectangular exhaust pipe. Two sets of rectangular thin blades are fitted at high angle of attack within the pipe Two options are available to compute the aeroacoustic noise: the direct noise computation and the so-called hybrid approach. By solving the compressible Navier-Stokes equations without any simplification, the direct noise computation outputs all aerodynamic and acoustic quantities (i.e. acoustic pressure fluctuations or density fluctuations). In many cases, the small Mach number constrains the time step to very small values, which makes this solution unpractical. We resort to the hybrid approach and employ the Lighthill acoustic analogy [4]. The compressible Navier-Stokes equations are combined into one acoustic equation:

02 Q' Ot2

c~V2p ' =

02T~j Ox~Oxj

(1)

o~ _c~V 2, The acoustic density fluctuation ~o~is propagated with the acoustic propagation operator b-~ where co is the speed of sound in the medium at rest away from the jet. Quadrupole source terms for t~' are on the right hand side. The Lighthill tensor

(2) depends on velocities ui, shear stresses ri 3 and entropy fluctuations p'-c02Lo'. In our case, entropy fluctuations can be neglected. Additionally, because of the very small Mach number, the velocity behaves incompressibly, so that T~j may be reduced to ~ou~uy. In the presence of walls, additional dipole source terms appear. They can be modeled using the the Ffowcs-Williams- Hawkings analogy [2]. We first need to compute the aerodynamic development of the flow. Note that the blades are set at high angles of attack, so that the flow is a priori massively separated. This results in a strongly unsteady turbulent activity, and consequently in a broadband noise. Resolving all source terms responsible for the broadband noise is yet an unsolved issue. Up to now, there is no known criteria or experience, how fine a computational grid should be designed to predict sufficiently accurately

711 the quadrupole source terms. We evaluate the ability of Large-Eddy Simulation (LES) to represent the acoustic sources. The next step consists in discretizing the equations on a computational mesh. Because of the complexity of our geometry, the generation of a body fitted grid would be very cumbersome. Besides, computations in curvilinear coordinates are less efficient than in Cartesian coordinates. We thus resort to a discretization on a Cartesian mesh. The body is taken into account via an Immersed Boundary Method. An up-to-date review of the different IBM flavors, along with their respective strengths, can be found in Iaccarino and Verzicco [3]. The method here retained will be explicited in the numerics section. The paper is divided as follows. In Section 2, the numerical method is explained. In Section 3, the aerodynamic development of the jet is analyzed. In Section 4, the volumic acoustic sources are presented. Influence of numerics is investigated. Conclusions and perspectives are drawn in Section 5.

2 2.1

METHOD Governing equations

We perform a LES. The governing equations are the incompressible filtered Navier-Stokes equations 0t

+~.V~=-

V/Y+~,V2~-V.~-

, V.~=0,

(3)

with _ being spatially filtered quantities. The subgrid scale tensor reads ~-~y= u-(dj - ~ . We use the constant Smagorinsky model [81 "rij = --2l,'TSij, which relates the subgrid scale stress tensor to the resolved strain rate S~j via an eddy diffusivity concept. The subgrid scale viscosity is uT = ( C A 2 ) ( 2 S ~ j S i j ) 1/2 (C = 0.1, A = ( A x A y A z ) I / 3 ) . No Van-Driest damping was used in near wall regions. Indeed, as will be later shown, the detached shear layers dominate the global flow dynamics. We thus considered that the exact prediction of flow behaviour in near wall regions is not a major concern.

2.2

Numerical integration

The LES equations (3) are solved in a Finite Volume formulation. The flow variables are defined on a non-equidistant Cartesian mesh in a staggered arrangement. A fractional step method is employed and time integration is performed by a Leapfrog scheme. The scheme is second order accurate in time and space. A detailed description of the numerical integration scheme is given in Manhart [5].

2.3

Immersed boundary method (IBM)

The computation is performed on an underlying Cartesian mesh. The immersed body is taken into account via a direct forcing IBM, originally introduced by Mohd-Yusof [6]. The direct forcing consists in setting boundary conditions on the Cartesian mesh in order to mimic the effect of the solid body. Compared to force-based forcing, the direct forcing has two main advantages: it does not have case dependent parameters and does not introduce any restriction on the time step

712 [1]. More precisely, let us consider the two dimensional example of Fig. 2. We derive boundary conditions for the u velocity. Cells cut by the geometry or within the geometry are not active for the computation. The boundary seen by the Cartesian solver is thus the dashed line and the velocity u must be set at point P. The direct forcing consists in interpolating the boundary condition from the body surface onto this dashed line. This interpolation is performed in a step preliminary to the usual Cartesian solver iteration. I

Ay-I-

p 4J

A.3

Ax Figure 2: Two dimensional interpolation The most elementary version of the interpolation consists in displacing the body boundary condition to the nearest Cartesian point, i.e. setting u(P) = 0. This Oth order interpolation results in a stepwise body whose surface is smeared because of the staggered variable arrangement. Higher order interpolations are achieved using body surface points (A0, B0) and interior fluid points. For instance, the value of u(P) can be computed from its neighbors Ai in x direction. For a Lagrangian interpolation of order n, points A0, A1 , ... An are used. The interpolation may equally well be performed in the y direction with the B neighbors. In order not to privilege arbitrarily a direction, a weighting of both interpolated values depending on A y / A x is performed [9]. The extension to three dimensional cases is straightforward. For our complex jet, Lagrangian interpolations proved difficult to handle. Even with linear interpolation, the computations blew up because of numerical instabilities in the near wall regions. This may be linked with a lack of clustering in these regions [1]. A flexible alternative was found with least-square interpolation. Successful computations were performed with 2nd order least square interpolation. It consists in fitting a parabola through four control points A0, A1, A2, and A3. The value at the wall point A0 is exactly zero. Note that because of the high order interpolation, no strong clustering is required at the wall. At the same time, the stability of the least-square interpolation prevents numerical oscillations [7].

2.4

Problem setup

The sige of the computational domain is 6h in streamwise direction x, 12h in spanwise direction y and 12h in normal direction z, where h = 53 is the nozzle height. The center of the nozzle exit plane is located at ( 8 7 0 , - 6 0 , - 1 4 0 ) . All distances are given in ram. Three computations with different resolutions have been performed. The order of the IBM has been varied from 0 (stepwise) to 2 with least square interpolation. The parameters are summarized in Table 1. As inflow condition, we set the velocity in the incoming channel to uin = U~f. At non-solid boundaries of the computational box, we allow for free outflow by setting the outside pressure to zero. This also enables the entrainment of the jet. In order to avoid instabilities coming from too

713

Table 1: Grid size (mm) 0.5 0.5

P A R A M E T E R S FOR T H E COMPUTATIONS

Number of points between blades ~6 ~ 12 ~ 12

Number of grid points 10.e6 40.e6 40.e6

IBM order 0 2

strong inflow at the free boundaries, we limited the incoming velocity to 1% of Urcf. Even with this limitation, problems were still observed at the exit boundary. Therefore, we implemented a sponge layer by doubling the domain length in main stream direction and linearly increasing the Smagorinsky constant towards the exit to 10 times its usual value. By progressively stretching the grid over the whole length of the sponge layer, the additional computational costs could be held low. The eddies were thus damped and swallowed by the grid, while no noticeable influence of the sponge layer on the structures upstream was observed. Computations were performed on the Hitachi SR8000 of the Leibniz Rechnen Zentrum Munich. CPU time per grid point and time step amounted to 0.12#s. An approximate 500 CPU hours to obtain a statistically converged flow were needed.

3

3.1

AERODYNAMIC

DEVELOPMENT

OF T H E J E T

P h y s i c a l i n v e s t i g a t i o n of t h e flow

The results of this section are obtained with least square 2nd order IBM on the 40 x 106 points grid. In Fig. 3, the instantaneous streamwise velocity u is plotted in an (x, z) plane at y = -44, i.e. close to the middle spanwise position of the exhaust pipe. The flow is characterized by strong recirculation regions, which are present on the leeward face of every blade. Their length varies in time up to one and a half blade length. Recirculation regions alternate with high speed zones related to the blockage effect of the blades, thus creating strong shear and leading to strong turbulent activity. On each side of the pipe, some fluid is engulfed, creating two widely spread back-flow regions. On the upper side of the inlet pipe, a recirculation bubble breathes on the concave inlet wall. This might be an effect of the inlet boundary condition, since the entrance velocity is fixed in time and the incoming pipe length is too short for the laminar flow to develop into a turbulent flow. A laminar profile being more prone to separation than a turbulent profile, the recirculation bubble might be an artefact of the entrance boundary condition. In future work, we will examine the influence of the inlet boundary condition by simulating the complete inlet pipe. Some marked differences with a classical jet (without obstacles within the pipe) can be quantified when considering time-averaged flow profiles. In Fig. 4, the profiles of the velocity magnitude I UI are plotted for different streamwise positions as function of the spanwise direction y (left) and normal direction z (right). In both profiles at x = 880, the blade wakes are recognizable. The peak at y ~ - 1 2 0 corresponds to the left most interblade channel (see Fig. 1 left). It is very narrow and causes a sharp over shooting. In y-profiles, the influence of the blades remains noticeable even after 4 pipe heights. Because of the numerous local maxima, no jet width can be properly computed. Nevertheless, it can be assessed to ~ 130. In z-profiles, the direct influence

714

Figure 3: Instantaneous streamwise velocity at y = -44. Thick black lines indicate the contour of the Cartesian computational domain. Thin black lines are u = 0 of the blades disappears within half a pipe height. The velocity profile then resembles a spreading skewed Gaussian. To further characterize the jet, the maximum of the time-averaged velocity magnitude is searched for in every (y, z) plane. Umax is plotted as function of x in Fig. 5 left. The maximal velocity continuously decreases, which indicates that the jet does not have a potential core as observed in a classical turbulent jet. From x = 870 to x = 880, the sharp drop of the maximal velocity corresponds to the release of the blade blockage. The location z where the maximal velocity is reached defines the jet centerline. When moving in positive z direction from the centerline, the point where I Ul(x,z) = Umax(X)/2 defines the upper width Zl/2. Similarly in the negative z direction, one finds the lower width. Centerline, upper and lower widths are plotted in Fig. 5, right. The jet spreads linearly. The angle of the centerline line with the horizontal blades is 7.4 ~ The opening angle is 4.8 ~. These results are in agreement with experimental flow visualizations performed at Audi AG.

3.2

I n f l u e n c e of n u m e r i c a l p a r a m e t e r s

Fig. 6 compares the time averaged streamwise velocity fields obtained from the three different simulations (see Table 1). Table 2 sums up the centerline jet angles and opening angles. The greatest differences are observed for the coarse grid simulation. Numerical wiggles are apparent and the separation regions are more restricted than in both fine grid computations. The opening angle is also much lower than in the fine grid simulations. The turbulence may not be correctly captured

715 -50 'i '

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'

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i

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lOOO IU[ (y)

200 880 900

1100

1000 I U I ( Z ) 1 1 0 0

Figure 4: Profiles of the time-averaged velocity magnitude at four streamwise locations (x = 880,900, 1000, 1100)

-80

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Figure 5: Left" maximum (within (y,z) planes) of the time-averaged velocity magnitude, as function of the streamwise coordinate x. Right" z-locations of the jet centerline, lower width and upper width by the coarse grid, thus limiting turbulent mixing and spreading. Both fine grid computations agree fairly well for the opening angle. But the jet angle is somewhat smaller for the stepwise body than for the 2nd order smooth body. One can relate this behavior to the smaller recirculation regions in the computation with stepwise body. Indeed, in the downstream part of the recirculation region of the lowest blade, the velocity vector points upwards. The larger the recirculation, the higher the upward velocity W. The higher values observed in the smooth body computation deflects the jet upwards. This phenomena is much more moderate in the stepwise body. It can be concluded that grid refinement has a more noticeable quantitative influence on the time-averaged flow fields than boundary treatment.

716

Figure 6: Time-averaged streamwise velocity. Thick black lines indicate the contour of the geometry. Thin black lines are U = 0. From left to right 9 2nd order IBM + fine ; 0th order IBM + fine ; 2nd order IBM + coarse. The grey scale is the same as in Fig. 3 Table 2: JET ANGLE AND OPENING ANGLE (IN DEG)

2nd order IBM + fine 0th order IBM + fine 2nd order IBM + coarse

4

4.1

FLUCTUATING

Centerline angle 7.4 6.6 5.3

Opening angle 4.8 4.6 2.7

FLOW ANALYSIS AND ACOUSTIC SOURCES

Physical investigation

The quadrupole acoustic source term on the right hand side of Eq. (1) can be related to vortex dynamics. In incompressible flows, it can be written as

02T~j Oui cgu~ = OxiOxj = c~ Oxi

Q (s~js~j

-

w~jw~j)

,

(4)

where Sij is the strain rate tensor and W~j the rotation rate tensor. Positive volumic sources are stretching regions, and negative volumic sources are rotating regions. The quadrupole acoustic source term is equivalent with the Q-criterion frequently used to identify vortices. Note that Eq. (4) can be very easily and accurately computed in our solver, because it is a by-product of the Poisson equation. Also note that for sound generation, only the fluctuating part of (4) is relevant. Fig. 7 shows a snapshot of the fluctuating quadrupole volumic source term in the exhaust pipe. See also Fig. 8 (left) for a cut through the pipe. Strong contributions are located in the blade wakes and in recirculation regions. They result from instabilities of the detached shear layers. Within one pipe height, the major sources have extinguished.

4.2

I n f l u e n c e of n u m e r i c a l p a r a m e t e r s

As shown in Fig. 8, both IBM order and grid refinement have a strong influence on the acoustic source terms. The coarse grid and the 0th order IBM (stepwise body) produce spurious volumic sources. The source terms take extremal positive and negative values from one grid point to

717

Figure 7: Volumic acoustic source terms. In white, negative sources (vortices) ; in black, positive sources (stretching zones) the neighboring grid point and no structure can be recognized. To capture the unsteady vortex dynamics and thus the acoustic sources, the solid body must be represented in a smooth manner, consistently with the overall 2nd order spatial accuracy. Even more critical is the grid resolution of the small details of the flow such as recirculation zones and shear layers produced by the blades and narrow gaps of the pipe. These results demonstrate that requirements for correctly predicting the mean flow and requirements for correctly predicting acoustic source terms are different.

5

CONCLUSIONS

The jet of our air-conditioning pipe shows some marked differences with a classical turbulent jet. The flow in the pipe is dominated by strong recirculation. Some widely spread back flow regions are also present at the orifice of the pipe. The jet does not have a potential core since it spreads linearly beginning right from the pipe exit. These features demonstrate the necessity to take into account the whole geometry of the pipe when trying to predict the jet development. The Immersed Boundary Method with direct forcing proved an efficient way of handling the problem of computing the flow around a very complex geometry. Least square interpolations revealed advantageous in comparison with Lagrange interpolations because of their numerical stability. The influence of grid resolution and IBM order is quantitative for time-averaged variables but qualitative for fluctuating variables. In particular, the stepwise discretization of the geometry and the smooth 2nd order representation produce analogous results for the mean flow. On the other hand, the stepwise representation leads to a qualitatively erroneous description of acoustic source terms. The description of fluctuating terms requires higher order boundary treatment and higher resolution than the description of mean quantities. Future work includes assessing the influence

718

Figure 8: Snapshot of the volumic acoustic source terms at y = -44. Left : 2nd order IBM + fine ; Top right : 0th order IBM § fine ; Bottom right : 2nd order IBM § coarse. Grey shades stand for the same values in all three plots. of the inlet boundary condition over the recirculation bubble at the pipe inlet. We will perform a simulation of the complete inlet pipe. Another problem of major interest concerns the wall contribution to the aeroacoustic noise generation. Because of the large recirculation regions in the pipe, one can expect strong pressure fluctuations. We will quantify the wall contribution with the Ffowcs-Williams- Hawkings analogy.

References [1] E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comp. Phys., 161:35-60, 2000. [2] J. E. Ffowcs-Williams and D. L. Hawkings. Sound generation by turbulence and surfaces in arbitrary motion. Phil. Trans. R. Soc. London A, 264:321-342, 1969. [3] G. Iaccarino and R. Verzicco. Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev., 56(3):331-347, 2003. [4] J. Lighthill. On sound generated aerodynamically. I General theory. Proc. Roy. Soc. London, 211 A:564-587, 1952. [5] M. Manhart. A zonal grid algorithm for DNS of turbulent boundary layers. Computers FLuids, 33(3):435-461, 2004. [6] J. Mohd-Yusof. Combined immersed-boundary / B-spline methods for simulations of flow in complex geometries. In Center of Turbulence Research, editor, Annual Research Briefs, pages 317-327, 1997. [7] N. Peller, A. Le Duc, F. Tremblay, and M. Manhart. Least-square vs Lagrange interpolations in the direct forcing immersed boundary method. International Journal for Numerical Methods in Fluids, In preparation. [8] J. Smagorinsky. General circulation experiments with the primitive equations. Monthly Weather Rev., 91:99-164, 1963. [9] F. Tremblay, M. Manhart, and R. Friedrich. DNS and LES of flow around a circular cylinder at a subcritical Reynolds number with Cartesian grids. In R. Friedrich and W. Rodi, editors, LES of complex transitional and turbulent flows, pages 133-150, Dordrecht, 2001. Kluwer Academic Publishers.

Engineering TurbulenceModellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

719

CHARACTERIZATION OF A SEPARATED TURBULENT BOUNDARY LAYER BY TIME-FREQUENCY ANALYSES OF WALL PRESSURE FLUCTUATIONS R. Camussi I , G. Guj 1, A. Di Marco1 and A. Ragni2 1Industrial and Mechanical Engineering Dept. (DIMI) University ~Roma 3', Rome, Italy 2Department of Experimental Aerodynamics Methodologies CIRA- Italian Aerospace Research Centre, Capua, Italy

ABSTRACT An experimental investigation of wall pressure fluctuations generated by turbulent boundary layers (TBLs) over surface irregularities has been conducted in a backward-forward-facing step geometry simulating a large aspect-ratio cavity. Measurements are conducted by means of microphones flush mounted at the wall. The present paper is focalized upon the characterization of the local properties of the auto and cross-spectra in terms of both space and time and in account for the effects of the flow separations near the steps. It is shown that the classical approach for the cross-spectrum modeling, based on a pure exponential approximation, correctly applies even within the regions close to the steps, provided the exponential decay coefficient is correctly evaluated. The convection velocity computed along the cavity reveals that the hydrodynamic contribution to the pressure fluctuations is always dominant except for the region close to the forward-facing step where acoustic effects are the most relevant. The effect of the steps is also reflected onto the Probability Distribution Functions (PDF) of the coefficient of the exponential decay of the coherence function, obtained through the wavelet analysis of the pressure time series.

KEYWORDS Separated boundary layer, pressure fluctuations, wavelet transform, spectral modelling, large aspect ratio cavities, Corcos-type models.

INTRODUCTION

Steps and geometrical irregularities on the exterior surface of modern high-speed passengers aircraft appear for example at skin lap joints or window gaskets, and are recognized as potential sources of

720 aerodynamically generated noise. It is known that the contributions of such aeroacoustic sources to the interior noise is significant and dominant at the front part of the fuselage. Similar aeroacoustic problems are also encountered in other fields of engineering interest, for example in the vehicles or trains aerodynamics. The primary motivation of the present work is to cover the lack of experimental results in the case of large aspect-ratio cavities and to explore the possibility of extending classical spectral modelling also in the case of flow separations. Even if the subject is of great interest from the viewpoint of practical and basic research applications, it has not been treated in detail so far and the results available in literature are limited and sometimes contradictory (see e.g. Efimtzov, 1999, and Leclercq et al., 2001). In the present study, the surface irregularities are modelled by a backward-facing step (BFS) followed by a forward-facing step (FFS) disposed in an incompressible turbulent boundary layer. The sketch of the surface irregularities model is exhibited in Figure 1, together with the main symbols adopted.

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Figure 1: Sketch of flow conditions over a backward-forward facing step. When the flow reaches the BFS, a detachment occurs and a reverse flow zone is generated just aider the step. After a distance of about 5-7H there is an oscillating reattachment point of the flow and the reattachment point unsteadiness is recognised as a strong noise source (e.g. Lee & Sung 2001, Lauchle & Kargus, 2001). After the reattachment, the flow encounters the FFS and a second flow detachment occurs at about 1H before the step. Finally a third separation bubble follows after the FFS; this recirculation is bounded downstream by a reattachment point, after which the TBL slowly recovers its characteristics. It is known that the largest pressure fluctuations occurs at the first reattachment position (x/H about 6) and close to the FFS. This is shown in Fig. 1 where explanatory results obtained in the present study are presented. In the present work attention is focalized towards the analysis of cross-correlations and cross-spectra along the cavity. As summarized by Graham (1997) several models have been proposed in literature to give suitable approximations of the wall pressure cross-spectra in the case of equilibrium TBL. Nevertheless, as pointed out recently by Leclercq & Bohineust (2002), the classical Corcos model (Corcos, 1963) is still recognized as the most accurate to reproduce the fluctuating pressure crossspectra in the case of equilibrium TBL. The extension of the Corcos-type approach to the non-equilibrium TBL developed in the configuration sketched in Fig. 1 is explored in the present work. In particular, the Corcos model is generalized to account for the effect of the steps both in terms of global quantities (e.g. the convection velocity Uo) and in terms of temporal statistics. Without entering into details, we remind only that within the framework of the Corcos-type modeling, the spatio-temporal streamwise cross-spectrum, Spe, , can be represented in terms of the auto-spectrum Sep through the following formula: t~

I--

e

(1)

721

where ~ is the streamwise separation, co = 2nfand a is the Corcos exponent. The coherence function, y, can therefore be approximated by an exponential function and the analysis of the spatio-temporal behavior of the coefficient R is one of the major objectives of the present work. The time-frequency analysis is extensively conducted by the application of specific post-processing tools based on the application of the wavelet transform that is able to extract the temporal location and the frequency content of highly energetic local events (see e.g. Camussi & Guj, 1997, Guj & Camussi, 1999). More details are given in the following, together with a brief description of the experimental set up and the main results exploitation. 130

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Figure 2: Sound Pressure Level (defined as SPL=201ogl0[~p/20BPa] with Gp the pressure signal standard deviation) along the cavity for two Reynolds numbers. The first maximum is located at the first reattachment position (see Fig. 1) while the second one is at the forward step position.

EXPERIMENTAL SET UP The tests were carried out in the closed-circuit low-speed wind tunnel of the aerodynamic laboratory of the DIMI and ENEA (Agency for New Technology, Energy and Environment) cooperation. The test section is 0.9x1.16 m 2 in cross-section and is 2.5m long. The maximum mean velocity along the centerline of the test section is 90m/s and the Relative Turbulence Level at 40m/s is 0.1%. The experimental setup is shown in Figure 3. The main element of the backward-forward-facing step model is an aluminium sliding plate assembled under the test section. { } The adopted instrumentation consisted of B~el&Kj~r 1/4" 4135 and 1/8" 4138 microphones, Falcon 2670 preamplifiers and Nexus 2690 signal conditioner. Two streamwise separated microphones were positioned on the cavity and flush mounted on the floor. Their separation distance was fixed to 2.5cm, sufficiently small to resolve the cross-spectra and cross-correlations. The sliding plate was translated by means of a numerically controlled micrometric traversing system driven by a step-by-step motor. In this way about 30 positions spanning the whole cavity opening were covered during the measurement campaign. The pressure signals were acquired using an 8 channel Yokogawa Digital Scope DL708E. Around 105 samples, with a sampling rate of 20kHz and a cut-off frequency filter of 5kHz, were acquired from each channel. The signals presently analyzed correspond to a small part of a large data base obtained in several flow conditions and geometrical configurations (details are given in Camussi et al., 2004, where the analysis of auto-spectra and auto-correlations is presented). In the selected cases, the incoming TBL,

722

characterized through hot wire anemometric measurements, is the one naturally developed along the wind tunnel wall while the free stream velocity magnitude U is fixed to 30 and 50m/s. The steps' height H is fixed to 15mm, of the same order of magnitude of the boundary layer thickness, while the cavity streamwise opening L is 640mm. The transverse cavity separation B is 800mm therefore sufficiently larger than the steps height so that the flow statistics can considered two dimensional at the cavity symmetry plane.

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_

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Figure 3: Experimental setup of backward forward-facing step.

RESULTS

Spatial evolution of Corcos model parameters In account of Eqn. (1) that represents the basic ingredient of the Corcos model, the parameters presently analyzed are U, and ct. The first quantity is computed from the temporal location of the peak of the cross-correlation while the second one by a linear fit of the semi-logarithmic representation of 7. Examples of the results obtained are reported in Fig. 4 and 5. The result of Fig. 5 has been obtained in a position very close to the second step. It is evident that in this case, on one side the exponential approximation of 7 still applies, but on the other the determination of a time delay from the cross-correlation is very difficult. This is due to the very large value of the convection velocity, as can be inferred from Fig. 6 and 7 where the U, and ot variations along the plate are reported respectively. It is shown that in the first reattachment region (up to about x/H=6), the Uc is very low as an effect of the recirculation. In the intermediate region, up to x/H~40, the convection velocity is about 70% the free stream velocity, in agreement with the results commonly obtained in equilibrium TBL (Lee & Sung, 2001). Also the magnitude of the exponent ct is, in this region, very close to the Corcos prediction (~0.12). For x/H>40, according to the result of Fig. 5, the convection velocity becomes very high. It should be pointed out that, in account of the temporal resolution of the acquired data, the computation of the time delay from the cross-correlation for x/H>40 is affected by a large uncertainty since it is very close to zero and thus the amplitudes of Uc reported for x/H>40 are only qualitative due to the large error bar (not reported). However, it can be argued that, in this separated region, the convection velocity is of the order of the velocity of sound, thus about one order of magnitude larger than the free stream velocity and independent on the velocity. This result is quite interesting from the physical viewpoint since it indicates that for about x/H<40, the wall pressure fluctuations sensed by the microphone, are not acoustic but hydrodynamic, as expected in equilibrium TBL. On the other hand, in the vicinity of the second step, the pressure fluctuations are

723 mainly acoustic and they are probably due to the impact of vortical structures shed from the upstream backward-facing step on the vertical side of the downstream forward step (this interpretation is supported by velocity-velocity and velocity-pressure cross-correlations presented in Camussi et al., 2004, and not reported here for brevity). I ,

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Figure 4 Cross-correlation between microphones (upper panel) and coherence function (lower panel) at x/H~20 and for U=30m/s. The straight line in the lower panel is an exponential approximation.

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Figure 7: Corcos exponent as a function of the normalized position along the cavity wall.

Wavelet analysis and temporal statistics The Wavelet analysis decomposes a signal into a set of finite basis functions thus uncovering locally transient characteristics obscured by the infinite sinusoidal basis functions used in Fourier analysis. In the present approach we are interested in exploring the local properties, in terms of temporal fluctuations, of the Corcos exponent a reported in Eqn. (1). The Wavelet coefficients W(t,r) are obtained through the convolution of the analyzed pressure signal with the dilated and translated counterpart of a so called mother wavelet ~ ( t )

725

W(t,r) =

x(t)~

dt

(2)

The mother wavelet used in this study is the Morlet one. The example reported in Fig. 8 indicates that at every instant it is possible to determine the local energy content of the signal as a function of the frequency, the so-called scalogram, and, thus, by combining two different signals, a Cross-spectrum local in time can be obtained. More specifically, the time-frequency wavelet-based auto-spectrum, cross-spectrum and coherence, follow the form of the Fourier-based definitions (Salvetti et al. 1999). As an example, the wavelet cross spectrum is given by:

Cfg(r) = ~Wf* (t,r)Wg(t,r)dr

(3)

T

wheref(t) and g(0 are two time series and Cfg is complex (incidentally, whenf(O and g(O coincide, the wavelet auto-spectrum is achieved). The localized time integration window (length T) is selected based on the desired time resolution in the resulting wavelet spectrum. An example of the 7 function obtained from the local wavelet coefficients is reported in Fig. 9 compared with the classical Fourier result from which the Corcos exponents are commonly retrieved. It is shown that by selecting a range of frequency of interest, a coefficient at of the exponential decay can be determined. The locality in time of the wavelet coefficients permitted us to analyze the temporal statistics of the coefficient at. The knowledge of the PDF of ott represents an important generalization of the Corcos model in particular if simple analytical approximations of the PDFs are achievable. It is found that, as expected, the mean value of ott coincides with the Corcos exponent ot introduced in Eqn. (1) while the fluctuations of at are large and are characterized through the PDFs, like the ones of Fig. 10 where examples in terms of reduced variables (zero mean and unitary standard deviation) are reported. It is found that in the middle part of the cavity, close to Gaussian curves are obtained. On the other hand, in the vicinity of the two steps, the PDFs exhibit exponential tails skewed towards positive values. This behaviour can be better appreciated from Fig. 11 where also a Gaussian reference curve is reported for comparison. It is worth noting that despite the physical differences evidenced in the previous paragraph, the backward-facing and the forward-facing steps induce the same statistical behaviour of the coherence exponent fluctuations. To the extent of the statistical modelling, analytical approximations of the above reported PDFs, can be obtained thus rendering more general the Corcos-type representation of the cross-spectra. This issue, as well as furher insight into the physical interpretation of the observed PDFs' shape variations, are left for future studies.

CONCLUSIONS An experimental characterization of the wall pressure fluctuations at low Mach numbers has been conducted in a shallow large aspect-ratio cavity in both the streamwise and transversal directions. The separation is much larger than the steps height. Several measurements were performed with high resolution microphones at the wall at different flow conditions. The main results obtained are the following: - The Corcos-type modelling of the cross-spectra can be successfully applied both in the region of the cavity far from the steps as well as in the separated flow regions. Moreover, the classical models commonly adopted for equilibrium TBL can be successfully applied in the middle of the cavity. This is not a trivial result since in this region, the TBL is still far to be considered fully developed. In the vicinity of the steps the Corcos coefficient cr have to be modified with respect

726 to the classical results but the exponential approximation of the coherence functions seems still plausible. The wall pressure fluctuation in the vicinity of the second step are dominated by acoustic effects rather than hydrodynamic, as expected in standard TBL. This result therefore suggest that the physical effects underlying the generation of the wall pressure fluctuations close to BFS and FFS are completely different. This point is important also for the purpose of noise control in shallow cavities since the pressure fluctuations at the FFS can be modified simply by acting on the BFS and in particular by controlling the vortical structures shed from it. The temporal statistics of the Corcos exponent is Gaussian in the quasi-equilibrium region of the TBL, i.e. in the middle of the cavity. The different nature of the pressure fluctuations close to BFS and FFS are not sensed by the temporal PDFs which, in both the regions close to the two steps, exhibit exponential tails. It is argued that simple analytical approximations of the PDF could be achieved and thus inserted in the coherence function definition. In this way, robust generalizations of the Corcos model can be obtained. Further issues concerned with the implementation of the statistical models and with deeper physical interpretations are left for future studies which have been currently undertaken by the authors. -

0

-

M. Corella and S. Neroni are acknowledged for their help during measurements and data analysis. D. Prischitch of ENEA is also acknowledgedfor his support during the experimentationat the ENEA wind tunnel. This work has been developedunder a EU grant GRDl-1999-10487 within the 5th FrameworkProgramme.}

Figure 8: Example of local energy distribution (lower panel) obtained from the scalogram (square of the wavelet coefficients) of the a portion of a pressure signal (upper panel). Lighter regions in the lower panel indicate high energy content. The case reported corresponds to a pressure signal obtained in the middle of the cavity at 30m/s.

727

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Figure 9: Classical Fourier (solid line) and local ~, function (solid-dotted line) reported together. The dashed straight line represents the pure exponential approximation of the local ~/function. The case reported corresponds to a pressure signal obtained in the middle of the cavity at 30m/s.

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",.. .... iii:-, ......... ::-,- ........ :-, ......... ::,, ....... :::'~, i

rail 4o -6

-4

-2

0

2

4

6

I%-<~:'1t%

Figure 10: PDFs of the local exponent at reported in terms of reduced variables and as a function of the spatial location along the cavity. The case considered corresponds to 30m/s.

728

-0.5

N

n

0

-2.~

?+ i

-3.5

,,,1"

-4

\ I

-3

I

'2

I

-1

I

0

I

1

[~-<(xt>]/oalpha

I

2

I

3

'~1

4

Figure 11" PDFs of the local exponent cxtfor some selected cases: in the vicinity of the first step (symbol 'o', position x/H=3), in the first re.attachment point ('n', x/H=6), in the middle of the cavity ('0', x/H=24), close to the second step ('.',x/H=42).

REFERENCES -

-

-

-

-

-

Camussi R. and Guj G. (1997). Orthonormal wavelet decomposition of turbulent flows: intermittency and coherent structures, d. FluidMech. 348, 177-199. Camussi R., Guj G. and Ragni A. (2004). Wall pressure fluctuations induced by turbulent boundary layers over surface discontinuities. Accepted for publication in the Journal of Sound and Vibration. Corcos G.M. (1963). The structure of the turbulent pressure field in boundary-layer flows. J. Fluid Mech. 18, 353-378. Efimtsov B.M., Kozlov N.M., Kravchenko S.V. and Anderson A.O. (1999). Wall Pressure Fluctuation Spectra at Small Forward-Facing Step. AIAA paper 99-1964. Graham, W. R. (1997). A comparison of models for the wave-number-frequency spectrum of turbulent boundary layer pressures. J. Sound Vibr., 206, 541-565. Guj G. and Camussi R. (1999). Statistical analysis of local turbulent energy fluctuations, d. Fluid Mech. 382, 1-26. Lauchle G.C. and Kargus W.A. (2001). Scaling of Turbulent Wall Pressure Fluctuations Downstream of a Rearward-Facing Step. Journal ofAcoust. Soc. Am., 107, L1-L6,. Leclercq D.J.J. and Bohineust, X. (2002). Investigation and modelling of the wall pressure field beneath a turbulent boundary layer at low and medium frequencies. J. Sound Vibr., 257, 477-501. Leclercq DJ.J., Jacob M.C., Louisot A. and Talotte, C., (2001). Forward-backward Facing Step Pair: aerodynamic flow, wall pressure and acoustic characterisation. AIAA paper 2001-2249. Lee I. and Sung H.J. (2001). Characteristics of Wall Pressure Fluctuations in Separated and Reattaching Flows over a Backward-Facing Step - Part I. Experiments m Fluids 30, 262-272. Salvetti M.V., Lombardi G. and Beux F. (1999). Application of wavelet cross-correlation technique to the analysis of mixing. AIAA J. 37, 1007-1017.

11. Turbomachinery Flows

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

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S T U D Y OF F L O W A N D M I X I N G IN A G E N E R I C G T C O M B U S T O R U S I N G LES B. Wegner, B. Janus, A. Sadiki, A. Dreizler and J. Janicka Institute for Energy and Powerplant Technology Dept. of Mechanical Engineering Darmstadt University of Technology Petersenstr. 30, 64287 Darmstadt, Germany

ABSTRACT Due to careful and deliberate trade-offs between often conflicting design requirements (low emission, high operating pressure and temperatures etc.), development and improvement of gas turbines nowadays relies heavily on advanced simulation methods. While the basic capability of LES has been amply demonstrated on a number of relatively simple academic configurations, there is still a lack of works applying LES to practical systems also performing detailed quantitative comparisons based on experimental data. This is mostly due to the lack of suitable reference data. In this paper, we present results of the isothermal LES of a generic gas turbine combustor rig studied in house by means of advanced laser measurement techniques. The results of these detailed measurements are used to thoroughly validate the LES results. Beyond a pure validation, the LES is used to analyse the influence of the precessing vortex core present in the studied configuration on the mixing of fuel and oxidizer and possible implications for reacting conditions are discussed.

KEYWORDS

large eddy simulation, generic GT combustor, coherent structures, mixing, validation

INTRODUCTION

The concern for our environment and the resulting tightening legal regulations as well as economical considerations force engineers to come up with new or refined technologies to reduce pollutant emissions and increase the efficiency of combustion devices. To this end it is essential to accurately control the mixing of fuel and oxidizer in both premixed and non-premixed combustion systems. In many modern combustion applications such as furnaces, industrial gas turbines or aero-engines, fuel injector nozzles with swirler devices play an important role due to the rapid mixing and flame stabilization they offer. Most of these swirl flows exhibit hydrodynamic instabilities called precessin9 vortex cores (PVC) which were first described by Chanaud (1965). Since

732

then, there have been numerous studies dealing with this phenomenon, which have recently been reviewed by Lucca-Negro and O'Doherty (2001). Although the underlying mechanisms are not well understood, there is some evidence, that these periodic coherent structures strongly influence mixing and combustion and can even lead to instable behavior of combustion systems by a coupling with acoustic modes of the geometry (Syred et al. (1997); Anacleto et al. (2001); Li and Gutmark (2003)). Hence, the influence of PVC on mixing processes deserves some attention. Since the PVC phenomenon is inherently three-dimensional and unsteady, the computational resources necessary for a numerical approach have become affordable only recently. Some attempts are documented in the literature to simulate swirl flow instability using unsteady RANS (URANS) type turbulence models with at least some success (Bowen et al. (1998); Guo et al. (2001); Wegner et al. (2004b)). The suitability of URANS for such unsteady phenomena is still a matter of principal debate though, see e.g. Iaccarino et al. (2003). Due to the complex unsteady flow and mixing processes involved, there is a wide spread agreement that the method of large eddy simulation (LES) has a high potential to predict such systems. The fact that the large energy carrying structures responsible for large scale mixing are directly simulated can also help to better understand the underlying fluid dynamical mechanisms and thus to improve new technologies such as LPP or RQL combustion systems. Some successful applications of LES to swirling flows have been presented by Pierce and Moin (1998); Derksen and den Akker (2000); Schliiter et al. (2001); Menon (2004) or Wegner et al. (2004a). A recent review on the state of art in the field of LES for combustion systems has been given by Janicka and Sadiki (2004). Based on the good experience made in the past, we use LES to study the flow and mixing in a non-premixed model GT combustor under isothermal conditions. This configuration, which features a single real industrial fuel injector nozzle operated at near-realistic aero-engine conditions, is investigated in house by means of advanced laser-based measuring techniques. These measurements, some of which have already been reported earlier in Janus et al. (2004a,b), have revealed the presence of a PVC for both isothermal and reacting conditions in this model combustor. Measurements by Midgley et al. (2004) on the same swirler nozzle in a water rig have also shown the existence of a PVC and will also be used as a reference for the present study.

G O V E R N I N G E Q U A T I O N S A N D LES M O D E L L I N G The filtered Navier-Stokes equations along with the filtered continuity equation describe the behavior of any Newtonian fluid. cOt + ~

0--V +

( ~ ~ ) = 0~

- 0

\0~j + 0 ~ j - ~

(1)

)

0~

(2)

A simple Smagorinsky model is employed for the subgrid scale stress-tensor ~i3 _ s a s , with Lilly's formulation of Germano's dynamic procedure for the determination of the model coefficient Lilly (1992). No special wall treatment is used. Instead, we resolve the boundary layer and rely on the ability of the dynamic procedure to capture the correct asymptotic behavior of the SGS model when approaching the wall, Lesieur and M~tais (1996). An additional filtered scalar transport equation is used to describe the evolution of a passive

733

scalar, namely the mixture fraction f.

O~f ~o---t--

(-~i f ) -

D

Ox,

_ j~GS

9

(3)

The mixture fraction which is commonly used for diffusion flame modelling can be viewed as a dimensionless fuel concentration. It is defined as f = 0 for pure oxidizer and f = 1 for pure fuel. It can be used for any two-feed system to describe the mixing ratio between the two different fluids. The diffusion coefficient of mixture fraction D is linked to viscosity via the Schmidt number (D = u/Sc), for which a value Sc = 0.7 is assumed. This relates to gaseous mixing. An eddy diffusivity model is used for the subgrid scalar flux j s a s , assuming a constant Schmidt number relationship between the turbulent diffusion coefficient and the turbulent viscosity

jsas =

NUMERICAL

ut O f . Sct Oxi '

Sct = 0 . 7 .

(4)

PROCEDURE

All the governing equations are integrated into the three dimensional finite-volume CFD code FASTEST-3D. The code features geometry-flexible block-structured, boundary-fitted grids with a collocated, cell-centered variable storage. Second-order central schemes taking into account the grid non-orthogonality by means of multi-linear interpolation (Lehnh/iuser and Schgfer (2002)) are used for spatial discretization except for the convective term in the scalar transport equation. Here, a flux-limiter with TVD (total variation diminishing) properties is employed to ensure bounded solutions for the mixture fraction, Waterson (1994). Pressure-velocity coupling is achieved via a SIMPLE similar procedure. As time integration scheme the second-order implicit Crank-Nicolson method is used. The resulting set of linear equations is solved iteratively using a 7-diagonal SIPsolver.The code is parallelized based on domain decomposition using the MPI message passing library. For details on the method refer to Durst and Sch/ifer (1996).

CONFIGURATION

AND NUMERICAL

SETUP

The swirler nozzle under investigation is a slightly simplified version of a nozzle which is operated in TURBOMECA aero-engines. Air is injected into an annulus through tangential swirler channels and gaseous fuel (natural gas) is injected through the central cone. The computational domain which has been used to model nozzle and rig is shown if figure 1. This nozzle is investigated in house in a test rig which is able to deliver preheated air of max. 773 K at pressures up to 10 bar. Both reacting and non-reacting cases have been investigated. For the isothermal conditions which are the subject of this paper, the natural gas was replaced by a helium-air mixture with the same density as the fuel to maintain Reynolds-similarity. Velocity measurements featuring one-point statistics as well as time series were obtained by means of laser Doppler velocimetry (LDV). Fuel concentration statistics were determined with planar induced fluorescence (PLIF) using acetone as fluorescence marker in the fuel stream. We will not describe the experimental setup in detail, but refer the reader to Janus et al. (2004b) and Janus et al. (2004a). The operating conditions used for the present study are summarized in table 1. The computational domain which can be seen in figure 1 consists of 137 grid blocks featuring an O-type structure. Both the swirler nozzle as a part of the outlet duct with cooling air injection are included in the computational model to reduce an effect of boundary conditions on the flow

734

Figure 1: Computational model of the TURBOMECA swirler nozzle and test rig (not to scale). The left picture shows a detailed view of the tangential swirler channels and the central fuel injection nozzle. The right picture shows the complete computational domain, indicating also the lines along which statistical profiles are compared. Table 1: Operating conditions used Combustor pressure p Air temperature T Air mass flow rate ?:Ftai r Fuel mass flow rate/nfuet Reynolds number Re

2 bar 623 h 30 g/~ 1.4 g~ 4600C

in the combustion chamber itself. The inclusion of the swirler has been found to be crucial in capturing precessing vortex cores in such geometries since they are linked with significant instantaneous back-flow into the swirler (Wegner et al. (2004a)). This is also the case for the current configuration as will be shown later. The total number of grid points on the actual or "fine" grid is 2.3- 106. For testing and verification purposes, this grid was coarsened by a factor of 2 in each direction. As inlet boundary conditions, the mass flows from the experiment were prescribed using homogeneous laminar profiles. No special measures had to be taken imposing artificial turbulence since the shear forces alone generated enough turbulence.

RESULTS The first computation was run on the coarse grid and the converged solution was then transferred to the fine grid as a good starting solution. On both grids, several flow through times were computed and samples for statistics were collected. In addition, time series of velocity and mixture fraction were recorded in selected points of the computational domain to perform spectral analysis. We start now with a qualitative description of the flow and mixing before presenting quantitative results. Q u a l i t a t i v e D e s c r i p t i o n of t h e F l o w S t r u c t u r e First, the time averaged stream lines in the combustor centre plane shown in figure 2 are used to get a global picture of the flow. It can be seen that a fairly complex flow pattern evolves featuring a central recirculation zone (CRZ) caused by the swirl into which the central fuel jet penetrates, resulting in a free stagnation point. A second recirculation zone is found in the shear layer between the fuel jet and the swirled air and a third, outer recirculation (not shown in the picture)is located

735

Figure 2: Time averaged stream line pattern in the combustor centre plane (y = 0).

Figure 3: Isosurface of the X2-criterion showing the precessing vortex core (left: top view; right: side view). in the corners of the combustion chamber. The second recirculation zone causes a separation of the flow from the central fuel injector cone and results in an average back-flow, even transporting fuel back into the swirler annulus. This picture is coherent with what was found in the experiments. When looking at animations from the LES, a highly unsteady behavior of both-the flow and mixing is noticed. Of course, this is partly due to turbulent fluctuations, but besides these, coherent structures, namely a precessing vortex core can be identified in the flow. This is done by using the A2-criterion proposed by Jeong and Hussain (1995) for the identification of vortices. Isosurfaces of A2 are shown in figure 3, clearly revealing a so called double-cell PVC - two helical vortices emerging from the swirler. These vortices, which were also observed by Midgley et al. (2004), revolve around the central fuel injector cone with the direction of the swirl.

Quantitative Description The samples taken from both the coarse and fine grid simulations were time averaged and profiles of statistical quantities were extracted on the radial lines shown in figure 1. In the following, we compare these profiles with experimental results. The average axial and radial velocities are plotted in figure 4 while the average tangential velocity and the mean mixture fraction are shown in figure 5. A very good overall agreement of the simulation results from both grids with the LDV velocity and PLIF concentration measurements can be observed. Especially the results close to the swirler exit at x = 1 m m match very well with the experiments. Here, it can be seen that there is back-flow into the swirler not only in a local, but in a mean sense due to the separation of the flow from the central fuel injector cone. Surely, this interaction between the nozzle and chamber

736

._

i

i

u

i

x=5Omtn

'

'

'

_.__.~..

x=4Omm .......... . ,.~ :,;,;:....;;:- ......... ;;:..... ..... x=3Omm ~ ' . '

,

x=2Omm ..... .,

,...::...

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, . ; ; , ~ .9. . . .

,,,,.,

2:---

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,

x=lOmm .......

x=5mm~

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1,

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=1 m

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m ......... -1

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0

0.5

1

I

I

I

I

-1

-0.5

0

0.5

y/D [ - ]

1

y/D [ - ]

Figure 4: Radial profiles of the mean axial (left) and radial (right) velocity component. Black symbols: LDV data; solid line: LES, coarse grid; dashed line: LES, fine grid. ....

|

i

i

!

~'1

x=5Omm ....

!

X=__...4;13m

',-,,oot,.ooooe

.- . . . .

. . . . . . . . . . .

.........

. . . . .

x=30mm ......

. . . . . . . . . . . . . .

x-20mm .......

x=15mm

................

.~',

x=lornrn i .......

T....._t.'.'t.t...

x=5 mm

2oS

i .......................

wo ~

:

,- ....

~

0 0 y/D [ - ]

~_

'_,' __ ~ - - 7 - ~

. -_

.__

...

..

~

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.

,

l,

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, "'''~176

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,-

i

.... ! .....--:

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~

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x=l mm

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y/D [ - ]

Figure 5: Radial profiles of the mean tangential velocity component (left) and the mean mixture fraction (right). Black symbols" LDV/PLIF data; solid line: LES, fine grid; dashed line" LES, fine grid. could not be described by neglecting the swirler geometry. The only major discrepancy between the LES and the experiments is in the region of the fuel jet which in the simulation is too slow. The reason for this is, that the temperature of the fuel was initially assumed to be equal to the

737

ambient temperature of 298 K and the according fuel density and mass flow rate were set in the simulations. It was suspected, that the fuel stream is significantly heated up by the preheated air due to heat conduction in the nozzle and inlet geometry. Thermocouple measurements confirmed this hypothesis showing a real fuel temperature at the nozzle exit of approximately 398 K. Hence, the velocity based on the mass flow and density is too low in the LES. As a consequence, the penetration of the fuel jet into the CRZ is not predicted deep enough, but the overall structure and shape of the CRZ is predicted to a satisfactory degree of accuracy. Also plotted in figure 5 is the mean mixture fraction which matches well the results from the PLIF measurements. In accordance with the velocity results, the mixing of fuel and air is somewhat to fast. Besides the mean velocity components, second moments, namely the velocity fluctuations were also compared. As can be seen in figure 6, the predicted resolved fluctuation of the axial and tangential velocity components in wide ranges show a remarkable degree of agreement with the measured profiles. Both the position a n d height of the peaks compare very well. In general, as expected, the fine grid results compare better to the measured data than the results obtained on the coarse grid. Even the fluctuation peaks stemming from the boundary shear layers in the swirler i

i

l

i

u

x=50mm

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x-.4..0.mm

.

.

.

.

.

.

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oo.~O I

,-....'",..~

,~.;.';;;;~"

-o

x=5 mm

~

o.~.oOOoo oo

x=10mm

~,o

m.~ ~

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x = l 5mrn..,o;.." "

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:. ..... ,~,-. .........

,~o~.. o o ~ o ~.n.,4.o-O ~

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I

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1..........

-1

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:.,-... I "'~

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Figure 6: Radial profiles of the resolved axial velocity fluctuation (left) and resolved tangential velocity fluctuation (right). Black symbols" LDV data; dashed line: LES, fine grid. are predicted correctly by the LES although (as mentioned above) no wall model was used. This is an indicator that the grid resolution in the wall near region was sufficiently fine. By evaluating the y+ distance of the wall next grid point which is y+ < 5 in most cells, it could be confirmed that indeed there is at least one point in the laminar sub-layer even on the coarse grid. It should be noted that compared to the mean velocity, the magnitude of the fluctuations is very large. This is another indicator for the presence of highly energetic coherent structures in the flow. To further analyse this, the time series recorded throughout the simulations were used to compute power spectra. Typical spectra obtained at a point located in the lower part of the CRZ 1 m m

738

........

,

.....

.

,

_ .......

i ....

1

9

O.O1

0.01

0.0001

0.0001

le-06

le-06

le-08

le-08

le-10

le-10 1000

1000

10000

10000 f [Hz]

f [Hz]

Figure 7: Normalized power spectral density of the axial velocity component (left) and mixture fraction (right).

above the nozzle exit are shown in figure 7. Both the spectra for the velocity as well as for mixture fraction exhibit strong dominant peaks at frequencies fl ~ 1090 Hz - and f2 ~ 2390 Hz. Power spectra obtained from the LDV data show similar peaks at slightly higher frequencies of 1370 Hz and 2900 Hz, respectively. With a factor between the higher and lower frequencies of 2.19 in the simulation and 2.12 in the experiments, it seems that the higher frequency is not exactly a harmonic of the lower one. Obviously, the frequencies belong to the precessing vortex core, clearly showing an influence of the PVC on the scalar field. In order to learn more about how the mixing is influenced by the PVC, phase averaging was used. To this end, the time series used for the spectral analysis were filtered with a zero phase-lag band pass filter around the higher of the two dominant frequencies. The resulting filtered time series was then used as a trigger signal for phase averaging. Figure 8 shows a comparison of the phase average and time average of mixture fraction in several cross sections. As can be seen, there are great differences between the results obtained by the two averaging procedures. The time averaging results in sightly tilted and squeezed, but otherwise symmetric contours, which are in accordance with the experimental findings. This is probably due to the fact that the combustion chamber is not rotationally symmetric because of the quartz windows necessary for optical access. In contrast, the phase averaging reveals a highly asymmetric distribution of the mixture fraction contours. This is due to the vortices forming the PVC, which by entraining air impose their double helical structure on the spreading fuel jet. Since the sense of rotation of the PVC vortices is such that it counteracts the direction of the fuel jet, the PVC imposes very high shear on the mixing layer of air and fuel. Thus, using the model for the filtered scalar dissipation rate ~ by Cook et al. (1997) = 2(D +

Dt) Of Of Oxi Oxi

(5)

one obtains values exceeding 2000 s -1 in the shear layer around the nozzle exit. From a flamelet concept point of view, this would surely lead to a quenching of the flame near the nozzle. In fact, the reacting case experiments show that the flame is lifted. It seems therefore, that the PVC has a significant influence on the mixing process and potentially also on combustion.

CONCLUSIONS Large eddy simulation was used to accurately predict flow and mixing in a model GT combustor.

739

Figure 8: Top: Time averaged contours of mixture fraction; Bottom: Phase averaged contours of mixture fraction. Left to right: Cuts in planes z - 0 , y = 0 and x = 10 rnrn. Comparisons between the simulation results and experiments show the high quality of the LES data. A double-cell precessing vortex core was found in the flow by means of the A2-criterion and spectral analysis. This PVC directly influences the process of mixing between fuel and air resulting in zones of high scalar dissipation rate. Thus, an influence of the PVC on combustion must be expected under reacting conditions. This implies, that the ability to capture PVC and related phenomena is crucial for any tool employed for the simulation of problems similar to the one applied here.

ACKNOWLEDGEMENTS This work was funded by the European Union through contract no. (MOLECULES) and by the Deutsche Forschungsgemeinschaft for financial support through SFB 568 "Flow and Combustion in Future Gas Turbine Combustion Chambers", project A4.

REFERENCES Anacleto, P., Fernandes, E., Heitor, M., Shtork, S., 2001. Characteristics of precessing vortex core in the lpp combustor model. In: 2nd Int. Symp. on Turbulence and Shear Flow Phenomena. pp. 133-138. Bowen, P., O'Doherty, T., Lucca-Negro, O., 1998. Rotating instabilities in swirling and cyclonic flows, part b: Theoretical analysis. In: Zhang, D., Nathan, G. (Eds.), Thermal Energy Engineering and the Environment. The University of Adelaide, Department of Chemical Engineering, Adelaide, pp. 197-220. Chanaud, R., 1965. Observations of oscillatory motion in certain swirling flows. J. Fluid Mech. 21 (1), 111-127. Cook, A., Riley, J., Koss G., 1997. A Laminar Flamelet Approach to Subgrid-Scale Chemistry in TurbulentFlows. Combustion and Flame 109, 332-341.

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Derksen, J., den Akker, H. V., 2000. Simulation of vortex core precession in a reverse-flow cyclone. AIChE Journal 46 (7), 1317-1331. Durst, F., Sch/ifer, M., 1996. A parallel blockstructured multigrid method for the prediction of incompressible flow. Int. J. of Numerical Methods in Fluids 22, 549-565. Guo, B., Langrish, T., Fletcher, D., 2001. Simulation of turbulent swirl flow in an axisymmetric sudden expansion. AIAA Journal 39 (1), 96-102. Iaccarino, G., Ooi, A., Durbin, P., Behnia, M., 2003. Reynolds averaged simulation of unsteady separated flow. Int. J. of Heat and Fluid Flow 24 (2), 147-156. Janicka, J., Sadiki, A., 2004. Large eddy simulation for turbulent combustion, proceedings of the 30th Symposium {International) on Combustion. Janus, B., Dreizler, A., Janicka, J., 2004a. Experimental study on stabilization of lifted swirl flames in a model gt combustor. Flow, Turbulence and Combustion. Janus, B., Dreizler, A., Janicka, J., 2004b. Flow field and structure of swirl stabilized non-premixed natural gas flames at elevated pressure. In: ASME Turbo Expo. No. GT2004-53340. Vienna. Jeong, J., Hussain, F., 1995. On the identification of a vortex. J. Fluid Mech. 285, 69-94. Lehnhguser, T., Sch/ifer, M., 2002. Improved linear interpolation practice for finite-volume schemes on complex grids. Int. J. Numerical Methods in Fluids 38, 625-645. Lesieur, M., M~tais, O., 1996. New trends in large eddy simulation of turbulence. Annu. Rev. Fluid Mech. 28, 45-82. Li, G., Gutmark, E., 2003. Geometry effects on the flow field and the spectral characteristics of a triple annular swirler. In: ASME Turbo Expo. No. GT2003-38799. Atlanta. Lilly, D., 1992. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids 4 (3), 633-635. Lucca-Negro, O., O'Doherty, T., 2001. Vortex breakdown: a review. Prog. Energy Combust. Sci. 27 (4), 431-481. Menon, S., 2004. CO emission and combustion dynamics near lean-blowout in gas turbine engines. In: ASME Turbo Expo. No. GT2004-53290. Vienna. Midgley, K., Spencer, A., McGuirk, J., 2004. Unsteady flow structures in radial swirler fed fuel injectors. In: ASME Turbo Expo. No. GT2004-53608. Vienna. Pierce, C., Moin, P., 1998. A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar. Phys. Fluids 10 (12), 3041-3044. Schlfiter, J., SchSnfeld, T., Poinsot, T., Krebs, W., Hoffmann, S., 2001. Characterization of confined swirl flows using Large Eddy Simulations. In: ASME Turbo Expo. No. ASME 2001-GT-0060. New Orleans, USA. Syred, N., Fick, W., O'Doherty, T., Griffiths, A., 1997. The effect of the precessing vortex core on combustion in a swirl burner. Combust. Sci. Tech. 125, 139-157. Waterson, N., 1994. Development of bounded higher-order convection scheme for general industrial applications. Project Report 1994-33, Von Karman Institute. Wegner, B., Kempf, A., Schneider, C., Sadiki, A., Dreizler, A., Sch/ifer, M., Janicka, J., 2004a. Large eddy simulation of combustion processes under gas turbine conditions. Prog. Comput. Fluid Dynamics 4 (3-5), 257-263. Wegner, B., Maltsev, A., Schneider, C., Sadiki, A., Dreizler, A., Janicka, J., 2004b. Assessment of unsteady RANS in predicting swirl flow instability based on LES and experiments. Int. J. of Heat and Fluid Flow 25 (3), 28-36.

EngineeringTurbulenceModellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

741

A N E V A L U A T I O N OF T U R B U L E N C E M O D E L S F O R T H E I S O T H E R M A L F L O W IN A GAS TURBINE COMBUSTION SYSTEM K. R. Menzies Rolls-Royce plc, P. O. Box 3, Filton, Bristol, BS34 7QE, UK

ABSTRACT Eddy viscosity models remain the most common form of turbulence modelling for complex industrial calculations but for complex flows it is not clear which of the many models available in the literature is most suitable. In this paper we compare the performance of five k-~ model variants (the standard high Reynolds number version, three alternative linear variants and a nonlinear form) for the isothermal flow in a model of a gas turbine combustor. The calculations are compared against detailed experimental data for the velocity and turbulence fields. It is concluded that for this flow there is little improvement to the quality of results in employing alternative k-c formulations, so improved predictions will require more sophisticated turbulence closures.

KEYWORDS Eddy viscosity models; gas turbine; combustor; validation; calculation

1

INTRODUCTION

The flow in the combustion system of a gas turbine includes many complex and three-dimensional features, including swirl, jet-crossflow interaction and jet-jet interaction along with fuel-air mixing and the effects of density variations through reaction. As the pressure to develop ever more efficient combustion systems has grown, the use of CFD in the design process has also increased to the point where it is now a routine tool in the design of new products. Most CFD calculations for combustion systems employ Reynolds Averaged Navier Stokes methods coupled with closures of the k-e type (see for example Anand et al (1999) or Menzies (2001)). The standard k-e model has known shortcomings for flows with complex strains and a number of extensions have been proposed in the literature. In this paper we compare the performance of the standard high Reynolds number k-~ model of Jones & Launder (1972) with three alternative linear models and one nonlinear model in calculating

742

the isothermal flow in a model gas turbine combustion chamber. The alternative models evaluated are: the stagnation point modification of Durbin (1996); the Renormalisation Group (RNG) k-e model of Yakhot et al (1992); and the realisable model of Shih et al (1995a). The nonlinear model tested was the quadratic form of Shih et al (1995b). These modifications have all been proposed to overcome deficiencies in the standard k-e model and have shown some promise in certain flows; the objective of this work is to evaluate the predictive capability of each model in the complex flow of a combustion chamber, determining whether the extensions to the k-c model deliver improved results in this flow.

2

GOVERNING

FLOW EQUATIONS

The basic equations governing the flow in a gas turbine combustor which are to be solved are assumed to be the Reynolds Averaged Navier Stokes equations. Employing Favre (density weighted) averaging we may express the continuity and momentum equations in Cartesian tensor notation as:

o~

0

0-7 +

= 0

OXy In equations 1 and 2 the tilde denotes Favre averages and the overbar denotes conventional Reynolds averages; ~;j denotes the mean velocity in the xy direction, while u~ is the (Favre) fluctuation from gj. The unknown Reynolds stresses pu~u~ in Eqn. 2 are closed using the k-e model variants.

3

3.1

TURRBULENCEMODELS

S t a n d a r d k-c M o d e l

In the standard high Reynolds number k-e model of Jones & Launder (1972) we employ the Boussinesq hypothesis to relate the Reynolds stresses to the mean rate of strain Sij via a turbulent viscosity #r:

2

0s

--pu i uj = 2#TSij -- -~--fikSij = #T

~

Os -~ OXi

2 Os 5ij 3 Oxk

- -~-fikS~j

(3)

We then solve modelled transport equations for the turbulence kinetic energy k and its dissipation rate e: 0-t(~k)+~(~gik)-~

0--t (~) -~- ~X/ (P?~i5) -- ~X/ = G

kT

p+~

~

ff -Jr-~

~X/

=V-~e

--" Celt - Cc2-~

(4)

(5) (6)

The turbulence kinetic energy production term 7' appearing in equations 4 and 5 is defined as = -ff"~ ~J Ozj

(7)

743

and the turbulence timescale T is defined as k T = -

(S)

s

The model constants are given the values C, = 0.09, ak = 1.0, ae = 1.3,

3.2

Gel : 1.44 and Ce2 = 1.92.

Durbin Timescale Modification

The modified k-c model proposed by Durbin (1996) employs a revised time scale to avoid unphysical growth of turbulence energy at stagnation points; this should be important in the flow considered here because of the presence of impinging jets in the combustor. This variant of the k-c model employs equations 3-7 with a modified definition of the turbulence timescale: T=min(

where S

3.3

=

k- 2 4 ~ ~ ) c'3-~

(9)

~/2SijSij. The model constants are the same as for the standard model.

R N G k-c Model

The RNG k-e model of Yakhot et al (1992) employs equations 3-8 but with modified values of the constants, using C, = 0.085, ak = 0.7179, ae = 0.7179 and Ce2 = 1.68. These values are all derived from the RNG scale elimination process. The coefficient Gel is no longer a constant but becomes a function of the normalised strain rate r / = Sk/c; this step does not come from RNG theory but is an additional modification to sensitise the ~ equation to mean strains. Cel = 1.42 - ~7(1 - T//4.38) 1 + 0.01577a

(10)

The RNG k-c model has been shown to deliver improved predictions compared to the standard model in a number of flows, especially through the generation of lower levels of turbulence kinetic energy; the aim here is to examine whether the model still performs well in the complex strain field of a combustor.

3.4

Realisable Model

A realisable k-c model was formulated by Shih et al (1995a) to guarantee that the calculated stresses implied by the Boussinesq relationship Eqn. 3 satisfy Schwartz' inequality. This model employs equations 3, 4, 7 and 8 with a reformulated e equation

O---t(-~e) + ~

(pg~e) - ~

# + --ae ~

= Ce~-~Se- Ce2-~k + v/--~/_~

(11)

The model constants in the new c equation are given by G e l - - max (0.43, r//(5 + r/)), Ce2 = 1.9 and ae = 1.2. The turbulent viscosity is again given by Eqn. 6 with a modified definition of C, to ensure realisability: 1 C, = Ao + AsU*T (12) In Eqn. 12 we define

Ao = 4.0;

U* -- ~SijSij -~- ~ij~ij;

cos

/

w

=

v~S~jS~k&~

S3

744

with fhj the vorticity tensor: ghj = ~1 (Ofti/Oxj Oftj/Oxi). While satisfying realisability is theoretically desirable it is not clear that this will improve predictions in such a complex flow.

3.5

N o n l i n e a r k-e M o d e l

The nonlinear version of the k-e model considered here is the quadratic model proposed by Shih et al (1995b). In this model the Boussinesq relationship Eqn. 3 is extended to include higher powers of the stress and vorticity tensor invariants. The specific form adopted by Shih et al (1995b) is 7'I

pu:'u~ = --2pTSiy + Z-PkSiJ6

+ 2C2-P-~2 (f~ikSkj -- Sikakj)

(13)

The standard k-c model equations 4-8 are employed (using Eqn. 13 in the turbulence energy production term) but with the definition of C u from Eqn. 12; the constant A0 is assigned the value 6.5 here. The standard k-e model constants from section 3.1 are used, while the additional model parameter C2 is defined as /

C2 = l/1

9C~$2T2 /2

1.0 + 3 S f ~ T 2

where ~ the stress linear k-e attractive

=

4

(14)

12~-~ij~ij. By extending the linear constitutive relationship to include higher powers of and vorticity invariants the model should predict a wider range of phenomena than the model. If this results in improved predictions for complex combustor flows the model is as an alternative to second-moment closure.

THE MODEL COMBUSTOR

To compare the performance of turbulence models for a flow as complex as that in a gas turbine combustion chamber it is necessary to have detailed experimental data in a realistic geometry which exhibits the important features of combustor flows. The geometry chosen is shown in Figure 1, which is a plexiglass model of a cannular combustion chamber run in a water flow facility. This model combustor, and its metal counterpart (used for reacting flow studies) have been used for previous combustor CFD validation of models of various levels of complexity, such as Alizadeh et al (1996) and di Mare et al (2004). The combustor comprises a hemispherical head with a 45 ~ single stream swirler, a circular crosssection barrel with two rows of ports, and a circular to rectangular cross-section discharge nozzle. There are six primary holes of diameter 10mm located 50mm downstream of the swirler exit, equally spaced around the combustor, with six dilution holes of 20mm diameter located 130mm downstream of the swirler exit. The dilution holes are staggered circumferentially from the primary holes, giving primary holes at 0 = 30 ~ 90~ and dilution holes at 0 = 0 ~ 60~ where 0 = 0 ~ is vertical. The swirler, primary and dilution holes are individually fed as is evident from Figure 1, allowing different mass flow splits to be used. The combustor model was run in a water flow facility, allowing detailed LDA measurements to be made as reported in da Palma (1988). Mean and fluctuating velocity components were measured at a number of stations in the combustor allowing axial and radial profiles to be built up. Although the use of a water rig does not allow the effects of density variation to be investigated it does permit detailed comparisons of predictions with experimental data for a relevant flow with complex strains and three dimensional features as found in production combustors. It also allows assessment of the turbulence model without the complication of potential inaccuracies through a combustion

745

DILUTION 6 bores ~'1o

PRIHARY z

l l x l

x

ix

i!

li'i

I

i

jl~i

i

i

!

11

6~

~

-

rxl

view I

view 2

.

50

z 2o

j : . /

__

60

..,

..,'-

/

/

8o

/

.~,- . ~

I.,-

-,- !

3

Figure 1: Model Combustor Geometry and Rig (all dimensions in mm) model; validation against reacting flow measurements for a combustor followed as a second phase of work.

5

COMPUTATIONAL

METHOD

The turbulence models described in Section 3 have been implemented in a three dimensional finite volume CFD solver. This code employs a structured three dimensional nonorthogonal computational grid with a colocated velocity storage arrangement. The interpolation method of Rhie & Chow (1983) is used to prevent velocity and pressure decoupling. The code is based on a low Mach number approximation and employs a pressure correction algorithm to couple the momentum and pressure equations. The TVD (Total Variation Diminishing) scheme of van Leer (1974) is employed for convection discretisation for all variables, implemented in the deferred correction form of Zhu (1991). This ensures numerically accurate and bounded results for all quantities. All diffusive fluxes are approximated using second order central differencing. In the case of the nonlinear model of Shih et al (1995b) some additional numerical practices were required in the implementation of the model. The linear contribution in the generalised Boussinesq relationship, Eqn. 13, is treated implicitly as for the linear k-c models. The nonlinear contribution is treated explicitly in the momentum equations and incorporated into the source terms. The nonlinear fragments of Eqn. 13 are also under-relaxed at each iteration to enhance stability of the overall calculation.

6

CASE CONSIDERED

A number of different mass flow distributions were tested in da Palma (1988); the chosen configuration corresponds to the massflow split shown in Table 6. In this configuration there is no flow through the fuel injector. For the calculations reported here the boundary conditions were

746

Feature

Swirler

Primary Ports

Dilution Ports

Massflow (kg/s)

0.543

1.040

1.6000

k (m2/s 2)

0.015

0.124

0.0112

(m~/~ ~)

30.600

4.370

0.0596

BOUNDARY CONDITIONS FOR CHOSEN CONFIGURATION

Figure 2: Velocity Vectors on Plane through Dilution Ports determined from the measurements and prescriptions detailed in da Palma (1988), as shown in Table 6. Converged solutions were obtained for all of the turbulence model variants, although the nonlinear model did display more problems in convergence than the linear models even with the treatments described in Section 5. A sensitivity study showed that the results were generally insensitive to plausible perturbations in the boundary conditions.

7

RESULTS

A general view of the flowfield is shown in Figures 2 and 3 where we plot flow vectors in the vertical plane through the dilution ports and the horizontal plane through the primary ports respectively. We can see how the primary jets impinge on the centreline of the combustor, creating a significant flow upstream towards the head. This flow combines with the swirler flow to create the recirculations in the primary zone. We can also see how the dilution jets are deflected downstream by the crossflow from the primary zone. In Figure 4 we plot the axial velocity along the combustor centreline for each of the models along with the experimental data from da Palma (1988); the velocity values have been normalised by the bulk velocity through the combustor, Ub - 0.74m/s. Axial velocity on a number of radial planes along the axis of the combustor are in Figure 5, while radial profiles of the turbulence energy, normalised as v/-k/Ub, are in Figure 6. We may see that while each of the models reproduces the general trend of the measurements, none correctly captures the depth of the velocity trough on the centreline within the primary zone. Of the models tested, the Durbin variant predicts the lowest (most negative) axial velocity in the

747

Figure 3: Velocity Vectors on Plane through Primary Ports 3 _ 2.5 2

'

I

'

I

'

I

'

'

I

'

I

'

I

'

j

q

--

- "7..-. 7..-.: .-..-...

,:.."

_

1.5

"

-/;f:/

_

0 ,.'-

-0.5

-

"':_~--:.:...: . . . . . .

[] !/,'

1 -

0.5

I

........

~ - : "

........

.i

-_ -

_ -

-1 u"

-1.5

"'\'.,.,,,,.

',

~'/

-2 -2.5 -3 --3.5 -4

Off]]

-

0

,

.

I

0.5

,

00 I

1

,

I

1.5

,

I

2 z/Rc

[ ..... I .... [ [] , . . 2.5

Realisable [ Nonlinear ] Experiment i . . I 3 3.5

--~ _._7

,I

4

Figure 4: Normalised Axial Velocity on Combustor Centreline primary zone region (consistent with a lower turbulent viscosity in the impingement region), but is still significantly at variance to the experimental value. In the absence of impingement this model performs as the standard model as expected. The RNG model does, as suggested above, reduce the levels of turbulence energy compared to the other models in the primary zone and just downstream of the primary jets but this leads to an impingement which is too strong and which results in the minimum axial velocity moving off of the primary zone centreline. The standard, Durbin and nonlinear models perform somewhat better for the turbulence field in the primary zone. The realisable model can be seen to perform poorly in this flow, suggesting that the modified e equation (11) results in too little dissipation of turbulence energy. While satisfying realisability

748

1 'l'i'l'l'i'~'!.'.~'i"i'...

'1'1'1'1'1'1'1'1'1'1'1 ' z = 8mm

9

_

0.5

,..~,

_

[]

-

-

[]

0.5

z = 40ram .,...,r

~

-9 -.. i ""

0 ~.:. _~

!:i. i

_

--

;..,e~

-

...- ~

i

-0.5

-1

,I,lli,i,iii,l,l,l,l,I, -3-2.5-2-1.5-1-0.5

0 0.5

I 'I'I'I'I'I'I'I'~'I'I' I' 0.5

0 0.5

1 1.5 2 2.5 3

1 'I'I'I'I'I'L~I'I'I'I'I' z = 60mm 0.5

~ ~-""~"

I I ;'Y%,I ~1, I, -3-2.5-2-1.5-1-0.5

1 1.5 2 2.5 3

-

~t:l-"%

~

~''.

o

-0.5

~

-1 , i , l , I

,I,1,1

-3-2.5-2-1.5-1-0.5

0 0.5

1 1.5 2 2.5 3

'l'l'i'l'l'q

'1

-0.5

-I , I j I, I, I, I ,~h~l, I, I, I, I, -3-2.5-2-1.5-1-0.5

""-.

--

1 1.5 2 2.5 3

1 'l'l'l'l'l'~l'l'

z = 70mm 0.5

0 0.5

z=

85mm

0.5

9

~/

-0.5

-0.5

,],l,],],i,

-1

,i

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-1 , I , I , I , I , I , I N , I , I , I , I -3-2.5-2-1.5-1-0.5

0 0.5

1 1.5 2 2.5 3

U/Ub

1 '1'1'1'1'1'1'~1'1'1' -

z = 130mm

Standard

0.5

i ""..'. i I

o

i~

Durbin

....

RNG

.....

Realisable

.....

Nonlinear

n

-0.5 1

.......

,I,I,I,I,I,I,

-3-2.5-2-1.5-1-0.5

0 0.5

Experiment

I,I,I,

1 1.5 2 2.5 3

U/Ub

Figure 5: Normalised Axial Velocity Within Combustor is conceptually important this more rigorous approach may simply be highlighting deficiencies in the linear eddy viscosity approach for complex flows. The nonlinear model can be seen to perform generally no better than the best linear models suggesting that the quadratic stress-strain relationship is not sufficiently general to deliver benefits in these complex strain fields. Overall therefore all of these versions of the k-e model have reproduced some of the features of the combustor flowfield. However from the comparison with the measurements the realisable model has performed poorly and cannot be recommended for use in calculations of this type. The poor

749

-

'-,..' ~',.,

i

" t ' i , z = 40mm -

~-

0.5

k...~

I

I

'

I

'

I

z-5Omm -I

0.:5

\~..;.

,'.. ? o , ,'*

~:l 0

9 I,~o

_

,' _.o",/7

-0.5

_

0

0.5

1

~~

1.5

2

z

60mm

-~,.

0.5

2.5

0.5

1

~,,

Io ' .on

o.5 ",. ~di !

I'

-0.5

/ " .."''" /,." 0

1

"~],t~

'

I

~5-\.

'

I

--

2

'

i

2.5

-0.5 _~ ~ 2 2 . .

0

.~/; Imt 0.5

i

,'"

__

I I I, 1

1.5

,&mb

I

2

I

2.5

i"

85mm

-

....... .... .... ..... ca

0.5 ca.~. "\ 8 ~JiJ -0.5

--

/"

~4

Z=

2.5

-

~./"

1.5

2

, , , ~~.,~.,,.,,.i ." \

ca

0.5

I

~

I' I ' z = 70mm _

]9. " ,~..',,"/.

f ...'""

q

~o

1.5

isi ", ~-1~. ",~ "~.

~ " "'-.,

o

?

,,-;" d'or

i /" i.~..

-0.5

'

Standard Durbin RNG Realisable Nonlinear Experiment

g// ca/"

o

,~,ca 0.5

I , I , 1 1.5

4kmb

t ,] 2 2.5

Figure 6' Normalised Turbulence Kinetic Energy W i t h i n C o m b u s t o r p e r f o r m a n c e in the p r i m a r y zone would significantly affect the prediction of fuel and air mixing in the reacting case. T h e R N G model has, as expected, reduced levels of t u r b u l e n c e energy, b u t to a value below the m e a s u r e m e n t s , which has not resulted in any i m p r o v e m e n t to the velocity field predictions. T h e s t a n d a r d , D u r b i n and nonlinear models have generally performed to an a c c e p t a b l e level in this flow.

8

CONCLUSIONS

A number of variants of the k-c turbulence model have been compared against experimental data for the isothermal flow in a model combustion chamber. Of the models tested the realisable model

750

performed poorly, while the RNG model did not deliver any benefits to the velocity predictions. The standard and Durbin models delivered acceptable results in comparison with the measurements. The nonlinear model also produced acceptable results but with additional convergence problems. Given the generally small differences between this model employing a quadratic stressstrain relationship and the standard model it is questionable whether the additional complexity is justified. Use of nonlinear models with cubic stress-strain relationships is also possible which offers theoretical advantages for swirling flows and is under investigation as a further phase of this work. From the results here though we conclude that for modelling combustion chamber flows at the eddy viscosity level of closure then the standard or Durbin models appear to be the most appropriate.

9

ACKNOWLEDGEMENTS

The author would like to thank Rolls-Royce plc for permission to publish this paper. REFERENCES

Alizadeh S., Askari A., Benodekar R., Ghobadian A. and Sanatian R. (1996). CFD Simulation of Combustion in a Model Gas Turbine Combustor. Third International Conference on Computers in Reciprocating Engines and Gas Turbines, 79-87, Mechanical Engineering Publications, Institution of Mechanical Engineers Anand M.S., Zhu J., Connor C. and Razdan M. K. (1999). Combustor Flow Analysis Using an Advanced Finite Volume Design System, Proceedings of ASME TURBO EXPO 99 (44th ASME Gas Turbine and Aeroengine Technical Congress), Indianapolis, USA. Durbin P. A. (1996). On the k-e Stagnation Point Anomaly. International Journal for Heat and Fluid Flow 17, 89-90. Jones W. P. and Launder B. E. (1972). The Prediction of Laminarisation with a Two Equation Model of Turbulence. International Journal of Heat and Mass Transfer 15, 301-314. van Leer B. (1974). Towards the Ultimate Conservative Differencing Scheme II: Monotonicity and Conservation Combined in a Second Order Scheme. Journal of Computational Physics 14, 361 di Mare F., Jones W. P. and Menzies K. R. (2004). Large Eddy Simulation of a Model Gas Turbine Combustor. Combustion and Flame 137, 278-294. Menzies K. R. (2001). Computational Fluid Dynamics for Gas Turbine Combustion Systems Where Are We Now and Where Are We Going? in Computational Fluid Dynamics in Practice (N. Rhodes, ed.) 99-111, Professional Engineering Publishing, UK. da Palma J. M. L. M. (1988). Mixing in Non-Reacting Gas Turbine Combustor Flows Ph.D Thesis, University of London, UK. Rhie C. M. and Chow W. L. (1983). Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation. AIAA Journal 31, 1525-1533. Shih T.-H., Liou W. W., Shabbir A., Yang Z. and Zhu J. (1995a). A New k-c Eddy Viscosity Model for High Reynolds Number Turbulent Flows. Computers and Fluids 24:3, 227-238. Shih T.-H., Zhu J. and Lumley J. L. (1995b). A New Reynolds Stress Algebraic Equation Model. Computer Methods in Applied Mechanics and Engineering 125, 287-302. Yakhot V., Orszag S. A., Thangam S., Gatski T. B. and Speziale C. G. (1992). Development of Turbulence Models for Shear Flows by a Double Expansion Technique. Physics of Fluids A 4:7, 1510-1520 Zhu J. (1991). A Low Diffusion and Oscillation Free Convection Scheme. Communications in Applied Numerical Methods 7, 225-232

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

751

LARGE EDDY SIMULATIONS OF HEAT AND MASS TRANSFERS IN CASE OF NON ISOTHERMAL BLOWING G. Brillant 1,2, S. Husson 2 and E Bataille 2 1 CEA/Grenoble DEN/DER/SSTH/LMDL, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France 2 INSA - Centre de Thermique de Lyon (UMR CNRS 5008), Bat. Sadi Carnot, 69621 Villeurbanne Cedex, France Corresponding author: E Bataille, INSA - Centre de Thermique de Lyon, Bat. Sadi Carnot, 69621 Villeurbanne Cedex, France, Ph: (33) 4 72 43 64 27, Fax: (33) 4 72 43 88 19, Email: Francoise.Daumas @Insa-Lyon.fr

ABSTRACT The impact of non isothermal blowing on a turbulent boundary layer is studied using Thermal Large Eddy Simulations with the Trio U code. We are interested in the application of blowing to the thermal protection of a plane wall, above which a hot fluid flows. Therefore, the present study focuses on the thermal aspect. The configuration and the numerical approach are presented. Our results are compared with experimental data. The numerical and experimental profiles show a good agreement. We observe that blowing dramatically decreases the mean temperature, permitting to protect the wall from the main hot flow. Blowing is shown to increase the velocity and temperature fluctuations. At the same time, their maxima are shifted away from the plate. The level of the temperature fluctuations is thus shown to remain low at the wall. Blowing modifies the velocity-velocity and velocity-temperature correlations as well. Finally, the study of the Q criterion shows that blowing increases the eddy size and the mixing process, which is directly related to the growth of the heat transfers between the main hot fluid and the cold injected fluid.

