Direct and Large-Eddy Simulation VII (ERCOFTAC Series)

Direct and Large-Eddy Simulation VII ERCOFTAC SERIES VOLUME 13 Series Editors R.V.A. Oliemans, Chairman ERCOFTAC, Del...

Author: Vincenzo Armenio | Bernard Geurts | Jochen Fröhlich

42 downloads 981 Views 25MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Direct and Large-Eddy Simulation VII

ERCOFTAC SERIES VOLUME 13 Series Editors R.V.A. Oliemans, Chairman ERCOFTAC, Delft University of Technology, Delft, The Netherlands W. Rodi, Deputy Chairman ERCOFTAC, Universität Karlsruhe, Karlsruhe, Germany

Aims and Scope of the Series ERCOFTAC (European Research Community on Flow, Turbulence and Combustion) was founded as an international association with scientific objectives in 1988. ERCOFTAC strongly promotes joint efforts of European research institutes and industries that are active in the field of flow, turbulence and combustion, in order to enhance the exchange of technical and scientific information on fundamental and applied research and design. Each year, ERCOFTAC organizes several meetings in the form of workshops, conferences and summerschools, where ERCOFTAC members and other researchers meet and exchange information. The ERCOFTAC Series will publish the proceedings of ERCOFTAC meetings, which cover all aspects of fluid mechanics. The series will comprise proceedings of conferences and workshops, and of textbooks presenting the material taught at summerschools. The series covers the entire domain of fluid mechanics, which includes physical modelling, computational fluid dynamics including grid generation and turbulence modelling, measuring-techniques, flow visualization as applied to industrial flows, aerodynamics, combustion, geophysical and environmental flows, hydraulics, multiphase flows, non-Newtonian flows, astrophysical flows, laminar, turbulent and transitional flows.

For other titles published in this series, go to http://www.springer.com/series/5934

Direct and Large-Eddy Simulation VII Proceedings of the Seventh International ERCOFTAC Workshop on Direct and Large-Eddy Simulation, held at the University of Trieste, September 8–10, 2008 Edited by Vincenzo Armenio Università di Trieste, Italy Bernard Geurts University of Twente, Enschede, The Netherlands and Jochen Fröhlich Technical University of Dresden, Germany

123

Editors Dr. Vincenzo Armenio Università di Trieste Dipto. Ingegneria Civile e Ambientale Via Valerio 34127 Trieste Italy [email protected]

Prof. Dr.-Ing. Jochen Fröhlich Institute of Fluid Mechanics Technical University of Dresden George-Bähr-Str. 3c 01062 Dresden Germany [email protected]

Prof. Bernard Geurts University of Twente Fac. Mathematical Sciences 7500 AE Enschede The Netherlands [email protected] [email protected]

ISSN 1382-4309 ISBN 978-90-481-3651-3 e-ISBN 978-90-481-3652-0 DOI 10.1007/978-90-481-3652-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010922449 c Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface In an era of continuously increasing computer power and increasing quality of algorithms and software Direct Numerical Simulations and Large Eddy Simulations receive continuously increasing interest and face widespread usee. Nevertheless, developments of are still needed and performed by researchers in various directions. The ERCOFTAC series of workshops on Direct and Large Eddy Simulation organized since 1994 by various people reﬂect this activity and serve as catalyst to stimulate scientiﬁc exchange on this topic. During September 8–10 2007, the seventh workshop of this series was held in Trieste, Italy. As the earlier workshops it aimed at establishing the state-of-art in turbulence modelling and numerical methodologies of Direct and Large Eddy Simulation as well as their use in fundamental research and applications. Six plenary sessions, 16 parallel sessions and 2 poster sessions have been run during the three days. The nine keynote lectures have dealt with diﬀerent ﬁelds of fundamental, industrial and environmental ﬂuid mechanics. Prof. S. Pope (Cornell University, USA) has discussed recent modelling techniques (LES/FDF) for reacting ﬂows, like ﬂames in combustors, and new methodologies to treat the transport of scalars in the ﬂow ﬁeld. Prof. G. Pedrizzetti (University of Trieste, Italy) has shown recent numerical results of the unsteady ﬂow ﬁeld within the left ventricular, and comparison with data obtained in real cases by means of eco-Doppler analysis. Prof. C. Meneveau’s talk (Johns Hopkins University, USA) has dealt with LES modelling of environmental ﬂuid mechanics, discussing large scale LES of pollen dispersion in the atmosphere, and comparison between numerical and ﬁeld data. Prof. U. Piomelli (Queen’s University, Canada) has given a summary of the state-of-art of wall-modelling techniques within the LES framework for high Reynolds number ﬂows. Prof. S. Elghobashi (UC Irvine, USA) has discussed Lagrangian techniques for multiphase ﬂows, focussing on new formulations of the particle motion equation and on two-way coupling between the Eulerian ﬁeld and the Lagrangian techniques. Prof. R. Verzicco (University of Rome, Tor Vergata, Italy) has shown recent results regarding natural convection in conﬁned ﬂows, aimed at explaining disagreements among experimental data and numerical simulations. Prof. T. Hughes (University of Texas at Austin, USA) has discussed a novel approach to LES methodology, based on the multi-scale methodology. Two

VI

Preface

sets of equations are considered, one for the large scales of motion and one for the small ones. Results of simulations on simple validation cases prove the reliability of the technique. This methodology is far from the classical ones where the small scales of motions are parameterized in a statistical sense using a SGS model, ant it can be a good alternative to classic SGS techniques in cases of very complex-physics ﬂows, where the direct knowledge of the small scales of motion is required. Prof. R. Friedrich (TU Munich, Germany) has discussed recent results of boundary layer dynamics in compressible ﬂows, considering both subsonic and supersonic conditions. Finally, Dr. Flohr (Alstom, Switzerland) has discussed results of high-Reynolds number large eddy simulation of gas turbine combustors, giving a nice example of the superiority of the LES approach when compared to classing RANS methodologies. The regular sessions focussed on diﬀerent topics. Environmental applications were shown, from the laboratory-scale numerical experiments carried out using both DNS and wall-resolving LES to applicative full-scale simulations carried out through LES with wall-layer models. Also fundamental studies of turbulence were discussed in a wide range of basic ﬁelds, from the canonical boundary layer to the isotropic turbulence. A session of compressible ﬂows ranged from basic studies of interaction between compressible eﬀects and turbulence to wall-layer modelling for LES of compressible ﬂows. The multiphase ﬂows session covered classical particle laden ﬂow analyses as well as the study of bubbles and featured state-of-art approaches where the dispersed phase is treated as an ensemble of ﬁnite-size particles. The session on aerodynamics and wakes treated classical aerodynamics problems, approached through the use of state-of-the-art numerical techniques and SGS models. Also, more challenging problems where discussed such as turbulent wakes aﬀected by stratiﬁcation. The session on evolution of active scalars, comprised both reactive ﬂows (for combustion problems) and classical natural convection problems. A session of analysis and quantiﬁcation of modelling errors in LES discussed and rigorously quantiﬁed the numerical errors associated with the use of low order numerical schemes in conjunction with SGS models. Finally, in the LES modelling session new ideas for roust and accurate SGS models were discussed. The number of oral presentation was 74. Short presentations of 15 posters completed the program. The number of participants was about 120, of which 37 were undergraduate and graduate students. As a whole, based on the scientiﬁc contributions presented during the workshop, LES seems to be a robust and reliable technique for practical applications in numerous high-end technological applications. However, many problems still remain open and were discussed during the workshop, e.g. regarding SGS closure in special applications, validation of numerical results against experimental, real-life data, and accuracy of the algorithms currently in use for the numerical integration of the governing equations. A general trend in ﬂuid mechanics conferences is that the amount of work presented on very fundamental issues such as subgrid-scale modelling is observed to persist, while complex physical phenomena and applications are teated to an

Preface

VII

increasing extent. In this scenery, the series of DLES workshops sees itself as a host for fundamental research on DNS and LES and will continue to be that in the future. The present book summarizes the written contributions to the seventh DLES workshop. As usual for a DLES workshop, the quality of the presentations was very high, and the same applies to the corresponding papers. This was furthermore ensured by an extensive peer-review process. The organizers were delighted about the stimulating atmosphere and the constructive discussions at this workshop. This is to a large extent due to the participants and the organizers are greatful for that. They ﬁnd it rewarding that the work of organizing this meeting turned into such a success. May this book be of much use for the reader and ﬁnd widespread distribution. Vincenzo Armenio Jochen Fr¨ ohlich Bernard J. Geurts

Contents Part I Fundamentals Wall-Modeled Large-Eddy Simulations: Present Status and Prospects Ugo Piomelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

A Study of the Inﬂuence of the Reynolds Number on Jet Self-Similarity Using Large-Eddy Simulation Christophe Bogey and Christophe Bailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Direct Numerical Simulation of Fractal-Generated Turbulence S. Laizet and J. Christos Vassilicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Turbulent Oscillating Channel Flow Subjected to Wind Stress W. Kramer, H.J.H. Clercx, and V. Armenio . . . . . . . . . . . . . . . . . . . . . . . . 27 DNS of a Periodic Channel Flow with Isothermal Ablative Wall O. Cabrit and F. Nicoud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Diagnostic Properties of Structure Tensors in Turbulent Flows D.G.E. Grigoriadis, C.A. Langer, and S.C. Kassinos . . . . . . . . . . . . . . . . . 43 Development of Brown–Roshko Structures in the Mixing Layer Behind a Splitter Plate Neil D. Sandham and Richard D. Sandberg . . . . . . . . . . . . . . . . . . . . . . . . . . 51 DNS of Spatially-Developing Three-Dimensional Turbulent Boundary Layers Philipp Schlatter and Luca Brandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Direct Numerical Simulation and Experimental Results of a Turbulent Channel Flow with Pin Fins Array B. Cruz Perez, J. Toro Medina, N. Sundaram, K. Thole, and S. Leonardi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

X

Contents

New Experimental Results for a LES Benchmark Case Ch. Rapp, F. Pﬂeger, and M. Manhart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Direct Computation of the Sound Radiated by Shear Layers Using Upwind Compact Schemes M. Cabana, V. Fortun´e, and E. Lamballais . . . . . . . . . . . . . . . . . . . . . . . . . . 77 DNS of Oriﬁce Flow with Turbulent Inﬂow Conditions George K. El Khoury, Mustafa Barri, Helge I. Andersson, and Bjørnar Pettersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 The Mean Flow Proﬁle of Wall-Bounded Turbulence and Its Relation to Turbulent Flow Topology Vassilios Dallas, J. Christos Vassilicos, and Geoﬀrey F. Hewitt . . . . . . . . 87 Large Eddy Simulation of a Rectangular Turbulent Jet in Crossﬂow E.H. Kali, C. Brun, and O. M´etais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Numerical Simulation of a 2D Starting-Plume Cloud-Flow K. Aditya, S.M. Deshpande, K.R. Sreenivas, and R. Narasimha . . . . . . . . 97

Part II Methodologies and Modelling Techniques Variational Multiscale Theory of LES Turbulence Modeling Y. Bazilevs, V.M. Calo, T.J.R. Hughes, and G. Scovazzi . . . . . . . . . . . . . . 103 An Immersed Interface Method in the Framework of Implicit Large-Eddy Simulation M. Meyer, A. Devesa, S. Hickel, X.Y. Hu, and N.A. Adams . . . . . . . . . . . 113 Simulation of Gravity-Driven Flows Using an Iterative High-Order Accurate Navier–Stokes Solver R. Henniger and L. Kleiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Compact Fourth-Order Finite-Volume Method for Numerical Solutions of Navier–Stokes Equations on Staggered Grids Arpiruk Hokpunna and Michael Manhart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 An Accurate Numerical Method for DNS of Turbulent Pipe Flow J.G.M. Kuerten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Local Large Scale Forcing of Unsheared Turbulence Julien Bodart, Laurent Joly, and Jean-Bernard Cazalbou . . . . . . . . . . . . . . 141

Contents

XI

Large-Eddy Simulations of a Turbulent Magnetohydrodynamic Channel Flow A. Vir´e, D. Krasnov, B. Knaepen, and T. Boeck . . . . . . . . . . . . . . . . . . . . . 147 Development of a DNS-FDF Approach to Inhomogeneous Non-Equilibrium Mixing for High Schmidt Number Flows Florian Schwertﬁrm and Michael Manhart . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Multi-Scale Simulation of Near-Wall Turbulent Flows Ayse Gul Gungor and Suresh Menon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Explicit Algebraic Subgrid Stress Models for Large Eddy Simulation L. Marstorp, G. Brethouwer, and A.V. Johansson . . . . . . . . . . . . . . . . . . . . 167 Scrutinizing the Leray-Alpha Regularization for LES in Turbulent Axisymmetric Free Jets F. Picano, C.M. Casciola, and K. Hanjali´c . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Localization of Unresolved Regions in the Selective Large-Eddy Simulation of Hypersonic Jets D. Tordella, M. Iovieno, S. Massaglia, and A. Mignone . . . . . . . . . . . . . . . 181 An ADM-Based Subgrid Scale Reconstruction Procedure for Large Eddy Simulation M. Terracol and B. Aupoix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Large-Eddy Simulation of Turbulent Flow in a Plane Asymmetric Diﬀuser by the Spectral-Element Method Johan Ohlsson, Philipp Schlatter, Paul F. Fischer, and Dan S. Henningson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 h and p Reﬁnement with Wall Modelling in Spectral-Element LES S. Hulshoﬀ, E. Munts, and J. Labrujere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Error-Landscape Assessment of LES Accuracy Using Experimental Data Johan Meyers, Charles Meneveau, and Bernard J. Geurts . . . . . . . . . . . . . 209 The Role of Diﬀerent Errors in Classical LES and in Variational Multiscale LES on Unstructured Grids H. Ouvrard, B. Koobus, A. Dervieux, S. Camarri, and M.V. Salvetti . . . 215

XII

Contents

Part III LES Modelling Errors Practical Quality Measures for Large-Eddy Simulation S.E. Gant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 The Simplest LES M. Germano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Application of an Anisotropy Resolving Algebraic Reynolds Stress Model within a Hybrid LES-RANS Method M. Breuer, O. Aybay, and B. Jaﬀr´ezic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 LES Meets FSI – Important Numerical and Modeling Aspects M. Breuer and M. M¨ unsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 A New Multiscale Model with Proper Behaviour in Both Vortex Flows and Wall Bounded Flows L. Bricteux, M. Duponcheel, and G. Winckelmans . . . . . . . . . . . . . . . . . . . . 253 LES Based POD Analysis of Jet in Cross Flow ˇ c . . . . . . . . . . . . . . . . . . . . . 259 D. Cavar, K.E. Meyer, S. Jakirli´c, and S. Sari´ A Dissipative Scale-Similarity Model L. Davidson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Optimization of Turbulent Mixing Restricted by Linear and Nonlinear Constraints Sara Delport, Martine Baelmans, and Johan Meyers . . . . . . . . . . . . . . . . . . 275 Stochastic Coherent Adaptive LES of Forced Isotropic Turbulence Giuliano De Stefano and Oleg V. Vasilyev . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 An Improvement of Increment Model by Using Kolmogorov Equation of Filtered Velocity L. Fang, L. Shao, J.P. Bertoglio, G.X. Cui, C.X. Xu, and Z.S. Zhang . . 287 Symmetry-Preserving Regularization Modelling of a Turbulent Plane Impinging Jet O. Lehmkuhl, F.X. Trias, R. Borrell, and C.D. P´erez Segarra. . . . . . . . . . 295 Progress in the Development of Stochastic Coherent Adaptive LES Methodology Oleg V. Vasilyev and Giuliano De Stefano . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Contents

XIII

Part IV Scalars LES of Heat Transfer in a Channel with a Staggered Pin Matrix G. Delibra, D. Borello, K. Hanjali´c, and F. Rispoli . . . . . . . . . . . . . . . . . . . 311 Turbulent Channel Flow with Λ-Shape Turbulators on One Wall Jaime A. Toro Medina, Benjamin Cruz Perez, and S. Leonardi . . . . . . . . 317 Implicit Large-Eddy Simulation of Passive-Scalar Mixing in a Conﬁned Rectangular-Jet Reactor Antoine Devesa, Stefan Hickel, and Nikolaus A. Adams . . . . . . . . . . . . . . . 323 Direct Numerical Simulation of a Turbulent Boundary Layer with Passive Scalar Transport Qiang Li, Philipp Schlatter, Luca Brandt, and Dan S. Henningson . . . . . 329

Part V Active Scalars Numerical Experiments on Turbulent Thermal Convection Roberto Verzicco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Direct Numerical Simulation of Turbulent Reacting and Inert Mixing Layers Laden with Evaporating Droplets J. Xia and K.H. Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Large Eddy Simulation of a Two-Phase Reacting Flow in an Experimental Burner M. Sanjos´e, E. Riber, L. Gicquel, B. Cuenot, and T. Poinsot . . . . . . . . . . 355 Hybrid LES/CAA Simulation of a Turbulent Non-Premixed Jet Flame C. Klewer, F. Hahn, C. Olbricht, and J. Janicka . . . . . . . . . . . . . . . . . . . . . 363 LES/CMC of Forced Ignition of a Bluﬀ-Body Stabilised Non-Premixed Methane Flame A. Triantafyllidis, E. Mastorakos, and R.L.G.M. Eggels . . . . . . . . . . . . . . . 371 Large Eddy Simulation of a High Reynolds Number Swirling Flow in a Conical Diﬀuser C´edric Duprat, Olivier M´etais, and Guillaume Balarac . . . . . . . . . . . . . . . . 377 Direct Numerical Simulation of Hot and Highly Pulsated Turbulent Jet Flows V. Clauzon and T. Dubois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

XIV

Contents

DNS of Convective Heat Transfer in a Rotating Cylinder R.P.J. Kunnen, B.J. Geurts, and H.J.H. Clercx . . . . . . . . . . . . . . . . . . . . . 393 Numerical Simulations of Thermal Convection at High Prandtl Numbers G. Silano, K.R. Sreenivasan, and R. Verzicco . . . . . . . . . . . . . . . . . . . . . . . . 399 Inﬂuence of the Lateral Walls on the Thermal Plumes in Turbulent Rayleigh–B´ enard Convection in Rectangular Containers M. Kaczorowski and C. Wagner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 DNS of Mixed Convection in Enclosed 3D-Domains with Interior Boundaries Olga Shishkina, Andrei Shishkin, and Claus Wagner . . . . . . . . . . . . . . . . . . 411 LES and Hybrid RANS/LES of Turbulent Flow in Fuel Rod Bundle Arranged with a Triangular Array Stefano Rolfo, J.C. Uribe, and D. Laurence . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Large-Scale Patterns in a Rectangular Rayleigh–B´ enard Cell A. Sergent and P. Le Qu´er´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 LES and Laser Measurements of Dynamic Flame/Vortex Interactions V. Di Sarli, A. Di Benedetto, G. Russo, E.J. Long, and G.K. Hargrave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 3D Direct Simulation of a Nonpremixed Hydrogen Flame with Detailed Models G. Fru, D. Th´evenin, C. Zistl, G. Janiga, L. Gouarin, and A. Laverdant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Part VI Environmental and Multiphase Flows Large Eddy Simulation of Pollen Dispersion in the Atmosphere Marcelo Chamecki, Charles Meneveau, and Marc B. Parlange . . . . . . . . . . 441 Internal Wave Breaking in Stratiﬁed Flows Past Obstacles Sergey N. Yakovenko, T. Glyn Thomas, and Ian P. Castro . . . . . . . . . . . . 449 DNS of a Gravity Current Propagating over a Free-Slip Boundary Alberto Scotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Contents

XV

Large Eddy Simulation of Turbulent Mixing in an Estuary Region F. Roman, V. Armenio, R. Inghilesi, and S. Corsini . . . . . . . . . . . . . . . . . 463 Dispersion of (Light) Inertial Particles in Stratiﬁed Turbulence M. van Aartrijk and H.J.H. Clercx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 The Inﬂuence of Magnetic Fields on the Rise of Gas Bubbles in Electrically Conductive Liquids Daniel Gaudlitz and Nikolaus A. Adams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Large Eddy Simulation of a Turbulent Droplet Laden Mixing Layer W.P. Jones, S. Lyra, and A.J. Marquis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 The Diﬀuse Interface Method with Korteweg Approach for Isothermal, Two-Phase Flow of a Van der Waals Fluid A. Pecenko and J.G.M. Kuerten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Numerical Simulation of Air Flows in Street Canyons Using Mesh-Adaptive LES Dimitrios Pavlidis, Elsa Aristodemou, Jeﬀerson L.M.A. Gomes, Christopher C. Pain, and Helen ApSimon . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Part VII Aerodynamics and Wakes LES of the Flow Around a Two-Dimensional Vehicle Model with Active Flow Control S. Krajnovi´c and J. Fernandes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Wake-Vortex Decay in External Turbulence Bernard J. Geurts and Arkadiusz K. Kuczaj . . . . . . . . . . . . . . . . . . . . . . . . . 513 DNS of Aircraft Wake Vortices: The Eﬀect of Stable Stratiﬁcation on the Development of the Crow Instability G.N. Coleman, R. Johnstone, C.P. Yorke, and I.P. Castro . . . . . . . . . . . . 519 On the Download Alleviation for the XV-15 Wing by Active Flow Control Using Large-Eddy Simulation M. El-Alti, P. Kjellgren, and L. Davidson . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Turbulent Flow Simulations Around an Airfoil At High Incidences Using URANS, DES and ILES Approaches Bowen Zhong, Satish K. Yadav, and Dimitris Drikakis . . . . . . . . . . . . . . . . 533

XVI

Contents

Large Eddy Simulation of Flow Around an Airfoil Near Stall Jaber H. Almutairi, Lloyd E. Jones, and Neil D. Sandham . . . . . . . . . . . . 541 Large Eddy Simulation of Turbulent Flows Around a Rotor Blade Segment Using a Spectral Element Method A. Shishkin and C. Wagner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Part VIII Compressible Flows DNS of Compressible Turbulent Flows Rainer Friedrich and Somnath Ghosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Large-Eddy Simulation of Transonic Buﬀet over a Supercritical Airfoil E. Garnier and S. Deck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 Detached-Eddy and Delayed Detached-Eddy Simulation of Supersonic Flow over a Three-Dimensional Cavity V. Togiti, H. L¨ udeke, and M. Breuer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 A WALE-Similarity Mixed Model for Large-Eddy Simulation of Wall Bounded Compressible Turbulent Flows G. Lodato, P. Domingo, and L. Vervisch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Parametric Study of Compressible Turbulent Spots J.A. Redford, N.D. Sandham, and G.T. Roberts . . . . . . . . . . . . . . . . . . . . . . 587 Azimuthal Resolution Eﬀects in LES of Subsonic Jet Flow and Inﬂuence on Its Noise F. Keiderling and L. Kleiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 Large Eddy Simulations of Compressible MHD Turbulence in Heat-Conducting Fluid A.A. Chernyshov, K.V. Karelsky, and A.S. Petrosyan . . . . . . . . . . . . . . . . 601

Part I

Fundamentals

Wall-Modeled Large-Eddy Simulations: Present Status and Prospects Ugo Piomelli Department of Mechanical and Materials Engineering, Queen’s University, Kingston (Ontario), Canada, [email protected]

Abstract The most common techniques used to perform large-eddy simulations of high-Reynolds number wall-bounded ﬂows are reviewed. The main sources of error of this approach are discussed, and considerations on future directions for development are made.