KEYWORDS blowing, heat and mass transfers, Thermal Large Eddy Simulations, boundary layers

752

1

INTRODUCTION

The protection of walls submitted to important heat fluxes is of high industrial interest. For example, the engines performance is determined by their efficiency and specific power, which are directly linked to the increase of the maximum temperature. Nevertheless, the walls of the engines are not resistant enough to high temperatures and have to be cooled. Different techniques permit a thermal protection, such as discrete injection, impingement, ablation and blowing. Our study is here focused on blowing, which consists in a cold fluid injection through a porous element. Even if blowing is very efficient to reduce the mean temperature, it has also to be studied in terms of turbulent fluctuations to prevent the degradation of the materials, as high temperature fluctuations create craks and thus weaken the walls. Studies have been conducted on blowing both experimentally (Kays (1972); Michard et al. (2003); Moffat and Kays (1968, 1984)) and numerically (Eckert and Cho (1994); Landis and Mills (1972)). In particular, previous RANS (Reynolds Averaged Navier-Stokes) simulations permitted to obtain the mean values in the blowing configuration (Bellettre et al. (1999, 2000)). However, a finer method like LES (Large Eddy Simulation) is necessary to have a better knowledge of the turbulent values. LES is based on the filtering of the motion and energy equations so as to calculate the effect of large scales and model the effect of small ones. This tool has become more and more commonly developped and used since the 1990's (Germano et al. (1991); Lesieur (1997); Piomelli and Balaras (2002); Sagaut (2001)). Nevertheless, few of the studies involving LES consider non isothermal configurations. Here, we chose to develop and to use Large Eddy Simulation in the anisothermal case of blowing, in order to determine the impact of blowing on the boundary layers and to study the temperature turbulent fluctuations. This method is here referred to as Thermal Large Eddy Simulation. In a first part, we describe the physical and numerical configurations of our study, including the inlet conditions. Then, we present the results of our numerical study of non isothermal blowing.

2 2.1

CONFIGURATION AND NUMERICAL SETUP

Studied configuration

We are interested in the cooling of a porous plate in contact with a hot flow (Figure 1). The cooling is achieved by using non isothermal blowing through the porous element, which has a porosity of 30 %. The main turbulent flow is hot air (T -- 313 K), whereas the injected fluid is cold air (AT = T~ -T~nj 20 K). The main velocity is equal to 10 m.s-1, leading to a Reynolds number based on the mean velocity and the impermeable plate length of 750,000. The injection rate (or blowing rate) F, defined as the ratio of the injected fluid mass flow to the main mass flow, is given by" F = Pi~JU~j p~U~

(1)

where U is the velocity and p is the density. The subscript ~ refers to the main flow far from the wall and the subscript i,~j is related to the injected fluid. F varies from 0 to 2 %. Experimental measurements in such a configuration have been conducted and permit to validate some of our numerical results (Bellettre et al. (2000); Brillant (2004); Michard et al. (2003)).

753

Figure 1" Studied configuration.

2.2

Thermal Large Eddy Simulations

The filtered equations of the flow, using the Boussinesq approximation, are given by Eqn.(2) to (4), where the s u p e r s c r i p t - indicates the filtered variables.

O~j

(2)

Ogi Ogigj _ --~ + ' Oxy -

1 O-fi O2gi p Oxi + u n-f-27~xjO +

OT OTN_ O2T O -~ ~ Ox~ - ~-r + ~

13(T-

2 0__0_ T,-~f)9i + Oxj (usg~Sij)

e%~-~zJ

(3) (4)

ui and uj are the velocity components, x~ and xj are the coordinates, t is the time, T is the temperature, P is the pressure, 9 is the gravity, u is the kinematic viscosity, ec is the thermal diffusivity and/3 is the compressibility coefficient. To express the eddy viscosity usgs, we use the WALE model (Wall Adapting Local Eddy), developped by Nicoud et Ducros (1999) : d d 3/2

~'~ = ( c ~ ) 2

(s~j~j)

where Cw is a constant, -A- is the filter width and Siy d __ S i k S k j with 5ij the Kronecker symbol" 5# = 1 if i = j, 0 if i r j.

-F ~ i k ~ k j

(5)

-- -~ l (~ij ( ' ~ m n - ~ m n

~rnn~mn)

Sij is the filtered strain tensor, defined by" --

1. O ~

O-gj

(6)

Ox~ )

(7)

S{j = 2(O~zj + -~z~)

and ~ j is the filtered rotation tensor and is equal to:

a~j = -~( Oxj

754

For the eddy diffusivity nsgs, various tests have been previously carried out in a turbulent plane channel, with a Reynold number based on the friction velocity and the half-height of the channel of Re~ ~ 180 (Brillant (2004)). Two meshes were tested, a very fine one (non-dimensional resolutions: A+ ,-~ 36, y+ ~ 1 at the wall, A+ ~ 15) and a coarser one (non-dimensional resolutions: A+ ~ 71, y+ "-' 5 at the wall, A+ ~ 30). Using the WALE model or a dynamic model for the eddy viscosity, we considered five models for the eddy diffusivity. Four of them were dynamic models with different stabilization procedures (6 nodes average, average in homogeneity directions, eulerian average, lagrangian average). The hypothesis of a constant subgrid scale Prandtl number (Prs9~ = 0.9) was used for the last model. Two thermal configurations were studied: one with imposed temperatures at both walls (T2/T1 = 1.01) and one with an adiabatic upper wall and an imposed heat flux (~ = - 1 W . m -2) at the lower wall. The comparison of the mean and fluctuating temperature profiles as well as the velocity-temperature correlations profiles showed no significant difference between the five models tested for the eddy diffusivity. Since the mesh resolutions and boundary conditions used in the blowing configuration are similar to those of this channel, we considered, in a first time, a constant subgrid scale Prandtl number (Pr~g~ - 0.9).

2.3

Boundary conditions and blowing model

Simulations are carried out with the Trio U code that is developed at the CEA (French Atomic Agency). In order to reduce the domain dimensions, and thus the calculation cost, we consider only the last 2 centimeters of non-permeable wall. Therefore, the first step of the simulations is the set up of appropriate inlet conditions. In our study, we consider the turbulent inlet developed by Brillant et al. (2004), who partition the simulation into two domains. The first one is a periodic turbulent channel flow used to provide velocity and temperature fluctuations. These fluctuations are imposed at the entry of the second domain, which corresponds to the studied configuration. In addition, the experimental mean profiles are imposed at the entry of the second domain. To ensure the similarity of the profiles in both domains, we impose the same Reynolds number based on friction velocity (Re~ ~ 380). The channel dimensions are 27rhx2hxTrh, with h ,-~ 0.0149 the half-height of the channel. Its non-dimensional resolutions are: A+ ,.., 40, y+ '~ 1 at the wall and A + "~ 40. To model the blowing through the porous plate, we use a holes model, which is built as a succession of walls and holes (Bellettre et al. (1999)). The ratio between the walls and the holes is set by the plate porosity (30%). Consequently, the number of wall elements is twice the number of hole elements. This model has the advantage to reproduce the whole physical phenomenon, since it takes into account both the friction along the wall and the heat and mass transfer due to the injection of cold fluid. The size of each element is equal to 2 mm, corresponding to the size of one cell. Previous simulations using RANS methods were carried out to study the influence of the element size by Bellettre et al. (1999) and Mathelin et al. (2001). They varied the element size from the actual value ( ~ 30#m) to 5 mm and observed that their results were not affected. They also investigated the influence of the number of cells per element. They showed that increasing this parameter do not lead to significant differences. Nevertheless, in the future it would be interesting to carry out these tests using LES. The domain dimensions are chosen carrefully. The length corresponds to the actual domain dimension: Lx = 0.37 m. The height is L v - 0.12 m, so that the upper boundary does not influence the boundary layer. The transverse dimension is chosen so as to avoid correlation between the variable of each boundary (Lz = 0.05 m). We here consider 186 x 50 x 31 nodes, which permits to have a finely resolved simulation (A+ ~ 50, A+ ~ 40) with a reasonnable calculation cost. To avoid the use of wall laws approximations, we construct an irregular mesh, which is refined near the wall, to have y+ _~ 1 at the

755 wall. A no-slip condition is applied to the velocity along the non-permeable wall and along the wall elements of the holes model. The holes are associated to the cold fluid inlet, with a velocity depending on the injection rate. The temperature of the porous plate is set to the experimental one for both walls and holes elements. Periodic conditions are imposed to the transverse boundaries. The upper and exit boundary conditions are convective outflows. Time integration is carried out by a third order Runge-Kutta scheme. The convection scheme for the velocity is a second order centered scheme. Even if a higher order could be considered, previous tests showed that the results obtained with the second order are accurate enough (Ackermann and M6tais (2001); Brillant et al. (2004)). For the temperature, we use a third order quick scheme, as recommended by Chatelain et al. (2004). More details about the numerical methods can be found in Bieder et al. (2000) and Brillant et al. (2004).

3

RESULTS

Figure 2 presents the mean velocity as a function of the distance to the wall for different injection rates. The results are compared with the experimental data. We can observe that the thickness of the dynamic boundary layer increases with the blowing and that there is a very good agreement between the numerical simulations and the experiments. Figure 3 shows the mean temperature in the boundary layer. The blowing is very efficient to reduce the mean temperature in the near-wall area. At the wall (y = 0), the temperature is about 90 % lower for an injection of only 2 %. We can again note that the experimental and numerical data match very well. Since the velocity and temperature gradients at the wall are modified in the boundary layers when blowing is applied, we give in Table 1, the values of these gradients for several injection rates. We also give the friction factor 9 C f -- 2 " r ~ / p ~ U ~ , with ~-w - p u O U / O y l ~ the wall shear stress. We note that the blowing decreases the velocity gradient and the friction factor, which is in good agreement with the literature results. The temperature gradient increases with the injection rate in a first time (F < 1%) and then decreases. For weak injection rates, the wall is cooled but the fluid above the wall is still hot, resulting in a high temperature gradient. For higher injection rates (2 %), a cold fluid film is present on the wall, leading to a smaller temperature difference between the wall and the fluid at its vicinity.

_,,_-_-~. . . . . . . .

~.~20,8

~t

[~s 0,6 i

0

O0

,

2

0,02

0,04

~

0,06 y (m)

0, I

Figure 2: Mean longitudinal velocity.

0,12

~

"2 .~"

0,2

0,08

il t. .,~

.r 0,4

00

r - ~ -- -- . . . . . .

/, ~'

._ t-,

~ =

F = 0.5 % (exp)]

F 1.0 % (exp) I ..... F 2.0 % (exp)] * F 0.5 % (sim)] F 1.0 % (sim)] F 2.0 % (sire) I

~."

~ 0,02

i 0,04

,

~ 0,06 y (m)

I 0,08

,

~ 0,1

Figure 3: Mean temperature.

,

0,12

756

TABLE 1: VELOCITY AND TEMPERATURE GRADIENTS AT THE WALL. injection rate velocity gradient (s- 1) friction factor CI temperature gradient (K.m- 1)

F=0,5%

F =0% 10,600 0.0032 25

F=1% 1,300 0.0004 1,100

4,200 0.0013 750

F=2% 300 0.0001 560

Figures 4 and 5 give the longitudinal and vertical velocity fluctuations for different injection rates. We observe that the maximum of these fluctuations is greatly increased and that there is an important shift away from the wall when F increases. The same behaviour is observed for the transverse velocity fluctuations (not shown). In Figure 6, we present the temperature fluctuations. We note that the amplitude of these fluctuations increases with F. Consequently, when blowing is used, the mean temperature decreases but the turbulent temperature fluctuations increase. However, when blowing is applied, the maximum of the temperature fluctuations is also shifted. The temperature fluctuations remain low at the wall. We note discrepancies between the numerical and experimental fluctuations profiles. However, the profiles have the same shape, the same behaviour with the blowing rate and a level of the same order of magnitude. The differences in the intensity of the fluctuations are attributed to both experimental errors (the measures using a crossed hot-wires probes are very delicate) and numerical parameters. In particular, Chatelain et al. (2004) showed that the quick scheme is the most appropriate for the energy equation but that it induces an attenuation of the fluctuations level. In another hand, numerical tests were carried out with finer meshes (372x50x31 nodes and 372xS0x62 nodes). The results were unchanged, showing that the mesh resolution is adequate.

1

,

7

5

f

~

1~

1,5

/

~ . 1'251

1 o

'

I

'~'l~

0,02

|

II J, II

~

0,04

0,06 y (m)

'

I

'

I

I

..... --..... ,,

., "9 "i ! i

90,8i- ]

",.

[

; ,..,'.,

04I''.::':..:i :,,,

0,1

0,12

Figure 4: Longitudinal velocity fluctuations.

00

I

'

9 F=2.0% (sim)]

-

0,2 0,08

'

F = 0.5 % (exp)l F = 1.0 % (exp)] F = 2.0 % (exp)[ F=0.5%(sim)[

9 F=l.O%(sim)I

06k "

'

0,25

"

""'"-.. 0,02

0,04

0,06 y (m)

0,08

0,1

0 12

Figure 5: Vertical velocity fluctuations.

Figures 7, 8 and 9 present the velocity-velocity and velocity-temperature correlations. We can observe that, for all these correlations, there is an increase and a shift of the maximum. We again remark some differences between simulations and experiments but the global trends are similar. Furthermore, the effect of the blowing rate is the same in both cases. Taking into account the experimental and numerical errors, the discrepancies are acceptable. In Figure 7, the velocity-velocity correlation change of sign for F = 2 % comes from experimental errors. Figure 10 displays the temperature field for a blowing rate of 2 %. It shows that the blowing has an impact on the whole boundary layer and on all the plate length. Consequently, the wall is well protected both at the beginning of the plate and for weak injection rates. This result illustrates the excellent efficiency of the blowing and shows that a short length of porous plate is sufficient to have a good thermal

757

3,5 /

'

3~ I" 2,51--

i

'

I

'

I

""'""

i" '. ! 9 " 9. . . . ~,~ 2b.r162 ",

~"

/'".g'..,, 9

0

_

"1

'.L\' ,

00

' I ' / F = 0.5 % (exp)l~ F = 1.0 % (exp)]"] F = 2.0 % (exp)[ F = 0 . 5 %(sim) l-I F = l . 0 % ( s l m 9) ] ~ F = 2 . 0 % (sim)]-~

:,

9 '~

,.

' , ] ]. . . . [. . . . . [ ,, [ 9 [ 9

"

5

~

"

0,02

0,04

"

0,06 y(m)

0,08

0,1

0,12

0,4 0,3 -.~ 0,2

{::

0 -0,1

'

I 0,02

Figure 6: Temperature fluctuations.

41

,

,

'

,

,"

i

k 31 [ ~a~ [

!., ! ", / ~. ; \

II;..'r

l[,lav!li,,~

9

1

"F~" 9 \ 9

9

x

x

t

0,02

i

i

9

l~

''

F _ 0.5 b/o (exp). ] - 1.0 % (exp)[ F = 2.0 % (exp)[ F=0.5%(sim)] F=l.0%(sim)] F=Z'0%(sim)l

,

0,8~ 0,6

I

.

a_

,

0,06 y (m)

0,08

0,1

,

0,12

'

I

'

I 0,06 y(m)

,

I 0,08

-0 2 ' 0

,

I

9

~,

I 0,1

,

I

'

0,12

0,02

'

'

F - 0.5 ~o (exp)] F = 1.0 % (exp)| F = 2.0 % (exp)[ F=0.5%(sim)] F = 1.0%(sim)] F=2.0%(sim)]_

;, ",,:i"~-~-o

~

I

==--.... ..... * 9

~" 0 4 i ~ ' ~ " " ~ \

0 .

x

,•

I 0,04

," i i i i i ~.,

i

0,2[:.27

9

0,04

,

Figure 7" - u ' v ' correlations.

'.

"-2

0

!

__ .... ..... * 9 "

t

I ~i

-0 20 '

I 0,04

,~ ' . - . -,. ~ - , - . ,

I , I 0,06 0,08 y (m)

_ -_ ,

I 0,1

,

~0,12

Figure 9: - v ' T ' correlations.

Figure 8" u'T' correlations.

protection. This last point is particularly interesting from an industrial point of view, since the insertion of a porous material weakens the wall. Finally, Figures 11 and 12 represent the impact of the blowing on the Q criterion for two injection rates. The Q criterion represents the eddy structures (Dubief and Delcayre (2000)) and is defined as:

Q=7

(8)

These figures permit to visualize how the blowing disturbs the eddy structures. When blowing is applied, the size of the eddies is increased. Furthermore, the area where the eddy structures develop is larger. This observation is in agreement with the previous results, which have shown that the blowing increases the boundary layers and the intensity of the turbulent structures of the flow. Therefore, blowing intensifies the mixing, and thus the heat transfers between the two fluids.

758

Figure 10: Mean temperature Tm ( F -- 2.0 %).

Figure 11" Q criterion in the near-wall region, threshold at 50,000 s -2 ( F = 0.5 %).

Figure 12" Q criterion in the near-wall region, threshold at 50,000 s -2 ( F = 2.0 %).

759

4

CONCLUSIONS

The impact of non isothermal blowing on a turbulent boundary layer has been numerically studied. We have focused on the influence of the injection rate on the velocity and temperature fields. We presented our results concerning the mean and turbulent fields. We have showed that the blowing thickens the boundary layers and reduces the mean temperature in the near-wall area. We have investigated the influence of the injection on the velocity fluctuations, on the temperature fluctuations, but also on the velocity-velocity and velocity-temperature correlations. The blowing is found to increase all these turbulent intensities and to shift their maxima away from the wall. Consequently, blowing cools the material without damaging it, since both the mean temperature and the temperature fluctuations are low at the wall.

References Ackermann C. and M6tais O. (2001). A modified selective structure function subgrid-scale model. Journal of Turbulence 2:011, 1-26. Bellettre J. Bataille E and Lallemand A. (1999). A new approach for the study of turbulent boundary layers with blowing. International Journal of Heat and Mass Transfer 42:15, 2905-2920. Bellettre J. Bataille E and Lallemand A. (1999). Prediction of thermal protection of walls by blowing with different fluids. International Journal of Thermal Sciences 38:6, 492-500. 1999. Bellettre J. Bataille E Rodet J.C. and Lallemand A. (2000). Thermal behaviour of porous plates subjected to air blowing. AIAA Journal of Thermophysics and Heat Transfers 14:4, 523-532. Bieder U. Calvin C. and Emonot E (2000). PRICELES: object oriented code for industrial large-eddy simulations. In 8th Annual Conference CFD Society, Canada. Brillant G. Bataille E and Ducros E (2004). Large-eddy simulation of a turbulent boundary layer with blowing. Theorical and Computational Fluid Dynamics (to appear). Brillant G. (2004). Simulations des grandes 6chelles thermiques et exp6riences dans le cadre d'effusion anisotherme. PhD thesis, INSA de Lyon. Chatelain A. Ducros E and M6tais O. (2004). LES of turbulent heat transfer: proper convection numerical schemes for temperature transport. International Journal for Numerical Methods in Fluids 44:9, 1017-1044. Dubief Y. and Delacayre E (2000). On coherent-vortex identification in turbulence. Journal of Turbulence 1, 1-21. Eckert E. R. G. and Cho H. H. (1994). Transition from transpiration to film cooling. International Journal of Heat and Mass Transfer 37:1, 106-113. Germano, M., Piomelli, U., Moin, E, et Cabot, W. H., A dynamic subgrid-scale eddy viscosity model, Physics of Fluids A, 3(7), pp. 1760-1765, 1991. Kays W. M. (1972). Heat transfer to the transpirated turbulent boundary layer. International Journal of Heat and Mass Transfer 15, 1023-1044. Landis R. B. and Mills E (1972). The calculation of turbulent boundary layers with foreign gas injection. International Journal of Heat and Mass Transfer 15, 1905-1932. Lesieur M. (1997), Tubulence in Fluids, Kluwer Academic Publisher. Mathelin L. Bataille E and Lallemand A. (2001). Blowing models for cooling surfaces. International Journal of Thermal Sciences 40:11, 969-980.

760 Michard M. Ait Ameur M. Brillant G. and Bataille E (2003). Experimental study of mixing between a heated turbulent flow and a cold airstream transpirated through a porous wall. In 4th ASME JSME Joint Fluids Engineering Conference, Honolulu. Moffat R. J. and Kays W. M. (1968). The turbulent boundary layer on a porous plate: experimental heat transfer with uniform blowing and suction. International Journal of Heat and Mass Transfer 11, 1547-1566. Moffat R. J. and Kays W. M. (1984). A review of turbulent-boundary-layer heat transfer research at Stanford, 1958-1983. Advances in Heat Transfer 16, 241-365. Nicoud E and Ducros E (1999). Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion 62:3, 183-200. Piomelli U. and Balaras E. (2002). Wall-layer models for large-eddy simulations. Annual Reviews of Fluids Mechanics 34, 349-374. Sagaut E (2001), Large eddy simulation for incompressible flows: an introduction, Springer Verlag.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

761

TURBULENCE MODELLING AND MEASUREMENTS IN A ROTOR-STATOR SYSTEM WITH T H R O U G H F L O W S. Poncet, R. Schiestel and M.-P. Chauve I.R.P.H.E. UMR 6594 CNRS-Univ. Aix-Marseille I & II, Technop61e Chgtteau-Gombert, 49 rue F. Joliot-Curie, B.P. 146, 13384 Marseille, France

ABSTRACT The present work considers the turbulent flow inside a high-speed rotor-stator cavity with and without imposed throughflow. New extensive measurements made at IRPHE using a two component laser Doppler anemometry technique and pressure transducers are compared to numerical predictions based on one-point statistical modelling using a low Reynolds number second-order full stress transport closure (RSM). A study of the influence of the flow rate on the flow structure is performed and a better insight into the dynamics of rotor-stator flows is gained from this study. The advanced second-order model provides better predictions than the usual k-e model and proves to be the adequate level of closure to describe such complex flows.

KEYWORDS Turbulence modelling, rotor-stator system, second moment closure, LDA, Batchelor flow

INTRODUCTION The flow in a rotor-stator cavity is typical of many configurations encountered in turbomachinery. Experimental and theoretical studies performed by Daily & Nece (1960) showed that four different regimes can exist in closed cavities: two laminar and two turbulent regimes, each of them corresponding either to merged or separated boundary layers. When the boundary layers are separated, the flow is classically divided into three zones and belongs to the Batchelor type family. The boundary layer developed on the fixed disk is called the B6dewadt layer and the one on the rotor is called the Ekman layer. Between these layers, the core is characterised by a constant azimuthal velocity and an almost vanishing radial velocity. Many studies have been devoted to rotor-stator system in the literature but refined measurements of the turbulence field are however scarce. The experiment of Itoh et al. (1990) provides an important contribution to the understanding of the turbulent flow in a shrouded rotor-stator system with an aspect ratio of 0.08. They measured the mean flow and all the

762

Reynolds stress components and brought out the existence of a relaminarized region even at high rotation rates. Modelling the turbulent flow developing in this type of geometry is particularly delicate because the interacting phenomena at play are complex and the presence of walls introduces further complexities. One of the first numerical contributions is the study of Iacovides & Toumpanakis (1993), in which four turbulence models were tested. They showed especially that a Reynolds stress model was an appropriate level of closure to predict such complex flows. In the same way, Elena & Schiestel (1996) proposed a variant of Reynolds stress model (denoted RSM in the following), which takes into account the implicit effects of rotation on the turbulence field and compared it to more classical models. Only few results have been published on the turbulent Batchelor flow with an imposed throughflow. Daily et al. (1964) measured the average velocity profiles in the case of rotorstator systems with a centrifugal flux. Debuchy (1993) carried out a comparative study between experimental results relative to a rotating centripetal flow and the computations obtained with a numerical model developed from an asymptotic approach. But the limitations inherent to the turbulence models and to the representations of the boundary conditions, did not allow to get reliable predictions. Chew & Vaughan (1988) studied the flow inside a rotating cavity with throughflow using a model based on a mixing length hypothesis in the whole cavity. Their results were quite comparable to the experimental data of Daily & Nece (1960) and Daily et al. (1964) apart from a relaminarization area close to the rotation axis. The model of Iacovides & Theofanopoulos (1991) used an algebraic modelling of the Reynolds stress tensor in the fully developed turbulence area and mixing length hypothesis near the walls. It provided good results but some discrepancies remain on the Ekman layer thickness and the rotation rate in the central area. Schiestel et al. (1993) have used both a lowReynolds number k-e model near the walls and an Algebraic Stress Model (ASM) in the core of the flow. Later Elena & Schiestel (1995) proposed also some numerical calculations of rotating flows based on a zonal approach. Afterwise, they have used a new modelling of the Reynolds stress tensor derived from the Launder & Tselepidakis (1994) model. It provides a better prediction than the former model of Hanjalic & Launder (1976). But in this case also, the authors note a too high laminarization of the flow in comparison to the expected experimental results. A new set-up has been developed in order to get more extensive insight into the mean and turbulent fields in rotor-stator systems. The present study relates to the comparison of turbulence modelling and measurements in a rotor-stator system with and without imposed throughflow.

EXPERIMENTAL APPROACH

Experimental set-up The geometrical characteristics of the cavity are sketched in figure 1. It consists in a stationary disk (the stator) and a smooth rotating disk (the rotor) enclosed by a shroud. The rotor and the central hub attached to it rotate at the same constant angular velocity g2. The turbulent Batchelor type flow mainly depends on three control parameters: the aspect ratio G = h / R 2 , the rotational Reynolds number Re=f2 R22/v and the flow rate coefficient C w = Q / ( v R 2 ), where h is the inter-disk space, R 2 the outer radius of the rotating disk, Q the imposed throughflow and v the kinematic viscosity of water. In the present work, we keep the rotational Reynolds number and the aspect ratio constant: Re = 106, G = 0.036 . So a

763

study of the influence of the flow rate coefficient is performed. A negative value of C w corresponds to a centripetal throughflow. The accuracy on the measurement of the angular velocity and the flow rate is better than 1%. In order to avoid cavitation effects, the cavity is maintained at a pressure of 2 bars. The temperature is also maintained constant (23~ by a special water cooling device in order to keep constant the water properties. The specificity of the experimental set-up is the prerotation level imposed to the flow at the peripheral inlet thanks to a breakthrough crown mounted undemeath the rotor and linked to it.

Figure 1" Schematic representation of the experimental set-up and notations: R 1 = 38, R 2 = 250, R 3 = 253, h = 9, d = 55 (mm).

Measurement techniques The measurements are performed by means of a two-component laser Doppler anemometer (LDA) and using pressure transducers. The LDA technique is a non intrusive method used here to measure the mean radial velocity V r and tangential velocity V 0 and the associated

* -V'rV'o/(~r) 2 R~2 - V ] / ( ~ r ) Reynolds stress tensor components Rll* =V r'2 / (~r) 2 , R12

2 in a

vertical plane (r,z) at a given azimuthal angle. This method is based on the measurement of the Doppler shift of laser light scattered by small particles (30 ~tm) carried along with the fluid. The main defect of this method is to provide an integrated value on a probe volume, whose size in the axial direction (0.8 mm) is large compared to the interdisk space (9 ram). Pressure is measured using 6 piezoresistive transducers, which are highly accurate (0.05% in the range 10 to 40~ and, which combine both pressure sensors and temperature electronic compensations. They are fixed to the stator at the radial positions: 0.093, 0.11, 0.14, 0.17, 0.2 and 0.23 and located along two radii. Previous pressure measurements by embarked pressure gauges performed by Gassiat (2000) showed that the pressure on the rotor side and the one on the stator side are almost the same within 2.5%. This is in fact a direct consequence of the Taylor-Proudman theorem, which forbids axial gradients in rapidly rotating flows.

764

STATISTICAL MODELLING The differential Reynolds stress model (RSM)

The flow studied here presents several complexities (high rotation rate, imposed throughflow, wall effects, transition zones), which are a severe test for turbulence modelling methods. Our approach is based on one-point statistical modelling using a low Reynolds number secondorder full stress transport closure derived from the Launder & Tselepidakis (1994) model and sensitized to rotation effects (Elena & Schiestel, 1996). Previous works (Elena, 1994; Elena & Schiestel, 1996) have shown that this level of closure was adequate in such flow configurations, while the usual k-e model, which is blind to any rotation effect presents serious deficiencies. This approach allows for a detailed description of near-wall turbulence and is free from any eddy viscosity hypothesis. The general equation for the Reynolds stress tensor R ij can be written:

l~ij = Pij + Dij + (I)ij- ~;ij + Tij

(1)

where Pij, Dij, Oij, eij and Tij respectively denote the production, diffusion, pressure-strain correlation, dissipation and extra terms. The diffusion term D ij is split into two parts: a turbulent diffusion, which is interpreted as the diffusion due to both velocity and pressure fluctuations (Daly & Harlow, 1970) and a viscous diffusion, which cannot be neglected in the low Reynolds number region. In a classical way, the pressure-strain correlation term O ij can be decomposed as below:

(1)

tbij =Oij

(1)

O ij

(2)

+tgij

(w)

(2)

+Oij

is interpreted as a slow nonlinear retum to isotropy and is modelled as a quadratic

development in the stress anisotropy tensor, with coefficients sensitized to the invariants of anisotropy. This term is damped near the wall. The linear rapid part O ij(2) includes cubic terms. A wall correction O ij(w) is applied to the linear rapid part which is modelled using the classical form proposed by Gibson & Launder (1978) but using a weaker value of the numerical constant. However the widely adopted length scale k

3/2

/e is replaced by the

length scale of the fluctuations normal to the wall. The viscous dissipation tensor E ij has been modelled in order to conform with the wall limits obtained from Taylor series expansions of the fluctuating velocities (Launder & Reynolds, 1983). The extra term Tij accounts for implicit effects of the rotation on the turbulence field, it contains additional contributions in the pressure-strain correlation, a spectral jamming term, inhomogeneous effects and inverse flux due to rotation, which impedes the energy cascade. Numerical method

The computational procedure is based on a finite volume method using staggered grids for mean velocity components with axisymmetry hypothesis in the mean. A 140x80 mesh in the (r,z) frame proved to be sufficient in most cases considered in the present work to get grid-

765

independent solutions (Elena, 1994). A more refined modelling 180x100 has been tested but it does not change significantly the predictions. In order to overcome stability problems, several stabilizing techniques are introduced in the numerical procedure, such as those proposed by Huang & Leschziner (1985). The stress component equations are solved using matrix block tridiagonal solution to enhance stability using non staggered grids.

Boundary conditions At the wall, all the variables are set to zero except for the tangential velocity V 0 , which is set to f i r on rotating walls and zero on stationary walls. At the inlet and outlet areas, V 0 is supposed to vary linearly from zero on the stationnary wall up to f2r on the rotating wall in order to take into account the prerotation of the fluid. When a throughflow is enforced, a parabolic profile is then imposed for the axial velocity V z at the cavity inlet, with a given low level of turbulence intensity. In the outflow section, the pressure is fixed, whereas all the derivatives for the other quantities are set to zero if the fluids leaves the cavity, and fixed external values are imposed if the fluid re-enters the cavity.

RESULTS AND DISCUSSION

The entrainment coefficient K The predictions of the RSM model have been first validated on experimental data measured using the two-component LDA. Poncet et al. (2004) have shown analytically that the entrainment coefficient K of the rotating fluid (the ratio between the tangential velocity in the central core and that of the disk at the same radius) can be correlated, in the case of a centripetal throughflow, to a flow rate coefficient C q r = Q ( ~ r 2/v) 1/5/(2nr 3 ~2) according to a power law: K=2 (axCqr+b)5/7-1, where a and b are experimental constants. In figure 2, several points deduced from the modelling results are plotted against the mean experimental K-curve. The constants deduced from the model a = 5.3 and b = 0.63 are very close to those obtained by the measurements a - 5.9 and b = 0.63. 2,0 1,81,6- . . . .

experimental data K = 2 (5.9 Cqr + 0.63) 5'7 - 1 model results K = 2 (5.3 Cq + 0.63) 5/7 1 f .

o ""

1,4A

1,2-

z~.,'x"

1,0 0,8 0,6 0,4 0,~)0

'

0,~)5

'

0,'10

'

0,'15

0,20

Cq r Figure 2" The 5/7 power law correlation giving K versus Cqr.

766

Mean velocity profiles The profiles of the tangential and radial velocities (nondimensionalized by g~r) obtained with the RSM model are shown in figures 3 and 5 and compared with experimental data for a 6 * Reynolds number of 10 and at a given radial position r = r / R 2 = 0.56. Figure 4 displays 2

the corresponding streamlines ~* = ~ / (D R2 )" In a closed cavity (C w = 0), the structure of the flow (fig.3 and 4d) is clearly divided into three areas: a centrifugal boundary layer on the rotor (Ekman layer) and a centripetal one on the stator (B6dewadt layer) separated by a central core where the tangential velocity is nearly constant and the radial component is close to zero.

1

Z

Cw=-

Cw=O 9

Cw-'--5929 1

0.8

0.8

0.8

.0.6

0.6

0.6

9

Cw---9881 9

1 ,

--

.

0.8 ~

0.6

0.4

~

0.2

v;

.2. 1

1.5

1

Z

1976

1

0

0.2 0

v;

~

1.5

1

0

0.2 0

0.5

1

08

08

0.8

.0.6

v;

1

1.5

D

0 0

1

3

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

02

0.2

0.2

~.5

~

0.5

0

-0.5

~

0

0.5

0

-0.5

0

0.5

0.5

0

-0.5

v;

Y

1.5

~.~

%

0

Figure 3: Mean velocity profiles at r = 0.56, Re-10 6 with a centripetal flux: comparison between the RSM model (--) and the experimental data (o). By superimposing a centripetal throughflow (fig.3), the tangential velocity in the core increases. The size of the boundary layer on the stator also increases, whereas the one on the rotor decreases but the flow at the periphery still presents the properties (fig.4c) of a Batchelor flow without flux. The axial flux now flows through the B6dewadt layer, then compressing the core. For C w = -5929, the boundary layer on the rotor and the central core disappear. Then, 70% of the flow rotates at almost the same velocity as the rotating disk. When approaching the center of the disk, the Ekman layer, which was centrifugal, becomes centripetal at a certain radius (fig.4b). For a stronger centripetal flux (C w = -9881 ), the two boundary layers are both centripetal (fig.4a). The core of the flow rotates faster than the rotor (fig.3). All of these results are comparable to the qualitative representation of the pseudo-streamlines that Owen & Rogers (1989) observed from their numerical simulations. The RSM model gives results in good agreement with the experimental data.

767

Figure 4: Effect of the flowrate on the streamlines (RSM), 15 regularly spaced intervals" (a) C w =-9881,-0.045 <_g _<0.007 , (b) C w =-5929,-0.027_<11t (c) C w --1976,-0.009_<~

_<0.012,

___0.016,(d) C w = 0, 0___~t ___0.017,

(e) Cw = 1976, 0_<wit _<0.024,(f) Cw = 5929, 0_<~ _<0.032,(g) Cw = 9881, 0 <_~ <0.046. When a weak centrifugal flux is imposed (fig.4e and 5) , the flow keeps the same characteristics as in a closed cavity. The tangential velocity decreases but we still observe two boundary layers. By increasing the flow rate, the central core disappears. The profiles for the tangential velocity are then Stewartson profiles, it means that the tangential velocity is zero apart from the Ekman layer. The flow tends to be a Poiseuille type of flow with a parabolic profile for the radial component.

Figure 5" Mean velocity profiles at r* =0.56, Re=10 6 with a centrifugal flux: comparison between the RSM model (--) and the experimental data (o).

768

We can notice that for C w = 5929 a recirculation zone appears at the periphery of the stator (fig.4f). In these cases, the RSM model provides still better results compared to the experimental data than the k-e model, which fails to estimate correctly the mean velocity inside the boundary layers (Elena & Schiestel 1995). Turbulence statistics

Comparisons for three components of the Reynolds stress tensor are given in figure 6 for some relevant cases. The k-e model, which overestimates the levels of the three tensors, totally fails to mimic the right profiles in most cases. The RSM model provides good results even in the boundary layers and so permits the computation of accurate Reynolds stresses. The behavior of the R12 tensor is not as well predicted in the core in the case of the closed cavity. These three components of the Reynolds stress tensor are parallel to the wall and so, as already noticed by Elena and Schiestel (1995), are well predicted. Cw:-9881

R .1/2

/

~.~o

I~ o.os~o ~ ~

o 1

.

0:5 . . . . . .

~

%

I ~

0

Cw= 9881

""

~ o.o5~ . . . .

0.1 R .1/2 22

Cw:

o o5 ~ ~

~o:5

~

%

0.1

0.1

0.05

0.05

0:5

t" 0.05

0 L 0 x 10 .3

0

' 0.5

~ 1

0.5 . Z

1

0

0 x 10 .3

0.5

1

0 x 10 3

0.5

0

0.5 . Z

1

0

0.5 . Z

1

Figure 6" Profiles of the Reynolds stress tensors at r* =0.56, Re=l 0 6. comparison between the RSM model (--), the k-e model (--) and the experimental data (o).

Pressure distributions

To complement the comparisons, we performed also pressure measurements by means of 6 pressure transducers located on the stator along two rays because of geometrical constraints. We choose to take as a reference the pressure measured at the outer radial position r and we define the following pressure coefficient: C p(r ) = P

(r)-P

= 0.92

(0.92). The

dimensionless pressure is given by: P* =P/(0.5 p~2 R 2 ), where p is the density of water. In figure 7, is plotted the pressure coefficient versus the dimensionless radial position for the centripetal cases. As expected, the pressure decreases towards the center of the cavity: C p is

769

then always negative. At a given radius and for a given Reynolds number Re, it can be observed that Cp increases for increasing values of the flowrate C w (in absolute value).

-0.2

-04

C p -0.6

-1

0.3

0.4

0.5

06

0.7

0.8

0.9

r*

Figure 7: Influence of the flowrate on the radial pressure distributions in the case of a centripetal throughflow: comparison between (--) the RSM model, and the experimental data (o) C w = -1976, (A) C w = -5929, (0) C w = -9881. Note that, in the case of a centrifugal throughflow, the pressure coefficient is much less weaker and is very close to -0.05 for each case.

CONCLUDING REMARKS Turbulence modelling and measurements of the turbulent flow in a rotor-stator cavity is a great challenge even more when throughflow is superimposed. In the present work, we have compared the second order model results with new experimental data for a Reynolds number equal to 10 6 , a fixed geometry and several relevant flow rates. For weak throughflows, the flow still keeps the characteristics of a closed cavity. In the case of a strong centripetal throughflow, we showed that the flow is then divided into three areas but the central core rotates faster than the disk and both boundary layers are centripetal. When a centrifugal flux is imposed, the flow becomes a Stewartson type flow with only one boundary layer on the rotor. The RSM model which is a second moment closure proved to give a great improvement compared to the classical k-e model for the predictions of the Reynolds stress tensors. Because good agreement with the measurements has been found in most of the various cases under consideration, it is confirmed that the RSM model offers an adequate level of closure to describe the mean and the turbulent fields in such type of fows.

ACKNOWLEDGMENTS Numerical computations were carried out on the NEC SX-5 (IDRIS, Orsay, France). Financial support for the experimental approach from SNECMA Moteurs, Large Liquid Propulsion (Vernon, France). They are gratefully acknowledged.

770

REFERENCES

Chew J.W. and Vaughan C.M. (1988). Numerical Predictions of Flow Induced by an Enclosed Rotating Disk. 33 ~d Gas Turbine and Aeroengine Congress, ASME Paper 88GT- 12 7, Amsterdam. Daily J.W., Ernst W.D. and Asbedian V.V. (1964). Enclosed rotating disks with superposed throughflow. Report n~ M.I.T., Department of Civil Engineering. Daily J.W. and Nece R.E. (1960). Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks. ASME J. Basic Eng. 82, 217-282. Daly B.J. and Harlow F.H. (1970). Transport equation for turbulence. Phys. Fluids A 13:11, 2634-2649. Debuchy R. (1993). Ecoulement turbulent avec aspiration radiale entre un disque fixe et un disque toumant. Ph.D. thesis, Universit6 des Sciences et Technologies de Lille. Elena L. (1994). Mod61isation de la turbulence inhomog6ne en pr6sence de rotation. Ph.D. thesis, Universit6 d'Aix-Marseille I-II. Elena L. and Schiestel R. (1995). Turbulence modeling of confined flow in rotating disk systems. AIAA J. 33:5, 812-821. Elena L. and Schiestel R. (1996). Turbulence modeling of rotating confined flows. Int. J. Heat Fluid Flow 17, 283-289. Gassiat R.M. (2000). Etude exp6rimentale d'6coulements centrip6tes avec pr6rotation d'un fluide confin6 entre un disque toumant et un carter fixe. Ph.D. thesis, Universit6 d'AixMarseille II. Gibson M. and Launder B.E. (1978). Ground Effects on Pressure Fluctuations in the Atmospheric Boundary Layer. J. Fluid Mech. 86:3, 491-511. Hanjalic K. and Launder B.E. (1976). Contribution towards a Reynolds-stress closure for lowReynolds number turbulence. J. Fluid Mech. 74:4, 593-610. Huang P.G. and Leschziner M.A. (1985). Stabilization of recirculating flow computations performed with second moments closures and third order discretization. 5 th Int. Syrup. On Turbulent Shear Flow, Cornell University, Ithaca, NY. Iacovides H. and Theofanopoulos L.P. (1991). Turbulence modeling of axisymmetric flow inside rotating cavities. Int. J. Heat Fluid Flow 12:1, 2-11. Iacovides H. and Toumpanakis P. (1993). Turbulence Modelling of Flow in Axisymmetric Rotor-Stator Systems. 5 th Int. Symp. On Refined Flow Modelling and Turbulence Measurements, Presses de l'Ecole Nationale des Ponts et Chauss6es, Paris. Itoh M., Yamada Y., Imao S. and Gonda M. (1990). Experiments on turbulent flow due to an enclosed rotating disk. Proc. Int. Symp. On Engineering Turbulence Modeling and Experiments, 659-668, Ed. W. Rodi and E.N. Galic, Elsevier, New-York. Launder B.E. and Reynolds W.C. (1983). Asymptotic near-wall stress dissipation rates in a turbulent flow. Phys. Fluids A 26:5, 1157-1158. Launder B.E. and Tselepidakis D.P. (1994). Application of a new second-moment closure to turbulent channel flow rotating in orthogonal mode. Int. J. Heat Fluid Flow 15:1, 2-10. Owen J.M. and Rogers R.H. (1989). Flow and Heat Transfer in Rotating-Disc S y s t e m s Vol. 1. rotor-stator systems. Ed. Morris, W.D. John Wiley and Sons Inc, New-York. Poncet S., Chauve M.-P. and Le Gal P. (2004). Turbulent Rotating Disk Flow with Inward Throughflow. To appear in J. Fluid Mech. Schiestel R. and Elena L. (1997). Modeling of Anisotropic Turbulence in Rapid Rotation. Aerospace Science and Technology 7, 441-451. Schiestel R., Elena L. and Rezoug T. (1993). Numerical modeling of turbulent flow and heat transfer in rotating cavities. Num. Heat Transfer A 24:1, 45-65.

12. Heat and Mass Transfer

This Page Intentionally Left Blank

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

773

IMPINGING JET COOLING OF WALL MOUNTED CUBES M.J. Tummers l, M.A. Flikweert 1, K. Hanjali6 l, R. Rodink 1 and B. Moshfegh 2 1 Delft University of Technology, Faculty of Applied Sciences, Thermal and Fluids Sciences Section, P.O. Box 5046, 2600 GA, The Netherlands 2 LinkOping Institute of Technology, Department of Mechanical Engineering, S-581 83 Link6ping, Sweden

ABSTRACT This paper reports on the flow structure and the surface temperature distribution of a heated wallmounted cube from an in-line array, subjected to cooling by the combined action of a channel flow and an impinging jet. The impinging jet issues from a round nozzle at the top wall. The turbulent flow field around the cube (measured with Particle Image Velocimetry (PIV)) and the temperature distribution on the surfaces of the cube (evaluated from an infrared imaging system) were determined for two different relative positions of the jet nozzle and the cube. The local flow structure in terms of flow separation and reattachment, and the rolling up of separating shear layers, has a marked influence on the local temperature distribution on the surface of the cube.

KEYWORDS Heat transfer, turbulent structures, wall mounted cubes, electronics cooling, particle image velocimetry, infrared thermography

INTRODUCTION The continuous miniaturization of electronic elements leads to an increase in local specific power dissipation. Local overheating and hotspots have been reported more and more frequently as a major cause of equipment malfunction, posing more demanding requirements for efficient cooling and control of heat transfer. A major prerequisite for achieving an optimal electronic design is the understanding of the relation between the local heat transfer and the flow and turbulence structure in the cooling fluid. Detailed experiments using modern measuring techniques that enable the nonintrusive measurements of the turbulent flow field and wall temperature distribution can help in achieving this goal. In a collaboration of LinkOping Institute of Technology and Delft University of Technology a generic configuration, consisting of an in-line array of five wall-mounted cubes is studied. The cubes are

774

~ s

top wall

|

!

Figure 1" Flow configuration.

cooled by two mutually perpendicular flows: a channel flow parallel to the mounting wall, and an impinging jet that issues from a single nozzle at the top wall. The position of the nozzle relative to the center cube can be varied. The configuration is studied experimentally and numerically. We report here on some results of the experiments carried out at Delft University of Technology. Calculations for the same configuration are carried out at LinkOping Institute of Technology.

EXPERIMENTAL SET-UP

Flow configuration Figure 1 depicts the flow configuration that was considered in this study. An in-line array of five cubes was mounted on the lower wall of a straight channel at mid span. The channel has internal dimensions of 1000x30• mm 3. The flow entering the channel has a bulk velocity of 2.4 m/s. The mean velocity and Reynolds stress profiles measured 150 mm upstream of the front face of the first cube are shown in Figs. 2a and b, respectively. The boundary layers on the top and lower walls of the channel are both turbulent. The boundary layer on the lower wall is thicker than the one on the top wall, because it was tripped farther upstream. The distribution of the Reynolds stresses shows that the channel flow is far from a fully-developed state.

3

0.2

-+- H'U' -,~- V'V'

0.15

_ <>_

~

eq

0.1

-6

0.05


0

-0.05 0

10

20

30

y [mm]

Figure 2a: Mean velocity components in the xdirection measured 150 mm upstream of the front face of the first cube.

, 0

i 10

c 20

, 30

y [mm]

Figure 2b: Reynolds stresses measured 150 mm upstream of the front face of the first cube.

775

10

9

8

"~

6

~-" 6 ~"~"

4

3

~

-+- u ' u ' ~ V'V'

0

0

-2

-3 -6

-3

0

3

r [mml

Figure 3a: Mean velocity profiles of the round jet measured at 0.8 mm from nozzle.

6

-6

-3

0

3

6

r [mml

Figure 3b: Reynolds stress profiles of the round jet measured at 0.8 mm from nozzle.

Each cube has a height of H-15 mm, and the distance between the cube centers in the array is 4H. The round, sharp-edged nozzle in the top wall has a diameter of 12 mm. The jet shift S is the distance between the centerline of the nozzle and the center of the third cube. The bulk velocity of the jet is 6.5 rn/s. The nozzle inlet profiles for the mean velocity and the Reynolds stresses are given in Figs. 3a and b, respectively. From the mean velocity profiles it seems that the jet has a diameter of less than 12 mm. However, this is due to the initial contraction of the jet. Out of the five cubes, only the third cube is heated in this experiment. This cube (identical to the cube used in the experiments of Meinders [1998] and Meinders et al [1996]) consists of a heated copper core coated with a 1.5 mm thick, low-conducting epoxy mantle. Heat is generated in the cubical copper core by a dissipating source. During the experiments the temperature of the core is kept at a constant value of 70 ~ This temperature follows from a thermocouple inside the core. The temperature of the core is essentially uniform because the thermal resistance of the epoxy layer is much larger than that of the copper core. The epoxy mantle acts as a heat flux sensor, in the sense that if the temperature of the core and the temperature distribution on the cube's surface are known (the latter may follow from e.g. infrared thermography), then the temperature distribution in the epoxy mantle can be computed from Laplace' s equation, thus yielding the heat flux at the cube' s surfaces. Throughout this paper an orthogonal coordinate system will be used having its origin at the front face of the third cube at mid span, see Fig. 4. The x-coordinate is measured along the channel axis, taken positive in the streamwise direction. The y-axis is measured normal to the lower wall of the wind tunnel.

776

impinging jet & channel flow -+-impingingjet only 3.5

top leading edge~ direction

side

] ......:

leading %-~.... fi edges / ~ ,,/-.//;,

side edges

/ .o

top trailing edge side trailing edges

z

Figure 4: The x, y, z-coordinate system with the origin at the front face of the third cube.

:~

~

3

2.5

2

' -16

-8

' 0

' 8

16

jet shift S [mm]

Figure 5: Power dissipation as a function of the jet shift S.

Measurement techniques The velocity field was measured by using a PIV system that included a Continuum Minilite doublepulsed Nd:YAG laser with a pulse energy of 25 mJ. This laser was used to produce an approximately 1 mm thick light sheet that illuminated the seed particles in the flow. A PCO Sensicam camera with a resolution of 1280x1024 pixels recorded images of the seed particles in the light sheet. A 55 mm focal length lens with a numerical aperture of 8 was mounted on the PCO camera. The time separation between pulses was set to 14 its. PIV measurements were performed in seven horizontal planes (y/H=0.067, 0.267, 0.5, 0.733, 0.933, 1.067 and 1.267) and five vertical planes (z/H=0, 0.233, 0.567, 0.767 and 0.967) focusing on the third cube. The measurement region in the horizontal planes is located at -1.8 <x/H <2.9 and -2.0
777

The time-averaged temperature distribution on the faces of the cube was measured by using a lowwavelength (2~tm to 5.5~tm) infrared imaging system (Varioscan, Jenoptik). The Varioscan camera is equipped with a scanning mechanism to create images that are composed of 200 lines each having 300 pixels. The relation between the pixel intensity and the temperature is established in an in-situ calibration procedure in conjunction with an image restoration technique based on a Wiener filter using the two-dimensional Optical Transfer Function as described by Meinders et al. [1996]. The four cube faces on which the temperature was measured are denoted as rear, front, top and side, as illustrated in Fig. 4.

RESULTS

Total heat transfer The total heat transfer from the third cube to the flow as a function of the jet shift S is shown in Fig. 5. This graph reveals a maximum for S=8mm, i.e., when the impinging jet is displaced over 8 mm in the upstream direction with respect to the center of the third cube. Inspired by this result, the flow field and the temperature distribution of the third cube were studied extensively for two cases: S-0 and S=8mm. Also included in Fig. 5 is the total heat transfer in absence of channel flow. Clearly, in that case the total heat transfer should be symmetric in the jet shift S.

Mean flow field for S=O Figure 6 sketches the main features of the mean flow around the cube in case of zero jet shift. The mean flow is dominated by a large horseshoe-type vortex denoted by A. This vortex forms through the interaction of the channel flow and the high velocity wall jet that flows radially outward from the jet impingement point (B) on the cube' s top surface. From the mean velocity field in the symmetry plane (z/H=O) that is shown in Fig. 7a, it is seen that the core of this vortex is located at x/H=-0.36, y/D=l.37. Figure 7b also identifies the core of a smaller, counter-rotating horseshoe vortex at x/H=0.68, y/H=l.O0. This vortex is denoted by C in the sketch. On the front face of the cube, at y/H=0.50, #H=0, there is a stagnation point that separates fluid moving up towards vortex C and fluid moving down towards a third horse-shoe vortex E. The latter vortex is very weak and its influence is limited to the lower part of the cube's front face, as can be seen from the mean velocity field in horizontal plane y/H-O.067 shown in Fig. 7c. The flow in the wake region of the cube is dominated by an arc-shaped vortex (F). The core of the arc vortex in the symmetry plane is at x/H=l.45, y/H=0.59, while its imprint in the horizontal plane can be observed at x/H= 1.17, z/H=+0.38. The high velocity wall jet on the top surface separates at the trailing edge and reattaches again at x/H=l.27. Figure 7c indicates that fluid moves radially away from the reattachment point G on the lower wall. The fluid moving towards the rear of the cube is trapped by the arc vortex. This vortex does not appear to end at the lower wall. Instead it has a sharp bend just above the lower wall, and it continues along line H. This vortex causes the flow to separate at point I.

778

Figure 6: Artist's impression of the mean flow structure about the third cube for zero jet shift.

Figure 7a: Mean velocity field in vertical plane z/H=0 for zero jet shift.

Figure 7b: Mean velocity field in vertical plane z/H=0.567 for zero jet shift.

779

Figure 7c: Mean velocity field in horizontal plane y/H=0.067 for zero jet shift.

Mean flow field for S=8 mm The mean flow field around the cube for S=8mm is sketched in Fig. 8. For convenience the cube is now viewed from a different angle. Upstream of the cube there is a large horseshoe vortex denoted by A. From the velocity field in the symmetry plane (Fig. 9a) it can be seen that the core of vortex A is located at x/H=-1.50, y/H=0.32. Due to the jet shift, the jet impingement point B is now located close to the leading edge on the top surface. Just below the leading edge, on the front face, there is an arc vortex (C). The jet fluid that moves upstream after reattachment separates at the leading edge and then attaches on the front face. The fluid then moves upstream along the channel floor and rolls up into the large vortex A. The wake region of the cube is characterized by an arc vortex (D). In contrast to the case of zero jet shift, there is no evidence to suggest that this vortex continues along the channel floor. Vortex E is located in the direct vicinity of the side edge. From the mean velocity field shown in Fig. 9b (z/H=l.067) it can be seen that the core of vortex E is close to the upstream top corner of the cube at x/H=0.21, y/H=0.81. Vortex E rolls up the wall jet on the top surface when it separates at the side edge of the cube. Fluid moving towards the front face is diverted towards the side leading edge, and accelerates around the corner. Possibly due to the interaction with vortex E, the high velocity region remains in the direct vicinity of the side face of the cube, thus promoting heat transfer as will be shown later. Fig. 9c illustrates that the mean flow just above the mounting plate (y/H=0.067) is dominated by the radial outflow from the impingement point at the base of the cube, and the presence of the arc vortex in the wake region.

780

Figure 8" Artist's impression of the mean flow structure about the third cube for jet shift S-8 mm.

Figure 9a: Mean velocity field in vertical plane 7./H=0 for jet shift S=8 mm.

Figure 9b: Mean velocity field in vertical plane JH=0.567 for jet shift S=8 mm.

781

Figure 9c: Mean velocity field in horizontal plane y/H=0.067 for jet shift S=8 mm.

Heat transfer measurements The surface temperature of the cube was determined with the infrared imaging system. The four different cube faces (front, top, rear and side) are folded out and projected on a plane thus yielding the temperature maps for S=0 and S=8 mm that are shown in Figs. 10 and 11, respectively. The temperature map for S=0 shows that the front, side and rear faces are hot in comparison to the well-cooled top face. Interestingly, the mean temperature level of the front face and the rear face is nearly the same indicating a very weak effect of the cross flow. For S=8 mm, both the top face and the front face are cooled well by the impinging jet. However, an interesting result is that for S=8 mm the side face of the cube is also relatively cool, whereas the side face of the cube is hot for S=0. The temperature along a front-top-rear surface path that is characterized by z/H=0 is presented in Fig. 12 for S=0 and S=8 mm. In both cases, a peak in temperature appears on the front face at y/H=0.2. For S=O, this roughly coincides with the presence of a weak horse-shoe type vortex close to the front face (vortex E in Fig. 6). Such a vortex is absent in the mean flow pattern for S=8 mm, and the origin of the temperature peaks on the front face at y/H=0.2 cannot be explained without considering the effect of turbulent velocity fluctuations. However, this information is not available at this stage. Figure 13 depicts the temperature along a side-top surface path characterized by x/H=0.5. The results confirm that the top face is cooled well by the impinging jets for both jet shifts. On the side face large differences in temperature for S=0 and S=8 mm are seen. Recall from the sketch in Fig. 8 that there is a vortex on the side of the cube that rolls up the wall jet separating from the top face (vortex E in Fig. 8). It is believed that this vortex is responsible for the relatively efficient cooling of the side face for S= 8mm.

782

Figure 10: Surface temperature map for S=0.

Figure 11: Surface temperature map for S=8 nqlTl.

Figure 12: Temperature along front-top-rear surface path characterized by z/H=0.

Figure 13: Temperature along side-top surface path characterized by x/H=0.5.

CONCLUSION The mean flow field and the surface temperature distribution around one cube in an in-line array of five wall-mounted cubes were studied experimentally. The heated cube is cooled by a channel flow and a jet that issues from the opposite channel wall. The mean flow structure and the surface temperature distribution were mapped for two situations, i.e., when the jet nozzle is positioned above the center of the cube, and when the jet nozzle just upstream of the front face of the cube. The latter situation is relevant because the total heat transfer from the cube to the flow then has a maximum. REFERENCES Meinders, E.R., Experimental Study of Heat Transfer in Turbulent Flows over Wall-Mounted Cubes, Dissertation, Delft University of Technology, Delft, The Netherlands, 1998. Meinders, E.R., van der Meer Th. H., and Hanjalid, K., "Application of Infrared Restoration Technique to Improve the Accuracy of Surface Temperature Measurements," Proc. Quantitative Infrared Thermography 3, EUROTHERM No.50, September 2-5, Stuttgart, Germany, 1996

Engineering TurbulenceModellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

783

N U M E R I C A L AND E X P E R I M E N T A L S T U D Y O F T U R B U L E N T P R O C E S S E S AND

MIXING IN J E T M I X E R S

E. Hassel, S. Jahnke, N. Komev, I. Tkatchenko, and V. Zhdanov Lehrstuhl ftir Technische Thermodynamik, University of Rostock, Germany

ABSTRACT Velocity and passive scalar fields in a co-axial jet mixer have been investigated both experimentally (LIF and LDA) and numerically (LES) in a single phase incompressible flow at different flow rate ratios, flow densities, Reynolds numbers and temperatures. Two flow regimes with and without recirculation zone are considered. The flow in the recirculation zone is unsteady and contains coherent vortex structures which enhance sufficiently the mixing of scalars. Different SGS closure models have been tested by comparing with experimental data. The best results were achieved using the dynamic mixed model. The experimental and numerical investigations showed that the mixing is mostly influenced by the flow rate ratio. The effect of other parameters is negligible.

KEYWORDS Mixing, Turbulent flow, Co-axial jet mixer, LES, LDA, LIF.

INTRODUCTION Various jet mixers are applied in many engineering devices such as combustion chambers, injection systems, etc. The classification of the jet mixers are given by Henzler, (1978). In this paper the classical jet mixer consisting of a nozzle of diameter d which is coaxial arranged in a pipe of diameter D has been considered (Figurel). Jet mixers of such type are used in chemical engineering for homogenization of two streams of different fluids or as chemical reactors to control chemical reactions.

784

Figure 1: Sketch of a jet mixer The mixing process can be quantified by defining a degree of mixedness. Following Danckwerts (1952), this can be done for each cross- section of the mixer as

~(f(x,r,O)- fav (x)) 2 rdrdO a'=l -s

where f,~ ( x ) = S

if

,

(1)

f ( f (0, r,O)- f~v (o)) ~rdrdO S f(x, r,O)rdrdO. S is the area of the cross section and f is the mixture fraction.

From dimensional analysis it is seen that the characteristics of the jet mixers depend on the following dimensionless parameters: D ! d is the diameter ratio, Re d = dVd/v is the Reynolds number for the nozzle flow, Sc is the Schmidt number, 1)"o 11)"d is the ratio of the co-flow rate to the injected one from the nozzle, lad !l)o is the fluid density ratio. In non-isothermal cases the Prandtl number Pr and the temperature ratio (TD - T d) !T o are the additional parameters. Also, a very important criterion is the mixing length lp at which fl takes a certain value of ~ usually taken as/5' = 0.9 or fl = 0.95. The following two kinds of flow structures can be observed in an axial mixer depending on the flow rate ratio 1)~ !11d and the diameter ratio D!d (see also Barchilon, Curtet, (1964), Korischem, (1987) and Tebel et al. (1988)): 1) Regime with recirculation region (herein after referred to as regime A): D / d < 1 + 1)o 11)"d 2) Free jet regime (herein after referred to as regime B): D / d > 1 + 120 !l/d The aim of the present paper is the numerical and experimental study of turbulent processes in jet mixers at different Reynolds numbers, flow rate ratios, fluid densities and flow temperatures in incompressible single phase flow conditions.

MATHEMATICAL FORMULATION The theoretical part of the study is based on LES. As for LES the governing equations for incompressible flows considered here may be written in Cartesian form as:

b

Ot

OXj

~Xj

~ OXi

OXj

3 ~Xk

(2)

r~ij

p OXi

785

a as+ at ax,

0"--7+'~xj

a

vat - ~ c ; ~ ) -i

~ x j / P r ~)x, )

j ax,

~gxj

(4)

(5)

where ,o represents the density (for liquids is constant), u the velocity with its Cartesian components denoted by i, P the modified pressure, f the mixture fraction as a dimensionless scalar, v the kinematic viscosity. The unclosed termrisos is modelled according to four SGS models: the Smagorinsky (1963) approach, the dynamic procedure (DGM) proposed by Germano et al. (1991), the dynamic mixed model (DMM) by Zang et a1.(1993) and the dynamic mixed model (DMM1) with modifications proposed by Vreman _SGS et al. (1994). Other terms JjSGS and z-oj are modelled using the gradient diffusion assumption with the constant turbulent Prandtl number being equal to 0.7. The turbulent Schmidt number was calculated dynamically using the dynamic scalar model by Lilly (1992). Wall functions were applied for momentum (3) and the temperature (4) equations. The Neumann condition ,gf/~)n = 0 is applied for the mixture fraction at the pipes' wall. The temperature and mixture fraction both of the co-flow and the jet are assumed to be constant at the inlet. The velocity profile within the nozzle and the co-flow slices is generated by superimposing random noises having prescribed statistical properties on the mean statistical profile U(r,O) taken from auxiliary calculations of fully developed turbulent flows in a pipe and in a coannular pipe. This is expressed as

ux(O,r,O,t)=U(r,O)+ux(r,O,t), /

Ur(O,r,O,t)=Ur /(r,O,t), uo(O,r,O,t) = u o/ (r,O,t) ,

(6)

/(r,O,t) are generated using the method of random spots where fluctuations Ux/(r,O,t), Ur/(r,O,t), UO proposed by Kornev et al. (2003). At the outlet boundary, a simple Neumann condition with clipping is used for the momentum in order to avoid reflections back into the computational domain and to force vortices to be carried out of the domain by convection only. Periodic boundary conditions for all scalars and velocities are applied in the circumferential direction. The governing equations are discretized on a staggered grid using a 2 nd order finite volume approach which implicitly applies a top hat filter to the equations. Time marching is done explicit with the formal accuracy of third order using a Runge-Kutta method.

EXPERIMENTAL SETUP Experiments were conducted in a water channel (Figure 2). The test section 6 consists of a perspex rectangular box filled with water to reduce refraction effects. Within this box there is a perspex tube of 1000 mm length with 50 mm inner diameter. A horizontally located inner steel tube 7 (nozzle) of 600 mm in length (d = 10 mm) was placed coaxial along the axis of the perspex tube. Co-flow comes to the mixer from the tanks 1, 2, 4 through the steel tube 5 of 5000 mm length and inner diameter of 50 mm. A rhodamin 6G water solution (0.03 mg/1) was injected from vessel 9 through inner tube 7 by pump 8. The flow rates were controlled by flow meters 13 (ultrasound meter FLUXUS AM7207). The mixture

786 behind the test section was collected in vessels 10 and 11 to prevent changes in the flow background intensity due to substitution of water by the rhodamin solution.

Figure 2: The scheme of the water channel and the common view of the mixer. 1, 2, 4 - water vessels; 3,8 - pumps; 5 -steel tube; 6 - test section; 7 - inner tube of the mixer; 9, 10, 11 - rhodamin solution vessels" 12- electric heater; 13 - flow rate meters; 14 - outer tube of the mixer.

Laser-induced fluorescence (LIF) was used for measuring instantaneous rhodamin 6G concentration distributions across the mixer. The light source was an impulse ND: YAG laser (%=532 nm, Continuum, PREC II 8000). The emitted light was imaged onto an intensified CCD camera (PI-MAX, Roper Scientific) supplied with a lens (f = 50 mm, F/1.2, Nikkor) and a sharp cut-off filter (%=600 nm, Edmund Industrie Optic GmbH). The camera was triggered by the laser. LDA-2D Probe Systems (FiberFIow 60X, Dantec) has been used to measure the longitudinal velocity and rms fluctuations in cross sections downstream of the nozzle. The mixture fraction was determined by normalizing the local intensity to the averaged maximal intensity I o found at the first cross-section (x/D = 0.2). The experimental accuracy is below 3 % and it is mostly determined by the accuracy of the calculation of

Io.

RESULTS OF EXPERIMENTAL STUDY Concentration and velocity measurements were made for values of the ratio Q = l)z~/I?a = 1.3, 5.0 and for the Reynolds numbers Red = 10000, 15000. The first flow rate ratio l)z~/12a = 1.3 corresponds to the flow with a recirculation zone (regime A) whereas the second one relates to the flow of a jet type (regime B). The temperature influence has been investigated for both regimes at Red = 10000. The temperature ratio between the co-flow and the injected fluid was ( T z ~ - T a ) / T D = 0.0 and 0.12 ( T o - T d =35~

The development of the mixture fraction and velocity fields along the mixer axis is

shown in Figures 3 and 4. 1"01~

_,m_ fo' Re=10000, Q=1.3

a)

-n-

9 fo, Re=15000

o t'/t0

.k fo, Re=10000, T

_~~0.6

1.0

9

f'/fo

\oo4

~o

{

I--A-- UoIU ' Q=5.0

k.~. .

.

t'a o

-A-

f'lfo

o.21 ^.4,~a~2

f'lfo

o.ola~ ~. ,

--A-- fo, Re=10000, Q=5.0 v 0

1

2

3

4

5

6

x/D

7

8

9

1(:

9 fo, Re=15000, fo, Re=10000, T

o t'/to

0

, ]

.

, 2

.

, 3

\~~,,___~,, .

, 4

.

,

.

5

, 6

.

, 7

.

, 8

.

,

.

9

x/D

Figure 3" Development of the mean mixture fraction f0 = f ( x , r = 0 ) , the rms of the mixture fraction fluctuations f / ( x , r = 0)(a) and the mean axial velocity U o = U ( x , r = O )

and its rms u / ( x , r = 0 ) (b)

along the mixer axis. Experiments with the heated co-flow are denoted additionally with "T".

787

For both regimes the mixture fraction decays much faster than the mean velocity. Significant differences between regimes A and B can be observed for the decay of the mean velocity and the mixture fraction along the mixer axis at x/D > 2.

1.o ,%~,,~ee~,,~~,,~

~,,

-,-,- x/D=-0.2.

1.0

08

q%*. ,l~.\,

--o-- x/D=-1.2 --O-- x/D=-1.7 --'~'-- x/D=-3.2

0 . 8_Ji_C) .

~

"] a)

~

0.6 .go

0.4-t " ~

~ "

k

0

.

~

- - " - - x/D=-0.2,

~ ~\ \.

,

0.2

0.0

U/Uo

~'\~ \

o~

0

--A-- x/D=-0.7

.

0.4

,

y/D

-,,- x/o=-o7

".-1~~--

0.6

~o

b)

0.8

,-*/~"*'*"*_.

.v

0.2 0.1

0.0

0.0

012

0'.4

y/D

o'6

--*--x/D=3.2

0

018

--A--o---0--'~--,~ v -".--,&-o--e-*--~1--'w--

-.~" ~r ,-7o

j,r

0.2

x/D=-1.7

1.10

0.4

0.3

+

~~

o

,'--I~" ~ L,

~

oh/

. 0 0.0

~

\9~~ -, \ "" I~, "~"

I\

~

~

.~ oO~

\

\,

'~ ",, \.

.

\.,

--~-- x/D=0.2,

-o- x,o=_,.

--Or--x/D=-3.2 u/uo

-=-x/'o=(,

mm.=_=-|_ll__ a --*-- x/D=3.2 - - ~ - x/D=5.2

,

0.2

0.4

.

y/D

. . . .

0.6

0.8

-- I~- x/D=9.2 l0

1.

f'/fo x/D=0.2 t~t0 x/D=0.7 0.4 --t:]-- x/D=0.2, x/D=-1.2 --O-- x/D=1.2 x/D=-1.7 0.3 --la--- x/D=3.2 x/D=3.2 d) ~ i - ~ - x/D=5.2 x/D=-5.2 :bo 0.2 ~ -- 1:::~-x/D=9.2 x/~-7.2 u 'lUo x/~-0.2 _ ~'?i~*,,. . ,8:" '"~'/o,'~.~ -~, " ~ - " I -m- x/D=0.2, x/D=0.7 e' \ 0.1 * " * - * - - * - - x/D=3.2 x/D=1.2 _._9 ,.O , B \ ~ _..a~..I x/D=-1.7 0"0 ~ ; ~ ; ~ m : ~ Irrrrn ""-|--I1=11"" ~ -~-x/D=5"2-1I~-,D=9.2 x,/D=-3.2 i x/D=5.2 0.0 0Z2 0Z4 0:6 0Z8 ~.0 x/D=-7.2 y/D

,,...

Figure 4: Radial distributions of the mean mixture fraction f , axial velocity component U , rms of mixture fraction fluctuations f / and rms of velocity fluctuations in regime A (a,b) and B (c, d). Comparing the profiles of the velocity and the mixture fraction we can state a difference in their evolution along the mixer in the regime A. The distributions of the mean velocity profiles indicate that the mean position of the recirculation region starts at the cross section x/D=l.2 (appearance of the negative mean axial velocity (Figure 4a)) while a clear effect of the recirculation vortex on the distribution of the mixture fraction have been already observed earlier, at x/D=0.7 (increase of the mixture fraction near the pipe walls (Figure 4a)). Obviously, the flow is unsteady and followed by a periodic generation of coherent vortex structures which transport the mixture fraction against the mean flow direction towards the inlet of the mixer. On the contrary, the evolutions of the velocity and the mixture fraction profiles are very similar in the regime B (Figure 4c). The homogeneous state with respect to the mean mixture fraction is reached before the uniform distribution of the rms of fluctuations takes place. At the homogeneous state, the level of the rms mixture fraction fluctuations in the regime A is twice as

788

low as the one in the regime B. However, the level of the rms of velocity fluctuations is significantly lower in the regime B compared with that of the regime A. The behavior of the autocorrelation function R/ for mixture fraction fluctuations shows that the region of the extensive mixing due to recirculation zone occupies the interval 0.7 < x/D <2.7 (Figure 5).

1.0J - I - - r/D=-0.22I ] ~ I-o-- r/D=-O.52IA 0.8 I == - - * , r/D=-0.841~ 0.6t: 0.45

-0.2 0.0

k

,,, o~'l 9

0.2

"

0.4

Ot

A~. && ~

I

~

i

9

y/D 0.6

0'

0.8 0.6

0.8

L,

rr 0.41

I

.f,.

1

o :%

I~

~.'~.~ ~... oO ~ ~o..,~~

-0.21

.

0.0

1.0

.

.

0.2

.

0.4

.

y/D

.

.

0.6

.

.

0.8

I

I

1.0

Figure 5" Distributions of the autocorrelation function for different points of the flow in the regime A: a)x/D=0.7; b)x/D=2.7 The analysis of the autocorrelation function R/ in the regime A indicates that the integral length scale

L z of the scalar fluctuations across the mixer is maximal in the area of strong separations and large vortices near the pipe walls. Downstream, at large x/D there is almost an uniform distribution of L/ across the mixer. In the regime B, the integral length scale L/ monotonically increases behind the nozzle being maximal at the jet axis. At large distances from the nozzle x/D >7, the integral length scale is nearly the same in both regimes. The Danckwerts criterion (1) shows that the homogeneous state of the flow is attained at the distance x/D = 3.2 downstream of the nozzle for the regime A and at the distance x/D > 7.2 for the regime B. There is no essential influence of the Reynolds number Re,~ and the temperature ratio (TD - T d ) / T ~ on the mixing within the range of the investigated flow parameters.

1.0 0.8 /~~--I--Q= 1.3, Re=10000 o Q=1.3, Re=15000 .JA- ~1I m Q=1.3, Re=10000, T /J I-*-Q=5.0, Re=10000 0.2 /In,/~ I ZX O=5.0, Re=15000 /EA" I ~x Q=5.0, Re=10000, T 0.0 0 1 2 3 4 5 6 7 8 9 10 x/D 0.6

84

0.4

/

9

i

i

9

i

9

|

9

i

9

i

9

i

9

|

9

|

9

i

9

I

Figure 6: Axial distribution of the degree of mixedness at different Reynolds numbers Re d for the regime A (Q=l.3) and the regime B(Q=5.0). Experiments with the heated co-flow (TO - T d ) / T D =0.12 are denoted additionally with "T".

789

INFLUENCE OF DIFFERENT PARAMETERS ON MIXING Table 1 summarizes numerical and experimental investigations of the influence of different parameters on the overall mixing (see also Jahnke et a1.(2004)). The numerical simulations were performed for single phase flows with both liquid (Sc=1000) and gases (Sc=0.7). Symbols in the table 1 characterize the influence of the increase of different parameters on the mixing: 0 - negligible influence, + weak enhancement, ++ strong enhancement of the mixing. The symbol "NI" stands for "not investigated". As can be seen from the numerical results, the Reynolds number Re d and the density ratio PD / Pd have an effect on the rate of mixing at small x/D. However, the homogenous state is attained approximately at the same length /0.9 independent on Re d and / 9 o / p d . In measurements, the range of the change of the Reynolds number was rather small (104-1.5.104) and its effect on the mixing was negligible for all x/D.

TABLE 1 INFLUENCE OF DIFFERENTPARAMETERS ON THE MIXINGENHANCEMENT

C O M P A R I S O N O F E X P E R I M E N T A L AND N U M E R I C A L DATA The validation of LES models and comparison with experimental data were carried out for both regimes A and B. The computational domain is an eight diameter long section of a pipe in which the flow is assumed to be turbulent. In terms of D the computational domain has the size 8•215 in a cylindrical coordinate system (x, r, 0). The grid is nearly equidistant in x and r directions. The computational grid contains N x • Nq, • N o • N a nodes in axial, circumferential, radial directions of the pipe and radial direction of the nozzle. The grid N x = 256, N~0 = 6 4 , N o = 1 6 , N d = 3 was chosen for further calculations because it provides results which are very close to those obtained for the finest grid N x = 512, N~0 = 6 4 , N o = 30,N a = 6 tested in a preliminary methodical investigations.

790

The numerical investigations have shown that parameters of the flow obtained by the classical Smagorinsky model strongly depend on its constant which is known to be flow type dependent. Therefore we only compare models with dynamic determination of the constant, i.e. the dynamic model by Germano, 1991 (DGM), the dynamic mixed model by Zang et al., 1993 (DMM) and dynamic mixed model with changes proposed by Vreman et al., 1994 (DMM1). To increase the numerical stability of the solution in the case of the DMM and DMM1 models a special clipping procedure based on the asymptotic analysis of the velocity field was applied. In the case of the regime B, all models show a good agreement with experimental data for the profiles of the mixture fraction (Figure 7) at all x/D. a) ' 1.0 , ' "i"~,,, ".% 9 0.8. mk~

'1 i Experiment, mean value I O Experiment, rms

"

I--German~

I- A - Germano, rms I- m - DMM, mean value

9X

0.6' O

r,... 0.4 7:) 0.2.

)o._.~---o-~-~u - o

-0.2

9

l

9

0.1

0.0

l

0.2

J

~

-""~-~-1~~--~ ] 9

l

0.3

r/D

.

l

0.4

,

9 --0-L --m-, ~ 9 --n--

--m"--m_

_~=b0.4.0"6"

]

~ ~

x/D=3.2

9 _

0.8.

9Im~l--D--.l~,l~DMM, rms

0.0

b) 1.0.

0.2-

Experiment, mean value Experiment, rms DMM1, mean value DMM1, rms

\"\'~m

9 x/D=3.2 ~m--~m~."

D~D~D~_

l.

~O--O~-T--q~-EL-CELm~ -

~'~,,

0.0"

,

0.5

0.0

0:1

0'.2

r/D

0:3

0'.4

0.5

Figure 7" The mixture fraction (a) and axial component of the velocity (b) for the regime B, Vo/Va = 5.0. On the contrary, in the case of the regime A, the obtained profiles of the mixture fraction show strong discrepancies with experimental data in the separation area at 1<x./D<3 (Figure 8). Comparing the SGS models one can conclude that the dynamic mixed models DMM and DMM1 reproduce the flow pattern better than the DGM model both qualitatively and quantitatively. For instance, near the wall (r/D>0.3) the DGM model predicts zero mixture fraction whereas measurements and dynamic models indicate the presence of the admixture (DO) in this area. The DGM model therefore gives a wrong size of the separation area and a wrong profile of the mixture fraction. Further down in streamwise direction, away from the separation area all models are in a good agreement with the experiment. b)

1.2 ~ 1.0~

Experiment, mean value /" e n t l ~ ~ Id. I 1m rms - -

.

_~.~0.6 b 0.40.01

vaJue

0.8.

~lil I--~-- Germano, rms "A'%~ I--mm--DMM, mean value \ ",,, 9 --[3-- DMM rms

0.8 ~

0.2 ~

I--A- Germano, me,,.

9.

9 --iOOi

9 9 9 9 9 9 9 9

"~m---m...m_m-m---m-m .

~ . , ~ ~ 4 ~ ~

0.0

o 0.6 .

-'1

X

o o o o o o o o

i

1.0. ' % \~. \,\ 9 ,\9

E, 0.4 :3 D---DJ~'~n~ 0.2, ,--ops163

9 Experiment, mean value --O-- Experiment, rms --A-- Germano, mean value --Z~--- Germano, rms --ram-- DMM1 mean value

--o--DMM1, r m

9

~

x/D=1.7

o 0.0'

0.1

0.2

0.3

r/D

0.4

0.5

0.0

011

0'.2

0'3

014

0.5

r/D

Figure 8: The mixture fraction (a) and axial component of the velocity (b) for the regime A, Vo/Vd = 1.3.

791

The following additional investigations directed at improving the mathematical model as well as the agreement between the theory and measurements within the recirculation area have been performed: global and local refinements of the grid, enlargement of the computational domain by including a part of the nozzle, small inclination of the nozzle axis to simulate the asymmetry caused by possible errors of the nozzle installation in the mixer, implementation of different wall functions, URANS calculations using various turbulence models, etc. All these efforts either failed or led to rather poor results. As mentioned above, the best improvement was attained by using the dynamic mixed similarity model, although the discrepancy remained rather large. A further detailed analysis of the numerical results in the recirculation area allows one to suppose that the reason of this discrepancy are the coherent structures generated in the flow within the recirculation area. Figure 9 shows the time history of the mixture fraction fj, j = 1, K , where K is the total number of time steps, obtained at different points of the jet mixer (see the snapshot of the mixture fraction field inside of the Figure 10) using LES. As can be seen, the oscillation of the mixture fraction from 0.05 to 0.55 in the recirculation area (the point at x/D=l.7, r/D=0.25) has a periodic character with pronounced dominating modes. If the frequency of LIF measurements is much lower than the frequencies of these modes and the number of samples is limited, the Reynolds averaging of the signal at this point can lead to quite different results depending on the initial sample and the step of averaging. Figure 10 shows the mixture fraction averaged accord1 ~t ing to the formulae f = ~-~7~ fN0+~i-~)M " Here N is the number of samples, N Ois the number of the inii=l

tial sample and M is the step of the averaging, To involve the points f k , k > K the time series

fk,k = 1,K was repeated many times by a simple shifting fk+m~= fk" At the point x/D-1.7, r/D-0.25 results of the averaging depend strongly on the number of the initial sample. Taking into account the fact that the frequency of the present LIF measurements was only 1 Hz at N=300 we can explain the discrepancy between the theory (DMM model) and experiment in the recirculation area by the insufficient time resolution of the LIF measurements.

Figure 9: Time history of the mixture fraction at different points in the jet mixer: 1- x/D=0.031, r/D=O.031; 2- x/D=1.7, r/D=O.031;3- x/D= 1.7, r/D=0.25; 4- x/D=5, r/D=0.25. LES Simulations with the DMM 1.

Figure 10: Averaged mixture fraction versus the number of the initial sample. M= 1000,N=300.

792 CONCLUSIONS The experimental and numerical investigations of mixing in a co-axial jet mixer were performed at different values of flow rate ratios, densities of media, Reynolds numbers and temperatures. The investigations showed that the mixing is mostly influenced by the flow rate ratio. The effect of other parameters is negligible. Two flow regimes with and without recirculation zone are investigated and compared. The flow in the recirculation zone is unsteady and followed by generation of coherent vortex structures. They influence the velocity and scalar fields and enhance the mixing drastically. The mixing length in the regime with the recirculation zone is three times less compared with that of the free jet regime. Different SGS closure models have been tested by comparing with experimental data. All models investigated provide good results for the free jet regime. For the regime with the recirculation zone there exists a discrepancy between the theory and measurements in the area of the flow separation. The best results are achieved using the dynamic mixed model (DMM). A possible reason of the discrepancy between the DMM model and experiment is the low time resolution of the LIF measurements which is not sufficient to cover the evolution of the coherent structures in the recirculation zone. ACKNOWLEDGEMENT The study was supported by the Deutsche Forschungsgemeinschaft within framework of the program SPP 1141.

REFERENCES

Barchilon M., Curtet R. (1964). Some details of the structure of an axisymmetric confined jet with backflow. J. Basic Engineering, 777-787. Danckwerts P.V. (1952). The definition and measurement of some characteristics of mixtures. Appl. Sci. Res., Sect. A3, 279-296. Germano M., Piomelli U., Moin P., Cabot W.H. (1991). A dynamic subgrid-scale eddy viscosity model. Phys. Fluids, A3, 1760-1765. Jahnke S., Komev N., Tkatchenko I., Hassel E. and Leder A., (2004). Numerical study of influence of different parameters on mixing in a coaxial jet mixer using LES, accepted for publication in

Heat and Mass Transfer. Henzler H.J. (1978). Investigations on mixing of fluids. Dissertation. RWTH Aachen. Komev N., Hassel E. (2003). A new method for generation of artificial turbulent inflow data with prescribed statistic properties for LES and DNS simulations. Schiffbauforschung, 42:4, 35-44. Korischem B. (1987). Homogenisieren yon Flassigkeiten mit Dasen. Diploma thesis, University of Dortmund. Lilly D.K.,(1992) A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4(3), 633-635. Smagorinsky J., (1963). General circulation experiments with the primitive equations. I: The basic experiment. Month. Weth. Rev. 91:3, 99-165. Tebel K.H., May H.O. (1988). Der Freistrahlreaktor - Ein effektives Reaktordesign zur Unterdrtickung von Selektivitatsverlusten durch schnelle, unerwtinschte Folgereaktionen. Chem.-Ing.Tech., 60:11, MS 1708/88. Vreman B., Geurts B., Kuerten H. (1994) On the formulation of the dynamic mixed subgrid-scale model. Phys. Fluids, 6:12, 4057-4059. Zang Y., Street R.L., Koseff J.R., (1993). A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids, A5:12, 3186-3196.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

793

EFFECTS OF ADVERSE PRESSURE GRADIENT ON HEAT TRANSFER MECHANISM IN THERMAL BOUNDARY LAYER

T. Houra I and Y. Nagano 2

1Department of Environmental Technology, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan 9Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan

ABSTRACT Characteristics of turbulent boundary layer flows with adverse pressure gradients (APG) differ significantly from those of canonical boundary layers. We have experimentally investigated the effects of APG on the heat transfer mechanism in a turbulent boundary layer developing on the uniformly heated plate. It is found that in the APG boundary layer the Stanton number follows the correlation curve for a flat plate, although the skin friction coefficient decreases drastically in comparison with zero-pressuregradient flow. The mean temperature profiles in APG flows lie below the conventional thermal law of the wall in the fully turbulent region. Moreover, the quadrant splitting and trajectory analyses reveal that the effects of APG on the thermal field are not similar to those on the velocity field. The structural change in APG flow causes the non-local interactions between the temperature fluctuations and the wall-normal motions. However, the situation is fairly complex because the heat transport is mainly determined by the ejection motions, which are not significant contributors to the momentum transport in the APG flow. KEYWORDS Turbulent Boundary Layer, Heat Transfer, Adverse Pressure Gradient, Hot- and Cold-Wire Measurement, Turbulent heat flux, Coherent Structure INTRODUCTION The efficiency of industrial machinery, such as heat exchangers and turbine blades, is often restricted by the occurrence of separation due to a pressure rise in the flow direction. It is well-known that the skin friction definitely decreases in adverse-pressure-gradient (APG) flows. However, it is still unresolved whether the heat transfer in APG flows is enhanced or not, though it is crucial in order to design highperformance heat exchangers with small drag. From the viewpoint of turbulence modeling, Bradshaw and Huang (1995) and Volino and Simon (1997) have discussed the behavior of the thermal field in the APG turbulent boundary layer. They pointed out that the velocity law of the wall is more robust

794 TABLE 1 FLOW PARAMETERS (U0=10.8 m/s, ~e= 10 K) z [mm] 935 (ZPG) 535 (APG) 735 (APG) 935 (APG)

Ue [m/s] 10.8 9.1 8.3 7.6

3~ [mm] 19.9 16.2 24.6 34.2

3t [mm] 21.0 19.4 25.3 37.1

R62 1620 1330 1930 2730

Rzx2 1920 1160 1280 1730

P+

/~

0 0.91 x l 0 -2 1.93x10 -2 2.56x10 -2

0 0.77 2.19 3.95 .

_

.

under streamwise pressure gradients than is the thermal law of the wall, although the energy equation does not contain the pressure-gradient-related term. They have investigated such important behavior of the APG thermal boundary layer by using the conventional mixing-length model and the two-equation turbulence model based on the equilibrium APG flow data (e.g., Blackwell et al., 1972). Therefore, it is very important to elucidate the effects of more realistic (non-equilibrium) APG in order to improve existing turbulence models and/or theory. In the present study, we have experimentally investigated the effects of non-equilibrium APG on a turbulent thermal boundary layer developing on a uniformly heated plate with a specially devised triple-wire probe, consisting of two symmetrically bent V-shaped hot-wires (Hishida and Nagano, 1988) and a fine cold-wire. We attempt to clarify the relationship between velocity and thermal fields with quadrant splitting (Lu and Willmarth, 1973), trajectory analyses (Nagano and Tagawa, 1995), and bispectrum measurement of turbulent transports (Nagano and Houra, 2002).

EXPERIMENTAL APPARATUS

The experimental apparatus used is the same as described in our previous studies (Nagano et al., 1998' Houra et al., 2000). For the heat transfer experiment, a flat plate, on which an air-flow turbulent boundary layer develops, is constructed with the 2-mm thick copper plate and a.c. heated from the back of the plate. The wall temperature distribution is monitored through embedded thermocouples and kept uniform during the experiment. The test section is composed of the heated flat-plate and a roof-plate to adjust pressure gradients (see Nagano et al., 1998). The aspect ratio at the inlet to the test section is 13.8 (50.7 mm high • 700 mm wide). Under the present measurement conditions, the free-stream turbulence level is below 0.1%. To generate a stable turbulent boundary layer, a row of equilateral triangle plates is located at the inlet to the test section as a tripping device. The important flow parameters are listed in Table 1. In the APG flow, the pressure gradient dC;/dz [Cp = (-P- -Po)/(p-U~/2)] keeps a nearly constant value of 0.6 m -1 over the region 65 mm _< z _< 700 mm, and then decreases slowly (z is the streamwise distance from a tripping point). The pressure gradient parameter normalized by inner variables P+ [= u(dP/dcc)/pu~] and the Clauser parameter/3 [= (3l/7-w)d-P/dz] increase monotonously, thus yielding moderate to strong APG. In order to neglect buoyancy effects, the temperature difference between the wall and ambient fluid is kept at about 10 K, which is sufficient to resolve by using a low-noise high-gain instrumentation amplifier and a 14-bit A/D converter. The variation in the wall temperature along the measurement length is within + 0.5 K of the mean. Velocity and temperature fluctuations were simultaneously measured by the hot- and cold-wire technique. The probe consisted of fine Pt (90%)/Rh (10%) wire, the upstream wire serving as the constant current cold wire (diameter 0.625 #m; length 1.0 mm) and the downstream one as the constant temperature hot wire (diameter 5 #m; length 0.8 mm), which is composed of two symmetrically bent V-shaped hot wires (Hishida and Nagano, 1988).

795 5•

'

I

'

'

St=O.O125Rff2~ pr -~ [] ZPG

10_3

9 APG

9

CI =0.025R52-0"25 9 APG

o ZPG |

[

i

103

,

R52, Rz32

4x103

Figure 1" Skin friction coefficient and Stanton number plotted against momentum thickness Reynolds number and enthalpy thickness Reynolds number

1-

'

'

'

'

'

'

'

'

'

'

1

'

'

'

I

'

'

'

'

I

'

,5" 0.5

I~ 0.5

o[] p+=P+=9.12x102 0 3 I P+= 1.93• 9 P+= 2.56x10.2 I

0

i

i

i

[

0.5

i

i

i

y/Su

i

]

~ |

o p + = 9.12• 9 P+= 1.93•

~

$ p+__2.56xlO-22

i

1

Figure 2: Mean velocity profiles in ZPG and APG flows in outer coordinates

0

0.5

Y/St

1

Figure 3" Mean temperature profiles in ZPG and APG flows in outer coordinates

RESULTS AND DISCUSSION

Statistical Characteristics Figure 1 shows the skin friction coefficient and the Stanton number plotted against the momentum thickness Reynolds number R62 and the enthalpy thickness Reynolds number Rzx2, respectively. The friction velocities in the APG flows are determined with the method of Nagano et al. (1998). As shown in Fig. 1, the skin friction coefficients decrease drastically in APG flows. The wall heat flux qw was measured from the mean temperature gradient near the wall, and the Stanton number St [= q~/(pcpU~O~)] was calculated. The measured Stanton number in ZPG flow is well correlated with the correlation curve for a flat plate (Kays and Crawford, 1993). From Fig. 1, it is found that in the APG flows, the Stanton numbers also follow this curve in a fully developed regime. Figure 2 shows the mean velocity profiles normalized by the free-stream velocity U/U~. With increasing P+, the defect in the mean velocity U from the free-stream velocity U~ becomes larger. Thus, in the outer coordinates, the mean velocity profile does not maintain self-similarity under the non-equilibrium condition. Figure 3 shows the mean temperature profiles normalized by the temperature difference between the wall and the ambient, i9/O~[= (Tw - T)/(Tw - T~)]. The abscissa is the distance from the wall normalized by the 99% thickness of the thermal boundary layer. As clearly seen from this figure, there are few effects of APG on the mean temperature profiles on the basis of this normalization. Figure 4 shows the mean velocity profiles normalized by the friction velocity u~-. It is clearly seen from this figure that the velocity profiles in APG flows lie below the following "standard" log-law profile for

796 '

+

'

3O _

'

' ' ' " l

'

'

'

' ' ' " l

'

'

'

' ' ' " 1

n P+=O

..m

p+= 9.12x10-3 a, p+= 1.93x10-z 9 p+= 2.56x10-2

30

-

........

n o . "

o

2O

9

1

9

10

_

0y/ '

|

i

,

,/

~

l|J,

........

I

10

~

~

v

,

10

I

,.,Oo o o

0 ? ~*~ ~ --~'~"

1

20 ~-~

........

(ID

~

^00

0 []

9

" ..51/

6" j

4

I

P+=O p + = 9.12x10 -3 p + = 1.93x10-2 p + = 2.56x10-2

,

,

,

O-Oi

,,,,l

,

102

,

,

y+

,

0

,,|,I

103

Figure 4: Mean velocity profiles in ZPG and APG flows in wall coordinates

1

~ ........

~ . . 9~.0~51I~~J I

10

, ....

,,ll

102

, , , .... iI

y+

10s

Figure 5: Mean temperature profiles in ZPG and APG flows in wall coordinates

ZPG flows: U+ = 2.44 in y+ + 5.0.

(1)

Moreover, in the outer region, the deviation from the log-law, i.e., the wake component, is very large. Since at the outer edge of the layer, the mean velocity conforms to the free-stream velocity, and is ex--+ pressed in wall units as U~ = v/'2/Cf, the increase in the wake component is due to the significant decrease in the skin friction coefficient (see Fig. 1). These important characteristics of the APG flows conform to our previous results (Nagano et al., 1998), and are also confirmed by direct numerical simulation (DNS) (Spalart and Watmuff, 1993) and actual measurement (Debisschop and Nieuwstadt, 1996). Figure 5 shows the mean temperature distribution O normalized by the friction temperature 0~-. In this figure, the broken and solid lines indicate the following distributions:

-0+ = Pry+,

N+

= Nt 1 in y +

+ Ct,

(2)

where ~t and Ct are 0.48 and 3.8 (for Pr -- 0.71), respectively. In the ZPG flow, the universal loglaw region for thermal fields definitely exists as previously reported by many researchers. The turbulent Prandtl number, estimated in the log region, is Prt (= ~/~ct) = 0.85. On the other hand, in the APG flows, the temperature profiles lie below the log-law profile, and the increase in the wake region generally seen in the mean velocity profiles of APG flows (see Fig. 4) disappears. These important characteristics of APG flows conform to previously reported experimental results in equilibrium APG flows (Perry et al., 1966; Blackwell et al., 1972) and well correspond to the theoretical conjecture by Bradshaw and Huang (1995), i.e., the law of the wall for a thermal field is more affected by pressure gradient than that for a velocity field. It should be noted that this reduction in the mean temperature at the outer edge of the

-+(

layer, i.e., O e

=

)

v/Cf/2/Xt , is mainly due to the reduction in the skin friction coefficient, because

the Stanton number is unchanged by imposing an APG. Figure 6 shows the profiles of Reynolds shear stress, -g-~, normalized the friction velocity, u~-. The abscissa is the distance from the wall normalized by the 99% thickness of the boundary layer, 5u. It includes

797 '

'

'

'

ZPG []

3

I

'

'

'

'

I

1.5

'

.

9

'

'

'

I

'

'

'

'

ZPG [] Present(P+=0) . O Verriopoulos (1983) 9

APG

APG 9 Present (p+= 2.56x10-2)

,,

I

00

I

'

Present(P~=0) Verriopoulos (1983) Exp. Spalaxt (1988) DNS

O

[]

0.5

Present (P+= 2.56x10 2)

[]

9

-

[]

@ []

0

y/Su

0.5

1

0

Figure 6: Reynolds shear stress in ZPG and APG flows 0.03

.

.

.

,

.

l/t,

0

~

. . . .

O O

..-

,

'

P+: 0

--

'

I

'

'

'

'

I

....

1

0.85 i~ ~

'

ZPG [] Present(P+=0) . 0 Verriopoulos (1983) APG 9 Present (P+= 2.56x10"2)

O --

~I1, 7 6 ~ ,0 1

Figure 7: Wall-normal turbulent heat flux in ZPG and APG flows

ut, 9 c~t P+=2.56x10 -2 0.02

/io

0'.5'

...........

,---,--,--;--.--;

....

o.ol

I

0 Figure 8: Eddy diffusivities for momentum and heat in ZPG and APG flows

0.5

l

i

,,

ylS~,

,

I

i

1

Figure 9: Turbulent Prandtl number distributions in ZPG and APG flows

the experimental and numerical results in ZPG flows (Verriopoulos, 1983" Spalart, 1988). With increasing P+, -~--g/u~ drastically increases in the outer region. Thus, the constant-stress-layer relationship - u v / u ~ "~ I observed in the ZPG flows is no longer valid. This, too, may account for the non-existence of the universal law of the wall in APG boundary layers. Figure 7 shows the wall-normal heat flux, -vO, normalized by the friction velocity, u,, and temperature, 0~, in ZPG and APG flows. As seen from Fig. 7, the wall-normal heat flux in APG flow is kept unchanged over the entire region compared with Reynolds shear stress in Fig. 6. In our previous experiment (Nagano et al., 1998), the intensity of the wall-normal velocity component, v, normalized by free-stream velocity, U0, at the inlet to the test section was not affected. Because the heat transfer is determined with the wall-normal motions, little effect of APG on wall-normal velocity component results in correspondingly little change in the thermal field. Figure 8 shows the eddy diffusivities for momentum and heat, ut and st, respectively, normalized by the free-stream velocity, Uc, and the displacement thickness, 51, defined as:

-u--~ /]t-

0U/0y'

-vO Ott-

0~/0y

(3)

As seen in Fig. 8, the eddy diffusivity for the momentum, ut, in the APG flow decreases in the large part of the boundary layer (y/5~, > 0.1) in comparison with the ZPG flow. On the other hand, because the mean temperature gradient has not been affected, though the wall-normal heat flux is slightly changed, the eddy diffusivity for heat, c~t, results in a small amount of decrease in APG flow in comparison with ZPG flow (see Fig. 8). Thus, as shown in Fig. 9, the turbulent Prandtl number Prt = ut/oLt in APG flow decreases in the outer layer (y/Su > 0.3), because the ut strongly decreases.

798 l '''l

'

'

'

'

''''I

'

'

'

'

''''1

"i~''I

,

,

,

'

'~'~1

'

'

'

'

r ' ' ' l

(b) P + = 2.56 X 10 -2 -

9 Q1 ~

9 Q2

.)

9

USeD

9

I~~~ ~

9 oo ~

o Q3

{D

1-

ID

or) Q3Q2

9 Q4

t)

*~Im

9 Q4

0 ,

,,,I

,

,

,

L

|

,|,|

10

t

102

l

I

,

,

y+

,,Jl

I

10a

III[

I

I

I

I

10

l l ' I [

l

*

102

,,,|[

'y4

10a

Figure 10: Fractional contributions to Reynolds shear stress from different motions ' ' ' ' I

'

'

'

' ' ' ' ' I

'

'

'

'

' ' ' ' I

' ' ' ' I

(a) p + = 0 aQ1

~

r

oQ3

'

" ' ' ' ' ' I

~

~

'

' ' ' W ' l

Q4

9

, , ,,,,,1102

,)

i~~~ ! Q2 ~IIIIII~,...$.Q3 9

,

''

]~1

"'''.

9 Q4

,,,,110

'

(b) P + = 2.56 • 10 .2

.

o

'

t~+ ,,,,,110 a

,,,,110

,

, , ,,,,,102

, Y+', ,,,,,110 a

Figure 11" Fractional contributions to wall-normal heat flux from different motions Fractional C o n t r i b u t i o n s

To understand the above features of the APG flows in more detail, we have investigated the fractional contributions of the coherent motions to Reynolds shear stress, -~--g, and wall-normal turbulent heat flux, -vO, as shown in Figs. 10 and 11, respectively. In the ZPG flow [Figs. 10 (a) and 11 (a)], Reynolds shear stress and wall-normal heat flux are dominated mainly by the Q2- and Q4-motions in comparison with the interaction motions (Q1, Q3). Moreover, in the log region the contribution of Q2-motions to both Reynolds shear stress and wall-normal heat flux is larger than that of Q4-motions. On the other hand, in the APG flow [Figs. 10 (b) and 11 (b)], the contribution of sweep motions (Q4) to Reynolds shear stress - u v become larger than that of ejection motions (Q2) in the log region. However, the contributions to the wall-normal heat flux, -vO, from the ejection motions (Q2) and sweep motions (Q4) show smaller differences, thus indicating that the effects of the APG on heat transfer are smaller than on momentum transfer. Trajectory A n a l y s i s

To clarify the changes in coherent motions in APG flows, we have conducted the trajectory analysis developed by Nagano and Tagawa (1995) and extracted the key patterns and flow modules contributing to the momentum and heat transports, which consist of three successive quadrants in the (u, v)-plane. Figure 12 shows the ensemble-averaged velocity and temperature fluctuations, <~>, <~> and <0>, and its correlations, - < ~ > , and - <~0>, of the identified sub-patterns (Q4-Q1-Q4 and Q2-Q3-Q2) in the log region. A caret denotes the normalization by the respective r.m.s, value. The time on the i-

abscissa is normalized by the Taylor time scale ~-e [= 1 2 - ~ / ( 0 u / 0 t ) 2 , which is the most appropriate L

for scaling the period of the coherent motions, irrespective of pressure gradients (Nagano et al., 1998). The mean burst periods have nearly the same value of 10 "re in both ZPG and APG flows at this location. In the APG flow, the amplitude of the ensemble-averaged fluctuations decreases in Q2-motions, and the duration of Q2-Q3-Q2 patterns becomes larger. On the other hand, the amplitude of Q4-Q1-Q4 pattern

799 ^^

( a ) l ~

<~2>

-

1~ , ~ , , . / ~ , . -

t/r E

1~ o -1

".'~ "'" .i

,

'~

,

1

1 ~ , \

,

V ,,u

t/rE

-

o1i 1

o

,

.

-...

.

t

[ v,.--

8

IrE

-~/"'~J7/3':.,', ,',s

o[,, :--,'~,, "'s

t/r E

-1

~

t ITE

<~>

- <~3~i> ,

,

t/-,-~

---"----

~"..<~~5>

-<,)~>

~...

o

,

10~ :,/~:

<~0>

<0> ~~~,

~

<5>

~

_



(b)l~__. ~

s

t/,-~

-1

tSTE

-

--Q4-Q1-Q4 . . . . Q2-Q3-Q2

--Q4-Q1-Q4 . . . . Q2-Q3-Q2

Figure 12" Ensemble-averaged characteristics of velocity and temperature fluctuations ( Q 4 - Q 1 - Q 4 and Q 2 - Q 3 - Q 2 patterns) in the log region (y+ '~ 50): (a) P+ = 0, (b) P+ = 2.56 x 10 -2 [xlO-4] 1.0~ . . . .

'

(~)

E .

05~

. . . .

'P+'

0 '2.56• '2

[xlO-4] 1.0[v (L)' t'-"

o

vu2

9

0.5 ' . .y /.S u.

1' '

'

'

'

'P+'

'0 '2.56• -2 ~

vuO

-" 9

F

~

t

~i

_0.50. . . . . . 0.5 .I . . y ./ S u

o

:

9

-

1,

, "

Figure 13: Distributions of turbulent transport in ZPG and APG flows: (a) velocity field, (b) thermal field. Velocity and temperature fluctuations are normalized by Uo and Oe, respectively becomes larger, but the duration becomes shorter. Moreover, the contribution of Q 4 - Q 1 - Q 4 becomes larger than that of Q 2 - Q 3 - Q 2 in the APG flow (Houra et al., 2000). On the other hand, for the thermal field, both of the sub-patterns ( Q 4 - Q 1 - Q 4 and Q 2 - Q 3 - Q 2 ) contribute to the wall-normal heat flux in the APG flow. Thus, both of these sub-patterns are identified as important flow modules for the heat transfer.

Turbulent

Transports

The structural differences in quasi-coherent motions reflect on higher-order turbulent statistics, especially third-order moments (Nagano and Tagawa, 1988). The distributions of turbulent transport in the velocity and thermal fields are presented in Fig. 13 (a) and (b), respectively. The definite effects of the APG are clearly seen in the velocity field, i.e., as shown in Fig. 13 (a), the positive region o f - v u v , v u 2, and v v 2 in the ZPG flows disappears at the inner region as P+ increases. Since third-order moments are predominately sensitive to the change of coherent structures such as ejections and sweeps (Nagano and Tagawa, 1988), this result indicates that internal structural changes do occur in the velocity field of APG boundary layers. Negative values of v u 2 in the near-wall to outer regions demonstrate the existence of turbulent energy transport toward the wall from the regions away from the wall. This important characteristic of the APG flows conforms to our previous result (Nagano et al., 1998), and is also consistent with the results of Bradshaw (1967), Cutler and Johnston (1989), and Skgire and Krogstad (1994). From Fig. 13 (b), it can be seen that a similar inward transfer takes place in the turbulent transport of the thermal field. However, the change in the thermal field is relatively small compared with that in the velocity field.

800

Bispectra of Turbulent Transports Because the turbulent transports consist of non-linear interactions in the turbulent motions, it is important to investigate these interactions in the Fourier space. The bispectrum is defined as the Fourier transform of the triple moment (Nagano and Houra, 2002). For example, the bispectrum B _ ~ ( f 2 , f3) of the triple moment - v u v is written as follows:

B-v~,v(f2, fa) =

-v(t)u(t + t2)v(t + ta)e-i>~(f2t2+/sta)dt2dta

(4)

(x)

and

-v(t)u(t + t2)v(t + ta) =

B-~uv(f2, fa)ei2"(f2t2+fat3)df2dfa 9

(5)

Thus, the bispectrum analyzes the frequency-dependent interactions between the frequency components at f2, fa, and f2 + fa. If we put t2 = ta = 0, the following relation is obtained:

-vuv =

B-~,~v(f2, fa)df2df3.

(6)

oo

In practice, the bispectrum is calculated using the following relation:

B_~.~(f~, f~) = -~.(f~ + A)a(f~)~(f~),

(7)

where fi(f) is the Fourier transform of a velocity fluctuation u(t). An asterisk '*' denotes a complex conjugate. Since the mean transport by turbulent motions is expressed in real values, we discuss the real part of the measured bispectrum. The time-series data analyzed are real values, and so the bispectra are symmetric at the origin in the (f2, f3)-plane. Thus, half of (f2, fa)-plane is sufficient to describe the bispectrum. Figures 14 and 15 show the measured bispectra, B _ ~ ( f 2 , f3), of the turbulent transport of the Reynolds shear stress, -vuv, and the bispectra, B-~vo(f2, f3), of the turbulent transport of the wall-normal heat flux, -vvO, respectively, in the ZPG flow (y/6~ ~ 0.2). In a similar way, Figs. 16 and 17 show the measured bispectra, B_v~,~(f2, f3) and B-v~o(f2, f3), respectively, in the APG flows (y/5~ "~ 0.2). The left figures, labeled (a), show the sum-frequency interaction fl = f2 + f3, i.e., non-linear interaction between ~?(f2 + f3), ~(f2) and ~?(f3) in Figs. 14 and 16. On the other hand, the right figures, labeled (b), show the difference-frequency interaction fl - I f 2 [ - If3], i.e., non-linear interaction between v(]f2l ]f31), ~(f2) and 5(f3) in Figs. 14 and 16. Because we need to show the net contribution to the bispectra on the frequency axes at the log-scale, the bispectra multiplied by frequencies, e.g., If2f3lB-~(f2, f3), are shown. The frequencies in the axes are normalized by the Taylor time scale, f' = fTE. The normalized frequency of the bursting events in the near-wall region corresponds to about 0.1 (Nagano et al., 1998). In the figures, the solid and broken contour lines show positive and negative values, respectively. In the ZPG flow, strong positive regions for the difference-frequency interactions in both velocity and thermal fields [Figs. 14 (b) and 15(b)] are seen, along the same frequency, i.e., on the diagonal line of the (f2, f3)-plane over the frequency range from 0.1 to 1.0, and the positive values (B-v~v > 0 and B_v~o > 0) indicate that non-linear transport from the wall to the outer regions is dominant. In Figs. 14 (a) and 15 (a), slight positive regions are discernible within the low frequencies; however, the transport associated with the sum-frequency interaction is very small. On the other hand, for the velocity field in the APG flow, non-local interactions occur that consist of streamwise fluctuating velocity with low frequency and wall-normal velocity with high frequency, as

801 (a) 1

t

-A 0.1

t . . . . . . .

|

0.1

9

0.1

0.1

0"11

7~.....i

0.1

f~

1

IAAIB . . . . (A,A) Figure 14: Bispectra o f - v u v in ZPG flow (y/Su ~ 0.2): (a) sum-frequency interaction, (b) difference-frequency interaction

(a) ] ........ ..::::::;....... 1 .......;;;..........:;;;-..;;"::.-.-:

-A ........ , 0.1

f~

1

0.1

f~

1

IAY~lB-,,vO(A, A) Figure 15: Bispectra o f - v v O in ZPG flow (y/5~, ~_ 0.2): (a) sum-frequency interaction, (b) difference-frequency interaction

(b) -...........:%1:::....... .........

f,"3 ~:ii i!i""::i(:i;!:!i ]:i:i:~i!i."i:.-.i:i.i_: .,~L. .: ;! ii o.,

......

0.1

f~

1

~ :'

.......

........: 9

0.1

..

f~

1

IAAIB . . . . (A, f4) Figure 16: Bispectra o f - v u v in APG flow (y/5~, '~ 0.2): (a) sum-frequency interaction, (b) difference-frequency interaction

0.1

:!' f~ 1

0.1

f~

1

IAf~lB-vvo(A, A) Figure 17: Bispectra o f - v v O in APG flow (y/Su "~ 0.2): (a) sum-frequency interaction, (b) difference-frequency interaction

seen in Fig. 16. The fluctuation fi(f~) with low frequency f~ _~ 0.06 and the fluctuation ~3(f~) with high frequency f~ "-~ 0.6 interact with both ~(f~ + f~) and v ( [ f ~ l - [f~[), and wallward turbulent transfer (B-v~,v < 0) occurs. This negative contribution to the bispectra corresponds to the sweep motions associated with high-frequency wall-normal fluctuations (Q4-Q1-Q4 pattern) in the near-wall region of the APG flow. A similar wallward transfer (B_v~o < 0) with high-frequency wall-normal velocity fluctuation ~(f~ _ 0.6) is observed for the thermal field in the APG flow (Fig. 17). However, along the diagonal line of the (fg, fa)-plane over the higher frequency range (f~ _~ f~ > 0.2), both positive and negative contributions are seen in Fig. 17. Thus, for the thermal field, the turbulence transport is fairly complex.

CONCLUSIONS Experimental investigation has been made on non-equilibrium turbulent boundary layers subjected to adverse pressure gradients developing on the uniformly heated flat wall. The results can be summarized as follows: (1) In the APG boundary layer, the Stanton number follows the correlation curve for a flat plate, although the skin friction coefficient decreases drastically in comparison with ZPG flow. The conventional thermal law of the wall does not hold in the fully turbulent region. Moreover, turbulent Prandtl number decreases in the fully turbulent region, i.e., the eddy diffusivity for heat becomes much larger than that for momentum. (2) The quadrant splitting and trajectory analyses reveal that the effects of APG on the thermal field are not similar to those on the velocity field. Both the ejection- and sweep-motions contribute significantly to the heat transport in the APG flow, though the sweep motions whose durations become shorter are the main contributors to the momentum transfer.

802 (3) In the APG flow, the turbulent transports for the momentum and heat occur in the direction toward the wall from the region away from the wall. The structural change in APG flow causes the non-local interactions between the temperature fluctuations and the wall-normal motions. The situation is fairly complex. This work was partially supported by the Nitto Foundation and Japan Society for the Promotion of Science (JSPS) through Grant-in-Aid for both Scientific Research (B) (No. 13450083) and Young Scientists (B) (No. 14750136).

REFERENCES Blackwell B.E, Kays W.M. and Moffat R.J. (1972). The Turbulent Boundary Layer on a Porous Plate: An Experimental Study on the Heat Transfer Behavior with Adverse Pressure Gradients. Thermosciences Div., Dept. of Mechanical Engineering, Stanford Univ., Rept. HMT-16, Stanford, CA. Bradshaw E (1967). The Turbulence Structure of Equilibrium Boundary Layers. J. Fluid Mech. 29, 625-645. Bradshaw E and Huang EG. (1995). The Law of the Wall in Turbulent Flow. Proc. R. Soc. Lond. A 451, 165-188. Cutler A.D. and Johnston J.E (1989). The Relaxation of a Turbulent Boundary Layer in an Adverse Pressure Gradient. J. Fluid Mech. 200, 367-387. Debisschop J.R. and Nieuwstadt ET.M. (1996). Turbulent Boundary Layer in an Adverse Pressure Gradient: Effectiveness of Riblets. AIAA J. 34, 932-937. Hishida M. and Nagano Y. (1988). Turbulence Measurements with Symmetrically Bent V-Shaped HotWires. Part 1: Principles of Operation. Trans. ASME, J. Fluid Engineering 110, 264-269. Houra T., Tsuji T. and Nagano Y. (2000). Effects of Adverse Pressure Gradient on Quasi-Coherent Structures in Turbulent Boundary Layer. Int. J. Heat Fluid Flow 21,304-311. Kays W.M. and Crawford M.E. (1993). Convective Heat and Mass Transfer, Third Edition, New York, McGraw-Hill, Inc. Lu S.S. and Willmarth, W.W. (1973). Measurements of the Structure of the Reynolds Stress in a Turbulent Boundary Layer. J. Fluid Mech. 60, 481-511. Nagano Y. and Houra, T. (2002). Higher-Order Moments and Spectra of Velocity Fluctuations in Adverse-Pressure-Gradient Turbulent Boundary Layer. Exp. Fluids 33, 22-30. Nagano Y. and Tagawa M. (1988). Statistical Characteristics of Wall Turbulence with a Passive Scalar. J. Fluid Mech. 196, 157-185. Nagano Y. and Tagawa M. (1995). Coherent Motions and Heat Transfer in a Wall Turbulent Shear Flow. J. Fluid Mech. 305, 127-157. Nagano Y., Tsuji T. and Houra, T. (1998). Structure of Turbulent Boundary Layer Subjected to Adverse Pressure Gradient. Int. J. Heat Fluid Flow 19, 563-572. Perry A.E. Bell J.B. and Joubert EN. (1966). Velocity and Temperature Profiles in Adverse Pressure Gradient Turbulent Boundary Layer. J. Fluid Mech. 25, 299-320. Skfire EE. and Krogstad E-A,. (1994). A Turbulent Equilibrium Boundary Layer near Separation. J. Fluid Mech. 272, 319-348. Spalart E R. (1988). Direct Simulation of a Turbulent Boundary Layer up to R0=1410. J. Fluid Mech. 187, 61-98. Spalart ER. and Watmuff J.H. (1993) Experimental and Numerical Study of a Turbulent Boundary Layer with Pressure Gradients. J. Fluid Mech. 249, 337-371. Verriopoulos C.A. (1983). Effects of Convex Surface Curvature on Heat Transfer in Turbulent Flow. Ph.D Thesis, Imperial College. Volino R.J. and Simon T.W. (1997). Velocity and Temperature Profiles in Turbulent Boundary Layer Flows Experiencing Streamwise Pressure Gradients. Trans. ASME, J. Heat Transfer 119, 433-439.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

803

STOCHASTIC MODELLING OF CONJUGATE HEAT TRANSFER IN NEAR-WALL TURBULENCE

Jacek Pozorski 1 and Jean-Pierre Minier 2 1 Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Fiszera 14, 80952 Gdafisk, Poland 2 Research & Development Division, Electricit6 de France 6 quai Watier, 78400 Chatou, France

ABSTRACT The paper addresses the conjugate heat transfer in turbulent flows where temperature is assumed to be a passive scalar. The Lagrangian approach is applied and the heat transfer is modelled with the use of stochastic particles. The intensity of thermal fluctuations in near-wall turbulence is determined from the scalar probability density function (PDF) with externally-provided dynamical statistics. A stochastic model for the temperature field in the wall material is proposed and boundary conditions for stochastic particles at the solid-fluid interface are formulated. The heated channel flow with finite-thickness walls is considered as a validation case. Computation results for the mean temperature profiles and the variance of thermal fluctuations are presented and compared with available DNS data.

KEYWORDS near-wall turbulence, conjugate heat transfer, thermal fluctuations, stochastic modelling

INTRODUCTION The issue of heat transfer in turbulent flows is of utmost practical importance in many engineering problems. Yet, physically-sound turbulence modelling at an acceptable computational cost is inherently difficult (Launder 1996, Nagano 2002). Detailed physics includes a complex evolution of the temperature field due to the near-wall vortical flow structures but the statistical approach is still very useful in practical computations. As discussed in the literature (Sommer et al. 1994, Kong et al. 2000, Tiselj et al. 2001 a,b), the type of wall boundary condition (b.c.) for temperature (isothermal, isoflux, conjugate heat transfer) directly influences the intensity of thermal fluctuations at the fluid-solid interface. Physically,

804 the intensity depends also on the molecular Prandtl number and on the wall material characteristics. At the level of statistical averages, the temperature variance, the turbulent heat flux, and the near-wall temperature spectra are of direct engineering interest. A motivation to undertake the present work originated from some industrial situations of conjugate heat transfer where a fairly detailed estimation of dynamical and thermal statistics of turbulence in the near-wall region is desirable for optimal design purposes and avoiding thermal fatigue of wall material, since the mechanical integrity of certain solid structures can be impaired by large and/or rapid temperature changes induced by neighboring fluid. The conjugate heat transfer problem has to be formulated in a coupled way, with the flow solution and the temperature field on the fluid side coupled to an unsteady heat conduction in the solid. The experimental and numerical studies of this general case are quite rare. Mosyak et aI. (2001) carried out an experimental study of the wall temperature fluctuations under different b.c.; they also examined coherent structures of the temperature field (thermal streaks). Tiselj et al. (2001 b) performed a DNS of the turbulent heat transfer with unsteady heat conduction in the solid; they reported a number of results, including the dependence of the r.m.s, temperature fluctuations on several relevant parameters. Concerning theoretical studies, Polyakov (1974) solved a simplified problem for temperature fluctuations in the viscous sublayer of turbulent flow next to a heat-conductive wall; therefrom, he derived analytical expressions for the near-wall frequency spectrum of temperature and predicted the wall level of temperature fluctuations, 0rms. The inverses of the dominant frequencies in the temperature spectrum (characteristic time scale) were found to fall roughly in the interval of 100-300 in wall units. To the best of these authors' knowledge, statistical (RANS) modelling attempts for turbulent flows with the wall boundary conditions different than isothermal are rare and, apparently, none resolved the temperature fluctuations in the wall material itself. Sommer et al. (1994) studied the sensitivity of near-wall thermal fluctuations in fluid to the type of boundary; yet, they took the prediction of Polyakov (1974) as wall b.c. for the <02) equation in the iso-flux case. In the present paper, the conjugate heat transfer problem is formulated in the stochastic Lagrangian approach. One building block of the modelling scheme proposed here is the scalar probability density function (PDF) method (cf. Pope 2000) with down to the wall integration (Pozorski et al. 2004), applied in fluid with the use of externally-provided turbulence statistics; the temperature is assumed to be a dynamically passive scalar. Another building block are model developments to estimate temperature fluctuations within the wall material. The reason for exploring the Lagrangian stochastic approach here is twofold. First, it allows us to obtain quantitative results for the temperature statistics both in fluid and in solid material; as a spin-off, closed Eulerian equations for the temperature variance are obtained. Second, the methodology developed here is readily extended to the case of structural turbulence models with instantaneous flow structures resolved (cf. conclusion).

GOVERNING EQUATIONS Under suitable conditions, the internal energy balance in the flow reduces to the diffusion-advection equation obeyed by instantaneous fluid temperature as an additional, passive scalar variable: OT

+ uj

OT

-

02T

( )

The instantaneous temperature in the solid (the wall material) obeys the unsteady heat equation OT 02T Ot = ~2 0xjOx----~ "

(2)

The subscripts 1 and 2 refer to fluid and solid, respectively; in particular, al and c~2 stand for the respective thermal diffusivity coefficients, o~ = )~/(pcp).

805 The geometric configuration considered in the paper is a flat fluid-solid interface, i.e. the plane y = 0, with solid material located in the region y < 0. Consistency conditions at the interface are the continuity of the instantaneous temperature and of the heat flux T[v:o+ - T y:o-

OT

and

OT

At-El2.v=o+ : A2-k-Ej.Iv:o- 9 uy uy

(3)

For future use, the so-called thermal activity ratio is introduced as:

K= i(pcpA)~__ a~ (pcp~)2 A2

(4)

The mean and fluctuating temperatures in fluid and solid are introduced through T = (T) + 0. In the statistical description of turbulent fluid motion, details of the (otherwise deterministic) relevant fields are skipped and only averaged variables are preserved. Notwithstanding the zero velocity field, the same concept of averaging is applied to the temperature in solid and proves useful upon a closer examination. There, the stochastic character of the temperature field originates from a random forcing exerted by the fluid on the separating interface. The meaning of ensemble averages (-) in solid is possibly best explained as the time average, like in the streaky-structure model of Kasagi et al. (1989). Concerning the notion of fluctuations in fluid, averaging over realisations is best understood in the PDF approach, the time average or averaging over directions of homogeneity are mainly used in DNS/LES, while only statistics of fluctuations make sense as variables in RANS. Although the moment (or mean) equations are not solved in the stochastic approach, they will be recalled here for the sake of completeness. Due to linearity of Eq. (2), the mean temperature equation in the solid requires no specific closure. However, this is not the case of higher-order moments. The exact equation for the fluctuating temperature variance (02) is derived from Eq. (2) in the same way as done for turbulent flow; yet, in solid it has a simpler form

o(o ot

2 (oo oo =

o jo j

) .

(5)

The first RHS term of (5) represents the molecular diffusion whereas the second (unclosed) term stands for the destruction rate of {02).

STOCHASTIC M O D E L L I N G OF T H E R M A L FLUCTUATIONS IN SOLID Before a stochastic approach to the conjugate heat transfer is detailed, we consider first the issue of temperature fluctuations within a layer of solid material only; the fluctuations are driven by a variable temperature at the boundary. To start with, let us recall the analytical solution of the following problem: determine the time-evolving temperature field T(y, t) in a semi-infinite solid (here: y > 0) with the initial condition T(y, t - 0) - 0, driven by the time-periodic changes of temperature (with a magnitude O0) at its surface T(y - 0, t) = ~ [(90e-~t] = O0 cos w t . (6)

(NB: although only real parts of relevant quantities have a physical meaning, the use of complex variables simplifies the notation and the solution procedure.) The task is in fact formally identical to the classical problem of semi-infinite viscous fluid adjacent to an oscillating wall. We assume that the oscillation period 27r/w is long enough so as not to defy the Fourier hypothesis on heat transfer. The solution of (2) is readily found; it is written for the fluctuating temperature since the mean is zero everywhere:

0(y, t ) -

-

cos Y

-

(7)

806 where 6 (2Ct2/OJ)1/2 is the penetration depth and k = (1 + i)/6 is the corresponding (complex) wave number. Two other related problems of interest consist in temperature propagation in a finite layer of solid of thickness d with the same b.c. on the surface y = 0 as before and with the b.c. on the backside stated as either isothermal or adiabatic: =

T(y=d,t)

=0

(OT/Oy)(y=d,t)=O.

or

(8)

As the mean temperature in solid is constant, the respective solutions for the fluctuating temperature are:

[

O(y, t) = N O0

sink(d y) e_iOot sin kd

and

O(y, t) - ~

[

@o

cos kd

(9)

Let us focus now on the semi-infinite case for detailed analytical expressions. The time averaging of 0 2 over the oscillation period 2Ir/co yields the variance of thermal fluctuations. It is computed from Eq. (7): (0~(y, t)} -

10~e-2V/a

(10)

Analogously, the destruction rate of (0 2)/2, cf. Eq. (5), is computed:

( 00 00 a2

a2

~zj~zj} -- O~e-2v/S(~ 2

(02) = d2/2ct 2 .

(11)

It is readily verified from Eqs. (10) and (1 1) that the balance of Eq. (5) is identically satisfied. Moreover, as it transpires from (11), the destruction rate can be expressed in the form (0 2}/70, with the characteristic time scale being 7-o = T0,~ where 7-0,~ = 1/w (the subscript oc denotes the value obtained for the semiinfinite layer of solid). This observation may seem trivial since in the problem considered cJ is the only parameter that can determine the time scale; however, the form of the destruction term directly suggests a possible model for particle simulation, i.e., a relaxation of the instantaneous temperature towards the mean (here: zero) with a time scale 7-0, cf. Eq. (13) below. In the Lagrangian particle approach, the motion of discrete portions of internal energy (particles) is tracked, cf. Pozorski et al. (2004) for a detailed explanation. In a statistically 1D setting, particles are ascribed the instantaneous temperature T and location Y; formally, the one-point scalar PDF fT~" (O, y, t) can be introduced, cf. Pope (2000). Roughly speaking, the heat equation in solid is solved by the random walk of particles and the thermal fluctuations (due to forcing at the solid-fluid interface) are simulated by changing the particle temperature. The locations and temperatures of stochastic particles evolve as:

dY

=

2x/~2 d W ,

(12)

dT

=

_ T - (T_~__~)dt To

(13)

where d W is an increment of the Wiener process. To the system of particle equations corresponds a closed evolution equation for the PDE The mean heat equation and a balance of the temperature variance (also higher-order momentes, if needed) are retrieved from it upon suitable integration. The Monte Carlo computation of the temperature inside the wall material is performed for the finite layer d of solid. The inlet boundary conditions at the interface y - 0 are formulated for particles that leave the computational domain on this side and are reflected back. The b.c. are meant to yield the correct temperature variance there (equal to O~/2)" this is achieved by setting the temperature of incoming particles to 00 Tin - ---~~N or Tin - O0 cos(27r~u) . (14) v'v,

807

The first expression assures that the temperature PDF of incoming particles is Gaussian (~x is a random number from the standard normal distribution). As an alternative, the second expression uses the forcing with a random phase (~u is a random number from the standard uniform distribution). We have tested in Monte Carlo computations that the resulting profile of {02) (y) is not affected by either choice of b.c. from formula (14). On the other boundary (y - d) two variants of boundary conditions, isothermal and adiabatic, have been imposed. In particle variables, this is respectively done as: or

Tin - - 0

Tin

-- Tou t .

(15)

For both boundaries, y -- 0 and y - d, the reflected Brownian motion has been accounted for by applying the b.c. to the particles that left the domain at a given time step but also (with a non-zero probability) to some of those that did not (i.e., those that left out and returned during the time step). To get a better insight, the simulations have been performed for three different cases: d >> 5, d ~ 6, and d << 6. Preliminary results (not shown here) for the decay of fluctuating temperature variance with depth present a growing discrepancy with respect to analytical predictions with the decreasing ratio of d/5. They have been obtained with a constant time scale ~0,~ in the scalar mixing model (otherwise correct in the case d - ~ ) . A remedy proposed here consists in taking a variable time scale, T0 = 7-o(y), determined as follows. The temperature relaxation term in the evolution of stochastic particles, Eq. (13), produces a modelled destruction term -2(02)/To in the transport equation for the temperature variance. Comparison with the exact equation for (02), Eq. (5), leads to the prescription of the relaxation time: To-

00 00

<0

]-i

(16)

For the isothermal and adiabatic b.c. on the back surface (y = d) the respective formulae are found from Eq. (16) with the use of Eq. (9)" 1 sink(d-y)]2 7-0- ~ a2 sin kd ]

kcosk(d-y) sin kd

-2

and

To=

1

cosk(d-y)

o~2

cos

kd

2

ksink(d-y) cos kd

-2

(17) The resulting profiles of the time scale To(y), normalised by the forcing frequency ~, are shown in Fig. 1 for the isothermal and adiabatic condition on the back surface and for three different ratios of the layer thickness d to the penetration depth 6. It is readily noticed that "r0 considerably departs from 1/~, specially for small ratios of d/5. At the backside wall (i.e., in the limit y --+ d) the time scale tends asymptotically either to zero (isothermal wall, rapid attenuation of any fluctuations there) or to infinity (adiabatic wall, non-zero fluctuations can persist there). The prescription of Eq. (16) makes it now possible to close also the Eulerian equation for the temperature fluctuations in solid, Eq. (5), as

0(0 Ot

0 (0

2

OXjOXj

TO

= oz2- -

(02>

(18)

where the time scale from Eq. (17) has the form T0 = To(w, y/d, d/6), shown in Fig. 1 and dependent also on the thermal b.c. type. Basically, Eq. (18) can be used in RANS, specially for routine computations (when the temperature PDF is not needed) that certainly are less CPU-intensive than the Monte Carlo method used here. The PDF computation results of the model (12)-(13) with To(y), Eq. (17), are presented in Fig. 2. The theoretical predictions of (02) resulting from Eqs. (9) are quite well matched. We emphasize that the spatially-variable relaxation time scale for temperature in solid material is meant to assure an approximately correct attenuation of thermal fluctuations that are due to external forcing; arguably, it is a necessary ingredient of a local (one-point) model. Otherwise, the position-dependent relaxation rate in a

808 10 2

/

101

|

10 0

3 10 -1

10 -2

0.0

0.2

0.4

y/d

0.6

0.8

1.0

Figure 1: Analytical profiles of the position-dependent time scale 7o (normalised with the forcing frequency co). Various ratios of the layer thickness d to the penetration depth 3:d/~5 = 0.5 (dashed lines), d/6 = 2 (solid lines), did = 25 (dotted lines), d/5 = oc (horizontal dash-dotted line). Ascending lines: upper boundary (y = d) adiabatic, descending lines: upper boundary isothermal. homogeneous material has no sound physical explanation in terms of underlying small-scale phenomena in solid. Yet, our main aim here has been to construct a model that deals with the solid part of a coupled fluid-solid system. The model will be given a posteriori support through the conjugate heat transfer computations.

0.8

0.8

0.8

0.6

0.6

0.6

r 0.4

~o,1 0.4

0.2

0.2

0.0

0.2

0.4

y/d

0.6

0.8

1.0

0.0

0.0

,

9

,

1.0

1.0

0.0

.

. . . . .

1.0

~e4 0.4

, \ \ ~ : x

0.2

~"~... 0.2

0.4

y/d

|

0.6

0.8

.0

0.0

0.0

'

J

0.2

0.4

y/d

0.6

0.8

1.0

Figure 2: Fluctuating temperature variance in the solid layer with forced lower boundary (y = 0). Computation with the position-dependent time scale 7-o and various ratios of the layer thickness to the penetration depth: a) d/6 = 5, b) d/6 = 2, c) d/(~ = 0.5. Simulation results with boundary ( y = d ) isothermal (solid lines) or adiabatic (dashed lines). Symbols (-, l): respective theoretical predictions.

STOCHASTIC PARTICLE A P P R O A C H F O R C O N J U G A T E HEAT T R A N S F E R A concept proposed here is to solve the conjugate heat transfer problem in a coupled Monte Carlo approach. Stochastic particles carry a certain amount of internal energy and represent both the heat transfer (convective and conductive) in fluid and the heat conduction in solid. This point is illustrated in Fig. 3. The building blocks of the approach are: the scalar PDF method for near-wall turbulence (Pozorski et al. 2004), a stochastic method for unsteady heat conduction in solid (cf. previous section), and the statement of appropriate thermal consistency conditions for particle temperature at the wall (at the solid-fluid interface).

809

Figure 3: Schematic picture: stochastic particle method for the coupled heat transfer at the solid-fluid interface (shaded area: solid material). The model for thermal fluctuations in the solid, introduced in the previous section, needs a forcing frequency as an external parameter. It characterises near-wall vortical structures in turbulent flow. The DNS of Tiselj et al. (2001b) for Pr = 7 provides some indications as to the choice of w. For given thermal activity ratio K -- 1, and the thickness of solid layer d ++ = 20, we have compared the profile of the thermal fluctuation intensity 0rms - - (02)1/2 in the solid, as given by the DNS, with the prediction resulting from Eq. (9) where 6 is uniquely determined by w and a2; in wall units (5++) 2 =

2 Pr w + "

(19)

(NB: the non-dimensional length in solid is introduced as y++ = y+ V/a~/a2.) The comparison gives the correct level of (02) at the backside wall for d/5 ~ 3.2-3.8. Hence r + = 1/w + ,,~ 100-140. Finally, we went for the time scale (that is an external input data for the stochastic model) of-l-+ - 1/w + - 120. The corresponding penetration depth results from Eq. (19): 5 ++ = 5.75, keeping in mind that this particular value is charged with uncertainty of about 10%. The scalar PDF approach for the fluid side is applied with further developments for the near-wall region as detailed in Pozorski et al. (2004). Stochastic particles in fluid are governed by equations dXi

=

(Ui) dt + ~ -

+

dt + V/2(a~ + at) dWi ,

1)

( r - (T)) dt + d T a ,

TM

(20) (21)

where % is the scalar mixing time scale, "rM accounts for the near-wall molecular transport effects, and at is the turbulent heat diffusivity. In the scalar PDF approach, at = ut/Prt is determined from the turbulent viscosity (externally provided) and the profile of the turbulent Prandtl number that depends on the type of the wall boundary condition, cf. Pozorski et al. (2004). Otherwise, no external input is needed in the standalone, joint velocity-scalar PDF approach to turbulent flows with heat transfer (Pozorski et al. 2003a, 2003b, 2004). Stochastic particles in solid evolve according to dX~

=

dT

=

2x/~2 dW~ , _ T - (T______~dt ) + dTx 7o

(22) (23)

where T0 is determined from Eq. (17b) for the case of the adiabatic backside wall considered here. In both Eqs. (21) and (23) the term dTx in homogeneous media (constant ,~1 and )~2) matters only for particles

810

that cross the interface at a given time step. It can be derived from the constant heat flux condition at the interface (details not presented here). In particular, for the 1D conjugate case with a fiat solid-fluid interface, the jump of particle temperature at the interface is determined as

K---~

dT),=-~

--~-y d~l

(24)

where g) is the scaled wall-normal coordinate: ~) = yv/ozl/o~2 in solid y < 0, while ~) = y in fluid y > 0. The boundary condition at the lower part of the solid, y = - d say (the bottom horizontal line in Fig. 3), is implemented here as the constant flux condition.

COMPUTATION RESULTS F O R HEATED CHANNEL F L O W The near-wall region of a fully-developed, heated channel flow at Rer- = 180 (based on the friction velocity and the channel half-width) bounded by solid walls of a finite thickness is considered as the validation case. The computational domain extends from the channel centerline (y+ = 180) through the fluid-solid interface (y = 0), down to the outer wall at g++ = - 2 0 (the lowest horizontal line in Fig. 3). Scalar PDF computations of conjugate heat transfer in near-wall turbulence have been performed for several choices of relevant physical parameters; the second-order thermal statistics (the fluctuating temperature variance) are of particular interest. The sensitivity of results to the molecular Prandtl number, the thermal boundary condition on the interface (determined by the thermal activity ratio of fluid and solid), and the solid wall thickness has been studied. Some qualitative results are reported first. Figure 4(a) shows computed non-dimensional mean temperature profiles in solid and fluid for some values of the thermal activity ratio K. Obviously, the temperature varies linearly in solid, and the behaviour in fluid is alike that previously computed for limiting (isothermal and isoflux) cases. This fact is well supported by the DNS. Next, Fig. 4(b) presents a profile of the fluctuating temperature r.m.s. (normalised by the wall level) for K = 10. In turbulent flow, the profile is similar to that computed previously for fluid-only case. On the other hand, the temperature r.m.s, in solid decays roughly exponentially with depth and then levels off (because of the isoflux thermal condition at the external wall). i

i

1

~-- 2~--~--5i

/"'

i

..... i./,

0.8

1.5

'

. . . . i

r

i

':

i

i

i

.......... i ........... i ............ ; ......... i ........... i ........... i ........ i .......... 1

0.6

l lS

.

/,i It." I--

0.4

t ~

'f

I

0.2

0

.

.

.

)

.

:

: .....

,

i

:

!

~. "

Ii

i

i

i

,

~

I!

i

;

~

!

i

!

0

20

40

60

80

100

120

/

I

:

:

':

i

:

.

=

i

0.5

z.ii ....... i ........ i ........... i ............ i ............. :........... i .................... i ........

/i -20

y§

140

160

180

0

-20

0

20

y§

40

60

80

Figure 4: Heated turbulent channel flow at Re~. = 180 bounded by finite thickness (d ++ - 20) solid walls, a) Mean temperature profile: isothermal wall (dotted line), finite thermal activity ratio: K = 1 (dashed), K = 5 (solid), K = 10 (dash-dotted); b) fluctuating temperature r.m.s, in fluid and solid (K = 10). Results made nondimensional with the external wall temperature TE, channel centerline temperature TeL, and the r.m.s, fluctuating temperature at the wall 0w, respectively. As far as quantitative comparison is concerned, computational results of our conjugate heat transfer model are validated against the data of Tiselj et al. (2001 b) from the DNS in fluid coupled with unsteady

811 heat conduction in solid. The comparison indicates that there is a fair agreement between the two; yet, some discrepancies persist. The r.m.s, temperature fluctuations in fluid and in solid are shown in Fig. 5 for Pr -- 7, a given wall depth (d ++ -- 20), and varying K. On the fluid side, the maximum in the buffer layer is shifted towards the wall. In the solid, the decay of thermal fluctuations is too fast as compared to the DNS; yet, the solid-only stochastic model shown previously has produced very good agreement. The reason for the discrepancy may be twofold. Contrary to the previous case, there is now a mean temperature gradient in the wall material that may have some influence. Moreover, all particles that penetrate the solid from the fluid side enter with their own instantaneous temperature; in the previous stochastic model for the solid only, the temperature of those particles was prescribed from a certain distribution law to preserve correct r.m.s, fluctuating temperature level at the wall. Both these issues need a further insight and are currently worked on. 2

........

,

........

10 8 .6

|

........

~

...y_

l,

6

~ ~

E

~4 4

" "'" 0

.

10

0

-1

'

1

' ' ' ' ''"00

........

y+

1

01|

.......

1

02

0

0

.

.

.

t

5

.

o

.

.

.

I~

i

9

10 Y + +

9 ~ .

.

.

,

O" I

15

,

,

,

,

20

Figure 5' Heated turbulent channel flow at Re~-= 150, Pr = 7, and the wall thickness d ++ = 20. The r.m.s, temperature fluctuations: (a) in fluid, (b) in solid. Symbols: DNS data (Tiselj et al. 200 lb) for the thermal activity ratio K = 5 (0), K = 1 (o). Lines with corresponding open symbols" PDF computation.

CONCLUSION New modelling proposals for turbulent flows with the temperature field have been advanced in the paper. A stochastic particle model has been proposed for thermal fluctuations in solid material; they are generated by a forcing on its surface. As a spin-off of the analysis, a modelled equation for thermal fluctuation variance in solid has been obtained; it can be solved, together with a statistical turbulence model for fluid, in the Eulerian approach to conjugate heat transfer, alleviating thus the need of prescribing the level of temperature variance at the solid-fluid interface. Yet, since the objectives of the present contribution are different, such a RANS computation has not been attempted here. Instead, the stochastic approach has been followed both in modelling and for numerical computations. Computation results for a solid layer compare favourably with analytical predictions of the decay of fluctuating temperature variance with depth. The model represents a first building block of a general stochastic approach to the conjugate heat transfer. In the one-point scalar PDF method proposed here, the external input parameter to the model, i.e. the forcing frequency (or the appropriate near-wall time scale of thermal fluctuations) has to be estimated. The DNS data of conjugate heat transfer (Tiselj et al. 2001b), in particular the decay of the r.m.s, fluctuating temperature in solid material, provide a convincing support for the time scale and to the stochastic modelling with a single forcing frequency that is also a dominant frequency in temperature spectrum at the wall. Interface conditions (i.e., consistency conditions for temperature statistics on the fluid-solid dividing surface) have been formulated in particle setting. The complete model for fluid and

812 solid yields reasonable results (compared to recent DNS) for the mean temperature and the fluctuating temperature variance in a coupled case for the heated channel flow with finite-thickness walls. Inevitably, for the temperature considered as a passive scalar the temperature frequency spectrum at the wall heavily depends on that of the wall-normal velocity. Any stochastic model able to provide an estimation of the temperature spectrum must contain some information on turbulence structure and thus goes beyond the one-point closure exploited here. The departure point for further studies in that direction may be an interesting structural turbulence model in unsteady 2D formulation with streamwise streaky structures (Kasagi et al. 1989). In a similar vein, a stochastic approach with large-scale or the scalar filtered density function (FDF) model combined with the dynamics of the large-eddy velocity modes (such as POD) in the near-wall region has recently been proposed (Waciawczyk & Pozorski 2004). It provides insight into instantaneous structures of flow and thermal fields at varying Prandtl numbers and can hopefully be refined for conjugate heat transfer phenomena. To take a full advantage of the PDF approach (exact representation of convective and chemical source terms), further developments include the FDF formulation with velocity dynamics computed from LES and the modelling of heterogeneous chemical reactors in turbulent flow, e.g., surface reactions of catalytic type.

References Kasagi N., Kuroda A. and Hirata M. (1989). Numerical investigation of near-wall turbulent heat transfer taking into account the unsteady heat conduction in the solid wall. J. Heat Transfer 111,385-392. Kong H., Choi H. and Lee J.S. (2000). Direct numerical simulation of turbulent thermal boundary layers. Phys. Fluids 12, 2555-2568. Launder B.E. (1996). An introduction to single-point closure methodology. In: Simulation and modeling of turbulentflows (Eds. T.B. Gatski, M.Y. Hussaini and J.L. Lumley), Oxford Univ. Press, pp. 243-310. Mosyak A., Pogrebnyak E. and Hetsroni G. (2001). Effect of constant heat flux boundary condition on wall temperature fluctuation. J. Heat Transfer 123, 213-218. Nagano Y. (2002). Modelling heat transfer in near-wall flows. In: Closure strategies for turbulent and transitionalflows (Eds. B.E. Launder and N.D. Sandham), Cambridge University Press, pp. 188-247. Polyakov A.E (1974). Wall effect on temperature fluctuations in the viscous sublayer (in Russian). Teptofizika Vysokikh Temperatur 12, 328-337. Pope S.B. (2000). Turbulent flows, Cambridge University Press. Pozorski J., Wactawczyk M. and Minier J.E (2003a). Full velocity-scalar probability density function computation of heated channel flow with wall function approach. Phys. Fluids 15, 1220-1232. Pozorski J., Waciawczyk M. and Minier J.E (2003b). PDF computation of heated turbulent channel flow with the bounded Langevin model. J. Turbul. 4, art. 011. Pozorski J., Wactawczyk M. and Minier J.E (2004). Scalar and joint velocity-scalar PDF modelling of near-wall turbulent heat transfer. Int. J. Heat Fluid Flow 25, 884-895. Sommer T.E, So R.M.C. and Zhang H.S. (1994). Heat transfer modeling and the assumption of zero wall temperature fluctuations. J. Heat Transfer 116, 855-863. Tiselj I., Pogrebnyak E., Li C., Mosyak A. and Hetsroni G. (200 l a). Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume. Phys. Fluids 13, 1028-1039. Tiselj I., Bergant R., Mavko B., Bajsid I. and Hetsroni G. (2001b). DNS of turbulent heat transfer in channel flow with heat conduction in the solid wall. J. Heat Transfer 123, 849-857. Wactawczyk M. and Pozorski J. (2004). Conjugate heat transfer modelling using the FDF approach for near-wall scalar transport coupled with the POD method for flow dynamics. In: Advances in Turbulence X (Eds. H.I. Andersson & EA. Krogstadt), pp. 253-256, CIMNE, Barcelona.

813

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

STUDY OF THE EFFECT OF F L O W P U L S A T I O N ON THE F L O W FIELD AND HEAT T R A N S F E R O V E R AN INLINE C Y L I N D E R A R R A Y USING LES Chunlei Liang and George Papadakis Department of Mechanical Engineering, King's College London, Strand, London, WC2R 2LS, UK

ABSTRACT This paper aims to study the effect of pulsating approaching flow, u(t) - uo~ + Au / 2 9sin(2nfpt) on the flow field and convective heat transfer over an inline cylinder array at a sub-critical Reynolds number Red,g = 3,400 (based on cylinder diameter and the gap velocity across the minimum section) using the Large Eddy Simulation (LES) technique. The LES computational results (mean and rms velocities) are directly compared with the LDA measurements of Konstantinidis et al. (2000) and good agreement is obtained. For steady approaching flow, a single mode of alternate vortex shedding with a constant Strouhal number (around 0.145) is observed behind all central cylinders. Flow pulsation at a frequency around twice the natural vortex shedding frequency ( Stp = f p D / Ugap 0.28 ) affects the cylinder wake -

-

by reducing the recirculation bubble size behind the first row cylinder and increasing the transverse mixing. In agreement with Konstantinidis et al. (2003), when the external pulsation frequency is around three times the natural vortex shedding frequency (Stp = 0.45), a symmetrical vortex formation mode is predicted successfully by LES. The heat transfer rate in the front part of the second row cylinder is greatly enhanced due to the vortex shedding lock-on behind the first cylinder. LES computations of six-row cylinders demonstrated that heat transfer around the third row and downstream cylinders are not influenced much by the external pulsation.

KEYWORDS Large eddy simulation, Dynamic model, Vortex shedding, Heat transfer, Pulsating crossflow, Inline cylinder array, Unstructured mesh, Finite Volume method.

1. INTRODUCTION Rollet-Miet et al. (1999) presented the first 3D LES calculations using the finite-element method of the turbulent flow across a staggered tube bundle. Since then, few publications have appeared although tube bundles are widely employed in cross-flow heat exchangers, as they combine ease of construction

814

with good thermal and mechanical efficiency. Rollet-Miet et al. (1999) pointed out the superiority of the LES technique because it is better suited to flows where the size of eddies (integral length scale of the turbulence) is comparable to the size of the obstacles of the flow. Benhamadouche and Laurence (2003) performed similar LES calculations for the turbulent flow across the staggered tube bundle using the finite-volume method on a collocated unstructured grid. They found that the type of the subgrid scale models (whether the standard or the dynamic Smagorinsky) is not critical for this type of application. Recently, Hassan and Barsamian (2004) performed three-dimensional LES of the turbulent flow across a five-row staggered tube bundle with pitch-to-diameter ratio 1.58 in curvilinear coordinates. The flow development from the first to the fifth row was simulated and both u'u' and v'v' had a good agreement with measurements. Liang and Papadakis (2004) performed 3D LES simulations with around 700K cells of the steady approaching flow over both inline and staggered tube bundles. The flow field around the inline tube bundle is more challenging for LES since the turbulent intensities are not as high as that of the staggered tube bundle and the effect of inlet turbulence level is significant on the development of rms level along the flow lane. In the first few rows in an intermediate spacing inline tube bundle as the one examined in this paper, a steady recirculation region consisting of a pair of counter-rotating vortices exists in the gap as found by Ziada and Oeng6ren (1992). This regime is called reattachment regime. Turbulence intensities at this regime are small compared to the alternate vortex shedding regime. However, Liang and Papadakis (2004) predicted a clear vortex shedding frequency behind the first cylinder and this is one reason that the LES calculations for the inline tube bundle over-predicted the rms level behind that cylinder. Konstantinidis et al. (2000) examined experimentally the effect of inlet flow pulsation in a cross flow over a tube array with an external frequency around twice that of the natural alternate vortex shedding. They found that the flow pulsation activates the flow field behind the first cylinder and increases the turbulence intensities for the first three cylinders. In a later study, Konstantidis et al. (2003) also observed a symmetrical vortex formation mode when the external frequency is around triple that of the natural alternate vortex shedding. In this paper, the effect of pulsation on the flow field and heat transfer in a six-row inline tube array is investigated using a 3D LES technique. 2. DETAILS OF THE NUMERICAL METHOD

2.1. Computational domain and boundary conditions The computational domain is described in a Cartesian coordinate system (x,y,z) where the x axis is fixed as the inlet flow direction (streamwise), the z axis is parallel to the cylinder axis (spanwise), while the y axis (transverse or cross-wake) is perpendicular to both x and z axis. Tube arrangement 6-row inline

SL

ST

Armin

rexp

N eff

Nz

N gap

N,otat

2.1D

3.6D

4.3•

1.2

112

20

36

702,720

Table 1" Grid parameters for inline tube bundle arrangement. N y is the number of cells along the periphery of the middle column cylinders, Ngap is the number of cells used along the streamwise direction between the first and second cylinders, N~ is the number of layers in the spanwise direction, rexp is the expansion factor in the radial direction and Armin is the radial thickness of the first cell on the

cylinder wall.

815

The grid parameters are shown in Table 1 for the inline tube bundle and the computational mesh is shown in Figure 1. The spacing-to-diameter ratios are S L / D x S r / D = 2.1 x 3.6 for the streamwise and transverse directions respectively and correspond to the geometry used by Konstantinidis et al. (2000). The smallest cell spacing in the radial direction is Armin / D = 4.3x 10 -3 and provides a coarser resolution than the finest mesh (Arm~n / D = 1.25 x 10 -3 ) used in Beaudan and Moin (1994) for a single cylinder at a very similar Reynolds number. In the spanwise direction (whose length is L z = reD) 20 equally-spaced layers are used, the same number as the one used by Benhamadouche and Laurence (2003).

Figure 1: (a) Computational mesh of the six-row inline tube bundles (x-y view). (b) Detail of the mesh around the first two cylinders. A convective boundary condition Ou / c3t + U cc3qk/ c3t = 0 is used for the exit boundary, where U c is the convective velocity normal to the outlet boundary, and ~b is any physical variable convected out through the outlet. Zero velocity boundary conditions are used for top, bottom and cylinder walls. Periodic boundary conditions are applied in the spanwise direction. The normal derivative tbr the pressure correction is set to zero at all boundaries. The inlet streamwise velocity is u(t) = uo~ + A u / 2 . sin(2nfpt), where uo~ = 0.2456m/s and f p is the extemal pulsation frequency and Au is the peak-to-peak amplitude. Details for their values are provided in section 3. For the heat transfer calculations, a free-stream temperature of 25~ is assumed, while a constant heat flux q"= 2149W / m z was imposed on all the central full cylinders concurrently.

816 2.2. Discretization schemes and subgrid scale model

The Finite Volume method applied on a collocated grid system is used to discretize the governing equations. All terms in the momentum equations are discretized using the 2 nd order central difference scheme (CDS) in space and the 2 nd order Crank-Nicolson scheme is employed to advance them in time, apart from the pressure term which is treated fully implicitly. The PISO algorithm is used to deal with the pressure coupling between the momentum and the continuity equations. In order to avoid the check-board pressure field, the interpolation method proposed by Rhie and Chow (1983) is employed. The bounded Gamma scheme, proposed by Jasak et al. (1999), can remove over- and under-shoots generated by central schemes, and is used to discretize the convective term in the energy equation. The top-hat filter is used to get the filtered Navier-Stokes and energy equations, m

OUi

-

"

O, i ~ {1,2,3},

(1)

OX i

Olgi

C~Uiblj

Op

Ot

Ox j

~x i

O|

~+u/ Ot

-

~g~ + o Or U Ox j

O|

OQ /

Oxj

Oxj

. . . .

~

Oui + Ouj

~x j

ax j

02 |

+ a ~ .

Oxj Oxj

(2)

Ox i

(3)

where the incompressible form of the subgrid scale Reynolds stress v~sgs = u i u j - u i u /

is modelled

using both the standard model and the dynamic Smagorinsky model proposed by Chester et al. (2001). The subgrid scale scalar flux Qj = u j | 1 7 4

is modelled using the same dynamic model for the sgs

Reynolds stress, assuming a constant value for the turbulent sgs Prandtl number Pr, - 0.9.

3. RESULTS AND DISCUSSION Table 2 provides the pulsating flow conditions for 3 cases as well as the corresponding mean Courant numbers using the dynamic model at Reynolds number Re d.g = 3,400. The lift coefficient of the first cylinder for all cases is shown in figure 2. Clearly, case 2 has the highest rms amplitude of the lift coefficient and case 3 the smallest one due to two symmetrical low-pressure regions associated with a symmetrical vortex formation pattern as explained below. The amplitudes also provide a good indication of the strength of the shed vortices. Case

Arrangement

f p (HZ)

St p = f pD / u gap

Au / u|

CFL . . . .

1 2 3

Six-row Six-row Six-row

0 10 15.5

0 0.28 0.45

0 0.1 0.1

0.29 0.30 0.31

Table 2: Pulsating flow conditions and Courant numbers for the 3 cases investigated. In the following section, the instantaneous flow and temperature patterns at different phases will be presented. It is not possible to compare the instantaneous velocity and vorticity patterns in the wake of cylinders for steady and pulsating approaching flow without some sort of common reference signal. For the pulsating flow, the external velocity provides such a reference, but this is absent for the steady flow. In order to compare the flow patterns, the variation of the lift coefficient is used as a reference signal. This coefficient has also been used in the past, for example by Beaudan and Moin (1994) in order to calculate the contributions of the random and coherent components of the Reynolds stresses.

817 In the present work, it also provides a useful common reference for comparing the instantaneous flow patterns for steady and pulsating approaching flows. The corresponding phases (A to E) for all cases are shown in figure 2. The time-averaged flow fields as well as the heat transfer rates around different cylinders will be presented next while the validity of the computational results for mean and rms velocities will be assessed through comparison with available measurement data.

Figure 2: The lift coefficients for the first cylinder and definition of 5 phases for cases 1, 2 and 3.

3.1. Instantaneous flow and temperature field

Figure 3 shows the velocity vectors colour-coded with the values of the spanwise vorticity for the first three cylinders for cases 2 and 3 at the 5 phases depicted in figure 2. When the flow is pulsed with twice the natural shedding frequency (case 2), a short vortex is formed behind the first cylinder, which folds up smoothly in front of the second one. LES successfully captured the symmetrical vortex formation behind the first cylinder when the pulsation frequency is around three times the natural shedding one (case 3). This is in agreement with the findings of Konstantinidis et al. (2003). Comparing the different phases for cases 2 and 3, it can be seen clearly that the transverse movement of the alternating vortices behind the second cylinder for case 2 is more intense and therefore leading to higher Vr, ~ in this region. For comparison, figure 4 shows the instantaneous velocity plot for case 1. It is clear that vortex shedding does take place behind all cylinders. The shedding frequency corresponds to Strouhal number 0.145. However, vortex shedding was not observed behind the first cylinder by Konstantinidis et al. (2000) although it was clear for the rest of the cylinders. This difference can be attributed to the insufficient grid resolution and possibly the size of the spanwise length. For example, calculations around two cylinders in tandem by Liang (2004) using a much finer mesh reveal that the flow is bistable (i.e. either an alternating vortex shedding pattern or a stable recirculation zone can develop depending on the initial flow conditions). This result agrees with the findings of Jester and Kallinderis (2003) for the examined Reynolds number and tube spacing. A possible explanation is that when the mesh is coarse, the dispersive errors of the central differencing scheme can trigger instability in the gap between the cylinders that always lead to vertex shedding. On the other hand, when the mesh is refined, the dispersive errors are reduced and a stable recirculation zone can also form. It is also likely that the spanwise length rd) may not be wide enough to obtain the correct vortex formation length behind the first cylinder (a shorter length makes vertex shedding easier). For instance, Xu and Zhou (2004) found that the vortex formation length is highly dependent

818

on the aspect ratio used in the experiments for the flow over two cylinders in tandem at similar Reynolds numbers to the present one. Figure 5 shows the temperature field at phase E for all three cases. The temperature distribution has a strong correlation with the spanwise vorticity field shown in figures 3 and 4. For example, the temperature pattern behind the first row for case 3 results from the symmetrical vortex formation observed in the velocity plots. The results also show that case 2 has the strongest transverse mixing, particularly behind the first two rows.

Figure 3: Instantaneous velocity vector plots colour-coded with the values of the spanwise vorticity for different phases of the shedding cycle. The values of vorticity are shown in figure 4.

819

Figure 5: Instantaneous temperature contour plots for Phase E. (a) case 1; (b) case 2; (c) case 3. 3.2. Time-averaged flow field and heat transfer

Figure 6 shows the time-averaged streamwise mean velocity as well as the rms velocity adjusted using the experimental inlet turbulent level (U~m~ = Urn s + 0.09) for the pulsating cross-flow over the inline tube bundle (case 2). Both the Smagorinsky model and the dynamic model predict good mean velocity profiles in agreement with the LDA measurements. The dynamic model predicts nearly identical results to the standard Smagorinsky model for both mean and rms streamwise velocity. Behind the first cylinder, the mean velocity is slightly over-predicted at the location x / D = 0.8. As mentioned earlier, the inlet turbulence level of the measurement was added to the rms velocity field in order to have a quantitative comparison. However, this practice over-estimates the effect of the inlet turbulence on the

820

rms velocity of the cylinder wakes. It is believed that this is one of the reasons that the predicted rms values shown in figure 6(b) are generally higher than the experimental results. (b) tlrrns+O.O9

(8)

":i

1.--,4-

~

,4

~ : ; ;

,4

.....~..-~

I a law~w_w_n

i~

"~-.~,;.

.....~ ! .

a aaaaa

a a

o__......~

~,~-.-.-~,.....,.~

-I ~i~ , - u a . i i .