1 Statement of the problem The fundamental assumption underlying large-eddy simulations (LES) is that the integral scales of motion, which are responsible for most of the momentum transport and mixing, are resolved by the grid. Only the smaller eddies, which are more isotropic, are modelled. One key feature that determines the accuracy of LES is whether, and how well, the integral scale is resolved, and the main limitation to this methodology comes from situations in which its accurate resolution becomes prohibitively expensive. As observed by Pope [23], LES have been very successful in free shear layers, in which the eddies that control momentum transfer and mixing can be resolved very well, even at high Reynolds number. Conversely, whenever the rate-controlling processes have to be modelled, LES has been less successful. Near-wall ﬂows are an example of the latter class: there, the size of the eddies that govern momentum transfer depends strongly on the Reynolds number; simulations that resolve those eddies have resolutions approaching those of Direct Numerical Simulations (DNS). Alternatively, these scales of motion may be modelled, but a strong dependence on the model chosen is then introduced. In this article, some issues related to the large-eddy simulation of wallbounded ﬂows will discussed. A brief summary of existing approaches to the problem will be presented (more extended discussions of these techniques can be found in [5,19,20]). Next, we will discuss some of the problems and sources of error encountered when the near-wall eddies are modelled. Some indications for future work will conclude the paper. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 1, c Springer Science+Business Media B.V. 2010

3

4

U. Piomelli

2 Near-wall modelling The resolution required by LES has been derived in various references [6, 20, 25, 28]. In wall-bounded ﬂows the integral scale, which must be resolved in accurate LES, is proportional to the boundary-layer thickness, δ, away from walls. A reasonable estimate of the grid spacing in each direction is then Δxi δ/25 − δ/15. For boundary-layer ﬂows, in which δ ∝ Re−0.2 , this results in a number of grid points required to resolve the outer layer proportional to Re0.4 (where the Reynolds number is based on outer parameters of the ﬂow such as the free-stream velocity and the boundary-layer thickness), and the cost scales like Re0.6 . In the inner layer region,1 the Re-dependence of the resolution is much steeper, since the there eddies that need to be resolved scale with wall units. As the Reynolds number is increased, the physical dimensions of these eddies decrease much more rapidly than the boundary layer thickness, resulting in more stringent grid requirements. Chapman [6] estimated that the number of points required to resolve the inner layer is proportional to Re1.8 and the cost to Re2.4 . This cost estimate is shown in Fig. 1, in which the CPU time required to compute a turbulent boundary layer is estimated for three diﬀerent codes used by the author: a Cartesian code with a staggered, second-order discretization; a co-located ﬁnite-volume code suited for body-ﬁtted grids, also second-order in space and time; an unstructured, second-order accurate, ﬁnite-volume code. The cost per time-step and grid point was measured on a 2.4 GHz AMD Opteron processor for a

Fig. 1. Cost, in CPU seconds, of the LES of ﬂat-plate boundary-layer ﬂow. The calculations were performed on an AMD Opteron, using two in-house codes and an open-source unstructured one. Actual mileage may vary. 1

The inner layer is deﬁned here as the lower 10% of the boundary-layer thickness. For a high-Reynolds number boundary layer, this would include the viscous sublayer, the buﬀer region and part of the logarithmic layer.

Wall-Modeled Large-Eddy Simulations: Present Status and Prospects

5

calculation at a given Re, and the scaling laws derived above were used to extend the cost estimate to higher Re. Even at Reynolds numbers o(104 ) over 50% of the resources are used to resolve only 10% of the ﬂow. For Re > 5×104 a vanishing fraction of the grid points is used in the outer layer. As a consequence of this unfavorable scaling, wall-resolved LES are limited to very moderate Reynolds numbers. Only if massive computational resources (clusters with thousands of processors) are available, calculations at Reynolds numbers of marginal engineering interest, Re = o(105 ), are possible. Resolved LES is clearly not suitable for design, in which substantially more rapid throughput is required to evaluate and compare possible designs within strict time limits, or to study aerodynamic or geophysical ﬂows at Re = o(106 − 109 ). This limitation was recognized from the earliest applications of LES, and attempt to bypass the inner layer and model its eﬀects in a global sense have been used. The basic consideration, in simulations in which the inner layer is modeled (Wall-Modeled LES, or WMLES), is that the grid near the wall is too coarse to resolve the eddies that contribute to the momentum transport. If the mixing due to the near-wall eddies is not computed, and no-slip conditions are applied, an incorrect velocity proﬁle results, and under-prediction of the wall stress. Therefore, one must model the transport of momentum by the innerlayer scales either implicitly, by relating the outer-layer velocity to the wall stress through an assumed velocity proﬁle, or explicitly, by parameterizing their eﬀect in the Reynolds-averaged sense. The critical assumption that must be made is that the near-wall grid is so coarse that it contains a very large (→ ∞) number of near-wall eddies, and that the time-step is much larger than their time-scale, so that the ﬁltering operation becomes equivalent to long-time averaging. Early calculations [7, 26] bypassed the inner layer by using approximate boundary conditions similar to the wall functions applied in ReynoldsAveraged Navier–Stokes simulations with turbulence models. These boundary conditions assume the existence of an equilibrium layer in which the stress is constant. This results in the existence of a logarithmic velocity proﬁle that can be used to relate the velocity in the outer layer to the wall stress. The development of more powerful computers resulted in wall-resolved LES of wall-bounded ﬂows at low or moderate Reynolds numbers (for instance, Moin and Kim [17]), which allowed more detailed studies of the turbulence physics. Approximate boundary conditions, however, remained in widespread use in environmental and geophysical ﬂows. In these ﬂows the geometry is generally quite simple, and the Reynolds number extremely high, so that the equilibrium layer assumption holds in most situations; corrections can be made for buoyancy, roughness and other eﬀects common in these applications. In engineering ﬂows the use of approximate boundary conditions based on the equilibrium-stress layer is less applicable. In the presence of strong favorable or adverse pressure gradients, in separated ﬂows or in ﬂows in which the mean velocity is highly three-dimensional, the assumption of the existence

6

U. Piomelli

of a logarithmic law does not hold (see for example [2, 27]). This prompted ﬁrst the derivation of corrections for models based on the logarithmic law, and then the development of hybrid models in which simpler transport equations are solved in the inner layer, weakly or strongly coupled to the outer-ﬂow LES. The ﬁrst model of this type was the Two-Layer Model (TLM), originally proposed by Balaras and coworkers [1,2]. A ﬁne one-dimensional grid is embedded between the ﬁrst grid point and the wall, and a simpliﬁed set of equations (generally, the Reynolds-Averaged turbulent boundary-layer equations) is solved in the embedded mesh. The outer-layer LES provide the boundary condition for the inner layer, whereas the inner-layer calculation provides the wall stress required by the LES. This model can account for somewhat more complex physics than the wall-boundary conditions based on the logarithmic law, but still has weak coupling between the inner and outer layers: it works well when the inner layer is forced by the outer one, but tends to be inaccurate when the ﬂow is driven by the wall region. Other models of this type were proposed by Cabot and Moin [5] and Kemenov and Menon [13]. Hybrid simulations in which the Reynolds-Averaged Navier–Stokes equations are used in the inner layer, while the ﬁltered Navier–Stokes equations are solved in the outer layer, may have become the most common method to perform WMLES. Several strategies can be used to switch between RANS simulations and LES, such as changing the length-scale in the model from a RANS mixing length to one related to the grid size, or using a blending function to merge the SGS and RANS eddy viscosities. An extensive review of previous work in this area is beyond the scope of this article, given the widespread use this method is acquiring; an overview of recent approaches, however, can be found in Piomelli [19].

3 Sources of error in WMLES The modeling of the near-wall layer introduces several sources of error in an LES. The three main ones are the inaccuracy of the wall-layer modelling assumptions, the inapplicability of the SGS models in the near-wall region, and the increased numerical errors that appear when the resolution becomes marginal, or the grid is suddenly reﬁned near a solid boundary. These errors interact with each other, but a global treatment is diﬃcult: Nicoud et al. [18] used suboptimal control theory to supply a wall stress that forced the outer LES to the desired mean velocity proﬁle. With this method the approximate boundary condition compensates for subgrid-scale modeling and numerical errors. Nicoud et al. [18] obtained improved results over standard approximate boundary conditions. This approach, however, is infeasible as a predictive tool, since it requires the solution to be known a priori to generate the desired cost function.

Wall-Modeled Large-Eddy Simulations: Present Status and Prospects

7

Fig. 2. Wall stress as a function of phase. From [24].

The ﬁrst class of errors is particularly signiﬁcant in simulations that use the equilibrium-stress hypothesis to relate the outer-layer velocity to the wall stress. Despite the modiﬁcations of the logarithmic law that can account for many non-equilibrium eﬀects (mild pressure gradients, roughness or stratiﬁcation, for instance), in many cases this approach is not accurate. In some cases the errors can be acceptable: Radhakrishnan and Piomelli [24], for instance, studied an oscillating boundary layer; during approximately one quarter of the cycle the logarithmic law does not hold, but the error is small, perhaps because the equilibrium assumption fails around the phase where the wall stress vanishes (as can be seen in Fig. 2). In other cases, especially when the ﬂow has separation, the error may be more signiﬁcant. In simulations of the ﬂow in an S-shaped duct, for instance, Silva Lopes et al. [27] observed that the approximate boundary conditions based on the logarithmic law overestimated the deceleration in regions of curvature and adverse pressure gradient. In cases with mild separation, the separation region may be entirely below the ﬁrst grid point (which in calculations with approximate boundary conditions may be 1/10th of the boundary layer thickness from the wall); in that case the logarithmic law will not predict the skin friction accurately. Another issue that is important in the simulations that use approximate boundary conditions is the accuracy of the SGS model itself. Near the wall, the integral scale of the ﬂow is proportional to the distance from the solid boundary itself, κy, and Δy = y. Thus, the basic assumption that the grid size is ﬁne enough to resolve the integral scale of the ﬂow, fails. Near the wall, then, the subgrid-scale model is required to resolve all the scales of motion, i.e., to behave like a turbulence model for the RANS equations. This consideration has suggested splitting the model for the SGS stresses into two parts [17, 26, 29]: τij = −2νt S ij − 2νT S ij . (1)

8

U. Piomelli

Fig. 3. Contours of the streamwise velocity ﬂuctuations in planes parallel to the wall. Plane channel ﬂow at Reτ = 5,000. From [21].

Here, the ﬁrst term is the “ﬂuctuating” eddy viscosity, which represents the subgrid eddies, while the second is the “average” term that depends only on mean-ﬂow quantities and plays the same role as the eddy viscosity in a RANS model. Hybrid methods do not suﬀer from this problem, since the RANS model is used to parameterize all the scales of motion in the near-wall region, and the LES zone is restricted to the region of the ﬂow where the grid size is suﬃciently smaller than the integral scales to ensure its proper resolution. While this technique allows to study problems in which the wall-layer is far from equilibrium, a new source of error is introduced at the interface between the RANS and LES regions. Here the disparity between the long time- and length-scales imposed by the inner-layer RANS and the smaller eddies required by the LES to support the momentum transport may result in errors. This is shown in Fig. 3. “Super-streaks” can be observed in the velocity-ﬂuctuation contours. In the RANS region (z + < 225) the ﬂow is not entirely smooth: signiﬁcant ﬂuctuation levels exist, but the size of the eddies in this region is unphysical. Also, one can observe a strong correlation between the velocity ﬁelds for z + < 1500. Only above this value shorter-scale eddies are formed, and the more isotropic eddies that are expected in the outer ﬂow can be observed. This error usually results in an overestimation of the velocity gradient in the interface region, and an underestimation of the wall stress. To remove them, turbulent ﬂuctuations are usually added in the interface region to accelerate the generation of momentum-transporting eddies; several methods have been proposed (see [19] for a review). None of them seems to have the robustness that would make it generally applicable. Germano [8] derived the governing equations obtained when a hybrid RANS/LES ﬁlter is applied to the Navier–Stokes equations. Given a RANS ﬁltering operator E, an LES ﬁltering operator F and a hybrid ﬁlter

Wall-Modeled Large-Eddy Simulations: Present Status and Prospects

9

H = (1 − γ)E + γF (where γ is some explicitly or implicitly applied blending function), the momentum equation takes the following form H ∂τijH ∂(uH ∂uH ∂ 2 uH 1 ∂pH j ui ) i i + =ν − − ∂t ∂xj ∂xj ∂xj ∂xj ρ ∂xi ∂γ ∂ui F δij pF + ui F uj F − ν + ∂xj ∂xj ∂ui E . − δij pE + ui E uj E − ν ∂xj

(2)

The additional terms that appear on the right-hand side are localized in the interface region (where ∂γ/∂xi is non-zero) and have the form of scale-similar models. The term ui F uj F − ui E uj E , for instance, represents the ﬂuctuations of the resolved velocity correlation. This term is unclosed, and deriving a more rigorous model for it might be more robust then the present heuristic approaches. One last issue that needs to be considered is the numerical error in the wall layer. Discussions of the eﬀect of numerical errors in LES of wall-bounded ﬂows can be found in several references [10,11,14,16]. These papers considered structured grids, in which the spacing is constant in planes parallel to the wall, and smoothly increased or decreased in the direction normal to the wall. Such grid is not optimal for the simulation of wall-bounded ﬂows: the inner layer need to be reﬁned in all directions, since the near-wall structures are much ﬁner than the outer-layer ones in the streamwise and spanwise directions. The use of embedded grids in LES was proposed by Kravchenko et al. [15], who performed wall-resolved simulations of channel ﬂows at very high Reynolds numbers. One of the grids they used is shown in Fig. 4. They remark that the ratio of grid spacing between successive layers had to be fairly close to one (in other words, the grid had to be nearly continuous between successive layers). In recent application of LES to complex geometries that use local mesh reﬁnement, however, the grid is generally reﬁned by a factor of two in each direction. The discontinuity of the grid spacing, in LES, has two consequences:

Fig. 4. Examples of discontinuous grids for LES. Left: channel (from [15]); right: cylinder inside a channel (from [22]).

10

U. Piomelli

ﬁrst commutator errors may be signiﬁcant; second, the SGS model may give unphysical jumps in the eddy viscosity. Commutator errors occur when ﬁltering and diﬀerentiation do not commute: ∂f /∂xi = ∂f /∂xi . They have been studied extensively by Geurts and co-authors [3,4,9], who found that they are of the same order as the SGS stress divergence, and proposed explicit models to account for this error. If the ﬁlterwidth is proportional to the grid size (as is the case in most LES), commutator errors can be expected to be very signiﬁcant when local reﬁnement is used. Using a ﬁlter-width proportional to the grid size also leads to discontinuities in the eddy viscosity, which is usually proportional to the square of the ﬁlter width. Recent studies of LES on discontinuous grids [22, 30] indicate that the explicit ﬁltering of the non-linear term is beneﬁcial when coupled with a ﬁlter width that is decoupled from the grid size. This ensures that the size of the eddies crossing a grid discontinuity is large enough that they can be resolved on the coarse side, commutator errors are reduced.

4 Conclusions The purpose of this article was to alert the reader to the many challenges faced by large-eddy simulations in which the wall-layer is modelled. Given the complexities of the ﬂow in the inner layer, and its importance in establishing the production cycle [12], one cannot expect that modeling its eﬀects on the outer ﬂow may be easy or painless, or that equal accuracy can be achieved compared with wall-resolved calculations. It is, nonetheless, critical to develop reasonably accurate wall-layer models, and also to estimate the errors that can be expected from WMLES. Despite the diﬃculties WMLES ﬁnd in predicting wall-layer related quantities (some of them, such as the skin-friction coeﬃcient, quite critical), the outer-layer physics are predicted quite accurately (as long as enough resolution is used – of the order of 20 points per integral length scale). Several methodologies exist, but none of them is generally applicable or robust enough to be clearly superior to the others. The logarithmic law is accurate if the ﬂow has only mild perturbations from equilibrium, and if the inner layer is driven by the outer one. In those cases it should be the method of choice, because of its simplicity, low computational cost, and because it does not introduce additional length and time scales. Zonal models are accurate when the perturbations are stronger, but still assume that the inner layer is driven by the outer one. They are somewhat more diﬃcult to implement than the logarithmic law, but their cost is also negligible compared with the rest of the calculation. Hybrid RANS/LES approaches may include more complex physics, and correct the SGS modelling errors discussed above. However, the transition between RANS and LES regions is not trivial. Several methods have been proposed to facilitate the generation of resolved eddies in this region; they all

Wall-Modeled Large-Eddy Simulations: Present Status and Prospects

11

work in the test cases, but none seems generally applicable. Perhaps, a better understanding of the stability characteristics of the hybrid problem may lead to more eﬃcient and robust forcing methods.

References 1. Balaras, E., Benocci, C.: Subgrid-scale models in ﬁnite-diﬀerence simulations of complex wall bounded ﬂows. In: AGARD CP 551, pp. 2.1–2.5. AGARD, AGARD, Neuilly-sur-Seine, France (1994) 2. Balaras, E., Benocci, C., Piomelli, U. (1996) Two layer approximate boundary conditions for large-eddy simulations. AIAA J. 34:1111–1119 3. van der Bos, F., Geurts, B.J. (2005) Commutator errors in the ﬁltering approach to large-eddy simulation. Phys. Fluids 17(035108):1–20 4. van der Bos, F., Geurts, B.J. (2005) Lagrangian dynamics of commutator errors in large-eddy simulation. Phys. Fluids 17(075101):1–15 5. Cabot, W.H., Moin, P. (2000) Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number ﬂows. Flow, Turb. Combust. 63:269–291 6. Chapman, D.R. (1979) Computational aerodynamics development and outlook. AIAA J. 17:1293–1313 7. Deardorﬀ, J.W. (1970) A numerical study of three-dimensional turbulent channel ﬂow at large Reynolds numbers. J. Fluid Mech. 41:453–480 8. Germano, M. (2004) Properties of the hybrid RANS/LES ﬁlter. Theor. Comput. Fluid Dyn. 17(4):225–231 9. Geurts, B.J., Holm, D.D. (2006) Commutator errors in large-eddy simulation. J. Phys. A: Math. Gen 39:2213–2229 10. Ghosal, S., Moin, P. (1995) The basic equations for the large-eddy simulation of turbulent ﬂows in complex geometries. J. Comput. Phys. 118:24–37 11. Gullbrand, J., Chow, F.K. (2003) The eﬀect of numerical errors and turbulence models in large-eddy simulations of channel ﬂow, with and without explicit ﬁltering. J. Fluid Mech. 495:323–341 12. Jim´enez, J., Pinelli, A. (1999) The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389:335–359 13. Kemenov, K.A., Menon, S. (2007) Explicit small-scale velocity simulation for high-Re turbulent ﬂows. Part II: Non-homogeneous ﬂows. J. Comput. Phys. 222:673–701 14. Kravchenko, A.G., Moin, P. (1997) On the eﬀect of numerical errors in large eddy simulations of turbulent ﬂows. J. Comput. Phys. 131:310–322 15. Kravchenko, A.G., Moin, P., Moser, R.D. (1996) Zonal embedded grids for numerical simulations of wall-bounded turbulent ﬂows. J. Comput. Phys. 127: 412–423 16. Lund, T.S. (2003) The use of explicit ﬁlters in large-eddy simulations. Comput. Math. Appl. 46:603–616 17. Moin, P., Kim, J. (1982) Numerical investigation of turbulent channel ﬂow. J. Fluid Mech. 118:341–377 18. Nicoud, F., Baggett, J.S., Moin, P., Cabot, W.H. (2001) Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation. Phys. Fluids 13:2968–2984. DOI 10.1063/1.1389286

12

U. Piomelli

19. Piomelli, U. (2008) Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci. 44(6):437–446 20. Piomelli, U., Balaras, E. (2002) Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34:349–374 21. Piomelli, U., Balaras, E., Pasinato, H., Squires, K.D., Spalart, P.R. (2003) The inner-outer layer interface in large-eddy simulations with wall-layer models. Int. J. Heat Fluid Flow 24:538–550 22. Piomelli, U., Kang, S., Ham, F., Iaccarino, G.: Eﬀect of discontinuous ﬁlter width in large-eddy simulations of plane channel ﬂow. In: Studying turbulence using numerical databases XI, pp. 151–162. Stanford University (2006) 23. Pope, S.B. (2004) Ten questions concerning the large-eddy simulation of turbulent ﬂows. New J. Phys. 6(35):1–24 24. Radhakrishnan, S., Piomelli, U. (2008) Large-eddy simulation of oscillating boundary layers: model comparison and validation. J. Geophys. Res. 113(C02022):1–14 25. Reynolds, W.C.: The potential and limitations of direct and large eddy simulations. In: J.L. Lumley (ed.) Whither Turbulence? Turbulence at the Crossroads, Lecture Notes in Physics, vol. 357, pp. 313–343. Springer-Verlag, Berlin (1990) 26. Schumann, U. (1975) Subgrid-scale model for ﬁnite diﬀerence simulation of turbulent ﬂows in plane channels and annuli. J. Comput. Phys. 18:376–404 27. Silva Lopes, A., Piomelli, U., Palma, J.M.L.M.: Large-eddy simulation of a ﬂow with curvatures using wall-models. In: H.I. Andersson, P.˚ A. Krogstad (eds.) Advances in Turbulence X, pp. 249–252. CIMNE, Barcelona (2004) 28. Spalart, P.R. (2000) Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 21:252–263 29. Sullivan, P.P., McWilliams, J.C., Moeng, C.H. (1994) A subgrid-scale model for large-eddy simulations of planetary boundary layers. Bound.-Lay. Meteorol. 71(2):247–276 30. Vanella, M., Piomelli, U., Balaras, E. (2008) Eﬀect of grid discontinuities in large-eddy simulation statistics and ﬂow ﬁelds. J. Turbul. 9(32):1–23

A Study of the Inﬂuence of the Reynolds Number on Jet Self-Similarity Using Large-Eddy Simulation Christophe Bogey1 and Christophe Bailly2 1

2

Laboratoire de M´ecanique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, 69134 Ecully Cedex, France, [email protected] Laboratoire de M´ecanique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, 69134 Ecully Cedex, and Institut Universitaire de France, [email protected]

Abstract Three round jets at Reynolds numbers 1,800, 3,600 and 11,000 are computed using Large-Eddy Simulations based on low-dissipation numerical schemes and explicit selective ﬁltering, in order to study the inﬂuence of the Reynolds number on jet self-similarity. At lower Reynolds number, the jet ﬂow achieves self-similarity more rapidly, and then develops at a higher rate. The eﬀects of the Reynolds number on the velocity moments as well as on the budget for the turbulent kinetic energy across the self-similar jet ﬂow are however found to be weak.