~

ii_;-,,

'i.//;;~,.i;. .....ii2,

-8 ~ ~

;,/i:.i.2ii_-~

~~~:*_7t.....

~

_n _ la _!

-8 ~'~'"S,:; ;:,;;__~j 9 ~ -9

0

I

0.5

1

1.5

0

Figure 6: Mean and rms velocity predicted for case 2. ( Smagorinsky model; (taE]nDE]n) Konstantinidis et al. (2000).

a i~_~l i~ _1~

0.5

1

)

1.5

Dynamic model;

(. . . .

)

The following figure 7 shows the time-averaged local Nusselt numbers around the second and fourth cylinders for cases 1, 2 and 3. It can be seen that case 2 has the highest heat transfer rate in front of the second cylinder due to the vortex shedding lock-on and case 3 has the lowest one due to symmetrical vortex formation which results in the smallest Vrm~ and the lowest transverse mixing. The heat transfer

821

rate of case 1 lies in the middle due to the fact that the coarse grid resolution and possibly the size of the spanwise length, result in a fictitious alternate vortex shedding as already explained. The heat transfer rate around the first cylinder increases only after the shear layer separation point by the pulsation (Liang (2004)). Figure 7(b) demonstrates that the heat transfer rate around the fourth cylinder is about the same for all three cases. Therefore, the flow pulsation has a clear effect over only the first two cylinders, a result that is in agreement with the findings of Konstantinidis et al. (2000,2003). Second row cylinder 65 / 60]-

'

'

6O

- - ~ 'Case 1 ---,,- Case 2

.~"%.-~

v ~'L./

Fourth row cylinder ,

,

^ 55

~so g ~4o ~ 30

25[ 200

25

~---'-50

(a)

0

100

200

150

50

()

100

150

(b)

Figure 7: Time-averaged local spanwise-averaged Nusselt number around cylinders in the six-row bundle. (a) Second cylinder; (b) Fourth cylinder. 4. CONCLUSIONS The LES technique with the standard and a dynamic Smagorinsky model is employed for the simulation of the flow inside a six-row inline tube array with two different pulsation frequencies at the inlet. The results produced by the two sgs models have no significant differences with the current grid resolution. The LES approach yielded satisfactory results with the LDA measurements when the pulsation frequency is around twice that of the natural vortex shedding frequency. The LES also captured the symmetrical vortex formation pattern behind the first cylinder when the pulsation frequency is around triple the natural one. The predicted heat transfer rate in the front of the second cylinder is enhanced significantly due to pulsation. However, the heat transfer rate around the third and downstream cylinders is not much affected by the external pulsation.

Acknowledgement Chunlei Liang gratefully acknowledges the financial support of EPSRC (GR/R04256/01) and the ORS award of Universities UK. REFERENCES Beaudan S. B. and Moin P. (1994) Numerical experiments on the flow past a circular cylinder at subcritical Reynolds number. Thermosciences Division, Department of Mechanical Engineering, TF62, Stanford University. Benhamadouche S. and Laurence D. (2003) LES, coarse LES and transient RANS comparisons on the flow across a tube bundle. Int. J. Heat Fluid Flow 24, 470-479.

822 Chester S., Charlette F. and Meneveau C. (2001). Dynamic turbulence model for large eddy simulations without test filtering: quantifying the accuracy of Taylor series approximations. Theor. and Comput. Fluid Dynamics 15, 165-181. Hassan Y. A. and Barsamian H. R. (2004). Tube bundle flows with the large eddy simulation technique in curvilinear coordinates. Int. J. Heat and Mass Transfer 47, 3057-3071. Jasak H., Weller H. G. and Gosman A. D. (1999). High resolution NVD differencing scheme for arbitrarily unstructured meshes. Int. J. Numer. Meth. Fluids 31, 431-449. Jester W. and Kallinderis Y. (2003). Numerical study of incompressible flow about a fixed cylinder. Journal of Fluids and Structures, 17, 561-577. Konstantindis E., Castiglia D., Balabani, S. and Yianneskis, M. (2000). On the flow and vortex shedding characteristics of an inline tube bundle in steady and pulsating crossflow. Transactions of IChemE, Chemical Engineering Research and Design 78, 1129-1138. Konstantindis E., Balabani, S. and Yianneskis, M. (2003). Relationship between vortex shedding and heat transfer: implications for tube bundles in cross-flow. Transactions of IChemE, Chemical Engineering Research and Design 81,695-699. Liang C. (2004) Large Eddy Simulation of steady and pulsating cross-flow and heat transfer over inline and staggered tube bundles. Ph.D. thesis, University of London. Liang C. and Papadakis G. (2004) Large eddy simulation of cross flow over inline and staggered tube bundles. Proceedings of the 8th international conference on Fluid-Induced Vibration, Paris, France, 1, 247-252. Rhie C. and Chow W. (1983). Numerical study of turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21, 1525-1532. Rollet-Miet P., Laurence D. and Ferziger J. (1999). LES and RANS of turbulent flow in tube bundles. Int. J. Heat Fluid Flow 20, 241-254. Xu G. and Zhou Y. (2004). Strouhal numbers in the wake of two inline cylinders. Experiments in Fluids 37, 248-256. Ziada S. and Oeng6ren A. (1992). Vorticity shedding and acoustic resonance in an inline tube bundle. Part I: vorticity shedding. J. of Fluids and Structures 6, 271-292.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

823

LARGE EDDY SIMULATION OF SCALAR MIXING M. Dianat, Z.Yang, and J.J. McGuirk Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, LE 11 3TU, UK.

ABSTRACT The paper describes the implementation of a passive scalar transport equation into an existing LES code, including assessment/testing of alternative discretisation schemes to avoid over/undershoots and prevent excessive smoothing. The numerical schemes are evaluated using a simple test case involving pure scalar convection in a specified velocity field. Both 2 nd order accurate (away from extrema) TVD and higher order accurate DRP schemes are assessed for use in solving the scalar conservation equation in order to optimize numerical behaviour in terms of both dispersive and diffusive errors. Although the best performance is displayed by a DRP method, this is only true on reasonably fine meshes, it produces similar (or even slightly larger) errors to a TVD scheme on coarser meshes, and at present the TVD approach has been retained for LES applications. The unsteady scalar mixing performance of the LES code is then validated in turbulent flow against published Direct Numerical Simulation (DNS) data for a slightly heated channel flow. Excellent agreement between current LES predictions and the DNS data is obtained, for both velocity and scalar statistics. Finally, the developed methodology is applied to scalar mixing in a co-annular jet mixing configuration for which detailed experimental data are available for both velocity and scalar fields. Agreement with measured scalar variance fields is good, and the LES predictions demonstrate the prediction of low turbulent Prandtl numbers (-0.7) in the free shear regions of the flow, as well as higher values (-1.0) in the wall affected regions.

KEYWORDS

LES scalar prediction, convection discretisation, scalar variance, turbulent Prandtl number, jet mixing. INTRODUCTION

The Large Eddy Simulation (LES) approach represents the most promising future methodology for modelling the unsteady and intermittent behaviour of large-scale turbulent eddies present in many industrial flows involving both complex physics and complex geometry, such as gas-turbine engine applications, Moin (2002). To date, however, most effort has been focused on the dynamic field with relatively little attention paid to the modelling of the scalar field, an area of central significance to gasturbine engine components such as low emission combustion systems involving lean premixed burner designs (Tang, Yang and McGuirk (2001)), or the mixing of jet engine exhaust plumes. The present

824

paper focuses on the implementation of a passive scalar equation into an existing LES code (Tang et al (2004)), including assessment/testing of appropriate convection discretisation practices for scalar transport to avoid over/undershoots and prevent excessive smoothing. The numerical implementation is first assessed using a simple 2D test problem that emphasizes accuracy of scalar transport. The unsteady scalar mixing performance of the LES code is then validated by application to the standard test case of a fully-developed channel flow with heat transfer, as studied recently using Direct Numerical Simulation (DNS) by Debusschere and Rutland (2004). In the premixed combustion applications mentioned above, it is scalar mixing across free shear layers which is the fundamental problem, so the final application of the LES code is to a co-annular jet mixing flow for which Lima and Palma (2002), have recently reported detailed data using LDA for the velocity field turbulence and LIF instrumentation for the scalar field.

MATHEMATICAL FORMULATION The LES equations can be obtained by applying a spatial filter to the instantaneous transport equations. For an incompressible flow in Cartesian coordinates, this will result in the following filtered NavierStokes equations (Sagaut (2002)):

Opu,

~ at + where

0

0-p 0a,, + - 0xj )= - - 0x,

I

Or,~ 3x,

u and fi are the resolved velocity and pressure, p is the density, o',j is the resolved stress

tensor defined in terms o f / t the fluid molecular viscosity, Sv is the resolved strain tensor, and r v is the unknown residual or subgrid-scale-stress tensor: --

-

-

cr o = 2ITS,,

1 (Ou,

So = ~ Ox j

+

Ouj --~x. )

r,, = p ( u, u , - e f t , )

g

In the present code, the sub-grid-scale stress is modelled using the simple Smagorinsky SGS model (Sagaut (2002))" -

2

r o = -2fl, S,, +-~6,jpk

/u, = p ! 2 S

i

S=(2S,,S,,) 2

The length scale is calculated from l.,. =C,.A r , where the filter width As _

is obtained from

1

A s = (AxAyAz) 3 withAx, Ay and Az being the local cell size in the x, y and z directions respectively. Close to a wall, a Van Driest damping function further modifies the length scale: I.,. = C , . ( 1 - e x p ( - y + / A+ ))A/ When a similar filtering operation is applied to the transport equation for a conserved scalar ~, the following equation is obtained:

Op~ - + 0 (pu~ ~) = 0 ( / / 0 ~

c3Jk

where Sc is the fluid molecular Schmidt number (0.7 for air), Jk is the subgrid flux of the scalar ~ , calculated here using a gradient model, with Sc, the Schmidt number for the SGS scalar flux being set equal to 0.7 for the calculations reported here:

J~ = p ( u ~ r 1 6 2

/x, J~ . . . .

0~

Sc, Ox k

825

The LES code used for the present study (LULES) is based on solving a transformed version of the Cartesian transport equations using a staggered grid arrangement. Details of the transformed equations for the momentum and scalar fields have been described elsewhere (Tang et al (2004)), but this leads to the following canonical transport equation for a general variable 0"

+so where Vk is the divergence operator and F~ is the total (molecular + SGS) diffusion coefficient. Note that this equation has essentially the same form as its Cartesian counterpart except that the velocities u~ now represent the contravariant components, the spatial derivatives are with respect to the curvilinear coordinates, and the divergence operator is defined by: V

= k

hk

3

],hh2h3 ~)xk

hlh2h 3 hk

where hk are the scale factors representing the ratio of curvilinear and computational distances.

NUMERICAL PROCEDURE Basic discretisation s c h e m e

The spatially filtered transport equations described above are discretised using the finite volume method. For the momentum equations, spatial derivatives for both convective and diffusive terms are calculated using a second order central differencing scheme. For the scalar transport equation, a central differencing scheme is used for the diffusion term while a range of schemes described below have been tested for convection discretisation. Integration over a control volume of (for example) the xl-direction convective term in the transport equation for the scalar property ~ leads to the following discretised form: V) (pur = g x z 6 x 2 6 ~ 1

where fl is cell face area, P,+~/2 = ~(P, + P~+~), u,+~/2 = u;+~ (due to the mesh staggering) and ~,+~ 2 is calculated according to a TVD or DRP scheme as described below. Similar expressions are obtained for the i - 1/ 2 cell face, and the y and z derivative terms. A similar approach is used for the diffusion terms, where the cell-face derivatives are approximated using centred differencing. For time discretisation, the Adams-Bashforth second-order explicit scheme is used. To illustrate this scheme, the transport equation for the scalar is written in the following form: 3pr = C + D = H 3t

where C and D represent the convective and diffusive approximations described above and H is the discretised version of fluxes due to all transport processes. The final form of the Adams-Bashforth scheme adopted yields: 3pO _ 1 I[[oP_~dxdydz~-J Ot - Ax, Ay /Az k x~, y , z ot

[

, \n+l

n

= [P~),,j,k r

_ 3 H,,/,k_ 1 ,_~ - -2 -2 H''/'k

where the superscripts n-l, n and n+ 1 represent values at the previous, current and next time step. Scalar convection d i s c r e t i s a t i o n - T V D s c h e m e

In many LES methods developed so far that include solution of a scalar transport equation (e.g. Branley and Jones (1997)), Total Variation Diminishing (TVD) type schemes have been used for the

826

convective term. This is because many scalar properties (e.g. mass fractions, mixture fraction etc.) are bounded variables with physically imposed limits (e.g. 0 < ~ < l ) and preserving the bounded, nonnegative nature of the scalar is important. It is well known, however, that the use of central differencing for the convective terms in a strictly conserved scalar equation does not guarantee the boundedness requirement, although it is often the preferred discretisation for LES of the dynamic field due to its non-dissipative and energy conserving properties. An alternative approach involves the use of a TVD type scheme (Sweby (1984)), which can be viewed as being based on central differencing, but with a limiting mechanism added to ensure that the scalar bounds are not violated. To illustrate this approach, consider a transient one-dimensional pure convective flow with positive velocity, u: ~+u--=0

0t

0x

The use of a simple explicit, forward time, central differencing scheme for this equation may be written:

#+' =07-Co where

O," +-~(0,"+,-O;) - 0,"_, +-{(O;-O,"-,)

'

IF

'

]}

u6t

is the Courant number. Here, the first term within each square bracket may be 6x viewed as a 1st order upwind contribution while the second term may be viewed as an anti-diffusive component. With TVD type schemes, the above form is maintained but the anti-diffusive component is limited in order to avoid instabilities and maintain boundedness. The above expression is then written: C O

=

{[

r

=O,"-Co{EO," +~v/,~,,~(O,~,-O,') ' ][-

07-,

+~vq-,,~(O,'-O,"_,) '

]}

The particular form of the limiter depends on the TVD scheme used. The limiter function ~ is intended to detect the formation of spurious local extrema and suppress these; the function is therefore made sensitive to the ratio of the scalar gradient across the cell (r), e.g. for the i+ 1/2 face and positive cell face velocity, with the gradients at cell faces calculated using central differencing:

k ax ),+,

k ax ),+,,~ x,+,- x,

Similar expressions may be written for other cell faces, and other co-ordinate directions; if the cell face velocity is negative, alternative expressions for the cell face ~ value and gradient ratio r are appropriate, but easily derived by analogy. Table 1 shows several schemes suggested in the literature (omitting subscripts for clarity): TABLE 1 Limiter functions for TVD schemes Scheme Central Upwind Roe ( 1981) minimod Roe ( 1981) superbee Van Leer (1974) Branley and Jones (1997)

Limiter function ~=1 ~=0 V - max (0, m!n(r, 1)) = max (0, min (2r, 1), min(r, 2)) = (r + mod(r))/(1 + mod(r)) Nt - max (0, min (2r, 1))

The particular form of the limiter function controls the range of values ofr for which the scheme is 2nd order accurate, or reverts to a 1st order scheme. For example, the Branley and Jones (1997) scheme results in upwind differencing for ri"_+~/2< 0 and central differencing for r" > 0.5 while within the --

tztzl / 2

--

827

n

range 0 < l'i+l/2

<( 0 . 5 ,

the value of the limiter defines the weighting between 1st and

2 nd

order accurate

schemes. Scalar convection discretisation - D R P scheme

Dispersion Relation Preserving (DRP) schemes, Tam and Webb (1993), Bogey and Bailly (2004), are high-order schemes designed to minimise the effects caused by numerical dispersion. The first order spatial derivatives in the convective terms (e.g. 0~/0x at node i) are approximated by a symmetrical central 2n+ 1 point stencil:

i=-n

where the coefficients a j are such that a / - - a / ,

providing a scheme without dissipation. In the

standard approach, the a j coefficients are determined purely by using a Taylor series expansion of r around i. In the DRP approach, as well as these constraints, the coefficients are optimised to minimise the dispersion properties of the scheme over the widest possible range of spatial wavenumbers. Tam and Webb (1993) and Bogey and Bailly (2004) give details and specific coefficient values for their particular methods. The two approaches differ principally in that Tam and Webb change the scheme to be increasingly one-sided as a boundary is approached, whereas Bogey and Bailly choose to reduce the order of accuracy of the scheme progressively as a boundary is approached whilst always maintaining a centred scheme. For example, in the l 0 th order Bogey and Bailly method used here, the size of the stencil is reduced progressively from l l points in the interior of the domain to 9, 7, 5 and 3 points closer to boundaries. Calculations are also reported using 8th order and 6th order versions of this method. The DRP approaches outlined above retain the classical defect of all centred schemes that they do not resolve grid-to-grid oscillations and this can lead to numerical instabilities. Artificial damping or selective filtering must therefore be used to remedy this undesirable property. The following symmetrical 2n+ 1 point operator is used to filter the DRP predicted ~ distribution in order to suppress spurious short wavelength oscillations:

r ....~(x):O(x)-a~D~(x)

D~(x): ~ d,C~(x+ j,~x) i=-n

The coefficients d j are such that d~ =d_j ensuring no dispersion and tyd is a constant that determines the strength of damping. In the current calculations, a minimum value for the damping coefficient is used to avoid excessive dissipation. In the calculations reported below, it was found that, for the Tam and Webb scheme, the solution diverged when damping was not introduced and on finer meshes, the damping factor had to be increased to retain stability. It is believed that this behaviour was mainly caused by the use of non-centred stencils near the boundaries. For consistency, a fixed damping coefficient of t y - 0.07 was used for all grid sizes with the Tam and Webb scheme. All versions of the Bogey and Bailly approach were found to be significantly more robust, with a damping factor ty = 0.01 being sufficient to provide stable solutions on all mesh sizes. It should be noted that this is significantly smaller than the value of 0.1 - 0.2 suggested by Bogey and Bailly in their study of acoustic propagation problems, but such values lead to excessive dissipation with substantially increased error levels in the present scalar transport problem.

RESULTS Evaluation o f Scalar Convection Discretisation Schemes

To evaluate the suitability of the scalar convection discretisation procedures described above, and to select an optimum methodology for incorporation into the LULES code, a simple test case involving

828

pure scalar convection in a 2D domain is considered. The velocity field is fixed so that it convects a scalar profile specified over the lett lower half of the solution domain boundary through 1800 to exit from the lower right half of the domain boundary. The rectangular region defined by - 1 < x < 1 and 0 < y < 1 defines the solution domain. In this situation, the predicted scalar contours must coincide with flow streamlines and any spreading of the scalar on its passage through the domain is attributable to numerical diffusion; similarly any over/undershoots in the predicted scalar are a consequence of numerical dispersion. This problem has otten been used to assess numerical diffusion effects, and a full specification (flowfield, scalar boundary conditions etc) is provided by Jiang et al (2001). Calculations were made on four progressively refined grid sizes (x, y) of 22• 42• 82• 42 and 162x82. Figure 1 illustrates the type of solutions that are obtained with different schemes and on different meshes.

Figure 1 Predicted scalar contours on coarse mesh: upwind (top lei~), central (top right) Branley and Jones (1997) TVD solution on coarse mesh (bottom left), finer mesh (bottom right) Results from 1st order upwind differencing on a coarse mesh illustrate the high numerical diffusion property of this scheme, which produces large spurious spreading of the high scalar gradient region, with a large overall error that decreases only slowly with mesh refinement (see below). Central differencing displays its second-order non-diffusive behaviour as expected, with little smoothing of the scalar gradient, but clear evidence of high dispersion. The TVD approach (the Branley and Jones scheme is used in Figure 1) is marginally dissipative on the coarse mesh, but all oscillations have been suppressed and bounded solutions are obtained. With only a small amount of mesh refinement (Figure 1), the scheme returns an accurate prediction of the scalar field. Since the exact solution for the scalar is known, a global error can be calculated from the following expression for any of the predictions obtained (Jiang, McGuirk, and Page (2001)):

Figure 2 provides a comparison of the accuracy of several TVD and DRP methods. As indicated in this figure, the DRP schemes of Bogey and Bailly lead to lower global errors than TVD schemes, particularly when fine meshes are used (an order of magnitude reduction at the highest mesh density). However, the schemes do not completely eliminate dispersion errors and the scalar field fails to remain bounded. The dispersion effects are more significant on coarser grids leading to higher levels of the overall error than in the 2 nd order TVD scheme. At this stage it seems that the optimum method for introducing damping into these DRP schemes has not been identified.

829 lO0 _

1#-

%

"-.,,.,~entral

,._10 "1

tll

uJ

,n

x'x.::. -."_-.~r ,. Superbee ._.... -',, . . . . -. - . .

o 010.2

9......

Branley a. 0

,.,,~

1

02

.

.

.

.

!,,,I

"~

~~%.."%""" N"~i].~i."'~Branley'~"""

~10"2

and

O

9

"%

;~id'~~

2nd order ~ 10-3

""" " - . . . ,

~

10.3

""...".,, Van Leer

4

B o g e y a n d Bailly "~.. 10th, 8th and 6th order ~,..

~

Jone s -, "'~ .~

.... ..',,

103

. . . . . . . .

Mesh S i z e

I

10-s

10 4

102

.

.

.

.

.

.

.

.

I

103

.

.

Mesh S i z e

.

.

,

,,,I

104

,

(a) (b) Figure 2 Accuracy of (a) TVD and (b) DRP schemes In most LES calculations likely to be encountered in engineering applications, the use of a coarse mesh is inevitable, at least in certain regions of the domain, for example where a thin scalar gradient layer is resolved by only a small number of cells. Furthermore, the accuracy of higher-order DRP schemes has been reported (Bogey and Bailly (2004) to be reduced when non-uniform grids with large stretching rates are used. In view of these factors, whilst the performance of DRP schemes is sufficiently promising to warrant further testing and development, the TVD scheme of Branley and Jones has been adopted for the current LES calculations.

Validation of Scalar Equation Solver in LULES code To validate the numerical performance of the scalar transport equation introduced into the LULES code, a slightly heated fully developed plane channel flow has been simulated. This low-Reynolds number flow (Re = 3000 based on the bulk velocity, Urn, and half channel height,6, or m

Re~ = u~6= 186 based on the friction velocity) was investigated using Direct Numerical Simulation v (DNS) by Debusschere and Rutland (2004) and focuses on passive heat transfer as a special case of scalar transport. Isothermal, but different wall temperatures are specified on upper and lower walls of the channel. For normalisation purpose, the wall friction temperature has been defined as:

P r by -y=o t, U r

The solution domain was 12•215 2 in the x (streamwise), y (transverse), and z (spanwise) directions. A uniformly spaced mesh was used in the x and z directions, while a non-uniform grid was used in the y direction to ensure that wall effects were fully resolved; the wall-adjacent node was located at y+ = 1. The results presented are based on the calculations carried out on a 72x58• 41mesh; further calculations on a finer 120•215 mesh showed little dependence on grid size. It should be noted that Debusschere and Rutland used a much finer mesh of 231x200• nodes in their DNS simulation over the same solution domain. The time step used in the calculation was V & = 0.12 .-5- = 0.01. In the results presented below, velocities and lengths are normalised by the mean Ur velocity and channel half-height respectively. Reynolds stresses and temperature variance are normalised by the wall variables u~ and t~ respectively; turbulent heat fluxes are normalised by uJ~.

830

Figure 3 shows contours of the instantaneous non-dimensional temperature on a horizontal plane close to the lower wall. The unsteady nature of the flow is apparent from the streaky regions of cold and hot fluid packets. Structures are formed near the wall and then spread away from it before being convected by the mean flow.

Figure 3 Instantaneous contour map of non-dimensional temperature at y+ = 25. Figure 4 shows transverse profiles of turbulent and total normalised shear stress and turbulent and total normalised transverse heat flux. The sub-grid-scale contribution (dashed line) is seen to be negligible as the Reynolds number is low. Also implied in this figure is the molecular contribution to the shear stress. Near the walls, as expected, the viscous stresses are significant. Similar remarks may be made about the heat flux, the sub-grid-scale contribution is again negligible but the molecular effects are now significant (10%) even in the central region of the channel. 1.5-

1"I ~ . . ~

(b)

(a)

Total

0.5

Turbulent

-

0.6

-0.6

i -1.5

0

.

.

.

o~6

.

4

f , 1.5

. . . . .

y

i

|

i

2i

~

'

'

I

I

I 0.5

,

~ i

|

I 1

|

~

,

i

I , 1.5

,

2

Figure 4 Profiles of (a) shear stress and (b) transverse heat flux For a fully developed flow, the production term for the transverse turbulent heat flux is

1;'7,=-v23T/3y. In the central region of the channel, the profile of v 2 is uniform and the mean temperature varies linearly with distance (see below). This yields a uniform production term for the turbulent heat flux leading to the uniform vt distribution as shown in Figure 4. Figure 5 shows the profiles of axial turbulent heat flux, mean velocity and temperature, and the three normal stresses and temperature variance. The comparison between LES (dashed lines) and DNS solutions (solid lines) is excellent for both mean properties and turbulence statistics. The agreement with temperature statistics 10 1.4

8

~

12.

I o.8 0.6 0.4

6 ~ 4 2 0

, 0

|

|

0.2

0.4

~ 0.8

t' 2

T

1

,

. .

02. 0

i

0.6

i

U

I

0 0

0.2

0.4

Y

0.6

0.8

1

0

1

1

0.2

0.4

Y

,

,

0.6

0.8

I

Y

Figure 5 LES vs. DNS for axial heat flux, mean profiles, normal stresses and temperature variance.

831

is taken as complete validation of the scalar equation implementation. Finally, Figure 6 shows the LES predicted variation of turbulent Prandtl number defined as shown: Two LES solutions are shown, on 1.4 LES, Coarse

1.2 1

- -9- , , . ~ S ,

DNS

0.8

Fine

0.6 0.4 09 0

|

|

|

i

0.2

0.4

0.6

0.8

Y

1

Figure 6 Turbulent Prandtl number predicted by LES coarse and fine grids. This is the one parameter that did alter on grid refinement, but only in the central channel region as shown; this is related to the high accuracy needed to resolve the ratio of small numbers involved in the Prandtl number definition. The important message is that the scalar LES solution is predicting a turbulent Prandtl number close to 1.0 in this wall-dominated flow, as expected from the DNS solutions.

Application of LES Scalar code to co-axial Jet Mixing The measured data of Lima and Palma (2002) were used to test the performance of the LES scalar predictions in a co-axial confined jet flow. The geometry is shown in Figure 7, with the working fluid used in the experiments being water. The velocity ratio Ua/Uin was 3.2 and dye was mixed into the core jet to act as a tracer; velocity and concentration fields were measured simultaneously using a combined LDA/LIF technique. The grid consisted of 380,160 cells with 7 2 x 6 6 x 8 0 grid nodes (x, r, 0). [ZIYlIZ~7-77-Z--A.___:T7-7-777-7-77-)I

o: ir

L

~Z.ffZZZLZJ

,..

Figure 7 Flow configuration.

J Figure 8 Instantaneous scalar contours

Figure 8 shows a contour plot of the instantaneous scalar on an x-r plane. The shear layer is clearly visible showing intense mixing between the central and annular jets close to the inlet. The potential core of the central jet extends to around x~ d o = 1.75 with the roll-up of vortices clearly captured by the simulation in the early region of the mixing layer. To concentrate on the scalar mixing performance of the simulation, two detailed comparisons are now shown. The first is the radial profile of fluctuating concentration, Figure 9, at three axial stations (x/d0 = 1, 2, 8) excellent agreement is seen, with discrepancies at the first station probably caused by the inlet condition assumptions, which were at present taken as small random perturbations superimposed on the measured mean profiles at x/d0= 0.14. Finally, Figure 10 shows radial profiles of the calculated turbulent Schmidt number at two axial locations. The value of around Sc, = 0.7 across the free shear layer part of the flow is consistent with observations for the round jet, with the simulation returning higher values as the wall is approached.

832

x/do =

~

1

x/do =

x/do = 8

2

1

1

4

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

I~ 0.4

0.2

0.2

0.2

0 0

e'/C~

05

|

0

v

,

c'/Cin

,

0.:

0

1.6 1.2

~

0.8

................

~

0.4 0 o

c'/Cin

Figure 9 Profiles ofc'/Cin LES: line, symbols: data.

o.1

0.3

0.36

0.4

0.45

0.6

r/R

Figure 10 Predicted Schmidt No.

CONCLUSIONS This paper has described the implementation of a passive scalar equation into an existing LES code, including assessment/testing of discretisation to avoid over/undershoots and prevent excessive smoothing. A simple 2D test case was used to select a TVD scheme for implementation, although a DRP scheme also gave good results. The unsteady scalar mixing performance of the LES code was validated for a heated channel flow. The agreement between the LES simulation and DNS data was excellent for both dynamic and scalar fields. Finally, the developed methodology was applied to scalar mixing in a confined co-axial jet configuration for which experimental data were available. The agreement for scalar turbulence statistics was very good. In particular, it was demonstrated that LES could return correct levels of predicted turbulent Prandtl number for near wall and free shear flows.

REFERENCES Bogey C. and Bailly C. (2004). A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. of Computational Physics 194, 194-214. Branley N., Jones W.P. (1997). LES of a turbulent non-premixed flame. Proc. 11th TSF Symposium. Debusschere B. and Rutland C.J. (2004) Turbulent scalar transport mechanisms in plane channel and Couette flows. Int. Jnl. of Heat and Mass Transfer 47, 1771-1781. Jiang D., McGuirk J.J., and Page G.J. (2001). Influence of mesh type and grid quality on spurious mixing in unstructured mesh predictions of scalar mixing layers. Proc. of ECCOMAS Conf. Lima M.M.C.L. and Palma J.M.L.M. (2002). Mixing in coaxial confined jets of large velocity ratio. Proc. of 11th Int. Symp. on Appl. of Laser Techniques to Fluid Mechs. Moin P. (2002). Advances in Large Eddy Simulation Methodology for complex flows. Int. J. of Heat and Fluid Flow 23, 710-730. Roe P.L. (1981). Numerical algorithms for the linear wave equation. Royal Aircraft Establishment Technical Report 81047. Sagaut P. (2002). Large Eddy Simulation for Incompressible Flows, Springer, Berlin, Germany. Sweby P.K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal for Numerical Analysis 21,995-1011. Tam C.K.W. and Webb J.C. (1993). Dispersion Relation Preserving finite-difference schemes for computational acoustics. J. Computational Physics !07, 262-281. Tang G., Yang Z. and McGuirk J.J. (2001). LES predictions of aerodynamic phenomena in LPP combustors. ASME paper 2001-GT-465. Tang G., Yang Z., McGuirk J.J. (2004). Numerical methods for Large Eddy Simulation in general coordinates. Int. Journal for Numerical Methods in Fluids 46, 1-18. Van Leer B. (1974). Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a second order scheme. J. Computational Physics 14, 361-370.

13. Combustion Systems

This Page Intentionally Left Blank

Engineering TurbulenceModelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

835

Experimental characterization and modelling of inflow conditions for a gas turbine swirl combustor R. Palm, S. Grundmann, M. Weismiiller, S. Sarir, S. Jakirli61 and C. Tropea Chair of Fluid Mechanics and Aerodynamics, Darmstadt University of Technology Petersenstr. 30, 64287 Darmstadt, Germany

ABSTRACT One of the most important processes in a gas turbine combustor, influencing to a large extent the efficiency of the entire combustion process, is the mixing between a swirling annular jet (primary air) and the non-swirling inner jet (fuel). To study this fundamental flow geometry an experimental facility has been built which allows independent flow rate adjustment of the central (mean stream) and coaxial jet flow V c / V m and furthermore, good optical access for laser-based flow measurement techniques. Further important flow parameters include the Reynolds number (Re.,), the swirl intensity (S) and the combustor confinement, expressed in terms of an expansion ratio (ER). The work presented focuses on the features of the swirling flow in the concentric annuli being the part of the inlet system. The laser Doppler technique has been used to measure the velocity profile and the gradient of the velocity in the annular cross section. The circumferential velocity profile follows the so-called free-vortex flow type, being characterized by an increase in the mean angular momentum by radius of curvature. The outcome of the experiment is that the axial velocity profile becomes increasingly asymmetric with increased swirl intensity. The velocity increases from the inside to the outside of the annular flow (with a decreasing gradient) corresponding to an intensified radial movement towards the outer wall due to imposed swirl. The numerical investigations, especially those accounting for the complete swirl generator system and using a second-moment closure reproduced all important mean flow and turbulent features in good agreement with available experimental data. Both the modelling and the Large Eddy Simulations of an equilibrium, fully developed swirling flow, performed with the method for the swirling inflow data generation due to Pierce and Moin (1998), revealed some interesting departures with respect to the opposite sign of the axial velocity gradient. KEYWORDS

Swirl combustor, swirling annular inflow, laser Doppler measurements, Large Eddy Simulation, turbulence modelling, Second-Moment Closure.

[email protected]

836

1 INTRODUCTION A general interest to reduce fossil fuel consumption and recent regulatory measures to limit combustion emissions, create demands to increase the efficiency of combustion chambers. For a clean and controlled combustion a fine and homogeneous spray is necessary. This can be achieved by using a swirling free shear layer, see Fig. 1. The free swirl-induced recirculation area leads to a stable flame and creates a homogeneous spray inside the flue. The flow is characterised by a high level of turbulence and strong streamline curvature. This complex flow geometry is also a good example of where experimental and numerical investigations can be complementary. The experimental results can be used as verification data for turbulence models and numerical simulations, which are more economical for parametric investigations.

Figure 1: Schematic of air flow in an idealized combustion chamber The common practice when computing the combustor flows numerically is that the inlet cross-section of the solution domain typically coincides with the first available measurement location, usually being situated in the interior of the combustor, e.g. Lai (1996). In that case the inlet plane crosses both the comer bubble and the large recirculation zone in the core flow. The measurement data commonly exhibit a certain scatter in this flow region and are often incomplete. Such a computational practice is especially unsuitable when specifying the dissipation rate of the kinetic energy of turbulence and the scalar profiles along the inlet boundary. Furthermore, the flow configuration is reduced to pipe geometry, unlike the geometry of the sudden expansion being typical for a combustor. Hence, the influence of the flow expansion- adverse pressure gradient- cannot be accounted for. Another possibility encountered in the literature is to extrapolate the experimental results from the first available measurement location situated in the interior of the combustor to the combustor inlet, which usually coincides with the inlet cross-section of the solution domain, by preserving the mass flow rate and angular momentum, e.g. Wennerberg and Obi (1993). This method is however wrought with a lot of uncertainties. The most direct way is to compute the flow in the swirl generator. However, it is numerically too demanding, especially if the higher-resolved computational schemes (e.g. LES) are to be applied. The generation of the inflow data by doing separate calculation is regarded as a better choice. Pierce and Moin (1998) proposed a simple, but efficient method for the swirling inflow data generation. The method is based on the computation of the fully developed, annular, swirling flow, imposing a uniform profile of the fictitious azimuthal pressure gradient (azimuthal body force). The method proved its credibility in a Large Eddy Simulation of the Roback and Johnson (1983) case. However the uncertainty in the data structure obtained computationally is still present. In general it is

837

much more desirable to have experimental reference data for comparison. Numerous experimental investigations, serving for years as benchmarks for computational methods and turbulence model validation 2 are known in the open literature, e.g. Roback and Johnson (1983), So et al. (1984), Nejad et al. (1989), Dellenback et al. (1988), etc. However, none of them offered information about the flow within the inlet section. These circumstances motivated the systematic experimental investigation of an isothermal, incompressible flow and mixing in a single tubular gas combustor (Fig. 2) within a range of mean flow Reynolds numbers, volume flow rate ratios, swirl intensities and expansion ratios, Table 1. Special attention was devoted to the inflow characteristics. The present work provides the profiles of both the axial and circumferential velocity and all six nonzero Reynolds stresses for three different swirl numbers S = 0.0, 0.55 and 1.2 and two different mass flow rates 0.1 and 0.25kg / s, corresponding to the Re c numbers (based on the hydraulic diameter) of 49.530 and 132.100 respectively, in the cross-section of the concentric annulus. Complementary to the experimental investigations, a computational study covering both configurations the entire swirl generation system (including swirl generator and annular pipe) and the flow in the concentric annulus, assuming a fully developed flow, was conducted. Hereby, a two-layer version of the basic high-Re second moment closure model due to Gibson and Launder (1978) and its near-wall extension due to Hanjalic and Jakirlic (1998) have been used. In addition, the Large Eddy Simulation of some selected flow cases in the concentric annulus (Re c = 49.530) was performed.

2 E X P E R I M E N T A L FACILITY For this investigation of unsteady flows and mixing, a single tubular swirl combustor (Fig. 2) has been constructed. The modular assembly allows velocity measurements directly after the swirl generator. This was realised by mounting an extended annular tube after the swirl generator, before the flow enters the flue.

Figure 2: Combustion chamber model Table 1 summarizes the operating parameters of the experimental facility, which have been chosen to represent values typical of a real combustor.

2.1 Design of the swirl generator The swirl generator is based on the "movable block" design, Leuckel (1969), see Fig. 3. By rotating an inner and an outer annular block relative to each other, varying degrees of tangential and radial channels will be created. With a pure radial inlet, a non-swirling flow will be obtained (lower left) and with a pure tangential inlet the maximum swirl is generated (upper left). The swirl generator (Fig. 4) works in the range 0 < S < 1.2. The swirl number S is defined, in Eqn. 1 and is a bulk quantity 2 See e.g., Hogg and Leschziner (1989), Wennerbergand Obi (1993), Lai (1996) and Jakirlic et al. (2002).

838

describing the swirl intensity. The swirl number is the ratio of the angular momentum flow (/9 ) to the product of the axial impulse flow ( ] ) and the radius (R) as a characteristic length. Ro

D

S - --

R.'I

-

2 zc l p g ~ r 2 dr R, R.

(1)

R.2~IPg2rdr Rs

TABLE 1" OPERATINGPARAMETERSOF THE COMBUSTIONCHAMBER Range

Parameter i

i

Reynolds number (Main flow)

22.500 < Rein < 112.750

Mass flow rate (Main flow)

0,01 kg/s < m < 0,05 kg/s

Reynolds number (Annular flow)

49.530 < Ree < 132.100

Mass flow rate (Annular flow)

0,1 kg/s < m < 0,25 kg/s

Swirl intensities

0 < S <1,2

Expansion ratio

1,5 und 2,0 TABLE 2: DIMENSIONS OF THE FACILITY Inner (Di)/ outer (Do) Diameter

Identification of the Tube Main flow

36 mm/40mm

Coaxial flow

40mm/100 mm

Flue (ER=2)

200 mm

Flue (ER= 1.5)

150 mm '

Zero swirl

,

Maximum swirl

Middle swirl intensity

Figure 3: Principle of the movable block swirl generator

839

Figure 4: Assembly of the swirl generator (left) and integration inside the entrance chamber (right) 3 MEASERUMENT DETAILS A two-velocity component laser Doppler instrmnent operated in backscatter was used to measure velocity profiles across the annular entrance channel before the flue. Optical access is provided through appropriate windows, Fig. 5. Both a horizontal and a vertical optical access of the laser Doppler probe to the annular flow is available, allowing all three components of velocity (two at a time) to be measured, whereby the axial velocity is measured once redundantly, useful for checking the reproducibility of the results. All velocity profiles were taken 40mm upstream of the sudden expansion. The facility was operated at a matrix of the following parameters: Swirl generator block position:

~= 0~

15 ~ and ~= 30 ~

Mass flow of central (non-swirled): mm = 0.010 kg/s (Rem = 22.500) mm = 0.025 kg/s (Rein = 56.300) mm = 0.050 kg/s (Rem = 112.700) Mass flow of swirling co-flow:

rnc = 0.10 kg/s (Rec = 49.530) rnr 0.25 kg/s (Rec= 132.100)

Figure 5: Optical access for the LDV probe head Seeding particles were introduced into the flow upstream of the blowers. Oil particles/droplets in the

840

diameter distribution were used. The mean diameter is 1.545/zm. The diameter range is from 0.75/zrn to 4/zm. Spectral processors (BSA) were used for signal processing, yielding data rates between 5002000 Hz. Subsequent processing, especially for estimation of the Reynolds stress terms, demanded coincidence of each particle in the two volumes. 4 RESULTS AND DISCUSSION 4.1 Numerical simulations Before commencing with the measurements a detailed numerical investigation has been performed using commercial CFD-software (Fluent). 3-D computations of the swirl generator system were performed (only one eighth of the configuration, meshed by ca. 150.000 cells, was accounted for), Grundmann (2003). The geometry includes an annular section of the entrance chamber, allowing the initial condition to be well defined. At the outlet of the co-axial part of the inlet section the pressure boundary conditions were adopted. Fig. 6 shows the computational grid for the maximum swirl number S = 1.2, for which only tangential channels are present in the swirl generator. The velocity results have been computed for Re = 345.205 (mass flow rate 0.5 kg/s). The streamlines in the coaxial part of the inlet section have been colored by Mach number, exhibiting a low compressibility as well as regions with local separation. The calculations are very demanding of the turbulence model. A Reynolds stress model (basic RSM due to Gibson and Launder - GL, 1978, with variable model coefficients due to Launder and Shima, 1989) with enhanced wall treatment (two-layer approach employing a one-equation turbulence model based on the solution of the equation governing the kinetic energy of turbulence) was used.

Figure 6: Numerical grid of the swirl generator (left) and streamlines in the co-axial part of the inlet section coloured by Mach number (right) Fig. 7 shows profiles of the axial velocity component in the co-axial part of the inlet section (left) and the development of the velocity profiles along the length (120mm) of the extended annular tube (fight). The flow was nearly fully developed at a length of 80mm for high swirl conditions. The inlet data have been taken 40mm upstream of the expansion.

Figure 7: The axial velocity component in the co-axial part of the inlet section (left) and the downstream development of the velocity profile (right).

841

4.2 Experimental Results The measured velocity and Reynolds stress profiles (at the cross-section located 40mm upstream of the expansion) are shown fi'om the inner side (left) to the outer side (right) in across the annular inlet in Fig. 8. All results are normalised using the bulk axial velocity Ub. The results shown in Fig. 8 are taken at a Reynolds number in the annular channel of 132.100 and a swirl intensity of S=1.2. Both the axial and tangential velocities are relatively insensitive to the inner jet flow velocity. The major result from these measurements is that the axial velocity increases from the inside to the outside walls. Contrary, the gradient of the axial velocity decreases from the inside to the outside of the annular flow. The maximum tangential velocity is formed approximately in the middle of the annular channel. 1.4

16

1.2 ..o

0

0

O

.~2

0

flO o

1.5

6 6 o 13

0 0 6 o

13

60 o

13 13

13

0.8 a o &

0.6 0.4 0.2

Re ==22.500 Re ~ 56.300 Re ~ 112.700 0.5

@

E~2 o.o25 o.o3 o.~35 o.~, o.c;4s o.o5

r[m]

o O 6,

i

~02

0.025

0.03

Re ~ 22.500 Re ,= 56.300 Re ~, 112.700

o.O+Srtm+.O, o.o4+ o.os

Figure 8: Normalised axial ( u / U b ) and tangential ( w / U b ) velocity in the annular flow for a Reynolds

number of Rec = 132.100 and swirl intensity S=1.2 in relation to the Reynolds number of the main flow All of the Reynolds stresses u'u', v'v', w'w', u'v', v'w' (not measured) and u'w' are non-zero.

Selected results of the Reynolds stress measurements are shown in Figure 9 and 10. 0.5

r

0.45 ~0.4

0.5

13 B

13 Re ~=22.500 O Re ,= 56.300 zx Re ,= 112.700

0.35

0.45 0.4 0.35

~1 n

n

0.3

0.3

0.25

0.25

0

~

i E.m o.ms om 0.c)35[0.04rm] 0.~,5 0.05

O O O &

13

o i,-i

Re,= 22.500 Re,= 56.300 Re ~ 112.700

o

0

@

o

o

o

o

o

0.625 0.03 0.035 0.04 0.045 0.05 r[m]

Figure 9: Reynolds stresses u'u'/Ub 2 (left) and w' w'/Ub 2 (right) in the annular flow for a Reynolds number of Rec = 132.100 and swirl intensity S=1.2 in relation to the Reynolds number of the main flow Fig. 11 shows the normalized axial velocity and the turbulent kinetic energy in the annular channel as a function of swirl number. For a non swirling flow (S---0) the profile is quite flat at a value U /U b -- 1. With increasing swirl number the formation of the swirl can be understood quite easily and give information about the development of the axial velocity and the sign of the gradient of the inner flow.

842

0.01

0.1 o

0.005

O ZX

Od t..~

~9

o

Re ~ 2 2 . 5 0 0 Re ==5 6 . 3 0 0 Re ,= 1 1 2 . 7 0 0

0

o

0.05

a o

o

zx

g

-0.005

zx o

-0.01

o

-0.015

O o

-0.02 -0.0205. 02

8

~, o 0.03 O.c6s 0.~4

o.02s

rl'ml

Re=22.500 Re == 5 6 . 3 0 0 Re J= 1 1 2 . 7 0 0

o

j]

-0.05

~

o.cJ4s 0.05

o.02s

0.03

o.c6s

0.~;= o.c;4s 0.05

r[m]

Figure 10: u'v---7/Uoz and u'w'/Ub 2 in the annular flow for a Reynolds number ofRec = 132.100 1.4

.

.

.

[]

2

.

A

z~

1.2

0.8

.

o

8

A 0

O o

O o

O o

O

0.6 0.4

0.2

o

A

S ~=0.06

0 S,= 0,6 zX 8~1,2

180/

A

O o

'

/

z~

A

1601-

o

/

o

,_ 140~-/

~

120

zx

o

S ~= 0.06

S=0,6 S ~ 1,2

A

100

0

0

"~ 80

0

0

0

0

o

6o 40

o

o

o

o

o

o

0

~.,,2 o.02s 0.03 o.03s 0.~4 0.d45.... o.os ~C)2 0.I)25 0.()3 0.()35 0.04 0.()45 0.05 r[m] rim] Figure 11: Axial velocity U/Ub (left) and turbulent kinetic energy in the annular flow (right) 4.3 Comparison of experimental and numerical results In addition to the computations of the complete inlet system including the swirl generator and the inlet pipe (Section 4.1, Figs. 12 and 133), separate computations of the swirling flow in the annular pipe section (geometry of a concentric annulus) were performed. The latter computations assumed fully developed flow conditions (periodic inlet/outlet boundary conditions were applied), whereby the (equilibrium) swirling motion was created by introducing a fictitious pressure gradient into the momentum equation governing the circumferential velocity. The magnitude of the pressure gradient (with constant value over the cross-section) was iteratively adjusted until the computed U- and Wvelocity fields satisfied the prescribed swirl intensity (Eq. 1). This method was successfully applied for generating the swirling inflow in a LES of Roback and Johnsons model combustor, Pierce and Moin (1998). In the present work, the method was applied on both the RANS computations (both a 2D axisymmetric and a 3-D geometry were considered using the previously described high-Re SMC GL, Section 4.1 and a near-wall SMC due to Hanjalic and Jakirlic - HJ, 1998; here only the results of the HJ model are shown) and the LES simulations (a second-order central differencing scheme for spatial discretization; the second-order Crank-Nicolson method for time discretization; dynamic Smagorinsky model; solution domain length: Lx=2.67~r(Ro-Ri); Cartesian grid with NxNrN o = 64x49x128 ; CFLm~x = 0.85 ). Whereas the computed (RANS and LES) swirling part of the flow (circumferential velocity and two additional shear stresses, Figs. 12 and 14 right) agrees reasonable with experimental results (the response of the velocity field on the strongest swirl was of a 3 The abbreviation "Comp.-Swirler" denotes the computational results obtained for the complete swirl generator system obtained by using a two-layer version of the GL model (Section 4.1). The denotation "Comp.-Annuli" is related to the computations of the "fully-developed"swirlingflow in the annular pipe by using the HJ model.

843

somewhat weaker intensity), the axial velocity component retains a profile form corresponding to the non-swirling flow in a concentric annulus, exhibiting the opposite (negative) gradient in the large portion of the cross-section, see Figs. 12 left and 14 left. Accordingly, the turbulent stresses agree poorly with the reference data (Fig. 13). The effect of the wrong gradient of the axial velocity is mostly visible on the uv-profile (Fig. 13 left) displaying a zero value at the point of the maximum axial velocity, which is in large disagreement with the experimental results. The possible reason is the weak influence of the circumferential velocity on the axial velocity, whose transport equations include no common variables for the fully-developed flow conditions. The only (indirect) connection between U- and W-equations occurs through the equations for the shear stress components u v and v w . It is obvious (see also Fig. 7)that the experimental flow conditions depart significantly from the fully developed conditions assumed. Pierce (2001) introduced a further correction being equivalent to adding an appropriately defined body force into the U-component momentum equation in order to obtain the proper U-velocity prof'de. This practice assumes the mean statistical properties having been specified. The latter fact lessens the attractiveness of this method for generating the swirling inflow conditions when using RANS methods. Nevertheless, some further investigations are necessary. As expected, the SMC results obtained by accounting for the swirl generator system reproduced all important mean flow and turbulent features in reasonable agreement with available experimental data (solid lines in Figs. 12 and 13).

1.2 oa

L

'

/"

0-... 4

' ' ................. O

0

'0

O'

,.,,,,., . . . . . . . .

0 -.

. .....~............

"

.........................

....~

"'"~

[]

'

~0.0 0.0

9

Comp.-Annuli ........ 0.025

0.03

0.035r, m0.04

0.045

........

O

o ..... ~,.

O

o

0.05

O

o

O

o

o

1.5

=31

]

0.5

Swirling flow annuli I~176in concentric ric anngolimp. ? sS= O ~1.2 B _0_ o

0.0 0.02

:"~

0 0.02

l

0.025

i

0.03

I

Exp.: S=1.2 o Comp.-Swirler - Compi-Annuli .........

0-035r, m 0.04

I

0.045

0.05

Figure 12: Axial velocity profiles for range of the swirl numbers (left) and circumferential velocity profiles for the case with strongest swirl: comparison between experiments and computations 5 CONCLUSIONS The flow structure in the annular chamber section of the inlet system of a swirl combustor with respect to the Reynolds number and swirl intensity influence was investigated experimentally and numerically. The velocity field in the inlet section is measured using the laser Doppler measurement technique. The RANS-SMC results obtained by computing the complete inlet system accounting for the swirl generator agree reasonable with the experimental results for both mean flow and turbulence quantities. The separate RANS-SMC and LES computations of the swirling annular pipe flow imposing the fully developed flow conditions in line with the method for generating the swirling inflow proposed by Pierce and Moin (1998) were also performed. Whereas the (concentrated-vortex type) circumferential velocity field and associated shear stress components were computed in good agreement with the experimental results as well as in good mutual agreement, the axial velocity profile, obtained by both the RANS-SMC and LES methods, exhibits the wrong shape indicating the negative velocity gradient, typical for the non-swirling flow in a concentric annulus. Further investigations related to appropriate corrections are necessary. Acknowledgements

The financial support of the German Ministry for Education and Science (BMBF) through the grant 03TRA2AC (R. Palm) and the German Scientific Foundation (DFG) through the grant JA 941/7-1 (S. ~ari6) is gratefully acknowledged.

844

0.03

'

0.02

0.7 Swirling'flow in COncentric' ~finuli

Exp.': S--1.2 ' o Comp.-Swirler Comp.-Annuli ........

0.6

.......................

o -O.Ol

0

0.02

l 0.025

.! ! 0.035r, m0.04

,

o

o

"N~

>0.3

o

! 0.03

o

~.4

"O ....

0.2 i~xp;;'~'-:~;'~ ..... ~,..... 0.1 .......... ............................... Comp.-Swirler

-0.02 -0.03

o

o.5 f'"'"~

0.01 o~.~

o

! 0.045

0.05

0

0.02

! ........

0.025

!

0.03

I

.

Comp(-Annuli ........ .

.

.

O.035r(m) 0.04

I

......

0.045

0.05

Figure 13: Shear stress (left) and normal-to-the-wall stress components (right) in the concentric annulus of the inlet section of the present model combustor 1.2 0.8

0.4

~0.0 0.0 0.0

_a__~ Swirling flow in concentric annuli

1.6

| Empty ~qmbols:'LES ' 1.4 ~- Full symbols: experiment I-

1.21 I

Low-Re RSM '-------_~1 W/U,,b t3 ~'l 5ouw/U% o

5ow/u~

" All

0.8 0.6 0.4 0.2 o -0.2 0.02

0.02 0.025 0.03 0.035r, mO.04 0.045 0.05 0.025 0,03 0.035r,toO.04 0.045 0.05 Figure 14: Axial velocity (left), circumferential velocity and shear stress profiles (right) in the concentric annulus of the inlet section of the present model combustor

References

[1]

Grundmann, S. (2003): Numerische Untersuchung drallbehafteter StrOmungen in einem realitatsnahen Drallbrennermodell. Diploma thesis, Darmstadt University of Technology (presented at the DGLR Kongress, 2004) [2] Jakirlic, S., Jester-Ztirker,R., and Tropea, C. (2002): Report on 9thERCOFTAC/IAHR/COST Workshop on Refined Turbulence Modelling. October, 9-10, 2001, Darmstadt University of Technology, ERCOFTACBulletin, 55, 36-43 [3] Lai, Y.G. (1996): Predictive capabilities of turbulence models for a confined swirling flow. AIAA J., 34:8, 17431745 [4] Leuckel W. (1969): Swirl intensities, swirl types and energy losses of different swirl generating devices. IFRF Doc. Nr. G02/a/16 [5] Dellenback, P.A., Metzger, D.E., and Neitzel, G.P. (1988): Measurements in turbulent swirling flow through an abrupt axisymmetric expansion. AIAA Journal, 26:6, 669-681 [6] Hanjalic, K., and Jakirlic, S. (1998): Contribution towards the second-moment closure modelling of separating turbulent flows. Computers and Fluids, 27, 137-156 [7] Hogg, S., and Leschziner, M.A. (1989): Computation of highly swirling conf'med flow with a Reynolds stress turbulence model. AIAA J., 27:1, 57-63 [8] Nejad,A.S., Vanka S.P., Favaloro, S.C, Samimy, M. and Langenfeld C. (1989): Application of Laser Velocimetry for Characterization of Confined Swirling Flow. ASME J. Eng. For Gas Turbines and Power, 111, 36-45 [9] Pierce,C.D. and Moin, P. (1998): Method for Generating Equilibrium Swirling Inflow Conditions. A/AA Journal, 36:7, 1325-1327 [ 10] Pierce, C.D. (2001): Progress-variable approach for Large-Eddy Simulation of turbulent combustion. PhD Thesis, Stanford University [11] Roback R., and Johnson B.V. (1983): Mass and Momentum Turbulent Transport Experiments with Confined Swirling Coaxial Jets. NASA Contractor Report 168252 [12] So, R.M.C., Ahmed, S.A., and Mongia H.C. (1984): An Experimental Investigation of gas Jets in Confined Swirling Air Flow. NASA CR 3832 [13] Wennerberg, D., and Obi, S. (1993): Prediction of Strongly Swirling Flows in Quarl Expansions with a NonOrthogonal Finite-Volume Method and a Second-Moment Turbulence Closure, Engineering Turbulence Modelling and Experiments, 2, 197-206

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

845

ON THE S E N S I T I V I T Y OF A FREE A N N U L A R S W I R L I N G JET TO THE L E V E L OF SWIRL AND A PILOT JET M. Garcia-Villalba and J. Frrhlich SFB 606, University of Karlsruhe, Kaiserstr. 12, 76128, Karlsruhe, Germany

ABSTRACT The paper presents large eddy simulation of unconfined swirling jets. In the first part, an unconfined annular jet is investigated for swirl numbers ranging from 0 to 1.2. The impact of the swirl on the mean flow and the precessing vortex structures in this flow is analysed. In the second part of the paper, a co-annular pilot jet is introduced near the axis. The investigations show that the additional swirl near the axis has a stronger effect than the pilot jet itself, leading to an almost entire removal of coherent structures.

KEYWORDS swirling jets, annular jets, large eddy simulation, coherent structures, pilot jet

INTRODUCTION

Swirling flows are widely used in many engineering applications. In combustion devices, they are often used to stabilize the flame by means of a recirculation zone. Swirling flows, however, are prone to instabilities which can trigger pronounced unsteadiness of combustion degrading the performance of the whole system. Lean premixed burners in modern gas turbines often contain a richer pilot flame typically introduced near the axis of the swirl burner raising the question of how this additional jet modifies the fluid mechanical behaviour of the system. This issue is addressed in the present paper for a model system corresponding to an experimental setup. Swirling flows are difficult to model with Reynolds-averaged methods (Jakirlid et al. 2002) due to the effects of streamline curvature of the mean flow. Large eddy simulations (LES) of such flows not encountering this problem such as Wegner et al. (2004), Wang et al. (2004), etc., are still scarce. The present paper aims at using LES to investigate these flows in a physical perspective and in particular to analyse their large scale instantaneous vortex structures. In Garcia-Villalba et al. (2004a, 2005) the present authors performed LES of an unconfined annular swirling jet and validated the simulation method by means of detailed comparisons with experiments for the same configuration. Large scale

846

coherent helical structures precessing around the symmetry axis at a constant rate were identified in these computations. The first goal of the present paper is to investigate the influence of the swirl parameter on these structures. Second, the impact of an additional co-annular pilot jet near the axis is investigated.

NUMERICAL METHOD The simulations have been performed with the code LESOCC2 (Hinterberger 2004), which is a successor of the code LESOCC (Breuer & Rodi 1996). It solves the incompressible Navier-Stokes equations on curvilinear block-structured grids. A collocated finite-volume discretization with secondorder central schemes for convection and diffusion terms is employed. Temporal discretization is performed with a three-stage Runge-Kutta scheme solving the pressure-correction equation in the last stage only. The computations for the two issues addressed in this paper have been performed with two very similar configurations. These match two experiments performed by Biachner and Petsch (2004) which are used for validation (the first configuration had also been investigated previously by Hillemanns 1988). For each sensitivity study one of the computations corresponds to an experiment so that these data can be used for comparison. The variation of the swirl number was investigated using the geometry shown in Fig. l(a) with an inner diameter of the annular jet of 0.5D where D is the outer diameter of the jet. The geometry of the second configuration only differs with respect to the inlet and is detailed below. The rest of the computational domain is the same (Fig. l(a)). The block-structured mesh consists of about 2.5 million cells in both cases. The grid is stretched in both the axial and the radial direction to allow for concentrations of points close to the jet exit and the inlet duct walls, while 100 grid points are used in the azimuthal direction. The stretching factor is everywhere less than 5 %. The minimum axial spacing appears at the jet outlet and is Ax = 0.02 R. Close to the walls, the minimum radial spacing is Ar = 0.012 R. In the first part of the paper, the subgrid-scale model used is the Smagorinsky model with Van Driest damping and a model constant Cs=0.1. In the second part, the dynamic model of Germano et al. (1991) has been employed, with least squares averaging and threedimensional test filtering. The eddy viscosity in the latter case is smoothed by temporal relaxation. In what follows, R=D/2 is the reference length and capital letters are used throughout the paper to indicate values averaged in time and circumferential direction.

(a)

~,

i...! ~ ~ l...!...q ~ ~ O. D I- /U/f ///A ....... I ~! l"--. "

/ / .......

1t

..............

[

(b)

,,

4D

I

1

12D

t ......

/,/"

..

,,., ............

. .......

...

....

- .........................................................

2,6D x

.......... i / '

' 0

D

16 D

0,5

0.8

07

,~

0.8

09

Figure 1: (a) Sketch of the computational domain and boundary conditions for the swirl study. (b) Inflow conditions. Mean tangential velocity imposed at x/R=-2. The line styles are defined in Table 1. The inflow conditions are obtained by performing simultaneously a separate periodic LES of swirling flow in an annular pipe using body forces to impose swirl and flow rate as described in Pierce & Moin (1998). This approach is illustrated in Fig. 1(a) and has been validated in Garcia-Villalba et al (2004b). No-slip boundary conditions are applied at the walls. The entrainment of outer fluid into the jet is

847 simulated using a weak co-flow in the outlet plane x/R=O remote from the jet. Free-slip conditions are applied at the open lateral boundary which is placed far away from the region of interest (see Fig. 1(a)). A convective outflow condition is used at the exit boundary.

SENSITIVITY TO THE LEVEL OF SWIRL An overview of the simulations performed is shown in Table 1. The Reynolds number of the flow based on the bulk velocity U0=25.5 rn/s and the outer radius of the jet R=50 mm is Re=81500. The swirl parameter is defined at the inflow plane x/R = -2 as

(1)

where u and w are the axial and azimuthal velocities, respectively. The range covered by the simulations is very wide, including a simulation without swirl, Sim. 1, another with a low level of swirl, Sim. 2, and three simulations with a high level of swirl, Sims. 3,4,5. TABLE 1 OVERVIEW OF THE SIMULATIONS PERFORMED TO INVESTIGATE THE IMPACT OF THE SWIRL NUMBER

Simulation 1 Swirl number S 0 Line style solid(thin)

2

0.4 dashed-dotted

3

4

0.7 1 dotted dashed

5

1.2 solid(thick)

Fig. 1(b) addresses the inflow conditions for the main simulation imposed at x/R=-2. It shows the mean azimuthal velocity resulting from imposing the desired swirl number in the precursor simulation. In fact, the mean azimuthal velocity increases with S, while the mean axial velocity (not shown here) is almost unchanged in all cases. Streamlines

First of all, a general view of the flow is presented. Fig. 2 shows the time-averaged streamlines computed from four of the simulations in Table 1. As the jet is annular, the flow characteristics differ from those of a usual round jet. In the non-swirling case, Fig. 2(a), a geometry-induced recirculation zone (GRZ) is formed due to the bluff-body effect of the cylindrical centre body. Fig. 2(b) shows the case of low swirl, Sim. 2. In this case, additional to the GRZ, a very thin central recirculation zone (CRZ) appears close to the axis. It extends up to about x/R=4. For this level of swirl, no CRZ is expected in a round jet (Gupta et al 1984), but in the present case the cylindrical center body introduces this feature. Increasing S leads to an increase in the size of the CRZ. For S=0.7, Fig. 2(c), the length of the CRZ is about 4R, and its width is increased to 0.6R, attained at x/R=2. The GRZ is still present at this level of swirl but substantially reduced in size and strength. For S=I, not shown here, the CRZ is longer reaching until x/R=8 and attaining its maximum width of 0.8R at x/R-1.5, i.e. further upstream compared to Sim. 3. Finally, Fig. 2(d) shows the case S=1.2, in which the CRZ has reached x/R=O, and the GRZ has been merged into the CRZ. The length of the CRZ is about 1OR and the maximum width of 0.8R is attained at x/R=l. Fig. 2 shows that with increasing swirl number the jet spreads further outwards in radial direction and the strength of the CRZ increases substantially. Let us finally address the slope of the streamlines in Fig. 2 remote from the jet, starting at x/R=O and r>R. Their shape is due to the co-flow boundary condition. Note, however, that the velocity at this location is only 5% of the jet axial velocity, so that the influence on the region of interest is negligible, as will

848 be seen in Figs. 3 and 4. This was also checked with different amounts of co-flow in Garcia-Villalba et al. (2005).

Figure 2: Streamlines of the average flow. (a) Sim 1. (b) Sim 2. (c) Sim 3. (d) Sim 5. Mean Flow and Statistics

Experimental data were available only for one flow condition, equivalent to S=1.2. The comparison of experiment and simulation was performed in Garcia-Villalba et al. (2004a) and is not repeated here. The agreement between experiment and simulation is excellent. Figs. 3 and 4 show mean velocity and turbulent intensity profiles at two axial positions in the near flow field of the jet. Fig. 3 shows profiles very close to the jet exit at x/R=0.2. Here, the jet forms two complex three-dimensional shear layers, the inner one with the recirculation zone, and the outer one with the surrounding co-flow. At this position, x/R=0.2, the inner one increases in thickness with S, reaching 0.5R for S=1.2, while the outer one remains thin and is just displaced radially outwards with increasing S. The axial fluctuations in Fig. 3(c) exhibit a peak in the region of the shear layer. The thicker the shear layer, the more pronounced and wider is the peak. The outer shear layer does not present these variations, but with increasing S, the velocity-difference is larger, and therefore the turbulence intensity is also larger. The velocity difference is generated by both axial and azimuthal velocity and hence complemented by Fig. 3(b) showing mean tangential velocity profiles. Similar conclusions as for the axial fluctuations hold for the azimuthal ones in Fig. 3(d). Fig. 4 shows the same quantities as Fig. 3 but at x/R=3. This position is located within the CRZ in the simulations with swirl. A qualitative difference between the simulations with S < 0.4 and the simulations with S > 0.7 is observed in all data. The spreading and decay rate is much lower in the former, Fig. 4(a). The profiles of mean velocity and fluctuations in Sims 3-5 do not present substantial differences at this location, i.e. as soon as the swirl is high enough to produce a strong recirculation zone, a kind of saturation of the profiles is reached. The shape of the turbulent intensities is also the same, Figs. 4(c) and 4(d), with a slight increase of intensity with S. At this position, the distinction between shear layers is not reflected by the profiles of the fluctuations. Only for the two low swirl cases, it is still possible to distinguish the peaks due to the fact that these flows develop slower in space than the others.

849 is

lS

9

f"

.i

o

-o2

,/-\\

(C)

05

1

o;

o4!

/

/.,.~ [=|

o25

oo~ 0.5

1

rZR

1.S

o.s

1

rm

i .s

Figure 3" Profiles at x/R=0.2 (a) Mean axial velocity. (b)Mean tangential velocity. (c) RMS values of axial velocity. (d) RMS values of tangential velocity. The line styles are defined in Table 1.

or

o

o.5

1

~z

0.5

1

1.5

,

.

2

25

3

35

-o,i!

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2

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s

35

2

25

3

35

o

os

~

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2

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35

03

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o

rm

Figure 4: Profiles at x/R=3 (a) Mean axial velocity. (b)Mean tangential velocity. (c) RMS values of axial velocity. (d) RMS values of tangential velocity. The line styles are defined in Table 1.

Flow visualization and Spectra In Garcia-Villalba et al. (2005) large scale coherent structures were identified and their evolution and interaction described for a high swirl number case, equivalent to Sim. 5. It was shown that two families of structures appear, which are best visible in Fig. 5(b). The outer, spiralling structure is located in the outer shear layer, see Fig. 3(a), where Oux /Or < 0, the darker one in the inner shear layer between the

850 annular jet and the recirculating fluid, where Ou x / Or > 0. In the cited reference it has been shown that these structures result from Kelvin-Helmholtz instabilities as they are perpendicular to the average streamlines. It is now interesting to study how these structures are modified when varying the swirl number. This is reported in Fig. 5. Due to lack of space, only one snapshot is included but further views and animations were produced upon which the following comments are based. In the literature on the subject the inner structure is usually called 'precessing vortex core' (PVC), (Gupta et al 1984). In the case of low swirl, it is not expected to be observed. However, as in the present case a very thin CRZ is produced, a thin elongated structure can be seen in Fig. 5(a). Larger values of p '<0 do not show any larger scale structure in these data. In the case of strong swirl the structures mentioned above are observed. For S=0.7, a single inner and a single outer structure are present. Animations of the flow show that their rotation is in phase and very regular. Upon increasing S, Figs. 5(c) and 5(d), the irregularity of the flow grows in the sense that the PVC change in number during their evolution and for S-1.2, up to three of them can co-exist at certain instants. Furthermore, with increasing S, the inner structures enter the inlet duct, slightly for S=I, and in a more pronounced way for S=1.2. At the same time the PVC is displaced off the symmetry axis since the shear layers are shifted outwards as discussed above. As a consequence, the tangential fluctuations near the axis increase for S--0 to S=I while decreasing for the larger value S=1.2. The radial fluctuations behave similarly. The outer structures consist of a long spiral for S=0.7 and also become more irregular with increasing S. Another interesting feature is that the separation between the inner and the outer structures decreases with increasing S, to the point that in absence of colour it is difficult to distinguish them when S =1.2. The reason for this behaviour is that the two shear layers, identified as the origins and locations of both types of structures, approach each other with increasing S, as seen in Fig. 3(a).

Figure 5: Instantaneous coherent structures visualized using an iso-surface of the instantaneous pressure deviation p '=p-P= - 0.2. (a) Sire 2. (b) Sim 3. (c) Sim 4. (d) Sim 5. The colour is computed according to the sign of the radial gradient of the mean axial velocity.

Figure 6: PSD of axial velocity fluctuations at x/R=0.1, r/R=0.7. Arbitrary units are used in the vertical axis and the curves have been shifted vertically for readability. (a) Diagram with logarithmic axes. (b) The same diagram with linear axes. The line styles are defined in Table 1. Fig. 6 shows the power spectrum density (PSD) of axial velocity fluctuations at x/R=O. 1, r/R=0.7, i.e. very close to the jet exit and in the region of the inner shear layer for all cases. In the cases of low swirl the spectra do not show a pronounced peak. When the level of swirl is high, i.e. S > 0.7, a dominant

851

peak and its higher harmonics appear in the spectrum. This peak reflects the precessing structures observed in Figs. 5(b)-5(d) (Garcia-Villalba et al. 2004a). For S=0.7 and S=I, the same frequencyfpeak = 0.24 Ut,/R is observed, changing only slightly to fpeat = 0.28 Ut,/R for S=1.2. Note that although the mean tangential velocity increases by about Uo in the inlet section (Fig. 1(b)), this increase is reduced near the outlet of the annular pipe (Fig. 3(b)) and further downstream (Fig. 4 (b)). Nevertheless, the fact that Aeak does not change much with S over a broad range is an important result of the present study.

I

04

0

j 05

r/R

r/R

Figure 7: Profiles of mean velocity at x/R=O.1 (a) Mean axial velocity. (b) Mean tangential velocity. The line styles are defined in Table 2.

INFLUENCE OF A PILOT JET

Configuration In the second part of the paper, the effect of an additional inner jet is studied. The inflow part of this configuration differs from the one described in Fig. 1 while the rest of the computational domain is identical. Here, the precursor simulation for the main annular jet is replaced by a duct reaching to x =3.82R and bending radially outwards to r=2.18R. At this position, where in the experiment the radial swirl generator is located, steady swirling flow is imposed in the simulations. It undergoes a rapid pseudo-transition and reproduces the experimental profiles of the main jet in the plane near the outlet quite well (cf. Fig. 7 below), provided that the swirl is adjusted appropriately. This procedure was studied and compared to the one of the previous section in Garcia-Villalba et aL (2004b). To allow comparison with the simulations of the previous section, the resulting swirl number in the main jet at x=-2R, has been determined. Its value is S=1.05. In the experiment, a co-annular pilot jet was introduced featuring an axial swirl generator ending flush with the outlet plane at x=O. This jet is modelled with a precursor simulation similar to Fig. 1 which also ends at x=O. The direction of swirl is co-rotating with the main jet. In this configuration the inner and outer diameter of the pilot jet is 0.27D and 0.51D, respectively, while the inner and outer diameter of the main jet is 0.63D and D, respectively. The flow conditions are very similar to the previous ones. The Reynolds number based on the bulk velocity of the main jet only Ub=22.1 m/s and the outer radius of the jet R--55 mm is Re=81000 while the swirl number at x=-2R is almost exactly the same as for Sim 4. The different width of the main jet, however, precludes direct comparison with the above computations. The mass flux of the pilot jet is 10% of the total mass flux and the swirl number for the pilot jet alone is 1. TABLE 2 OVERVIEW OF THE SIMULATIONS PERFORMED TO INVESTIGATE THE INFLUENCE OF A PILOT JET

Simulation Inner jet Inner jet with swirl Line style

6 7 No Yes No solid dashed

8 Yes Yes dotted

Exper. Yes Yes O

852 The purpose of these simulations is to clarify the impact of the inner jet on the stability of the entire flow. This is investigated by consideration of three cases which are summarized in Table 2. Fig. 7 shows the mean axial and azimuthal velocity profiles near the outlet at x/R=O.1 for the three cases, in order to visualize the strength of the pilot jet. Thedeviation with respect to the experimental data in this range results from the model for the pilot jet. Its inflow condition is imposed at x=0, while in the experiment the flow is less stiff. Moreover, the guide vanes of the axial swirl generator are not represented since this renders grid generation very complicated. Nevertheless, the present data allow very well to assess the impact of the pilot jet on the flow as will be demonstrated below.

Mean flow The streamlines for the three cases investigated are displayed in Fig. 8. Since S is in the range of high swirl, the flow is very similar to the ones in the previous section. Fig. 8(a) showing the case without pilot jet indeed is very close to Fig. 2(d). As before, a CRZ starting directly behind the cylindrical centre body occupies a long region near the symmetry axis. The streamlines with pilot jet of Sim 7 in Fig. 8(b) are very close to those of Sim 6, with just the streamlines emanating from the pilot jet entering the inner shear layer. In the case of the swirled pilot jet, Sim 8, the shape of the recirculation zone is slightly modified in the region of its maximum width but still remains mostly unchanged. Fig. 9 presents a comparison of velocity profiles at x/R=3 from the simulations together with experimental data. Fig. 7, 8, and 9 show that the influence of the pilot jet on the average flow is only small.

Figure 8: Streamlines of the average flow. (a) Sim 6. (b) Sim 7. (c) Sim 8.

Figure 9: Profiles of mean velocity at x/R=3. (a) Mean axial velocity. (b) Mean tangential velocity. The line styles are defined in Table 2.

Flow visualization and Spectra The instantaneous flow is now visualized in exactly the same way as in the previous section using isosurfaces of pressure fluctuations, Fig. 10. Obviously, there are important differences between the three cases. Without pilot jet, Fig. 10(a), the structures have the characteristics described in the first part of the paper. They are very coherent, precess at a quasi-regular rate and persist over long time intervals. When the non-swirled pilot jet is introduced, Fig. l O(b), it is still possible to recognize similar structures as in the case without pilot jet. These are however substantially less coherent, much thinner,

853 and do not persist that long. In particular, the PVC are smaller and are more numerous along the circumference. In this case, four or five small PVC can co-exist at certain instants. Finally, the addition of swirl to the pilot jet has a dramatic impact on the flow, Fig. 10(c). The regularity is completely lost and the appearance of the structures is more random. Here, even the outer structures are affected and have almost vanished or, when they appear, exhibit only small coherence The addition of near-axis swirl hence is observed to have a strong influence on the instantaneous flow characteristics. Recently, in a different context, the addition of near axis swirl has been proposed as a strategy to control vortex breakdown, Husain et al. (2003).

Figure 10 : Instantaneous coherent structures visualized using an iso-surface of the instantaneous pressure deviation p '-

-- - 0.2. (a) Sim 6. (b) Sire 7. (c) Sire 8. Colour is given by the sign of the radial gradient of the mean axial velocity. The previous analysis of the coherent structures is confirmed by analysing the PSD of the radial velocity fluctuations at two points close to the outlet at x/R=0.4. Fig. 1 l(a) shows this data on the symmetry axis. Note that in spite of the difference in geometry with respect to the configuration studied in the previous section, the peak in the spectrum for Sim. 6 also appears at a frequency fpeak = 0.24 Uy/R. The difference in geometry hence does not have much influence in the precessing rate of the structures. A pronounced peak can also be observed for Sim. 7. The spectrum of Sim. 8 also shows a peak but at substantially larger time scales. These are hardly resolved by the present integration time and deserve further investigation. Fig. 11 (b) shows the PSD at r/R--0.6, i.e. in the inner shear layer of the main jet. The spectrum of Sim. 6 shows a pronounced first and second harmonic (both label A). The spectrum of Sire. 7 also exhibits peaks at these frequencies, but their energy content is smaller. Instead, more energy is displaced to the next harmonic, which shows that the PVC are more irregular in this case. For Sim. 8 the energy is contained in substantially higher harmonics (the most dominant ones with label B) and much less in low-frequency modes.

Figure 11: PSD of radial velocity fluctuations at x/R=0.4. (a) location on the symmetry axis, r/R=0. (b) r/R=0.6. The line styles are defined in Table 2.

854 CONCLUSIONS The computations performed for an annular jet show dominating spiralling structures in the two shear layers present in this flow. With increasing swirl number and S beyond 0.7, with the definition employed here, their shape becomes more complex but the precessing frequency remains almost constant over a wide range. When a pilot jet is introduced close to the axis the average flow is only little affected. Visualizations and spectra however demonstrate that although axial and angular momentum of this jet are small, it has a dramatic effect on the instantaneous vortex structures. This is an important result for the consideration of mixing effects in this type of flow. Based on the data presented here more analyses of the flow field will be performed in the future. It would also be interesting to further modify the parameters of the pilot jet and, e.g., investigate the consequences of counter-rotating swirl. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the German Research Foundation (DFG) through the collaborative research center SFB 606 'Unsteady Combustion'. The calculations were performed on the IBM Regatta of the Computing Center of Garching (RZG) and on the VPP5000 of the Forschungszentrum Karlsruhe (FZK). REFERENCES Breuer M. and Rodi W. (1996). Large eddy simulation of complex turbulent flows of practical interest, In Hirschel E. (ed.), Flow simulation with high performance computers II, vol 52 of Notes on Numerical Fluid Mechanics, pages 258-274. Vieweg, Braunschweig, Germany. Btichner H. and Petsch O. (2004). Private communication. Garcia-Villalba M., Fr6hlich J. and Rodi W. (2004a). Unsteady phenomena in an unconfined annular swirling jet. In Andersson H.I. and Krogstad P.A. (eds.) Advances in Turbulence X. 515-518. Cimne, Barcelona, Spain. Garcia-Villalba M., Fr6hlich J. and Rodi W. (2004b). On inflow boundary conditions for large eddy simulation of turbulent swirling jets. In Proc. 21st Int. Congress of Theoretical and Applied Mechanics. Warsaw. Poland. Garcia-Villalba M., Fr6hlich J. and Rodi W. (2005). Large eddy simulation of the near field of a turbulent unconfined annular swirling jet. In preparation Germano M., Piomelli U., Moin P. and Cabot W. (1991). A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3, 1760-1765 Gupta A.K., Lilley D.G. and Syred N. (1984). Swirl Flows, Abacus Press, Kent, USA. Hinterberger C. (2004). Dreidimensionale und tiefengemittelte Large-Eddy-Simulation von Flachwasserstr6mungen. PhD thesis, University of Karlsruhe. Hillemanns R. (1988). Das Str6mungs und Reaktionsfeld sowie Stabilisierungeigenschaften von Drallflamen unter dem Einfluss der inneren Rezirkulationszone. PhD thesis, University of Karlsruhe. Husain H.S., Shtern V. and Hussain F. (2003). Control of vortex breakdown by addition of near-axis swirl. Phys. Fluids 15:2, 271-279 Jakirli6 S., Hanjali6 K. and Tropea C. (2002). Modeling rotating and swirling turbulent flows: a perpetual challenge. AIAA J. 40:10, 1984-1996 Pierce C.D. and Moin P. (1998). Method for generating equilibrium swirling inflow conditions. AIAA J. 36:7,1325-1327 Wang P., Bai X.S., Wessman M. and J. Klingmann (2004). Large eddy simulation and experimental studies of a confined turbulent swirling flow. Phys. Fluids 16:9, 3306-3324 Wegner B., Kempf A., Schneider C., Sadiki A., Dreizler A. and Janicka J. (2004) Large eddy simulation of combustion processes under gas turbine conditions. Prog. Comp. Fluid Dyn. 4,257-263

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

855

PREDICTION OF PRESSURE OSCILLATIONS IN A PREMIXED SWIRL C O M B U S T O R FLOW AND C O M P A R I S O N TO MEASUREMENTS P. Habisreuther, C. Bender, O. Petsch, H. Btichner and H. Bockhom Engler-Bunte-Institute, Division of Combustion Technology University ofKarlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, GER