1 Introduction The self-similarity region of jet ﬂows has been investigated extensively over the last ﬁfty years. Reference solutions for round jets have for instance been provided by the experimental works of Wygnanski and Fiedler [1], Panchapakesan and Lumley [2] and Hussein et al. [3]. Discrepancies between the solutions are however observed. They are expected to result from diﬀerences in the measurement methods and in the jet initial conditions, the diameter-based Reynolds numbers ranging in particular from 1.1 × 104 to 105 . The eﬀects of the Reynolds number on jet ﬂow development have indeed been shown to be signiﬁcant up to Reynolds numbers around ReD = uj D/ν 104 , where uj and D are the jet inlet velocity and diameter, and ν is the kinematic molecular viscosity. They have been examined experimentally specially ¨ ugen [5], Kwon and Seo [6] and by Lemieux and Oostuizen [4], Namer and Ot¨ Deo et al. [7], as well as numerically by Bogey and Bailly [8]. In experiments, variations of the Reynolds number might however result in modiﬁcations of other ﬂow initial conditions, whereas the latter computational study only dealt with the transitional jet region. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 2, c Springer Science+Business Media B.V. 2010

13

14

C. Bogey and C. Bailly

It appears therefore now worthwhile to study the inﬂuence of the Reynolds number on jet self-similarity using simulations. Computations of self-similar jets have been performed previously by Boersma et al. [9], Freund [11] and Uddin and Pollard [12] for instance. In Bogey and Bailly [10], a round jet at a Reynolds number 11,000 has also been calculated by a Large Eddy Simulation (LES) using an approach combining low-dissipation schemes [13] and relaxation ﬁltering [8]. Jet self-similarity was reached on the computational domain, and described in detail. A good agreement with the data obtained by Panchapakesan and Lumley [2] at the same Reynolds number was found. Given these results, two jets with the same initial conditions as the jet at Reynolds number 11,000 except for the diameter, providing Reynolds numbers of 1,800 and 3,600, have been simulated by LES with the aim of characterizing the inﬂuence of the Reynolds number on self-similar round jets. Preliminary comparisons are reported here.

2 Simulation parameters Three round jets at Mach number 0.9 and at Reynolds numbers 1,800, 3,600 and 11,000 are computed by LES using the same numerical set-up, on grids containing from 18 to 44 millions of points, extending, respectively up to 90, 120 and 150 jet radii r0 in the downstream direction, see in Table 1 for additional parameters. The jet inﬂow conditions such as the mean ﬂow proﬁles, the shear-layer thickness or the forcing used to seed the turbulent transition are identical except for the diameter. They are described in detail in a previous paper [10]. The LES are performed using low-dispersion low-dissipation ﬁnite-diﬀerence and Runge–Kutta schemes [13], in combination with the application of an explicit ﬁltering to the ﬂow variables in order to remove the smaller scales discretized without appreciably aﬀecting the larger scales. This LES methodology has been used successfully for transitional jets [8,14]. It does not lead in particular to an artiﬁcial decrease of the eﬀective Reynolds number of the ﬂow, as it might be the case with eddy-viscosity-based LES modellings. In addition the budgets for the turbulent kinetic energy are also computed directly from the ﬂow-governing equations. All the energy terms are estimated explicitly [10]. Table 1. Number of grid points and simulation time, as well as location of the end of the potential core xc , decay constant B, spreading rate A and centerline ﬁltering activity sf ilt in the self-similar jets. ReD

nx × ny × nz

T uj /D

xc

B

A

sf ilt

1,800 411 × 211 × 211 0.77 × 10 24.1r0 5.8 0.096 0.082 3,600 531 × 261 × 261 0.77 × 105 17.1r0 6.1 0.091 0.153 11,000 651 × 261 × 261 1.35 × 105 13.5r0 6.4 0.087 0.334 5

A Study of the Inﬂuence of the Reynolds Number on Jet Self-Similarity

15

3 Results In the jets, mean ﬂow and turbulence properties, including the second-order and third-order velocity moments, and the energy budgets, have been evaluated from the LES ﬁelds over a long time period to ensure statistical convergence. Some comparisons are provided here to give some insight into the Reynolds number eﬀects on jet self-similarity. A ﬁrst illustration is provided by the vorticity ﬁelds presented in Fig. 1. The transitions from laminar shear layers toward fully-developed turbulence can be seen. As the Reynolds number increases, the presence of ﬁne scales is more visible. The initial development of the jet occurs also more rapidly, leading to a decrease of the length of the potential core, in agreement with experimental ﬁndings. The core lengths xc , deﬁned by uc (xc ) = 0.95uj where uc is the centerline mean axial velocity, are consequently 24.1r0 , 17.1r0 and 13.5r0 at Reynolds numbers 1,800, 3,600 and 11,000, refer to Table 1. To determine the axial locations at which the simulated jet ﬂows achieve self-similarity, proﬁles of turbulence intensities along the centerline are presented in Fig. 2. The establishment of self-similarity, obtained when constant values are observed on the jet axis, is shown to take place more rapidly at lower Reynolds number, i.e., closer to the end of the potential core, in agreement with experimental and numerical results [8,15]. More quantitatively, selfsimilarity seems to be reached around x = 60r0 , 70r0 and 120r0 at Reynolds numbers 1,800, 3,600 and 11,000. The eﬀects of the Reynolds number of the mean ﬂow development are considered from the variations of the centerline mean axial velocity uc . The

Fig. 1. Snapshots of vorticity norm |ω| × x/uj , in the plane z = 0, for the jets at Reynolds numbers: (a) 1,800, (b) 3,600 and (c) 1,1000. The color scale ranges for levels from 4 to 20.

C. Bogey and C. Bailly 0.32

0.32

0.24

0.24

[v′v′]1/2 / uc

[u′u′] 1/2 / uc

16

0.16 0.08 0

0.16 0.08

0

30

60

90

120

0

150

0

30

60

x / r0

90

120

150

x / r0

12

0.25

10

0.2

8

0.15

1/B′

uj / u c

Fig. 2. Variations along the jet centerline of turbulence intensities [u u ]1/2 /uc (left) and [v v ]1/2 /uc (right), for Reynolds numbers 1,800 (dotted lines), 3,600 (dashed lines) and 11,000 (solid lines).

6 4

0.05

2 0

0.1

0

30

60

90 x/r0

120

150

0

0

30

60

x/r0

90

120

150

Fig. 3. Axial variations of the inverse of centerline mean velocity uj /uc (left) and of the inverse of local decay constant 1/B (right), for Reynolds numbers 1,800 (dotted lines), 3,600 (dashed lines) and 11,000 (solid lines).

inverse of uc is represented in Fig. 3 for the three jets. After the transitional region, it seems to vary linearly as expected. In self-preserving round jets, the mean velocity evolves indeed as uc /uj = B × D/(x − x0 ) where B is the decay constant and x0 denotes a virtual origin. The establishment of mean ﬂow self-similarity is then investigated by plotting the inverse of the local decay constant deﬁned as 1/B = d(uj /uc )/d(x/D) in Fig. 3. This mean ﬂow parameter tends to asymptotic values in the downstream direction, indicating self-similarity and providing decay constants B = 5.8, 6.1 and 6.4 at ReD = 1,800, 3,600 and 11,000, respectively. In the same way, values of 0.096, 0.091 and 0.087 are reported in Table 1 for the spreading rates A governing the variations of the jet half-width δ0.5 . The self-similar mean ﬂow therefore develops at a higher rate at lower Reynolds number, in agreement with the recent experimental data obtained for plane jets by Deo et al. [7]. To characterize the turbulent ﬁelds in the self-similar jets, the proﬁles of second-order and third-order velocity moments are calculated across the jets, and plotted as functions of y/δ0.5 . Illustrations are given in Fig. 4 with the

A Study of the Inﬂuence of the Reynolds Number on Jet Self-Similarity 0.002

0.3

0.001

0.2

[u′u′v′] / uc3

[u′u′]1/2/uc

0.25

0.15 0.1

0 −0.001

0.05 0

17

0

0.5

1

1.5

2

−0.002

2.5

0

0.5

y/δ0.5

1

1.5 y/δ0.5

2

2.5

Fig. 4. Proﬁles of second and third-order velocity moments [u u ]1/2 /uc (left) and [u u v ]/u3c (right) across the self-similar jets, for Reynolds numbers 1,800 (dotted lines), 3,600 (dashed lines) and 1,1000 (solid lines). 0.015

convection

0.01

production

budget for k

0.005 0

pres diff

−0.005

turb diff

−0.01 −0.015 −0.02

dissipation 0

0.5

1

1.5

2

2.5

y/δ0.5

Fig. 5. Budget for the turbulent kinetic energy k across the self-similar jets at Reynolds numbers 1,800 (dotted lines), 3,600 (dashed lines) and 11,000 (solid lines): convection, production, dissipation, turbulence diﬀusion and pressure diﬀusion.

proﬁles obtained for [u u ]1/2 /uc and [u u v ]/u3c . For the former quantity, the curves fairly collapse, whereas for the latter, the magnitude of the velocity moment only slightly increases when the Reynolds number decreases. The inﬂuence of the Reynolds number therefore appears weak on the ﬂow features considered, over the given Reynolds number range. Finally, the budgets calculated for the turbulent kinetic energy in the selfsimilarity regions of the three jets are represented in Fig. 5. Dissipation is here the sum of the viscous and ﬁltering dissipations, whose relative contributions vary with the Reynolds number. On the centerline for example, the ﬁltering dissipation is 33% of the total energy dissipation at ReD = 11,000 but only 8% at ReD = 1,800, as indicated in Table 1 by the centerline ﬁltering activity sf ilt . The shapes obtained for the diﬀerent energy terms, namely for mean ﬂow convection, production, dissipation, turbulence diﬀusion and pressure diﬀusion,

18

C. Bogey and C. Bailly

are very similar. The magnitudes of the energy terms only appear to be higher at lower Reynolds number. This is particularly the case for production, dissipation and mean ﬂow convection, whereas turbulence diﬀusion is surprisingly nearly unchanged. The present results suggest that the balance between the diﬀerent turbulent mechanisms in the self-similar jets does not depend significantly on the Reynolds number.

4 Conclusion The present LES show that in round jets the distance required to achieve selfsimilarity as well as the development of the self-similar jet mean ﬂow depend on the Reynolds number over the range 1,800 ≤ ReD ≤ 11,000, but that the eﬀects are quite weak on the turbulence features. Further analyses will be done to support this contention.

Acknowledgments The authors gratefully acknowledge the Institut du D´eveloppement et des Resources en Informatique Scientifique of the CNRS and the Centre de Calcul Recherche et Technologie of the CEA for providing CPU time on Nec computers and for technical assistance.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Wygnanski I, Fiedler H (1969) J Fluid Mech 38(3): 577–612 Panchapakesan NR, Lumley JL (1993) J Fluid Mech 246: 197–223 Hussein HJ, Capp SP, George WK (1994) J Fluid Mech 258: 31–75 Lemieux GP, Oosthuizen PH (1985) AIAA J 23: 1845–1847 ¨ ugen MV (1988) Exp. Fluids 6: 387–399 Namer I, Ot¨ Kwon SJ, Seo IW (2005) Exp. Fluids 38: 801–812 Deo RC, Mi J, Nathan GJ (2008) Phys. Fluids (20): 075108 Bogey C, Bailly C (2006) Phys Fluids 18(6): 065101 Boersma BJ, Brethouwer G, Nieuwstadt FTM (1998) Phys Fluids 10(4): 899–909 Bogey C, Bailly C (2009) J Fluid Mech 627: 129–160 Freund JB (2001) J Fluid Mech 438(1): 277–305 Uddin M, Pollard A (2007) Phys Fluids 19: 068103 Bogey C, Bailly C (2004) J Comput Phys 194(1): 194–214 Bogey C, Bailly C (2006) Int J Heat and Fluid Flow 27(4): 603-610 Pitts WM (1991) Exp. Fluids (11): 135–141

Direct Numerical Simulation of Fractal-Generated Turbulence S. Laizet and J. Christos Vassilicos Turbulence, Mixing and Flow Control Group, Department of Aeronautics and Institute for Mathematical Sciences, Imperial College London, London SW7 2AZ, United Kingdom, [email protected]; [email protected] Abstract The ﬂow obtained behind a fractal square grid is studied by means of direct numerical simulation. An innovative approach which combines high order schemes, Immersed boundary method and a dual domain decomposition method is used to take into account the multiscale nature of the grid and the resulting ﬂow.

1 Introduction Recently at Imperial College London, experiments of turbulence generated by fractal grids (see Fig. 1) placed at the entrance of a wind tunnel have shown some unprecedented properties. For example, a fractal square grid can generate a far downstream decaying homogeneous isotropic turbulence with broad power law (approximately −5/3) energy spectra but laminar-like dissipation [1, 7]. Although the wind tunnel measurements have provided invaluable time-resolved informations on the unique properties of multiscale generated turbulent ﬂows, understanding the spatial structure of these ﬂows is also necessary to discover the origins of these properties. In this paper, with the help of the new generation of parallel-architecture platforms, it is suggested that Direct Numerical Simulation (DNS) can provide the spatiotemporal information of the ﬂow and be an important complement to advanced experimental techniques, despite the very large simulations required. As a preliminary study, a DNS of the turbulent ﬂow generated by one of the fractal square grids used for the experimental measurements is presented here.

2 Numerical methods A numerical code fully based on sixth-order compact ﬁnite-diﬀerence schemes and a Cartesian mesh is used to solve the incompressible Navier–Stokes equations. The incompressibility condition is ensured via a fractional step V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 3, c Springer Science+Business Media B.V. 2010

19

20

S. Laizet and J.C. Vassilicos

Fig. 1. Scaled diagrams of a fractal cross grid (left), a fractal square grid (middle) and a fractal I grid (right).

method introducing a Poisson equation for the pressure. An original characteristic of the present code, called Incompact3d, is that this equation is directly solved in the framework of the modiﬁed spectral formalism. More precisely, our Poisson solver is only based on 3D Fast Fourier Transforms (FFT3D) despite the use of inﬂow/outﬂow boundary conditions. This very direct solving technique is obviously possible for periodic and free-slip boundary conditions, but also when Dirichlet conditions for the velocity are combined with homogeneous Neumann conditions for the pressure (see for instance [8] for the basic principles of the spectral solution of a Poisson equation based on cosine expansions). Note that the Poisson equation’s solution cost is less than 10% of the overall computational expense. Concerning the pressure mesh, we follow here conclusions previously reached by [2, 3] and use a staggered pressure mesh in order to avoid spurious pressure oscillations particularly when the code is combined with an immersed boundary method. More details about the present computational methodology can be found in [3]. The fractal grid is modelled using an immersed boundary technique where the speciﬁc direct forcing method of [6] is employed. The basic principle is to adapt the forcing which replaces and models the eﬀects of the immersed solid grid in a way which yields the no-slip condition at the boundary between grid and ﬂuid. The parallel version of Incompact3d (with MPI implementation for running on massive parallel platform) is used here because of the multiscale nature of the fractal grid. The code’s dual domain decomposition strategy oﬀers two major advantages: parallelisation is possible without reducing the order of our schemes, and scalability is excellent (up to more than 1,000 processors) because, even though our schemes are implicit in space, there is no data communication (overlapping) at the boundaries of each subdomain [4].

3 The fractal square grid For this preliminary numerical study, only one family of fractal grids is considered, which consists of diﬀerent sized squares with four rectangular bars (see Fig. 1, middle). This fractal grid is completely characterised by

Direct Numerical Simulation of Fractal-Generated Turbulence

• •

•

21

The number of fractal iterations N = 4. j The bar lengths Lj = RL L0 and thicknesses tj = Rtj t0 (in the plane of the grid, normal to the mean ﬂow) at iteration j, j = 0, . . . , N − 1 (0.19Meﬀ for the thickness in the direction of the mean ﬂow, see below for Meﬀ ). RL = 1/2, Rt = 0.584, L0 = 0.5Ly , L3 = 0.125L0, t0 = 0.4 Meﬀ and t3 = 0.08Meﬀ for the grid considered here. The number B j of patterns at iteration j (B = 4).

By deﬁnition, L0 = Lmax , LN −1 = Lmin , t0 = tmax and tN −1 = tmin . The blockage ratio σ of this grid is the ratio of its total area to the area T 2 of the tunnel’s cross section (T = Ly = Lz ) and is equal to 25%. It is also of interest to deﬁne the thickness ratio tr = tmax /tmin = 5. Unlike classical grids, the grid considered here does not have a well-deﬁned mesh size. [1] introduced 2√ an eﬀective mesh size for multiscale grids Meﬀ = 4TP 1 − σ where P is the grid’s perimeter length in the (y −z) plane. Note that the fractality of the grid inﬂuences Meﬀ via its perimeter P which can be extremely long in spite of being constrained to ﬁt within the area T 2 . For the grid considered here, Meﬀ = 26.5 mm. The governing equations are directly solved in a computational domain Lx × Ly × Lz = 36.5Meﬀ × 18.25Meﬀ × 18.25Meﬀ discretized on a Cartesian mesh of nx × ny × nz = 2305 × 1152 × 1152 mesh nodes. Inﬂow/outﬂow, and periodic boundary conditions are used in the x direction and y-z directions respectively. Based on a preliminary study [5], the streamwise position of the grid (5tmin from the inﬂow boundary of the computational domain) has been carefully chosen to avoid any spurious interactions between the modelling of the grid and the inﬂow boundary condition. Note that the parameters of the targeted grid correspond to one of the grids used in the laboratory experiments of [1] (Fig. 32b in [1]). Indeed, one of the imperatives of this preliminary numerical study is to be as close as possible to the experimental set-up of [1]. However, the DNS has one signiﬁcant diﬀerence compared to the U M experiments: the Reynolds number ReMeﬀ = ∞ν eﬀ is reduced from 20800 to 3785. Consequently, only a qualitative agreement with experiments can be expected. Due to the multiscale nature of the grid, the simulation requires state-ofthe art top-end parallel computing and therefore the number of mesh nodes is of crucial importance. A preliminary study [5] has shown that ﬁve mesh nodes is enough to discretize the smallest thickness tmin of a fractal grid for the Reynolds number considered here.

4 Results One of the most interesting experimental results is that, for the fractal square grids, two regions exist downstream from the grid: a turbulence production region where the turbulence intensity continuously increases till it

22

S. Laizet and J.C. Vassilicos

Fig. 2. Instantaneous streamwise velocity in the (x − z) plane, for y/T = 0, 0.25, 0.125, 0.0625 and 0.03125.

reaches a peak, and a turbulence dissipation region where the turbulence is approximately homogeneous and isotropic [1]. Remarkably, the integral and Taylor length-scales remain constant during decay far downstream of fractal square grids (as opposed to regular grids where they markedly increase). For our simulation, we can observe a considerably complex turbulent ﬂow due to the multiscale nature of the grid. An illustration of the ﬂow is presented in Figs. 2 and 3 through instantaneous streamwise velocity visualisations. These visualisations clearly suggests that there are interactions between the diﬀerent wakes, at diﬀerent scales and at diﬀerent locations. The production of turbulence observed in [1] could result from these sequential interactions between wakes, from small-scale ones all the way up to the large-scale ones. It should be noted also that the inﬂuence of the shape of the fractal grid remains important far from the grid and strongly inﬂuences the ﬂow as shown in Fig. 3. Indeed, at the end of the computational domain, the turbulence reached is still far from isotropic even if u /v and u /w are roughly the same (see Fig. 4). Unfortunately the big wakes generated by the bigger square only start to interact at the end of the computational domain. For this reason, it was not possible to observe the peak in turbulence intensity on the centreline (y/T = z/T = 0) and a subsequent decay as in the experimental measurements [1]. These initial conclusions can be conﬁrmed with some turbulence intensity statistics presented in Fig. 4. We can observe a production of turbulence for u at diﬀerent locations in the vertical direction behind the grid. It should be

Direct Numerical Simulation of Fractal-Generated Turbulence

23

Fig. 3. Instantaneous streamwise velocity in the (y − z) plane for x/Meﬀ = 0.5, 1.5625, 3.125, 6.25, 12.5 and 25. 1.4

y/T=0 y/T = 0.25 y/ T = 0.125 y/T = 0.0625 y/T = 0.03125

1.3 1.2

7 6

u’/U∞ (%)

1.1

U/U∞

y/T=0 y/T=0.125 y/T=0.0625 y/T=0.03125

8

1 0.9

5 4

0.8

3

0.7

2

0.6

1

0.5

0

0

5

10

15

20

25

30

35

40

5

0

x/Meff u’/U v’/U w’/U

20 x /Meff

25

30

35

40

u’/v’ u’/w’

1.4

6 5 4 3 2

1.2 1 0.8 0.6

1 0

15

1.6

u’/v’ and u’/w’

u’/U∞, v’/U∞ and w’/U∞ (%)

8 7

10

0.4

0

5

10

15

20 x/Meff

25

30

35

40

–0.5 –0.4 –0.3 –0.2 –0.1

0 0.1 0.2 0.3 0.4 0.5 y/T

Fig. 4. Streamwise evolution of the mean velocity for diﬀerent y/T and for z/T = 0 (top, left); streamwise evolution of u /U∞ for diﬀerent y/T and for z/T = 0 (top, right); streamwise evolution of turbulence intensities for y/T = 0.125 and for z/T = 0 (bottom, left); large-scale isotropy indicators at x/Meﬀ = 31 (bottom, right).

noted that this production of turbulence seems to be smaller for v and w . Our simulation suggests that, it is possible for a new production region to exist downstream of the ﬁrst one (see for instance, the streamwise proﬁle of u

24

S. Laizet and J.C. Vassilicos

for y/T = 0.125 on Fig. 4, top right). These two peaks were clearly identiﬁed in a movie of the numerically simulated streamwise velocity and the second peak seems to result from the interaction with the largest bar’s wake which occurs at about x/Meﬀ = 27. This represents a clear prediction for future experimental measurements to be made at diﬀerent lateral locations. Due to the limited size of the computational domain in the streamwise direction, it is not possible to observe the decay region which follows the production region, especially on the centreline. Note also that the location of the peak of turbulence may have a dependence on the Reynolds number as this peak was found to be much closer in the experimental results (where the Reynolds number was more or less ﬁve times bigger). The experimental results suggest that, by increasing the thickness of the biggest square (increasing tr ) it should be possible to observe a peak of turbulence much closer to the grid which will be useful for future simulations.

5 Conclusion In spite of the diﬀerence in Reynolds number, the present DNS results are very encouraging. A long post-processing and more comparisons with the experimental results are now required in order to better understand the properties and dynamics of this new turbulent ﬂow. It will also be necessary to increase the computational domain in the streamwise direction in order to study the turbulence in the decay region. A large range of tr will also need to be investigated. Unfortunately, it will not be possible to increase the Reynolds number due to computational considerations and future comparisons between experiments and simulations will have to take this into account. Indeed, because of the multi-scale nature of the grid, already more than ﬁve billions mesh nodes were needed for this DNS on one of the state-of-the art parallel-architecture platform.

Acknowledgments We acknowledge the EPSRC grant EP/F051468 and the UK Turbulence consortium for the CPU time made available to us on HECToR without which this paper would not have been possible. The authors are grateful to Roderick Johnstone for help with the parallel version of Incompact3d. We also thank Eric Lamballais for very useful discussions and acknowledge support from EPSRC Research grant EP/E00847X.

Direct Numerical Simulation of Fractal-Generated Turbulence

25

References 1. D. Hurst and J. C. Vassilicos. Scalings and decay of fractal-generated turbulence. Phys. Fluids, 19:035103, 2007. 2. S. Laizet and E. Lamballais. Compact schemes for the DNS of incompressible ﬂows: in what context is the quasi-spectral accuracy really useful? In Proc. IV Escola de Primavera de Transi¸cao e Turbulˆencia, Porto Alegre, RS, Brazil, 2004. 3. S. Laizet and E. Lamballais. High-order compact schemes for incompressible ﬂows: a simple and eﬃcient method with the quasi-spectral accuracy. J. Comp. Phys., Submitted, 2008. 4. S. Laizet, E. Lamballais, and J.C. Vassilicos. A numerical strategy to combine high-order schemes, complex geometry and massively parallel computing for the dns of fractal generated turbulence. Computers and Fluids, Submitted, 2008. 5. S. Laizet and J.C. Vassilicos. Multiscale generation of turbulence. Journal of Multiscale Modelling, 1:177–196, 2009. 6. P. Parnaudeau, E. Lamballais, D. Heitz, and J. H. Silvestrini. Combination of the immersed boundary method with compact schemes for DNS of ﬂows in complex geometry. In Proc. DLES-5, Munich, 2003. 7. R. E. Seoud and J. C. Vassilicos. Dissipation and decay of fractal-generated turbulence. Phys. Fluids, 19:105108, 2007. 8. R. B. Wilhelmson and J. H. Ericksen. Direct solutions for Poisson’s equation in three dimensions. J. Comput. Phys., 25:319–331, 1977.

Turbulent Oscillating Channel Flow Subjected to Wind Stress W. Kramer1 , H.J.H. Clercx1 , and V. Armenio2 1

2

Fluid Dynamics Laboratory, Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]; [email protected] V. Dipartimento di Ingegneria Civile e Ambientale, Universit` a degli studi di Trieste, Trieste, Italy, [email protected]

Abstract The channel ﬂow subjected to a wind stress at the free surface and an oscillating pressure gradient is investigated using large-eddy simulations (LES). A slowly pulsating mean ﬂow occurs, with the turbulent mechanics essentially being quasi steady. Logarithmic boundary layers are present at both the bottom wall and the free-surface. Turbulent streaks are observed in the bottom and free-surface layer. The mean velocity and the structures emerging in the ﬂow have been discussed.

1 Introduction In addition to tidal forcing, the wind stress exerted on the free surface might play a role in driving the ﬂow in estuary regions. Therefore, we estimate the amplitude of both eﬀects based on data for the Westerschelde estuary. Here, the ﬂow is strongly tidal driven with a typical ﬂuid velocity during high tide of U = 0.2 – 1.0 m/s. The Reynolds number based on the depth (h = 10 m) is Reh = U h/ν =106 –107 with ν the kinematic viscosity. For this range of Reh the ﬂow is turbulent for all phases of the tidal cycle. The estimated maximum wall stress due to the tides is 0.08–1 N/m2 . A typical wind velocity of 7 m/s yields a wind stress of 0.17 N/m2 . The wind stress and the wall stress are thus of the same order. The turbulent ﬂow in the Westerschelde estuary is characterized by a large Keulegan–Carpenter number, KC = 100–600, which indicates that the time scale related to the turbulence is much shorter than the tidal period. Turbulence is known to accelerate mixing and transport of particles. Hence, it is of relevance to sand sedimentation and dispersion of plankton formations. This work is part of a project aimed at modeling dispersion of plankton in geophysical ﬂows. The approach is to ﬁrst investigate particle dispersion at the smallest turbulent scales. This knowledge can then be used for modeling dispersion statistics in large-scale geophysical ﬂows. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 4, c Springer Science+Business Media B.V. 2010

27

28

W. Kramer et al.

To inquire the type of turbulence in such a geophysical ﬂow we performed large-eddy simulations of an oscillating ﬂow subjected to a wind stress at the free surface.

2 Problem description In this work, we study the turbulent oscillating channel ﬂow subjected to a wind stress by means of large eddy simulations (LES). The largest structures in the ﬂow are obtained solving the ﬁltered Navier–Stokes equation, ∂ ui ∂ j ∂ p i ui u ∂2u ∂τij + = −ρ−1 +ν + fi − . ∂t ∂xj ∂xi ∂xj ∂xj ∂xj

(1)

Here, u i , p and fi are the ﬁltered velocity, the ﬁltered pressure and the ﬁltered external forces, respectively. The density ρ is assumed to be constant and, hence, the velocity ﬁeld is divergence free. The stresses τij the small scales of the ﬂow exert on the large scales are modeled using a dynamic eddy-viscosity model combined with a scale-similarity model [1]. The channel domain is periodic along the horizontal x- and y-direction and is bounded in the vertical direction by a no-slip bottom (z = 0) and a free-surface layer at the top (z = 1). The horizontal dimensions are larger than the channel height to capture the largest eddies in the domain (lx × ly × h = 2 × 1.4 × 1). To mimic a tidal ﬂow an oscillating pressure gradient fp = −U ω cos ωt with frequency ω = 1/80 and velocity amplitude U = 1 is applied over the x-direction. Along the positive x-direction a constant wind stress τwind = 10−3 acts on the free-surface layer. All quantities are made dimensionless using the height of the channel and the velocity amplitude of the tidal oscillation. Equation (1) is solved using a ﬁnite-volume method based on the method by Zang et al. [8]. The Reynolds number and Keulegan–Carpenter number are decreased to Reh = U h/ν = 5 × 104 and KC = 80 to make the simulations feasible. As no wall-model is available for this kind of ﬂow, the wall stress must be resolved by the LES. For the no-slip boundary layer to be resolved the ﬁrst grid cell is set to be equal to one wall unit, which is deﬁned as Δz + = Δz/z ∗ with z ∗ = ν/uτ and uτ = τw,max /ρ . The maximum wall stress τw,max can be estimated in advance using the maximum wall stress for the purely oscillating case and adding the surface stress. Using data from Jensen et al. [3] this yields an estimated maximum of τw,max = 3.0 × 10−3 . Resolving the free-surface layer proved to require a ﬁner resolution. Here, the ﬁrst grid cell height is Δz + = 1/2. The horizontal grid spacing required in the boundary layer for resolved LES is Δx+ ≈ 60 for the streamwise direction and Δy + ≈ 30 for the spanwise direction. These requirements are reached with a resolution of 48×64×128 if grid stretching is applied for the vertical. For these grid resolutions the numerical model has been used successfully to simulate a turbulent oscillating channel ﬂow as is typical for the Gulf of Trieste [5].

Turbulent Oscillating Channel Flow Subjected to Wind Stress

29

3 Mean velocity The obtained ﬂow ﬁeld is strongly turbulent during the complete cycle. The combination of the tidal and wind forcing results in a pulsating mean ﬂow in the x-direction (Fig. 1). The mean ﬂow is obtained by plane and phase averaging. The wind stress drives a constant mean ﬂow in the interior with strong shear layers at the bottom and free-surface layer. A similar velocity proﬁle is observed, although with a larger velocity, when the oscillating pressure gradient is absent. The pressure gradient will accelerate and decelerate the ﬂow in positive x-direction in the ﬁrst half period. While in the second half period it will accelerate and decelerate the ﬂuid in the negative x-direction. The mean velocity reaches a maximum velocity of about Umax = 1.4 in the interior at ϕ = 100◦ . The ﬂow then decreases, subsequently reverses and reaches a value of Umax = −0.6 at ϕ = 250◦ . Investigation of the bottom boundary layer reveals that a clear log layer, u+ = κ−1 log z + + C, is present with κ = 0.41 and the constant C ranging between 5 and 7 for the phases 30–150◦ . The same increased values of C were found by Salon et al. [5] for the purely oscillating case. For the phases, when the mean velocity is either small or reversing, no log layer is present. The wind stress, which is no opposite to the wall stress, disturbs the formation of a bottom log layer for 210–240◦ . When the wall stress amplitude reaches a maximum again at the phases 270◦ a log layer reappears. A thinner ‘viscous sublayer’ for u − uz=h is observed at the free surface. Tsai et al. [7] argued that presence of horizontal ﬂuctuations at the free-surface has a similar eﬀect as surface roughness, which leads to a decrease of the viscous sublayer. Starting 0◦ − 60◦

90◦ − 150◦

180◦ − 240◦

270◦ − 330◦

a

b

1

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0

0 0.5 1 1.5 2 2.5

u

0

c

0 0.5 1 1.5 2 2.5

d

0 −1 −0.5 0 0.5 1 1.5

0 −1 −0.5 0 0.5 1 1.5

u

Fig. 1. The mean streamwise velocity for oscillating channel ﬂow subjected to a wind stress for diﬀerent phases (a) 0–60◦ , (b) 90–150◦ , (c) 180–240 ◦ and (d) 270–330 ◦ . The darker colors represent the later phases, e.g., in (a) the proﬁles are light grey, dark grey and black for 0◦ , 30◦ and 60◦ , respectively.

30

W. Kramer et al. x 10−3

x 10–2

4 1.5

τwall

1 u2i 2

2

0

−2

1

0.5

0

90

180

270

360

0

0

phase

90

180

270

360

phase

Fig. 2. The phase averaged wall stress (left) and the turbulent kinetic energy (right). Integrated over a period the wall stress matches the wind stress, τwind = 1 × 10−3 .

at 210◦ the thickness of the viscous layer seems to increase to Δz + = 10 pushing the log layer further away from the free surface. The wall stress is given in Fig. 2. Integrated over a period the wall stress matches the applied wind stress. The typical Reynolds number Reω = 2 Umax /ων = 4 × 106 in the ﬁrst half cycle and 7 × 105 in the second half cycle. The wall stress proﬁles in the ﬁrst and second half cycle are in agreement with the proﬁles Jensen et al. [3] observed for diﬀerent Reω (see their Fig. 9). Now following their observations, the Reynolds numbers indicate that the ﬁrst half cycle is in the fully turbulent regime leading to an increased wall stress and increased levels of turbulence. The second half cycle is in the intermittent turbulent regime. The small increase of the wall stress amplitude and turbulent kinetic energy around ϕ = 270◦ is most likely due to a burst like production of turbulence occurring in the deceleration phase, as was observed by Hino et al. [2].

4 Structures of the turbulent ﬂow The varying wall stress has an impact on the intensity and structure of turbulence. The turbulent structures are visualized using the ﬂuctuations in the streamwise velocity. In the ﬁrst half cycle the strong shear causes the formation of turbulent streaks in the bottom layer (Fig. 3). When the ﬂow decelerates the streaks are smoothened by viscosity. Then in the second half cycle wall stress builds up again and turbulent streaks reappear at ϕ = 270◦ . The observed streak spacing is identical to the typical spacing in canonical wall turbulence. The streaks at the free surface have similar spacing but some distinct low-speed spots are present throughout the cycle. These features were also observed by Tsai et al. [7] for a stress-driven free-surface ﬂow.

Turbulent Oscillating Channel Flow Subjected to Wind Stress 0◦

90◦

180◦

270◦

180◦

270◦

31

free-surface layer 0◦

90◦

bottom layer Fig. 3. Gray-scale plots of the streamwise velocity ﬂuctuations (a) at the free surface (z = 1) and (b) in the bottom boundary layer (z = 10−2 ). Values are ranging between −0.3 (black) and 0.3 (white).

To study the turbulent structure in the interior we constructed Lumley triangle or turbulence triangle (Fig. 4). In each horizontal plane the second and third invariants, IIb and IIIb , respectively, of the anisotropic Reynolds stress tensor, bij = ui uj /uk uk − δij /3, are calculated. The combination of the two invariants relate to an anisotropic state of the Reynolds stresses. For most phases we observe one-component turbulent structures in the bottom boundary layer, where the streamwise ﬂuctuations are dominant over the spanwise and vertical ﬂuctuations. The structures become more 3D isotropic away from the wall. This speciﬁc signature in the Lumley map, compares well with the one observed for steady boundary layer ﬂow [4]. In the free-surface layer turbulence is also nearly one component throughout the cycle. The laminarization of the ﬂow in the bottom boundary layer in the deceleration phases leads to 2D Reynolds stresses or pancake-like structures at 180◦ . The region where two-component turbulence is observed then further protrudes into the domain at 240◦ . In the wall layer a return to one-component turbulence is driven by the increasing amplitude of the wall stress.

5 Conclusion The oscillating pressure gradient together with a ﬁxed stress at the free surface in the same direction results in a pulsating ﬂow. The ﬁrst half cycle is in the turbulent regime with a strongly increased wall stress. Due to the decreased mean velocity in the second half cycle turbulence falls back to the intermediate regime with a burst-like production of turbulence in the deceleration phase. However, in real estuary ﬂows it can be expected that both half cycles are in the turbulent regime because of the overall higher Reynolds number.

32

W. Kramer et al. 0◦

90◦

180◦

1/3

1/3

1/3

1/6

1/6

1/6

0

0

1/3

0

(IIIb /2)1/3

0

1/3

(IIIb /2)1/3

0

0

1/3

(IIIb /2)1/3

240◦ 1/3

1/6

0

0

(IIIb /2)

1/3 1/3

Fig. 4. The anisotropy of the Reynolds stresses is mapped to the turbulence triangle (grey). The lower edge (0,0) relates to 3D isotropic structures, the left edge (−1/6, 1/6) to 2D isotropic structures and the right edge (1/3,1/3) to 1D structures. The black line quantiﬁes the structure of the turbulence at diﬀerent depths within the ﬂuid column. Marked are the ﬁrst twenty grid points above the bottom wall (plus) and below the free surface (star).

At the bottom a logarithmic layer with strong turbulent streaks develops in the acceleration phases. In the deceleration phase a relaminarization takes place as the streaks are smoothened by viscosity. High-speed streaks are present at the free surface throughout the cycle. The observed pattern is in agreement with the ﬁndings of Tsai et al. [7]. Shear production leads to intense streamwise ﬂuctuations in the boundary layer. Spanwise and vertical ﬂuctuations are less intense and overall one-component turbulence is observed. The relaminarization of the bottom boundary layer gives rise to two-component turbulence that protrudes into the interior. The large Keulegan–Carpenter number results in turbulent ﬂow where the turbulence can be considered quasi-stationary for most phases. This is conﬁrmed by the presence of log layers during acceleration, which are related to steady boundary-layer turbulence. This steady-state behavior is in agreement with the observations by Scotti and Piomelli [6] for pulsating ﬂows. In future work we will investigate the eﬀect of wind stress orientation and density stratiﬁcation. A stable stratiﬁcation can suppress turbulent motion in the vertical direction. This might decouple the interaction between the bottom and free-surface layer turbulence.

Turbulent Oscillating Channel Flow Subjected to Wind Stress

33

References 1. Armenio V, Piomelli U (2000) Flow Turb Comb 65:51–81 2. Hino M, Kashiwayanagi M, Nakayama A, Hara T (1983) J Fluid Mech 131: 363–400 3. Jensen BL, Summer BM, Fresoe J (1989) J Fluid Mech 206:265–297 4. Kim J, Moin P, Moser R (1987) J Fluid Mech 177:133–166 5. Salon S, Armenio V, Crise A (2007) J Fluid Mech 570:253–296 6. Scotti A, Piomelli U (2001). Phys Fluids 13:1367–1383 7. Tsai WT, Chen SM, Moeng CH (2005) J Fluid Mech 546:163–192 8. Zang Y, Street RL, Koseﬀ JR (1994) J Comp Phys 114:18–33

DNS of a Periodic Channel Flow with Isothermal Ablative Wall O. Cabrit1 and F. Nicoud2 1

2

CERFACS, 42 avenue Gaspard Coriolis, 31057 Toulouse, France, [email protected] Universit´e Montpellier 2, Place Eug`ene Bataillon, 34095 Montpellier, France, [email protected]

1 Introduction Ablative surface ﬂows often arise when using thermal protection materials for preserving structural component of atmospheric re-entry spacecrafts [1] and Solid Rocket Motors (SRM) internal insulation or nozzle assembly [2]. In the latter application, carbon–carbon composites are widely used and exposed to severe thermochemical attack. The nozzle surface recedes due to the action of oxidizing species, typically H2 O and CO2 , which is an issue during motor ﬁring since the SRM performance is lowered by the throat area increase and the apparition of surface roughness. Full-scale motor ﬁrings are very expensive and do not provide suﬃcient information to understand the whole phenomenon. Numerical simulations can then be used to generate precise and detailed data set of generic turbulent ﬂows under realistic operating conditions. Many studies have already proposed to couple numerically the gaseous phase and the solid structure [3–5]. However, most of them are dedicated to the structural material characterization by predicting the recession rate or the surface temperature and few are dealing with the ﬂuid characterization. Hence, the objective of the present study is to increase the understanding of changes of the turbulent boundary layer (TBL) when thermochemical ablation occurs. In this framework, the present study aims at presenting the results obtained from a DNS of turbulent reacting multicomponent channel ﬂow with isothermal ablative walls. Another simulation led under the same operating conditions but with inert walls will constitute a reference case. Results of the preliminary study [6] have been improved thanks to an ensemble average procedure giving more details on mass/momentum/energy conservations. The full 3D compressible reacting Navier–Stokes equations are solved by using the AVBP code developed at CERFACS. This third-order accurate solver (in both space and time) is dedicated to LES/DNS of reacting ﬂows and has been widely used and validated during the past years [7, 8]. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 5, c Springer Science+Business Media B.V. 2010

35

36

O. Cabrit and F. Nicoud Lx Lz

y Ly = 2h z

DNS

x

flow

Reτ Lx/h Lz/h

wall B.C.