ABSTRACT The most common and reliable technique used for flame stabilization of industrial combustors with high thermal loads is the application of strongly swirling flows. In addition to stabilization, these swirl flames offer the possibility to influence emission characteristics by simply changing the swirl intensity or the type of swirl generation. Despite of these major advantages, swirling flows tend to evolve flow instabilities, that considerably constitute a significant source of noise. In general, noise generation is substantially enhanced, when such a swirling flow is employed for flames. Thus, the minimization of the resulting noise emissions under conservation of the benefit of high ignition stability is one major design challenge for the development of modem swirl stabilized combustion devices. The present investigation makes an attempt to determine mechanisms and processes to influence the noise generation of flames with underlying swirling flows. Therefore, a new burner has been designed, that offers the possibility to vary geometrical parameters as well as the type of swirl generation, typically applied in industrial devices. Experimental data has been determined for the isothermal flow as well as swirl flames with 3-D-LDV-diagnostics comprising the components of long-time averaged mean and rms-velocities as well as spectrally resolved velocity fluctuations for all components. The noise emission data was acquired with microphone probes resulting in sound pressure levels outside the zone of perceptible fluid flow. In addition to the experiments, numerical simulations using RANS and LES have been carried out for the isothermal case. The results of the measurements show a distinct rise of the sound pressure level, obtained by changing the test setup from the isothermal into the flame configuration as well as by varying geometrical parameters, which is also resembled by the LES simulation results. Additionally, a physical model has been developed from experiments and verified by the LES simulation, that explains the formation of coherent flow structures and allows to separate their contribution to the overall noise emission from ordinary turbulent noise sources.

KEYWORDS Swirl combustion, Combustion Noise, LDV-Measurement, Microphone Measurement, LESSimulation

856 OBJECTIVE Due to high turbulence intensity the most outstanding properties of swirl flows is their capability of providing a fast mixture of fuel and air and the phenomenon of forming a central recirculation zone when swirl intensity exceeds a critical value. Fast mixture is the basis to obtain short flames with high thermal loads, and, therefore, is a key feature when designing modern combustion systems. In addition, a central recirculation zone enhances stability due to the back-flow of hot gases which serve to ignite the cold mixture (Beer & Chigier (1972), Leuckel & Fricker (1972)). For industrial application, the ability to vary swirl intensity with relatively small effort offers a relatively cheap possibility to influence the emission characteristics of the combustion system. An overview of basic research and state-of-the-art investigations of swirl flames is given e.g. in Lilley (1977), Gupta et.al. (1984), Keck et.al. (2002). Besides the mostly positive properties, swirl flames also tend to evolve coherent flow instabilities, which may in some cases again lead to flame stability problems and increasing noise emissions. In the recent decades many investigations have been carried out on the topic of combustion instabilities using pulsed inflows (Laverdant & Candel (1989), Btichner (1992), Bai et.al. (1993)). An other academic system, that at least has the existence of a recirculation zone in common with technical swirl flames, deals with combustion, stabilized by the recirculation after a backward facing step (e.g. Cohen & Anderson (1996)). A major difference of these experiments to swirling flames is nature of the central recirculation zone, that usually exists in swirl stabilized flames. Investigations of combustion instabilities in swirl flames are published e.g. in Schadow et.al (1989), Ktilsheimer & Btichner (2001), Btichner & Lohrmann (2003) and state the importance of coherent structures in the vicinity of the stabilization zone. The coherent structures represent a phenomenon that carefully has to be taken into account in modem burner development and, therefore, is a major topic in present investigations. With increasing computational speed also the calculation of coherent structures, often formed under acoustic forcing, has come into play. Currently, the large eddy simulation method is the most powerful technique for calculation of this type of combustion instability (Poisnot et.al. (1987), Angelberger et.al. (2000), Fureby (2000)). Direct numerical simulation still is restricted to low Reynolds numbers or spatially small regions of interest (Ruith et.al. 2003).

EXPERIMENTAL SETUP In the presented work, the measured flow fields of isothermal swirling flows as well as of the corresponding swirl flames are presented for a new burner, which was especially designed for the variation of typical geometrical parameters of industrial swirl combustor devices. A sketch of this burner is shown in Figure 1. This burner covers two flows, a central fuel flow (pilot) and a concentrically aligned air flow, which may both separately loaded with swirl. The inner diameter of the burner nozzle Figure 1: Sketch of the new burner. (air flow) is D0=110 mm and the inner diameter of Swirl generation: pilot flow-axial the fuel-lance is d0=57mm. In addition to this swirler; main flow-tangential configuration, the burner may be operated in channels. premixed mode as well, in which case both flows are substituted with already premixed fuel/air mixtures. To preserve the combustible volume as small as possible, premixing of the outer annular flow is carried out close by the burner mouth with fuel fed by the double-walled lance. For the generation of swirl in the central pilot lance, axial swirlers

857

Figure 2: Experimental setup. with blades at various angles of attack may be applied. The swirl in the annular concentric main flow on one hand may be generated using axial swirlers. On the other hand, as shown in Figure 1, rectangular channels may also be used to feed the inflow. These channels can be rotated to gain variable tangential orientation, allowing to adjust continuously swirl intensity freely without re-installation effort. Together with the possibility to arrange the axial position of the pilot lance, these design patterns resemble a wide spectrum of burner designs used in industrial applications. To provide a reproducible wall boundary condition for the appropriate numerical simulations, all experiments were carried out in an open surrounding with the burner assembled in the center of a horizontal circular bottom-plate. At first, isothermal experiments were carried out substituting fuel with air, supplied by a compressor. The following experiments covered flame-configurations in premixed operational mode with overall equivalence ratios from dp=0.55 to 1. The volume flux of the air and fuel flows were balanced using a calibrated metering orifice and loaded with solid particles (MgO, particle size < 1 ~tm) from a particle generator. The measured data of the turbulent flow fields were acquired with a three channel LDA (LDA, laser Doppler anemometry) diagnostic system. This system uses a water-cooled Ar-Ion-laser radiating three characteristic wave lengths (~=514 nm; ~2=488 nm; ~3=476 nm). The emission and reception optics were operated in back-scattering mode and aligned rectangular in a way that the coordinate systems of burner and diagnostics matched and, thus, a coordinate transformation of the acquired data could be omitted. Fine adjustment of the three laser-beams was done using an adjuster bit. The whole diagnostic system could be traversed over three axes to a specified location of interest, using an electronic control unit (Figure 2). The experimental assessment of the raw, stochastic data covered the calculation of long term averages as well as statistical RMS-values for all three velocity components. To get quantitative information about the expected and found occurrence of coherent structures, a spectral analysis using the Fast-Fourier-Transform (FFT) of the statistically distributed data was carried out for the velocity components at several monitoring points. To gain information about the acoustic radiation of the flow fields and the flames, the sound pressure level (SPL) has been measured with a microphone probe at various axial and circumferential positions. Again, the experiments were performed in an open large environment to minimize influence of noise reflections. The probing unit was a B&K condenser microphone. The microphone signals were processed to get both, the wavelength-integral sound pressure level and the spectral resolved amplitudes of the acquired noise signal. From several operational conditions and configurations investigated, three isothermal flows and one premixed flame are presented below. The isothermal cases employed a total inflow

858 of air of 200 mN3/h (flow

ratio" '~rmain / *Qpilot -- 9/1) with and without withdrawn pilot lance (Xpilot: -40 mm / 0 mm) and 609

mN3/h (Xpilot: -40 mm). The premixed flame measurements where acquired using a total thermal load of 135 kW (air equivalence ratio:

~main=l.7 / ~pilot=l.05). These conditions lead to Reynolds numbers based on the average axial nozzle velocity and Do of mN3/h), 59000 (200 180000 (609 mNa/h) and 67000 (premixed flame).

Figure 3: Section of the computational grid representing the interior of the double annular burner with twisted region at the mouth of the pilot lance.

NUMERICAL METHODS In the present work the isothermal flow field was calculated in parallel with the experiments described above, using the computational fluid dynamics solvers CFX-TASCflow 2.12 and CFX 5.7. As Version 5.7 of the CFX code is a further development of the CFX-TASCflow solver, both codes use a coupled strategy to solve the mass and momentum equations. For turbulence closure two different two-equation models in a RANS context as well as the LES strategy were applied. The two equation models used were the k-g model in its standard formulation (Launder & Spalding 1974) and a k-o~ model according to Menter (1994). For turbulence closure within the subgrid scale of the LES procedure Smagorinsky's model (Smagorinsky (1963)) was used in its standard formulation with a fixed constant Cs=0.16. The LES calculation employed second order spatial (central difference) and temporal (second order backward euler) schemes. The computational grids used, were built up to represent main parts of the interior flow domain of the burner as well as a cylindrical zone up to 1.2 m axial distance from the burner mouth with a diameter of 1.9 m. Three block stn~ctured grids with increasing resolution (total number of nodes: 0.35, 0.7 and 1.3 million) were used for previous RANS simulations (Habisreuther et. al. (2004)). to prove grid independence of the results obtained. On basis of these results, all simulations shown in this work were obtained using only the 0.7 million nodes grid. Although being coarse compared to LES calculations shown in literature, this grid is relatively fine in the vicinity of the burner mouth, where typical cell lengths are of about 1 mm in each direction and, additionally, provides comparability with the RANS simulations. A further comparison with results of LES simulations using a more time consuming, finer grid will be the next thing to do. A very important feature of the grids used was the treatment of the axial swirler blades in the pilot lance. These blades were approximated using wall boundary conditions between circumferential parts in a twisted grid section at the end of the pilot lance (Figure 3). Using this approach, a pure axial inflow condition for the LES simulation could be applied, omitting a circumferential component for the time-mean and fluctuating velocities. For the chosen inflow boundary conditions a fixed mass flow with a given direction for both inlets (axial for pilot and axial/tangential for main) of the RANS and LES calculations was given. The turbulence at the inlets was specified assuming the degree of turbulence Tu and the turbulent macro length scale It for the RANS calculations (both inlets: Tu=0.1; pilot inlet: lt=lmm; main inlet: lt=8mm). The turbulent fluctuations for the LES calculations were assumed to evolve themselves in the domain upstream of the burner outlet. This

859 advance is assisted by the narrow channels between the swirler blades in the central lance and by the strong radial acceleration of the flow from the main inlet to the annular burner outlet of the main flow. The boundary conditions used at the outer borders of the calculation domain was a free slip condition at the radius of 0.85 m and an outlet condition at the burner distance if 1.2 m. As the goal of the LES-simulation was to check whether coherent structures can be calculated within the turbulent swirling flow, all simulations were carried out in a first step applying only the incompressible formulation of the balance equations for momentum and mass assuming constant density (Habisreuther et.al. (2004)). Using the so calculated velocity and pressure fluctuations within the framework of an acoustical perturbation equations (APE) approach (Ewert et. al. (2002)), serves as a first approach to numerically access the noise radiated from the flow and will be the next step to do. But, before this hybrid procedure is being engaged, the proper representation of the coherent structures found in experiments has to be checked.

Figure 4: Axial velocity profiles of the isothermal flow at four traverses in the vicinity of the burner. The lines denote calculations, circles are measured data ('Q = 200 mN3/h; lance position XL=0mm). TIME-AVERAGED ISOTHERMAL FLOW In the following section, experimental and numerical results are shown for the double concentrical swirl-burner setup described above. The first section of the examinations concentrate on the description of the isothermal flow field connected with the reacting case, which is discussed later. The experiments were carried out under variation of several operational and geometrical parameters like swirl intensity (pilot and main flow), flow rate ratio (pilot to main), and position of the central lance. The appropriate numerical simulations up to now only cover one of these configurations (Wtot =200m3/h,X~nce =0mm,S0,th,main =0"9,S0,th,pilot =0.79) concentrating on the variation of the applied turbulence closure models and their validation against measured data. Hence, numerical and experimental results are plotted against each other, whenever available. Figures 4 and 5 give a first overview over the isothermal flow field displaying the time averaged radial profiles of the axial, and tangential component of the flow velocity at four axial positions in the vicinity of the burner (x/D0=0.05, 0.5, 1.0 and 1.5), which is indicated at the bottom of the plots. Streak lines, calculated using the k-o turbulence model, are plotted in grey in the

860

Figure 5: Tangential velocity w and axial turbulence intensity u'u' of the isothermal flow at four traverses in the vicinity of the burner. The lines denote calculations, circles are measured data (V = 200 mN3/h; lance position XL=0mm). background of each figure to provide a better orientation. On the axis a central, inner recirculation zone (IRZ) with the corresponding negative values for the axial velocity can be identified. The center of this IRZ is calculated at x/D0=0.67 and the recirculation zone closes at x/D0=3.64 (not displayed). The broken lines show the velocity results of the two RANS calculations using the k-~ and k-o~ model, respectively. The straight line shows the time averaged results of a large eddy simulation obtained using a temporal resolution of At - 90 ~ts. This relatively coarse time resolution was chosen in a first approximation to collect a large time-span for averaging (9 seconds). The CFL numbers resulting from this temporal resolution is typically below 0.1 in the vicinity of the burner mouth with maximum values below 1 in the nozzle flow. The circles display averages of several thousands (up to 60000) single individual data acquired with the LDA diagnostics. In general, a very good agreement of calculation results with the measured data can be stated for all models, RANS and LES. In general, the results using the k-~ model show the biggest differences to the measured data, while results obtained using the k-co model or the LES / Smagorinsky model almost perfectly resemble the measured data. Despite the good representation, some characteristic differences can be found and will be discussed below. In the diagrams for all three components two peaks can be distinguished in the first traverse, representing pilot and main flow. Those peaks vanish until the second traverse at one nozzle-radius axial distance, indicating the fast mixture, which is characteristic for swirling flows. The remaining single swirling jet then spreads radially and entrains ambient, non-rotating fluid while propagating further downstream. Here, the k-~ model results show a to small spreading rate, which results in radially inward displaced maxima of the velocity maxima. The profiles of the radial velocity shows the greatest deficiencies from measured data. The most interesting difference can be seen looking at the profiles of x/D0=0.05 and x/D0=0.5: Although the peak values, predicted using the k-~ model, in the first traverse are approx. 30% smaller than the measured data, the profile at the second traverse shows a better agreement with experiments than the other models. The prediction of a too low radial speed of the k-~ model result consistently yields in a displacement of the jet-peak for the axial and tangential velocity. Although this cannot be stated a general conclusion, the excellent results, in particular of the LES-procedure shown above, in retrospect give a justification for the selection of the coarse time resolution. The right side of Figure 5 pictures the axial turbulence intensity URMS. The turbulence intensities in radial and tangential direction show similar behavior and, thus, are not explicitly displayed. For the RANS calculations, the turbulence intensity was calculated using the eddy viscosity concept:

861

. . . ~tt - - , (1) 9 9 0x with k standing for the specific turbulent kinetic energy, P denoting the density of the fluid and l.tt the turbulent dynamic viscosity as a result of the turbulence closure model used. For comparison of the LES results, the square root of the averaged statistical Reynolds-stress component u'u' was used: .

.

.

UX,RMS,LES ~u'u'. (2) An overall comparison of the calculated with measured values shows the best agreement using the k-e0 model. In contrast, LES and k-e model results in particular show deficiencies representing the peak values, connected with the shear layers of the swirling jet. Going into details, the first traverse is calculated in relatively good agreement with measurement for all models used (RANS and LES). This fact especially justifies the assumptions made for the inflow boundary conditions. Both peaks, arising from the steep gradient near the burner nozzle are reproduced very well. The only deficiency can bee seen looking at the peak over the burner lance, which could not be reproduced by any calculation. As this local deviation is shown for all calculations, it may result from deficiencies in grid resolution. The calculated values at x/D0=l and 1.5 do not agree in detail with the measured values, showing a general deviation in the overall level of the turbulence intensities calculated with the different models. Generally, the k-e model tends to under predict, the LES results to over predict the peak turbulence intensities. This behavior can be seen in particular at x/D0-0.5. Besides the peak values, an other important difference can bee seen, looking at the turbulence intensities at Y/D0 > 1. Here, the k-e results show a completely different behavior than the k-c0 results: As the k-co results reveal a slow decay, the k-e results show a sharp border of the entrainment with (almost) zero values outside. As the time mean values of the velocities are very small, statistical averaging is very (simulation-) time consuming and, thus, the statistical analysis of the fluctuating velocities from the LES results in this region obviously is not converged and can only be judged as preliminary. =

DYNAMIC STRUCTURES IN THE ISOTHERMAL F L O W To look more into details of the dynamic behavior of the isothermal flow sound pressure data, acquired with a microphone probe in the ambient of the swirling flow, were analyzed with respect to their spectral representation. As clearly demonstrated in Figure 6, the formation of periodic coherent structures (red line) in dependence of the chosen burner outlet geometry (Xlance= -40 mm) 0.2

sound pressure amplitude / Pa ~--- fmax= 56 Hz

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0.05

0

100

200

300 400 500 frequency f / H z

600

700

800

Figure 6: Measured spectral distribution of sound pressure amplitudes and wavelengthintegral sound pressure levels for two lance positions in the isothermal swirlin~g flow. Blue line: Xlance--0 mm; red line: Xlance=-40 m m . ('~rmain = 190m 3 / h , V p i l o t - 1 0 m ~ / h , probe position: x/D0 =1, y/D0=4.55).

862

Figure 7: Two snap shots of the calculated isothermal axial velocity field distribution in m/s in the vicinity of the burner mouth. Time difference between the snap shots is 2,8 ms. does not effect at all the shape of the sound level spectra, but adds a considerable contribution at 56 Hz, resulting in an increase of the sound pressure level (SPL) from 75.7 dB to 80.7 dB. This is equivalent to an absolute rise of 80% in comparison to the undisturbed turbulent noise emission. Figure 7 shows results of another LES calculation using the same computational solution procedure, but utilizing a total volume flow of Vtotal = 650m3 / h . It displays two calculated, typical meridian cuts of the axial velocity fields in the vicinity of the burner mouth, obtained using the LES simulation. Both fields represent snap shots with a time delay of At-2.8 ms. In this picture, a small recirculation zone (negative axial velocities shown in blue) can be observed in the annular main flow. These recirculation zones periodically move around in the annular channel attached to the inner wall, making one turnaround in approx. 2 x 2.8ms. The effect of these coherent structure is shown in the corresponding Figure 8, where a spectral representation of the calculated static pressure at a monitoring point in the ambient flow is compared to an according measured normalized sound pressure level. The experiments are not directly comparable, as they were performed utilizing only Vtota I = 609rn~/h, which turned out to be the maximum inflow, that could be supplied by the compressor. Anyhow, the observed preferential frequencies compare very well. This indicates that the coherent structures result from an axial displacement of the

Figure 8: Comparison of spectral resolved static pressure amplitude and normalized sound pressure level in the ambient flow of the isothermal swirling flow.

863

Figure 9: Spectrally resolved periodical part of the measured tangential velocity in the isothermal flow (square symbols) and a premixed flame (diamond symbols) in m/s. central recirculation zone, which is characteristic for swirl flows. This supports the mechanism for the generation of coherent structures in swirl flames earlier found in literature (Lohrmann & Btichner). DYNAMIC STRUCTURES IN THE PREMIXED FLAME The experimental investigations additionally covered premixed swirling flame configurations. In addition to time averages and statistical moments of the velocities, the acquired data were processed by an FFT analysis to yield the spectra of the velocities at specified monitoring points. As an example, the spectra of the tangential velocity is shown in Figure 9 for both, the isothermal flow and the corresponding premixed flame. Both, isothermal flow and flame, were operated with a withdrawn lance (Xlance'---40 mm). As already shown for the isothermal cases, again preferential frequencies could be recorded for both cases. The comparison shows that the frequency in the isothermal flow was smaller than in the reacting case and the frequency distribution for the flame configuration covered a wider frequency band. In order to compare both cases properly, the Strouhal number St has to be taken into account for both preferential frequencies, which almost exactly match (0.74 vs. 0.72). Thus, the effect has to be considered to originate from the same physical phenomenon. To prove this finding, large eddy simulations are currently carried out. The results of this simulations will be available in the near future and analyzed with respect to the phenomenon described. CONCLUSIONS The investigations shown in the present work covered detailed velocity measurements and simulations using turbulence closure models at various levels of complexity. The simulations, up to now, only were performed for the isothermal, strongly swirling flow while the experiments covered the isothermal flow field as well as corresponding premixed swirl flames for various operational conditions. A comparison was shown between the results of RANS and LES procedure. A first analysis showed characteristic differences between the results of the approaches used. The very good representation of the calculated velocity field, with respect to time-averaged values encourages a further analysis with the LES approach using a finer computational grid and time resolution to check for the reason of the deficiencies observed. In addition, a spectral analysis of the flow field as well as of pressure fluctuations, acquired with microphone probes, showed distinct preferential frequencies for the isothermal and the reacting case. As a reason, coherent

864 structures observed with LES-calculations at the burner mouth could be identified. A comparison of measured with calculated preferential frequencies showed good agreement. The recorded sound pressure levels of the swirl flame exceeded the corresponding isothermal sound pressure level by approx 40%.

REFERENCES

Angelberger, C., Veyante, D., and Egolfopoulos, F. (2000). LES of chemical and acoustic forcing a premixed dump combustor, Flow Turbul. Combust. 65, 205-222. Bai, T., Cheng, X.C., Daniel, B.R., Jagoda, J.I., and Zinn, B.T. (1993). Vortex shredding andperiodic combustion processes in a Rijke type pulse combustor, Combust. Sci. Technol., 94, 245258. B~er, J. and Chigier, N. (1972). Combustion Aerodynamics. Applied Science Publishers. Btichner, H. (1992). Entstehung und theoretische Untersuchungen der Entstehungsmechanismen selbsterregter Druckschwingungen in technischen Vormisch- Verbrennungssystemen, Dissertation, University of Karlsruhe (T.H.), Biichner, H., and Lohrmann, M. (2003). Coherent Flow Structures in Turbulent Swirl Flames as Drivers for Combustion Instabilities, Proc. Intern. Colloquium on Combustion and Noise Control. Cohen, J.M. and Anderson, T.J. (1996). Experimental Investigation of Near-Blowout Instabilities in a Lean Premixed Step Combustor, AIAA paper 96-0819 Ewert, R., Schrfder, W., Meinke, M., E1-Askary, W. (2002) LES as a basis to determine sound emission. AIAA Paper 2002-0568. Fureby, C. (2000). A computational study of combustion instabilities due to vortex shredding, Proc. Combust. Instit., 28, 783-791. Gupta, A. K., Lilley, D. G. and Syred, N. Swirl Flows, Abacus Press, (1984), Kent, U.K. Habisreuther, P., Petsch, O., Biichner, H., and Bockhorn H. (2004). Berechnete und gemessene StrOmungsinstabilitaten in einer verdrallten Brennerstr6mung, G A S W ~ E International 53, 326-331. Keck, O., Meier, W., Stricker, W. and Aigner, M. (2002). Establishment of a confined swirling natural gas~air flame as a standard flame: Temperature and species distributions from laser raman measurements, Combust. Sci. Technol. 174:8, 117-151. Kfihlsheimer, C. and Btichner, H. (2002). Combustion Dynamics of Turbulent Swirling Flows, Combust. Flame Vol. 131:1-2, 70-84. Launder, B. E., Spalding, D. B. (1974). The numerical computation of turbulent flows. Comp. Meth. Appl. Mech. Eng. 3 269-289. Laverdant, A.M. and Candel, S.M. (1989). Computation of diffusion and premixed flames rolled up in vortex structures. J. Propul. Power, 5, 134-143. Leuckel, W. and Fricker, N. (1976). The characteristics of swirl-stabilized natural gas flames. Part I: Different flame types and their relation to flow and mixing patterns, J. Inst. Fuel 49, 103-112. Lilley, D. G. Swirl flows in combustion: A review. (1977). AIAA J. 15, 1063-1078. Lohnnann, M., and B~ichner, H. (2000) Periodische StOrungen im turbulenten StrOmungsfeld eines Vormisch-Drallbrenners, Chem. Ing. Tech. 72.512-515. Menter, F. R.: (1994). Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA-J. 32:8. Poisnot, T., Trouve, A., Veyante, D., Candel, S. and Espitito, E. (1987). Vortex driven acoustically coupled combustion instabilities. J. Fluid Mech., 117, 265-292. Ruith M. R., Chen P., Meiburg E., and Maxworthy T. (2003). Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331-378. Schadow, K.; Gutmark, E.; Parr, T.; Parr, K.; Wilson, K. and Crump, J. (1989). Large-scale coherent structures as drivers of combustion instability, Combust. Sci. Technol., 64, 167-186. Smagorinsky, J. S. (1963). General circulation experiments with the primitive equation. Monthly Weather Rev, 91, 99-164.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

865

INTERACTION BETWEEN THERMOACOUSTIC OSCILLATIONS AND SPRAY COMBUSTION Wajid A. Chishty, Uri Vandsburger, William R. Saunders and William T. Baumann Virginia Active Combustion Control Group Department of Mechanical Engineering Virginia Polytechnic Institute & State University Blacksburg, VA-24061-0238, U.S.A.

ABSTRACT Unsteady heat release and pressure oscillations are inherent in many gas tm'bine combustors, at least for some operating conditions. It is the in-phase coupling of these quantities that leads to thermoacoustic instabilities. Combustors burning liquid fuel sprays with high-energy release rate are susceptible to this kind of instability because of the inherent unstable processes associated with spray combustion. Any effort towards development of effective active fuel modulation methodologies for control of thermoacoustic instabilities is therefore, dependent upon the understanding of the possible coupling between the spray dynamics and the combustor acoustic field. In this paper, we present some aspect of the ongoing efforts in our laboratory to gain knowledge of this interaction. Experimental and modeling results from both reacting and non-reacting flow studies are presented. Observations made on a kerosene fuelled LDI combustor suggest that the onset of the thermoacoustic instability is a function of the energy gain of the system, while the sustenance of instability is due to the in-phase relationship between combustor acoustics and unsteady heat release driven by acoustic oscillations. The presence of a two distinct unstable operating combustor regimes is observed and discussed as well. It is also shown that the intensity of the acoustic field in comparison to other flow field phenomena in the combustor is the dominant factor in influencing the combustor operation i.e., reacting flow field and spray characteristics. KEYWORDS Thermoacoustic oscillations, Spray combustion, Spray modeling, Combustor dynamics, Lean Direct Injection, Phase Doppler Anemometry INTRODUCTION Demand for higher performance gas turbine engines especially for propulsion, required to operate at high-temperature/high-pressure, combined with low emission regulations has posed many challenges for combustor designers. One of the challenges is the control of thermoacoustic instabilities. Presently, Lean Pre-mixed, Rich-bum Quick-mix Lean-bum, Lean Pre-vaporized Pre-mixed and Lean

866

Direct Injection combustors have been the focus of many studies addressing the occurrence of instabilities, their modeling and instability attenuation via active combustion control. The later two lean combustion concepts have mostly been applied to liquid spray fuelled combustors. Spray combustion is inherently an unstable phenomenon because it comprises a number of unsteady processes like liquid injection, atomization, vaporization, mixing and chemical kinetics (Sirignano, et al, 1995), which result in unsteady heat release rate. At the same time pressure oscillations are inherent in many gas turbine engines, at least for some operating conditions (McManus et al, 1993). However, it is the in-phase coupling of unsteady heat release rate and combustor acoustic oscillations that lead to self-sustained thermoacoustic instabilities, according to the well-cited Rayleigh criterion (e.g., Clanet at all, 1999). The sustenance occurs because the acoustic oscillations in turn influence the spray characteristics leading to oscillatory combustion. These instabilities have resulted in loss of performance and hardware damage in propulsion systems. Combustion instabilities, including the thermoacoustic kind, have been extensively examined since 1950's. However, a recent surge of interest has emerged from attempts to actively control the levels of thermoacoustic instabilities (or complete cancellation) via modulation of liquid fuel supply (McManus et al, 1993). The cause-and-effect relationship between spray dynamics and combustor acoustics is still a gray area in spray combustion, understanding of which is essential in designing effective active fuel modulation methodologies for control of thermoacoustic instabilities. In this paper we present some of the ongoing effort in our laboratory to gain knowledge of this interaction. These comprise experimental investigation as well as modeling studies conducted under reacting and non-reacting conditions. The organization of the paper is as follows: The reacting-flow experimental setup is presented first together with the discussion on investigation methodology and experimental observations of the combustor operational characteristics. This is followed by the results from non-reacting experiments, conducted to study the effects of acoustic excitation on sprays. Finally results from a modeling effort to track droplet trajectories and secondary breakup are presented. These results qualitatively verify some of our hypotheses about the combustor dynamic characteristics observed over the operating range of investigation. REACTING F L O W INVESTIGATION The experiments were performed on a cylindrical 75 kW lean direct injection, swirl-stabilized combustor rig, fuelled with kerosene. To date, the investigations have been conducted under atmospheric pressure conditions, which is considered as the obvious first step towards providing valuable insights into more intense processes in actual gas turbine combustors. The combustor rig comprises three main sections: the combustor, the burner-plenum and the flow-conditioning section that houses the fuel and air supply lines. The sudden expansion combustor section was 1270 mm long with an internal diameter of 127 mm, while the flow-conditioning section was 864 mm long with an internal diameter of 76 mm. Optical access was provided on the combustors through three rectangular fused-silica windows. Combustion air was introduced in the combustor through a 45 ~ axial swirler, which gives the air a geometric swirl number of 0.81. Kerosene fuel was introduced into the combustor via a full-cone pressure-swirl simplex atomizer. Pressure measurements were obtained using dynamic pressure transducers, which were mounted throughout the length of both the combustor and the flow-conditioning sections. The heat release rate measurements were made using a photo multiplier tube with suitable optics to capture the OH* chemiluminescence. This chemiluminescence has been found to be a good indicator of heat release rate over a wide range of fuel-to-air ratios (Haber and Vandsburger, 2003), and served in the experimental setup as an indicator of the total heat release rate of the flame within the viewed volume.

867

Figure 1. Stability mapping of combustor's operating range (0: boundary of 1st unstable regime for 0.47 FN injector; ll: boundary of 1st unstable regime for 0.63 FN injector; 9 : boundary of 2 nd

unstable regime for 0.47 FN injector; 4,: boundary of 2 nd unstable regime for 0.63 FN injector).

Combustor stability was mapped over the operating regime and the results for two flow number (FN) atomizers are shown in Figure 1. At low Global Equivalence Ratio (GER, defined as the fuel-to-air ratio normalized by the stoichiometric fuel-to-air ratio), the combustor was found to operate in a stable mode characterized by a well-mixed and compact flame. As the GER was increased the combustor became thermoacoustically unstable entering what we define as the 1st unstable regime, characterized by a poorly mixed, luminous flame surrounding diffusion burning of individual droplets or droplet groups. An abrupt transition to a 2 na unstable regime was encountered when the GER was further increased towards stoichiometric. The flame appeared highly stretched and resembled a lean prevaporized pre-mixed flame. This characteristic of the combustor's unstable response was found to be independent of the manner in which the GER was varied, i.e., either via changes in airflow rate (with constant fuel flow rate) or changes in fuel flow rate (keeping air flow rate constant) or both. It may also be noted from Figure 1 that the 1st unstable boundary was independent of the injector FN, while a distinct relationship was observed at the 2 "a unstable boundary. This behavior was attributed to a secondary atomization caused by the presence of a strong acoustic field and is discussed later in the paper. Figure 2 shows the evolution of thermoacoustic instability in the combustor with increasing GER. The pressure signals were obtained at the dump plane and 200 mm upstream of the dump plane (in the flow-conditioning section), while the global heat release rate was measured via the OH* chemiluminescence intensity. It is seen that at a GER of 0.4 although there was no preferred oscillations in the heat release rate, the mean thermal energy was high enough to excite the combustor at its quarter wave resonance frequency (and 1st odd harmonics). A coupled oscillation in the upstream pressure can also be observed in Figure 2(b). However, the combustor acoustic losses from radiation at the open end, from heat diffusion at the wall and due to the presence of spray fuel prevent the combustor from going unstable (Clanet at all, 1999). As the mean thermal energy contents into the system was increased by increasing the GER to 0.45, the amplitude of the acoustic oscillations in the combustor increased and the first sign of acoustic coupling with heat release was observed (spike indicated in Figure 2(c)) at the combustor resonance frequency. Any further increase in GER caused

868

the acoustically driven heat release rate oscillations to satisfy the phase relationship with combustor pressure oscillations, thus making the combustor thermoacoustically unstable and exhibiting even larger amplitude pressure oscillations. We define this GER limit as the boundary of 1st unstable regime. Measurements taken well into this unstable mode (GER of 0.5) show the sharp peaks in pressure and acoustically driven heat release rate spectra. The presence of distinct even harmonics indicates the presence of strong non-linear effects (Dowson and Fitzpatrick, 2000). Excitation frequencies then shift to the limit-cycle values, which were different (lower) from the resonance frequencies (see Figure 3(a)). It may also be noted from Figure 2 that there also exists a strong nonlinear coupling of the combustor with the upstream section as evident by the high amplitude even harmonics in the power spectra of upstream pressure (shown in Figure 2(b)).

Figure 2. Power spectra of acoustic pressures and heat release rate. (a) Combustor pressure, (b) Pressure upstream of the dump plane and (c) Heat release rate.

Before combustor's transition to the 2 nd unstable regime is discussed, it is appropriate at this stage to shed light on the mechanism through which the high amplitude pressure oscillation manifests itself in causing thermoacoustic instability. Studies have shown that acoustic pressure effects the combustion processes via acoustic velocity oscillations, which: effect the surface density of the flame and the reaction rate (Ducruix et al, 2003), cause formation of periodic vortical structure (Yu et al, 1991), cause oscillations in fuel spray vaporization (Tong and Sirignano, 1989), and prompt droplet breakdown (Anilkumar et al, 1996). The intensity of these interactions was found to depend upon the magnitude of flow oscillations (Toong, 1983).

869

Figure 3. Results of combustor characterization. (a) Limit-cycle frequencies and pressure amplitudes, (b) Corresponding acoustic velocities and droplet relative convective velocities, (c) Mean heat release rate. Limit-cycle frequencies and acoustic amplitudes are shown in Figure 3, as a function of GER. Also shown are the corresponding measured magnitudes of acoustic velocities and calculated relative droplet velocities in the combustor. As seen in Figure 3(b), at high GER, the combustor acoustic velocities can reach high amplitudes in comparison to the droplet relative velocities. We propose that these amplitudes were high enough to cause secondary droplet breakup, which result in the abrupt transition to the 2nd unstable regime. In this regime the high amplitude acoustic velocities non-linearly interact with the processes highlighted in the succeeding paragraph giving the flame the strained appearance. This hypothesis was validated by our modeling effort discussed later in the paper. Experimental verifications are at present in progress. Mean values of the measured OH* chemiluminescence are shown in Figure 3(c). Although the validity of these global measurements in the 1st unstable regime is questionable because of the diffusion type characteristics of the flame in this regime, however the measurements in stable as well as in 2nd unstable regimes are considered authentic because of the well-mixed nature of the flames in these regimes. A qualitative validation of the effect of acoustic velocities on reaction rate is evident here from the sharp drop in measured OH* intensity in the 2 nd unstable regime, which is indicative of transition to the distributed and strained reaction zone.

870

NON-REACTING SPRAY INVESTIGATION Non-reacting spray experiments were performed using 1-D Phase Doppler Anemometry (PDA) to study the influence of one-dimensional acoustic field on droplet trajectories. An isothermal acoustic rig was constructed for this purpose. Details of the experimental setup and measurement methodology can be reviewed in Chishty et al, 2004. A full-cone pressure-swirl atomizer (FN=0.63), similar to the one used in combustor tests, was used with water as the working fluid. Acoustic forcing was achieved using a 40-watt speaker. Phase-locked swept-sine measurements for 70-500 Hz were performed under varying acoustic forcing conditions and spray feed pressures. Measurements were conducted both in quiescent environment and with co-flowing swirled air. Measurements made at four locations in the spray were related to these variations in mean and unsteady inputs. A typical droplet velocity response to forced acoustic oscillations, measured in these experiments is shown in Figure 4. The data was normalized by the ratio of mean droplet velocity to mean acoustic pressure. The droplet dynamics shows a second order response with a cut-off frequency of about 150 Hz, beyond which the droplet velocities show no response. The corresponding acoustic period to this cut-off frequency commensurate with the relaxation time of a 25 ~tm water droplet, which was the mean droplet size measured during our experiments. 6

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It was also found from the dynamic analysis of PDA measurements, that substantial variations in droplet movement can be achieved depending upon the acoustic field strength, the excitation frequency and droplet size and relative velocity (Chishty et al, 2004). At maximum acoustic excitation (40 watts speaker power), the peak-to-peak displacement in droplet movement were found to be almost 4 mm, which is significant keeping in mind the acoustic velocity of only 0.125 m/s achieved at this excitation setting. Compared to this, the acoustic velocities in the combustor are in the order of 15 to 30 m/s in the unstable regimes and thus their effect on the droplet oscillatory excursions is expected to be substantial. Another important conclusion drawn from the non-reacting flow experiments was that under the influence of an acoustic field the droplets tend to migrate radially outwards. The effect was more pronounced on smaller size droplets as compared to the larger size droplets. This was found to modify the spatial drop size distribution in the spray. SPRAY M O D E L I N G STUDIES A deterministic separated-flow (DSF) approach was used to model the spray dynamics in a flow field with superimposed acoustic oscillations. The technique utilizes a Lagrangian-drop/Eulerian-flow approach to track droplets as they traverse in the flow field. The method has been found to be fairly accurate and less computationally intensive compared to stochastic approaches also used for trajectory

871

modeling (Zhang, 2002). In the model, the spray was treated as a group of spherical droplets, each group having its own fixed size, velocity and history, and not interacting with the other groups. The droplet was injected in a 1-D flow field, with droplet initial velocity, co-flow air velocity and the acoustic velocity all having different directions relative to each other. The drag force acted in the direction opposite to that of the droplet relative displacement (with respect to its initial velocity and the acoustic field). Gravity effects were considered negligible in the model. The gas velocity was updated at each time step using gas phase continuity equation. The model can be adopted for both reactive and non-reactive flows. The set of differential equations, which pose an initial value problem, were solved using Runga-Kutta fourth-order method. The initial values for droplet diameter and droplet velocity were taken from the PDA experimental results. For the non-reacting cases the acoustic amplitude and frequencies were also taken from the PDA experiments, while for the reacting flow studies the actual combustor limit-cycle values were used. Droplet deformation and secondary atomization due to acoustics were ignored in the trajectory modeling. However, these effects were separately studied using the Taylor Analogy Breakdown (TAB) Model (O'Rourke and Amsden, 1987), which has been used in earlier studies to investigate the droplet breakup due to harmonic resonance. The model is based on an analogy between an oscillating droplet and a spring-mass-damper system, where the spring restoring force and the damping force are analogous to the droplet surface tension forces and the liquid viscous forces respectively. The aerodynamic drag force on the droplet due to the droplet relative velocity and the acoustic force substitute the external forces on the mass. The secondary breakup was assumed to occur when the steady state value of the oscillatory deformation at the north and south poles of the droplet was equal to half the droplet diameter.

Figure 5. Trajectory modeling results for non-reacting flows. (a) Spray behavior under 80 Hz acoustic excitation in quiescent environment, (b) Spray behavior when acoustic velocity and swirl co-airflow velocity are comparable (3 m/s at 80 Hz and 2m/s respectively). The non-reacting flow results are discussed first and compared with the PDA experimental results. For the purpose of this study, evaporation of water droplets was ignored. The results discussed here and shown in Figure 5, were based on a spray taken as an ensemble of three groups of droplets, with mean diameters of: 10, 15 and 25 microns. These values were reasonably close to the droplet size distribution observed in PDA experiments. To investigate the influence of acoustic field on droplet trajectories, the initial velocity of each droplet was assumed 3 m/s. This was once again consistent with the values observed in PDA experiments at 1034 kPa feed line pressure. The result of acoustic

872

field excitation on the spray is shown in Figure 5(a). An 80 Hz cosine wave was used to introduce the acoustic velocity oscillation, with peak amplitude of 0.125 m/s, which was the same as that obtained during PDA measurements at the same frequency. For this simulation, the injection of the droplets was phase-locked with the start of the acoustic cycle. As shown, the smaller droplets are the first to get affected. The droplets are forced to migrate radially outwards and their axial propagation is seen halted by the acoustic field. The maximum axial distance that the droplets can travel is proportional to their size. The effect of this radial movement is to modify the drop size distribution in the spray. Also shown (inset Figure 5(a)) are the excursions in droplet displacement. The behavior of the spray as discussed here, qualitatively validates the PDA experimental findings. Simulations were also performed for the case when 2 m/s swirling co-flow air was introduced, through a 45 ~ axial swirler, to the spray in an acoustic field. Keeping the excitation frequency constant at 80 Hz, the acoustic velocity amplitude was increased to 3 rn/s (comparable to the swirl airflow velocity). As can be observed in Figure 5(b), this resulted in spatial bands of high droplet densities. It may also be noted from Figure 5(b) that the dense pockets of droplets appear in intervals, which correspond to the acoustic wavelength (approximately 19mm) at an excitation frequency of 80 Hz. This result along with a comparison of acoustic and droplet velocities in the 1st unstable regime (Figure 3(b)), explains the group-burning appearance of the flame in that regime. 100, E

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unstable operational modes are represented by l , A and 9 respectively.

Figure 6 shows the modeling results for reacting flows. Once again, a cosine wave was used to introduce the acoustic velocity oscillations, with amplitudes corresponding to the limit-cycle amplitudes in the combustor. 50 pxn droplets were injected in a uniform temperature bath of combustion gases, at phase lags of 0, 90, 180 and 270 degrees relative to the start of the acoustic cycle. The temperature values from combustor tests were used for these simulations and effects of temperature, convective medium and acoustic oscillations on the evaporation of droplets were accounted for by using empirical correlations (Okai et al, 2000 and Sujith et al, 1999). Results for three GER values: 0.40, 0.60 and 0.90, corresponding to the three stability regimes of the combustor are shown in Figure 6. It is seen that for half of the acoustic cycle at high levels of acoustic amplitudes the axial traverse of the droplets is considerably retarded (and even reversed). This retardation of the droplets in a preheated environment would lead to a high density of fuel that gets pre-vaporized, premixed and possibly dilute before it is convected into the reaction zone during the other half of the

873

acoustic cycle. This extreme condition would cause the flame behaviors observed in the 2nd unstable regime of the combustor.

Figure 7. TAB model result showing the occurrence of secondary droplet breakup due to acoustic oscillations To account for the presence of high acoustic levels in the combustor a secondary droplet breakdown analysis was conducted as described earlier. A non-dimensional deformation of the droplet was defined by normalizing the actual deformation by the droplet initial diameter. The result of this analysis is shown in Figure 7, where for a range of droplet sizes the deformation is plotted versus the GER. The GER values here, only serve as indicators for the corresponding values of the sound pressure level, acoustic velocity and relative droplet convective velocities (shown in Figure 3). The droplet breakup criterion is the non-dimensional deformation value of 0.5. The influence of combustor acoustics on the droplet breakup, relative to the convective flow is evident. The plot shows that the smaller the mean size of the droplets, the higher the critical value of GER at which the breakup occurs. These results qualitatively verify our experimental f'mdings shown in Figure 1, where, for a smaller FN injector (which produce smaller mean size droplets) the 2nd unstable regime (characterized by droplet secondary breakup) is reached at a higher value of GER. CONCLUSIONS Interaction between thermoacoustic oscillations and spray combustion was presented. Three stability regimes of a kerosene fuelled lean direct injection combustor were reported, over a range of operating conditions. Experiments both in reacting and non-reacting flows and modeling studies were conducted to explain the transitions to two unstable regimes and sustenance of thermoacoustic instability in the combustor rig. It was found that the onset of instability (transition to 1st unstable regime) was a strong function of the system's thermal gain, relative to the radiative, diffusive and viscous acoustic losses in the combustor. At a certain critical gain the acoustic oscillations were high enough to cause corresponding oscillations in the rate of heat release. The combustor became thermoacoustically unstable when these two oscillations were brought in phase. The non-reacting flow and modeling investigations indicated that the unsteadiness in the heat release rate was attributable to the changes in drop size distribution brought about by the influence of the acoustic field. It was also found that when the acoustic velocities and the relative convective droplet velocities were of the same order of magnitude, bands of high droplet densities were formed which favored the individual drop or group burning seen during the 1st unstable regime of the combustor. Modeling efforts also indicated that the presence of high levels of acoustic amplitude can cause secondary breakup of the droplets and can

874

retard the droplet movement to an extend that they get pre-vaporized and pre-mixed before being convected to the flame zone. This latter mechanism explains the transition to the 2nd unstable regime.

REFERENCES Anilkumar, A. V., Lee, C. P. and Wang, T. G. (1996). Studies of the Stability and Dynamics of Levitated Drops. NASA Conference Publication 3338, 559-564. Chishty, W. A., Vandsburger, U., Saunders, W. R. and Baumann, W. T. (2004). Effects of Combustor Acoustics on Fuel Spray Dynamics. Proceedings of 2004 ASME International Mechanical Engineering Congress, Anaheim, CA 2004-61325. Clanet, C., Searby, G. and Clavin, P. (1999). Primary Acoustic Instability of Flames Propagating in Tubes: Cases of Spray and Premixed Gas Combustion. Journal of Fluid Mechanics 385, 157-197. Dowson, S. and Fitzpatrick, J. A. (2000). Measurement and Analysis of Therrnoacoustic Oscillations in Simple Dump Combustor. Journal of Sound and Vibration 230 (3), 649-660. Ducruix, S., Schuller, T., Durox, D. and Candel, S. (2003). Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms. Journal of Propulsion and Power 19 (5), 722-734. Haber, L. C. and Vandsburger, U. (2003). A Global Reaction Model for OH* Chemiluminescence Applied to a Laminar Flat-Flame Burner. Combustion Science and Technology 175, 1859-1891. McManus, K., Poinsot, T. and Candel, S.M. (1993). A Review of Active Control of Combustion Instabilities. Progress in Energy and Combustion Sciences 19:2, 1-29. Okai, K., Moriue, O., Araki, M., Tsue, M., Kono, M., Sato, J., Dietrich, D. L. and Williams, F.A. (2000). Combustion of Single Droplets and Droplet Pairs in Vibrating Field under Microgravity. Proceedings of the Combustion Institute 29, 977-983. O'Rourke, P. J. and Amsden, A. A. (1987). The TAB Method for Numerical Calculation of Spray Droplet Breakdown. SAE Technical Paper Series 872089. Sirignano, W. A., Delplanque, C. H., Chiang, C. H. and Bhatia, R. (1995). Liquid-Propellant Droplet Vaporization: A Rate-Controlling Process for Combustion Instability. Progress in Astronautics and Aeronautics 169, 307-343. Sujith, R. I., Waldherr, G. A., Jagoda, J. I. And Zinn, B. T. (1999). A Theoretical Investigation of the Behavior of Droplets in Axial Acoustic Fields. Journal of Vibration and Acoustics 121,286-294. Tong, A. Y. and Sirignano, W. A. (1989). Oscillatory Vaporization of Fuel Droplets in Unstable Combu:;tor. Journal of Propulsion and Power 5 (3), 257-261. Toong, T. Y. (1983). Combustion Dynamics: The Dynamics of Chemically Reacting Fluids. McGrawHill Inc. Yu, K. H.. Trouve, A. and Daily, J. W. (1991). Low-Frequency Pressure Oscillations in a Model Ramjet Combustor. Journal of Fluid Mechanics 232, 47-72. Zhang, 11. Q., Yang, W. B., Chan, C. K. and Lau, K. S. (2002). Comparison of Three Separated Flow Models. Computational Mechanics 28, 469-478.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

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Dynamics of Lean Premixed Systems: Measurements for Large Eddy Simulation D. Galley 1.2,A. Pubill Melsi6 2, S. Ducruix 2, F. Lacas 2 and D. Veynante 2 Y. Sommerer 3 and T. Poinsot 3 1SNECMA Moteurs, D6partement YKC, 77550 Moissy Cramayel, france 2 Laboratoire EM2C, CNRS UPR 288 - Ecole Centrale Pads, 92295 Chatenay-Malabry,France 3CERFACS, CFD team, 42 Av. G. Coriolis, 31057 Toulouse CEDEX, France

ABSTRACT Lean Premixed Prevaporized (LPP) injection systems have been designed to offer a minimum NOx and soot emissions. The basic principle of LPP systems is to optimize combustion through an efficient mixing of fuel and air. This can be achieved by vaporizing the initially liquid fuel and then mixing it with the air before combustion using for example a swirling flow. It is well known that premixed combustion can reduce pollutant emissions more than non-premixed combustion [1]. Moreover, a lean mixture allows to control the flame temperature and then NOx production since it increases with temperature. However, LPP systems are known to be very sensitive to couplings leading to many kind of unstable behaviors. This work is a contribution to the understanding of the dynamical phenomena occurring in a LPP combustor, using advanced laser diagnostics. This paper presents an experimental and numerical study of a Laboratory-scale gas turbine combustion chamber designed and operated at laboratoire EM2C. These results are compared with large eddy simulations (LES) performed at CERFACS.

KEYWORDS Gas Turbine, Lean Premixed Prevaporized (LPP) burner, Planar Laser Induced Fluorescence (PLIF), Large Eddy Simulation (LES), Thickened Flame model, Flashback, Precessing Vortex Core (PVC).

876

EXPERIMENTAL FACILITY The facility is a lean premixed burner operated at atmospheric pressure, using gaseous propane.

Figure 1: Experimental Setup

The experimental setup is fed with dry compressed air and propane (see Figure 1). The flow rates are monitored through two electronic mass flow meters. Air and propane are injected in the premixing tube of the combustion chamber presented in Figure 2. Premixing tube and combustion chamber are made in high quality quartz (fused silica) allowing visible and UV optical access. Main dimensions are provided on Figure 2. Characteristic numbers of the combustion facility are summarized in Table 1. The Reynolds number given in Table 1 is based on the bulk velocity and the diameter of the premixing tube.

TABLE 1 CHARACTERISTICNUMBERSOF THE COMBUSTIONFACILITY Max. Air mass flow rate Max. Propane mass flow rate Max. Reynolds Number .... Rated therma ! power

300 m3/h 15 m3/h From 40,000 up to 280,000 300 kW

Figure 2 9Swirler, Premixing tube, Combustion Chamber (dimensions in mm)

877

Mixing is enhanced using a radial-type swirl generator. Air and propane are introduced separately in the premixing tube. The 6 mm diameter jet of propane is injected axially and sheared by the surrounding swirling flow of air. Tangential air velocity in the premixing tube is produced using a radial-type swirl generator (radial guide-vane cascades). Eighteen constant-section vanes, which impart a helicoidal movement to the airflow, compose the swirl generator. The detailed geometry and dimensions of the swirl generator are given in Figure 3. The mixture is ignited in the combustion chamber using a spark plug.

Figure 3: Characteristics of the swirl generator (dimensions in mm)

COMBUSTION REGIMES Depending on the swirl and Reynolds numbers, large-scale spatial fluctuations of the swirling flow are coupled with a Central Toroidal Recirculation Zone (CTRZ) [2]. This recirculation zone plays an important role in flame stabilization as it locally supplies the flame front with hot burned gases to sustain combustion [3]. An example of swift-stabilized flame is displayed in Figure 4 (a). Figure 4 shows OH* spontaneous emission of the flame, obtained with a ICCD camera using UV filters WG305 and UG5. In this case, the flame is stabilized in the combustion chamber by the central recirculation zone created by the swirl ("compact flames"). In some situations, the flame propagates upstream in the premixing tube as shown in Figure 4 (b). This phenomenon, called flashback, can lead to catastrophic failure in real gas turbine, but is a stable regime of the facility ("flashback flames"). Since both swirl stabilized flame and flashback can be safely investigated, this facility gives the opportunity to understand the dynamics of partially premixed swirling flames and the phenomena leading to flashback. Flame regimes mainly depends on air and propane mass-flow rates, and are summarized in Figure 5. For low and intermediate equivalence ratios (and intermediate air flows), the flame is stabilized in the combustion chamber due to the CTRZ ("compact flames"). For higher values of the equivalence ratio, the structure of the flame can be either flashback or compact depending on initial and transient conditions ("hysteresis region"). The transition from flashback to compact flame takes place at approximately the same equivalence ratio, 9 = 0.68, whatever the air flow rate. Considering the compact flame situation, decreasing equivalence ratio leads to a detached flame, stabilized downstream in the combustion chamber. Then, for a lower equivalence ratio, the flame is spread all over the combustion chamber. Further decrease of the equivalence ratio leads to blow off [4].

878

Figure 4 (a): Compact flame

Figure 4 (b): Flashback flame

Figure 5: Burner Regimes

Laser Induced Fluorescence Imaging and Measurements The phenomena occurring in LPP devices, such as flashback, are intrinsically unsteady. Diagnostics used to understand these phenomena must take this into consideration. The key points of LPP behavior are mixing efficiency and flame dynamics, which can be linked to acoustic couplings. Planar laser induced fluorescence (PLIF) gives an instantaneous insight of these two aspects. The mixing is quantified by seeding the propane flow with acetone vapor. PLIF of acetone as tracer, under restrictive and well-controlled conditions, provides quantitative measurements of fuel mass fraction [5]. OH radical displays the instantaneous position of flame front and burned gases. These two diagnostics allows to study the flow dynamics either in the combustion chamber or in the premixing tube. The imaging plane may be parallel or perpendicular to the symmetry axis of the experiment. In the last case, as shown in figure 6, a cooled mirror is placed in the burned gases to transmit the fluorescence

879

signal to the camera. Longitudinal images have already been studied in [4]. In the present paper, we focus on the transversal case. Both OH or acetone vapor PLIF can be carried out in this situation. Quantitative results in the longitudinal situation can be found in [4].

Experimental setup The whole experimental setup, including lasers and acetone seeding, are given in Figure 6.

Figure 6: Experimental setup and diagnostics

Results

The mixing process is first analyzed. Propane is seeded with acetone (10% in mass of acetone vapor in propane) and a tranverse cut is made 5 mm downstream the exit plane of the premixing tube. Examples of PLIF images of acetone is displayed Figure 7. These images show a very coherent structure: a "comet plume" of fuel rotating in the same direction as the swirl movement created by the blades. This offset structure seems to turn in the combustion chamber, feeding the flame front. As the laser frequency is limited to 10 Hz, the images are not temporally connected. The direction of rotation is deduced from the shape of the propane core, since the plume is at the rear part of the structure due to the rotating movement. The rotation center is also slightly rotating (as can be deduced from the mean field). Such coherent structure, known in the field of non reactive swirling flows, is called Precessing Vortex Core (PVC) [2]. Figure 7 emphasizes the importance of unsteady structures. The average image (top left) does not present any of them: from the mean point of view, the fuel concentration field is isotropic in the radial direction. Nevertheless, instantaneous images show anisotropic structures that control the flame behavior. As a consequence, the OH instantaneous images exhibit a similar behavior. Indeed, even 2.5 cm downstream the premixing tube, the reacting zone is not uniform. In each image (Figure 8), OH signal presents a zone of weak signal, which also rotates from one image to another. This is because the

880

flame is stabilized on the PVC, the only region where fuel concentration exceeds the lean extinction limit. This gives us information on the stabilization process of partially premixed swirled flame in this kind of configuration. Due to the swirl movement of the airflow, a vortex is created in the premixing tube and convected by the flow. This vortex presents a decreasing fuel concentration profile along its radius. The inner core is fuel rich whereas the outer cell is lean [4]. Due to the swirl effects and the sudden expansion in the combustion chamber, this vortex precesses in the combustion chamber. The flame is then stabilized in a "precessing way". This mode of stabilization has been confirmed using a high-speed ICCD camera, recording spontaneous emission of the flame up to 10,000 images per seconds. The precessing movement of the reactive zones has been confirmed, and a rotating frequency has been estimated to 660 Hz. The combination of these two diagnostics, OH and acetone PLIF, has permitted to explain the stabilization mechanisms of swirled turbulent flame in this particular configuration. The mechanisms controlling the flame stabilization are non-stationary. As a consequence, simulations of such burner must be intrinsically unsteady. Reynolds Average Numerical Simulations (RANS) could only give results corresponding to the mean propane concentration profile (Figure 7 top left) whereas the reality is quite different as shown in Figure 7. Large Eddy Simulations (LES) resolves the structures of the flow and thus is an adequate tool to simulate these phenomena.

Figure 7: Acetone LIF, transversal visualization of propane mass fraction 5 mm downstream the premixing tube. First image: mean image obtained over 100 images. Regime: 120 m3/h of air and 3 m3/h of propane; Equivalence ratio: ~ = 0.6, compact flame.

881

Figure 8: OH LIF, transversal visualization of reactive zones, 2.5 cm downstream the premixing tube. Regime: 60 ma/h of air and 1.5 m3/h of propane; Equivalence ratio: ~= 0.6, compact flame.

LARGE EDDY SIMULATIONS The numerical solverfor turbulent reactingflows

The calculations are carded out with the LES parallel solver AVBP developed by CERFACS [6]. The full compressible Navier Stokes equations are solved on structured, unstructured or hybrid grids allowing the simulation of reactive turbulent flows on complex geometries by using refined grid cells only in the mixing and reactive regions of the flow. The numerical scheme provides third-order spatial accuracy on hybrid meshes [7]. This point is important because high order numerical schemes are particularly difficult to implement on hybrid meshes but required to perform precise LES. The time integration is done by a third order accurate explicit multistage Runge-Kutta scheme. The Navier Stokes characteristic boundary conditions (NSCBC) have been implemented [8] to ensure a physical representation of the acoustic wave propagation. The objective of LES is to compute the large scale motions of the turbulence while the effects of small scales are modeled. The WALE model [9] is chosen to estimate subgrid scale stresses, whereas the flame-turbulence interaction is described by the dynamic thickened flame model [10-12] which was found relevant to accurately predict partially premixed flames. The grid mesh used for this simulation is very fine in the mixing tube in order to resolve weakly thickened flame. The thickening factor has been set to F = 5 (i.e. the thickness of the resolved flame front is about five times the unstretched laminar flame thickness). This low value is required to allow flashback since a too thick flame would not be able to penetrate in the mixing tube due to the quenching distance. A non premixed flame is expected near the injector nozzle because mixing zones

882

between fuel and air are too small, while a well-premixed flame should occur in the combustion chamber. The use of a small thickening factor increases the accuracy of the thickened flame model and reduces the importance of the subgrid scale model. In such a case the model handles accurately both mixing and perfectly premixed combustion, but also correctly reproduces pure diffusion flames. For the present study, an hybrid grid combining hexahedral, prismatic and tetrahedral elements is used with a total of about 600,000 cells (Figure 9). The walls are assumed to be adiabatic, and the gaseous fuel injected is propane. A single step chemistry is used. The total physical time simulated for each transition is about 0.05s corresponding to 3000 hours CPU time on a SGI 03800 R 14000 500Mhz. The computations are typically performed on 32 processors.

Figure 9: 3D view of the mesh Numerical results The simulations are carded out for the regimes explored experimentally [13]. Snapshots of a compact and flashback regimes are given in figures 10 and 11. The burner dynamics are well reproduced. Both compact and flashback flames can be simulated. Moreover, transitions between these regimes are also well reproduced. Details are given in [ 13] and focus is put in the present paper on the mixing process. Figure 12 compares propane mass fraction from simulations (left) and acetone LIF signal (fight). The coherent structure of mixing is well reproduced by the simulation. Moreover, Figure 10 reveals the high dynamics of reactive zones (symbolized by the white temperature iso-surface) which is observed experimentally in Figure 8 ([4]).

Figure 10: Instantaneous visualization of the compact flame. Iso-surface: temperature (T=1600K); vertical plane: axial velocity; black iso-line: zero axial velocity (U=0); gray iso-line: stoechiometric mixture fraction. Regime: 120 m3/h of air and 3 m3/h of propane; Equivalence ratio: ~= 0.6

883

Figure 11: Instantaneous visualization of the flashback flame. Iso-surface: temperature (T=1600K); vertical plane: axial velocity; black iso-line: zero axial velocity (U--0); gray iso-line: stoechiometric mixture fraction. Regime: 21 m3/h of air and 0.75 m3/h of propane; Equivalence ratio: ~= 0.89

Figure 12: Numerical (left) and experimental visualizations of propane mass fraction, 5mm downstream the combustion chamber. Regime:120 m3/h of air and 3 m3/h of propane; Equivalence ratio: ~= 0.6, compact flame. CONCLUSION We have presented a numerical and experimental combined study. Advanced diagnostics have been used to improve our understanding of the phenomena occurring in lean premixed prevaporized (LPP) burners. Laser induced fluorescence of OH radical shows the high dynamics of the flame and its chaotic behavior due to high turbulence levels. Laser induced fluorescence of acetone demonstrates the presence of highly coherent structures in the mixing process (PVC). These structures are due to the swirl movement imparted to the airflow. These unsteady phenomena, which explain the stabilization process of swirled burners, are well reproduced by Large Eddy Simulations (LES). Further calculations are presently carried out and close comparisons between experiments and simulations will be presented.

884

REFERENCES

1.

Williams, F.A., Combustion Theory (2nd ed.). 1985: Addison-Wesley.

2.

Gupta, A.K., D.G. Lilley, and N. Syred, Swirlflows. 1984: Abacus Press.

3.

Beer, J.M. and N.A. Chigier, Combustion aerodynamics. 1983, Malabar, Florida: Krieger.

.

Galley, D., Pubill Melsi6, A., Ducruix, S., Lacas, F., Veynante, D., Experimental Study of the Dynamics of a LPP injection System. in 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. 2004. Fort Lauderdale, CA. Thurber, M.C., Acetone Laser-Induced Fluorescence for Temperature and Multiparameter Imaging in Gaseous Flows. 1999, PhD Thesis, Stanford University. Sch6nfeld, T. and M. Rudgyard, Steady and Unsteady Flows Simulations Using the Hybrid Flow Solver AVBP. AIAA Journal, 1999. 37(11): p. 1378-1385. Colin, O. and M. Rudgyard, Development of high-order Taylor-Galerkin schemes for unsteady calculations. Journal of Computational Physics, 2000. 162(2): p. 338-371. Poinsot, T. and S. Lele, Boundary conditions for direct simulations of compressible viscous flows. Journal of Computational Physics, 1992. 101(1): p. 104-129. Nicoud, F. and F. Ducros, Subgrid-scale stress modelling based on the square ofthe velocity gradient. Flow Turbulence and Combustion, 1999. 62(3): p. 183-200.

10.

Angelberger, D., et al. Large Eddy Simulations of combustion instabilities in premixed flames. in Summer Program. 1998: Center for Turbulence Research, NASA Ames/Stanford Univ.

11.

Colin, O., et al., A thickened flame model for large eddy simulations of turbulent premixed combustion. Physics of Fluids, 2000. 12(7): p. 1843-1863.

12.

L6gier, J.-P., T. Poinsot, and D. Veynante. Dynamically thickened flame Large Eddy Simulation model for premixed and non-premixed turbulent combustion, in Summer Program 2000. Center for Turbulence Research, Stanford, USA.

13.

Sommerer, Y., Galley, D., Poinsot, T., Ducruix, S., Lacas, F., Veynate, D., Large Eddy Simulation and Experimental Study of Flashback and Blow-Off in a Lean Partially Premixed Swirled Burner. Journal of Turbulence, 2004. 5(037).

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

885

W H I T E IN T I M E S C A L A R A D V E C T I O N M O D E L AS A TOOL FOR SOLVING JOINT COMPOSITION PDF EQUATIONS" DERIVATION AND APPLICATION V. Sabel'nikov and O. Soulard, ONERA, DEFA/EFCA, Chemin de la Huni~re, 91761 Palaiseau Cedex, France

ABSTRACT A new light is shed upon Eulerian Monte Carlo (EMC) methods and their application to the simulation of turbulent reactive flows. A rapidly decorrelating velocity field model is used to derive stochastic partial differential equations (SPDE) allowing one to compute the modeled onepoint joint probability density function of turbulent reactive scalars. Those SPDEs are shown to be hyperbolic advection/reaction equations. They are dealt with in a generalized sense, so that discontinuities in the scalar fields can be treated. The EMC method thus defined is coupled with a RANS solver and applied to the computation of a turbulent premixed methane flame over a backward facing step.

KEYWORDS Turbulence simulation and modeling, Chemically reactive flow, Probability density functions, Hyperbolic Stochastic Partial Differential Equation

INTRODUCTION In turbulent reactive flows, phenomena of interest, such as pollutant production, soot formation or extinctions/ignitions, mainly arise from a conjunction of rare physical events (peak temperature, weak mixing conditions, ...) and finite rate chemistry effects. Predicting these phenomena thus requires a precise knowledge of the statistics of the species concentrations and temperature, as well as an accurate description of chemical reactions. Regarding both aspects, the one-point joint composition probability function (PDF) appears as a promising tool: it contains the detailed one-point statistical information of the turbulent scalars and allows chemical source terms to be treated exactly. These advantages are nonetheless counterbalanced by a severe numerical constraint. Owing to the presence of many species in practical applications, the composition PDF possesses a potentially high number of dimensions. This in turn induces heavy computational costs. In particular, the

886 finite methods traditionally employed in computational fluid dynamics (CFD) cannot be used, as their cost increases exponentially with dimensionality. Monte Carlo methods, on the other hand, yield a linearly growing effort and are more adapted to solve PDF equations. So far, in the field of turbulent combustion, Monte Carlo methods have mostly been considered under their Lagrangian form, following the impulsion given by the seminal work of Pope (Pope, 1985). However, Lagrangian Monte Carlo (LMC) methods yield inherent difficulties for controlling statistical convergence and also induce complex couplings with RANS or LES solvers. These results appear as strong incentives to use Eulerian Monte Carlo (EMC) methods. EMC methods are based on stochastic Eulerian fields, which evolve from prescribed stochastic partial differential equations (SPDE) stochastically equivalent to the PDF equation. With the notable exception of Valifio's work (Valifio, 1998), EMC methods have scarcely been used in the field of turbulent combustion. In this article, a new path to derive SPDEs allowing one to compute a modeled joint composition PDF is proposed. This new approach has its foundation in the rapidly decorrelating velocity field model first proposed by Kraichnan (Kraichnan, 1968) and Kazantsev (Kazantsev, 1968). Starting from this model, advection/reaction hyperbolic SPDEs are obtained. These SPDEs are stochastically equivalent to a Fokker-Planck PDF equation with a diffusion term in physical space arising from the delta-correlated velocity field. These SPDEs are treated in a generalized sense. Indeed, discontinuous stochastic fields are likely to appear due to the influence of boundary conditions, even for continuous and differentiable initial solutions. Finally, due to their hyperbolicity, the SPDEs are shown to be intimately connected to Lagrangian methods through the notion of stochastic characteristic. The remaining of the paper is organized as follows. First, a modeled equation for the composition PDF is given. SPDEs stochastically equivalent to this PDF are derived, and a simple example is proposed to illustrate their basic features. Then, the EMC method is coupled with a RANS solver and applied to the computation of a turbulent premixed methane flame over a backward facing step.

P D F E Q U A T I O N OF A T U R B U L E N T

REACTIVE SCALAR

Without loss of generality, the composition PDF, and the subsequent derivation of the SPDEs allowing to compute it, is detailed for only one turbulent reactive scalar c. This scalar evolves according to an advection/diffusion/reaction equation:

Oc Oc O---t+ Uj Oxj _

10Jj pOxj f- S(c)

(1)

The left-hand side describes the advection of the scalar field by the turbulent velocity U while the two terms on the right-hand side respectively describe the effects of molecular diffusion and chemical reaction. For variable density flows with low Mach numbers, working with density-weighted (Favre) statistics is a widespread technique. If pc is the one point PDF of the scalar c, then the Favre one-point PDF fc is defined by: (p) fc(C) = p(c) pc(c) (2) where p is the density. Favre averaged quantities are noted z while Reynolds averages are noted (.). The Low-Mach number assumption is here necessary for expressing the density as a function

887 of the scalar concentration p = p(c). In this work, it will be supposed that this assumption is verified. Using standard techniques (Pope, 1985), one can derive from equation (1) the transport equation of the Favre PDF f~. In this equation, the effects of molecular mixing and turbulent advection appear in an unclosed form and require modeling. Molecular mixing is here modeled by an operator noted Yt4. This general notation will be kept in the derivation of the EMC method. In the practical applications described afterwards, the IEM model (Villermaux and Devillon, 1972) will be used: (3) where (w~} is the mean mixing frequency. As for turbulent advection, it is modeled with a gradient diffusion assumption (Pope, 1985). As a result, the following modeled transport equation is obtained for f~, the Favre one-point PDF of c : 0

0 (P) lrO~zj

- Occ ((p} 3Arc) - Oc ((p} S(c)f~)

(4)

where FT is a turbulent diffusion coefficient. Note that if Adfc does not include derivatives in composition space (as with the IEM model), equation (4) is parabolic in space and hyperbolic in composition space, so that it is a degenerate hypo-elliptic Fokker-Planck equation.

DERIVATION

OF T H E S P D E S

In this section, an SPDE stochastically equivalent to the PDF equation (4) is derived. This SPDE governs the evolution of a stochastic scalar field hereafter denoted 0. In devising such an SPDE, the major difficulty does not stem from the influence of mean advection, chemical reactions or micromixing. Mean advection and chemical reactions appear under an exact form in the PDF equation (4) and will also be present under an exact form in the stochastic field equation. As for micromixing, stochastic processes yielding a model A4 in the PDF equation (4) have already been devised in the frame of LMC methods and can be readily applied to our case. These processes corresponding to the operator A4 are further noted M(O; x, t), and are added as source terms in the stochastic field equation. For the IEM model, M is deterministic and is defined by (Villermaux and Devillon, 1972): M(O; ~, t) = - ( ~ 3 (0 - O) (5) The last and main question that now remains to be answered is: how can one account for the influence of turbulent advection on the scalar field statistics ? To try and figure out this problem, an equation for the stochastic field 0 is looked for under the following form: 00

O--t

00

+ uj-d-- = F(O; x, t) ozj

(6)

where F(O, x,t) - -Uj-g-~x M(O; x, t ) + S(O) accounts for mean advection, micromixing, and chemical reaction as explained above. Equation (6) is a first order SPDE. In this equation, u is a stochastic velocity which needs to be precised. It does not correspond to the Favre fluctuating velocity u" and in particular does not necessarily respect the continuity constraint and does not necessarily average to zero. The only feature which is of interest for our purpose is that u should yield, in the PDF equation derived from equation (6), a diffusion term similar to the one present in equation (4).

888 It is known that such a term is obtained with the Kraichnan-Kazantsev (Kr-Ka) velocity field model (Kraichnan, 1968; Kazantsev, 1968), which accounts for the influence on a passive scalar of a delta correlated Gaussian velocity field. Thus, the key idea of our approach consists in modeling u as u = u d -4- U g where ' I t d is a deterministic component and u 9 is a Gaussian random component of the velocity. Then, in equation (6), we let the correlation time of u g tend to zero, in the same way Stratonovitch did to give a meaning to his stochastic integral. As a result, we obtain the following SPDE with Stratonovitch interpretation for the stochastic field (see (Gardiner, 1985) for more details on Stratonovitch interpretation): 00

e 00

9

ot +

+

00 o

-

F(o,

(7)

The symbol o is used to denote the Stratonovitch interpretation of the stochastic product. The t'), where the tensor Aij velocity u g is delta correlated: ( u ~ ( x , t ) u ~ ( x ' , t ' ) } = 2 A i j ( x , x ' ) 5 ( t accounts for the spatial structure of the velocity field. It is essential to note that this equation is a hyperbolic advection/reaction equation. The Stratonovitch calculus is identical to the classical one, so that ujg o o0 has the same physical advection properties as if ug was deterministic. In particular, if F(O, x) = 0 then the stochastic field 0 is simply advected alongside a stochastic path. Except for the influence of boundary conditions, initial profiles are strictly preserved and do not undergo any kind of diffusion process. This advection properties would be lost if an Ito interpretation was used (see section ). The last step in the derivation of an equation for the stochastic field consists in precising U d and u 9 so that the P D F of 0 is identical to fc. This can be achieved by expressing the PDF equation of the stochastic scalar field 0 and by identifying it to the P D F equation (4) of c. This procedure is not reproduced here for the sake of brevity and will be published elsewhere. It yields the following constraints on u d and ug:

1 ( d u ~ ( x , t ) d u ~ ( x , t ) } - VT6ijdt

uJdt -- -- 21 { Od--~ug(x' ~)dug

(8)

, ~) } __ -~1 O_~OxjFTdt

Solutions fulfilling these constraints are not unique. The simplest one is given by: -

2

gr-Tewj

d _ 10rT uj - - 5 ox~

10_~FT (p-~Ox~

(9)

where the Wj are independent standard Brownian processes (zero mean, variance equal to dt). With this solution, the following SPDE is obtained:

OO dt +

o-7

1 OFT

1 0 (p} FT

(pl

dt +

o d W j ( t ) = dM(O" x, t) + S(O)dt

(10)

This equation is an hyperbolic advection/reaction equation, stochastically equivalent to the PDF equation (4). In its derivation, as opposed to Valifio (1998), no hypothesis on the smoothness and differentiability of the stochastic fields was required so that it has a generalized sense. Besides, the velocity advecting the stochastic field is formed by mean quantities, so that its length scale is also that of a mean quantity. This, however, does not imply that the scalar field also evolves on a mean length scale. Equation (10) is also driven by a chemical source term which, in practice, possesses

889

stiff gradients in composition space. These in turn can generate strong gradients in physical space for the stochastic fields.

ILLUSTRATION

OF THE PROPERTIES

OF H Y P E R B O L I C

SPDES

The concepts developed in the previous section are more clearly presented in the context of pure turbulent advection, i.e. with zero mean velocity and no micromixing nor reaction. For the sake of simplicity, only one dimension is considered and a constant coefficient FT ---- F is chosen. The corresponding abridged version of equation (4) is:

OA 02f~ Ot = F OX 2

(11)

Two cases will be considered. In the first case, the physical domain is chosen unbounded. This case allows to gain more insight into the connection between PDF equations and hyperbolic SPDEs and to illustrate the advecting properties of hyperbolic SPDEs. In the second example, a finite domain is considered. This case discovers the impact of boundary conditions on the regularity of the solutions of hyperbolic SPDEs.

Turbulent advection acting alone : u n b o u n d e d d o m a i n PDF equation and hyperbolic SPDEs Let the initial condition of equation (11) be given by:

f(~; x, t = 0) = f0(c; ~)

(12)

The solution of the parabolic equation (11) is then : f(c; x, t) =

f

1 _~Ldy 4Ft

fo(c; X -- y) ~ ( ~

(13)

By definition, the function ~ e1- 4 r t L is the PDF of the Brownian process y = x/~FW (W is the standard Brownian process). Hence, the integral on the right-hand side of equation (13) can be interpreted as the mean of fo(c; x - ~ W ( t ) ) :

f(c;x,t) - ( f o ( c ; x - x / ~ W ) } W

(14)

where (')w denotes averaging over the Brownian process. Now, one can identify the function f(c; x, t) = f0(c; x - x/~FW(t)) as a first integral of the following SODEs :

dc - 0

(15)

dx - v ~ d W ( t )

The term first integral means that ] is constant alongside the trajectory given by the SODEs (15). As a result, knowing the initial condition f0 and the trajectories given by (15) allows to compute the PDF f, through equation (14).