Inert

300 3.2 1.25 isothermal/no-slip homogeneous directions: x, z, t ablation 300 3.2 1.25 isothermal/blowing homogeneous directions: x, z Fig. 1. Computational domain.

The classical channel ﬂow conﬁguration [9] is used in this study. As shown in Fig. 1, periodic boundary conditions are used in both streamwise and spanwise directions. A no-slip isothermal boundary condition is imposed for inert case while a blowing isothermal boundary condition is set for the ablation case (see below for its description). Moreover, the streamwise ﬂow is enforced by adding a source term to the momentum conservation equation while a volume source term that warms the ﬂuid is added to the energy conservation equation in order to sustain the mean temperature inside the computational domain. Realistic gas ejected from SRM nozzles contains about a hundred gaseous species. Only the ones whose mole fraction is greater than 0.001 are kept to generate a simpler mixture, nitrogen being used as a dilutent. Hence, seven species are kept for the simulation: H2 , H, H2 O, OH, CO2 , CO and N2 . To simulate the chemical kinetics of this mixture, one applies a kinetic scheme based on seven chemical reactions extracted from the Gri-Mech elementary equations (see http://www.me.berkeley.edu/gri mech). For simpliﬁcation reasons, the questions of two-phase ﬂow eﬀects, mechanical erosion and surface roughness will not be discussed in the present work. Hence, inspired by the wall recession model proposed by Keswani and Kuo [3] and the work of Kendall et al. [5] a reacting gaseous phase model is merely coupled with a boundary condition for thermochemically ablated walls. A full description of the boundary condition is given in Ref. [6]. It is based on the mass budget of the species k at the solid surface: ρw Yk,w (Vinj + Vk ) = s˙ k

(1)

where subscript w refers to wall quantities, ρw denotes the density of the mixture, Yk,w the mass fraction of species k, Vinj the convective velocity of the mixture at the surface, Vk the diﬀusion velocity of k in the direction

DNS of a Periodic Channel Flow with Isothermal Ablative Wall

37

normal to the wall and s˙ k the production rate of k. One obtains the injection velocity (also called Stephan velocity) by summing over all the species: Vinj =

1 s˙ k ρw

(2)

k

Moreover, making use of the Hirschfelder and Curtiss approximation with correction velocity for diﬀusion velocities [10], it is possible to relate Yk,w to its normal gradient at the solid/gas interface:

∇Yi,w s˙ k cor = − Dk ∇Yk,w + Yk,w Vinj + V + Ww Dk (3) W ρ i w i where Dk is the diﬀusion coeﬃcient of k in the mixture, Wk the molecular weight of k, Ww the mean molecular weight of the mixture at the wall, Xk,w the mole fraction of k and V cor a correction diﬀusion velocity that ensures global mass conservation. The latter equation is used as a boundary condition for species k while (2) is used for the normal momentum equation. Note that because the production rate of species k depends on the concentration of the oxidizing species at the wall surface, the Stephan velocity is both space and time dependent which diﬀers from the classical blowing surface DNS [11]. One single chemical reaction is accounted for at the interface, namely C+H2 O → H2 +CO. However, more complex oxidation schemes can be used within the framework described above.

2 Results The DNS with ablative walls is statistically unsteady because the oxidation reaction at the wall consumes the H2 O species initially present in the computational domain. Thus, the data cannot be averaged over time which makes harder the statistics convergence (the computational box size should be about a hundred times wider to obtain converged statistics at a given time). Moreover, the time advancement of the DNS is dependent on the initial condition. For these reasons, to look at relevant data we have performed an ensemble average from twenty diﬀerent DNS’s with ablative walls, diﬀering because of the initial conditions. To insure time decorrelation, the initial conditions are chosen from the inertwall simulation with a separating time taken to the diffusion time, τd = h/ τw /ρw . The statistics convergence with this procedure is illustrated by Fig. 2. Another question now arises: how do one has to look at into the generated data to observe a representative behavior of a generic TBL with wall ablation? Indeed, the main diﬀerence between the simulated and the real cases is that the oxidizing species are continuously consumed in the simulated case (which means that no ablation would be observed for an inﬁnite simulated time)

38

O. Cabrit and F. Nicoud

a

8.05

1.0

b

7.95

7.90

7.85

signal of one DNS (ablated walls)

0.8

Vinj/V 0inj

ρ[kg.m−3]

8.00

1

2

3

4

0.6

0.4

signal of one DNS (ablated walls) ensemble average from 20 DNS

0

ensemble average from 20 DNS

0.2

5

0

1

2

τs/τd

3

4

5

τs/τd

Fig. 2. Evolution of the density (a) and of the injection velocity (b) for one probe situated at the wall during the ablative wall simulation. The velocity is scaled by 0 , and the simulated time, τs , is scaled by τd . the initial injection velocity, Vinj

b 1.005

a 0.258 no scaling

0.257

proper scaling

0.256

1

Xo\Xo,c

Xo

0.255 0.254

0.995

0.253 inert wall ablative wall - τs < τd ablative wall - τs < τd

0.252 τs

0.251 0.25

0

0.2

0.4

0.6

y/h

0.8

inert wall ablative wall - τs < τd ablative wall - τs > τd

0.99

1

0.985

0

0.2

0.4

0.6

0.8

1

y/h

Fig. 3. Atomic mole fraction proﬁles of O: (a) without scaling; (b) scaled by its centerline value XO,c .

whereas in the real case the combustion products passing through the nozzle continuously bring oxidizing species. This implies to ﬁnd the appropriate scalings of the diﬀerent observed variables to render them time independent. For instance, Fig. 2b illustrates that the statistics cannot be performed before τs = τd which is the necessary time for the ﬂow to adapt from the inert wall initial condition. For τs > τd , the oxidation mechanisms is mainly led by the diﬀusion of oxidizing species towards the wall and unsteady terms tend to constant values which facilitates the research of a time-autosimilar behavior of the TBL. Figure 3 shows that a proper scaling is found by normalizing the atomic mole fraction proﬁles by their centerline value. Looking at Fig. 3b it is clear that the proﬁles do not collapse for τs < τd . This plot also illustrates the signiﬁcant changes of atomic composition arising inside the ablative wall TBL which explains that contrary to the inert wall case the equilibrium state is not only dependent on the temperature proﬁle when ablation occurs [6]. This is presented in Fig. 4 for the oxidizing species H2 O. The presented results are also worthwhile for other atoms and species but not shown herein.

DNS of a Periodic Channel Flow with Isothermal Ablative Wall

39

0.14 0.13 0.12

Y H2 O

0.11 0.1 inert wall - DNS inert wall - equilibrium state ablative wall - DNS after 5τd ablative wall - equilibrium state

0.09 0.08 0.07

0.06

0

0.2

0.6

0.4

0.8

1

yh Fig. 4. Mass fraction proﬁles of H2 O: comparison between the DNS and the chemical equilibrium state assessed from the chemkin software using the mixture composition at y = h and the DNS temperature proﬁle. 0

1

–0.4

0

q/ |qw|

q/ |qw|

2 –0.2

–0.6

–1 –2

–0.8 –1

–4 0

0.2

0.4

y/h

0.6

ablative walls after 5τd (v~¹ 0)

–3

~ 0) inert walls (v= 0.8

1

0

0.2

0.4

0.6

0.8

1

y/h

Fig. 5. Heat ﬂux balance scaled by the modulus of the ﬂux at the wall |qw |. h + v : ﬂux of sensible enthalpy, qhs = ρ (v hs ); : ﬂux of s

0 chemical enthalpy, qhc = ρ Yk )Δhf,k ; : Fourier heat ﬂux, k (v Y k + v

qF ourier = −λ dT ; · · · · · · · · : species diﬀusion ﬂux, q = ρ k {hk Yk Vk,y }, spec dy : full total heat ﬂux, qtot . ({hk Yk Vk,y } is a Favre average quantity);

For the present channel ﬂow conﬁguration, the speciﬁc enthalpy conservation equation can be written as d (qh + qhc + qF ourier + qspec ) = Q dy s

(4)

qtot

where qhs is the heat ﬂux of sensible enthalpy, qhc the heat ﬂux of chemical enthalpy, qF ourier the Fourier heat ﬂux, qspec the heat ﬂux of species diﬀusion, and Q the enthalpy source term that warms the ﬂuid to sustain the mean temperature. Figure 5 presents each term of the total heat ﬂux balance for the inert wall case and the ablation one after τs = 5τd . One ﬁrst see that according to the speciﬁc enthalpy balance, equation (4), the total heat ﬂux is linear through the boundary layer in both cases, the slope being related to the source term Q which is constant in space. However, strong diﬀerences are

O. Cabrit and F. Nicoud 6

qh s qspec

4

q/ | qw|

2 0 –2 –4

q F ou ri er qh c

–6 0

1

2

3

4

5

surface heating surface cooling

40

6

τs/τd

Fig. 6. Time evolution of the heat ﬂuxes at the wall during the ablative wall simulation.

visible, notably because of the blowing eﬀect of the ablation process. Indeed, for inert walls the no-slip boundary condition at the wall combined with the continuity equation imposes that v = 0 ( v being the Favre average wall normal velocity). This is not the case for ablative wall DNS. In addition, the diﬀusion velocities are not null at the ablative wall. As a consequence, none of the terms of the heat ﬂux balance are null (neither negligible) at the ablative wall whereas the total heat ﬂux is recovered by the Fourier heat ﬂux at the inert wall. This is emphasized by Fig. 6 that shows the time evolution of the diﬀerent heat ﬂuxes at the wall during the ablative wall simulation. This ﬁgure shows that the heat ﬂuxes of sensible enthalpy and of species diﬀusion are responsible for surface cooling which explains that the total wall heat ﬂux is lowered by a factor 3.5 in the ablative wall DNS compared to the inert one. Moreover, this plot shows that the scaling of the heat ﬂux by the total wall heat ﬂux modulus converges after τs = 4τd . This result justiﬁes that observing the data at τs = 5τd (as in Fig. 5) gives a representative state of the ablative wall TBL. Concerning the momentum conservation balance, the wall ablation makes appear a convective term, namely −ρ u v. However, the mass ﬂux ratio, F = ρw Vinj /ρb ub (b-subscripted variables refer to bulk value), of the current simulation is too weak to change the shear stress balance. Indeed, F ≈ 0.02% in the DNS which is very low compared to the values found in classical blowing surface studies [12]. This convective term is thus negligible, leading to the same shear stress conservation mechanism for both inert and ablative wall TBL.

3 Conclusion This study brings some keys in performing a DNS of periodic channel ﬂow with ablative walls. It is shown that with the framework describes above, the generated data cannot be analyzed for τs < τd and that the time dependency

DNS of a Periodic Channel Flow with Isothermal Ablative Wall

41

could be cut oﬀ by using appropriate scalings. In this way it is possible to analyze the behavior of a generic ablative wall TBL. The analysis of the shear stress and heat ﬂux balances reveals the negligible terms and explains the surface cooling eﬀect of wall ablation.

References 1. J. Zhong, T. Ozawa, and D. A. Levin. Modeling of stardust reentry ablation ﬂows in near-continuum ﬂight regime. AIAA J., 46(10):2568–2581, October 2008. 2. J. H. Koo, D. W. H. Ho, and O. A. Ezekoye. A review of numerical and experimental characterization of thermal protection materials - part I. numerical modeling. In 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Sacramento, California, 9-12 July 2006. 3. S. T. Keswani and K. K. Kuo. An aerothermochemical model of carbon-carbon composite nozzle recession. AIAA Paper 83-910, 1983. 4. P. Baiocco and P. Bellomi. A coupled thermo-ablative and ﬂuid dynamic analysis for numerical application to solid propellant rockets. AIAA Paper 96-1811, June 1996. 5. R. M. Kendall, R. A. Rindal, and E. P. Bartlett. A multicomponent boundary layer chemically coupled to an ablating surface. AIAA Journal, 5(6):1063–1071, 1967. 6. O. Cabrit, L. Artal, and F. Nicoud. Direct numerical simulation of turbulent multispecies channel ﬂow with wall ablation. AIAA Paper 2007-4401, 39th AIAA Thermophysics Conference, June 2007. 7. P. Schmitt, T. J. Poinsot, B. Schuermans, and K. Geigle. Large-eddy simulation and experimental study of heat transfer, nitric oxide emissions and combustion instability in a swirled turbulent high pressure burner. J. Fluid Mech., 570: 17–46, 2007. 8. S. Mendez and F. Nicoud. Large-eddy simulation of a bi-periodic turbulent ﬂow with eﬀusion. J. Fluid Mech., 598:27–65, 2008. 9. J. Kim, P. Moin, and R. Moser. Turbulence statistics in fully developed channel ﬂow at low Reynolds number. J. Fluid Mech., 177:133–166, 1987. 10. J.O. Hirschfelder, F. Curtiss, and R.B. Bird. Molecular theory of gases and liquids. John Wiley & Sons, 1964. 11. Y. Sumitani and N. Kasagi. Direct numerical simulation of turbulent transport with uniform wall injection and suction. AIAA J., 33(7):1220–1228, July 1995. 12. R. L. Simpson. Characteristics of turbulent boundary layers at low Reynolds numbers with and without transpiration. J. Fluid Mech., 42(4):769–802, 1970.

Diagnostic Properties of Structure Tensors in Turbulent Flows D.G.E. Grigoriadis, C.A. Langer, and S.C. Kassinos University of Cyprus, 75 Kallipoleos Avenue, P.O. Box 20537 Nicosia, 1678, Cyprus, [email protected]; [email protected]; [email protected]

Abstract The behavior of turbulence structure tensors based on Large-Eddy Simulations (LES) in a wide range of turbulent channel ﬂows is presented. The structure tensors provide signiﬁcant physical information on the character of turbulent ﬂows, since they provide an accurate description of the energy containing turbulence structure. LES is ideally suited for their computation since these tensors are quantities representing the larger – energy containing – turbulent scales. The basic aims of the present work are to (i) demonstrate the diagnostic properties of structure tensors in turbulent ﬂows, (ii) report turbulence quantities which would be useful to develop and assess structure-based turbulence models, (iii) demonstrate the capability of LES to accurately compute structure tensors in a variety of ﬂows. Structure tensors have been computed in the presence of complicated physical phenomena like frame rotation or MHD eﬀects. Comparisons with available DNS solutions, conﬁrm the capability of LES to accurately predict such quantities with fundamental signiﬁcance in turbulent ﬂows.

1 Introduction Structure tensors are closely linked with the coherent structures existing in turbulent ﬂows. While they were originally proposed by Kassinos et al. [1] in the context of turbulence modelling, they can also be used as a diagnostic tool for the quantiﬁcation of a turbulence ﬁeld. The calculation of structure tensors is based on the deﬁnition of the turbulence stream function ﬁeld Ψi , deﬁned by, ui = its Ψs,t , Ψi,i = 0, Ψi,nn = −ωi , (1) where ωi is the turbulence vorticity. Note that Ψi is a divergence-free ﬁeld that satisﬁes a Poisson equation and carries important non-local information. 1.1 Deﬁnitions Using the deﬁned stream function ﬁeld (equation (1)), the Reynolds stress tensor Rij , its associated normalized tensor rij , and its anisotropy tensor r˜ij can be computed from, V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 6, c Springer Science+Business Media B.V. 2010

43

44

D.G.E. Grigoriadis et al.

Ψ , Rij = ui uj = ipq jts Ψq,p s,t

r˜ij = rij − δij /3. (2) Correlations of the stream function gradients can then be used to deﬁne properties like the level of two-dimensionality of the turbulence from the dimensionality structure tensor Dij , Reynolds [6, 7] and its associated normalized and anisotropy tensors, Ψ , Dij = Ψn,i n,j

rij = Rij /Rkk = Rij /q 2 ,

dij = Dij /Dkk ,

d˜ij = dij − δij /3.

(3)

When turbulence for example, is nearly independent of xα , then dαα 0 because the stream function components vary slightly in that direction (where no summation is implied over repeated Greek indices). That is normally an evidence of existence of near-wall streaks in turbulent boundary layers [1]. As it will be shown later, regions with relatively low values of dαα , also indicate preferential alignment of turbulent structures along xα axis. The large-scale structure of the vorticity ﬁeld can be illuminated by the deﬁnition of the circulicity tensor Fij deﬁned by, Ψ , Fij = Ψi,n j,n

fij = Fij /Fkk ,

f˜ij = fij − δij /3,

(4)

which is directly related with concentrated large-scale circulation in speciﬁc directions. For example, in areas where fαα 0, no large-scale circulation is expected around xα axis. Finally, the relative importance of ﬂow inhomogeneity can be qualitatively expressed by the inhomogeneity tensor Cij , Ψ , Cij = Ψi,n n,j

cij = Cij /Dkk ,

c˜ij = cij − δij /3,

(5)

where the common trace of the dimensionality and the circulicity tensor Dkk = Fkk is used to normalise the tensor.

2 Results In order to demonstrate the diagnostic properties of structure tensors, a wide range of turbulent channel ﬂows has been considered including rotation, or MHD eﬀects as presented in Table 1. The numerical method is based on Table 1. Computational parameters for the test cases presented in a plane turbulent channel ﬂow, Ro denotes the rotation rate and N the interaction parameter. Case

Ro,N

Reτ

Grid

Domain

A, Arx, Ary, B, Bnx,

Ro = 0, N = 0 Rox = 5.0, N = 0 Roy = 1.16, N = 0 Ro = 0, N = 0 Ro = 0, Nx = 0.2

180 180 180 140 140

128×96×90 256×192×90 128×96×90 256×192×90 256×192×90

4π×2π×2 8π×4π×2 4π×2π×2 8π×4π×2 8π×4π×2

Diagnostic Properties of Structure Tensors 1

1

Reτ = 180, Kim DNS (1992) CaseA, Reτ τ= 180, Cs,1 28x96x90 Reτ = 180, Rox = 5.0, M.Oberlacketal. (1998) Case Arx, Reτ = 180, Rox = 5.0, Cs, 256x192x90

0.8

0.8

r11

0.6

0.6

rr11

0.4 0.2

0.2

0

0.5

rr22 1

1.5

dr33

d22

rr33 r22

dr22

d33

0.4

r33

0

45

d11

dr11

0

y/h

2

0

0.5

1

1.5

y/h

2

1 0.4 0.8

f12

0.2

rr12

0.6

f22

0

d12

fr22

0.4

f33 f11

fr12

r12

–0.2

fr11

fr33

0.2

–0.4

0 0

0.5

1

1.5

y/h

2

0

0.5

1

1.5

y/h

2

Fig. 1. Computed structure tensors for a plane turbulent channel ﬂow for a ﬁxed frame (Case A, full lines) and streamwise rotation mode (Case Arx, dotted lines) at Rox = 5.0. Comparison against the reference DNS data of Kim [2], and Oberlack et al. [5].

a time-splitting scheme for the temporal discretization, and a second order ﬁnite-diﬀerences approximation on Cartesian grids for the spatial discretization. Sampling for the calculation of the structure tensors was acquired concurrently, i.e. during the actual computation itself. The identiﬁcation of coherent structures was based on the widely applied Q criterion [8], while similar conclusions can be drawn by examining the isosurfaces of pressure ﬂuctuations. Turbulent channel ﬂows under frame rotation at diﬀerent orientations and intensities were closely examined. Figure 1 shows the computed structure tensors for a turbulent channel ﬂow at Reτ = 180 under a streamwise rotation rate of Rox = 5.0, compared against reference DNS solutions [5]. The eﬀect of streamwise rotation is more profound in the variation of the dimensionality tensor dij which is associated with the alignment of turbulent structures. Due to the streamwise rotation, the value of d11 remains small close to the walls, and reduces signiﬁcantly in the core. Such a variation implies that in both the rotating and the non-rotating cases coherent structures remain aligned along the streamwise direction in the near-wall regions. Under streamwise frame rotation they tend to elongate further close to the core region as shown in Fig. 2 using the Q-criterion. Spanwise rotation induces a non-symmetric ﬂow distribution, having a much stronger eﬀect on the morphological characteristics of turbulence, as outlined by the computed structure tensors in Fig. 3 compared against the DNS data of Kristoﬀersen and Anderssen [3]. Both wall regions seem to

46

D.G.E. Grigoriadis et al.

Fig. 2. Eﬀect of streamwise rotation on coherent structures. Iso-surfaces of Q = 0.025 at the central part of a turbulent channel (y = 0.5δ − 1.5δ), (a) non-rotating case A, (b) rotating case Arx. Colors correspond to distance from the lower wall. 1

1

Reτ =180, Kim DNS (1992) Reτ =194, Roy = 1.156, Andersson and Kristoffersen, (1993) Case Ary, Reτ =180, Roy = 1.16, Cs, 128x96x90

0.8

r11

0.6 0.4

0.8 0.6

rr11

r33

dr33

d33

rr22

0.4

d22

0.2

d11

dr22

rr33

0.2

r22 0

dr11

0 0

0.5

1

1.5

y/h

2

0

0.5

1

1.5

y/h

2

y/h

2

1 0.4 0.8 0.2 0.6

fr11

f22 0.4 0.2

0

fr22

f33

rr12

f12

d12 r12

–0.2

fr12

fr33

f11

–0.4

0 0

0.5

1

1.5

y/h

2

0

0.5

1

1.5

Fig. 3. Computed structure tensors for a turbulent channel ﬂow for a spanwise rotation case (Case Ary, solid line), compared against the reference DNS data of Kristoﬀersen and Andersson [3] and Kim [2].

share similar characteristics with the non-rotating case in terms of streamwise structure alignment. However, within the stable part of the channel, reduced values of the spanwise dimensionality tensor d33 were computed, signifying a tendency for the structures to align along the spanwise direction also, as shown in Fig. 4.

Diagnostic Properties of Structure Tensors

47

Fig. 4. Eﬀect of spanwise rotation on the coherent structures (iso-surfaces of Q = 0.025) in a plane turbulent channel ﬂow from Case Ary, (a) y = 1.5δ − 2.0δ, (b) y = 0.5δ − 1.5δ, (c) y = 0 − 0.5δ. Colors correspond to distance from the lower wall. 1

1

r11-MHD r11

0.8

0.8

0.6

d33-MHD

0.6 D.Lee & H. Choi DNS (2001), Reτ =140, Nx = 0.2 Case Bnx, LES, 256x192x90, Reτ =140, Nx = 0.2 CaseB, LES, 256x192x90, Reτ =140, N = 0

0.4

r33

0.2

r22

0

0.5

0.4

1

1.5

d22

0.2

r33-MHD r22-MHD

0

d33 d22-MHD

d11

d11-MHD

0

y/h

2

0

0.5

1

1.5

y/h

2

1 0.4 0.8 0.2 0.6

f22-MHD

f22

f12

d12

f33-MHD

0.4

f33

f11-MHD

f11

f12-MHD

r12

–0.2

0.2

r12-MHD

0

–0.4

0 0

0.5

1

1.5

y/h

2

0

0.5

1

1.5

y/h

2

Fig. 5. Computed structure tensors for a purely hydrodynamic turbulent channel ﬂow (Case B, dotted lines) and an MHD case with a magnetic ﬁeld applied along the streamwise direction (Case Bnx, full lines) against the reference DNS data of Lee and Choi [4](bold lines).

It is well known that the action of Lorentz forces can signiﬁcantly alter the dynamics of turbulent MHD ﬂows. Figure 5 presents the computed structure tensors for such a MHD ﬂow, when a streamwise magnetic ﬁeld is applied at Nx = 0.2, where Nx denotes the interaction or Stuart parameter. In terms of structure alignment, a drastic reduction of the streamwise dimensionality tensor component d11 is noticed, implying increased correlation lengths along the ﬂow direction due to the action of the Lorentz forces as shown in Fig. 6.

48

D.G.E. Grigoriadis et al.

Fig. 6. Eﬀect of Lorentz forces on coherent structures (iso-surfaces of Q = 0.025) from Case Bnx. Colors correspond to distance from the lower wall.

In an unexpected manner, the streamwise circulicity tensor component f11 is drastically reduced throughout the channel’s width, indicating stronger “jetal” motions. This eﬀect can be explained by the action of the Lorentz forces which tend to destroy the coherent structures with maximum variation along the streamwise direction such as vortex sheets along the x2 –x3 planes.

3 Conclusions The diagnostic properties of the structure tensors have been demonstrated for a variety of plane channel turbulent ﬂows using LES. From all the computed cases, the variation of these tensors seems to be clearly associated with the characteristics of the instantaneous coherent structures in terms of size, alignment and rotation preference.

Acknowledgements This work has been performed under the UCY-CompSci project, a Marie Curie Transfer of Knowledge (TOK-DEV) grant (contract No. MTKD-CT2004-014199) funded by the CEC under the sixth Framework Program. Partial support by a Center of Excellence grant from the Norwegian Research Council to Center for Biomedical Computing is also greatly acknowledged.

References 1. Kassinos SC, Reynolds WC, Rogers MM (2001). One-point turbulence structure tensors. J. Fluid Mech., 428, 213–248. 2. Kim J (1992). Personal communication. 3. Kristoﬀersen R, Andersson HI (1993). Direct simulations of low-Reynolds-number turbulent ﬂow in a rotating channel J. Fluid Mech., 256, 163-197.

Diagnostic Properties of Structure Tensors

49

4. Lee D, Choi H (2001). Magnetohydrodynamic turbulent ﬂow in a channel at low magnetic Reynolds number, J. Fluid Mech., 439, 367-394. 5. Oberlack M, Cabot W, Rogers MM (1998). Group analysis, DNS and modeling of a turbulent channel ﬂow with streamwise rotation Proceedings of the Summer Program 1998, CTR. 6. Reynolds WC (1989). Eﬀects of rotation on homogeneous turbulence. In Proc. 10th Australasian Fluid Mechanics Conference. University of Melbourne, Melbourne, Australia. 7. Reynolds WC (1991). Towards a structure-based turbulence model. In Studies in Turbulence, (ed. T. B. Gatski, S. Sirkar & C. Speziale) pp. 76–80. Springer. 8. Yves D, Franck D (2000). On coherent-vortex identiﬁcation in turbulence J. of Turbulence, 1:1, 1–22.

Development of Brown–Roshko Structures in the Mixing Layer Behind a Splitter Plate Neil D. Sandham and Richard D. Sandberg Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected]; [email protected] Abstract A direct numerical simulation has been performed of the near ﬁeld of a mixing layer formed between a high-speed turbulent stream and a low-speed laminar stream. For improved realism, the simulation includes the splitter plate in the computational domain. The shear layer develops initially from the viscous sublayer of the upstream turbulent boundary layer. A trend towards two-dimensional or weakly oblique structures is observed immediately downstream of the splitter plate. The structures do not show very high coherence levels, suggesting that additional twodimensional forcing may be present in experiments.

1 Introduction The experiments of Brown and Roshko [1] can be said to have initiated a new era of research in turbulence, emphasising the role of coherent structures, that continues to the present day. Their experiments illustrated the development of large scale spanwise-coherent rollers. Importantly these structures did not disappear as the Reynolds number increased and thus could not be easily dismissed as a transitional or low Reynolds number phenomenon. Despite the subsequent research, a convincing explanation of the origin of the striking two-dimensionality of Brown–Roshko structures has been elusive. A simple explanation might be that the primary inviscid instability is most unstable as a two-dimensional disturbance and hence the mixing layer, which is convectively unstable, will naturally tend to amplify such structures in preference to oblique waves. However, when one looks closer at the ampliﬁcation rate curves it is clear that the curves are almost ﬂat for wave angles near zero degrees. Indeed, the temporally-developing simulations of Rogers and Moser [2] only showed clear Brown–Roshko structures when additional two-dimensional forcing was included in the initial condition. The subject has re-emerged recently since direct numerical simulations of subsonic temporally-developing mixing layers are rather notable for the

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 7, c Springer Science+Business Media B.V. 2010

51

52

N.D. Sandham and R.D. Sandberg

absence of Brown–Roshko structures [3]. This is important because in this situation entrainment proceeded more by the old idea of ‘nibbling’ (small turbulence eddies entraining ﬂuid at the edge of the turbulent region), rather than by ‘engulfment’ of large lumps of unmixed ﬂuid into the centre of the shear layer, as would be the case if Brown–Roshko structures are active. Other explanations for the presence of spanwise-coherent structures focus on the presence of a splitter plate in the experiments. This has two interesting consequences. Firstly there will be a small wake behind the trailing edge and such wakes may be absolutely unstable. The presence of a spanwise-coherent resonator due to such an instability may lead to spanwise coherency, although this may only happen for thick splitter plates. Secondly, the trailing edge of the splitter plate is a spanwise line and a receptivity process along that line might provide a bias towards two-dimensional structures, at the expense of oblique modes. Such a receptivity mechanism is only poorly understood at present, and suggests the need for simulations that include the trailing edge itself. One previous simulation of a subsonic mixing layer that includes the trailing edge is that of Laizet and Lamballais [4], who considered the diﬀerences between a bevelled and blunt trailing edge, ﬁnding a self-excited near wake for the latter conﬁguration. In the present contribution we modify the trailing-edge aeroacoustics simulation of Sandberg and Sandham [5] to study the initial development of the mixing layer formed after an inﬁnitely thin trailing edge. A structural parameter is used to quantify the change in the ﬂow from more streamwise-oriented structures in the turbulent boundary layer upstream of the trailing edge to spanwise-coherent structures immediately downstream of the trailing edge. Results are interpreted in the light of the above discussion.

2 Methodology The code used for the present study employs fourth order accurate ﬁnite differences in space and a fourth order Runge–Kutta time advance with entropy splitting and other conditioning of the compressible Navier–Stokes equations [6]. When run as a direct numerical simulation, as in the present contribution, the code does not use upwinding, ﬁltering or additional dissipation. Inﬂow conditions are based on a synthetic turbulence approach, here combined with an inﬂow fringe zone. A turbulent boundary layer under a freestream at Mach 0.6 is developed along the upper side of a ﬂat plate, reaching a well-developed state with a Reynolds number of 2,300 based on the displacement thickness at the trailing edge. At the end of the plate this turbulent boundary layer encounters a laminar boundary layer with a velocity a tenth that of the upper stream, but with the same temperature and density. All lengths have been normalized with the displacement thickness of the turbulent boundary layer at the trailing edge. The simulation uses 462 million grid points and has a spanwise domain

Mixing Layer with Splitter Plate

53

width equal to 18 with spanwise-periodic boundary conditions. The simulation was run on 2,048 processors of the UK national high performance computer facility, HECToR.

3 Results Figure 1 shows a side view of the vertical density gradient for the developed ﬂow. Although the simulation domain extends upstream to x = −113.3 and downstream to x = 200.5 we focus attention in this paper on the region of ﬂow immediately downstream of the trailing edge. The domain for Fig. 1 starts at x = −20, where the thick turbulent boundary layer on the upper side of the plate can be seen. The mixing layer originates in the viscous sublayer of the upstream turbulent boundary layer. The vortical ﬂow from the upper side of the plate is gradually entrained into the developing mixing layer. The change of the ﬂow in the vicinity of the trailing edge is illustrated by means of velocity proﬁles on Fig. 2a and the streamwise evolution of shear layer thickness on Fig. 2b. Velocity proﬁles are shown at the trailing edge and at locations in the early development of the mixing layer. It can be seen that the ﬂow immediately behind the plate smoothes out rapidly, while it takes some distance downstream for the out region of the turbulent boundary layer to be aﬀected. The vorticity thickness can be seen to rise rapidly from its 15 10

y

5 0 –5 –10 –15 –20

0

20

40

60

80

100

x

Fig. 1. Contours of vertical density gradient (∂ρ/∂y), illustrating the ﬂow structure. 1 0.8

DNS Curve fit Experiment (trend)

10

0.6

δω

15

x = 0.0 x = 8.5 x = 17.4 x = 26.5

0.4

5

0.2 0

−5

0 y

5

0 −20

0

20

40 x

60

80

100

Fig. 2. Velocity proﬁles in the initial mixing region (left) and thickness of the shear layer (right), including the experimental trend (dδω /dx = 0.15).

54

N.D. Sandham and R.D. Sandberg

low value in the upstream turbulent boundary layer. This plot also shows the expected growth rate based on experiments of dδ/dx = 0.15 [1]. A limited statistical sample towards the end of the computational domain prevents a more detailed study of the later evolution towards a self-similar state. The turbulence development is illustrated by contours of turbulence kinetic energy k = 0.5 ρui ui on Fig. 3. A wedge of high turbulent energy develops just downstream of the trailing edge and spreads downstream. The outer part of the turbulent boundary layer is only entrained by x = 45, after which the rate of growth of the mixing layer increases signiﬁcantly. Figure 4 shows a close up of the trailing edge region, extending from −20δT∗ E upstream to 20δT∗ E downstream. Surfaces of constant pressure perturbation, with the light surface locating low pressure and the dark surface high pressure, reveal the presence of coherent structures. Upstream 15 10 y

5 0 −5 −10 −15 −10

0

10

20

30

40

50

60

70

80

x

Fig. 3. Contour plot of turbulence kinetic energy.

Fig. 4. Pressure ﬂuctuations (dark surfaces are high pressure, light surfaces are low pressure), showing a close-up of the trailing edge and the early formation of spanwise-coherent structures. Flow is from lower left to upper right.

Mixing Layer with Splitter Plate

a

b

1 Upper Lower

0.03 0.025

0.5 p’

Γ

0.02

0

0.015 0.01 0.005

Upper Lower

−0.5 −20

55

−10

0

10 x

20

30

40

0 −20

−10

0

10 x

20

30

40

Fig. 5. Structure parameter Γ (a) and root-mean-square pressure ﬂuctuations (b).

of the trailing edge we see hairpin-type structures typical of the buﬀer region of a turbulent boundary layer. Within a very short distance of the trailing edge we can see evidence of spanwise organisation. The ﬁrst high pressure zones downstream of the trailing edge are already preferentially oriented in the spanwise direction, with signs of dislocations across the span. These structures do not yet occupy the whole thickness of the turbulent shear layer and may perhaps be considered precursors of the later Brown–Roshko rollers, already containing the essential structure. The change in structure near the trailing edge can be quantiﬁed by considering a parameter Γ deﬁned by Γ =

px − pz , px + pz

(1)

where a subscript denotes diﬀerentiation. This measure is equal to 1 for purely spanwise structures and −1 for purely streamwise structures. Figure 5a shows the development of Γ in the vicinity of the trailing edge and Fig. 5b shows the corresponding root mean square levels of the pressure. On the upper side of the plate (solid lines) the ﬂow changes from a steamwise orientation of ﬂow structure in the turbulent boundary layer to a marked spanwise orientation Γ ≈ 0.5 after the end of the plate. Thus spanwise coherency in the mixing layer develops very close to the trailing edge. The high turbulence kinetic energy developed in the mixing layer immediately after the trailing edge may have a signiﬁcant eﬀect on the downstream structure. The spanwise-coherent pressure ﬂuctuations seen on the lower side of the plate upstream of the trailing edge are long wavelength acoustic waves propagating upstream.

4 Conclusion Direct numerical simulations have been conducted of a spatially developing mixing layer at a high velocity ratio, including the trailing edge in the computational domain. A turbulent boundary layer on the upstream plate undergoes

56

N.D. Sandham and R.D. Sandberg

a rapid change in structure as it passes the trailing edge. Associated with this is the development of a region of high turbulence kinetic energy and a change from streamwise- to spanwise-oriented ﬂow structures. The region near the trailing edge appears to provide rapid ampliﬁcation of a broad band of preferentially spanwise-oriented structures. These may have a strong inﬂuence on the subsequent downstream development of the ﬂow. However, the strong spanwise coherency, seen experimentally from early in the shear layer development, is not reproduced. Two possible mechanisms not included in present simulations, but which may be active in experiments, have been identiﬁed. The ﬁrst is a ﬁnite thickness splitter plate, which may act as a two-dimensional resonator. The second is the possible presence of two-dimensional acoustic waves in the experiments, which may interact with the splitter plate trailing edge to force a two-dimensional vortical response in the shear layer.

Acknowledgement Computer time for the present study was provided via the UK Turbulence Consortium (EPSRC grant EP/D044073/1) and the simulation was run on the UK high performance computing service HECToR.

References 1. Brown, G.L. & Roshko, A. (1974) Density eﬀect and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816. 2. Laizet S. & Lamballais, E. (2006) Simulation num´erique directe de l’inﬂuence de la forme aval d’une s´eparatrice sur une couche de m´elange. Comptes Rendus Mecanique 334, 454–460. 3. Mathew, J., Mahle, I. and Friedrich, R. (2008) Eﬀects of compressibility and heat release on entrainment processes in mixing layers. Journal of Turbulence, 9(14), 1–12. 4. Rogers, M.M. and Moser, R.D. (1994) Direct simulation of a self-similar turbulent mixing layer, Phys. Fluids 6(2) (1994), 902–923. 5. Sandberg, R.D. & Sandham, N.D. (2008) Direct numerical simulation of turbulent ﬂow past a trailing edge and the associated noise generation. J. Fluid Mech., 596, 353–385. 6. Sandham, N.D., Li, Q. & Yee, H.C. (2003) Entropy splitting for high-order simulation of compressible turbulence. J. Comp. Phys. 178(2), 307–322.

DNS of Spatially-Developing Three-Dimensional Turbulent Boundary Layers Philipp Schlatter and Luca Brandt Linn´e Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden, [email protected]; [email protected]

1 Introduction and ﬂow conﬁguration Many experimental, numerical and theoretical studies are available dealing with the physics of turbulent boundary layers (TBL) with a two-dimensional mean-ﬂow proﬁle, i.e., the velocity component in the spanwise direction is vanishing. Moreover, external pressure gradients (e.g., favorable, FPG) are usually introduced to be aligned with the mean ﬂow. However, most relevant ﬂows in nature and technical applications tend to occur on curved surfaces, and they usually include a certain crossﬂow component. Studies of 3D boundary layers can be found in, e.g., Refs. [3, 4]. In 3D boundary layers diﬀerent factors may be responsible for the various changes in statistical quantities with respect to the unswept case. In particular, the three-dimensionality may be introduced via streamwise vorticity generated at the wall (e.g., via a impusively started moving wall leading to a transient, non-equilibrium state) or via pressure gradients not aligned with the free-stream velocity. Despite its relevance, it appears that there is a lack of highly-resolved numerical data for general turbulent 3D boundary layers under various conditions (crossﬂow magnitude, pressure gradient). Such data, apart from increasing the physical understanding of turbulence, may also be instrumental in developing and improving turbulence closures (RANS, LES) for industrial applications. The aim of the present contribution is thus to consider fully turbulent, spatially evolving 3D boundary layers via direct numerical simulation. As opposed to, e.g., Refs. [2–4] an inﬁnitely wide ﬂat plate with a swept leading edge is considered. This conﬁguration, sketched in Fig. 1, leads to a non-orthogonal alignment of the free-stream velocity and the direction of homogeneity, which will in turn induce three-dimensionality in the TBL. In addition, a streamwise (i.e., normal to the swept leading edge) pressure gradient can be introduced, which will lead to curved streamlines. Various well-resolved spatial DNS are performed corresponding to diﬀerent combinations of (favourable) streamwise pressure gradients (characerised by the m factor in the FSC equations) and crossﬂow magnitudes W0 /U0 , see V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 8, c Springer Science+Business Media B.V. 2010

57

58

P. Schlatter and L. Brandt leading edge W0

U0

U0

computational domain z

x α streamline

pressure gradient

x

z

Fig. 1. Sketch of the problem setup including the inﬂowing velocity (U, W ), the leading edge and the orientation of the computational domain. Table 1. DNS runs for the 2D/3D boundary layers with W0 /U0 indicating the strength of the cross ﬂow at the inlet and m being the acceleration parameter. Case

W0 /U0

m

Comment

Resolution

Reθ

A

0/1

0

ZPG

3,072 × 301 × 256

300–2,500

B B∗

0/1 0/1

0.2 0.2

FPG FPG

2,048 × 241 × 256 2,048 × 241 × 256

430–720 600–1,400

C D

1/1 2/1

0 0

Skewed ZPG Skewed ZPG

2,048 × 241 × 256 2,304 × 241 × 256

700–1,500 1,000–2,100

E F

1/1 2/1

0.2 0.2

Skewed FPG Skewed FPG

2,304 × 241 × 256 2,304 × 241 × 256

500–1,200 1,000–2,400

Table 1. The simulation code [1] is based on an accurate and eﬃcient spectral method with Fourier decomposition in the wall-parallel (x/z) directions and Chebyshev discretisation in the normal y direction. The periodic domain in x is combined with a spatially developing ﬂow via the fringe method. The inﬂow is located at comparably low Reynolds number to allow for a natural ﬂow development after turbulent transition is forced by a trip forcing. The threedimensionality and pressure gradients in the ﬂow are induced by appropriately modifying the free-stream boundary condition, i.e., by prescribing varying U0 and W0 at the upper boundary. The resolution for the present DNS is always chosen as Δx+ ≈ 10, Δz + ≈ 5 and Δy + < 7.

2 Results Zero pressure gradient boundary layer (Cases A, C and D) The mean velocity proﬁle is shown in Fig. 2a for the 2D ZPG turbulent boundary layer up to Reθ ≈ 2,500, case A, see Ref. [7]. The agreement with the experimental data ¨ provided by Osterlund [6] is excellent at the comparably high Reθ = 2,500, however, deviations when compared to Spalart [8] are noticeable, which are most probably due to the spatio-temporal approach used by the latter.

Three-Dimensional Turbulent Boundary Layers

b

25

Reynolds stresses

a

U+

20 15 10 5 0 100

59

3 2 1 0 −1

101

102 y+

103

100

102 y+

Fig. 2. ZPG turbulent boundary layer, cases A,C and D. (a) Mean velocity proﬁle of case A in wall units at Reθ = 670 1,410 and 2,500 ( ), compared ¨ Spalart [8] at Reθ = 670, 1410 and • Osterlund [6] at Reθ = 2,500. to ) compared (b) Reynolds stresses at Reθ = 1, 410 and 2,000 for cases C and D ( to ZPG (case A, ).