890 In the deterministic case, ODEs similar to (15) are named characteristic curves of hyperbolic advection PDEs. In the stochastic case, it is logical to name the SODEs (15) stochastic characteristic curves of the following hyperbolic advection SPDE:

O--~dt+ ~

o dW - 0

(16)

Equation (16) is the abridged version of equation (10). As it was stated above, it describes the pure turbulent stochastic advection of the concentration field by the white in time velocity field

v~dW. The Stratonovitch interpretation arises from considering a limit of a short correlated velocity field. This interpretation is essential: if one erroneously chooses the Ito interpretation then one gets after averaging equation (16) with Ito interpretation:

0r =0 Ot

(17)

since for the Ito interpretation (X/2-F~dW} = 0. This result is incompatible with the scalar mean equation deduced from the PDF equation (11)"

O (c> = F 0~~ r

Ot

(18)

Ox2

With the Stratonovitch interpretation, this correlation is not zero. Indeed, the Furutsu-Novikov formula (Gardiner, 1985) gives:

(~~x

0~ r o dW} = -V~ Ox2

(19)

This in turn is compatible with the equation of the averaged scalar (18).

Advecting properties of the SPDE (16) The Stratonovitch interpretation preserves the classical differential calculus (Gardiner, 1985), so that equation (16) is an hyperbolic advection equation as would be the case if the coefficient in the advection term had a non-zero correlation time. The solution of equation (16) is given by:

(20)

~(~, t) - ~o(~- 45-~w(t))

where Co is the initial condition of the stochastic field c. Equation (20) is another way of expressing the fact that equation (16) preserves the shape of the initial solution and advects it along Brownian paths. In particular, even an initial discontinuous profile such as the Heaviside function H(x) is transported without alteration. It can also be checked that the solution (20) yields correct evolutions for the moments. For instance, with a Heaviside initial condition co(x) - H(x), and knowing that the PDF fw of W is a centered gaussian of variance t (fw(W) 1 e-W2/2t ), one obtains for the scalar mean"

(c) (x, t) -

~ H(x - v ~ F W ) f w d W = ~ 1 + e r f ( 2 ~

)

This expression is also the one obtained directly from the scalar mean equation (18).

(21)

891

B o u n d e d domain: impact of boundary conditions Let us consider now a bounded domain in order to illustrate the impact of boundary conditions on the solution of (16). The domain is chosen to be [0, 1] and the boundary conditions for the PDF fc are chosen to be fc = 5(c) at x = 0 and f~ = 5 ( c - 1) at x = 1. This corresponds for the stochastic field of equation (16) to the boundary conditions c = 0 at x = 0 and c = 1 at x = 1. However, if the simultaneous specification of two boundary conditions is necessary for the diffusion equation of the PDF f~, it is not the same for the stochastic field c(x, t), due to the advective nature of equation (16). For instance, at x = 0, the c = 0 inflow condition is only effective when dW is positive and it becomes an outflow condition when dW is negative, and reciprocally at x = 1. What might seem more surprising is that with any arbitrary initial conditions, the limit when t -+ oe of the solution of (16) can be shown to be a step, whose position is moved randomly by the Brownian motion in intervall [0, 1]. Thus, initial profiles, even continuous, are transformed into discontinuous ones due to the influence of boundary conditions. This process can be loosely explained as follows: when dW is positive, part of the initial profile is advected beyond the x = 1 boundary. When dW becomes negative, this initial information is lost, as it is replaced by the inflow value at x = 1 boundary. The same also happens at the x = 0 boundary, where initial information is replaced by the inflow value at x = 0. This process is then repeated at both boundaries until eventually, with probability one, the initial information is lost and only the information given by both boundaries remains.

NUMERICAL

ASPECTS

Numerical scheme The numerical analysis of SPDE (10) is considered in terms of weak convergence and accuracy. Temporal integration is addressed by recasting equation (10) in an SODE form; this allows the use of traditional SODE techniques (Gardiner, 1985). An explicit first order scheme is chosen, with a predictor-corrector procedure generalizing the Heun scheme (Carrillo et al., 2003). As for spatial discretization, scalar fluxes are interpolated with a second order Essentially NonOscillatory (ENO) scheme and a decentered procedure is used for the advection term. Decentering derivatives yields a correlation between the white noise and the discretization error (Soulard and Sabel'nikov, 2003). As a result, despite the second order interpolations, the resulting scheme is only first order in space.

N u m e r i c a l tests The efficiency and accuracy of the EMC method have been tested on a simplified one-dimensional version of the scalar P D F equation (4), with constant density and constant mean velocity. Details of these calculations can be found in Soulard and Sabel'nikov (2003). Except for a symptomatic case, which does not correspond to practical calculations, it is checked in Soulard and Sabel'nikov (2003) that statistical and spatial convergence rates are conform to the theoretical ones. Besides, the EMC method is also compared against its Lagrangian counterpart. It is shown that, in general, both methods attain a given precision for an equivalent CPU time.

892

S I M U L A T I O N OF A B A C K W A R D

FACING STEP WITH A RANS/EMC

SOLVER

In this section, the SPDEs derived and illustrated in the previous sections are used to construct a RANS/EMC solver. This solver is then applied to the simulation of a premixed methane flame over a backward facing step.

RANS/EMC

solver

In the expression of the PDF equation (4), it has been supposed that the mean density (p), the Favre averaged velocity U, the turbulent diffusion coefficient FT and the turbulent mixing frequency (we) were known. A RANS solver can be used to compute these quantities. Namely, the Favre averaged continuity and momentum equations are solved and a standard k - e model is used to compute the turbulent stresses, with k the turbulent kinetic energy and r the turbulent dissipation.

Continuity

ot

Momentum

ot

Turbulent kinetic energy

9

~

)__ 0 (

o(

Turbulent dissipation

(22)

--ozj + ozj

O{p)k_.~_ 0

)_ o(

Pr~

aii models the turbulent stresses with an eddy viscosity hypothesis-

05j 2 oGs,j) Oxi

(23)

30xk

Pk and dk (respectively P~ and G) are the production and dissipation terms of the turbulent kinetic energy (resp. dissipation). Standard expressions are used for these terms, as found for instance in Pope (2000). The eddy viscosity is given by u t - C,@. Standard values are chosen for the k - e model constants, as given in Pope (2000). The statistics of species mass fractions Yk and total enthalpy ht are computed with an EMC solver using SPDEs derived with the procedure detailed in section 9

Mass fraction

9

1 OFT

10---~-FT~ ~xjdt -Jr-~ ~ x j

o dWj(t)

= --wk(Yk -- fZk)dt + S(Y, ht)dt Total enthalpy

9

1 0FT

L~

(24)

~

+ ~ ~

o dWj(t)

= --COh(ht- ]tt)dt In the enthalpy equation, a unity Lewis number assumption has been made and acoustic interactions, viscous dissipation, and body forces were neglected under a low Mach number assumption. In particular, the material derivative of pressure has been neglected. In these equations, the turbulent diffusivity is defined by FT = ut/Sct, where Sct is a turbulent schmidt number supposed to be of order unity. The mixing frequencies are defined by wk = cob - wc = Cr where Cr is a constant supposed to be equal to 0.7. The Favre averaged values of the species mass fraction and ~ 1 total enthalpy are computed by: ])k - ~1 ~ Yk and ht = -~ ~_, ht, where the sums are taken over N realisations of the stochastic fields. (Favre - and not Reynolds - averages are obtained from these sums, because the stochastic fields directly represent the Favre PDF.)

893

Information is transmitted from the RANS solver to the EMC solver via (p), f), FT and coc. The influence of the EMC solver on the RANS solver is achieved through the mean pressure" the Favre averaged temperature is computed from IS'k, ht and U'. Then, the mean equation of state is used to compute the mean pressure ( P ) = (p)fT. From a numerical point of view, the RANS equations (22) are solved with an explicit Osher procedure and a second order Essentially Non-Oscillatory interpolation. The stochastic equations (24) are solved as described in the previous section.

Results The physical domain is L = l m horizontally and H = 0.1m vertically. The height of the step is h = 0.035m. At inlet, a stoichiometric methane/air mixture is injected at U0 = 55m/s and To = 525K. The inlet values of the turbulent quantities are k0 = 30rn2/s 2 and ~0 = 500m2/s a. At oulet, the pressure is fixed: Ps = 1bar. The upper and lower walls are considered adiabatic and wall functions are applied. The domain is discretized by a 40 x 20 mesh. N = 50 stochastic fields are used for the EMC solver. Chemistry is accounted for by a 5 species-1 reaction scheme. Convergence was obtained for a physical time of t = 8 NL after 30000 time steps. The corresponding CPU time is 4 hours on a single 1.6 Ghz processor. This was found to be on the order of the CPU time required by a LMC solver with a number of particles per cell equivalent to the number of stochastic fields. Mean and rms temperature vertical profiles at different locations are shown on figure 1. They are compared against the experimental results obtained by Magre and Collin (1994). The main point of this comparison is the good agreement found between the calculation and the experiment. In particular, the peak location of the calculated rms temperature is close to the experimental one. Nonetheless, several differences can be observed. Near the lower wall, the computed temperature x=460mm x=250

mm

.......... y. "\

.

.+ j"q2%.

"\

:',k\

\..

X ,

,

,

|

20

.

.

.

.

i

40

.

.

.

.

, 6O

.

.

.

.

,

80

.

.

.

.\ '.,

.

(a)

Y

x--460mm

x=250mm

[ ~,,"/

/

//

/

/

/

/

x

\ ",,

',,,

......

~_2 . c ...... ]

)A,j'~/- "

/;2

"2,]",,,

"'",,,,

/ / //// / / j ~ / , / /,

.

\

- _ _

~..>-

_ _ ~

..........> ~ "

/ /

. . . .

2'0 .

. . . . .

;:o

v

6o

~o

(b)

Figure 1: Mean (a) and rms (b) temperature profiles is higher than the experimental one. This can be explained by the fact that walls are adiabatic in the simulation, but not in the experiment. Other differences, as the low levels of computed rms

894 temperature, can be attributed to numerical diffusion due to a coarse grid and to the simplicity of the models used. It should be noted that standard constant values were used in these models and that no optimization was performed.

Conclusions

A new path is proposed to derive SPDEs allowing one to compute the modeled one-point FokkerPlanck PDF of turbulent reactive scalars. A rapid decorrelating velocity field (Kraichnan, 1968; Kazantsev, 1968) is used to model the turbulent advection of stochastic scalars. The obtained SPDEs are then shown to be hyperbolic advection/reaction equations and are dealt with in a generalized sense. They allow one to establish a connection between Eulerian and Lagrangian Monte Carlo approaches, through the notion of stochastic characteristic. The EMC method is then applied to the calculation of a premixed methane/air flame over a backward facing step. Comparison against experimental data yields a qualitatively good agreement between mean temperatures and temperature variances. These promising results are now studied further and finer calculations are being carried out.

REFERENCES

Carrillo, O., nes, M. I., Garcia-Ojalvo, J., Casademunt, J., Sancho, J., Feb 2003. Intrinsinc noiseinduced phase transitions: beyond the noise interpretation, arxiv:cond-mat. Gardiner, C., 1985. Handbook of Stochastic Methods, 2nd Edition. Springer. Kazantsev, A., 1968. Enhancement of a magnetic field by a conducting fluid. Sov. Phys. JETP 26, 1031. Kraichnan, R., 1968. Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11,945. Magre, P., Collin, G., 1994. Application de la drasc ~ l'opSration a3c. Tech. rep., ONERA/DEFA. Pope, S. B., 1985. Pdf methods for turbulent reactive flows. Progress in Energy and Combustion Science 27, 119-192. Pope, S. B., 2000. Turbulent flows. Cambridge University Press. Soulard, O., Sabel'nikov, V., septembre 2003. M~thode stochastique eul~rienne pour la r~solution des ~quations pdf et application/~ la simulation des 5coulements turbulents r~actifs. Tech. rep., ONERA/DEFA. Valifio, L., 1998. A field Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent flow. Flow, turbulence and combustion 60, 157-172. Villermaux, J., Devillon, J., 1972. Representation de la redistribution des domaines de s~gr~gation dans un fluide par un module d'interaction ph~nom~nologique. In: 2~e Int. Syrup. Chem. React. Engng. Amsterdam. Vol. B-1-13.

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

895

T H E E F F E C T S OF M I C R O M I X I N G O N COMBUSTION EXTINCTION LIMITS FOR MICRO COMBUSTOR APPLICATIONS C. Dumand and V.A. Sabel'nikov Fundamental and Applied Energetics Department, Onera, 91761, Palaiseau, France

ABSTRACT A preliminary estimation of combustion characteristics in micro combustors is performed with the Perfectly Stirred Reactor model. This model allows us to qualitatively assess the range of stable combustion, as well as the extinction limits for different fuels, inlet temperatures, equivalence ratios and heat loss rates. Linked to low turbulence rates, the weak mixing effects on chemistry are studied with the Partially Stirred Reactor model. Two micromixing models (Vulis and Interaction by Exchange with the Mean) are compared and the crucial influence of mixing at molecular scale on combustion extinction limits is pointed out. Then, practical cases with estimated operating conditions are tested and results are discussed. KEYWORDS

Turbulence-chemistry interaction, Perfectly and Partially Stirred Reactors, Micromixing, Micro-machines, Combustion extinction limits

1

INTRODUCTION

A Japanese company presented this summer a micro helicopter weighting only twelve grams and totally autonomous. This new record is representative of the proliferation of small portable electronics and micro-systems: computers, cell phones, digital assistants, micro-machines, etc. Those micro devices require compact energy supplies. Micro engines appear as a promising solution. Indeed, hydrocarbon fuels have a specific energy content about two orders of magnitude greater than the best available batteries. Thus, even with a 5% conversion efficiency, hydrocarbon fuels provides about five times higher energy storage density than batteries. This simple observation led many research groups to consider using micro-scale engines for converting fuel to electricity. We can note, for example, several new concepts of micro-machines as micro gas turbine (Epstein, 2003), toroidal "Swiss Roll" counterflow heat exchanger (Ahn et al., 2004) or combustor coupled with thermoelectric converter (Vican et al., 2002).

896

All those new concepts of micro-generator are based on combustion in a small volume (< 1 cma). However, scale-down of macro-scale combustion devices implies new problems concerning flame stabilization and combustion efficiency. Various technical difficulties, which may not be significant or well known in macro scale combustors, become crucial for combustion in micro volumes. The most specific difficulties are : small mean residence time (~ 1 ms) which is comparable with chemical times (DamkShler number ~ 1), weak turbulent mixing linked to modest Reynolds number (Re ~ 500), and significant heat losses on walls due to large ratio between combustor surface and volume. A theoretical analysis of combustion in small volumes has to be led to design an adapted geometry of micro combustor and prepare future experiments. In a first step, simple models of reactor are used to qualitatively evaluate the range of intensive and stable combustion. Those models allow to study separately the influence of each parameter. Then, in this paper, a parametric study is performed with two models of reactor to prepare more quantitative 3-D calculations. In order to get a preliminary estimation of the low DamkShler number influence on the overall characteristics (extinction limits, combustion efficiency, mean temperature), a simplified approach is used to model the combustion chamber. At first, we suppose that the mixing inside the combustor is instantaneous. Then, the Perfectly Stirred Reactor (PSR) model, widespread in the combustion community (Beer & Lee, 1965), is considered to perform a parametric study on relevant parameters (temperature of inlet gases, type of fuel, equivalence ratio, heat losses rate). However, due to the very small residence time in small combustor, variations linked to non-perfect mixing become considerable and play a prominent role on combustion. To study these turbulencechemistry interactions, the Partially Stirred Reactor (PaSR) model (Correa , 1993) is used :inside the combustion chambers, the fluid is broken up into discrete fragments or "pockets" with a very small size and these fragments are uniformly dispersed throughout the reactor. The molecular exchange between pockets is controlled by micromixing. In this study, two different approaches of micromixing modelisation are compared. These models of molecular mixing modify the finite rate chemistry used in our calculations. Finally, practical cases are tested for relevant operating conditions. Those conditions are closely linked to technologies and materials used for manufacturing the micro systems. For example, the studied range of equivalence ratios is under-stoichiometric to respect the upper temperature limit of wall material. Inlet temperature and pressure of gases are taken from the MIT micro turbine configuration (Epstein , 2003). The temperature of inlet gases is very important for combustion efciency. In MIT micro-turbine, inlet gases are heated-up by compression and heat exchanges in the mixing channel. Then, reactive gases are injected at a temperature between 700K and 1000K. We can note that the same range of temperature is reached in other concepts as micro heat exchanger. Two fuels are tested to study the best case in term of combustion stabilization; hydrogen, and for a more realistic case, methane. Furthermore, mixtures of those two fuels in several concentrations are also studied. Due to small size of devices and the high surface to volume ratio, a large range of heat loss rates is considered (from an adiabatic case to a 50% heat losses case). All the studied ranges and the reference values used in this study are reported in table 1.

2

MODELISATION

First of all, to understand the effect of small DamkShler number, the Perfectly Stirred Reactor model is used. This model allows us to assess the range of stable combustion and to obtain the extinction limits for different operating conditions of micro systems.

897

TABLE 1 P A R A M E T E R RANGES AND R E F E R E N C E VALUES Studied ranges

Reference values

700- 1000

800

Pressure (Pa)

300000

300000

Equivalence ratio

0,5 - 0,8

0,6

Fuel composition

//2 with CH4

H2 or CH4

0 - 0,5

0

Inlet temperature (K)

Heat loss rate coefficient A

2.1

P e r f e c t l y Stirred R e a c t o r model

The combustor is modeled by an open reactor in which inlet gases are instantaneously well mixed after injection. Mixing between fresh and burnt gases is perfect for all turbulence scales. There are no fluctuations and mean values are assumed to be instantaneous ones. Furthermore, concentrations of chemical species, temperature and the others thermodynamics properties are considered uniform throughout the reactor. The inlet gas is a premixed air-fuel reactive mixture and different ratios between hydrogen and methane are considered in this study. A detailed chemical kinetic mechanism is used for the air-H2-CH4 combustion with 79 reactions and 21 species (Davidenko et al. , 2002). The instantaneous production rate W of species n is given by the law of mass action

Wn-Mn~

l]nr--l]nr) r=l

kfr

C; i r - k b r i=1

I I c~ :'r

,

(1)

i=1

where Mn is the molecular mass of species n, N~ the number of reactions involved, kfr and kb~ are the reaction rates of reaction r, u~r and v~'r are stoichiometric coefficients, and Ci (i = 1, ..., Ns) is the concentration of species i. The last concentration CNs+l represents third-body species and ti,~ the third-body efficiency for reaction r (this term is fixed to unity in our calculations)

Ns

PYi

CNs+l,r = E t i r ~ i=1 Mi

(2)

For stationary state, the classical balance equations for species, enthalpy and mass conservation are

W~ (Y~, T) Yo,i - Yi Tr ho - h Q "r~ p pV 7-r ~ rh

(a) (4)

(5)

where Yi and Y0# are the reactor and the inflow mass fraction of species i, rr is the mean residence time, p the density, h and h0 are the reactor and the inflow mass enthalpy, Q is the heat loss rate, V the volume of combustor a n d / n the mass flow rate. The mean residence time inside the combustor depends on the mass flow rate, the pressure of gases and the volume of reactor. Eqn. 4 respects the enthalpy conservation between the system entry and outlet for an adiabatic case (Q = 0). In the non-adiabatic case, heat losses are modeled by a linear function of the sensible

898

enthalpy creation during combustion T

Q = _A f c p ( T ) d T Tr To

,

Cp(T) - ~ Y~Cpi(T)

O <_ A < 1

(6)

(7)

,

i=1

where Cp is the specific heat of the mixture at constant pressure. Thermodynamic properties and mass fractions of the reactive mixture are determined for a stationary state and a fixed residence time. For extinction limit study, calculations are started with an ignited combustion and a high residence time in comparison with the chemical time. Then, the balance equations are solved for a stationary state with smaller and smaller residence times until a combustion extinction is observed. This critical residence time is compared for different conditions of inlet temperature, fuel composition, equivalence ratio, and heat loss rates previously presented. All results of this parametric study are presented and discussed in section 3.

2.2

P a r t i a l l y Stirred R e a c t o r models

Partially Stirred Reactor (PaSR) models have been developed to take into account finite micromixing times win, that is, the presence of scalar fluctuations at the molecular scale inside the combustor. This model have been used previously by Sabel'nikov and Figueira da Silva to show the role played by the ratios between the residence time and the micromixing time and chemical times, respectively on combustion (Sabel'nikov & Figueira da Silva , 2002). Fluctuations of temperature and species concentrations have a great influence on the highly non-linear term Wi presented in Eqn. 1. Two different micromixing models are studied and compared : Vulis model (Vulis, 1961) and Interaction by Exchange with the Mean (IEM) model (Villermaux & Devillon , 1972).

2.2.1

Vulis model

Vulis model is a global and much simpler approach than the other model presented in the next paragraphs. From a physical point of view, it is supposed that fresh gases injected in the burning zone are well mixed with hot gases after a finite time rm representative of micromixing intensity. Mean mass fractions inside the reactor are determined with chemical production rates and, in Vulis model, those production terms are calculated with "virtual" mass fractions Yv,i,

Y~ - ~,o r,.

w~(Yv#, T) =

_

p

(8)

.

Furthermore, those mass fractions Yv,i are linked to micromixing time and production rate by the following equation

gv,~ - g~ w~(Yv,~, T) = _ Tm p

.

(9)

If we define x as the dimensionless ratio between 7r and Tin, the combination of Eqn. 8 and 9 leads to the following relation between "virtual" and mean mass fractions inside the reactor ( 1 ) 1 rr Yvi-Y/ 1+-xY/,0 , x(10) '

X

Tm

Consequently, when the residence time is much greater than the micromixing time, the virtual fractions tend to the value of mean mass fractions and results for stationary state are the same as

899

in the PSR case. Conversely, when x is small, the virtual fractions, Yv,i, tend to the inflow values, Y~,0, and the stationary solution is a slow combustion regime.

2.2.2

Interaction by Exchange with the Mean (IEM) model

The IEM model introduced by Villermaux in 1972 takes into account a particle age repartition inside the combustor. Injected particles are heated by mixing with older ones in burnt pockets of gases. Furthermore, mixing between all particles depends on the micromixing time scale Tin. Thus, the instantaneous mass fraction values in reactor Y/(t) for each given residence time 7 are determined by dY/ _- Y / - Y /

dt

rm

t

W~(~,T) -fi

,

0
,

Y/(t-0)=Y/,0

.

(11)

In the general case, the mean concentration Y/ is the integral of Y/ weighted by the PDF p(Y/) which is representative of the species concentrations repartition in the reactor 1

- f Y~p(Y~)dY~

(12)

o

When the combustion is stationary, an ergodicity assumption is made between the concentration repartition for a given time and the particle age repartition inside combustor p(Yi)dYi - f (t)dt , (13) and thus Eqn. 12 yields

-Y~ /=

oo

Y~(t) f (t)dt

(14)

o

The particle age PDF f(t) corresponds to a Poisson distribution" particles are randomly added to the reactor while other ones are randomly removed. This repartition has a characteristic form described by e-~/'T-r-----:-~' where Tr is the mean residence time.

3

3.1

RESULTS AND ANALYSIS

P a r a m e t r i c s t u d y with the P S R model

Several parameters having influence on combustion in small volumes are studied in ranges corresponding to possible operating conditions for a micro power generator. This parametric study represents a data base obtained with a full kinetic chemistry mechanism and shows the respective importance of each parameter on combustion extinction limit. Those data should be compared with more accurate calculations results of the complete combustor geometry performed with a CFD code. Figure 1 a. shows the temperature evolution inside the reactor according to different residence times, from 10 seconds to extinction limit, for several fuel compositions. Those results are obtained for an inlet gases temperature of 800K and with adiabatic conditions. We can notice that the critical residence time of extinction strongly depends on the fuel properties. Those extinction limits are presented on figure 1 b. for several temperatures from 700K to 1000K of-injected gases. Considering those results, it appears that inlet temperature plays a great role on combustion extinction limits. Thus, the critical time can be divided by two when the inlet temperature increases by 300K. The inlet gas heat-up in heat-exchangers before their injection in combustor must be considered for the combustion stabilization. As expected, a hydrogen-air mixture has a smaller

900

2200

"" "" ~

~ -

TO = 8 0 0 K ]

j

i ~ . ~ -.,~.. " " ~~ _ "".9_

.~."

... -.- -

5E-05~ ~

~ . _ - ----."

4E-05

..... -

-

-.

-

3E-05 _

_

2000

2E-05

1800

~

"" "" "" "- "-- ~ . . . .

,~1E-05

I - - 18o0

./~/ // // J

1400

~

"

. . . . . . . . .......

e

/

1 0 0 % C H 4 0 % H2 9 5 % C H 4 5 % H2 7 5 % C H 4 2 5 % H2 ;0% CH4 50%H2 0%CH4 1 0 0 % H2

.........~. ~. ~.~.~...._ ~'~'~"--.~...~ n --

I

,00% c . 4 o , : - , ~ - - ~ - c - 4 - c - a - ~ _ _

--~--

-

--->-.----4"----- -~-- -

1200 1 0 -e

1 0 -s

1 0 -+

10-3

1 0 .2

1 0 -1

i

100

Tr(s)

I

I

95% CH4 5% H2 75% CH4 25% H2 50% CH4 50% H2 0% CH4 100% H2 i

i

700

~

800

i

i

I

i

i

i

i

i

900

TO (K)

a.

___.~

"" "- ~...~

I

1000

b.

Figure 1" a. Evolution of temperature for different fuels ; b. Critical residence time for different fuels and inlet temperatures critical residence time than a methane-air mixture. We can notice that only 5% of hydrogen added to methane allows to divide by two the critical residence time of the reactive mixture. Moreover, a Norton and Vlachos study (Norton & Vlachos, 2004) shows that the propane-air combustion is more robust than the methane-air one. Propane could be a good compromise between combustion stability, observed with hydrogen, and storage facilities offered by hydrocarbon fuels. Another important parameter is the equivalence ratio of the inlet reactive mixture. For reference values in operating conditions, the change of the equivalence ratio from 0.5 to 0.8 can lead to multiply the critical residence time by three. The temperature level of combustion decreases strongly with the equivalence ratio for both fuels studied. Furthermore, a difference of 400K on stationary state temperature is observed in the range considered. Figures 2 a. and b. show the crucial influ-

2200 2100

J

f

[]

Micromachines me

10 a

.

.

.

.

.

.

.

.

.

.

.

2000 1900

y

o .... C H 4 A = 0 o . CH4 A=0,5 ----e--H2 A = 0 ----~--H2 A=0,5

1800

~1700

/

I- 16oo

/

1300 1200 110~q

~

/

104

,_

/

1500 1400

/

I--

,~/

~

o

---O-_

+ d .............. ,0+ ~"0-,' ................ ,0-3 +0+' ................. lO-, 1oo' Tr(s)

10

i 0

a.

CH4 H2

[]

10 s

--

i

--

i

e

, I 0.1

--

--

,

i

.,-"

._ --(3~ --

i

i

I 0.2

i

i

i

, I 0.3

,

,

,

, I

Heat Losses Rate A

0.4

i

,

,

~ I 0.5

b.

Figure 2: Evolution of temperature a. and critical residence time b. for different fuels and different heat loss rates ence of wall heat transfers on combustion temperature and extinction critical time. A high heat loss rate leads to a strong decrease of the temperature level, the combustion efficiency and then

901

the global efficiency of the micro system. In order to limit those transfers, a thermal insulation of combustor walls can be imagined. In the case of the MIT micro turbine, heat losses have been estimated in a range of 20% to 30% (Ribaud, 2003). The critical residence time increases dangerously for a methane-air combustion and reaches the same order of magnitude as the characteristic residence time of micro systems (mr = 5 x 10-4s, (Epstein, 2003)), and thus, the flame stability is threatened. Finally, we can notice that the hydrogen combustion is less affected by heat losses than the methane one.

3.2

Micromixing models comparison and P a S R model

The above micromixing models are tested with our reference conditions shown in table 1 and results are exposed on figure 3 a., for the methane-air combustion, and figure 3 b., for the hydrogen-air combustion. In the studied range and for the two fuels, the Vulis models and the IEM model give

22oo~

ooo

, oo

]

ooo

1800

1800 ~-

~'1700

;~

~1600

A S

v

1500 1400

.]]'/i l' i I /ll ,'J / I7 I Lr

t~

=

L/

];

II

,,

t~ ~i" I ~

]~f

[]

~

I: / .~ [ I ~ I ~r~

O ~ +

;SIRs PSR le-35 Vull$1e-4$ Vulll le-5 $

~1700~[ ~'1600~" --

~-.~~-,EM-le:3-$- - - e - - - IEUle-4$ .... IEMle-5I .... ,EM 1e-6 $

1400p ~

l/

1100 ~- . . 1 100(~0 a.

' ir I /

//r il

if/ //

13~176 13 1200~ ' ~

' 1301~0.s ' .... 7~-,' .... ;~)-a ' .... 7~-:' .... ;'10., ' ..... ;'0o' ..... ;'~l'

Tr(s)

H.', . ' , //# ~ /

II II I =1 ! I //

I

.//

I~

I1

I=/ s

o

o

:

~

PSR VuII$ le-3 $ vu,,,,-,, Vulis le-5 $

vul 'l,':l:

-----O--- IEMle-3$ .... IEMle-4I .... iEMle-5I .... ,EM le-6 $

I/

]

~

j/

~ ..... t~ ......................................

Tr(s)

10' b.

Figure 3: Evolution of temperature for different micromixing times with a methane-air combustion a. and a hydrogen-air combustion b. very close results. Moreover, the Vulis model is simpler than the IEM model and has a low CPU cost. Then, the sensitivity of combustion extinction limit is studied with this micromixing model and results are presented below on figure 4. It is relevant to notice that the critical extinction limit is closely linked to the molecular mixing characteristic time Tin. Actually, when 7m is very small (Tin < 10-5S), the micromixing is good and combustion is not affected. Then, the solution is quite close to the PSR case. For this configuration, combustion is governed by the chemical kinetics of reactions. If the micromixing time becomes larger, the critical residence time of extinction increases strongly and tends to Tin. In this case, extinction of combustion is governed by the turbulencechemistry interactions inside the combustor. Based on the MIT micro turbine geometry, mean velocity is evaluated and, in a first approximation, the characteristic time of molecular mixing is estimated to be 0,1 ms. This value is very close to the characteristic residence time of micro systems previously mentioned. Figure 5 shows calculations for a practical case based on MIT micro turbine operating conditions. Those curves indicate, in a first approximation, that maximum temperature level reached for a hydrogen-air combustion is near 1800 K, and near 1650 K for a methane-air combustion. Furthermore, we can notice that temperature decreases quickly for residence times smaller than 10-as. Thus, to guarantee a high intensity combustion and flame stabilization, the mean residence time inside the combustor must be greater than 10-as, especially for methane-air

902

101

:

/x~./ 4,/./

i .I

10 .2

,/

/

I

Combustion

/

Zone

/z( /

10.3

*/~ ////

//~/./ //

/

lO-~

~e~ - I--1 / / t.5

/

9,

, 10 .5

,

H2

/

/ /

Extinction Zone

/

.......

CH4

---O---

//

./ 10.5

13

/~

/

,,,,,~,~

-4

,

,,,,,,,

,

10 3

........ =

10 .2

.......

, 10 1

T m (s)

Figure 4: Evolution of extinction limits for hydrogen-air and methane-air combustion according to micromixing time combustion. Nevertheless, as shown previously, this limit can be improved with a better mixing inside the combustor. Then, 3-D reactive flow calculations will be performed and several geometries of combustion chamber with large recirculation zones will be proposed.

I

1800

/

/

I I

/ /// 1600

/ /

1500

....

/

Methane

//

i i illll{I

10.5

Air Air

"E "~ '~

I l 10 -e

Hydrogen

E

.~= .~ "~ "~ i i Illllll

10-4

I [ IIIII1|

10.3

Tr(s)

I I IIlllll

10.2

I I Illllll

10-~

I I II[lltl

100

Figure 5: Evolution of stationary combustion with operating conditions of MIT micro turbine (Tin- 10-4S, A - 0, 3) according to residence time

4

CONCLUSIONS

A parametric study is performed with a Perfectly Stirred Reactor model to determine the sensitivity of combustion limits to significant parameters of micro systems and to different fuels. It appears that heat losses, increased by scale-down, are very significant for combustion efficiency. With realistic values of heat losses, the critical time of extinction limit can rise dangerously and reaches the same order of magnitude as the characteristic residence time of the combustor, especially for hydrocarbon fuels. Those results will be compared with complete CFD calculations and with experimentation results.

903

Furthermore, in micro volumes, the time required for having a complete mixing at molecular scale can be of the same order of magnitude as the mean residence time. Thus, influence of partial mixing on combustion is particularly studied with the Vulis and IEM models of micromixing. Those models lead to comparable results in our ranges of study. The cheaper one in CPU cost (Vulis model) will be chosen to be included in a CFD code for global calculations previously mentioned. The micromixing time appears as a crucial parameter and sets the minimum combustor volume for a given mass flow rate. Particle ages inside combustor must be improved in order to support exchange between fresh and hot gases, and thus, maintain a stabilized flame. For example, recirculation zones allow to keep burnt gases in a hot region of the combustor and favour the combustion stabilization. This work is limited to the study of stationary states. It could be interesting to study the time needed to reach high intensity combustion at ignition and the time needed to return to lower temperature when extinction is observed. If the time needed to extinguish combustion is long, the critical limit can be overstepped without an immediate extinction of combustion. A security factor on extinction limit must be respected in order to prevent residence time fluctuations and a strong temperature decrease. In future works, a same parametric studies will be performed with a direct injection of fuel inside the combustor. Results will be compared with the premixed case. Then, several geometries of combustor will be studied experimentally to determine the effect of residence time distribution on mixing efficiency and combustion stabilization. In parallel, combustion in those geometries will be studied numerically with the modified CFD code.

REFERENCES

Ahn J., Kuo J., Eastwood C., Sitzki L. and Ronney P.D. (2004). Excess enthalpy combustion for microscale power generation. International Conference on Combustion and Detonation, W2-11, Moscow Beer J.M., Lee K.B. (1965). The effect of the residence time distribution on the performance and efficiency of combustors. 10th International Symposium on Combustion, 1187-1202 Correa S.M. (1993). Turbulence-chemistry interactions in intermediate regime of premixed combustion. Combustion and Flame, 93:41-60 Davidenko D., GSkalp I., Dufour E., Gaffi~ D. (2002). Kinetic mechanism validation and numerical simulation of supersonic combustion of methane-hydrogen fuel. AIAA 2002-5207 Epstein A. (2003). Millimeter-scale, MEMS gas turbine engines. ASME, GT-2003-38866, Atlanta Norton D.G., Vlachos D.G. (2004). A CFD study of propane/air micro-flame stability. Combustion and Flame, 138:97-107 Ribaud Y. (2003). Internal heat mixing and external heat losses in an ultra micro turbine. International Gas Turbine Congress, OS-109, Tokyo Sabel'nikov V. A., Figueira da Silva L. F. (2002). Partially Stirred Reactor : Study of the sensitivity of the Monte-Carlo simulation to the number of stochastic particles with the use of a steadystate, semi-analytic, solution to the PDF equation. Combustion and Flame, 129:164-178 Vican J., Gajdeczko B.F., Dryer F.L., Milius D.L., Aksay I.A., Yetter R.A. (2002). Development of a microreactor as a thermal source for microelectomechanical systems power generation. 29 th International Symposium on Combustion, 909-916 Villermaux J., Devillon J.C. (1972). Representation de la coalescence et de la redispersion des domaines de segregation dans un fluide par un modele d'interaction phenomenologique (In French). 5th European Symposium on Chemical Reaction Engineering, Amsterdam Vulis L.A. (1961). Thermal regime of combustion. McGraw-Hill, Chapter 3

This Page Intentionally Left Blank

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

905

Joint RANS/LES approach to premixed flames modelling in the context of the TFC combustion model V.L. Zimont 1 and V. Battaglia CRS4 Research Center POLARIS, 09010 PULA (CA), Italy

Abstract

We present an original time-saving joint RANS/LES approach to simulate premixed combustion simulation in strong turbulence, which is intended mainly for industrial applications. It is based on successive RANS/LES numerical modeling where part of the information (stationary average fields) is achieved by RANS simulations and part (instantaneous nonstationary image of the process) by LES. This is performed using the fields of turbulent energy and dissipation rate, obtained from the RANS simulations, to model turbulent scalar transport and combustion rate. This approach has been developed using the TFC combustion model [9]-[12]. This is based on a generalization of Kolmogorov's idea (equilibrium not only of small-scale turbulent structures but also of small-scale structures of the reaction zones), which is employed to express the combustion rates both for RANS and LES problems in terms of turbulent characteristics (described by the model equations) and the chemical time. It permits to overcome in some way well known fundamental difficulty of modeling (so called "challenge of turbulent combustion") connected with the fact that the reaction rates strongly depend on unresolved small-scale coupling of turbulence and chemistry. In this work we present results of simulations carried out employing the joint RANS/LES approach described above and compare them with predictions obtained from a classic LES approach where the sub-grid scale terms are closed according to the Smagorinsky model. Keywords: Turbulent premixed combustion, RANS, LES, joint RANS/LES modeling

I. Introduction

The aim of this work was to develop an original joint RANS/LES approach based on the TFC model for simulation of premixed combustion in strong turbulence (mainly applying to clean gas turbine and boiler combustors). We try to develop an economical tool for prediction of mean and instantaneous fields by combining the properties of both RANS and LES approaches: 1.RANS is more economical in terms of computational resources and directly yields the required average fields on sufficiently refined meshes. RANS modelling of the mean combustion rates, which are controlled by unresolved, small-scale coupling of turbulence and chemistry is easier to justify from a theoretical point of view. In fact, our approach aims at describing such terms using Kolmogorov's ideas of small-scale equilibrium, and expressing them in terms of large-scale turbulence parameters (obtained from a turbulence model) and a characteristic chemical time. For this purpose we assume equilibrium small-scale structures of turbulence and reaction zones. 2.In LES, large-scale structures as well as thin and strongly wrinkled reaction sheets, are explicitly computed with no need for modelling. Also, it is possible to explicitly predict the occurrence of counter-gradient transport. The commonly observed combustion regime where the turbulent flame in an uniform flow has nearly constant speed and, at the same time, increasing width can be directly predicted by LES computations in contrast to RANS where it is modeled.

Corresponding author. Tel.: +39-070-9250-239; fax: +39-070-9250-216. E-mail address: zimont(a)crs4.it.

906

In this work we present results of simulations where LES is not viewed as an alternative to RANS, but is used conjointly with it. Such approach can be seen as the combination of two stages. During the first step of the procedure, an average solution is produced using RANS; during the second stage, the fields of turbulent kinetic energy and turbulent dissipation obtained from the RANS computations is used to estimate the subgrid viscosity (from the Kolmogorov theory) and subgrid combustion intensity (from the LES version of the TFC combustion model). In the paper we briefly describe the TFC model which is, in fact, based on extended Kolmogorov general assumption of equilibrium in the small-scale hydrodynamics structures. Conceptually, the TFC model is similar to Kolmogorov original "k-co" turbulence model[ 1]. Assuming equilibrium of the turbulence small-scale structure, he first expressed the dissipation rate e ~ u '3/L (controlled by unresolved, small-scale coupling of turbulence and molecular viscosity) in terms of resolved, large scale parameters [2], which can be obtained from modelled transport equations. He then used this expression to close balance equations describing nonequilibrium turbulence[I]. Following a similar way of reasoning, we express the turbulent flame velocity (controlled mainly by unresolved smallscale coupling of turbulence and chemistry) in terms of the combustion chemical time rch and resolved turbulent parameters (integral u',L for RANS and sub-grid u),,L A for LES). This expression and the turbulent diffusion law for the brush width due to nonequilibrium large sheet wrinkles are then used to close an equation for the progress variable 5 , hence the name Turbulent Flame Closure (TFC). The results of RANS and joint RANS/LES simulations for several standard situations are also presented. In these, the turbulent parameter u' Lt has been obtained by "k-g' model for the RANS simulations, whilst the subgrid u'~x, LA are obtained directly from the Kolmogorov theory for LES. For comparison we also show numerical results obtained following the traditional approach (LES instead of RANS), using the TFC combustion model and the Smagorinsky model for the subgrid turbulence.

2. Underlying assumptions and governing equations of the TFC premixed combustion model The TFC model describes an intermediate combustion regime whereby the structures responsible for the small-scale wrinkles (mainly controlling the turbulent flame speed Ut ) are already in statistical equilibrium while the structures responsible for the large-scale wrinkles (mainly controlling the turbulent flame width 6t) are still far from equilibrium, so that the turbulent flame displays nearly constant Ut and increasing flame brush width ~ ,controlled by turbulent diffusion. Such transient flames dominate in experiments [3-5]. In fact, a short initial stage (when both small- and large-scale wrinkles are far from equilibrium) usually is not significant, whilst the final stage (both large- and small-scale wrinkles have reached statistical equilibrium) is not usually attainable in practical situations [6]. We called this specific regime "intermediate steady propagation (ISP)" [6]. In the case strong turbulence the observed instantaneous flame is not laminar but it is thickened by small-scale turbulence [7]. Our model describes namely this mechanism ofpremixed combustion. The TFC model therefore can be illustrated by the following procedure : 1. For the hypothetical case of constant density we deduced expressions for the 1D ISP thickened flamelet speed UU and width 6f, the dimensionless flamelet sheet area ( F / F 0 ) and hence the turbulent flame velocity U t - U / ( F /Fo) is a function of u',L (or k,e ) and r ch [6, 8]. 2. We introduced the expression for Ut in 3D unclosed kinematic equations describing the ISP flame for the general case of variable density. The thermo-chemical and hydrodynamic fields are therefore governed by a modeled transport equation for U , by the " k - e " turbulence model and the appropriate Reynolds equations. Validations of the RANS equations is presented in [9-12].

907

3. Complete solution of the problem includes prediction of the progress variable transport pff'c' and source l~c . This meets with the difficulty connected with the existence of the so called counter-gradient transport phenomenon, which occur when average flux takes place in the opposite direction to turbulent diffusion. This phenomenon is related to different acceleration experienced by the cold and hot volumes of fluid in the flame. An original gasdynamic model which allows to predict this phenomenon was developed in [13, 14]. We isolate and describe separately this gas dynamics effect, and this justifies using " k - g ' model in simulations where it describes only the gradient turbulent component of the flux. We should especially stress that at RANS approach the ISP regime and the counter-gradient phenomenon need modeling while LES demonstrates ISP flames and the counter-gradient transport phenomenon (as we see below) without modeling. 2.1 I n s t a n t a n e o u s reaction z o n e a n d p a r a m e t e r s o f the turbulent f l a m e

The instantaneous thickened flamelet speed Uf and width 6f (which are larger than the normal laminar flame speed and width) can be derived from dimensional analysis [4, 5]" Uf~u'.Da

-1/2 > S L ,

6f~L.Da

(1)

-3/z > 6 L.

D a = r t / rch is the Damkohler number where

r t = L / u'

and rch = Z / S~ are

the turbulent and

chemical time scales ( Z is the molecular transfer coefficient of the mixture). The instantaneous flame broadening takes place at 6L > r/= L Re[ 4/3 (/7 being the Kolmogorov turbulence microscale and Re t = u ' L / v

the turbulent Reynolds number). At the same time the

thickened flamelet sheet is strongly wrinkled by turbulence when L >> 6/ and hence the condition for this combustion mechanism is the following [6]" (2)

D a 3/2 >> 1 >> D a 3/2 Ret 4/3

So the model is valid at large Re-102-103 and moderately large Da-~10-20; these conditions are commonly encountered in large scale gas turbine combustors. For faster chemistry or weaker turbulence we have wrinkled laminar flame sheet. Eqs.(1,2) are valid also for final stage of combustion, 1D stationary flame, while expressions for m ( F / F o ) (following from some general properties of the random surface and the Buckingham theorem) and hence U t are valid only for ISP flames[6, 8] (transient flames with increasing width):

(F/Fo)~ Oa3/4, Ut =Uf(ff /Fo) ~uroal/4 "~u3'/4VI/2x-1/2LI/4"-L'

, 6t ~ ~/crZ = ( 2 D , t) 1/2

(3)

In accordance with estimates in [6] the ISP regime takes place, and Eqs. (3) is valid, for (4)

t < r, D a

and ISP flames are observed in real flames as the residence time usually is less than r, D a . For comparison at t >> r, D a when both the large and small scale wrinkles are in equilibrium (running 1D stationary flame) we have the following: U, ~

U r

,

6, "~ L" D a

1/2

.

(5)

908

The former expression is presented in the pioneering works on turbulent premixed combustion by Damkohler and Shchelkin [ 15, 16], the latter has been derived by the authors from the assumption that the stationary width is controlled by u',L and rch due to the small-scale equilibrium. Fundamental differences between the flames in these two stages can be observed: for the transient regime the flame speed depends on the turbulence and chemistry and the increasing flame width is controlled directly only by the turbulence whilst in the stationary regime the situation is reversed.

2.2 RANS equations of the TFC combustion model We analyze premixed flames adopting the flamelet formalism in terms of the progress variable using known bimodal approximation ofPDF p(c). Similar to known BML model we have the following:

p / p, = ( l + ~ ' ( p , / p b - - 1 ) ) - ' , where pu,T,, Ci,. and lOb,To,Ci, b

are

T = T, (1- ~) + Tb'~ , Ci = C,,, (1--'g) + C,,b'g

(6)

the density, temperature and species concentrations in unburned

(reactants ~" = 0 ) and burned (equilibrium products ~ = 1) gases. The turbulent combustion front, moving with speed Ut and having increasing brush width controlled by Dt is described by the following transport equations:

+,~

O-fig~at + V-(,~ff~') = - V.pu,"c" CGT a N / at + v . ( N ~ )

= v . (,~DtV~')

~

[ + p,s, lv l

, Actual source

(7)

Model source

(8)

I

Describing a "joint closure" procedure we should keep in mind the difficulty of directly modeling the unknown terms in Eq. 7: 9 The source ,~W depends on both large-scale and small-scale processes as combustion takes place in thin flamelets with unresolved small wrinkles but their global distribution in space is controlled by large-scale wrinkles. In the model source these processes are separated as U, is controlled mainly by the small-scale turbulence-chemistry coupling and is described by Eq. (3) while ]V?I is controlled by large-scale dispersion and is controlled by Dt. 9 The transport p~"c" in premixed flames as a rule is counter-gradient. The physical reason for this is that there are two opposite tendencies: the gradient turbulent diffusion (GTD) connected with random velocity pulsation and the gasdynamical pressure-driven countergradient transport (CGT) connected with different acceleration of hot and cold volumes of fluid in the flame. The transport term of the model equation contains only the turbulent component controlled by D, ~ k 2 / c while the pressure-driven component is included in the model source ( "joint closure"). Model equation predicts in particular (as we will see below) flames with increasing brush width in spite of the counter-gradient transport as it is observed in experiments. The complete solution of the RANS combustion problem includes the prediction not only of ?(Y,t) but also of pui.'c"(Y,t ) and l~(Y,t). To find the actual source,aI~, we must know the pressure-driven CGT term and subtract it from the model source given by Eq. 8. Also, to find the "real" pff"c"

we

909

must add CGT term to the computed GTD. Luckily this could be done at the post-processor stage using a very simple gas-dynamics model for reasonable prediction. The flux is as follows" PZlin c n = P ( f t u i --

U-'-bi)5(1 -- 5) = (,~(Aft.,)5(1 - 5)) arz~+ (,~(Aftip. )5(1 - 5)) car

(9)

where Aft, is the difference in the conditional averaged velocities of unburned and bumed gases associated to turbulent fluctuations and Aftpd is the contribution in the gas-dynamics pressure driving term (different acceleration of hot and cold volumes due to the pressure gradient generated by heat release). Estimation of Aft, follows directly from the approximation(,~(Aft,)~(1-5))crz~=

- ~ D , 05/Ox i where Dt is estimated using " k - e "

model. The quantity Aft,.pd can be estimated

following a simple reasoning. We assumed that the conditional pressures fi, = fib = fi and the total pressure of unbumed gas fi', =const. Simulations yield ,~(Y), p(Y) and z7;(.~), from the previous assumption fi', =const it is known ~,.(.~); therefore, using ffi(Y) we can find ~bi(Y). Hence we know pul;c" (Y). If we subtract the CGT component from the simulated model source term p,U, IV'~I or insert pul;c"(.~ ) in Eq. 7, we can calculate the actual source term p W ( Y ) . In the case of the 1D stationary flame the analytical expression could be written in a general form such as the one Aftpd/U, = f ( P , / P b ,5) we used in [13] to explain the transition of the turbulent flux from gradient to counter-gradient (observed in open flames in [ 14]). In [ 14] a similar expression was adopted to explain the occurrence of counter-gradient diffusion in the impinging flame.

2.3 Joint RANS/LES formulation of the problem In LES of premixed flamelet combustion we describe obviously not the actual instantaneous thickened flamelet sheet but an instantaneous smoothed (averaged on the subgrid level) flame sheet. The turbulence controlling this flame characteristics is the subgrid turbulence and its parameters depend also on the mesh size A. LES has obvious advantage over the RANS modelling as the largescale turbulence (described in our RANS simulations below by the " k - e " model) as well as the counter-gradient phenomenon (described by the original gasdynamic model) do not need modelling. But simulations of complete industrial combustor characterized by complicated geometry (including modelling of the premixed chamber and air cooling jets at the walls), are unaffordable. Therefore the RANS approach is generally chosen for industrial applications. At the same time some information about the instantaneous flow field and unsteady characteristics is needed in many applications such as unsteady combustion in gas turbine and cycle variations in SI engines to mention a few. For these cases we propose a joint RANS/LES approach. The main idea of this approach is to combine LES and RANS in a two-stage process. The first step consists of the RANS simulation which yields the averaged flow field; the second step entails LES using the turbulent energy k (Y,t) and dissipation ?(Y,t) obtained from RANS to estimate the subgrid turbulence and subgrid flame speed. The latter staged gives a nonstationary image corresponding to the former stationary one. Assuming the existence of Kolmogorov inertial interval E ( k ) = Cc2/3k -5/3 [2] we can directly estimate the subgrid turbulent velocity and scale

u/, ,~

(k)dk ,~

, LA ~

A

k E(k)dk /

A

E(k)dk ,~ A

and hence the subgrid turbulent transport coefficients are equal to

(10)

910

D6 ~ vA ~ ZA ~ UALA ~ ~1/3A4/3 t

(11)

Using these turbulent parameters to estimate the subgrid flame speed and inserting them in Eq.3 we have the expression for the flame speed in the model LES equation U tA = U f ( PA / Fo ) "-- eel"l*U A ' J~ L

[ ~,~ U t ( A / L )

1/2 ]

(12)

We stress that Eq. 12 is less rigorous in comparison with Eq. 3 due to lower subgrid Damkohler number. The modelled LES equation (ignoring sugrid counter-gradient phenomenon) can be then cast as follows:

o(pAu~)/ot + v.(pA~;,~;,)= V'(pDAV~A)+ puU,~]V'~]

(13)

obviously we can use Eq.12 to express the subgrid flame speed in the context of traditional LES approach, but as in this case we do not know the fields of e and k to use in our "pure" LES simulations, we adopt the Smagorinsky model [ 17]: u a = A IS l, DA ~ Va = (c, A)

I,gl,

IS l- (2suso.) '/2.

(14)

3. Numerical simulations

We present the results of computations carried out using both the RANS and joint RANS/LES approach for different configurations. For one of the cases examined we also present comparison between LES (where the Smagorinsky model has been adopted to estimate the subgrid turbulence) and the joint RANS/LES. The code used for the simulations is the latest release of Fluent, Fluent6 [ 18], a finite volume code that allows to customize the turbulent and combustion models implemented. The size of the meshes generated for the different test cases vary from 250000 to 900000 nodes. An IBM-SP3 with 16 processors was used to perform the simulation. Statistics of CPU time requirements are not crucial to the discussion to follow and have not been reported. The average field for LES was calculated with the specific post processing tool for flow statistics available in the Fluent code. 3 . 1 0 N E R A standard burner

The first test case is the standard Moreau burner [ 19], a rectangular section burner with the flame stabilized by a burned gas flow. The fuel is a methane-air mixture with equivalence ratio equal to 0.84. The averaged flame has increasing brush width and constant speed, as can be seen by the constant angle with respect to the main flow (Fig.l). The comparison with experimental data shows that both RANS and joint RANS/LES simulations capture well the average width and slope of the flame (Fig. 2). The instantaneous results of LES show effects of the inlet turbulence on the flame surface (Fig. 1). In this test case LES results are very sensitive to inlet boundaries conditions for turbulence. In the present work we introduced a disturbance into the average inlet velocity with amplitude and length derived from upstream flow analysis reported in [ 13]. To better understand transport phenomena we post-processed RANS results on the base of Eq.9. The ~ - transport is counter-gradient in x-direction, where due to significant fall in pressure in the axial direction the gasdynamics effect prevails. In the ydirection, however, a transition occurs from gradient to counter gradient transport, due to balance of GTD and CGT in Eq. 10. Validity of these results was confirmed by comparison with experimental data in open flames in [13]. LES also captures the transition occurring in y-direction without need for modeling ( Fig. 9 ). It is worth remarking upon the fact that, at the same time, the transport of a non-

911

uniform passive scalar z is predicted by LES to take place following the gradient in the y-direction

(Fig. 3).

I

~ ~ - ~ - _

0~ ~ ~ : : I

"I

[

'

I c=.s

,.,,

_L__________L; "'=

0

Tb=2200K

RANS

0,5

Ub=120m/s 1

.m~ 00 ~ '

CH4+air

~=0.8

u' =8m/s L.=5.4mm u'==23mls L~=l.6mm

o=

~ o / ~ cZ= " ~

" I

x(m)

1

x(m)

Figure 1" Average and instantaneous fields of progress variable Y,rY,1 0.9

....

"

" ---~~ \~..

l

0.8 0.7

EXP-DATA --m--U,:120(mls)

i

0.6 0.5 0.4 0.3 0.2 0.1

~'.~

,

,

,

1i

'

'

'

1'.5

u~.

=

i 0.5 I . . . .

0,175

T,% 1

Figure 2" Comparison of axial velocity (left) and temperature (fight) with experimental data Stn~rnlme

y,~.-a 03rn - - -

gradient tranlport

9

g m dlonl tnlrmport

....I.~- .........

.-~

0

.

_ ~ - 0

0.0(3 i v

0.07 ~9

m

~2~

N

o o6

o o~1~ o.o. IF

0 03

I:

-1

"~o~ 0

.~

0.25

~- counterorad~nt(ac/ax~o) 0.5

c

0 75

~-~-----~" _ ~

1~-~'-

"-~

-

-o 1 1

; ~

- - -

...... -8 005

0

,

x..o.o=m x-o.1 m

X==O.6m x=o.o=m

X: ml~ .6,111

0.005

Figure 3" Mean scalar ~ - and 2'- fluxes. Left: RANS along streamline, right LES x-section 3.2

Volvo standard c a s e

The second case investigated here is the Volvo test rig described in [20]. A triangular section bluff body anchors the flame inside a rectangular section channel. The fuel is a propane-air mixture with equal to 0.65. The flow displays vortex-shedding behind the bluff body. In this test case inlet turbulence is less relevant than in the previous one, the main generation of turbulence occurring behind the bluffbody. Periodic boundary conditions were applied in spanwise direction to simulate 1/3 of the burner span. Agreement of predictions with experimental data is good for both the RANS and LES approach, Fig.5(left). The extent of the recirculation zone is predicted with reasonable accuracy as shown by the profile of axial velocity at sections at 15mm and 150mm. Fig. 4 shows comparisons of results obtained from the LES computation performed by the Fluent 6 code, using the code implementation of the TFC-LES and Smagorinsky models. It appears that the joint RANS/LES allows to resolve the unsteady phenomena connected with large scale structures starting from a RANS results more quickly, whilst the LES-Smagorinsky approach requires longer to develop the starting turbulent field, without the possibility of using previous RANS field.

912

RANS-TI=C

Model

oomour-lhes

of progress verlable

o o6

RANS-TFC

~. ~ - - - - ~ . ~

o

'

o.~5

,

,

=

,

I

, ~

Test Rig Propane-air (:1)=065 Uin=l 8m/s Tin=288K U'in=3%

)

0.06

RANS/LES-TFC

o._~

go -o.o8

I

~

o

I

l

ol

I

o2

l

I -

)

o3

x(rn)

0.06

I

,

o,4

! I

-

LES Smag.-TFC

~

go -oo8

I

J

o

I oI

~

I

02

~

I

~

o3

xlm)

I

o4

Figure 4: Contour-lines of progress variable from RANS and instantaneous LES x.aols~

, ~ = . x.o

1~,,, 0

06

x ~ .032 rn

-'i <<..

x=O.O37m

x==0.048m

transition

"~

x=0.082 m

x=0.250 m

0 03

o=

~

o' u/u,

'

'"

'

i ~ j

a~-J~y(+)

counter-gradient

go

-0

9

.

;

9

;

,

:

,

;

,

,

03

-0.1

0

i

i

0.1

i

i

i

i

1

I

I

Figure 5: Comparison with experimental data (left) and transport in y-direction (fight) Figure 5(right) shows the transport term for progress variable in y-direction

pu;c"calculated directly

during LES simulation at different section behind the bluff body (x is the distance from the rear of the bluff body. The flux in y-direction undergoes transition from gradient to counter gradient. As for the previous case, the reason for this is that at relatively small width fi, the gradient turbulent diffusion prevails on the gasdynamics effect while the situation is reversed downstream where the width 6, becomes larger.

913

3.3 Impingent flames and the ISP flame concept Previous TFC-RANS simulations based on the concept of the transient ISP flames and LES modeling in the context of our RANS/LES and traditional LES approach demonstrate the existence of a transient regime of combustion. Our TFC-RANS simulations of impinging flame appear to show that whilst displaying some characteristics of 1D stationary flames, these are in fact also transient with dependence on chemistry Ut and increasing width 6t (due to turbulent diffusion) which is compensated for by the divergent flow. Fig. 6 shows results of simulations for different configurations of impinging flames characterized by flow and mixture parameters as in the ONERA combustor (Tin=300K; Uin=60m/s; Pop=105pa; p,=l.lKg/m3; pb=0.157kg/m3). Fig. 7 shows the effect of the mixture combustibility (characterized by the speed SL) on the flame configuration and the turbulent, gasdynamics components and the total transport term. Validation of our gasdynamics model for the counter-gradient component of the transport was performed against experimental data for impinging flames in [ 14]. o.12 ~-

.,/- ' / t I

o.~2 L-

0 08

0 08

0

01

02

03

-

0

0 08

0 08

0 o4

~ o4

'

,

'

,

0.1

1

,

,

1

I 02

l

,

l

t

I 0.3

i

i

i

i

unburned

O--

I

0

I

I

I

I

,

01

~

;

,

1

02

,

i

,

~

I

03

x(m)

t

,

i

~

O F

~

0

i~

0.1

02 x(m)

03

Figure 6: Different configuration tested with RANS/TFC 8 /t S,---0.4m/s

~

'

~

"

o.~

-

S,=2m/s

0.o4

/]

/

o.o

i

o.8

0o6 06

o~

o

006 oo4

oo2 '

[

I

02

.

.

.

.

.

0.3

0.4

O

X(m)

-O.O4 '

;

I

02

'

'

[

03

~

I

I

04

,

x(m)

~.~'

027

o.~

0.~

o:,~

X(m)

Figure 7" The flame configurations and the transport (pu"c"/lOuUt ) in front of the obstacle (total flux: solid line, CGT component: dotted line, GTD components: dashed line). 4. C o n c l u s i o n s

We propose a joint RANS/LES approach for numerical modeling of premixed combustion intended mainly for industrial applications, as, whilst the RANS approach is often not sufficient to capture important features of the flow field, performing LES computations of real geometries is still prohibitively expensive.

914

In our approach the first step consists of the RANS formulation of the TFC combustion coupled with the " k - ~" turbulence model, which yields the averaged flow field. In the second stage the LES formulation of the combustion and turbulence model is introduced, using RANS k and c to estimate the subgrid combustion rate and transport. The second step yields the instantaneous flowfield which can be used for analysis, for example, of the thermo-acoustic interaction and possible combustion instability or for vibration analysis. The present simulations of the ONERA and VOLVO configurations demonstrate that a. RANS and LES modeling in the context of the TFC model are consistent as the ISP transient flames and counter-gradient transport phenomenon, described by the RANS TFC model, are predicted by LES without modeling; b. Comparison (for the VOLVO case) of the traditional LES approach with the LES step of our approach shows that the latter is less time consuming. It can be speculated that this is due to the use of the Komogorov theory for the estimation of the subgrid viscosity, and that being does pulsations of the latter absent, a larger time step can be afforded during the simulations without incurring in convergence problems. This LES approach allows to start the simulation directly from the RANS field, with no need for lengthy initial stages as usually required in classic LES modeling. The LES step generally speaking needs a coarser and more uniform grid with respect to the RANS steps. In fact, the LES step is devoted only to resolving largescale vortex structures and it does not need all the detail of the RANS simulation grid.

Acknowledgments This work has been financially supported by Sardinian Regional Governament. Special thanks go to Dr. Francesca di Mare for discussions and help.

References [1 ] Kolmogorov, A. N., Izvestia Academy of Sciences, USSR; Physics, 6 (1942) 56-58. [2] Kolmogorov, A. N., Doklady AN. SSSR 30 (1941) 299-303. [3] Kobayashi, H., Tamura, T., Maruta, K., Niioka, T., Williams, F., Proc. Comb. Inst .26 (1996) 389-396. [4] Dinkelacker, F., H61zler, S., Combust. Sci. and Tech., Vol. 158 (2000) 321-340. [5] Lipatnikov, A., Chomiak J., Progress in Energy and Combustion Science 28 (2002)1-74. [6] Zimont, V.L., Experimental Thermal and Fluent Science 21 (2000) 179-186. [7] Chen, Y., and Mansour, M.S., Proc. Combust. Inst. 27 (1996) 811-818. [8] Zimont, V.L., Combust. Expl. and Shock Waves 15 (1979) 305-311. [9] Zimont, V.L., Lipatnikov, A.N., Chem. Phys. Reports, vo1.14(7) (1995) 993-1025. [ 10] Karpov, V. P., Lipatnikov, A. N., Zimont V. L., Twenty-Sixth Symposium (International) on Combustion, the Combustion Institute, 1996, pp. 249-261. [ 11 ] Maciocco, L., and Zimont, V. L./20-th Annual Meeting of the Italian Section of the Combustion Institute "Frantic '97", (1997) X-2.1 - 2.4. [ 12] V. Zimont, F. Biagioli, K. Syed, International Journal "Progress in Computational Fluid Dynamics", Vol. 1, Nos. 1/2/3, 2001, pp. 14-28. [13] Zimont V. L., Biagioli F., Combustion Theory and Modeling, v. 6, (2002) 79-101. [14] Biagioli F., and Zimont V., Twenty-Nine Symposium (International) on Combustion, the Combustion Institute, (2002) 2087-2096. [15] Damk6hler, G., NACA Tech. Memo. 1112 (1947). [16] Shchelkin, K.I.,NACA Tech. Memo. 1110 (1947). [ 17] Smagorinsky J., Mon. Weather Rev. 91 (1963) 99. [18] Fluent user's guide (version 6.0), Fluent Inc (2003). [ 19] Moreau P., AIAA paper No 77-49 (1977). [20] Sjunnesson A., Henrikson P., Lofstrom C., Proc. 28 th Joint Prop. Conf. AIAA-92-3650 (1992).

Engineering Turbulence Modellingand Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

915

OPTICAL OBSERVATION AND DISCRETE VORTEX ANALYSIS OF VORTEX-FLAME INTERACTION IN A PLANE PREMIXED SHEAR FLOW Norio Ohiwa and Yojiro Ishino Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, JAPAN

ABSTRACT To obtain practical schemes of the vortex-flame interaction, a series of organized eddies formed in the plane shear flow is employed in this investigation, instead of a single vortex ring or a single vortex tube. The plane shear layer is first formed between two parallel uniform propane-air mixture streams. After an electric spark is discharged at the midpoint between two neighboring organized eddies, a premixed flame is initiated and is observed using the simultaneously two-directional and high-speed schlieren system. The outline of propagating flame after the spark ignition is simulated combining the discrete vortex method with the Huygens' Principle. According to detailed comparison of the results simulated with those optically observed, it is found that the simulated flame profiles after the midpoint ignition agree well with those of schlieren images, and that the lateral flame propagation velocity depends almost linearly on the vortex tangential velocity and the mixture equivalence ratio.

KEYWORDS

Premixed flame, Plane shear flow, Organized eddy, Vortex-flame interaction, Discrete vortex method

INTRODUCTION

Recently much attention has been paid on the vortex bursting, since it is considered to be one of the most important elements of the vortex-flame interaction and constitutes one of the most probable enhancement mechanisms of the flame propagation velocity. Many models for the vortex bursting mechanism have been proposed based on the active experimental and numerical investigations [Chomiak (1976), Ashurst (1996), Asato et al. (1997), Umemura & Tomita (1998) and Ishizuka et al. (1998)]. However, these studies have been carried out using a single propagating premixed flame either in a single vortex tube or in a single vortex ring. This means that the proposed models do not include the concrete information about the vortex-flame interaction under the practical combustion conditions,

916 and that they can not always be used for modeling turbulent premixed flames generated in various practical flow fields. To obtain practical schemes about the vortex-flame interaction, it is necessary to make detailed observations and analyses of those flames which are initiated and propagate in the practical flow fields; such as jets, shear flows and wakes. From the point of view that the threedimensionality of flame phenomena makes thorough understanding of the key processes of the vortexflame interaction extremely difficult, the plane shear layer and the wake behind a rod-array are considered to be most suitable and useful for the objective flow fields. This is because that the former is well-known and characterized by a series of large scale organized eddies [Brown & Roshko (1974) and Ishino et al. (1996)], while the latter is well-known by two-dimensional Karman vortex streets [Ohiwa et al. (1994) and Ishino et al. (1997)], and that the two-dimensionality of their vortex structure makes optical observations and analyses extremely easy, leaving the generality almost undamaged. In this investigation the plane shear flow has been selected as an objective flow field. In the preceding experimental studies [Ishino et al. (2001), Ishino et al. (2002) and Ohiwa et al. (2003)], the plane shear layer was first formed between two parallel uniform propane-air mixture streams by varying the equivalence ratio and the velocity difference as two main parameters, an electric spark was discharged at the prescribed point in the fully developed plane shear flow, and a propagating flame after the spark ignition was optically observed using the simultaneously two-directional and high-speed schlieren system. Acoustic excitation of the shear flow was employed to make the organized eddy formation extremely regular and to enable synchronized ignition with a high-speed video camera. Two kinds of ignition points were prescribed; one at the center of an organized eddy and the other at the midpoint between two consecutive organized eddies. Based on detailed picture processing of a series of highspeed and two-directional schlieren photographs, it was found that two kinds of vortex-flame interaction exist; one was the well-known vortex bursting in the axial direction of the organized eddy, the other the vortex boosting in the lateral flow direction. It was also found that two kinds of interaction exert almost same influence on the enhancement mechanism of flame propagation. In this paper, by varying the equivalence ratio and the velocity difference as two main parameters, the propagating flame after the spark ignition at the midpoint between two adjacent organized eddies in the plane premixed shear flow is first observed using the simultaneously two-directional and highspeed schlieren system, and then simulated numerically using the discrete vortex method [Ishino et al. (1992)]. In the numerical simulation the effect of the acoustic excitation is replaced by the small perturbation of the inlet flow velocity [Inoue (1992)], whereas the flame propagation is expressed using the Huygens' Principle. Detailed comparison of the results numerically simulated with those experimentally obtained is then made.

EXPERIMENTAL ARRANGEMENT AND METHODS The schematic diagram of the experimental arrangement is shown in Figure 1, where the structural details and dimensions of the measuring section are also shown as an inset. Air supplied by a blower and propane supplied from a cylinder are regulated and metered to a prescribed equivalence ratio, are distributed into two definite flow rates for the higher and lower velocity mixture streams, and flow into the combustion tunnel. After being acoustically excited using a pair of speakers on each side of the duct, two mixture streams are issued from the exit section. As shown in the inset in Figure 1, the exit section is divided by a thin splitter plate into two square sections of 140 mm • 40 mm, from which propane-air mixture streams having the higher velocity Uj and the lower velocity U2 are issued form the right- and left-hand side sections, respectively. A sheet of stainless steel wire gauze of # 100 is installed in the exit section to prevent flame from flashing back into the duct.

917

TABLE 1 SETTING CONDITIONS OF FLOW~ VORTEX AND FLAME

Flow number

(a) .(b) (c)

Flow and vortex conditions

Flame conditions and velocity ratios

U1 m/s

U2 m/s

Um m/s

v0 m/s

4.0 5.0 6.0

2.5

3.25 3.75 4.25

0.75 1.25 1.75

~b= 0.7 (SL = 0.23 m/s) 3.26 5.43 7.61

~b= 0.8 (SL = 0.31 m/s) 2.42 4.03 5.65

Vo/SL

~b= 0.9 (So = 0.3 7 m/s) 2.02 3.39 4.73

The simultaneously two-directional and high-speed schlieren system [Ishino et al. (2002) and Ohiwa et al. (2003)] is reproduced in Figure 2. A part of the parallel light from the first concave mirror is reflected by the flat mirror 1 at a right angle, goes through the measuring section in the axial direction of the organized eddy, is reflected once again by the flat mirror 2 at a right angle, and joins the original parallel light to focus on the high-speed CCD camera after reflecting by the second concave mirror. The reflected and original parts of the parallel light give the front and side views of each schlieren image, respectively. To superimpose outlines of organized eddies without combustion on the two-directional schlieren image, a weak density gradient is introduced into the shear layer by heating the lower velocity stream by about 10 ~ using an electric heater, as depicted in Figure 1. Two starting motions of the igniter and the high-speed video camera are synchronized with the organized eddy formation using the voltage waveform applied to a set of speakers as a reference signal. On detecting the timing signal indicating that the midpoint between two neighboring organized eddies reaches the fixed ignition point at y - 50 mm, the igniter is discharged and the high-speed video camera is simultaneously started. The flame propagation process after the spark ignition is then recorded on the high-speed video camera with a framing speed of 1000 frames/s and an exposure time of 50 laS. In the experimental observation the higher mixture stream velocity is varied from Ul = 4.0 m/s to 6.0 m/s at 1.0 m/s intervals with the lower mixture stream velocity kept constant at U2 = 2.5 m/s, and the equivalence ratio is varied from ~b= 0.7 to 0.9 at 0.1 intervals. The frequency of acoustic excitation is fixed at f = 125 Hz, since the maximum sensitivity of the plane shear layer to the external sound

Oscillator

Wire gauge ~

(#~~ - ~

'\,,\

I

Ignition point

I

Front view

,'TE

o

A.C. amplifier

/

//

jo ~

o) h= ~ I"~

FFT analyzer

o>, 0

Details of the measuring section

I Inverter I Valve

,,

~

Blower

Figure 1" Schematic diagram of the experimental apparatus

918 application is obtained at the frequency. The flow and flame conditions are summarized in TABLE 1, where Um= (Ul + U2)/2, vo - (U1 - U2)/2 and SL mean the average convection velocity, the vortex tangential velocity and the laminar burning velocity, respectively. TABLE 1 shows that the vortex tangential velocity takes a value in the range of v0 = 0.75 - 1.75 m/s and the velocity ratio varies in the range of vo/SL = 2.02 -- 7.61. In this paper the point of spark ignition is limited to the midpoint between two adjacent eddies, being abbreviated to the midpoint ignition. The relation between the ignition point and the outline of the organized eddy structure is illustrated in Figure 3(a).

O U T L I N E S OF D I S C R E T E V O R T E X SIMULATION Flow Model and Assumptions Introduced

In this investigation the reactive flow after the spark-ignition in the two-dimensional confined duct is numerically analyzed by combining the panel method and the discrete vortex method [Ishino et al. (1992)]. The two-dimensional flow model is presented in Figure 3(b). The higher and lower uniform mixture streams having an equal equivalence ratio are separated by the splitter plate and issued from the right- and left-hand side sections having an equal width of H/2, respectively, and meet at the downstream end of the splitter plate, at which the two-dimensional shear layer starts and the origin of the two-dimensional coordinate system is set, Z = (X, Y) -- (0, 0); Z = X + iY. To simplify the numerical simulation, the following assumptions are introduced; (1) the objective flow is two-dimensional and incompressible, (2) the flame zone is so thin compared with the turbulence scale as to be considered as a front of discontinuity, (3) the laminar burning velocity is constant, (4) the vortices entering into the burnt gas region become ineffective due to intense viscosity, and (5) the vorticity generation due to the baroclinic effect and the three-dimensionality of vortex-flame interaction are neglected.

Boundary Conditions Boundary conditions are also schematically shown in Figure 3(b). The shear layer is expressed by introducing a series of discrete vortices having intensity of ~ from the vortex source point at Y = Yin. To stabilize the vortex movement, the shear flow is confined by a pair of side walls having ordinates of X = + H/2. Since the slip condition is applied to the side walls and the splitter plate, the velocity component normal to these surfaces should be zero. To realize the velocity compensation and to prevent the induced velocity from diverging due to the singular point, the discrete compensator sources and discrete compensator vortices are arranged at hew intervals on both outside virtual planes, hew apart from the side walls, and at h~s intervals on the splitter plate, respectively. The total number of compensator sources and vortices are 30 x 2 and 10, respectively. The velocity inspection points are

Lamp

Condenser lens Pin hole

Concave mirror 2 ~lVlirr~

i ~ i " " % ~ " ~ ~--.. ~

~" ~~

Meas~"" usection ,,,

/'Mirror 2 r, ....... mirror

Schlieren sto

CCD camera

Figure 2: Simultaneously two-directional and high-speed schlieren system

919

also set on the side walls at hew intervals and on the splitter plate at hcs intervals, the latter being placed at the midpoint between two adjacent discrete compensator sources. The intensity of the discrete compensator vortex ~ j and the discrete compensator source Acj are determined at every time step so that the induced velocity by the discrete vortex offsets that by the discrete compensator source at every velocity inspection point and the resulting velocity component in the X-direction becomes zero. The acoustic excitation is replaced by the weak sinusoidal perturbation of the flow velocity having a frequency of 125 Hz [Inoue (1992)]. A series of discrete flow sources having intensity of Asj and a total number of 20 are placed on the inlet section of the duct, as shown in Figure 3(b), and are sinusoidally vibrated at a frequency of f = 125 Hz. The higher and lower mixture streams, whose velocities are adjusted by controlling the intensity of discrete flow sources at the inlet, join at the downstream end of the splitter plate and form the plane shear layer. In order to avoid the initial unusual flame growth and the local energy concentration to the mixture, ignition is realized by the abrupt introduction of a flame kernel at the prescribed ignition point, instead of the spark discharge. The flame propagation is expressed by adding the flame movement normal to the flame surface due to the laminar flame velocity and the flame convection induced by a series of discrete vortices. The former flame movement is performed using the Huygens' Principle. Consequently the flame region is bounded by a series of line surfaces.

Governing Equations for Discrete Vortex and Source Conditions The complex velocity potential W(Z), which describes the flame and discrete vortex movement and determines the induced velocity, is given by the equation; W(Z) = tZ,[(/-~/2n)ln(Z- Zj)] + iZz[(~/2no')(Z- Zj)]

+ Zl[(Ao/2n)ln(Z-Zj)] + Ez[(ao/2nos)(Z-Zj)] + Zl[(asj/2n)ln(Z- Zj)] + s

Z~)]

+ tZ1 [(F~j/2n)in(Z- &)]

(1)

where the subscript j indicates the j-th discrete vortex or source, /-~ means the circulation of the discrete vortex, A0 means the intensity of the discrete flame source, and Fcj means the intensity of the discrete compensator vortex on the splitter plate. The first and second terms on the right hand side of Equation (1) indicate the induced velocity terms due to the discrete vortices, the third and fourth terms

Ignition point

o Main source 9Compensator source e Compensator vortex x Inspection point

,.

Ignition

!Y

Vortex source point Y = Y,.

x "1~ I--

40

0',

=j=

40

(a) Eddy appearances and ignition point (b) Flow model and boundary conditions Figure 3: Experimental and numerical setups and conditions

920

TABLE 2 VARIATIONS OF BURNING VELOCITY AND DENSITY RATIO

Equivalence ratio ~b Burning velocity SL [ m / s ] Density ratio Pu/Po 0.7 0.23 2.43 ........................................................................................................................................................................... 0.8 0.31 4.74 ....................................................................................................................................................................... 0.9 0.37 6.08 present the induced velocity terms due to the discrete flame sources, the fifth term shows the main stream velocity term, and the sixth and seventh terms mean the velocity compensation terms given by the discrete compensator sources on the virtual outside planes and the discrete compensator vortices on the splitter plate, respectively. To prevent the induced velocity from diverging, the cut-off radii of the discrete vortex and source, cr and os, are introduced, and the induced velocities are set to be constant in the cases of or, O's > I Z - Z]I. El and g2 indicate that the summation is performed for each case of or, ~ < ] Z - Z]] and or, Ors> [ Z - Z~I, respectively. Movement of the i-th vortex due to the mean convection is performed in the Lagrangian manner after estimating the induced velocity from the complex velocity potential, whereas the Euler method is applied for the integration with respect to the time, as given by the following sets of expressions, dZi*/dt

= [dW(Z)/dZ]z

= Zi-- Ui-

Xi (t + At) = Xi (t) + uiAt,

ivi

Yi (t + At) = Yi(t) + viAt

(2)

(3)

where the asterisk (*) indicates the complex conjugate. The flame movement due to the Huygens' Principle is also included in the calculation. A discrete vortex is introduced every time step of At into the shear layer from the fixed point on the Y-axis, which is defined by Yin = At (U~ + U2)/2 = At Urn.

(4)

The circulation of discrete vortex/-j is set equal to the intensity of the compensator vortex at the end of the splitter plate. The cut-off radius of the discrete vortex is then set equal to En, namely cr = Yin. Flame propagation is accomplished by the combined effects of the laminar burning velocity SL and the expansion velocity due to the thermal expansion us, and is expressed by the following equation.

Vf= & + u~

(5)

Movement due to the laminar burning velocity is made according to the Huygens' Principle, whereas the expansion velocity is replaced by a series of discrete sources set on the flame front at hs intervals. The intensity of discrete flame source is defined by Afj = [(Pu/po) - 1]SLhs,

(6)

where p~ and Pb indicate the density of unburnt and burnt gases, respectively. The cut-off radius of the discrete source is then defined as

92I

O's - AO/rc[(pu/Pb) - 1]Sb

(7)

Variation of the laminar burning velocity [Yamaoka & Tsuji (1985)] and the density ratio with the equivalence ratio are summarized in TABLE 2.

RESULTS AND DISCUSSIONS Simultaneously Two-Directional Schlieren Images o f Flame Profiles after the Midpoint Ignition

A series of simultaneously two-directional and high-speed schlieren photographs of a propagating flame taken after the midpoint ignition are shown in Figure 4 (a) - (c) for the flow conditions of U1 = 5.0 m/s and U2 = 2.5 m/s, where the equivalence ratio is varied as a parameter; (a) ~b= 0.7, (b) ~b= 0.8 and (c) ~b= 0.9, respectively. The number written in the left-hand side indicates the time after the spark ignition. The front views in Figure 4 show that the spark-ignited flame grows obliquely along the boundary region of two Uniform mixture streams in the shear layer, pushes its way toward neighboring organized eddies, and reaches them. The side views indicate that, on the other hand, the spark-ignited flame propagates in the up- and down-stream lateral directions and forms a dumpy rod-shaped body. By taking into account of the fact that the aspect ratio of the oblique slender flame is much greater than unity, it is found that there exists another type of enhancement mechanism of the flame propaga-

Figure 4: Simultaneous and two-directional high speed schlieren photographs of a propagating flame taken after the midpoint spark-ignition for Ul = 5.0 m/s and U2 = 2.5 m/s