In the laminar case, the ﬂow under consideration can be accurately calculated via the family of the Falkner–Skan–Cooke solutions, in which the eﬀect of sweep is only relevant with streamline curvature. The inﬂuence of pure skew on the statistics is shown in Fig. 2b (cases C and D). Physically, this corresponds to a TBL whose leading edge is non-orthogonal to the mean ﬂow, but rather forming an angle of α = 45◦ and α = 63.4◦ , respectively. Consequently, the direction of homogeneity z is not orthogonal to the meanﬂow direction x. Thus, the same boundary-layer development is experienced at diﬀerent distances downstream of the leading edge. Nevertheless, it is conﬁrmed in Fig. 2b that the statistics do not change with respect to the 2D ZPG case A. Favourable pressure gradient (Cases B and B∗ ) The FPG is imposed via a varying free-stream velocity, which is characterised by the Hartree parameter βH = 2m/(m+1). Relaminarisation might occur for larger pressure gradients, 3 e.g., K = −ν/(ρU∞ )(dp/dx) > 3 · 10−6 . For the present DNS a maximum of −6 K = 2 · 10 has been imposed at the inﬂow when setting m = 0.2. Inside the domain, an essentially constant value of β = δ ∗ /τw (dp/dx) ≈ −0.2 is maintained. Statistics of the turbulent boundary layer under FPG (cases B/B∗ ) are given in Fig. 3. The shape factor H12 is decreasing slightly from 1.51 to 1.47, indicating a fuller proﬁle. This is also seen in the mean proﬁle, where U + is slightly higher in the buﬀer region, and lower in the wake. The skin friction cf , however, is virtually unchanged for a ﬁxed Reθ . Similarly, the ﬂuctuations closer to the wall (y + ≈ 100) are suppressed due to the pressure gradient, and increased in the outer part (y + > 200 for the present Reθ ). It is interesting to note that all ﬂuctuations are also decreased in the buﬀer layer, except for the streamwise ﬂuctuations. The vanishing diﬀerence between cases B and B∗ obtained in independent simulations at diﬀerent inﬂow Re allows to assess the statistical quality of the present data.

P. Schlatter and L. Brandt

a

20

U+

15 10 5 0 0 10

10

1

10

2

b

3

Reynolds stresses

60

2

1

0

−1

3

10

0

100

y+

200

300

400

500

y+

Fig. 3. FPG boundary layer, cases B ( ) and B∗ ( ) compared to ZPG (case A, ). (a) Mean velocity proﬁle, and (b) Reynolds stresses at Reθ = 670.

b

25

Reynolds stresses

a

U+

20 15 10 5 100

3 2 1 0

−1

101

y+

102

103

0

100

200

300 400 y+

500

600

700

Fig. 4. Skewed FPG boundary layer, cases E ( ) and F ( ), compared to ). (a) Mean proﬁle and (b) Reynolds stresses at Reθ = 1100. ZPG (case A,

Streamline curvature (Case E and F) The eﬀect of the three-dimensionality is quantiﬁed by the angle α as the change of mean-ﬂow direction as a function of both the streamwise and wall-normal distance. The angle α reaches its minimum value at the wall, followed by an essentially linear increase up to the boundary-layer edge. Further out, the mean-ﬂow direction is constant, given by the free-stream boundary condition. The changing mean-ﬂow direction throughout the boundary layer is on the order of a few degrees; while the total angle α is approximately 45◦ and 66◦ for W0 = 1 and W0 = 2, respectively. It can be concluded that the inﬂuence of the FPG – owing to the lower momentum of the ﬂuid – is felt more close to the wall leading to a partial realignement of the streamlines with the direction x. The ﬂow in the outer part is dominated by the cross-ﬂow component of the free stream. Turbulent statistics are presented in Fig. 4, rotated into a coordinate system (x, z) aligned with the local free-stream velocity. As opposed to the 2D cases A, C and D, in the 3D case all Reynolds stresses need to be considered. Therefore, in Fig. 4 the two additional stresses u w and vw are also plotted. Close to the wall (y + > 50) a negative peak of u w is visible, which is due to the vertical curvature of the mean ﬂow. Further away a weak contribution of vw is evident up to the boundary-layer edge.

Three-Dimensional Turbulent Boundary Layers

61

Close to the wall, both stresses and mean tend to approach the ZPG results with increasing cross ﬂow W0 . As indicated above, near the wall the eﬀect of the FPG is more evident, however, by increasing the cross-ﬂow amplitude, its relative importance compared to the cross-ﬂow shear is less dominant. Thus, the suppression of the ﬂuctuations due to the pressure gradient is gradually decreasing. On the other hand, in the outer region of the TBL (y + > 350) the ﬂuctuations are enhanced even more compared to the pure FPG case (see Fig. 3). It is thus concluded that both FPG and cross ﬂow act to increase outerregion ﬂuctuations. Furthermore, the comparison of cases A and B shows both a mean-ﬂow modiﬁcation, but also an increase of the turbulent ﬂuctuations close to the boundary-layer edge due to the FPG; similarly, case D shows a comparable inﬂuence due to the FPG even in the presence of cross ﬂow. These results indicate that for turbulent statistics the inﬂuence of the pressure gradient is more relevant than that of a cross-ﬂow component. A visulisation of the streamwise velocity component u in a cross-stream plane (y/z) aligned with the computational domain is shown in Fig. 5. Cases A and F are compared, showing the impact of the cross ﬂow on the instantaneous structures: Close to the wall, the ﬂow is essentially unchanged, dominated by streaky structures oriented in the mean-ﬂow direction. In the outer part of the TBL, the eﬀect of the skew is visible by an apparent inclination of the ﬂow structures according to the imposed velocity component in the z direction (arrow in the ﬁgure). In addition, qualitative evidence of the increased activity in the outer part can be inferred, mainly dominated by cross-ﬂow like vortices. To summarise the impact of both skew and acceleration on the mean velocity proﬁle, consider Fig. 6 featuring the indicator function Ξ = y + (dU + /dy + ) for the various cases A, B, E and F. The indicator function Ξ closely follows the composite proﬁle suggested by Monkewitz et al. [5], up to the (Redependent) wall-normal distance where the wake region is reached. The eﬀect of the pressure gradient is twofold; the near-wall peak at y + ≈ 10 is clearly increased above the ZPG value, and in the outer part the curve seems to follow longer on the values for higher Re as indicated by the composite proﬁle. Moreover, the resulting second peak is consistently lower than for ZPG. Adding cross ﬂow is gradually decreasing the inner peak, and the outer peak is

W0

y

y

Fig. 5. Instantaneous cross-stream (y/z) view showing contours of the streamwise (u) perturbation velocity pertaining to (a) 2D ZPG (case A) and (b) skewed FPG 3D boundary layer (case F). The cross-ﬂow direction is indicated by the arrow, Reθ = 1700, natural aspect ratio.

62

P. Schlatter and L. Brandt

a

b

6

5

5

4

4

3

Ξ

Ξ

6

3

2

2

1

1

0 100

101

102 y+

103

0 100

101

102 y+

Fig. 6. Indicator function Ξ for (a) cases A ( ) and B ( (b) Cases A ( ), E ( ) and F ( ) at Reθ = 1100.

103

) at Reθ = 670. Ref. [5].

also, albeit much slower, approaching the ZPG values. Common to all cases is that the boundary-layer thickness is seemingly increasing with both pressure gradient and cross ﬂow, compared to a ZPG TBL at the same local Reθ .

3 Conclusions Fully-resolved spatial DNS of zero pressure gradient, accelerated and skewed turbulent boundary layers up to Reθ ≈ 2, 500 are presented. The main aspect is to quantify the modiﬁcations of the turbulent statistics and ﬂow structures due to the presence of both (favourable) pressure gradients and cross ﬂow. The results obtained indicate that the pressure gradient directly inﬂuences both the mean proﬁles and ﬂuctuations, by producing a fuller velocity proﬁle (lower shape factor and higher von K´ arm´ an “constant”), and decreasing the turbulent activity close to the wall. The indicator function Ξ features a higher peak in the buﬀer region, and lower values in the wake part. If a cross-ﬂow component is added, it is observed that a purely skewed boundary layer is virtually indistinguishable from a two-dimensional ZPG boundary layer. An additional streamwise pressure gradient will lead to curved streamlines, which in turn give rise to a vertical non-alignement of the mean ﬂow. The near-wall region is more aligned with the pressure gradient, due to its lower momentum; consequently the turbulence-increasing eﬀect of the acceleration is felt less with increasing cross ﬂow. However, in the outer part of the boundary layer, both pressure gradient and cross ﬂow lead to more turbulent activity.

References 1. M. Chevalier, P. Schlatter, A. Lundbladh, and D. S. Henningson. simson - A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows. Tech. Rep. TRITA-MEK 2007:07, KTH Mechanics, Stockholm, Sweden, 2007. 2. G. N. Coleman, J. Kim, and P. R. Spalart. A numerical study of strained threedimensional wall-bounded turbulence. J. Fluid Mech., 416:75–116, 2000.

Three-Dimensional Turbulent Boundary Layers

63

3. C. Kannepalli and U. Piomelli. Large-eddy simulation of a three-dimensional shear-driven turbulent boundary layer. J. Fluid Mech., 423:175–203, 2000. 4. R. O. Kiesow and M. W. Plesniak. Near-wall physics of a shear-driven threedimensional turbulent boundary layer with varying crossﬂow. J. Fluid Mech., 484:1–39, 2003. 5. P. A. Monkewitz, K. A. Chauhan, and H. M. Nagib. Self-consistent highReynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids, 19(115101): 1–12, 2007. ¨ 6. J. M. Osterlund. Experimental studies of zero pressure-gradient turbulent boundary-layer ﬂow. PhD thesis, KTH Mechanics, Stockholm, Sweden, 1999. 7. P. Schlatter, Q. Li, G. Brethouwer, A. V. Johansson, and D. S. Henningson. Towards simulations of high-Reynolds number turbulent boundary layers. In XXII ICTAM, 25-29 August 2008, Adelaide, Australia, 2008. 8. P. R. Spalart. Direct simulation of a turbulent boundary layer up to Rθ = 1410. J. Fluid Mech., 187:61–98, 1988.

Direct Numerical Simulation and Experimental Results of a Turbulent Channel Flow with Pin Fins Array B. Cruz Perez1, J. Toro Medina1 , N. Sundaram2 , K. Thole2 , and S. Leonardi1 1

2

Department of Mech. Eng. University of Puerto Rico at Mayaguez, Puerto Rico, USA, [email protected]; [email protected]; [email protected] Department of Mech. and Nuclear Eng., The Pennsylvania State University, USA, [email protected]; [email protected]

Abstract Low aspect ratio pin ﬁns are used to improve the heat transfer in turbine blades. The present study compares experimental pressure drop measurements for several pin ﬁns arrays with DNS results. For arrays greater than 6 rows the friction factor becomes independent of the number of rows. DNS shows that this is due to the form drag accounting for approximately the 90% of the total drag.

1 Introduction Gas turbine engines are designed to operate at high turbine inlet temperatures. Typical gas turbine operating temperatures are higher than the melting temperature of the turbine blades. Due to this, advanced cooling techniques such as ﬁlm-cooling and internal cooling technologies are necessary to maintain a reasonable blade life. One of the internal cooling methods consists of placing an array of low aspect ratio pin ﬁns at the trailing edge of a blade. Several studies have been performed with the aim of quantifying how the heat transfer depends on the pin ﬁn array layout. Metzger et al. [1], and Brigham and VanFossen [2] measured the heat transfer diﬀerence between high aspect ratio pin ﬁn arrays and low aspect ratio arrays. Lyall and Thrift [3], Sparrow and Ramsey [4], and Chyu et al. [5] studied the eﬀects of pin spacing on the heat transfer coeﬃcients. They showed that tighter pin spacing augments the performance of the array and found that an in-line arrangement gave lower heat transfer coeﬃcients than staggered arrangements. Metzger et al. [1] reported a dependence of the heat transfer and the Reynolds number with diﬀerences between in-line and staggered conﬁgurations. Metzger and Haley [6] made sparse measurements of the ﬂow ﬁeld in arrays with low aspect ratio pin-ﬁns. Their measurements indicated that the ﬂow was highly three-dimensional near the end-wall. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 9, c Springer Science+Business Media B.V. 2010

65

66

B. Cruz Perez et al.

Fig. 1. Experimental setup.

Despite all these eﬀorts, the ﬂow physics associated with the pin ﬁn arrangement needs to be completely understood to better predict the heat transfer enhancements due to pin ﬁn arrays. The present study aims at ﬁlling this gap by presenting experimental and Direct Numerical Simulation results on a pin ﬁn array varying the Reynolds number (Re = Ub D/ν, where D is the diameter and Ub is the bulk velocity) from 3,000 to 15,000. The spanwise distance between the pin ﬁns is s/D = 2.5 while the streamwise spacing is x/D = 1.5.

2 Experimental set up A schematic representation of the experimental facility is shown in Fig. 1. A detailed account of its construction was previously described by Lyall and Thrift [3]. The test facility was closed-loop with the ﬂow in the clockwise direction. A blower forced the air through a plenum that consisted of a splash plate so as to prevent the air from propagating through the center of the plenum. The plenum also contained a heat exchanger that was used to maintain a steady inlet air temperature into the test section. The test section had a width to height ratio of 64:1 with the pin ﬁn arrays arranged in a staggered manner. Pressure taps were located 6.75 hydraulic diameters upstream and 8.75 hydraulic diameters downstream of the pin ﬁn row so as to measure the friction factors. An oriﬁce device was used to measure the ﬂow rate through the channel. Measurements of the pressure drop were made for staggered arrays of pin ﬁns ranging from 1 to 10 rows with h/D = 1.

3 Numerical procedure The non-dimensional Navier–Stokes, continuity and energy equations for incompressible ﬂows are ∂P 1 ∂ 2 Ui ∂Ui ∂Ui Uj + =− + + Πδi1 , ∂t ∂xj ∂xi Re ∂x2j

∇·U = 0 ,

(1)

Experimental Results of a Turbulent Channel Flow with Pin Fins Array

∂T ∂T Uj 1 ∂2T + = , ∂t ∂xj Re P r ∂x2j

67

(2)

Π is the pressure gradient required to maintain a constant ﬂow rate, δij is the Kronecker delta, Ui is the component of the velocity vector in the i direction and P is the pressure, T is the temperature, α the thermal diﬀusivity and P r = ν/α = 0.71 is the Prandtl number. The Navier–Stokes and energy equations have been discretized in an orthogonal coordinate system using the staggered central second-order ﬁnite-diﬀerence approximation. Here, only the main features are recalled since details of the numerical method can be found in Orlandi [7]. The roughness is treated by the eﬃcient immersed boundary technique described in detail by Orlandi and Leonardi [8].

4 Results Time averaged velocity vectors, in a horizontal plane at y = 0.5h, are shown in Fig. 2a. When the ﬂow impinges on a cylinder, a stagnation point occurs in the center. The ﬂow divides in two streams which are tilted towards the region between the two cylinders where, for continuity, the velocity increases up to 6 times the bulk velocity. On the leeward side of the cylinder, 2 counter rotating recirculations can be observed. The behavior of the ﬂow near a pin ﬁn is further illustrated by time averaged velocity vectors in a vertical section at the center of a cylinder (Fig. 2b). Reverse ﬂow regions occur near the upstream corners of the cylinder with the walls. Flow is convected toward the wall thus increasing the stagnation pressure. The form drag of the cylinder is not uniform in the vertical direction (not shown here for lack of space), but it

Fig. 2. (a) Time averaged velocity vectors superimposed to color contour of normal wall velocity in a horizontal section at y/h = 0.25; (b) time averaged velocity vectors superimposed to color contour of time averaged pressure in a vertical section at the center of a pin ﬁn.

68

B. Cruz Perez et al.

presents a maximum at the wall and a minimum in correspondence of y = 0.8 which is approximately the location of the center of the spanwise vortex. The total pressure drop was measured for several pin ﬁns arrays having a number of rows ranging from 1 to 10. The friction factor was calculated as f=

ΔP˜ , 2 N 2ρVmax

(3)

where ΔP is the pressure drop across the pin ﬁn array, ρ is the air density, N is the number of rows, and Vmax is the maximum velocity in the test section. The total pressure (ΔP˜ ) drop take into account both the drag due to the pin ﬁns (ΔP ) and the friction on the walls. Experimental measurements show that from row 1 to row 3 the friction factor tends to increase with the Reynolds number (Fig. 3a) while from row 4 to row 10, the friction factor decreases as Re increases and it is weakly dependent on the number of rows (Fig. 3b). Given that a periodic channel is similar to an array of pin ﬁns with an inﬁnite number of cylinders, numerical results compare well with experiment and conﬁrm that for a number of rows greater than 6, the friction factor does not depends on the number of rows. The dependence of the total drag on the Reynolds number is mostly due to the Reynolds number dependence of the frictional drag. Using the wealth of information of Direct Numerical simulations, the frictional (Cf ) and pressure drag (Pd ) contribution to the total drag could be calculated. Figure 4 shows the Reynolds number dependence of the frictional and form drag. By increasing the Reynolds number they both decrease, however, the frictional drag is reduced by about 50% while the form drag decreases about 10% only. Then, for higher Reynolds numbers the total drag is likely to be Reynolds number independent since the frictional drag contribution is very small with respect to the form drag. This is further illustrated in Fig. 5 where the ratio between the pressure drop due to the pin ﬁn rows and the total pressure drop is shown.

Fig. 3. Dependence of the friction factor on the number of rows in a pin ﬁn array and on the Reynolds number.

Experimental Results of a Turbulent Channel Flow with Pin Fins Array

Fig. 4. Dependence of frictional the Reynolds number.

and form drags

69

2 (normalised by 2ρUmax N ) on

Fig. 5. Contribution of pin pressure drop relative to the total pressure drop for varying pin ﬁn rows.

For a given Reynolds number, ΔPpin /ΔPtot increases by increasing the number of rows (Fig. 5). It was also seen that with the increase in the number of pin ﬁn rows, there was less dependence on the ﬂow Reynolds number. For the number of pin rows greater than 5, the increase in the Reynolds number had a negligible eﬀect on the pressure drop due to the pin ﬁns. These measurements are further corroborated by the DNS results showing that ΔPpin /ΔPtot ranges between 0.916 for Re = 4,000 to 0.955 for Re = 16,000.

70

B. Cruz Perez et al.

5 Conclusions DNS and experiments have been performed for a turbulent channel ﬂow with pin ﬁns array on the wall. The spanwise distance between the cylinder is s/D = 2.5 while the streamwise pitch to diameter ratio is 1.5. Experimental measurements and numerical results have shown that for a number of pin ﬁns rows greater than 6 the friction factor has a universal dependence with the Reynolds number. On the other hand, Reynolds number eﬀects are weak since the form drag, which is weakly aﬀected by viscosity, represent more than 90% of the total drag. These results will aid in the prediction of the friction factor for arrays of pin ﬁns with more than ﬁve rows.

Acknowledgments The authors gratefully acknowledge the support of the National Science Foundation for support of two undergraduate students to take part in the experimental portion of this work while at Penn State. Dr. Giorgio Amati is acknowledged for useful discussions on the parallel code optimization. Caspur is acknowledged for having provided computational time.

References 1. Metzger, D.E., Berry, R.A., & Bronson J.P., (1982). “Developing Heat Transfer in Rectangular Ducts With Staggered Arrays of Short Pin Fins.” ASME Journal of Heat Transfer, 104: 700–706. 2. Brigham, B.A., and VanFossen, G.J., (1984). “Length-to-Diameter Ratio and Row Number Eﬀects in Short Pin Fin Heat Transfer.” ASME Journal of Engineering for Gas Turbines and Power, 106, pp. 241–245. 3. Lyall, M.E., & Thrift A.A., (2007). “Heat Transfer from Low Aspect Ratio Pin Fins.” ASME Paper GT-2007-27431. 4. Sparrow, E.M., and Ramsey J.W., (1978). “Heat Transfer and Pressure Drop for a Staggered Wall-Attached Array of Cylinders with Tip Clearance.” International Journal of Heat and Mass Transfer, 21: 1369–1377. 5. Chyu, M. K., (1990). “Heat Transfer and Pressure Drop for Short Pin-Fin Arrays with Pin-Endwall Fillet.” ASME Journal of Heat Transfer, 112: 926–932. 6. Metzger, D. E. & Haley, S. W. (1982). Heat transfer experiments and ﬂow visualization for arrays of short pin ﬁns. American Society of Mechanical Engineers, International Gas Turbine Conference and Exhibit, 27th, London, England, Apr. 18–22, 1982. 7. Orlandi, P. (2000). Fluid ﬂow phenomena, a numerical toolkit. Kluwer Academic Publishers. 8. Orlandi P. & Leonardi S. (2006). DNS of turbulent channel ﬂows with two- and three-dimensional roughness. Journal of Turbulence, 7.

New Experimental Results for a LES Benchmark Case Ch. Rapp, F. Pﬂeger, and M. Manhart Fachgebiet Hydromechanik, Technische Universit¨ at M¨ unchen, Arcisstraβe 21, 80333 M¨ unchen, Germany, [email protected]; [email protected]; [email protected] Abstract In this paper 2D PIV and 1D LDA data of the ﬂow in a channel with periodic constrictions are presented. This ﬂow has been used for various benchmark studies for RANS and LES. The goal of this work is to provide experimental reference data for this case at high Reynolds numbers. Special emphasis has been placed on controlling the prerequisites of this case, namely the periodicity in the streamwise and the homogeneity in the spanwise direction and the validation of the 2D PIV ﬁeld measurements through pointwise 1D LDA measurements at certain positions in the ﬂow. New results for Re = 19,000 and Re = 37,000 are presented and discussed.