922 tion in the lateral flow direction. Since the rolling-up motion promotes mixing and boosts the flame convection, this type of flame enhancement was named the vortex boosting [Ishino et al. (2002) and Ohiwa et al. (2003)]. On reaching either of the adjacent organized eddies at both sides, the flame tip is rolled up into the organized eddy to form a reverse S-shaped profile, as pointed out by the white arrow in each front view. Once the flame tip goes deep into the organized eddy, the vortex bursting comes to exert promotive effect on the flame propagation in the axial direction of the organized eddy, forming a pair of ellipsoids superimposed on both ends of the dumpy rod-shaped flame in the side views of Figure 4. It is found that the two-staged enhancement constitutes the vortex-flame interaction in the plane shear flow; the vortex bursting enhances the flame propagation in the vortex axial direction, whereas the vortex boosting promotes the lateral flame propagation. Effects of the equivalence ratio on the flame appearances are found in every schlieren view, where the flame becomes longer and thicker with increasing the equivalence ratio from ~b= 0.7 to 0.9 and the laminar burning velocity from SL = 0.23 m/s to 0.37 m/s, leaving the fundamental flame movement described above almost unvaried.

Comparison of the Propagation Behavior Simulated with That Optically Observed In Figure 5 the numerically simulated flame propagation processes after the midpoint ignition are compared with those optically observed using the high-speed schlieren photography, where the flow conditions of Figure 5 (a) -~ (c) correspond to those listed in TABLE 1 (a) -~ (c), respectively, and the equivalence ratio is kept constant at ~ = 0.8. The left- and right-hand side pictures of Figure 5 present the temporal variations of simulated flame appearances and those of the schlieren images, respectively. The number written under each set of pictures indicates the time after the midpoint ignition. It should be mentioned here that the simulated flame profiles are not three-dimensional but two-dimensional, whereas the schlieren images express the integrated figures of the three-dimensional flame bodies. Figure 5 indicates that the appearances of the simulated flame agree well with those of the schlieren images of the spark-ignited flame. The following features can be observed with the exception that the schlieren images exhibit rather round and thick outlines due to their three-dimensionality. During the initial stage of flame propagation; t _< 10 ms, both simulated profiles and schlieren images propagate obliquely along the boundary of two mixture streams. At about t = 13 ms both ends of the slender flame are rolled up into adjacent organized eddies to form a slender reverse S-shaped flame. Once the flame tip is rolled into the organized eddy as can be seen in the schlieren images, however, the flame body becomes thicker than the simulated flame due to the combined effects of the baroclinic effect and the three-dimensionality of the vortex-flame interaction. Since these combined effects are not taken into account, the proposed simulation results in un-negligible underestimation of the flame propagation velocity after t > 13 ms. Concerning the influences of the flow conditions on the flame growth rate, it is found that the increase in the higher uniform mixture velocity elongates and makes the simulated flame more slender with the flame thickness almost unvaried, while it thickens and elongates the spark-ignited flame. Also found in the schlieren images is that the three-dimensional effects of coherent structure are concentrated on both end zones of the reverse S-shaped flame. Since the higher the vortex tangential velocity becomes, the stronger the combined effects of the baroclinic torque and the three-dimensionality become, the differences both in the flame appearances and the growth rate between those simulated and those experimentally observed become more prominent with increasing the vortex tangential velocity. The effect of the mixture equivalence ratio on the flame propagation characteristics, which is simply attributed to the variation of the laminar burning velocity in this simulation, is briefly discussed here. Since the effect of the laminar burning velocity is considered to be homogenous in all direction, the

923 faster the laminar burning velocity becomes, the thicker and longer the flame profile becomes, whereas the lower the laminar burning velocity, the slenderer and shorter the flame appearances become. These features of the simulated flame are found to agree well with those observed in the schlieren images, as already shown in Figure 4, where the effect of the equivalence ratio on the flame propagation behavior is clearly shown in the schlieren images.

Effect of the Vortex Boosting on the Flame Propagation Velocity In order to examine the contribution of the vortex boosting to the vortex-flame interaction, the lateral flame propagation velocity Vft is estimated by analyzing the increasing rate of the flame length in the flow direction with increasing the vortex tangential velocity v0. The results are summarized in Figure 6 in the Vft-vo coordinate system, where the equivalence ratio is varied as a parameter. The experimental data obtained in the preceding study [Ohiwa et al. (2003)] are also included and indicated by

Figure 5: Comparison of the simulated lateral flame propagation behavior with a set of schlieren images taken for a constant equivalence ratio of # = 0.8 with the flow conditions varied

924

8.0

= 0.70

7.0 6.0

0.80

0.90

9

9

9

[]

0

A

Experiment

Calculation

5.0 E

4.0 3.0 2.0 ~ 1.0

q

7-

....~r~"

!

0.0 0.0

U2 = 2.5 m/s

I

0.5

1.0

1.5

2.0

2.5

3.0

Vo m/s

Figure 6" Dependency of the lateral flame propagation velocity on the vortex tangential velocity and the equivalence ratio; an empirical expression for the vortex boosting the solid symbols, whereas the analytical data evaluated by the proposed discrete vortex method are expressed by the open symbols. Figure 6 shows that, although slight scattering in the data points for each equivalence ratio and slight overestimation of the simulated points are apparent, the simulated flame propagation velocity agrees reasonably well with that obtained from the schlieren images. This result verifies that the proposed two-dimensional discrete vortex method combined with the Huygens' Principle is useful for estimating the flame propagation velocity in the plane premixed shear layer, although many assumptions are introduced to simplify the numerical simulation. Dependency of the lateral flame propagation velocity on the vortex tangential velocity and the equivalence ratio is finally considered. As shown in Figure 6 by three straight lines having an equal gradient, the data points for each equivalence ratio are reasonably collapsed into an empirical single straight line; Vft (0) = Vo(qk) + vo. Vo(#) means the flame propagation velocity in the uniform mixture stream and is related to the laminar burning velocity and the density ratio as V0(~b)= (pu/po)SL [Ohiwa et al. (2003)]. It is concluded that the enhancement mechanism of the lateral flame propagation due to the vortex boosting is expressed by the following linear equation, Vet ( ~ ) : v 0 ( ~ + v0 : (f~/po)& + v0.

(8)

CONCLUDING REMARKS The flame propagation after the midpoint ignition between neighboring two organized eddies formed in the plane premixed shear layer is first observed with the simultaneously two-directional and highspeed schlieren system and then simulated with the two-dimensional discrete vortex method combined with the Huygens' Principle. Detailed analysis and comparison of the results numerically simulated with those experimentally obtained enable the following concluding remarks. In the initial stage after the spark-ignition the flame propagates faster in the flow directioh due to the vortex boosting than in the transverse direction and exhibits an obliquely slender profile. When the flame tips at both ends reach the neighboring organized eddies and are rolled up into the organized

925 eddy, the vortex bursting comes to exert promotive effect on the flame propagation in the axial direction of the organized eddy. This fact indicates that the two-staged enhancement constitutes the vortex-flame interaction in the plane shear flow; the vortex bursting enhances the flame propagation in the vortex axial direction, whereas the vortex boosting promotes the lateral flame propagation. It is verified that, in spite of assumptions introduced to simplify the numerical simulation, the outlines and the behavior of the propagating flame simulated agree reasonably well with those experimentally observed, and that the proposed discrete vortex method is extremely useful for analyzing the flame propagation in the initial stage after the midpoint ignition. According to detailed analysis of the increasing rare of the flame length in the flow direction, finally, the dependency of the lateral flame propagation velocity after the midpoint ignition on the vortex tangential velocity and the equivalence ratio is estimated by the superposition of each contribution and is simply expressed by the following linear straight empirical relation; Vft(~b) (pu/iOb)SL + VO. =

ACKNOWLEDGEMENT This investigation is supported in part by Grant-in-Aid for Scientific Research of Japan Society for the Promotion Science: (C) (2), No. 16560182, for which the authors express their great thanks.

REFERENCES

Asato K., Wada H., Hiruma T. and Takeuchi Y. (1997), Combustion and Flame 110, 418-428. Ashurst W. M. T. (1996), Combustion Science and Technology 112, 175-185. Brown G. L and Roshko A. J. (1974), Journal of Fluid Mechanics 64, 775-816. Chomiak J. (1976), 16th Symp. (Int.) Combust., The Combustion Institute, Pittsburgh, 1665-1673. Inoue O. (1992), Journal of Fluid Mechanics 234, 553-581. Ishizuka S., Murakami T., Hamasaki T., Koumura K. and Hasegawa R. (1998), Combustion and Flame 113, 542-553. Ishino Y., Yamaguchi S. and Ohiwa N. (1992), Trans. JSME (in Japanese) 58:547B, 653-660. Ishino Y., Kojima T., Ohiwa N. and Yamaguchi S. (1996),JSME Int. J., Ser. B 39:1,156-163. Ishino Y., Ohiwa N., Abe T. and Yamaguchi S. (1997), Combustion Science and Technology 130, 97-113. Ishino Y., Ohiwa N. and Yamaguchi S. (2001), 2nd International Symposium on Turbulence and Shear Flow Phenomena; TSFP-2, Vol. 3,327-332. Ishino Y., Yamaguchi S. and Ohiwa N. (2002), 2nd Mediterranean Combustion Symposium; MCS-2, Vol. 1,403-410. Ohiwa N., Ishino Y., Ikari M. and Yamaguchi S. (2003), 25th International Congress on High-Speed Photography and Photonics; ICHPP-25, SPIE 4948, 176-181. Ohiwa N., Ishino Y. and Yamaguchi S. (1994), Combustion and Flame 99, 302-310. Umemura A. and Tomita K. (1998), 25th Symp. (Nat.) Combust. (in Japanese), 311-313. Yamaoka I. and Tsuji H. (1985), 20th Syrup. (Int.) Combust., The Combustion Institute, Pittsburgh, 1883-1892.

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14. Two-Phase Flows

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Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

929

SIMULATION OF MASS-LOADING EFFECTS IN GAS-SOLID CYCLONE SEPARATORS

J.J. Derksen Kramers Laboratorium, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, Netherlands

ABSTRACT Three-dimensional, time-dependent Eulerian-Lagrangian simulations of the gas-solid flow in a cyclone separator have been performed. The Eulerian description of the gas flow is based on lattice-Boltzmann discretization of the Navier-Stokes equations, and a Smagorinsky subgrid-scale model. Through this largeeddy representation of the gas flow, solid particles with different sizes are tracked. By viewing the individual particles (of which there are some 107 inside the cyclone at any moment in time) as clusters of particles (parcels), we study the effect of particle-to-gas coupling on the gas flow and particle behavior at appreciable mass-loading (0.05 and 0.1). The presence of solid particles causes the cyclone to lose some swirl intensity. Furthermore, the turbulence of the gas flow gets strongly damped. This has significant consequences for the way the particles of different sizes get dispersed in the gas flow. It is anticipated that also the collection efficiency is significantly affected by mass-loading.

KEYWORDS cyclone separator, swirling flow, large-eddy simulation, gas-solid flow, turbulence, mass-loading

INTRODUCTION

In cyclone separators, a strongly swirling turbulent flow is used to separate phases with different densities. In gas cyclones solid particles are to be separated from a gas stream. A typical geometrical layout of a gas cyclone is the Stairmand high-efficiency cyclone as depicted in Figure 1. Dimensions and process conditions imply high Reynolds numbers (105 and up with

Re Uin.---~D,and Ui,, the

superficial inlet v velocity). The tangential inlet generates the swirling motion of the gas stream, which forces particles toward the outer wall where they spiral in the downward direction. Eventually the particles are collected in the dustbin, which is the lower, cylindrical part of the geometry. The cleaned gas leaves through the exit pipe at the top. Swirl and turbulence are the two competing phenomena in the separation process: the swirl =

930

induces a centrifugal force on the solids phase which is the driving force behind the separation; turbulence disperses the solid particles and enhances the probability that particles get caught in the exit gas stream. Both phenomena are related to the particle size, and the flow conditions in the cyclone. Prediction of the separation process therefore requires an adequate representation of the gas flow field (including its turbulence characteristics) in the presence of a particulate phase. On top of that, the way the solid particles interact with the each other could be influential to the process performance. In a previous paper (Derksen, 2003a), it was shown that with a large-eddy simulation (LES) the singlephase (i.e. gas) flow features of the Stairmand cyclone could be well represented: mean and fluctuating velocities agree well with experimental data. Furthermore, the LES was able to accurately predict vortex core precession, which is a low-frequency, coherent motion of the core of the vortex. Precession causes the velocity to coherently fluctuate in addition to the fluctuations induced by the turbulence. In the same paper, an Eulerian-Lagrangian description of the two-phase (gas-solid) flow was presented: the motion of solid particles on their way through the simulated gas flow field was modelled based on a one-way coupling assumption (the particles feel the gas flow, but the gas flow is not influenced by the presence of the particles). It was furthermore assumed that the particles did not interact with one another via e.g. hardsphere collisions or cohesive forces. These assumptions were justified by the fact that our main interest is in high-efficiency cyclones that are used in the final stages of gas cleaning and operate at very low solids mass-loadings. However, effects of mass-loading (0m) on the performance of cyclones have been reported, even for Om well below 1 (e.g. Ontko, 1996). Therefore, in the present paper we extend the simulation procedure as described in Derksen (2003a) by including two-way coupling effects. We do this by feeding the force that the gas exerts on the particles back to the gas phase. In order to get appreciable mass-loading with particles that have sizes around the cut-size* and at the same time limit the number of particles to some 10 million, we multiply the gas-to-particle force with a factor much bigger than 1 before we feed it back to the gasphase. This implies that we view the individual numerical particles as parcels, i.e. as assemblies of particles (see e.g. Elgobashi & Truesdell, 1993). The aim of this paper is to demonstrate the application of detailed modeling to a gas-solid flow system that has direct practical relevance, and employ some of the many notions that exist in the field of dilute (in terms of solids volume fraction), turbulent gas-solid flows. These notions were up to now mainly applied to flows in simple geometries such as fully periodic domains, or planar channels. This paper is organized in the following manner. First we briefly recapture the numerical set-up of the LES (from Derksen, 2003a) and the way particle motion is coupled to the LES. Then the various flow cases will be defined. Results are presented in terms of the gas flow field, the spatial distribution of particles in the cyclone, and the separation efficiency. Finally concluding remarks are presented.

SET-UP OF T H E S I M U L A T I O N S The incompressible Navier-Stokes equations that govern the motion of the continuous gas-phase were discretized by means of the lattice-Boltzmann method (see e.g Chen & Doolen, 1998). This method was chosen for its geometrical flexibility in combination with numerical efficiency. The lattice-Boltzmann solver was coupled to a standard Smagorinsky subgrid-scale (SGS) model (Smagorinsky, 1963) in order to perform large-eddy simulations (LES). Wall functions were applied at no-slip walls. dxp

Solid particles were released in the gas stream. Their location x v was updated according to

* The cut-size of a separation process is defined as the diameter of particles that are for 50% collected.

dt

-" V p ;the

931

Figure 1. Stairmand cyclone geometry. Gas enters through a tangential inlet with superficial velocity Uin, and exits through the exit pipe at the top. Solids are collected in the bin underneath the conical section.

Figure 3. Left three graphs: single realizations of the particle positions in a vertical cross-section (slice-thickness 0.04D) through the cyclone in the one-way coupled simulation at t=230Tint. From left to right: Stk=3.10 -4, 2.3.10 -3, 1.8.10 2. Outer right graph: single realization of the particle velocities for Stk=2.3.10 .3 (slice thickness 0.005D).

particle velocity Vp obeyed Newton's second law with the particle feeling Stokes drag and gravity: d v p = Uin (U-Vp)+g dt Stk D

(1)

2

The Stokes number Stk = Pp dp Uin is defined as the ratio between the particle relaxation time and the gas P~ 18vD flow integral time scale T/,,t = D . In an LES, the gas velocity u in Eqn. 1 is composed of a resolved part U/n

and a SGS part. The former was determined by linear interpolation of the velocity field at the grid nodes to the particle position, the latter is mimicked by a uniform random, isotropic process with zero average and f_

an RMS value U,.gs = ~ 2 ksgs . The SGS kinetic energy ksu., can be estimated from the SGS model and the assumption of local equilibrium (Mason & Callen, 1986). In Derksen (2003b) it was demonstrated that for

932

one-way coupled simulations the SGS motion of the gas had hardly any influence on the motion of the solid particles. We therefore conclude that the resolution of the LES is such that it largely resolves the scales relevant for the dynamics of the particles. The effect of the particles on the gas is modeled by the particle-source-in-cell (PSIC) method (Crowe et al., 1996). Boivin et al. (2000) adopted this method coupled to LES's of homogeneous isotropic turbulence and assessed the LES predictions by comparing them with DNS results. With the Smagorinsky model they obtained reasonable estimates of the SGS dissipation as a function of mass-loading. However, for correctly describing backscatter, they recommend more advanced SGS models. For single-phase, swirling flow we have experimented with e.g. mixed-scale models (Derksen, 2004) and showed improved levels of accuracy in cases with strong velocity gradients. In future works we will explore the promises of more refined SGS modeling for two-phase flow as well. In the simulations the particles did not interact with one another, i.e. a particle does not undergo collisions with other particles and therefore particles are allowed to overlap. Based on the space averaged solids volume fraction that is below 10 -4 this assumption is fair. However, since the bigger particles accumulate at the outer wall, locally the solids volume fraction becomes as large as 200 times the average value and particle-particle collisions may become relevant. Particle-wall collisions are assumed to be elastic and frictionless. In this respect we expect conservative estimates of collection efficiency: inelastic particles would have a stronger tendency to stay in the wall region and eventually end up in the dustbin.

O V E R V I E W OF CASES The gas flow conditions are fully characterized by the Reynolds number that was set to Re-2.8.105. The particles that were fed to the cyclone had a uniform size distribution (in terms of numbers of particles) with nine different Stokes numbers (Stk=3.0.10 -4, 5.0-10 -4, 8.3-10 -4, 1.4.10 -3, 2.3-10 -3, 3.9.10 3, 6.5.10 -3, 1.1.10 -2, 1.8-10-2). These values were chosen to lie around the cut-size Stkso=1.5.10 -3 as it was determined from previous simulations (Derksen, 2003a). The gravitational acceleration was such that the Froude number amounted to F r -

Uin2 --90.

qgl

4.10 7

N~, 3-10 7 ,p

2.10 7

10 7

0 150

i

175

i

200

i

225

!

250

275 t/Tint

Figure 2: Time-evolution of the number of particles in the one-way coupled simulation. Short dash: number of injected particles" long dash: particles exhausted at the top; dash-dot: particles collected at the bottom; solid: particles inside the cyclone.

933

In our previous paper (Derksen, 2003a), the particulate phase was treated in a transient manner: particles were injected in the fully developed cyclone flow during a limited time-window (18 T/,,t). The injected particles were then followed on their way through the cyclone. Grade-efficiency curves were based on the particles that were not exhausted after a certain amount of time. It was shown that we needed to simulate the system for some 102 integral time-scales after the start of the particle injection in order to get more or less converged grade-efficiency curves. Since we now want to investigate turbulence modification due to particles, we need a representative, quasi-steady distribution of the solids phase throughout the cyclone. In order to initialise such a distribution, we took a one-way coupled simulation as a reference case. This case was started with a fully developed flow and a cyclone without particles. From t=0 on, the particles were continuously fed into the cyclone at a rate of 1.24.105 particles per Tint. They were randomly distributed over the inlet area. In order to reach a steady state, particles not only got exhausted through the exit pipe at the top, but also through the bottom: it was assumed once a particles was below z/D=-I.9 (see Figure 1 for a definition of the coordinate system) it can be considered collected, and was no longer taking part in the simulation. The boundary condition for the gas flow remains unchanged at the bottom: at z/D=-2 there is a no-slip wall. It takes quite some time before a steady state is reached. Figure 2 shows a part of the timeevolution of the fate of the particles (exhausted, collected, still inside the cyclone). Even at t=275Tint we are not yet completely at fully steady conditions: the number of particles inside the cyclone is still slowly increasing. In this paper we will discuss next to the one-way coupled case two cases with finite mass-loading: ~m=0.05, and 0.1. These mass-loadings were achieved by assuming that our system mimicked an airchalkpowder (CaCO3) mixture as was used in the experiments by Hoekstra (2000). Hoekstra's cyclone had a diameter of D=0.29 m and was fed with air at ambient conditions and Uin=16 m/s. The density ratio was O--L - 2.5.103 . If we then assume that each computational particle represents a parcel containing 3.95.105, Pg and 7.90.105 real particles we obtain the two mass-loadings mentioned above. We realize that these multiplication factors are huge. However, we do not have much choice. The number of particles inside the cyclone in each simulation is of the order of 107. From a point of view of computer memory usage we could increase this number by a factor of say 3, which would slow down the computations with a factor of approximately 2. Such an increase would, however, not drastically reduce the multiplication factors.

RESULTS In Figure 3 the separation process is visualized for the one-way coupled simulation. Here it can be clearly seen that turbulence plays a crucial role. The small particles (low Stokes numbers) are dispersed throughout the cyclone, and are likely to get caught in the flow through the exit pipe at the top. The bigger the particles, the more they accumulate in the wall region and gradually move (due to gravity) to the dustbin. In the classical cyclone models (e.g. Barth, 1956), collection efficiency as a function of particle size is explained in terms of the competition between the centrifugal (outwardly directed) force, and the (inwardly directed) drag force due to an average radial velocity. From the vector plot in Figure 3 it is clear, however, that the average radial velocity component is of minor importance compared to the erratically fluctuating velocities due to turbulence. Inside the dustbin, the swirl is greatly reduced but definitely not absent. There is a strong interaction between the flow in the dustbin and the separation section of the cyclone (the latter being the part in between z=0 and z=3.5D) and therefore modeling the flow and particle motion in a cyclone should include the dustbin in order to get realistic predictions.

934

2.107

Np

1.5.107 1.107 5"106t 0230

i

i

I

i

I

i

i

i

I

t/Tint 330

Figure 4: Time-evolution of the number of particles inside the cyclone. Thick curve: one-way coupled; thin curve: two-way coupled with ~m=0.05; dashed curve: two-way coupled with ~)m=0.1. The two-way coupled simulations were started from the gas-solid field as it was at t=230Tint for the oneway coupled simulation. In Figure 4, we observe that the system strongly responds to switching on the particle-to-gas coupling. Initially the number of particles inside the cyclone strongly reduces. A closer look reveals that the number of particles inside the dustbin goes down by typically one order of magnitude, and also in the lower part of the conical section particle concentrations are lower than for one-way coupled simulations (see Figure 5). The reduced concentrations are related to the swirl intensity in the dustbin:

Figure 6: Profiles of the time-averaged tangential velocity in the dustbin (at axial location #/9=-1). Solid line: one-way coupled simulation; short-dashed line: ~m=0.05 averaged in the time-window t=230Tint260Tint (i.e. directly after switching on twoway coupling); long dashed line: ~m=0.05 averaged in the time-window t=272Ti,,tFigure 5. Single realization of the particle positions (irrespective of particle size, all particles are on display) in a vertical cross-section (slice-thickness 0.03D) through the cyclone. Left: the one-way coupled simulation; right: the ~m=0.1 simulation.

295Ti,,t.

935

0.1

0.5 0.05

-2

0

//L

,,.. i

/,

- :! |

-0.5

i

a

\',

,."]

0

J

..,,7,'. Y

l . -0.5

0.0

r/D

0.5

-0.5

0.0

r/D

0.5

-0.5

0.0

r/D

0.5

Figure 7: Radial profiles of the time-averaged tangential gas velocity (left), axial velocity (middle), and (resolved) turbulent kinetic energy (right) at axial location z/D=3.25 (top) and z/D=2.0 (bottom). The solid curves are the results with one-way coupling, the short-dashed curves have ~m-0.05, and the long-dashed curves (~m=0.1. shortly after the back-coupling force is switched on, the swirl velocity in the dustbin has reduced by a factor of two. Apparently, the flow in the dustbin cannot carry the large amounts of particles present. A new equilibrium is establishing itself with much less particles in the dustbin and a swirl intensity that has almost recovered to its original (one-way coupled) level (see Figure 6). From Figure 4 it can be clearly concluded that at this stage (early September 2004) the two-way coupled systems have not reached their (quasi) steady state yet; this will still need some 102 integral time scales (and typically 1 month of computer time). This should be kept in mind when interpreting the time-averaged profiles of gas velocity and particle concentration that are given below (Figures 7 and 8). These profiles have been obtained from time-averaging over the interval t=272Tint- 295Ti, t. In the separation section of the cyclone the gas flow field has significantly changed as a result of the presence of the solid particles, even at the low mass-loadings of 0.05 and 0.1 that are considered here, see Figure 7. The average tangential velocity reduces although the inflow of angular momentum increases with switching on two-way coupling: once solids and gas are fully coupled the particles contribute to the momentum of the gas stream and vice versa. The reduction of swirl is mostly felt in the free-vortex part of the swirl profile, since here the particle concentrations are much higher than in the core. The higher the solids loading, the more the swirl is reduced. The increased levels of tangential velocity close to the outer wall clearly are a two-way coupling effect. The particles carry tangential momentum with them when moving towards the wall and then partly transfer it to the gas. i ) reduces strongly as a result of the The kinetic energy contained in the gas velocity fluctuations (k = -~ 1 U,2 particles (Figure 7). Especially in the lower part of the cyclone, fluctuations hardly survive. As explained earlier (Derksen, 2003a), the elevated levels of k in the center are mainly due to vortex core precession" a coherent motion of the vortex core. The tangential velocity contributes most to the fluctuation levels

936

,oo[ 50 ~

f

0

A |

-0.5

I

0.0

|

r/D

0.5

Figure 8: Radial profiles of the time-averaged particle concentration at axial location JD=3.25 (top) and z/D=2.0 (bottom) at three different Stokes numbers: From left to right Stk-5.10 -4, 2.3.10 -3, 1.1.10 -2. The thick-solid curves are the results with one-way coupling, the dashed curves have ~rn=0.05, and the thin-solid curves ~m=0.1. related to precession since it is the component with the highest gradients. We observed (not shown here) that the amplitude and frequency of the vortex core precession are hardly affected by the presence of the particles. The reason for the central peak in the one-way coupled simulation being broader and higher than in the two-way coupled simulations is the vortex core* being broader, and the slope of the average tangential velocity profile being higher. Outside the vortex core, turbulence dominates k. In this region the reduction of k is most pronounced. The observations with respect to the average gas flow field correlate well with the particle concentrations profiles that are presented in Figure 8. The differences in terms of solids concentration between the three mass-loadings (0, 0.05 and 0.1) are largest in the bottom part of the cyclone. In that region, the case with ~)m=0.05 shows the highest concentrations near the wall: for this case the reduced turbulence more than compensates the loss of swirl due to the particles. Finally, we want to find out if the mass-loading affects the collection efficiency. As a reference we take the one-way coupled simulation. Figure 9 shows the time evolution of the number flux of particles with specific Stokes numbers through the top and bottom of the cyclone. We observe that the temporal fluctuations of the particle flux are much stronger for the higher Stokes numbers. Also for the bigger particles the sum of fluxes through top and bottom is on average slightly less than the influx, indicating that steady state is not yet fully reached. The latter is also reflected in the systematic difference between the collection efficiency curves that were obtained by means of the time-averaged flux of particles through the top exit pipe and the time-averaged flux through the bottom. The time series of Figure 10 show the strong response to switching on particle-to-gas coupling for the case ~m=0.05. From Figure 10 it is clear * The vortex core is defined as the region of the flow with radial positions smaller than the radius with maximumtangential velocity.

937

t

f

0

230

t/Tint

i

33

'

~

i,

i

10 -4

10 .3

Stk

10 .2

10 1

Figure 9: One-way coupled simulation. Left three graphs: time series of the particle flux (relative to the flux at the inlet) through the exit pipe (thin curve), through the bottom (thick curve), and the sum of the two (dashed curve) for (from left to right) Stk=5.0.10 -4, 2.3.10 -3, 1.1.10 ~ Far right graph: time-averaged collection efficiency as a function of Stk. The thick curve is based on the flux at the bottom, the thin curve on the flux at the top. that we still have not reached steady state: the sum of particle outflux through top and bottom for

Stk= 1.0.10 -2 is still well below the influx. Therefore, we cannot draw conclusions with respect to massload!ng effects on collection efficiency yet.

O/~in 4

,-,,,

230

t/Tint

330

Figure 10: Two-way coupled simulation with ~m=0.05. Time series of the particle flux (relative to the flux at the inlet) through the exit pipe (thin curve), through the bottom (thick curve), and the sum of the two (dashed curve) for (from left to right) Stk=5.0.10 -4, 2.3.10 -3, 1.1.10 2.

SUMMARY In this paper, the effect of mass-loading on the gas flow and solid particle motion in a Stairmand highefficiency cyclone separator has been studied numerically. It was shown that the separation process is an interplay between centrifugal forces induced by swirl, and dispersion due to turbulence. In our EulerianLagrangian simulations, both swirl and turbulence are affected by the presence of particles: the turbulent fluctuations strongly reduce, even at the relatively low solids loadings that we have considered so far. The swirl in the cyclone gets less intense, especially in the lower part of the cyclone. Both features have consequences for the way the particles distribute inside the cyclone. We observed increased particle concentration levels close to the wall for ~m=0.05, most pronounced in the lower part of the separation section of the cyclone, caused by reduced turbulence. At higher mass-loading (~=0.1) particles tend to be less concentrated near the wall due to reduced swirl. These changes in the way the particles are distributed will have consequences for the collection efficiency. We expect the latter to be clearly dependent on the mass-loading.

938

Also some sensitivity of the result on the particle-wall collision characteristics is expected. We plan to study cases with rough walls, as opposed to the smooth walls that were the subject of the present study.

References Barth W. (1956) Berechnungen und Auslegung yon Zyklonabschiedern auf Grund neuerer Untersuchungen. Brennstoff, Wiirme, Kraft 8, 1-9. Boivin M., Simonin O. and Squires K.D. (2000). On the prediction of gas-solid flows with two-way coupling using large eddy simulation. Physics of Fluids 12, 2080-2090. Chen S. and Doolen G.D. (1998). Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 30, 329-364. Crowe C.T., Troutt T.R. and Chung J.N. (1996). Numerical models for two-phase turbulent flows. Annual Review of Fluid Mechanics 28, 11-43. Derksen J.J. (2003a). Separation performance predictions of a Stairmand high-efficiency cyclone. AIChE Journal, 49, 1359-1371. Derksen J.J. (2003b). LES of swirling flow in separation devices. Proceedings of the 3rd International Symposium on Turbulence and Shear Flow Phenomena. Sendai, Japan. 911-916. Derksen J.J. (2004) Simulations of confined turbulent vortex flow. Computers & Fluids, in print (preprint available at http://kramerslab.tn.tudelft.nl/,-,jos/pbl/caf_preprint_sw.pdf). Elgobashi S.E. and Truesdell G.C. (1993). On the two-way interaction between homogeneous turbulence and dispersed solid particles I: Turbulence modification. Physics of Fluids 5, 1790-1801. Hoekstra A.J. (2000) Gas flow field and collection efficiency of cyclone separators. PhD Thesis, Delft University of Technology, The Netherlands. Mason P.J. and Callen N.S. (1986) On the magnitude of the subgrid-scale eddy coefficient in large-eddy simulations of turbulent channel flow. Journal of Fluid Mechanics 162, 439-462. Ontko J.S. (1996) Cyclone separator scaling revisited. Powder Technology 87, 93-104. Smagorinsky J. (1963) General circulation experiments with the primitive equations: 1. The basic experiment. Monthly Weather Review 91, 99-164.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

939

ON EULER/EULER MODELING OF TURBULENT PARTICLE DIFFUSION IN DISPERSED TWO-PHASE FLOWS R. Oroll* and C. Tropea Fachgebiet Stroemungslehre und Aerodynamik, TU Darmstadt, Petersenstrasse 30, 64287 Darmstadt, Germany

ABSTRACT Reynolds averaged, volume-fraction weighted momentum transport equations are solved in an Euler/Euler approach to numerically simulate the turbulent, dispersed two-phase flow in a twodimensional channel and a backward-facing step geometry. Particular attention is given to the modeling of turbulent diffusion, assuming local equilibrium but introducing individual terms for particle/fluid drag interaction, particle collisions and trajectory crossings. These influences have been quantified in terms of partial viscosities, a restitution power and a turbulence structure parameter. Boussinesq approximations have been used for each phase and their interaction. The results show improvements over conventional Euler/Euler modeling schemes, and advantages and disadvantages over Euler/Lagrange schemes are discussed. The new concept involves modeling the turbulent viscosity by the turbulent kinetic energy k ~: and the restitution power of the turbulence. This new kind of modeling is required, because the turbulence loss of' the continuous phase in form of eddy dissipation has no corresponding term in the transport equation of the turbulent kinetic energy of the dispersed phase. This energy appears as the velocity variance of the dispersed particles. Based on this approximation a consistent derivation of the turbulent kinetic energy is presented. KEYWORDS particle-laden flow, diffusion modeling, Euler/Euler approach, backward-facing step flow INTRODUCTION A computational model is presented, involving transport equations of momentum and turbulent quantities for a continuous and a monodispersed second phase, interacting with each other in the framework of a two-phase Euler/Euler approach. The underlying turbulence model uses eddy viscosity c,[ as the model parameter. In addition to the dynamical equations for tile turbulent kinetic energy k k and its dissipation rate c~, the model employs transport equations for ~he covariance of velocities of the dispersed and continuous phase qk, used to characterize the turbulence interaction between the phases. Corresponding author: gr~

940

The m o m e n t u m transport equation includes a turbulent diffusion term, which characterizes motion not resolved by the convective term. This turbulent diffusion depends on the turbulent kinetic energy and the characteristic turbulent time scale. Based on "Csanady's Approzimation" (Csanady, 1963), the time scale of the dispersed phase is coupled with the turbulent time scale of the continuous phase. This time scale quantifies the diffusion intensity and is influenced by the drag interaction of the particles with the viscous gas phase and inter-particle collisions.

T R A N S P O R T EQUATIONS The Reynolds averaged, volume-fraction weighted momentum transport equations for the continuous and the dispersed phase read:

k, k

akooj.r~,

k k)=

(1)

pk~-~ ]

T

where k stands for either C (continuous phase) or D (dispersed phase). A negation notation is used here, i.e. C = D, D = C. The drag force relaxation time scale T~ depends on the particle Reynolds number R@, based on the velocity difference between the continuous and the dispersed phase (Schiller et al., 1933), see e.g. Cvowe et al. (1998). The standard momentum equation is weighted with the volumetric fraction a k. Using the Reynolds time-averaged filter for averaging the equation, the correlation of volumetric fraction a k and velocity u~ has to be considered. The convective terms and volumetric fraction-velocity correlations a ku ik = ak < u ik >k are described with the help of a volumetric fraction weighted averaging operator < . >k (see, e.g. Politis, 1989). The fluctuation of the filter variable is w r i t t e n {~}~ = ~ _ < ~ >~. Collision and crossing trajectory terms were approximated by Grad (1949) and Csanady (1963). The underlying turbulence model for both phases is based on the Boussinesq analogy, employing eddy viscosity as the model quantity, whose formulation was provided in the framework of the standard k-c modeling concept, k c defines the turbulent kinetic energy of the continuous phase. The corresponding variable of the dispersed phase k D describes the particle velocity variance at a point.

((+ k}k

k

k

k_pk~-T~2

)

ok ]

,

(2)

1.0

(3) Velocity C o r r e l a t i o n The transport equations for the turbulent kinetic energy k k and its dissipation rate atk differ from those for a single-phase flow by several additional production terms and the modified dissipation. k in the present model arises from relative drag and The total turbulent kinetic energy loss G~ particle collision processes. (4) (7 e

u j } >~ cry-- 1.3

,

CI-

1.44

,

~k pk a---Z~ _ _ (qk_

k~

k

-C2 =

1.92

,

C3-

2kk)

1.2

The velocity covariance qk = < {u~}k{u~}k >k of He and Simonin (1993) represents the trace of

941

the velocity vector correlation tensor of both phases:

pkak

0, ( , ~ ~ ) + oj ( , ~ ~ < ~k >~) : y

k k_,o~ ~

- ~ ) q~ with + (9 ( pkak---~Ojqk _pk--~__ O'q

(2Zkk k +

_ p_~ _

2k~}~

(1 +

(5)

Zk)q k)

k

~

Z k-

Tc~

pk ozk

'

which completes the present three-equation model for each of both phases. Velocity correlations, representing the turbulent momentum diffusion are modeled by the following Boussinesq appro-

zimations: <{u~}k

-<

k

{~-"}~ >

k

{u/~}~{u~} ~ >~

23 k~6ij + t~t (Oj < ~ ~ >1~ +o~ < ~,.k > k -55~. 2 o~ < ~ >k )

=

lq~6ij+r 'k (Oj <

-

~ ~

~ >~

26

)

(6)

The diffusivity coefficients ut~ : "r2 9 k '~

k

and

k lqk

u~ = % ' 3

(8)

are given by the characteristic diffusion time scale 7~ (Eqn. 17). To close the present formulation of particle and carrier gas phase motion this time scale has to be modeled. Based on this model of m o m e n t u m diffusion, which depends on the velocity gradients of the diffusing phase, the characteristic diffusion time scales have to be defined by the velocity correlation and its associated loss rate % (Eqn. 14).

Local equilibrium Local equilibrium describes the equivalence of production and loss of turbulent kinetic energy. 01 ~ 0 yields an expression (i = 1, j = 2) for the non-diagonal elements of the Reynolds stress tensor. Assuming

}~ --

o~ < ~ >

~

=

(9) (10)

02 < Ulk > k

With the definition of the turbulence structure parameter C~ of the phase k

< : _ < {~}~{~}~ >~ ~kk

2

~

_ < {U~}k{u~}k > k =

(~C~kk)(~ 02 < u) >k

(11)

y=:Yr.k

the turbulent viscosity is calculated using the turbulent kinetic energy k k, its loss rate arid the turbulence structure parameter. Based on the diffusivity definition (Eqn. 8) the turbulent time scale is also defined by these values:

,-',~ =

~-2. ~k '~

~

o

~'~ = ~ k'rk - 3 -

kk

-2c~ ~7

(12)

With the definition of the restitution power 7c~, k the turbulent viscosity is determined by the turbulent kinetic energy and the restitution power.

~k_

,~-

_

k

~o ~

~_~ - c~ ~

_<{~}~{~}~>~:

(2

]

~k~j ~

~

::~

o~ < ~ >k

(13)

942

C 7r7

J o n e s e t al 9

d

Csanady

P

Schiller et at.

C

Jenkins et al.

E?C

I c ~I k-c _ ~9~,

~

0.

Tic

D 71-,),

0. 2 kD

0.

'

57

9. k D

-- 5 7

'd

5qC)

~

0.

E~D

qC)

~

a

~/1 - -

~-~) (3 k

0.

0 /1

5g- 5

.

T tJ~ t~r flC/-C

2 .

0,

--

5

)

qD)

~

(1

e~ 2)

7p~r~

Table 1: Diffusion rates and turbulence loss components

The power ~rk~ describes the restitution of turbulent shear forces based on the dissipation and structure of turbulence and reduces the turbulent diffusion.

Restitution Power This restitution power consists of the partial powers defined by four different effects (see table 1), which are described in the following subsections with the indices /3, p, c and d. The total turbulence loss rate e ak is given by the sum of individual loss rates (e~, see table 1)" k+

k

(14)

The different diffusion rates and turbulence loss rates are induced by the viscous turbulent shear stress (7c~,s~" Jones et al., 1972), crossing trajectory effects (Tr~,ek'd Csanady, 1963), drag forces (Trp,Cp k k. Schiller et al., 1933) and collision terms (7c),c c. k. Jenkins et al., 1985). Adding together these influences, the new restitution power term of the turbulent diffusion is modeled. k = "n-~ --t--'7r~ -t-- ~ pk + "n-ck '7to,

(15)

Simulating the general restitution power 7c~ k and energy loss %k the resulting turbulence structure parameter C~ is computed in the following way:

~-~ ~kk =

~

=

~'~ ~

c~~

~

C~=

.I ~

(16)

~~

The turbulent time scale T~ depends on the sum of all diffusion rates (Tr.y, k see table I )

~2 = ~ + ~: ~k~ + ~ - + ~k

~

~ : ~2 ~k ~' ~~ = ~~ ' 5 2q~

(17)

This way of calculation yields a deterministic method to compute the turbulent time scale, which is needed for the calculation of the general turbulent viscosity of both phases L,tk as modeled in equation (8). The particle diffusion describing components of the restitution power 7r~ are defined in the following chapter.

DIFFUSION

MODELING

Standard single-phase k< model (Index ~) Considering the standard k-c model of' Jones and Launder (1972) the turbulence structure parameter can be deduced from Eqn. 4 and is constant.

~=~t

k

;

C~-0.45

=r

ctk

7r~=C~ 2

(18)

943

The dissipation time scale, here the partial turbulent time scale r~ k, depends on the partial restitution power as follows:

kk

2Ck2 k ~

---- rc~ = 5

~/3 k-

• 65

=>

C~=,~/3

(19)

::1/r k

The corresponding turbulent viscosity of the viscous shear-stress diffusion is defined by the associated time scale and the turbulent kinetic energy.

--

with

C~ = ( ~C~)

= 0 .09

(20)

=:C~

In this way the numerical value of the coefficient C~ is determined.

Drag interaction (Index p) The drag dependent turbulent dissipation rate is given by the turbulence transport equation (Eqn. 4) under conditions of local equilibrium. 6pk :

~1

(2kk_qk)

"

C~ = V/~

(21)

Using the drag specific turbulence structure parameter C~ and the drag force formulation of Schiller and Naumann (1933) (see, e. g. Cwwe et al., 1998) the partial restitution power k __

~

6p

c~

__ ~

kk

~

D

_

_

5 q~

D2

0.687)

with

Dp (1 + 0.1 5Rere D1 r D = p 18#C

and

Rerel =

pCaC

D

t

acpc

(22)

-

defines the specific turbulent time scale ~2kk

,k Tp

--

k

1 --

/' 1

rpk qk

--

qk

(23)

which determines the partial drag induced turbulent viscosity:

~ = .,_;k 2k~ =

~

2k~

(24)

This diffusion is based on the drag dependent turbulence interaction of the dispersed and the continuous phase.

Collision interaction (Index c) The degree of particle velocity variance caused by particle collisions is characterized using the following variables:

Ub

=

~~kk

" ua'g,~ ,

=

k -ec'U~,'4~

(25)

The impact velocity before the collision is u~. The variable u ak defines the characteristic velocity after the impact. The parameter e ck characterizes the elasticity level. For purely elastic collisions

944

ect = 1. The dissipation rate due to collisions is obtained as the difference in kinetic energy before and after the collision: 1

1

2

=

9

" 2

1 (Ub -- Ua)(UV + ua)

lu2

1

+

~2

). ~kt

(26)

The resulting dissipation rate of the colliding particle phase is determined by elasticity, turbulent, kinetic energy and the characteristic collision r a t e Tck-l" kt cck _- <1k ' - 21G ,26 { j 6 { j - ( 1 - e) 2) ~

1 =

with

Tf =

and

at n t - [D~3 -

Nt ~

(

-

r?~-D~ 4 .

=~

2 t

_ 24 a t Tck 1 =

~D~ i

(27) 2

kk

The turbulence structure parameter C) depends on the restitution parameter c~ct defined by Jenkins and R i c h m a n (1985) C~=

31-c~ 2

with

t

1

t

(l c

the restitution power of collision, which influences the m o m e n t u m diffusion, and its turbulent time scale, calculated in a manner analogy to Eqn. 13:

~-C~~

~

ffk=

~.~: = <

Computing the partial viscosity corresponding to the minimum diffusion (max [a~] = 0.8) results for pure elastic collisions (e~t = 1) (Grad, 1949)

k = ,k. _2kk __ 7~ . _2k~ G T~ 3 or.t 3

(30) '

and no other mechanism is affecting the m o m e n t u m diffusion, because the restitution parameter m a x i m u m is reached when Gt = 1.

Crossing trajectories (Index d) The "Crossing Trajectories Effect" is defined by Csanady (1963). This kind of diffusion is induced by chaotic and unresolved particle motions of the di_spersed phase, which are algebraically coupled to the Lagrangian time scale of the carrier phase f2. W i t h o u t turbulent dissipation the turbulent time scales of the dispersed and the continuous phase are coupled in the following way:

e~=0

9 C~=0

;

rd = r

2

with

()=

~ c~kkt

(31)

The crossing trajectory parameter C) characterizes the turbulence specific turbulent intensity. The partial restitution power is defined by the turbulence characterizing quantities of the other phase k. _

~=~~ -- ~~ 1+c~<~ 2k t

2 .k V/

~

(32)

945

Analog to the other cases, the viscosity is calculated with the dedicated time scale. u~ = ,r.o~k 2 kk =

T~. ~k k

3

(33)

v/l+

The resulting momentum diffusion decreases with increasing velocity differences of the two phases. Special C a s e s The standard k-e-model is obtained by neglecting 7rCe, upc and r

2 kC C ~2 = c% S

2 k2 k C ~ c : 5c~ __ ~

~

of the continuous phase.

~ k c2 ~ : ~2 . k ~ : c , ~~

~

(34)

If, for example, 7r~ and 7r~) of the dispersed phase is neglected, the modeled diffusivity is equivalent to the value of He and Simonin (1993). 2kD 5

= 71"c~

=*

=~

--

71"p

Tf

--

kD .

TD 2 kD = TD2_kD

//D __

=~P 2kD

.

5 q~

.

D lqO =~3

.

+~-y 3

(35)

kD

" 9 0 cD TD 2kS

(36)

=~P (37)

1 + ~- ~'~ This definition of turbulent particle momentum viscosity considers influences of drag and collision in an averaged momentum transport equation of the dispersed phase. RESULTS

This model was validated using experimental data of 70#m copper particles in a fully developed channel flow [Exp.: Kulick et al., 1994] (Fig. 1, left) and a flow over a backward-facing step [Exp.: Fessler et al., 1999] (Fig. 1, right). The results using the present model (Eul./Eul. [2]) were also compared with the results obtained by an Euler/Lagrange scheme [Huber et al. 1994, Sim.: Kohnen 1997 (Eul./Lag.)] and a well-known Euler/Eulerian diffusion approach [He et. al. 1993, Sire.: Groll et al. 2002 (Eul./Eul. [1])].

200[nnl ! ~ ~hmm

Figure 1" Geometries of the channel flow (left) and the backward-facing step flow (right) Gravity acts in the positive z-axis direction. The channel flow Reynolds number, based on channel height (2h = 40ram) and single phase channel centerline velocity (U0c = 10.5m/s) is Re2h = 27600. The flow is regarded as fully developed after 125 channel heights and at this position it is assumed

946

that the particle velocity and particle turbulence has reached an asymptotic state. The step height (H = 4h/3) Reynolds number is .ReH = 18400. The copper particles have a density of pD = 8800kg/m3 and a diameter of Dp = 70pro. The inlet mass loading of particles is Z0D = 10% and the parameter of elasticity is e~ = 0.90. The results obtained using the present method were also compared with the computational results obtained by an Euler/Lagrange scheme (Kohnen, 1997).

Particle-laden, fully developed channel flow Fig. 2 shows the normalized particle stream-wise mean velocity uD/uoc, its normalized standard deviation values uD/uoc and the normalized standard deviation of the particle velocity magnitude D U,~ag/U ~C . Comparing the present model (Eul./Eul [2]) with a standard particle diffusion model (Eul./Eul. [1]), the velocity and the standard deviation of the velocity magnitude of the present model agree better with the Euler/Lagrangian results. Because of the assumed isotropy of the present model, the standard deviation of the stream-wise particle velocity does not agree as well with the predicted standard deviation of the particle velocity magnitude. I

0.8

I

9

Eul./EuI, [2] U ~ EuI./Eul. [1] U D Eul./Lag. U D . . . . . . . 9

9

9

I

'Eul/Eul. [2] u D EuI./Eul. [1] u D . . . . 9 Eul./Lag. u D

Ex

0.8

u

\':

0.4

..

0

.

.

.

0.2

.

0.4

0.6 uD/uC 0

0.8

1

o

~\

0.6

w

Eui./Eul. [2]' UDrnag Eul./Eul. [1] UOrnag Eul./Lag. u o ,., Dma~ Exp. u mag

_J

0.4

0.2

., .

i l O' ":i/ ,\ _

0.8

0.4

0.2 o

I"', ' '\i ~.,

"' 9 :'. 9

: 0

0.05

0.2

:;' ...... ~'~"

0.1 0.15 uD/uC0

0.2

0.25

o

0

:~'~"'-, 0.05

0.1 0.15 UDmag/UC0

0.2

0.25

Figure 2: Normalized stream-wise mean velocity, normalized standard deviation and normalized standard deviation of the particle velocity magnitude in the fully developed, particle-laden channel flow The characteristic model values describing diffusion: the energy loss rate of the turbulent kinetic k the restitution power 7r~ k and the turbulence structure parameter C'~ are shown in Fig. energy %, 3. The dissipation loss of the continuous phase and restitution power of the dispersed phase dominate over the respective values of the other phase. 1

1

1

D

0.8

C

D

08

D

C

0.6

0.6

0.6

O.4

0 4

0.4

0.2

0.2

0.2

0

....

0

" ' ' ' . . . . 50 100 150 200 250 300 350 400 450 Energy Loss [m2/s 3]

0

0

1000

2000

3000

4000

~

0.8

5000

Restitution Power [m2/s 3]

6000

0

:

0

0.1 0.2 0.3 04 Turbulence Structure Parameter

0.5

k and the turbulence Figure 3: Turbulent kinetic energy loss rate ck~, turbulence restitution power 7rc~ structure parameter C~ of the dispersed phase (D) and the continuous phase (C) As expected the turbulence structure parameter of the dispersed phase CO decreases near the wall, because of the decreasing ratio V/~D/TrD against the nearly constant turbulence structure ~/~ parameter of the continuous phase C~c. The diffusion properties of the continuous phase are influenced only by drag terms and the viscosity dependent dissipation. The partial restitution power 7r~, dependent on the viscous dissipation rate, dominates the diffusion process of the continuous phase, because it is much higher than c So the solutions of the mean and the turbulent velocity fields of the the corresponding term 7r~.

947

continuous phase correspond to those of a standard k-c model. Particle-laden flow over a backward-facing step

The normalized particle stream-wise mean velocity (Fig. 4) with stream-wise velocity fluctuations indicate that the shear layer of the backward-facing step flow (Fig. 5) is well predicted by the present model, unlike the other models used for comparison. 2.5

Eul.IEul. [2] U o

2

"

2.5

"'~-~

Eul./Eul.[11U . . . .

;'.':~X t ~' ) I

Eul./Lag. U . . . . . . . .

~u,,~u,I21u~:- : ~u,,~o,~, u~ . . . .

2

Eu,/,~g

u ~ .......

ex~o~

1.5

9

o,

..........;:i! ....

0

-0.2

0

0.2 0.4 0.6 x/H=2 U9 ~ 0

t

0.8

." ;Y

1

0

"./

9 9,/

-;'": i:' /,

,:...........//

0 -0.2

~

9

1 0.5

~-~ ";,~ t

0.2 0.4 0.6 x/H=5 UD/uC 9 o

I

t

t

1

0.8

2.5

EuI'.IEul. [i] U D '

~ '

9"'~ " 9

Eul.ILag.UO ....... Exp. U D

1.5

0~ -0.2

.~/:-:::: .,0.2 0.4 / < / 0.6

0

I

1

//~e." o'."i

9

99

o

1

--..'

Eul.IEul.[11 U . . . . . . . . .

2

.,-

4

t

,.

-J

0.8

1

x/H=9 " uD/uC 0

Figure 4: Normalized particle stream-wise mean velocity at several positions behind a backwardfacing step The is 9 modeled stream-wise particle velocity standard deviation (Fig. 5) agrees better with the experimental data inside the shear layer because of the decreasing anisotropy. 2.5

:- . ~ , ' , !/ ~..

2

.... E~ui.}Eul. [2i u D

'

u:

.li'" ;'i

2.5

Eul./Eul. [1] u D Eul./Lag. u o ""

1.5 1

-~":. y ~

2 9

15

-'@-"

0.5 0

0.05

' 0.25

0.1 0.15 0.2 x/H=2 : uD/uC 0

0

0.3

o-o

.

!/~,..: ~ -.S

2

I

,.,

1

..:..';": 0.05

i " 0.1 0.15 0.2 x/H=5 : uD/uC 0

' 0.25

-;"

/:,,;...

0.5 0.3

0

.... IE'u/.)Eul. [2i u ~

j

0

" ": " .,:""

0.05

'

Eul./Eul. [1] u D . . . . .~ul.ILag. u0

',i~ ~- s

1.5

'"' ""

0

..I,o,

2.5

9

9

[

0

'

Eul./Eul. [1] u D Eul.ILag. u O

::.:.~ ~~x~.u .: ,;

1

0.5

; .... I~ull;Eul. [2i u D #

,,,:x~.o~

9

9

9

9

9. 9 9

' 0.1 0.15 0.2 x/H=9 : uD/uC 0

' 0.25

Figure 5: Normalized particle stream-wise velocity standard deviation at several positions behind a backward-facing step Figure 6 shows several profiles of the restitution power of the dispersed and the continuous phase. Looking at the maxima of the restitution power in the field behind the step, the maximum o f restitution reflects the position of th( shear layer. 2.5

2.5

~_-

D - -

D O

2

2

.-

.

C

1.5

1.5 -r"

I 1

1

0.5

0.5

0

0

-

""

1000

.

2000

.

.

3000

.

4000

.

. . . . . . ::3 .,-

5000

x/H=2 ' Restitution P o w e r [m2/s 3]

6000

0

1000

2000

3000

4000

5000

x/H=5 ' Restitution P o w e r [m2/s 3]

6000

0

0

. . . . . 1000 2000 3000

4000

5000

6000

x/H=9 ' Restitution P o w e r [m2/s 3]

k of the dispersed phase (D) and the continuous phase (C) at several Figure 6' Restitution power 7r~ positions behind a backward-facing step CONCLUSION Using this kind of diffusion blending, simulations are able to give better results than conven-

948

tional Euler/Euler formulations for turbulent shear layers of the dispersed phase, including their turbulence production. The prediction of turbulent particle diffusion is limited by the quality of modeling of the momentum diffusion and the turbulence production of the dispersed phase. Compared to classical equilibrium models, which solve an additional differential equation for the energy loss. In the new model the viscosity independent energy loss of the dispersed phase is given by algebraic equations. The turbulence structure parameter remains nearly constant in the dissipative systems examined here. This is because of the nearly constant ratio of restitution and dissipation power of the involved submodels. For the non-dissipative, dispersed phase this parameter decreases and corresponds to a high restitution and locally low momentum diffusion inside the shear layer. This arises because of the additional restitution power without a corresponding loss rate induced by the crossing trajectories. Based on this new coupling scheme of the partial diffusion components, the predicted field values fit the experimental data much better than the commonly used Euler/Euler models. References Crowe C.T., Sommerfeld M. and Tsuji Y. (1998). Multiphase flows with droplets and particles. CRC Press LLC, Florida, Csanady G.T. (1963). Turbulent diffusion of heavy particles in the atmosphere. J. Atm. Sc. 20, 201 Fessler J.R. and Eaton J.K. (1999). Turbulence modification by particles in a backward facing step flow. J. Fluid Mech. 394, 97 Grad H. (1949). On the kinetic theory on rarefied gases. Communications on Pure and Applied Mathematics 2:4, 331-407 Groll R., Jakirlid S. and Tropea C. (2002). Numerical modeling of particle-laden flows with a four-equation model. 5th International Symposium on Engineering Turbulence Modeling and Measurements, Elsevier Science Ltd., 939 He J. and Simonin O. (1993). Non-equilibrium prediction of the particle-phase stress tensor in vertical pneumatic conveying. ASME-FED : Gas-Solid Flows 166, 253 Huber N. and Sommerfeld M. (1994). Characterization of the cross-particle concentration distribution in pneumatic conveying systems. Powder Techn. 79, 191 Jenkins J.T. and Richman M.W. (1985). Grad's 13-moment-system for a dense gas of inelastic spheres. Arch. Ration. Mech. Anal. 87, 355 Jones W.P. and Launder B.E. (1972). The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat and Mass Tran@ 15, 301 Kohnen C. (1997). Uber den Ei'nfluss der Phasenwelchselwirkungen bei t'urbulenten ZweiphasenstrSrnungen und deren nurnerische Erfassung in der Euler- Lagrange Bet'rachtungsweise, Shaker Verlag, Aachen, Germany Kulick J.D., Fessler J.R. and Eaton J.K. (1994). Particle response and turbulence modification in fully developed channel flow. J. Fluid Mech. 277, 109 Schiller L. and Naumann A. (1933). kraftaufbereitung. VDI 77, 318

Uber die grundlegenden Berechnungen bei der Schwer-

Politis S. (1989). Prediction of two-phase solid-liquid turbulent flow in stir'red vessels, PhD Thesis, Imperial College London

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

949

INFLUENCE OF THE GRAVITY FIELD ON THE TURBULENCE SEEN BY HEAVY DISCRETE PARTICLES IN AN INHOMOGENEOUS FLOW

B. Arcen, A. Tani~re and B. Oesterl6

LEMTA - UMR 7563 CNRS, ESSTIN, Universit6 Henri Poincar6-Nancy I 2, rue Jean Lamour, 54519 Vandoeuvre-l~s-Nancy, France

ABSTRACT The motion of heavy discrete particles in a fully developed horizontal channel flow is investigated by means of direct numerical simulation (DNS). The paper explores the influence of the gravity field on the decorrelation time scales of the fluid seen by the discrete particles comparing to those obtained without external forces (Arcen et al., 2004). As expected, the crossing trajectory effect introduced by the presence of gravity induces a decrease of such time scales in all directions. The data extracted from DNS also enable to test the ability of the famous expressions proposed by Csanady (1963) to model the time scales of the fluid seen in an inhomogeneous flow, under conditions which are somewhat far from his theoretical analysis. It is demonstrated that the time scale decrease is about two times more important in the directions perpendicular to the mean relative velocity, a result which is qualitatively conform to Csanady' s analysis although the mean relative velocity and the turbulent intensity are of the same order.

KEYWORDS Csanady's analysis, DNS, decorrelation time scales, fluid seen, crossing trajectory effect, inertia effect.

1. I N T R O D U C T I O N The objective of this paper is to discuss the introduction of the inertia and crossing trajectory effects in the two major gas-particle flow modelling approaches, i.e. in the two-fluid approach (He & Simonin, 1994) or in the framework of Lagrangian dispersed phase modelling, when the carrier phase is inhomogeneous. The crossing trajectory effect, due to the presence of an external force (e.g. gravity), induces a decorrelation of the velocity of the fluid seen by solid particles with respect to the velocity of fluid particles. Usually, the

950 decorrelation time scale of the fluid seen is modelled using Csanady's formulae (1963). By means of a horizontal channel flow DNS coupled with a Lagrangian particle tracking, we examine the use of Csanady' s formulae in an inhomogeneous turbulent flow. To that purpose, we study the influence of the gravity on the time scale of the fluid seen by solid particles, this time scale being a common parameter in the two above-mentioned modelling approaches. For instance, in the frame of Lagrangian approaches, it is necessary to know the velocity vector of the fluid seen, U*, appearing in the equation governing the motion of a solid particle. A stochastic differential equation is often chosen to describe the infinitesimal increment of the fluid velocity seen by a solid particle over a time-step dt [Minier & Peirano (2001), Reynolds (2004)], such as: dU* : A d t + B d W ,

(1)

where A is the drift vector and B is the diffusion matrix related to the fluid seen. W is a vector of independent Wiener processes which have mean zero and variance dt. This stochastic model has to be able to produce statistics of the fluid seen by heavy particles which differ from those following a tracer. The inertia effect, represented by the particle relaxation time v v' and the crossing trajectory effect, are both responsible for this divergence. Such effects are generally reflected by the use of integral time scales of the fluid seen in the modelling of A and B. The crossing trajectory effect is introduced via Csanady's formulae (1963) which are a function of the ratio between the mean relative velocity relative V r and the root mean square of the fluid velocity u ...... . These expressions are restricted to the case of low particle inertia, within the limits of V r / Urm., << 1 and V r / Urm~ >> 1. Due to the continuity effect, the parallel and perpendicular directions to the mean relative velocity are distinguished. Csanady's expressions are applicable if the mean relative velocity is aligned with one of the axes of the reference system. Therefore, the decorrelation time scales of the fluid seen parallel and perpendicular to the gravity T,*g and T~ g , are given by the following expressions: T, *~ ,

,

1 ~

l+

where f l = u

.....

Lr

Tf g ~

1

~

(2)

l+4(flVr /

T L , T L is the fluid Lagrangian decorrelation time scale and Ls is the integral length scale.

Note that the definition of /9 is not clear and its value is not well known. For instance, Minier et al. (2004) assumed that /3 is the ratio between T L and the moving Eulerian time scale T me. For many authors fl is called the time scale reduction parameter which can be modified according to the nature of the turbulent flows, fl is often chosen to be of the order of unity. In inhomogeneous flows such as a vertical turbulent pipe flow, Reynolds (2004) used Csanady's formulae with a time scale reduction parameter equals to 1.5, as did Wilson (2000) in atmospheric turbulence. In order to test a stochastic model in a bluff-body gas solid flow, Minier et al. (2004) supposed fl being equal to unity. Following a similar approach to that used by Csanady (1963) but taking the inertia effect into account, Wang & Stock (1993) gave semi-empirical expressions for estimating T,*g and T_~~ . It has to be noted that in the absence of external forces their expressions satisfy the limits identified in turbulent homogeneous isotropic flows (T* = T L for z p --+ 0 and T* T mE for rp --+ ~,). Using a different approach, as proposed by Deutsch =

(1992), the fluid Lagrangian decorrelation time scale (T '~) can be substituted by the time scale of the fluid seen by solid particles in the absence of gravity (T*) in Eqn. 2 in order to take the inertia effect into

951 account. In constrast, Iliopoulos & Hanratty (2004) determined the decorrelation time scales of the fluid seen without using Csanady' s expressions. They assumed that the ratio of the time scales of the fluid seen to the fluid Lagrangian decorrelation time scales c~ = Ti;~ (y)/T~) (y) is constant throughout the channel width but depends on the direction. Using DNS of a horizontal channel flow, they computed the ratio c~ in all directions at a fixed wall-normal location. Then, from the fluid Lagrangian decorrelation time scales computed by Mito & Hanratty (2002) using DNS, they estimated T~7g along the channel width just by multiplying these fluid Lagrangian time scales by the ratio oe. Their Lagrangian stochastic model adapted to the case of low Reynolds number flows was validated against DNS results. Good agreement was observed although crude assumptions were made. In order to study the inertia and crossing trajectory effects, we examine in the present paper the ratios Ti;g/T~; and Ti;~/T~. Then, these ratios are compared to those modelled using Csanady's formulae adapted to the case of inhomogeneous flows. The simulations were carried out at two fixed values of the + + dimensionless parameter rp g (the superscript + denotes normalization by the wall shear velocity u~ and the kinematic viscosity v ) where g is the gravitational acceleration and for two different dimensionless + + + particle relaxation times z'p . It has to be noted that in a quiescent flow, r p g is the ratio of the terminal velocity to the wall shear velocity.

2. E U L E R I A N - L A G R A N G I A N S I M U L A T I O N O V E R V I E W

2.1 Carrier Phase The Navier-Stokes and continuity equations governing the considered incompressible channel flow are:

Ou, 3t

"t- Uj

Ou, 3xj

Op 1 32ui -- - ~ + - - - 3x i R% 3 X j 3 X j '

3ui = 0 ,

0x,

(3) (4)

where u i are the instantaneous fluid velocity components, p is the pressure term and Re b = Ub8 / V is the Reynolds number. Variables are normalized with respect to the bulk velocity U b and the channel halfwidth b . The subscripts 1, 2 and 3 refer to the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively. The domain size in the streamwise, wall-normal, and spanwise direction is 2.5n~ • 2~ x 1.5n~ and the corresponding grid 192 x 128 • 160, respectively. Computations are operated at Re b = 2800 and the flow rate is kept constant (the Reynolds number based on the wall shear velocity is 184.5). The channel flow is homogeneous in the streamwise and spanwise directions, and periodic boundary conditions are applied in these directions. The second order finite difference DNS solver is based on the model of Orlandi (2000). The time discretisation is semi-implicit, i.e. the non-linear terms are written explicitly with the third-order Runge-Kutta scheme and the viscous terms are written implicitly by using a Crank-Nicolson scheme. In the wall-normal direction, the mesh is stretched according to a hyperbolic tangent law, whereas a uniform mesh is applied in the streamwise and spanwise directions. This time step is smaller than the Kolmogorov time scale which is of the order of unity in wall units. Choi & Moin (1994) suggested that a time step of this order is sufficient to give an accurate prediction of turbulence statistics. The capability of second order finite difference solvers to predict realistic turbulent flow statistics has been shown in many papers, see for instance Choi et al. (1992).

952 2.2 Dispersed Phase

The DNS code was modified to perform Lagrangian tracking of fluid or solid particles. The numerical simulation of solid particle trajectories is restricted to spherical particles smaller than the dimension of the smallest cell ( Ay + = 1 ) and consequently smaller than the smallest Kolmogorov length scale. Therefore, we made use of the point-force approximation. The solid particle volume fraction is assumed to be relatively small and particle-particle interactions are neglected. In addition, considering that the ratio between the particle and fluid density obeys pp / Pl >> 1, the particle equation of motion can be written without taking the added mass, history and spin induced lift forces into account. Consequently, under these considerations and taking the gravitational acceleration g into account, the equation governing the motion of a solid particle is: dv i

u ~ - vi ._

dt

F ..[_ d i 2

rp

LS

__

(5)

gdgz ,

mp

where mp is the mass of a single particle, vi are the particle's velocity components, u~ are the fluid velocity components interpolated at the solid particle's position defined by u~ = u i (Xp (t)), and 6i; is the Kronecker symbol. The aerodynamic forces considered here are the non-linear drag and the Saffman lift force FLs, both of them are corrected for wall effects, r p is expressed in terms of the drag coefficient C D and the magnitude of the relative velocity. C o is computed from Morsi-Alexander's correlation since the particle Reynolds number can exceed unity. In order to well differentiate the crossing trajectory and inertia effects, we chose to characterize the motion of the dispersed phase by the following dimensionless 2 +

quantities" dp, rp (with Vp defined in the Stokes regime as rp

__

+

ppdp ), and g+ The dimensionless 18~ rp .

+g+ for two parameters are summed up in Table I. The simulations were carried out at fixed values of 2"p +

different values of 2-p. The particle equation of motion was time-advanced by using a third order RungeKutta scheme. 640 000 solid particles were homogeneously injected in the channel and their initial velocity was set equal to the surrounding fluid velocity. Statistics on the dispersed phase were started after + a time lag of 4 2-p in order to get results independent of the imposed initial conditions, moreover this time lag is the time necessary for particle statistics (with the exception of the mean concentration) to reach a stationary state. Concerning the smooth wall boundary conditions of the dispersed phase, perfectly elastic collisions were assumed when the particle center was at a distance from the wall lower than one radius. Furthermore, as soon as particles moved out of the computational domain, they were re-introduced via periodic boundary conditions. TABLE 1 SOLID PARTICLES PROPERTIES. Case

dp

"/'+p

A

0.0925

1

0.26

15

+

0.0925 D

0.26

15

+

vp g 1

+

953 During the particle tracking, the time step of the Eulerian-Lagrangian simulation is the minimum between the Eulerian time step and Vp_m~,, / 5 , 7~'p_mi n being the lowest particle relaxation time. The computation of the trajectories of fluid or discrete particles necessitates to interpolate the instantaneous fluid velocity components at the particle's location. A 3D Hermite interpolation, which has the property to build a continuous velocity field with continuous derivatives when a particle is crossing a cell, was chosen.

3. PRELIMINARIES 3.1 D e f i n i t i o n O f The Time C o r r e l a t i o n

The computation of the fluid Lagrangian fluctuating velocity correlations and those related to the fluid seen was carried out according to the following conventional definitions, respectively:

(6)

!.;

(,))u;(x <,+

(7)

where Xf(t) and X p (t) respectively stand for the fluid and solid particle position vectors, u~ is the f

velocity fluctuation defined as u; = u i - u i , where u i and fi-; are the instantaneous and mean Eulerian velocity respectively. The notation (.) stands for ensemble averages based on particles being near the same * wall-normal position at time t. The computation of R0L and RiJ has been carried out from five and four

initial flow realizations respectively, and then averaged in order to avoid biased statistics. The turbulent flow being statistically stationary, we choose for simplicity t = 0. Thus, the present results are function of the time lag v and the initial particle position X(0). From these time correlations, we can determine a quantitative measure of how fast the fluid velocities seen by solid or fluid particles become decorrelated. The decorrelation time is simply the integral of the time correlation from v = 0 to v = oo. Due to the difficulty to numerically calculate such an integral, we defined, assuming that the time correlation has an exponential-like shape, the decorrelation time scale as the time at which the time correlation is equal to -1 e . The investigation of solid particles dispersion in a horizontal channel flow by DNS has led us to think about the reliability of our results, since in such a configuration, the solid particles tend to accumulate at the bottom wall. This means that at large times, the mean concentration is very high in the vicinity of the bottom wall and is approximately zero everywhere else. Consequently, it is not possible to investigate the particle dispersion under fully developed conditions. However, the effects of a developing mean concentration profile have been investigated using a coarser computational grid, and the results show that the non-fully developed character of the particulate phase flow does not affect the statistics presented in this paper about the turbulence seen by the particles. 3.2 T h e D e c o r r e l a t i o n T i m e Scales o f the F l u i d

The fluid Lagrangian decorrelation time scales are key parameters in the simulation of gas-solid flows since they represent the limit of the decorrelation time scales of the fluid seen in the case of zero inertia particles without any external forces. This is the reason why we present here the fluid Lagrangian decorrelation time scales calculated by DNS. These time scales are compared with those obtained by Choi

954 et al. (2004) and Iliopoulos et al. (2003) by spectral DNS. From Figure l(a), it can be observed that the comparison of T~ + with the DNS data of Iliopoulos et al. (2003) and Choi et al. (2004) is quite

satisfactory, even if some little differences can be noted for y + > 75. As indicated in Figure l(b), the present estimation of T2~+ is consistent with the DNS data. The integral time scale T3~+ is presented in Figure 1(c), the agreement with the DNS data is very good except near the channel centre where some discrepancies are noticed with the data of Iliopoulos et al. (2003). 100

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4. I N F L U E N C E OF THE G R A V I T Y FIELD ON THE DISPERSION This section is devoted to the decorrelation time scales of the fluid seen by heavy particles computed in the presence of a gravity field. Results have been obtained for two sets of particles and for two different values of the dimensionless gravitational acceleration. A comparison with the results of Arcen et a/.(2004) is carried out in order to clearly show the influence of the gravity on the time scales of the fluid seen. 4.1 D i s p e r s e d P h a s e Statistics

The dispersed phase statistics were obtained by time averaging over particles located in a same wallparallel slice. Only statistics for which a significant amount of data was collected are presented. In Figure 2, the mean relative velocity, i.e. Vr, i = (V i --U;), is presented as a function of y+. The bottom wall is located at y + = 0.

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Figure 2: Mean relative velocity in the streamwise and wall-normal directions. Solid lines, case A; dashed lines, case B. Symbols: I-1, case C; 9 case D. The mean velocity of the fluid in the wall-normal direction being zero, only two terms are involved in the mean relative velocity: the mean particle velocity and the drift velocity. As the latter is very small in comparison with the mean particle velocity (not shown here), the mean relative velocity Vr,2 is practically equal to the mean particle velocity. In a quiescent flow, the terminal velocity of a solid particle would be 2"p+g+. From Figure 2, it can be observed that whatever the particle inertia this terminal velocity is reached except for the highest inertia and rp§

= 2 (case D), in which the mean wall-normal velocity is 10%

lower than the theoretical terminal velocity. In addition to the mean wall-normal relative velocity, there is a mean relative velocity in the streamwise direction. This mean relative velocity is approximately zero in the case of low particle inertia with the exception of the near wall region, whatever the value of rp+g+. In the case of high particle inertia (cases B and D), this mean relative velocity is low everywhere in the channel except in the near wall region. The main mechanism responsible for the high value of the mean relative velocity in the streamwise direction is due to wall-particle interactions. Moreover, in the streamwise direction, it can be noticed that an increase in gravitational acceleration induces a higher mean relative velocity whatever the particle inertia. As can be seen in Figure 2, for low particle inertia, the mean relative velocity is aligned with the wall-normal axis except in the near wall region. In contrast, for high particle inertia the mean relative velocity vector is inclined with respect to the coordinate axes. However, w e c a n observe that [Vr.2l> ]Vr,~l almost everywhere in the channel. Therefore, in the remainder of the

paper we will consider that the crossing trajectory effect is only due to the mean relative velocity in the wall-normal direction.

4.2 Decorrelation Time Scales of the Fluid Seen In this section, the DNS computed decorrelation time scales of the fluid seen are compared to the predictions which would be obtained from the famous formulae of Csanady (1963). Taking into account only the mean relative velocity in the wall-normal direction, the following extended expressions of the decorrelation time scales of the fluid seen in the streamwise, wall-normal and spanwise directions may be proposed from Csanady's analysis, keeping in mind, however, that his analysis is restricted to isotropic turbulence with Vr, 2 / U2...... >> 1, an assumption which is not true in our case where Vr, 2 / U2,rm s ~- 1 to 3 :

956

rl g=

TIL , T~g'-" T2L , r3~ = T3L , I 1+4 ( ~l E2' / 2 I 1 + ( ,82 g r 2' / 2 I 1 + 4 / /~3 E 2 / 2 I~2, rms l'12, rms U2,rms

(8)

'

where fli = T'~ii[ TiimE ' T'mEii being the integral time scales of the fluid calculated in a frame moving at the local mean velocity; this ratio, determined by the present DNS (not shown here), evolves between [0.45, 1] along the channel width and whatever the directions, u 2..... is the root mean square of the fluid velocity in the wall-normal direction. Figure 3 displays the decorrelation time scales of fluid seen by heavy particles in the presence of a gravity field. These time scales are normalized by the fluid Lagrangian time scales as well as by the time scales of the fluid seen computed in the absence of gravity by Arcen et al. (2004). These two different normalizations enable to examine whether the formulae based on Csanady' s analysis can be applied for high inertia particle and for Vr, 2 = U 2 ..... . As expected, increasing the gravity acceleration induces a decrease in the decorrelation time scales of the fluid seen whatever the particle inertia. The decrease due to crossing trajectory effect is of the same order in the streamwise and spanwise directions, whereas it is much less important in the wall-normal direction. This is in accordance with the study of Csanady (1963) who showed that the decrease of the time scales of the fluid seen is less important in the direction parallel to mean relative velocity due to the continuity effect. From our DNS results, we can observe that the decrease in the direction parallel to the principal mean relative velocity is twice less important than in the perpendicular directions. For low particle inertia (cases A and C), it can be observed from the left plots in Figure 3 that the ratios T/7g / Tii (where T/i "-- T.iiL or T/i = T/7 ) are almost constant except in the near wall region. No similar trend is observed for high particle inertia (cases B and D, fight plots), where these ratios cannot be considered as constant. As can be seen from Figure 3, there is some disagreement between the DNS data and the values predicted by Csanady' s formulae, particularly in the wall-normal direction. Whatever the particle inertia, the predicted decrease of the time scales of the fluid seen (Eqn. 8) is in acceptable accordance with the computed ones in the streamwise and spanwise directions. However, there is a significant discrepancy in the wall-normal direction. According to the DNS results, it is possible to check whether Csanady's expressions give better results if the fluid Lagrangian decorrelation time scales in Eqn. 8 are substituted by the time scales of the fluid seen by solid particles in the absence of gravity. From Figure 3, the conclusion is not straightforward, since it is seen that using the fluid Lagrangian decorrelation time scales in Csanady' s expressions would provide better results in the streamwise direction, whereas in the spanwise direction it would be better to use the times scales of fluid seen in the absence of gravity.

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Figure 3" Decorrelation time scales of the fluid seen. Left plots: Case A and C. Right plots: Case B and + g+ = 1" dashed lines, Eqn. 8 for rp+ g+ = 2. Right triangles: normalization D. Solid lines, Eqn. 8 for ~'p +

+

by T/~" squares" normalization by T/~. Symbols filled in grey correspond to Z'pg = 1 and empty symbols to ~'p+ g + = 2 . 5. C O N C L U S I O N The influence of gravity upon the decorrelation time scales of the fluid seen by solid particles has been investigated by means of DNS of a fully developed turbulent channel flow in a horizontal configuration. The presence of a gravity field induces a mean relative velocity in the wall-normal direction whereas the inhomogeneity of the flow induces a mean relative velocity in the streamwise direction. The streamwise

958 mean relative velocity being lower than the wall-normal mean relative velocity, the analysis of the crossing trajectory effects has been carried out neglecting Vr., . As expected, the crossing trajectory effect induces an anisotropic decrease in the decorrelation time scales. This decrease is enhanced as the gravitational acceleration increases. In accordance with Csanady's analysis, owing to the continuity effect, the time scale reduction is higher in the directions perpendicular to the mean relative velocity than in the parallel direction. The DNS results suggest that the time scale decrease is about two times more important in the directions perpendicular to the mean relative velocity. This result is qualitatively conform to Csanady's analysis although Vr, 2 [ U2...... 1 to 3. Moreover, in the case of low and high particle inertia, the ratios =

T~;~/T,; and T~;g/T~ which quantify the decrease have been seen to be quite similar in the perpendicular directions. Although the DNS results reflect quite well Csanady' s theory, the modelled decorrelation time scales using Csanady' s expressions do not agree well with the DNS data. The discrepancies are seen to be more important in the wall-normal direction. We think that further investigations are necessary in order to improve Csanady' s formulae for inhomogeneous flows, especially in the wall-normal direction.

References Arcen B., Tani~re A. and Oesterl6 B. (2004). Numerical investigation of the directional dependence of integral time scales in gas-solid channel flow. 5 th International Conference on Multiphase Flow, ICMF'04, Yokohama, Japan, Paper No. 297. Choi H. and Moin P. (1994). Effects of the computational time step on numerical solutions of turbulent flow. J. Comput. Phys. 113, 1-4. Choi H., Moin P. and Kim J. (1992). Turbulent drag reduction : studies of feedback control and flow over riblets. Rep. TF-55. Department of Mechanical Engineering. Stanford University. Choi J., Yeo K. and Lee C. (2004). Lagrangian statistics in turbulent channel flow. Phys. Fluids 16, 779793. Csanady G.T. (1963). Turbulent diffusion of heavy particles in atmosphere. J. Atmos. Sci. 20, 201-208. Deutsch E. (1992). Particle dispersion in stationary isotropic homogeneous turbulence calculated by large eddy simulation. EDF Report n~ Laboratoire National d'Hydraulique, Chatou, France. He J. and Simonin O. (1994). Numerical modelling of dilute gas-solid turbulent flows in vertical channel. EDF Report n~ Laboratoire National d'Hydraulique, Chatou, France. Iliopoulos I. and Hanratty T.J. (2004). A non-Gaussian stochastic model to describe passive tracer dispersion and its comparison to a direct numerical simulation. Phys. Fluids. 16, 3006-3030. Iliopoulos I., Mito Y. and Hanratty T.J. (2003). A stochastic model for solid particle dispersion in a non homogeneous turbulent field. Int. J. Multiphase Flow 29, 375-394. Minier J.P. and Peirano E. (2001). The PDF approach to polydispersed turbulent two-phase flows. Phys. Rep. 352(1-3),1-214. Minier J.P., Peirano E. and Chibbaro S. (2004). PDF model based on Langevin equation for polydispersed two-phase flows applied to a bluff-body gas-solid flow. Phys. Fluids. 16, 2419-2431. Mito Y. and Hanratty T.J. (2002). Use of a modified Langevin equation to describe turbulent dispersion of fluid particles in a channel flow. Flow Turbul. Combust. 68, 1-26. Orlandi P. (2000). Fluid Flow Phenomena. A numerical toolkit, Kluwer Academic Publishers. Reynolds A.M. (2004). Stokes number effects in Lagrangian stochastic models of dispersed two-phase flows. J. of Colloid and Interface Science 275, 328-335. Wang L.P. and Stock D.E. (1993). Dispersion of heavy particles by turbulent motion. J. Atmosph. Sci. 50, 1897-1913. Wilson J.D. (2000). Trajectory models for heavy particles in atmospheric turbulence: comparison with observations. J. of Applied Meteorology 39, 1894-1912.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

959

MODELLING TURBULENT COLLISION RATES OF INERTIAL PARTICLES L. I. Zaichik 1, V. M. Alipchenkov ~, and A. R. Avetissian2 1Institute for High Temperatures of the Russian Academy of Sciences Krasnokazarmennaya 17a, 111116 Moscow, Russia 2All Russian Nuclear Power Engineering Research and Development Institute Cosmonaut Volkov Str. 6a, 125171 Moscow, Russia

ABSTRACT The objective of the paper is to present a new statistical model for predicting collision rates of inertial particles immersed in turbulent flow. This model is valid over the entire range of particle inertia (from the zero-inertia to the high-inertia limit) and accounts for two mechanisms influencing the collision rate, namely, the particle relative motion induced by turbulence and the accumulation effect that leads to an additional enhancement to the collision rate. The model is applicable to near-homogeneous two-phase turbulent flows with colliding or coalescing particles.

KEYWORDS Turbulence, two-phase flow, collision rate, coalescence, kinetic equation, statistical model.

INTRODUCTION The rate of coagulation due to collisions of solid particles or liquid droplets in multiphase turbulent media is of importance in many environmental and industrial processes. Examples of turbulence-induced particle collisions include the formation of rain drops in clouds, precipitation of aerosols, agglomeration of fme powders in gas flows, pulverized coal combustion, dust and spray burners, pneumatic conveying, and so on. Because of the practical interest, a number of theoretical studies of the collision rate induced by turbulence have been performed. Relatively simple solutions to this problem may be derived with the assumption of homogeneous and isotropic turbulence. Two solutions are most familiar in the literature, corresponding to the limiting cases of zero-inertia and high-inertia particles. The first solution is valid for fme zero-inertia particles, whose collision rates are determined only by their interaction with smallscale energy-dissipating turbulent eddies (Saffman & Turner, 1956). The second solution relates to coarse high-inertia particles, whose motion is statistically independent and governed by the interaction with large-scale energy-containing turbulent eddies (Abrahamson, 1975). However, of the most

960

practical interest is the case of particles of intermediate inertia, when the ratios of the particle response time to the turbulence microscale and macroscale are finite. In this situation, it is necessary to take into consideration the interaction of particles with the overall spectrum of turbulent eddies as well as to account for the correlation of the motion ofneighbouring particles and their preferential concentration. The collision kernel is defined as the product of the half-surface of a collision sphere by both the mean radial relative velocity, (] wr ]), and the particle radial distribution function, F, fl : 2nd2 ([ %(d)[)F(d)

(I)

where d is the radius of a collision sphere, which is equal to the particle diameter for identical particles. It is clear from ( 1 ) that the turbulence-induced collision rate is governed by the mean relative velocity as well as by the radial distribution function. Consequently, the interaction of particles with turbulent eddies causes two statistical mechanisms that contribute to the collision rate: (i) the relative velocity between neighbouring particles (the so-called turbulent transport effect) and (ii) the non-uniform particle distribution (the accumulation effect). The accumulation effect is measured by the particle radial distribution function that is the probability of observing a particle pair normalized by the corresponding value in a uniform suspension.

TWO-POINT

PDF

MODEL

Let us first consider the two-point probability density function (PDF) model. This model is suitable for predicting two-particle statistics and particle-pair dispersion in homogeneous isotropic turbulence. The particle volume fraction is kept small enough so that the two-phase system is quite good within the dilute limit; therefore, only double collisions between particles are taken into consideration. The density of particles is assumed to be much more than that of the carrier continuous phase (in this case, the drag force acting on a particle by the surrounding fluid flow is only of importance), and the particle size is small as compared to the Kolmogorov lengthscale. Equations for two separate particles provide the equations describing the relative motion of a particle pair drp

Au(rp, t ) - w p

dw p

dt

P'

dt

-

(2)

rp

where rp and Wp are the separation distance and the relative velocity between two particles, rp is the particle response time, and Au(rp,t) designates the increment in velocities at two points in which the particles are located. To proceed from stochastic equations (2) to the statistical description of the relative motion of two particles, the pair PDF is introduced Pw : ( P w )

(3)

: ( 5 ( r - r p (t))5(w - w p (t)))

The pair PDF, Pw(r,w,t), describes the probability of finding a pair of particles separated by a distance r , with a relative velocity w, at time t. Differentiating (3) with respect to time and accounting for (2), we derive the following equation for the pair PDF: ~+ cTt

%

+ c~rk

rp

~w k

(4) rp

cNt,k

961

To determine the correlation (Au'kPw) that describes eddy-particle interactions, the fluid relative velocity field is modelled by a Gaussian random process with known two-point correlation moments. Then, using the functional formalism and the Furutsu-Donsker-Novikov formula for Gaussian random functions, we can derive the following expression for the correlation between the fluid velocity increment and the particle-pair probability density:

, = -Si, (Auip.,)

Z'p.LI DStj (~p rp/,S~k c3AUj C3pw, ~DS v = ~~S+ v.A U k ( fr--~,j< + rpgr <1 ~l'j j + - -2 - - Dt Owjw _ O.rk (~wj Dt &

fr =1--~- q~r~(r)exp -

Tp

d r , gr = ~ - -

~'p

fr, frl = _----f TLr(r)rex p -

Tp

Here S o -(Aui(r)Auj(r)) and So.k =(Au,(r)Auj(r)Auk(r))

~'p )

c3S~. +_____ &~,,k c~rk &k

lr = g r -

are the second-order and third-order fluid

velocity structure functions. The coefficients fr, gr, frl, and lr quantify a response of a pair of particles, separated by the distance r, to velocity fluctuations of the turbulent fluid, T L~(r Ir) symbolizes a Lagrangian autocorrelation function that descn'bes the velocity increments of two fluid elements separated initially by the distance r, and TLr ~--I : T L~(r)dr" When using the exponential approximation. ~Lr "--exp(-r /TL~) , the response coefficients become as follows:

.fr--

zLr

-rp+TL----~ .

gr =

Z[r r p ( r p + TLr )

' frl =

ZL2r

T~, (rp + Lr)

lr:

ZL3r

rp(rp + TLr) 2

Substituting (5) into (4) provides the following kinetic equation for the two-point PDF of the particlepair relative velocity distribution in homogeneous isotropic turbulence:

cgP~ ?'Pw 1 c g ( A U k - % ) P ~ s DS~ 02p ( f r 02P t32P~ OAU c32p "] Wk + 4w - S ~ ~-------------~-W +g r +1r " w ) (6) Ot cTrk rp 3w k 2 Dt Ow,Owk rp c3wiOwk ~r.c3wk c~rk 8w~w

~+

Equation (6) describes the convective and diffusive transport in phase space (r,w) and resembles the one-point kinetic equation (e.g., Reeks, 1991; Zaichik, 1999; Derevich, 2000). However, the onepoint and two-point kinetic equations bear a superficial resemblance, because the former deals with the one-particle PDF and hence does not take into consideration the spatial correlation of the motion of two particles. In contrast, the two-point statistical model allows for the spatial correlation between the velocities of particle pairs and thereby can predict the effect of clustering. When DS v / Dt = AU, = 0, (6) reduces to the two-point PDF equation presented previously in Zaichik & Alipchenkov (2003). Equation (6) generates a set of balance equations governing the pair concentration, momentum, particulate stresses, or any appropriate statistical two-point moments of the relative velocity PDF. By this means the equations describing the particle-pair density, the mean relative velocity, and the secondorder two-point structure function are written as

or

--+ Ot 0~

c3W~

dSp,k

~ + W k ~ = - ~

dt

dr k

Ork

cr~

=0

Ork

(7)

AU,-~ +.

.

rp

.

D~,,k ~lnF .

rp

c3rk

(8)

962

~Sp ~t

v + Wk

~Sp Ork

1 cTSp Vk

V+__~ F

Ork

+ frl

DSv

OWa

= -(Sp + grS, k ) ~ - ( S p

--~

,k

Ork

OW, + )~ .,k g~S,k Ork

+lr(S,k~aU, 9+ Sak ~AU,) + ~v (frS,j -Sp ) q---(A~_~) 2 -Sp ) &k

Ork

,a

rp

(9)

'J

1

To close the infinite equation set stemming from (6) at the second-order closure level of (7)-(9). we invoke a gradient algebraic approximation for the triple particle fluctuating velocity correlations. This approximation follows from the corresponding differential equation for the third moments by neglecting time evolution, convection, and generation due to mean velocity gradients as well as by using a quasiGaussian approximation for the fourth-rank correlations (Zaichik & Alipchenkov, 20031)

-3 k

Spijlr = ! ( D r

OSp jk ""

4r-

dr,

OSp ,k

Drpj" Or.

+

dSp ,a.)

(10)

Drp~ Or.

PARTICLE SUSPENSION IN ISOTROPIC T U R B U L E N C E

In this section, the two-point PDF model is used for predicting pair dispersion in a steady-state suspension of particles immersed in homogeneous, isotropic, and stationary turbulence. In isotropic turbulence, due to spherical symmetry, the pair relative velocities and density distributions are independent of the orientation of the separation vector r and may be only dependent on r - ] r ]. Moreover, the mean convective transport in the fluid is supposed to be absent ( AU = 0 ), and the total number of particles is not changed in time. The latter infers that the balance between the net radial relative inward and outward fluxes takes place, and, therefore, the mean relative velocity, W,, is equal to zero. By this means the set of Eqns (7)-(9) along with (10) is constricted to the following system: w

_

2(Spu-Sp F

St 2 d

St2 3F2

F(~

+grS1,)

_

) dSplt

-

-

dlnF=0

(11)

" + dF + (Sp ll + g~Sll) dF

+Stfr,F

tr2r('~P + grSu) dP + 2 u

VF(Sp

(12)

+2F(frS,l-~ll)=O

~r,~ll

,,.

+ gfi,,,,)(Sp -Sp,,,) u

+ StfrlF-23r {d~rr(--2----dSnn]+2~rl TLrS-lI~r ) r~LrSnn(~l--Snn)]}+2r(frSnn--S--pnn):Or Here and hereinafter the overbar stands for normalization by the Kolmogorov microscales, St

(13)

= rp

/ r k iS the

Stokes number that specifies the particle inertia, r k is the Kolmogorov time microscale, and S p u and Sp.. are the longitudinal and transverse components of the second-order particle velocity structure function Spv"

963

Thus, in isotropic turbulence, the two-point model amounts to solving three nonlinear ordinary differential equations involving the radial distr~ution function and the second-order longitudinal and transverse structure functions. Equation (1 l) expresses the balance between the turbophoretic and diffusion forces in the separation direction between two particles. Relevant boundary conditions for (11)-(13) are given by

dSp.

dSp

~ = d~

d~

~" =0 for ?-=d

(14)

Spu = f,.Su, Sp,,,, = frS,,,,, F = 1 for V ~ oo

(15)

In (14), the case of ~ - 0 corresponds to elastic collisions of the so-called ghost (zero-volume) particles, which are flee to occupy any space in the suspension without being excluded by other particles (Reade & Collins, 2000; Zaichik & Alipchenkov, 2003). It seems likely that the approximation of ghost particles can provide prediction results, which are bound to be close to those obtained by Wang et al. (1988) and Wang et al. (2000) when using Scheme 1 for collision counting. In this collision-counting scheme, particles were allowed to overlap in the system and were not removed from the system after collision. The case of d = d appears to be appropriate for displaying the hard-sphere elastic collision model used by Sundaram & Collins (1997) and Reade & Collins (2000). Relations (15) point to the fact that the particle velocities become nearhomogeneous at large separation, whereas the particles are randomly distributed. The Eulerian longitudinal and transverse velocity structure functions of the fluid as well as the Lagrangian one-point and two-point timescales are given by the following approximations (Zaichik et al., 2003): 2(Re~+C1)

nt-(Cr-2/3)k ~2R~e~)'

'~-Sll ~ l

~r m

k

1

1

A: + (A:"'3)~ + ~ '

Jnn--~-~)

I 11/2

Re~= 15u'4 ~,,

,C=2,

4C?-z/3

2 Re~

'

151:ZCo~,

k = 2 0 , C0w=7, C 1=32,A 1=51/2,A 2=0.3

Equations (111)-(.13) along with boundary conditions (14) and (15) are solved numerically, and the results obtained are compared with DNS computations. In order to elucidate how the particle size affects pair dispersion and preferential concentration, we will also compare the prediction results obtained when using (14) with d = 0 and d = d . Figure 1 demonstrates the ratio between the particle transverse and longitudinal structure functions for Y = 1 versus the Stokes number. For fluid elements as well as for zero-inertia particles, this ratio is two. In accordance with the DNS of Wang et al. (2000), the predicted ratio of Spn n / Sp ~l that corresponds to zerovolume particles ( ~ = 0 ) drops quickly and monotonically towards one as St increases. The transverse-tolongitudinal structure function ratio predicted for finite-volume particles (~-= l) demonstrates a nonmonotonic variation with Stokes number: Sp,~ / Sp 11 decreases initially with St, reaching a minimum at St ~ 2, before increasing and finally approaching unity. In Figure 2 we compare the mean radial relative velocities predicted from (11)-(15) with those simulated by Wang et al. (2000). Under the assumption that the PDF of the relative velocity is Gaussian, the mean relative velocity magnitude is defined in terms of the longitudinal structure function as ([ w,-I) = (2S-pu / ~)1/2 Figure 2 illustrates the influence of particle inertia on (I w,-[) over a wide range of Stokes numbers. As is clear, the behaviour of ([ wr [), with respect to St, is characterized by the presence of a maximum. The initial rise in ( ~r [) is attributable to a decrease in the correlation of particle velocities

964

Sp,,,,(1)/Sp~t(1) 2"0t

'

'

o,, 87

1.6

" 9

1 " 2 ~ o o I ', o _2...... g'~' 0.8t '....,-'",i. 4,5,6 .

i

,

0

.

1

5

.

.

.

,

.

.

i

~ ,

10

~ !

15

St

Figure 1. Transverse-to-longitudinal structure function ratio for ?-- = 1' 1-3 - predictions for ghost particles ( ~ = 0 )" 4-6 - predictions for finite-volume particles ( h- = 1 ); 7-9 - DNS by Wang et al. (2000); 1, 4, 7 - Re x =24; 2, 5, 8 - Re x =45; 3, 6, 9 - Re x =75. (I ~r I) o 8

OO'o~$

10~

10-161 ~' _

~ ~_ ~_ 10~ 10~ 102 St Figure 2. Influence of particle inertia on the mean relative velocity magnitude for ?- = 1" 1-3 - predictions with ~ =0" 4-6 - predictions with ~ = 1" 7-9 - DNS by Wang et al. (2000); 1, 4, 7 - Re x =45; 2, 5, 8 - Re x =58; 3, 6, 9 - Re x =75. with rp. The subsequent decay of (] Wr [) beyond the maximum results from a decrease in the particle fluctuating velocities, since the particles become more sluggish and less responsive to the fluid turbulence. From Figure 2, it is also seen that using ~ = 0 in boundary conditions (141) leads to slight smaller values of mean relative velocities at small St as well as to better agreement with the DNS date as compared to the case of using d - 1. However, a difference in the mean relative velocities of zero-volume and finite-volume particles takes place only for relatively small Stokes numbers and vanishes at large St. Figure 3 shows the influence of particle inertia on the radial distribution function for separations and Reynolds numbers, which correspond to the DNS data of Sundaram & Collins (1997) and Wang et al. (2000). As expected, in the limiting cases of zero-inertia and high-inertia particles, the concentration field is statistically uniform, and therefore the radial distr~ution function is equal to unity. In accord with the computations, the predicted radial distribution function goes through a peak as the particle inertia time increases. Thus, there exists a critical particle response time which results in maximum preferential concentration. The value of this critical response time is of the same order as the Kolmogorov timescale, that is, the clustering is more considerable when tuning the particle parameters to the flow Kolmogorov scales. Figure 3 demonstrates a qualitative agreement between the predicted and simulated particle distributions, although the predicted maxima of F are slightly shifted towards particles with larger inertia. Figure 3 also exhibits that the effect of finite particle size results in decreasing preferential concentration.

965

F

i

,

!

|

----O--

9 lO

3/1 .

..

.

5

--m--6

4

!

....

-

-

1~176

,

0

I

I

2

4

|

I

J

6

St

Figure 3. Influence of particle inertia on the radial distribution function: l, 2, 3, 4 - predictions; 1, 2 - g = 0 ; 3 - ~ =0.36; 4 - ~ = l; 5 - DNS by Sundaram & Collins ( 1997); 6 - DNS by Wang et al. (2000); l, 3, 5 - F =0.36, Re~ =54; 2, 4, 6 - F =1, Re~ =58. Finally let us consider the effect of particle inertia on the collision rate that is derived from the relationship

fl = (8rcSr ,(d))~/2 d2F(d)

(16)

Figure 4 represents the collision kernel normalized with the Saffman-Turner one for free zero-inertia particles and compares predictions with the DNS data of Wang et al. (2000). As is seen, the model being developed properly captures the crucial trends of the DNS results, although the predicted maxima of f l , much like F in Figure 3, are slightly shifted towards particles with larger inertia. The collision kernels, predicted when using E = 1 in boundary conditions (14), are slightly greater in values and worse correspond to the DNS than those obtained for zero-volume particles. This fact apparently confirms that, as it has been mentioned above, the approximation of ghost particles is a quite good assumption for the treatment of DNS performed by Wang et al. (2000) using Scheme 1 for collision counting. It is also important to emphasise that the neglect of the accumulation effect leads to considerably less values of the collision rates of low-inertia particles than those given by DNS.

1,~L ~,."'-.]. ,2 3

~;

[] ll

J

10 5

0

10

20

30

St

Figure 4. Influence of particle inertia on the collision kemel for E =1: 1-3 - predictions with ~ = 0 ; 4-6 - predictions with E = 1 ; 7-9 - predictions with neglecting the accumulation effect; 10-12 - DNS by Wang et al. (2000); 1, 4, 7, 10 - Re~ =45; 2, 5, 8, 11 - Re~ =58; 3, 6, 9, 12 - Re~ =75.

966

DROPLET AGGLOMERATION DUE TO COALESCENCE

A major motivation for developing the collision model described in the preceding sections is the simulation of the droplet size evolution due to coalescence in aerosol reactors. Therefore, as an example of engineering application, we employ the collision model to evaluate the coalescing droplet size in a turbulent channel flow. We restrict the consideration to the channel core where the nearly homogeneous and isotropic flow conditions allow the application of the collision model developed. The droplets are regarded as a mono-size suspension, and, consequently, the droplet size growth is treated as a sequence of the formation of doublets due to the loss of singlets. In such an interpretation of the agglomeration process, the depletion of the number droplet concentration (number of droplets per unit volume) along the channel axis is governed by the equation dN

kN 2

dx

2

U. . . .

(17)

where U is the mean axial velocity of the carrier gas, and k is the coalescence kernel. The coalescence kernel incorporates both the kinematic collision rate caused by fluid turbulence and the effect of interparticle interactions (e.g., van der Waals attraction and hydrodynamics) which are responsible for the collision efficiency. In what follows, the collision efficiency is assumed as unity, and hence the coalescence kernel is inferred to be equal to the collision one, k - ft. Moreover, because coalescence does not result in a change of the volume fraction, the following equation relating the number droplet concentration to the droplet diameter takes place: N d 3 = Nod 3

(18)

where the subscript 0 denotes the inlet values. Let us consider the agglomeration process of droplets in the flow conditions, which correspond to DNS performed by Chen et al. (1998). In these simulations, the Reynolds number, Re~, at the channel centreline based on the Taylor length microscale and the turbulence intensity, averaged over the three coordinate directions, was 29.7. Figures 5 and 6 show the performance of Eqn(16) for the prediction of the collision rate at the channel core. As is seen, both the droplet-to-fluid density ratio and the droplet diameter significantly influence the collision kernel. They affect the collision kernel because Pd / P / and d affect the droplet inertia increasing the particle response time, r p. When the density ratio decreases, the particle response time decreases as well and, according to Saffman & Turner (1956), fl becomes independent of Pd / Py" For comparison, the results calculated from the Saffman-Turner and Abrahamson theories are also

plotted in Figure 6. It is clear that the Saffman-Turner collision rates, which do not take into account the droplet inertia, underestimate the DNS results and approach those only for the smallest droplets. Conversely, the Abrahamson theory, which does not take into consideration the velocity correlation between two colliding droplets, overestimates the DNS date. The collision rates based on the present model are in good agreement with the direct simulations. Thus, we can draw the conclusion that the model advanced for interparticle collisions in homogeneous isotropic turbulence may be successfully applied to predict droplet coalescence in turbulent channel flow. Figure 7 displays droplet size distribution obtained from Eqn (17) along with (18) for d o -1 lam. Three different curves denote results based on the collision rates given by Eqn (16), Saffman & Turner (1956), and Abrahamson (19751), respectively. As one might expect, the Saffman-Tumer and Abrahamson collision theories predict reasonable sizes of coalescing droplets only in the limiting cases of small and large droplets.

967

fl,

cm3/s

9 2

1r

10.s

10-6

. . . . . . . .

~

10'

. . . . . . . .

I

102

103

"Od/'Of

F i g u r e 5. Effect o f the droplet-to-fluid density ratio on the collision kernel ( d = 9.96 lam ): 1 - model prediction, 2 - DNS by Chen et al. (1998).

fl,

cm

3/s

9 4

....... . f - " ' " ....... """

10.3 ....-.-"

104 10.5 ............ """

.

p s

s

3

10 .6

10.7 100

9

....

.........t~""

" ,

,

J

101

d,l~m

F i g u r e 6. Dependence o f the collision kernel on the droplet diameter ( P a / ,or =713): 1 - present model, 2 - Saffman & Turner (1956), 3 - Abraharnson (1975), 4 - DNS by Chen et al. (1998).

, m . . . . . . . .

|

. . . . . . . .

i

. . . . . .

10-3

10.5

10~62

(~l 1

10~

,~c,rn

Figure 7. Variation in the droplet size along the channel centerline: 1 - present collision model, 2 - Saffman - T u r n e r theory, 3 - A b r a h a m s o n theory.

968

SUMMARY A statistical model for predicting the collision rate of inertial particles in homogeneous isotropic turbulence was developed. This model is based on a kinetic equation for the two-point PDF of the particle-pair relative velocity distribution. It accounts for two mechanisms influencing the collision rate: (i) the particle relative motion induced by turbulence and (ii) the preferential concentration that leads to an additional enhancement to the collision kernel. The model presented is valid over the entire range of Stokes numbers (,from the zero-inertia to the high-inertia limit). The effect of particle size on the relative fluctuating velocities, the radial distribution function, and the collision rate is investigated. A comparison between predictions obtained for zero-volume (ghost) and finite-volume particles is performed. On the basis of comparisons with DNS results, it is concluded that the model presented is able to capture the main t~atures of the collision rate of inertial particles in homogeneous isotropic and nearaxis channel turbulent flows. This collision model appears to be applicable to the evaluation of the turbulence-induced agglomeration process in aerosol reactors.

ACKNOWLEDGMENTS This work was supported by the Russian Foundation of Basic Research (grant numbers 03-02-16830 and 04-02-08190).

REFERENCES Abrahamson J. (1975). Collision Rates of Small Particles in a Vigorously Turbulent Fluid. (7hem. Engng. Sci. 30:11, 1371-1379. Chen M., Kontomaris K., and McLaughlin J.B. (1998). Direct Numerical Simulation of Droplet Collisions in a Turbulent Channel Flow. II. Collision Rates. Int. J. Multiphase Flow 24, 1105-1138. Derevich I.V. (2000). Statistical Modelling of Mass Transfer in Turbulent Two-Phase Dispersed Flows. I. Model Development. lnt. J. Heat Mass Transfer 43, 3709-3723. Saffman P.G. and Turner J.S. (1956). On the Collision of Droplets in Turbulent Clouds. d. FluidMech. 1, 16-30. Reade W.C. and Collins L.R. (2000). Effect of Preferential Concentration on Turbulent Collision Rates. Phys. Fluids 12:10, 2530-2540. Reeks M.W. (1991). On a Kinetic Equation for the Transport of Particles in Turbulent Flows. Phys. Fluids A 3:3, 446-456. Sundaram S. and Collins L.R. (1997). Collision Statistics in an Isotropic Particle-Laden Turbulent Suspension. Part 1. Direct Numerical Simulations. J. FluidMech. 335, 75-109. Wang L.-P., Wexler A.S., and Zhou Y. (1998). On the Collision Rate of Small Particles in Isotropic Turbulence. I. Zero-Inertia Case. Phys. Fluids 10:1,266-276. Wang L.-P., Wexler A.S., and Zhou Y. (2000) Statistical Mechanical Description and Modeling of Turbulent Collision of Inertial Particles. d. Fluid Mech. 415, 117-153. Zaichik L.I. (1999). A Statistical Model of Particle Transport and Heat Transfer in Turbulent Shear Flows. Phys. Fluids 11:6, 1521-1534. Zaichik L.I. and Alipchenkov V.M. (2003). Pair Dispersion and Preferential Concentration of Particles in Isotropic Turbulence. Phys. bTuids 15:6, 1776-1787. Zaichik L.I., Simonin 0., and Alipchenkov V.M. (2003). Two Statistical Models for Predicting Collision Rates of Inertial Particles in Homogeneous Isotropic Turbulence. Phys. Fluids 15:10, 2995-3005.

Engineering Turbulence Modelling and Experiments 6 W. Rodi (Editor) 9 2005 Elsevier Ltd. All rights reserved.

LARGE EDDY SIMULATION OF THE DISPERSION SOLID PARTICLES AND DROPLETS IN A TURBULENT BOUNDARY LAYER FLOW

969

OF

I. Vinkovic, C. Aguirre, S. Simoi~ns Laboratoire de M~canique des Fluides et d'Acoustique UMR CNRS 5509, Ecole Centrale de Lyon 69131 Ecully Cedex, France

ABSTRACT

In order to study the dispersion of droplets and solid particles in neutral atmospheric surface layers a large eddy simulation (LES) using the dynamic sub-grid scale model of Germano, was coupled with a Lagrangian stochastic model for particle trajectories. The flow dynamics are computed with a LES. The droplets are tracked by solving the modified version of the BBO equation of motion for a small rigid sphere in a turbulent flow. In order to take into account the subgrid-scale (SGS) motion of particles, the velocity of the fluid encountered by the particle is computed with two components. Namely, to the large scale velocity directly computed by the LES we added a fluctuating value obtained by a one particle, one time-scale Lagrangian stochastic model. A final coalescence/breakup model is used to take into account the interactions of droplets. Tracking droplets with a LES coupled with a stochastic model allows the computation of instantaneous information at the location of the particles, as well as the computation of SGS fluctuations which are essential for inter-particle interactions and chemical reactions. The specificities of this study reside in the use of a LES coupled with a stochastic model for the study of the dispersion of solid particles and droplets.

KEYWORDS

LES, stochastic model, solid particles, droplets, coalescence, dispersion, boundary layer

INTRODUCTION

During the last decades, a good understanding of atmospheric dispersion has been achieved in neutral boundary layers as a result of numerical, laboratory and field investigations. Previous studies have focused on passive, reactive and buoyant scalar dispersion. However, industrial stack

970 emissions contain also particulate matter, as well as inertial particles and droplets. The aim of this research is the modeling and simulation of dispersion from elevated sources of solid particles and droplets in a turbulent boundary layer. To study these problems we use an Eulerian-Lagrangian approach. A LES with the dynamic sub-grid scale model of Germano et aI. (1991) is used to compute the velocity field. The dispersed phase is computed with Lagrangian particle tracking. In addition to this, the LES is coupled with a Lagrangian stochastic model in order to take into account the SGS motion of particles. Finally, the coalescence and breakup of droplets are computed by a stochastic breakup/coalescence model that is an extension of the stochastic model for secondary breakup developed by Apte et al. (2003) Since the pioneering work of Deardorff (1970), LES has become a well established tool for the study of turbulent flows (Lesieur & M~tais 1996), the transport of passive scalars (Xie et al. 2004), the dispersion of reactive plumes (Meeder & Nieuwstadt 2000) as well as the computation of particleladen flows in a variety of conditions (Wang & Squires 1996, Shao & Li 1999). However, since only the motion of the large scales is computed, the effect of the small scales on particle dispersion, motion or deposition must be either modeled separately or neglected (Armenio et al. 1999). On the other hand, Lagrangian stochastic models are a suitable way of describing dispersion (Durbin 1983). Point sources and complex geometries can be treated with this approach without any new assumptions. In addition to this, Lagrangian tracking of particles allows implementation of chemical reactions and inter-particle interactions. In order to couple the stochastic modeling with the LES, the three-dimensional Langevin equation (Thomson 1987) is written in terms of the LES decomposition. This way, the stochastic model is expressed only as a function of the quantities directly computed by the LES. The results of the LES coupled with the stochastic model are compared with the experiments on passive scalar dispersion of Fackrell & Robins (1982). Then, the coupled simulation is applied to solid particle dispersion. The results are compared with the wind tunnel experiments of Nalpanis et al. (1993). Finally, a case of droplet dispersion from an elevated point source in a turbulent boundary layer is computed. Unfortunately, no in-situ measurements or wind tunnel experiments were found and the computed profiles of mean concentration and variances at different distances from the source are compared with the experimental results on passive scalar dispersion (Fackrell & Robins 1982). Two test cases are computed. In the first case there are no interactions between droplets while in the second one we introduced coalescence.

LARGE EDDY SIMULATION A turbulent boundary layer flow is computed using a LES of the incompressible Navier-Stokes equations. The equations governing transport of the large eddies obtained by grid filtering the Navier-Stokes equations are"

Ofti =0 Oxi Ot

~

(1) Ox~

Rer OxjOxj

ogxj

where ui is the fluid velocity, p is the pressure,/)ij is the Kronecker delta, and i = 1, 2, 3 refers to the x (streamwise), y (spanwise), and z (normal) directions respectively. The tilde denotes application of the filtering operation. The filtered Navier-Stokes equations have been made dimensionless using the boundary layer depth 5, the friction velocity u, and the kinematic viscosity u. The Reynolds number in Eqn. 1 is Re, = u,(5/u. The effect of the SGS on the resolved eddies in Eqn. 1 is

971 presented by the SGS stress, ~-i} = u ~ viscosity hypothesis:

-~2d2j. In this work ~-i~ is parametrized using an eddy

r

15 7_r =

where the eddy viscosity is ~,~. :

cZ,~.l~l,

(3)

the resolved-scale strain tensor is defined as l(Og, O~ty) S'J = -2 \Oxj +-~zi

(4)

and 'SI = V/2SijSij is the magnitude of Si3. The filter width/~ is defined as/~ = (/~1/~2/~3) l/a, where /~1, /~2, /~3 are the grid spacings in the x, y and z directions, respectively. The model coefficient C in Eqn. 3 is determined locally and instantaneously with the dynamic SGS closure developed by Germano et al. (1991) and modified by Lilly (1992). The dimensions of the computational domain in the streamwise, spanwise and wall-normal directions are, respectively, lx = 65, ly = 2.75 and lz = 25. The Reynolds number based on the friction velocity and the boundary layer depth is Re,- = 15040. A 72 x 32 x 32 grid is used. The grid is uniform in the xy-planes and stretched in the z-direction by a hyperbolic tangent function. The grid spacings are Ax = 0.0835, Ay = 0.0835 and 0.00255 < Az < 0.0835. The no-slip boundary condition is applied at the wall. On the top of the domain and in the spanwise direction the mirror free-slip and the periodic boundary conditions are applied, respectively. In the streamwise direction, at the end of the domain the wave-radiation open boundary condition is used, Klemp & Wilhelmson (1978). At the beginning of the domain, permanent forcing is applied.

SUBGRID-SCALE

VELOCITY

FLUCTUATIONS

The SGS motion of particles is taken into account by a one particle, one time-scale Lagrangian stochastic model. Generally, the Lagrangian stochastic model is obtained from Reynolds turbulent statistics. Here the coefficients of the stochastic model will be written in terms of filtered and SGS statistics.

Model Description The velocity of a fluid particle is given by: ~

= ~(e(t))+ ~

(5)

vi is the total Lagrangian velocity of the fluid particle in the x~ direction, u~ is the Eulerian velocity at the position 2~(t) of the fluid particle. The tilde denotes the grid filtering, v~ is the Lagrangian fluctuation around the Eulerian large scale velocity ~2i. A three-dimensional Lagrangian stochastic model adapted to the SGS decomposition will be used for modeling v~:

dv~ = (~i(s g, t) + aij(~, t) (vj - gy)) dt + ~j(~, t)vj(t)

(6)

~?j(t) is a Wiener process with zero mean and variance dt, satisfying < ~Ti(t')7]j(t") > = 5~jh(t'-t")dt. The velocity of a fluid element at a time t is given by a deterministic part ~/~ + a~yv~ depending

972 only on the value of the velocity at the moment t - dt and by a completely random part/3ijrb(t ). Since g is a Markovian variable, the subgrid Lagrangian probability density function (PDF) of the fluid velocity, Pc(g, t) satisfies a Fokker-Planck equation. By integrating this equation all the statistical moments of the velocity can be obtained. The LES gives only the filtered Eulerian moments, a~j, ~ij and 7i are determined by relating the stochastic model to the Eulerian conservation equations, using the Fokker-Planck equation and the Eulerian equations for the moments of fluid velocity. We proceed by analogy with van Dop et al. (1985), who determined the coefficients of the stochastic model for a Reynolds decomposition. By defining a subgrid Eulerian PDF 7)E(g; t), as Cook & Riley (1994) we adapt some of the assumptions made by van Dop et al. (1985) and determine a~j,/3ij and 7~ for a SGS decomposition. Finally, the Lagrangian velocity v~ is given by:

dv~ = ~ - ~

Oxj

~~

~ 2

E

3 dt

c~

dt+ v/Cog~rl(t)dt

(7)

where t) is the turbulent kinetic energy, g is the turbulent dissipation rate and Co is the Lagrangian constant. Prom Eqn. 7 each fluid particle has a large-scale velocity component and a SGS component. The large-scale component is computed by interpolating the values of the Eulerian velocity with a tri-linear quadratic Lagrange interpolation method using 27 nodes, while the SGS is obtained by computing the turbulent kinetic energy/), the turbulent dissipation rate g and the tensor ~-/~ in the grid where the particle is positioned. ~-/~is obtained with the SGS closure. In order to obtain/), the turbulent kinetic energy transport equation (Deardorff 1980) is resolved. Results and Validation The coupling between the LES and the Lagrangian stochastic model for the SGS displacement of fluid particles is validated by comparison with the wind tunnel experiments of Fackrell & Robins (1982). They measured profiles of mean concentration, concentration variances and concentration fluxes of a passive scalar spreading from an elevated point source in a turbulent boundary layer. Vertical profiles of the mean velocity and the turbulent kinetic energy are illustrated in Figure 1, showing close agreement with the experimental data. The authors do not provide measurements near the wall. The peak of the turbulent kinetic energy is maybe overestimated. Vertical profiles of mean concentration, concentration fluctuations c 2, and concentration fluxes ~ at five different positions along the plume are computed. Results for the third station are shown in Figure 2. The simulations match well with the experimental profiles.

DROPLET

AND SOLID PARTICLE DISPERSION

D i s p e r s e d P h a s e Trajectories For particles with a density much greater than the density of the carrier fluid (pp/pf >_ 10a), and with a diameter dp smaller than the Kolmogorov scale, a simplified equation of motion including only the drag and gravity forces can be considered"

ds dt d~(t)

dt

=~(t) ~ (i~(t), t) - ~ ( t )

(s) f (Rep) + g

973

1

1

L E S avg 1 4 s - 2 0 s Fackrell and Robins ( 1 9 8 2 )

=

0.9 0.8

0.8

0.7

0.7

0.6

0.6

~.o.5

"~o,5

0.4

N 0.4

0.3

0.3

0.2

0.2 -

N

LES avg 14s-20s obins ( 1 9 8 2 )

\

0.9

0.1

0.1 ::

00

I

I

L

I

I

0.25

,

0.5

U/Ue

00-, ,,,~ .... ~ .... ~,";

I

0.75

1

~ ;

.... _~

k/u.2

Figure 1" Vertical profiles of mean velocity and turbulent kinetic energy. Lines- LES. Squaresexperimental results of Fackrell & Robins (1982)

1

1

i

0.9

FR xstat=2.88 LES xstat=2.88

o8

i

FR xstat=2.88 LES xstat=2.88

0.8

9

E;;-

07

0.7 0.6

0.6

"~o.5

~.N0.5

N

=

0.9

0.4

0.4

0.3

0.3

0.2 01

0.1

0.3

N

0.2

0.2 ,,,I

....

l ....

I ....

I ....

I ....

l,,I,l

....

I ....

I ....

00 0.1 0.2 0.3 0.4 0.5 0.6 0 7 0 8 0 9 C/Cmax

I

1

iii

III

,I

....

I,

,,ll

....

I,,,,I

....

I ....

I ....

| ....

00 01 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Crms/Crms

max

I

1

o

l

-0.5

~

i

J

i

0

i

wc/u.Cmax

i

J

i

I

0.5

J

Figure 2: Vertical profiles of mean concentration, concentration fluctuations c2, and concentration fluxes w---dat z = 2.885. Lines - LES. Squares - experimental results of Fackrell &: Robins (1982)

Up is the velocity of the particle, ff (Jp(t), t) is the velocity of the fluid at the particle position and g is the acceleration of gravity. ~-p = ppd~/18pfu is the particle relaxation time and Rep = ~p-~ d/u is the particle Reynolds number, f (Rep) = 1 + 0.1 ot~ep ~n 06s7 as proposed by Clift et al. (1978). Eqn. 8 is appropriate for describing the motion of smooth rigid spheres. It neglects the influence of virtual mass and the Basset history force on particle motion. The volume fraction of particles is assumed small enough such that wind profile modification by the presence of particles is assumed negligible. From computation of an Eulerian velocity field, Eqn. 8 is integrated in time using second order Runge-Kutta scheme. The driving fluid velocity ~ (~Tp(t), t) is given by the velocity field of the LES and a fluctuating subgrid component determined by the Lagrangian stochastic model described above. To obtain the Lagrangian large scale velocities the same interpolation procedure as for the fluid particles is employed. The fluctuating subgrid component of ff (2p(t), t) is given by the closest fluid particle. Namely, in addition to tracking solid particles or droplets, we track also fluid particles. When the domain is full of fluid particles, solid particles or droplets are injected and the fluctuating subgrid component of g (2p(t), t) is given by the subgrid velocity of the closest

974 fluid particle. Properties of the dispersed phase are obtained by following 200.000 fluid particles and 50.000 solid particles or droplets.

Solid Particles The results for solid particle dispersion of the LES coupled with the Lagrangian stochastic model are compared with the wind tunnel experiments of Nalpanis et al. (1993). Even though we are essentially interested in the dispersion of particles from elevated sources, this test was conducted for validation purposes. Unfortunately, no experimental studies on solid particle dispersion from elevated point sources or stack emissions were found. Nalpanis et al. (1993) studied the dispersion of heavy sand particles in a rough turbulent boundary layer. Initially, the floor of the wind tunnel is covered with loose sand particles with a median diameter of dp = 188#m. Profiles of concentration and wind speed as well as impact, ejection and trajectory characteristics are measured. In our computations, the measurements of ejection velocity and angle are used to provide the initial conditions for the ejected particles. A rough turbulent boundary layer is computed. The vertical profiles of the mean velocity and turbulent kinetic energy of the computed flow are shown in Figure 3. The turbulent kinetic energy profile is compared to the profile measured by Fackrell & Robins (1982), because the corresponding profile was not published by Nalpanis et al. (1993). On the right-hand side of Figure 3, the vertical profile of the concentration 2 meters from the upwind edge of the sand bed is shown. The simulations match well with the experimental results.

Figure 3: Vertical profiles of mean velocity (left), turbulent kinetic energy(center) and mean concentration (right). Lines- LES. Squares- the experimental results of Nalpanis et al. (1993) for the mean velocity and mean concentration profile, the experimental results of Fackrell & Robins (1982) for the turbulent kinetic energy profile

Droplet Dispersion and Comparison with a Passive Scalar Plume In order to evaluate the impact of inertia and gravity we computed the dispersion of a plume of droplets from an elevated point source. Since we did not find any wind tunnel experience or insitu measurements of droplet dispersion in a turbulent boundary layer we compared the numerical results with the passive scalar plume dispersion of Fackrell & Robins (1982). In the numerical simulation, droplets with a Gaussian size distribution of mean dp = 60pro and variance 10pro were injected at the source. The mean concentration profiles at different stations compared to the mean concentration profiles for a passive scalar are shown on Figure 4. At the source height

975 the rate of t u r b u l e n t dissipation has an average value of e ~ 10u3/6 ~ O.055m2/s a, and the Kolmogorov timescale is 7-~ ~ 0.02s. Since the droplets have a relaxation time of ~-p = 0.012s, the Stokes n u m b e r is around St ~ 0.6. Even t h o u g h this n u m b e r is smaller t h a n one, droplets fall to the ground and as we can see from Figure 4 the highest concentration for the droplets is at the ground. Even in the case of small Stokes numbers the dispersion of plumes with particles is substantially different from passive scalar plumes.

1

9

0.9

Fackrell and Robins LES fluid particles LES drops

......... ,1

0.8

1~ 0.9~ 0.8~

[] ......... ~,

Fackrell and Robins LES fluid particles LES drops

0.7 0"7I,,..." 0.6

0.6

-I-~o.5-

-~o.s

0.3-

0.a~

""

0.2

......

~ ~ , , , l , J , , l , , , , l

-0

;.

o.4 ~--,~ .......... ,,

0.4-

....

I ....

o.2~-

-IB--'--

I,,HI

....

~%.

0.1 ~ I ....

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

C/Cmax

........... ,...

"'L -~-

----.

"'"""

m

,,."";

I

1

C/Cmax

Figure 4: Vertical profiles of m e a n concentration at z = 0.925 and x = 2.886. L i n e s - LES. Squares - experimental results of Fackrell & Robins (1982)

'F

9 ......... =

0.90.8

Fackrell and Robins LES fluid particles LES drops

0.7

1

-,

0.9

Fackrell and Robins LES fluid particles LES drops

......... o

0.8 0.7

0.6

0.6

"~0.5

-~ 0.5 ~......

0.4

0.4

0.3 i

.......

o~~ - 3 - . ' . o, : . _ ~ :

!~

9

,,.,...,.,

"=""

"--"

nvO; ~0.1 ~

" .............

.~

0.3

....................................1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Crms/Crms max

.........

02 0.1 O0

n...

f.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Crms/Crms max

=

. . . . . .s 1

Figure 5: Vertical profiles of concentration fluctuation at x = 0.925 and x = 2.885. Lines - LES. Squares - experimental results of Fackrell & Robins (1982)

Coalescence Model and Droplet Dispersion In the present section the coalescence of droplets is taken into account. T h e focus of the study is on how atmospheric turbulence affects droplet coalescence and the droplet size distribution. We are concerned with particle loadings t h a t are dilute and we ignore the effect of particles on the

976 fluid motion (two-way coupling) as in Reade & Collins (2000). On the other hand, the coalescence and breakup of droplets are taken into account by a stochastic subgrid model which is based on the same idea as the pairing particle model for scalar dispersion problems (Hsu & Chen 1991). Furthermore, this model contains some aspects developed by Apte et al. (2003) for secondary breakup of droplets. In this model, both breakup and coalescence of droplets are viewed as discrete random processes. The coalescence/breakup of droplets is considered in the framework of uncorrelated events, independent of the initial droplet size. With these assumptions, a differential Fokker-Planck equation for the PDF of droplet radii is obtained. At each time step, the size and number density of the newly produced droplets is governed by the evolution of this PDF in the space of droplet-radius. The occurrence of coalescence is determined as a function of a critical relative Weber number Wec~, based on the relative velocity between two droplets and a critical coalescence frequency l/coal. The domain is divided in boxes. In each box, at each time step, particles are randomly selected by pairs. If the lifetime of both selected droplets is bigger than 1/Ucoal and if the relative Weber number of the selected droplets is smaller Wec~ then the size of the droplets is modified based on the droplet-radius distribution function. Finally, mass conservation is applied inside each box. This coalescence/breakup model was first applied to the same case as Ho & Sommerfeld (2003) in order to fix the model parameters Wecr and l/coal. The model was introduced without any changes in the LES coupled with the Lagrangian subgrid stochastic model and applied to droplet dispersion within a plume. Since the coalescence model leads only to small changes in the mean diameter and in the size distribution of droplets (Figure 6) no important modifications appear in the mean concentration profiles (Figure 7) when compared to the case without coalescence.

Figure 6: Left - mean diameter as a function of distance from the source (triangles - with coalescence; squares- without coalescence). Right - Size distribution of droplets at x = 2.885 (plain line- initial size distribution; line with symbols- size distribution when coalescence takes place)

CONCLUSION A L E S coupled with a Lagrangian stochastic model has been applied to the study of solid particle and droplet dispersion in a turbulent boundary layer. The stochastic model usually given in terms

977

1

1

0.9 0.8

9 .........

0.7

9

Facl~rell and Robins LIES fluid particles LES coalescence LES droplets

0.9

9

.........

0.8 0.7

0.6

9

FackmM and Rcd31ns LES fluid particles LIES coalescence LES droplets

0.6

~0.5

~.,,.,

~0.5 0.4

0.3 .':-

0.3

0.2 .':-

...............

"

. . . . ,,~.,,

.

0.2 0.1

O. ....

i ....

i ....

i ....

i ....

i ....

C/Cmax

i ....

I ....

i ....

,,,,,

~ "0'i....i .........i ............. C/Cmax

Figure 7: Vertical profiles of mean concentration at x -- 0.925 and x -- 2.885. Plain line - LES with droplets with coalescence; dash-dotted line- LES with fluid particles; squares- experimental results of Fackrell & Robins (1982); diamonds - LES with droplets without coalescence

of the Reynolds decomposition is adapted here to a SGS decomposition, Vinkovic et al. (2004). The coupling is validated by comparison with the experimental results on passive scalar dispersion of Fackrell & Robins (1982). The numerical results are in good agreement with the experimental profiles. Furthermore, the dispersed phase is computed by Lagrangian particle tracking. The velocity of the fluid at the position of the solid particle or droplet has a large-scale and a SGS component. The large-scale component is obtained by interpolation of the velocity directly computed by the LES while the SGS component is given by the modified stochastic model. A test case for solid particles is computed and the numerical results are compared with the experiments of Nalpanis et al. (1993). Our results match well with the experimental profiles. Then, the simulation is applied to droplet dispersion from an elevated source. The results are compared with the experimental profiles for a passive scalar (Fackrell & Robins 1982). Finally, a coalescence/breakup stochastic model is introduced. There are no important differences in the mean concentration profiles between the cases with and without coalescence because only a slight change in the size distribution of droplets is introduced by the coalescence process. In this study, the dynamic conditions do not lead to droplet breakup. Further tests with various wind conditions will be lead in order to study the process of coalescence and breakup in the atmospheric boundary layer. We will introduce two-way coupling which could be crucial if the source of particles is close to the ground. By coupling the LES with the modified stochastic model we take into account the SGS movement of particles. This issue will be crucial for further studies of plume dispersion, droplet transport and sedimentation and also for evaluating the impact of instantaneous and local turbulent structures on inter-particle interactions as coalescence and breakup.

References Vinkovic I., Aguirre C., Simo~ns S. and Gence J.N. (2004). Coupling of a Subgrid Lagrangian Stochastic Model with Large-Eddy Simulation. submitted to C.R. Mdcanique. Apte S.V., Gorokhovski M. and Moin P. (2003). LES of Atomizing Spray with Stochastic Mod-

978 eling of Secondary Breakup. J. Multiphase Flow 29, 1503-1522. Armenio V., Piomelli U. and Fiorotto V. (1999). Effect of the Subgrid Scales on Particle Motion. Phys. Fluids 11:10, 3030-3042. Clift R., Grace J.R. and Weber M.E. (1978). Bubbles, Drops and Particles, Academic Press, New York. Cook A.W. and Riley J.J. (1994). A Subgrid Model for Equilibrium Chemistry in Turbulent Flows. Phys. Fluids 6:8, 2868-2870. Deardorff J.W. (1970). A Numerical Study of Three-Dimensional Turbulent Channel Flow at Large Reynolds Numbers. J. Fluid Mech. 41,453-480. Deardorff J.W. (1980). Stratocumulus-capped Mixed Layers Derived from a Three-dimensional Model. Boundary-Layer Meteorol. 18, 495-527. Durbin P.A. (1983). Stochastic Differential Equations and Turbulent Dispersion. NASA reference publication 1103, 1-69. Fackrell J.E. and Robins A.G. (1982). Concentration Fluctuations and Fluxes in Plumes From Point Sources in a Turbulent Boundary Layer. J. Fluid Mech. 117, 1-26. Germano M., Piomelli U., Moin P. and Cabot W.H. (1991). A Dynamic Subgrid-Scale Eddy Viscosity Model. Phys. Fluids A 3:7, 1760-1765. Ho C.A. and Sommerfeld M. (2002). Modelling of Micro-Particle Agglomeration in Turbulent Flows. Chem. Engng. Sci. 57, 3073-3084. Hsu A.T. and Chen J.Y. (1991). A Continuous Mixing Model for PDF Simulations and its Applications to Combusting Shear Flows. Paper 22-4, 8th Syrup. on turbulent shear flows, Terminal University, Miinich, 9-11 September. Klemp J. B. and Wilhelmson R. (1978) The Simulation of Three-Dimensional Convective Storms Dynamics J. Atmos. Sci. 35, 1070-1096. Lesieur M. and Me~tais O. (1996). New Trends in Large-Eddy Simulations of Turbulence. Ann. Rev. Fluid Mech. 28, 45-82. Lilly D.K. (1992). A Proposed Modification of the Germano Subgrid-Scale Closure Method. Phys. Fluids A 4:3, 633-635. Meeder J.P., Nieuwstadt F.T.M. (2000). Large-Eddy Simulation of the Turbulent Dispersion of a Reactive Plume from a Point Source into a Neutral Atmospheric Boundary Layer. Atmos. Environment 34, 3563-3573. Nalpanis P., Hunt J.C.R. and Barrett C.F. (1993). Saltating Particles Over Flat Beds. J. Fluid Mech. 251,661-685. Reade W.C. and Collins L.R. (2000). A Numerical Study of the Particle Size Distribution of an Aerosol Undergoing Turbulent Coagulation. J. Fluid Mech. 415, 45-64. Shao Y. and Li A. (1999). Numerical Modelling of Saltation in the Atmospheric Surface Layer. Boundary-Layer Meteorol. 91, 199-225. Thomson D.J. (1987). Criteria for the Selection of Stochastic Models of Particle Trajectories in Turbulent Flows. J. Fluid Mech. 180, 529-556. van Dop H., Nieuwstadt F.T.M. and Hunt J.C.R. (1985). Random Walk Models for Particle Displacements in Inhomogeneous Unsteady Turbulent Flows. Phys. Fluids 28:6, 1639-1653. Wang Q. and Squires K.D. (1996). Large Eddy Simulation of Particle-Laden Turbulent Channel Flow. Phys. Fluids 8:5, 1207-1223. Xie Z., Hayden P., Voke P.R. and Robins A.G. (2004). Large-Eddy Simulation of Dispersion: Comparison Between Elevated and Ground-Level Sources. J. of Turbulence 5, 1-16.

Engineering Turbulence Modelling and Experiments6 W. Rodi (Editor) 9 2005 ElsevierLtd. All rights reserved.

979

DYNAMIC SELF-ORGANIZATION IN PARTICLE-LADEN TURBULENT CHANNEL FLOW B.J. Geurts a'b and A.W. Vreman c aMathematical Sciences, Faculty EEMCS, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands bFluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5300 MB Eindhoven, The Netherlands cVreman Research, Godfried Bomansstraat 46, 7552 NT Hengelo, The Netherlands

ABSTRACT We study fundamental aspects of turbulent riser-flow which contains large numbers of interacting particles. We include particle-particle as well as particle-fluid interactions. These interactions and the flow-forcing are the source for the dynamic formation and destruction of large-scale coherent particle swarms in the flow. We establish the basic scenario of this self-organization and investigate the dominant aspects of the resulting turbulence modulation. Large-eddy simulations with different subgrid models, and large numbers of particles at a significant volume fraction and realistic mass load ratio, indicate the development of a thinner boundary layer and an accumulation of particles near the walls. At an average volume fraction of ~ 1.5% it was found that neglecting particleparticle interactions leads to an unphysical modulated flow.

KEYWORDS turbulence, particle-laden flow, large-eddy simulation, channel flow, inelastic collisions, coherent structures, four-way coupling, turbulence modulation

1

INTRODUCTION

Many flows of relevance to large-scale chemical processing involve solid catalyst particles at significant concentrations embedded in a carrying gas-flow. Control over the spatial distribution of these particles, especially its homogeneity, is essential in order to provide a chemical processing that is as complete and uniform as possible, that is consistent with modern environmental requirements and that does not constitute a strong safety hazard. This provides the main context for this study which is directed toward understanding the fundamental aspects of the dynamics of the embedded, interacting particles, and to develop a simulation strategy with which the central up-scaling of laboratory-scale experiments to realistic industrial settings can be supported.

980 The dynamics of the embedded particle-ensemble is quite complex and interacts nonlinearly with the carrying gas-flow. The particles are dragged along by this carrying gas-flow, exchanging momentum with it through a friction force-density. Moreover, the discrete particles interact among each other, e.g., through inelastic particle-particle collisions. In case only particle-fluid interactions are incorporated the description is referred to as 'two-way coupled' while a 'four-way coupled' formulation arises when also the particle-particle interactions are included, e.g. [1]. At sufficiently low particle volume fraction ~ the two-way coupling certainly will be adequate. However, with increasing ~ the collisions will become dynamically significant and the computationally more involved four-way coupling will become required. The main purpose of this paper is to demonstrate that at a realistic mass load ratio of 18 and a modest particle volume-fraction of about 1.5 % the collisions constitute a major dynamic effect that needs to be incorporated in order to retain a physically reliable flow description. Without this four-way coupling, dynamic self-organization of the embedded particles in coherent swarms is completely missed in the computational model. The two-phase gas-solid flow is governed by an interplay between the convective gas-flow nonlinearity, the particle-fluid and the particle-particle interactions. These effects may accumulate and significantly change basic turbulence properties. A large-scale dynamic flow-structuring may arise affecting the flow-statistics compared to the case with no or only weak interactions. These flowalterations constitute the so-called 'modulation of turbulence' which, e.g., seriously complicates the prediction of the up-scaling of flow-phenomena from laboratory-scale experiments to industrialscale settings. We will consider large-eddy simulations to support research in this problem-area and to help understand the fundamental modulation of the turbulent flow properties. The control of large-scale chemical processes is hindered by the lack of precise prediction-tools which characterize the dynamics in systems of realistic proportions. Since full-scale experimental research is costly and often not precise or not feasible, the development of accurate simulation tools is very important. A specific example is the cracking of oil which is facilitated by adding large numbers of catalyst particles to a carrying gas-flow. Basic to catalyst cracking is an understanding of the granular dynamics of large swarms of grains of sand. Specifically, it is important to investigate whether particle-particle collisions are dynamically important and lead to large clusters, thus contributing to spatial non-uniformities that may jeopardize safety and product-consistency and increase pollution. The particles interact mainly through inelastic collisions which by themselves lead to a granular clustering. In figure 1 an illustration of the clustering in a granular medium is provided [2] which may be loosely associated with the clustering behavior seen in large ensembles of catalyst particles in a riser flow. Such a dynamic 'blocking' of part of the flow-domain by itself induces an alteration of the overall flow. Moreover, the relative motion of the particles with respect to the fluid leads to considerable momentum transfer and flow-modulation. In this paper we adopt the large-eddy simulation approach to study possible flow-structuring phenomena at relatively high particle volume fractions. We extend some of our earlier work on this topic as reported in Ref. [3]. In the present paper we consider the formation of coherent particle structures in more detail and compare statistics for a number of subgrid models. Other computational studies found in literature involve quite modest particle volume fractions, typically 10 .4 or lower [4, 5, 6]. In these studies particle collisions and even particle-fluid interactions are often not taken into account. In contrast, in this paper we study dynamic consequences of both particle-particle as well as particle-fluid interactions at relatively high volume fractions. We incorporate three subgrid models in order to assess sensitivity of the predictions and include two dynamic models, both the standard [7] and a faster but approximate version [3], and a recently developed filtered multi-scale approach [8].

981

Figure I: Generation of a sand-jet after the impact of a heavy sphere in a loose sand-bed. A primary eruption follows from the initial impact from which the formation of a sand-jet arises. As this column of sand start to fall down, simultaneous inelastic collisions among the sand-particles give rise to the formation of granular clusters. Further details: http://tnweb, tn. utwente .nl/pof/. Before going into detail, the main findings may be summarized as follows. Incorporating particlefluid interactions induces a thinning of the boundary layer, leading to a higher skin-friction coefficient. Moreover, the inelastic collisions considered in the four-way description cause a striking dynamic self-organization. A cyclic clustering in the particle concentration may be observed. During a cycle, larger clusters are formed due to inelastic collisions and subsequently dispersed due to newly developing strong turbulent motions in the nearby 'clean' regions. This starts a new phase of cluster formation, followed by increasing turbulent fluctuation levels in the clean regions, leading to bursting of the clusters that formed etc. etc.. The clustering process occurs only in the four-way coupled description and leads to a flattening of the mean velocity profiles. The two-way model was found to be seriously flawed at these high volume fractions and induced a spurious high velocity jet in the center of the channel. The organization of this paper is as follows. In section 2 we present the mathematical model and provide the large-eddy formulation of the particle-laden flow. Subsequently, the dynamic flow-structuring is sketched in section 3. The corresponding turbulence modulation is discussed quantitatively in section 4. Finally, some concluding remarks are collected in section 5.

2

LARGE-EDDY

FORMULATION

OF PARTICLE-LADEN

FLOW

The computational model distinguishes a gas-phase and a solid-phase. The embedded solid particles are considered to be small compared to turbulent length-scales. This allows to effectively approximate the equations for the gas-phase in terms of flow through a (time- and positiondependent) porous medium [I]. We consider small spherical particles with radius a and volume

982 Vp = 47raa/3. The porosity ev(x, t) associated with a small volume V >> lip around the point x at time t can be defined as:

ev(x, t) = 1 - nv(x, t)Vp

V (1) where nv(x, t) is the number of particles in V at time t. In terms of ev we observe that ev --* 1 in case V does not contain any particles while ev --* 0 if the volume V contains no gas. In the continuum-limit in which V ~ 0 and nyVp/V --. constant the porosity approaches e. The Navier-Stokes equations that govern the flow in a porous medium specified by e read:

0~(p~) + oj(p~j) = 0 O~(p6~,) + O j ( p ~ , ~ ) = -o,(~p) + o j ( ~ , j ) + (p~9 - o,(sPm))5,~ + f, Ot(ee) + Oj((e + p)euj)=Oj(eaiju,) + (peg - 03(ePm))ua + f,u, - Oj(eqj)

(2) (a) (4)

where the symbols Ot and 0y denote the partial differential operators O/Ot and O/Oxj respectively. In these equations we selected the conservative formulation following, e.g., [9, 10]. This gives rise to pressure gradient and viscous stress contributions that incorporate the porosity e inside the spatial derivative operators. Alternative formulations may be found in literature in which, e.g., eOip arises instead [11]. The latter formulation is formally equivalent to the one selected here, provided a corresponding adaptation in the interpretation of the drag-force terms fi is included. The coordinate xa denotes the streamwise, x2 the normal and xl the spanwise direction. The domain is rectangular and the channel width, height and depth equal L2 = 0.05m, La = 0.30m and L1 = 0.075m respectively. No-slip boundary conditions are imposed in the x2-direction and periodic boundary conditions are assumed for the stream- and spanwise directions. Furthermore, p is the density, u the velocity, p the pressure and e = P / ( 7 - 1 ) + p u k u k / 2 the total energy per unit volume. The constant -y denotes the fraction of specific heats Cp/Cv = 1.4. The viscous stress ai3 is defined as the product of viscosity p = 3.47. lO-Skg/(rns) and strain-rate Sij = Oiuj + Oju~- ~25ijOkuk . The heat-flux qy is defined as -t~OjT where T is the temperature and ~ = O.035W/(rnK) the heat-conductivity coefficient. Pressure, density and temperature are related to each other by the equation of state for an ideal gas pRT = Mgasp where R = 8.314J/(rnolK) is the universal gas constant and Mgas = O.0288kg/mol is the mass of the gas per mole. The gravitation acceleration in the momentum equation equals g = -9.81rn/s 2 and fi denotes the force of the particles on the flow per volume unit. These are induced by an effective relative motion of the particles with respect to the gas which gives rise to drag forces on the fluid [1]. The equations formulated above are equivalent to the equations governing a compressible ideal gas with velocity u, temperature T, density/5 = ep, pressure i5 = ep, viscosity ~ = e# and heatconductivity k = e~. Therefore to solve this flow it is convenient to use a compressible flow solver with additional forcing terms representing gravity and the forces from the particles on the fluid. The flow is driven by a pressure gradient in the vertical direction, involving the imposed mean pressure Pm which is assumed a function of time only and its level is such that the total fluid mass flow is kept constant. For a channel flow without particles the pressure gradient corresponds to 7w = O.0625N/m 2, ur = 0.25m/s and Rer = 180. We will simulate a section of a riser flow with a vertical velocity of about 4rn/s. The parameters of the gas in the riser are close to those for air. The initial gas density equals Pl =l.0kg/rn 3 and we use a Mach number of .~ 0.2 for which the turbulent is effectively incompressible.

The discrete particle model calculates the motion of particles in the fluid and includes ineleastic particle collisions. The forces on a particle taken into account are gravity, pressure and the drag force resulting from the velocity difference with the surrounding fluid. The mean velocity of the riser is low enough to neglect the heat transfer during particle collisions and the heat transfer

983 between particles and fluid. The particle diameter equals 0 . 4 m m and the particle density is p~. = 1 5 0 0 k g / m 3. The number of particles equals Np - 419904. With these parameters the average volume fraction is 0.013 and the Stokes-response time Tp -- 0.4S. The numerical method applies a second-order finite volume method for the spatial discretization. Moreover, a second order explicit Runge-Kutta time-stepping method with time-step 2 . 1 0 - s s for the fluid phase and Euler forward time-stepping with time step 10-4s for the solid phase are adopted. Most of the computational effort is associated with the particle phase. The computational grid contains 32 • 64 • 64 cells. The grid is non-uniform in the wall-normal direction with the first grid point at y+ ..~ 1.5. The porosity : is determined on a uniform 'auxiliary' grid which is coarser than the particle diameter a. Linear interpolation is used to communicate information between grid-nodes and particle positions. The simulations run until at least t - 5s, while statistics are accumulated between t = 3s and t = 5s. In order to make large-scale turbulent flow simuations at high particle volume fractions feasible, the gas-phase is described using large-eddy simulation [12]. This is obtained by applying spatial filtering to the flow equations in order to reduce their dynamical complexity. In particular, we may consider convolution filtering in which

(5)

= L ( u ) = f G ( x - ~)U(~)d~ = G 9 u

where ~ denotes the filtered solution and G the filter-kernel. The filter is assumed to be normalized, i.e., L(1) = 1. If the spatial filtering is applied to the governing equations the result may be expressed in terms of the LES-template: NS(U) - R(U,-U) where the original and filtered statevector are defined by U = [~p, uj, ep]; U = [Uf, ~j,~pp] with Favre-filtered velocity ~ = p : u j / - ~ . The spatial filtering yields a 'closure-residual' R(U,-U) which contains, e.g., the filtered forcing term f i and the divergence of the turbulent stress tensor ~-~j = p : u ~ u j - p : u ~ p : u j / - ~

= -~{uT~j

- ~

(6)

}

We restrict attention to explicit modeling of the turbulent stress tensor and evaluate other closure terms by calculating the original formulation in terms of the filtered variables. For the representation of the sub-filter scales we will next introduce two subgrid modeling approaches. The filtered multi-scale model [8] can most transparently be formulated for the incompressible turbulent stress tensor ~'ij - uiuj - uiuj. We consider a second decomposition by introducing an extra filter (.~ which allows to define f ' - f - f, i.e., ~' = ~ - ~ and Tij -- ?ij + T~j. Correspondingly, the modeling problem for the turbulent stress tensor is composed of a large-scale (?ij) and a small-scale contribution (T~j). Within the context of LES, these may be modeled in different ways. If the filter-width of the second filter is sufficiently large then ~j may be modeled by the similarity or gradient models, or simply ~ij = 0. The small-scale contribution will be modeled by T~j -'+ mij(Ul), i.e., in terms of the small-scale velocity field ~ only. We will adopt the filtered multi-scale Smagorinsky model: ~-iy = ~j + ~i% ~ vi'j ~ m ~ j ( ~ )

where m i j ( v ) = - ( C s A ) 2 S ( v ) S ~ j ( v )

; S = IS~yI

(7)

By formulating the Smagorinsky model in terms of the filtered velocity fluctuations, the excessive dissipation associated with the Smagorinsky model in laminar and transitional flows is removed. This is particularly relevant close to the solid walls of the vertical channel. Next to the filtered multi-scale model we incorporate dynamic subgrid modeling. We begin with

984 the standard eddy-viscosity assumption in the basic model, expressed by m i j -- --

[Cd~--~/~2S(u)] S i j ( u )

;

i--

1

(A1/~2/~3)3

(8)

where As denotes the filter-width in the xi direction. The dynamic procedure [7] is based on the well-known Germano-identity and provides the possibility to calculate a 'Germano-optimal' coefficient which adapts itself to the evolving flow. In fact, after some calculation and the usual approximations the dynamic coefficient Cd may be obtained from

Cd = (MijLij) / (MijM~j)

(9)

with appropriately defined tensors Lij for the resolved turbulent stress tensor and a tensor Mij which collects subgrid model contributions. Here, (.) represents an averaging over homogeneous directions and 'clipping' is applied in case the right-hand side of (9) would return negative values. We distinguish two implementations of the dynamic procedure, referred to as the 'standard' and the 'approximated fast' procedure. These may be defined as follows: 9 Standard dynamic procedure: This requires explicit test-filtering and involves

Lij = [Kguiuj]

- ~i~j/~

(10)

Miy = - ~ ( x / 5 A ) 2 S ( - ~ / ~ ) S ~ j ( - ~ / ~ ) + [-FgA2S(g)S~j(g)] (11) 9 Approximated fast procedure (see [3] and its references)" For common second order filters @ = + O(A2), hence Lij ~-, 5 p e A k i ) k U i O k U j

Mij

(12)

This procedure does not require explicit test-filtering and is computationally much cheaper. In the next section we will consider the dynamic self-organization that arises due to the 'competition' between the structuring associated with the inelastic particle collisions and the bursting of particle-clusters due to the underlying tendency of the clean flow to develop strong turbulence.

3

DYNAMIC TRATION

PARTICLE-LADEN

FLOW STRUCTURING

AT HIGH CONCEN-

In this section we will show that the four-way coupling model gives rise to large-scale coherent particle swarms which are completely absent when the two-way coupling model is used. Moreover, the observed flow-structuring displays a 'cyclic' behavior which will be illustrated. In order to characterize the flow-structuring we concentrate on visualizing the particle volume fraction. For this purpose we introduce a uniform rectangular grid which contains nl x n2 x n3 cells respectively. The volume of each cell is denoted by V~cu. At time t we may count n particles in cell (i, j, k) and the corresponding volume fraction is r = 1 - evc~, = nVp/Vccu where V~u >> Vp = 47ra3/3. As point of reference, the basic riser flow was simulated using LES with the standard dynamic model and four-way coupling. The grid on which r is evaluated contains 32 x 25 x 64 cells. We show the particle volume fraction at different times in figure 2. From these snapshots one may infer the formation and destruction of large-scale coherent structures in a self-contained cyclic manner which will be described next.

985

Figure 2: Snapshots of the particle volume fraction showing iso-surfaces at r = 0.03 at t = 3.1 (a) with steps of 0.05 until t = 3.45 (h).

We start to describe the particle clustering cycle at an instant where r displays a rather fragmented distribution. In this state, the inelastic particle-particle collisions induce a distributed loss of energy from the flow. This reduces turbulent fluctuation levels in a fairly uniform manner in the domain. As a result, the fragmented distribution starts to form larger and larger particle patches, until all particles are contained in only a few clusters. In figure 2 this more 'organized' state may be seen in figures d-e-f for example. The region outside the larger particle clusters corresponds more closely to 'clean' flow. During the clustering phase the turbulence in these clean regions gradually builds up. This results in more structured stages with regions of somewhat intensified turbulence next to more calm particle-laden regions. The turbulence that grew in the clean regions becomes strong enough to destroy the clusters and leads to strong mixing and a new fragmented flow impression. The cyclic dynamics is certainly not periodic; sometimes the majority of the particles is grouped into a single swarm while in a next occurrence two or more larger swarms may be visible. The cyclic self-organization seen in figure 2 also arises when the approximated fast implementation or the filtered multi-scale model are used. As such, the dynamic self-organization of the particles is a robust phenomenon. At the particle volume fractions considered here, the use of the full fourway coupling is essential. This is illustrated in figure 3 in which we compare a structured particle field associated with four-way coupling, with a structure-less field arising in the two-way coupling model. These first, qualitative, impressions indicate that four-way coupling can not be replaced by the computationally more appealing two-way coupling.

986 TURBULENCE MODULATION PLED DYNAMICS

ARISING IN TWO- AND FOUR-WAY

COU-

In this section we will compare results obtained in 'clean' riser-flow with the particle-laden case, using three different subgrid models. We will separately consider the two- and four-way coupling models. The comparison between the clean and the two-way coupling case quantifies effects due to particle-fluid interactions. A comparison between the two- and four-way coupled cases quantifies effects due to particle collisions. First, we sketch results of clean riser-flow and turn to the particleladen case afterwards.

Figure 3: Granular clustering in coherent particle-swarms is strongly associated with the four-way coupling description. Snapshot of the particle volume fraction at t - 4 comparing the four-way coupling (a) with the two-way coupling (b). The iso-surfaces shown correspond to r = 0.03. The reference 'clean' riser-flow corresponds to Re~- = 180. The results for the mean streamwise fluid velocity (Uz) display only a very limited variation with the adopted subgrid model as shown in figure 4. A close agreement with the corresponding DNS results is observed. A more sensitive assessment of the quality of LES predictions is obtained by considering rms-fluctuation levels. These also showed limited dependence on the adopted subgrid model. This establishes the quality of these reference simulations. Compared to the unfiltered DNS data the large-eddy simulations yield a slight over-prediction of the fluctuation levels. Theoretically, LES should provide an underprediction of these fluctuation levels; the observed over-prediction is indicative of the modest level of discretization errors that remains in these coarse grid simulations. We next turn to predictions of mean fluid properties for the full particle-laden flow. In figure 4 we collected the mean streamwise fluid velocity. Relative to the clean channel we observe that both the two- and four-way coupling descriptions give rise to a strongly reduced boundary layer thickness. Variations in the predictions arising from changing the subgrid model are seen to be relatively modest with the two-way model slightly more sensitive than the four-way description. The prediction of the bulk flow away from the boundary layers is quite different when comparing the two-way and the four-way approaches. The two-way description is seen to give rise to a somewhat localized 'center-jet' in which the fluid velocity is up to about 60 % larger than the velocity at the edge of the boundary layer. In contrast, the four-way coupling gives rise to a slightly flatter velocity profile compared to the clean channel; the particle-particle collisions evidently allow to avoid the 'center-jet'. The effects of the embedded particles on the developing flow are seen more clearly in figure 4(b). For sake of comparison the velocity in the first grid-cell is scaled such as to correspond to the clean channel. Compared to the clean case an approximately logarithmic

987 5.5

I

,

,

51

,

t+- .+r~...~ ,, .

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./+...

'

~'/*

9

t

4

3.

3.5

2.

~ 2.5

o

/

o

~

O

O

0

.

0

~

0.01

0.5

0.02

0.03

0.04

0

0.05

10-a

107

(b)

Figure 4: Mean streamwise fluid velocity (uz): linear (a) and logarithmic (b) for a particle-laden flow, comparing clean with two- and four-way coupling and different subgrid models. The clean channel results are with pure lines and the two- and four-way results are labeled with + and , respectively. Filtered multi-scale results (solid), approximated fast dynamic (dashed) and standard dynamic (dash-dotted); reference DNS results of clean flow are indicated by (o). We scaled the velocity in (b) such that at the first grid point it coincides with the clean case. velocity profile develops for 10 -3 < x2 < 10 -2 but at a reduced yon Ks163 0.0151 .

2"~'~-,,

.

.

.

.

.

.

.

0.014~

/'P/'~~, 2.5

.

'constant'.

~176t

/'~1

0- 0.011[

>,~2

F

"~1.i

o~ I 0

.

0.007i, 0.5

0

0

8

~

'

t

-r. --1--T~-r----~-~ -

*

~ .

/ 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Y

[M]

(a)

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

YM

(b)

Figure 5: Mean streamwise particle velocity (vz) (a) and particle volume fraction (r (b). Filtered multi-scale (solid), approximated fast dynamic (dashed), standard dynamic (dash-dotted). We label two-way by + and four-way by ,. Notice that (r is shifted downward by 0.005 for the two-way results for clarity. The consequences of particle-particle interactions for mean particle-properties are shown in figure 5. The strong center-jet in (Uzl observed in the two-way coupling model, is even more pronounced in (Vzl. The sensitivity of the results with respect to the adopted LES model is again quite modest. The particle volume fraction distribution is shown in figure 5(b). A characteristic turbo-phoresis effect is visible in terms of an approximately 15 % higher concentration near the solid walls. This effect is well established experimentally and does not arise in case the two-way coupling description is adopted.

988 5

CONCLUDING REMARKS

In this paper we presented large-eddy simulation results of particle-laden turbulent flow in a vertical riser. We showed that already at a modest particle volume fraction of about 1.5 % the particle-particle interactions play an important role in the development of the flow. The computationally more accessible two-way coupling model proved to give rise to unphysical predictions, among others the absence of a turbo-phoresis effect and the occurrence of a fairly strong 'centerjet' which was not recorded in experimental studies. The presence of a large number of interacting particles leads to a strong modulation of the turbulence in the channel. Relative to the clean channel the boundary layer reduces in thickness and corresponds to a lower von Ks163 constant. Particle-particle interactions are responsible for cyclic, dynamic self-organization of the embedded particles in coherent swarms.

Acknowledgments Fruitful discussions with Niels Deen and Hans Kuipers are gratefully acknowledged. The illustrations in figure 1 were kindly provided by the Physics of Fluids Group, University of Twente. Computations were done under grant SC-244 of the Dutch National Computing Foundation (NCF).

REFERENCES

[1] Hoomans, B.P.B., Kuipers, J.A.M., Briels, W.J., and van Swaaij, W.P.M. (1996). "Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidized bed: A hard-sphere approach", Chem. Eng. Science 51, 99-118. [2] D. Lohse, R. Bergmann, R. Mikkelsen, C. Zeilstra, D. van der Meer, M. Versluis, K. van der Weele, M. van der Hoef, H. Kuipers, "Impact", http://xxx.lanl.gov/abs/cond-mat/0406368 [3] A.W. Vreman, B.J. Geurts, N.G. Deen and J.A.M. Kuipers (2004). "Large-eddy simulation of a particle-laden turbulent channel flow," in Direct and Large-Eddy Simulation V. Edited by R. Friedrich, B.J. Geurts and O. Metais (Kluwer, Dordrecht), 271-278. [4] Yamamoto, Y., Potthoff, M., Tanaka, T. Zajishima, T., and Tsuji, Y. (2001). "Large-eddy simulation of turbulent gas-particle flow in a vertical channel: effect of considering inter-particle collisions", J. Fluid Mech. 442, 303-334 (2001). [5] Squires, K.D., and Simonin, O. (2002). "Recent advances and perspective of DNS and LES for dispersed two-phase flow", Proceedings of the 10th workshop on two-phase flow predictions, Merseburg, 152-163. [6] Marchioli, C., Giusti, A., Salvetti, M.V., and Soldati, A. (2003). "Direct numerical simulation of particle wall transfer and deposition in upward turbulent pipe flow", Int. J. of Multiphase Flow 29, 1017-1038. [7] Germano, M., Piomelli, U., Moin, P., and Cabot, W.H. (1991). "A dynamic subgrid-scale model", Phys. Fluids A 3, 1760-1765. [8] Vreman, A.W. (2003). "The filtering analog of the variational multi-scale method in large-eddy simulation", Phys. Fluids 15, L61-64. [9] Whitaker, S. (1996). "The Forchheimer equation: a theoretical development", Transport in porous media, 25: 27-61. [10] Lakehal, D., Smith, B.L., Milelli, M. (2002). "Large-eddy simulation of bubbly turbulent shear flows", Journal of Turbulence, 3: 025. [11] Zhang, D.Z., Prosperetti, A. (1997). "Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions", Int. J. Multiphase Flow, 23: 425-453. [12] B.J. Geurts (2004). "Elements of direct and large-eddy simulation", Edwards Inc.

989

AUTHOR INDEX

Abe, S. 585 Aguirre, C. 969 Ahlstedt, H. 399 Alipchenkov, V.M. 959 Angel, B. 657 Antonia, R.A. 207 Arakawa, C. 689 Arcen, B. 949 Aupoix, B. 137 Avetissian, A.R. 959

Balabani, S. 451 Balaras, E. 349 Barhagi, D.G. 287 Barthet, A. 441 Bataille, F. 751 Battaglia, V. 905 Baumann, W.T. 865 Benabid, T. 471 Bender, C. 855 Benhamadouche, S. 117, 237 Benocci, C. 26 B6zard, H. 77, 147 Bockhorn, H. 855 Braud, P. 491 Braza, M. 441, 533 Brillant, G. 751 Brunn, A. 555 Biichner, H. 855 Bunge, U. 617

Cabrita, P.M. 667 Camussi, R. 719 Cao, S. 257 Carpy, S. 127 Cazin, S. 441 Chartrand, C. 107 Chauve, M.-P. 761 Chishty, W.A. 865 Cid, E. 441 Clayton, D.J. 247 C16on, L.-M. 657 Cokljat, D. 647 Collin, E. 491 Constantinescu, G. 277

Coustols, E. 637 Craft, T. 117 Craft, T.J. 157, 627

Dahlstr6m, S. 319 Daris, T. 77 Davidson, L. 287, 319 De Langhe, C. 329 De Prisco, G. 349 Dejoan, A. 97 Delville, J. 491 Deng, G.B. 389 Derksen, J.J. 929 Deutz, L. 511 Di Marco, A. 719 Dianat, M. 823 Dick, E. 329, 523 Djenidi, L. 207 Dreizler, A. 731 Ducci, A. 451 Ducruix, S. 875 Dumand, C. 895 Durbin, P.A. 167

Eliasson, P. 607 Elkins, C.J. 3

Faghani, D. 533 Falchi, M. 595 Ferr6, J.A. 411 Ferrey, P. 137 Fleig, O. 689 Flikweert, M.A. 773 Fr6hlich, J. 845 Fu, S. 227 Fujita, S. 501

Gailler, D. 875 Garcia-Villalba, M. 845 Gautier, P.-E. 657 Geurts, B.J. 979 Gr6goire, O. 195 Groll, R. 939 Grundestam, O. 607

Grundmann, S. 835 Guj, G. 719

Habisreuther, P. 855 Haire, S.L. 185 Han, G. 545 Hanjali6, K. 67, 369, 773 Harima, T. 501 Hassel, E. 783 Hattori, H. 175 Hellsten, A. 147 Hoarau, Y. 441,533 Horiuti, K. 585 Houra, T. 793 Huang, P.G. 31 Husson, S. 751 Hutton, A.G. 381

Iaccarino, G. 3 Iacovides, H. 157 Iida, M. 689 Ishino, Y. 915

Jahnke, S. 783 Jakirli6, S. 835 Janicka, J. 731 Janus, B. 731 Ji, M. 167 Johansson, A.V. 607 Jones, W.P. 247

Karlatiras, G. 87 Karlsson, R. 287 Karvinen, A. 399 Kassinos, S.C. 185 Kawaguchi, Y. 575 Keating, A. 349 Kenjere~, S. 369 Knowles, K. 667 Kobayashi, T. 297 Konstantinidis, E. 451 Kornev, N. 783 Lacas, F. 875 Langer, C.A. 185

990

Langtry, R. 31 Launder, B.E. 627 Laurence, D. 117, 237 Laurence, D.R. 67 Le Duc, A. 709 L~, T.-H. 565 Leonardi, S. 207 Leschziner, M.A. 97, 359 Li, Q. 227 Liang, C. 813 Liu, Y. 339 Lodefier, K. 523 Loh~isz, M.M. 267

Mahon, S. 461 Makita, H. 431 Manceau, R. 127 Manhart, M. 709 Mary, I. 565 Mathey, F. 647 McCoy, A. 277 McGuirk, J.J. 307, 823 Menter, F.R. 31 Menzies, K.R. 741 Merci, B. 329 Minier, J.-P. 803 Mockett, C. 617 Moradei, F. 441 Moshfegh, B. 773 Moulinec, C. 237 Mudde, R.F. 511

Nagano, Y. 175,793 Nait Bouda, N. 471 Nievaart, V.A. 511 Nitsche, W. 555

Obi, S. 481 Oesterl6, B. 949 Ohiwa, N. 175,915 Okamoto, M. 217 Okuno, A. 257 Oliemans, R.V.A. 49 Orlandi, P. 207 Osaka, H. 501

Page, G.J. 307 Palm, R. 835 Papadakis, G. 87, 813 Paradot, N. 657 Peller, N. 709

Perid, M. 237 Perot, B. 107 Perrin, R. 441 Petsch, O. 855 Pietrogiacomi, D. 595 Pinson, F. 195 Piomelli, U. 349 Poinsot, T. 875 Poncet, S. 761 Popovac, M. 67 Portela, L.M. 49 Pozorski, J. 803 Provenzano, G. 595 Pubill Melsi6, A. 875

Terracol, M. 699 Tessicini, F. 359 Thiele, F. 617 Thiery, M. 637 Tkatchenko, I. 783 Tokai, N. 481 Travin, A. 679 Tropea, C. 835,939 Tsubokura, M. 297 Tucker, P.G. 339 Tummers, M.J. 773

Uribe, J.C. 67 Usera, G. 411

Queutey, P. 389

Ragni, A. 719 Rambaud, P. 267 Revell. A.J. 117 Rey, C. 471 Robinson, C.M.E. 627 Rodink, R. 773 Romano, G.P. 595 Rosant, J.M. 471

Sabel'nikov, V. 885 Sabel'nikov, V.A. 895 Saddington, A.J. 667 Sadiki, A. 731 Sakai, K. 481 Sari6, S. 835 Sassa, K. 431 Saunders, W.R. 865 Schetz, J.A. 421 Schiestel, R. 761 Sevrain, A. 441 Shima, N. 217 Simo/~ns, S. 969 Simonin, O. 195 Sommerer, Y. 875 Soulard, O. 885 Spalart, P.R. 679 Strelets, M. 679 Suga, K. 157

Takagi, Y. 585 Tamura, T. 257 Tani~re, A. 949 Taniguchi, N. 297 Temmerman, L. 359

Van Den-Berg, M. 461 van Maanen, H.R.E. 511 Vandsburger, U. 865 Veloudis, I. 307 Ventikos, Y. 533 Vernet, A. 411 Verzicco, R. 17 Veynante, D. 875 Vinkovic, I. 969 Visonneau, M. 389 V/31ker, S. 31 Vreman, A.W. 979

Wachsmann, E.-P. 709 Wallin, S. 607 Wang, C. 97 Weber, L. 277 Wegner, B. 731 Weismtiller, M. 835 Williams, C. 461 Wygnanski, I. 545

Yang, Z. 307 Yang, Z. 823 Yaqobi, K. 117 Yianneskis, M. 451 Yu, B. 575

Zaichik, L.I. 959 Zhang, X. 461 Zhdanov, V. 783 Zhou, M.D. 545 Zimont, V.L. 905