1 Introduction As it features separation from a curved surface, natural reattachment and recirculation, the ﬂow over a periodic arrangement of 2D hills [4] has been used by various workshops and research initiatives as a test case for RANS and LES (e.g., [3]). Fr¨ ohlich et al. [2] investigated the ﬂow at Re = 10,600 (see equation (1)) in full detail whereas a compulsory study over a wide range of Reynolds numbers was published by Breuer et al. [1]. First experimental results including pressure measurements were published by Rapp and Manhart [5]. The geometry and the coordinates referred to are indicated in Fig. 1. The dimensions relate to the hill height h. The smoothly curved hill constricts the rectangular channel that is 3.035h high by about one third. The domain is 9h long whereas the symmetric hill has got a foot to crest extension of 1.93h. Computational domains apply periodic boundary conditions in the streamwise and in the commonly 4.5h wide spanwise direction. 3.035h ub h 1 Re = u (y) dy (1) ; ub = ν 2.035h h To make this ﬂow feasible as benchmark case for higher Reynolds numbers, an experiment has been thoroughly set up at the Fachgebiet Hydromechanik of the Technische Universit¨ at M¨ unchen. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 10, c Springer Science+Business Media B.V. 2010

71

72

Ch. Rapp et al. 3.035 2 y/h 1 0 0

1

2

3

4

5

6

7

8

9

x/h

Fig. 1. Geometry of the periodic hill testcase and streamlines at Re = 10,600.

Fig. 2. Experimental setup.

2 Experimental setup The water channel sketched in Fig. 2 was built in the laboratory for Hydromechanics of the Technische Universit¨ at M¨ unchen. The water channel is fed by a pump with a maximum discharge of 70 L/s which is suﬃcient to reach Re = 37,000. The hill height h was chosen to be 50 mm. Contrary to the computational domains the channel width was chosen to be constantly 18h to accomplish homogeneity in the spanwise direction. A 40h long channel module, partly ﬁlled with ﬂow straighteners, is mounted between the inlet reservoir and the foot of the ﬁrst hill. In total ten hills were chosen to achieve periodicity in the ﬂow, whilst the measurement area lies between the crests of hill seven and eight. The channel ends 34 hill heights downstream the last hill in an outlet reservoir. The water has been ﬁltered, decalciﬁed and chlorinated and subsequently seeded with hollow glass spheres. The maximum Stokes number is Stmax = 0.221 (Re = 37,000), hence it can be assumed that the particles follow the ﬂow. 1

t

The Stokes number of the particles is deﬁned as St = tkp , with the Kolmogorov ν d2 time scale tk = and the particle time scale tp = ρρPF 18ν ; max taken from a DNS at Re = 5,600 [1] and scaled by ∼ Re3 .

New Experimental Results for a LES Benchmark Case

73

3 Velocity measurements 3.1 PIV measurements The velocity ﬁeld measurements were carried out with a 2D PIV system from TSI. It consists of a 190 mJ Quantel BigSky Nd:YAG Laser emitting 532 nm pulses. The images taken with a 4 M camera were streamed through a RAID system so that 10,000 double frames could be recorded at a frame rate of 7.25 Hz. To avoid laser light reﬂections at the channel walls the camera position was adjusted in the six particular optical axis. The data of the six individual runs were merged into joint proﬁles through calibrating by the mean streamwise velocities in the overlapping parts of the images. The spatial resolution of the 32 × 32 interrogation areas that are overlapping each other by half of their length, is 0.045h. The ratio of the size of an interrogation area to the very smallest scale in the ﬂow ﬁeld range from IAL /ηmin = 29 for Re = 5,600 to IAL /ηmin = 121 for Re = 37,000.2 For each of the six diﬀerent frames 10,000 double images were taken in separate experimental runs resulting in a measuring time of tM = 1380 s. The measuring time, that was set constant for the Re investigated, led to ﬂowthrough times3 from tft(Re=5,600) = 300 to tft(Re=37,000) = 2,000. For saving computational time cross correlations of 32 × 32 interrogation areas have been conducted in Fourier space. Displacements were sub pixel reﬁned with a Gaussian three point estimator. The vectors were subsequently validated with a global range and a local median ﬁlter. Dismissed vectors (less than 3%) were interpolated. The displacements identiﬁed in this ﬁrst iteration step were used to initially shift the interrogation areas of a second iteration step. The procedure explained above was done all over again. The motivation for this time consuming method is that the interpolated shift of a dismissed vector leads to more particle image pairs and therefore a valid peak detection in the second iteration step. 3.2 LDA measurements To check the PIV results LDA measurements were conducted with a 1D system from ILA at certain positions in the ﬂow ﬁeld. The main components are a 75 mW Nd:YAG continuous wave laser with a wave length of 532 nm, a Bragg cell for the frequency shift and the beam division, a photo multiplier and a controller. The ellipsoidal measuring volume is maximal 0.18h wide in the z direction and maximal 0.005h long (in the x-) and high (in the y direction). The measuring points were accessed periodically with an overall measuring time of 15 min per point. 2 3

Kolmogorov scales taken from a DNS at Re = 5,600 [1] and scaled by Re−3/4 . tM tft = ub9h .

74

Ch. Rapp et al.

4 Conducted experiments The PIV measurements were validated through LDA measurements at positions x/h = 0.05 and x/h = 4.0 for the Re investigated. The periodicity of the ﬂow in the streamwise direction was controlled by measurements one hill pair upstream of the measurement section (hills 6–7). Figure 3 shows the mean streamwise velocity component u/ub and the mean RMS values u u /u2b from PIV and LDA measurements exemplarily at x/h = 4.0 for Re = 10,600 in comparison to LES data from Breuer et al. [1]. The LDA proﬁles are rougher. Nevertheless a very good compliance between PIV and LDA data could be found. Therefore it can be assumed that the PIV data are reliable. The deviations of the consecutive hill pairs are marginal. As ﬂows at higher Reynolds numbers reach periodic behaviour sooner it could be shown that the ﬂow is periodic for Re ≥ 10,000. Figure 4 shows very good compliance in PIV and LES RMS values and only minor deviations for u/ub at x/h = 8.0 for Re = 10,600. Spanwise structures The homogeneity of the ﬂow in the spanwise direction was checked with PIV measurements in horizontal planes. There are considerable deviations in the mean streamwise velocity component u/ub of up to 10% across the left channel half for Re = 5,600. This high value decreases for Re = 10,600 to ≈ 2% ub . x/h=4.00

x/h=4.00

3

3 PIV 7−8 PIV 6−7 LDA 7−8 LDA 6−7 LES M. Breuer

2 y/h 1

PIV 7−8 PIV 6−7 LDA 7−8 LDA 6−7 LES M. Breuer

2 y/h 1 0

0 −0.2

0

0.2

0.4 0.6 /ub

0.8

0

1

0.02

0.04

0.06

0.08

0.1

/u2b

Fig. 3. Left u/ub and right u u /u2b for Re = 10,600. x/h=8.00

x/h=8.00

3

3 2.5

PIV LES M. Breuer

2

2

y/h

y/h 1.5

1.5 1

1

0.5

0.5 0

PIV LES M. Breuer

2.5

0 0

0.2

0.4 0.6 /ub

0.8

1

−0.0 3 −0.02 −0.01

0

/u b2

Fig. 4. Left u/ub and right u v /u2b for Re = 10,600.

0.01

0.02

0.03

New Experimental Results for a LES Benchmark Case

75

0.03

0.01 ∂u/∂x ∂u/∂y

−0.01

∂u’u’/∂x ∂u’v’/∂y

−0.03

<w>∂u/∂z ∂u’w’/∂z

−0.05

−0.07

2

3

4 x/h

5

6

Fig. 5. Terms of the Reynolds equation in the x-direction. Ruu

1.2

x/h=3.03 x/h=4.01 x/h=4.99 x/h=5.96

1 0.8 0.6 0.4 0.2 0 −0.2 −9

−8

−7

−6

−5

−4 z/h

−3

−2

−1

0

Fig. 6. Two-point correlations across the left channel half; y/h = 1.53; Re = 10,600.

The inﬂuence of the spanwise variations on the momentum balance was checked at the intersection of the vertical and the horizontal PIV planes where, except for the pressure gradient, all terms of the Reynolds equation for the x component (equation (2)) could be computed. The results are shown in Fig. 5. uj

∂ui uj ∂ui 1 ∂p ∂ 2 ui =− +ν − ∂xj ρ ∂xi ∂x2j ∂xj

(2)

The terms that result from spanwise variations are by magnitudes smaller so that the ﬂow can be considered as homogeneous in the center part of the channel from Re ≥ 10,600. The correlation length was checked by computing two-point correlations as measurements are statistically independent if they vanish (see equation (3)). u u Ru1 u2 = 1 2 u1 u1 u2 u2

(3)

Figure 6 shows the development of the two-point correlations for the streamwise ﬂuctuations across the left channel half at y/h = 1.53 for Re = 10,600.

76

Ch. Rapp et al.

As can be seen from the plot, the two-point correlations become zero at about 6h apart from the center plane. The channel is wide enough for the large scale structures of the ﬂow. Reynolds number dependency The ﬂow does not change dramatically with increasing Re, which is generally stated for separated ﬂows by Fr¨ohlich et al. [2]. Nevertheless, the greatest changes can be found on the hill crest, where the streamwise velocity component tilts with increasing Re such that its maximum is shifted from the channel top to the region close to the hill. The mean point of reattachment travels upstream with increasing Re. It is x/h = 4.4 for Re = 5,600, x/h = 4.0 for Re = 10,600, x/h = 3.8 for Re = 19,000 and x/h = 3.6 for Re = 37,000.

5 Conclusions The ﬂow over periodic hills has been investigated experimentally for Re from 5,600 to 37,000. Periodicity could be proofed for Re ≥ 10,600 and the variations in the spanwise direction were identiﬁed to have minor eﬀects on the momentum balance. With increasing Reynolds number the recirculation zone ﬂattens and the point of reattachment travels upstream. The data will be made available as ERCOFTAC data base.

References 1. Breuer, M.; Peller, N.; Rapp, C.; Manhart, M.: Flow over periodic hills - numerical and experimental study in a wide range of Reynolds numbers. In: Computers and Fluids (2008) ¨ hlich, J.; Mellen, C. P.; Rodi, W.; Temmerman, L.; Leschziner, M.: 2. Fro Highly resolved large-eddy simulation of separated ﬂow in a channel with streamwise periodic constrictions. In: J. Fluid Mech. 526 (2005), S. 19–66 ´, S.; Jester-Zu ¨ rker, R. ; Tropea, C.: Report on 9th ERCOF3. Jakirlic TAC/IAHR/COST Workshop on Reﬁned Turbulence Modelling. In: ERCOFTAC Bulletin, No. 55. Darmstadt University of Technology, October 9-10, 2001 2002, S. 36–43 ¨ hlich, J.; Rodi, W.: Large-eddy simulation of the ﬂow over 4. Mellen, C. P.; Fro periodic hills. In: 16th IMACS World Congress. Lausanne, Switzerland, 2000 5. Rapp, C.; Manhart, M.: Experimental Investigations on the Turbulent Flow over a Periodic Hill Geometry. In: (Hrsg.) Friedrich, R.; Adams, N. A.; Eaton, J. K.; Humphrey, J. A. C.; Kasagi, N.; Leschziner, M. A.: Turbulence and Shear Flow Phenomena Bd. 2. Garching, August 2007 (International Symposium Fifth), S. 649–654

Direct Computation of the Sound Radiated by Shear Layers Using Upwind Compact Schemes M. Cabana, V. Fortun´e, and E. Lamballais ´ Laboratoire d’Etudes A´erodynamiques, CNRS† , Universit´e de Poitiers, ENSMA, Bˆ at. K, 40 avenue Recteur Pineau, France, [email protected]

1 Introduction It is well known that direct numerical simulation (DNS) of compressible ﬂow and computational aeroacoustics (CAA) require the use of accurate numerical schemes allowing the drastic reduction of dispersive and dissipative errors. Despite this agreement in the DNS/CAA community about the usefulness of highly accurate schemes, there is no consensus about the best combination of numerical techniques to solve the compressible Navier–Stokes equations, even for free-shock ﬂow. An important point in the various approaches lies in the control of spurious oscillations at marginal resolution, these oscillations being linked to the nonlinear nature of the governing equations and to the artiﬁcial treatment of boundary conditions. In the literature, the speciﬁc care about this numerical artefact is found to diﬀer signiﬁcantly from one simulator to another. As a ﬁrst condition, previous authors have shown that the formulation of governing equations can inﬂuence the robustness of the computational procedure. For instance, the improvement of conservation properties through relevant term splitting [11, 17] has been found to enhance numerical stability. As an additional condition to avoid spurious oscillations, the spatial discretization itself can be adapted via the grid arrangement (staggered schemes, see for instance [3,16]) or the introduction of upwinding in the spatial diﬀerentiation [2]. Alternatively, the use of an artiﬁcial damping term [19] or a ﬁltering procedure [7, 9, 10] have allowed some authors to simply use a collocated grid in conjunction with ﬁnite central diﬀerence schemes of high accuracy [5, 13]. Note that in the context of large eddy simulation (LES), some authors use also an explicit ﬁltering procedure as a subgrid scale modelling [6, 14]. To our knowledge, no extensive comparisons between the corresponding numerical †

The authors gratefully acknowledge the Institut du D´eveloppement et des Ressources en Informatique Scientiﬁque (IDRIS) for providing computing time.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 11, c Springer Science+Business Media B.V. 2010

77

78

M. Cabana et al.

techniques are reported in the literature in terms of stability domain, accuracy and computational cost. In this paper, we are interested in the use of upwind compact schemes for the direct computation of sound by DNS. More precisely, the goal of this paper is to evaluate the ability of high-order upwind ﬁnite diﬀerence schemes to avoid the occurrence of spurious oscillations without the need of any additional numerical treatment like artiﬁcial damping or ﬁltering. We have chosen to perform several simulations of a mixing layer ﬂow and its associated acoustic ﬁeld, a test case well-documented in the literature [4, 8, 15]. To analyse the actual beneﬁce oﬀered by the upwinding procedure, counterpart calculations only based on centred schemes (of similar accuracy) will be presented for two grid resolutions. The computational eﬃciency of each case will be evaluated through the spatial resolution requirement and through the cost associated with the upwinding. Preliminary three-dimensional results are also shown. A brief conclusion will be proposed through a comparative discussion of upwinding and ﬁltering.

2 Numerical techniques Computations are performed by solving the compressible Navier–Stokes equations using a non-conservative, characteristic-type formulation [15, 18], which involves in particular the acoustic, entropic and vorticity modes. Such a formulation is very well suited to implement upwind schemes. Time advancement is performed by a fourth-order Runge–Kutta scheme and spatial derivatives are evaluated using compact ﬁnite diﬀerences. Then two types of simulations are carried out, in which no explicit ﬁltering is employed. In the ﬁrst group of simulations, all derivatives in interior nodes are evaluated using compact sixth-order centred schemes [13]. In the second one, the ﬁfth-order Compact Upwind scheme with High Dissipation (CUHD) developed by Adams and Shariﬀ [1] is chosen and used to compute the ﬁrst derivatives involved in the acoustic, entropic and vorticity modes. In all simulations, the viscous and heat conduction terms, involving second derivatives via a Laplacian formulation, are solved with the compact centred schemes. Figure 1a shows a comparison of the resolution characteristics of both schemes. The CUHD scheme provides a good representation of the ﬁrst derivative for a larger range of wavenumbers than the centred scheme. Inherent dissipation of the CUHD scheme (see Fig. 1b) is introduced only for the highest frequencies, which are inaccurately evaluated by both schemes. Simulations of the spatial development of a mixing layer are performed in a square domain (Lx = 800δω , Ly = 800δω ) which gives a direct access to the radiated acoustic ﬁeld. δω is the initial vorticity thickness of the mixing layer. The Mach number is set to M = 0.25. The inﬂow mean longitudinal velocity is given by a hyperbolic-tangent proﬁle, while the mean transversal velocity is zero. To initiate the ﬂow transition, the mixing layer is forced by

Direct Computation of the Sound Using Upwind Compact Schemes

b

a 3

3.5

exact FD centred FD upwind CUHD

2.5

79

FD upwind CUHD 3 2.5 Im(km h)

Re(km h)

2 1.5

2 1.5

1 1 0.5 0

0.5 0

0.5

1

1.5 kh

2

2.5

3

0

0

0.5

1

1.5 kh

2

2.5

3

Fig. 1. (a) Modiﬁed wave number as a function of the wave number for the ﬁrst derivative approximation, obtained using exact diﬀerentiation (black curve), centred sixth-order scheme [13] (red curve) and upwind CUHD scheme [1] (blue curve). (b) Dissipation for the upwind scheme CUHD [1].

adding deterministic, incompressible velocity ﬂuctuations to the mean proﬁles. Non-reﬂecting boundary conditions, associated with sponge zones at the outﬂow, are applied to allow the evacuation of acoustic waves [15,18]. The grid is uniform in the physical domain in the streamwise x-direction and stretched in the transverse y-direction. Four simulations are carried out, based on centred and upwind compact schemes, using low (nx , ny ) = (1035, 393) and high (nx , ny ) = (2071, 785) grid resolutions. The ﬂow simulation based on the centred schemes with the highest grid resolution provides a reference solution, validated by Moser et al. [15].

3 Results and comments Figure 2 shows the vorticity inside the dynamic ﬁeld in the physical domain obtained from the simulations based on the centred/upwind schemes, using high/low grid resolutions respectively. We note that the results obtained from the upwind schemes with the low resolution are very similar to those of the reference solution. Figure 3 shows the comparison of acoustic results, in terms of dilatation ﬁelds, obtained from the four simulations. The acoustic ﬁeld is largely dominated by the radiation associated with the vortex pairing. We observe also that the acoustic ﬁelds resulting from both simulations with the high grid resolution match very well. We note that the acoustic ﬁeld obtained from the centred scheme with the low grid resolution is highly contaminated by wiggles. On the contrary, the use of the CUHD scheme, associated with the low grid resolution leads to a satisfactory evaluation of the radiated sound ﬁeld, in comparison with the reference solution.

80

M. Cabana et al.

Fig. 2. Vorticity ﬁelds in the mixing layer, Re = 400, obtained from (at top) the centred schemes using the high grid resolution (reference solution) and (at bottom) the upwind schemes using the low grid resolution. 0 ≤ x ≤ 450 and −15 ≤ y ≤ 15.

Fig. 3. Dilatation ﬁelds radiated by the mixing layer, obtained from centred (at left) and upwind schemes (at right) for the high / low grid resolution (at top / bottom).

4 Discussion and conclusion From the 2D mixing layer test-case, the upwinding procedure is found to be a very simple technique to automatically damp spurious oscillations due to aliasing errors or boundary condition treatment. We have shown that the optimized upwind schemes of Adams and Shariﬀ [1] provide reliable acoustics results in the context of direct computation of sound, even at marginal

Direct Computation of the Sound Using Upwind Compact Schemes

81

110 100 90 80 SPL

70 60 50 40 30 20 10

0.02

0.05 0.1

0.2 St

0.5

1

2 3

Fig. 4. Snapshots of the enstrophy and dilatation ﬁelds (at left) in a M = 0.9, Re = 1700, 3D round jet; sound pressure spectrum as a function of the Strouhal number at the observation point located at (18.5r0 , 45r0 , 0) (at right). Computational domain (50r0 , 46r0 , 46r0 ) with (311 × 295 × 295) grid points.

resolution. No over-smoothing of the resulting dynamic and acoustic ﬁelds is observed despite the use of a highly-dissipative version of the upwind schemes [1]. In addition, we have performed 3D simulations of a M = 0.9 round jet. Preliminary results about the dynamic and acoustic ﬁelds (see Fig. 4) conﬁrm the good behaviour of the CUHD scheme, because present upwind compact schemes allow realistic sound prediction in agreement with reference studies [7, 10]. We observe for instance the typical shape of an acoustic spectrum for a moderate Reynolds number jet, with a peak for a Strouhal number around St = 0.2. Note moreover that the compact centred schemes cannot lead to stable calculations at the same resolution. In terms of computational cost, the additional time associated with the use of upwind compact schemes is found to be moderate regarding its beneﬁce. More precisely, the upwinding procedure requires the double evaluation of each ﬁrst derivative in the governing equation, the cost associated with the CUHD scheme (pentadiagonal matrix inversion) being about 25% more expensive than with a sixth-order centred compact schemes (tridiagonal matrix inversion). Integrated into the full resolution of the Navier–Stokes equations, this extra-cost is found to lead to a 20% increase of the computational time compared with a fully centred code at the same spatial resolution. Because present tests suggest that the use of upwind schemes allows us to cut the number of grid points in half for each spatial direction, an overall reduction by a factor of about 6 of the computational time can be expected for a given accuracy, without taking into account additional gain due to the possible relaxation of the time step. From this viewpoint, the use of upwind compact schemes is found to be very eﬃcient.

82

M. Cabana et al.

Naturally, an analogous reduction of spurious oscillations could be possible with centred schemes through the use of a calibrated ﬁltering procedure, with probably similar results at marginal resolution for a comparable computational cost. In consequence, the upwinding technique can be seen as an alternative technique that avoids the introduction of any additional discrete operator (deﬁning the damping or ﬁltering procedure) and the related questions to their ad-hoc adjustment (numerical coeﬃcients, extraction of the stationary part of the ﬂow) and their actual working. It is perhaps in this last point that the upwinding diﬀers the most subtly from the ﬁltering. For the explicit ﬁltering approach, the most common practice is to ﬁlter directly the ﬂow variables at each time step. In this case, the actual cumulative eﬀects after a large number of time step (i.e. ﬁltering) are diﬃcult to address, especially for very small time steps, due to the loss of time-consistency caused by the use of a nonindempotent ﬁltering [12]. Practically, it can lead to an over-smoothing of the solution that is diﬃcult to evaluate a priori. The use of an artiﬁcial dissipation concentrated at very small scale through upwinding avoids these cumulative eﬀects. Even for a very small time step, the actual dissipation remains well controlled through its convergence toward a prescribed level given by a semi-discrete analysis (free from time discretization error). Due to this interesting property, in addition to the simplicity of the use of a single operator able to spatially diﬀerentiate and to damp spurious oscillations simultaneously, we think that high-order upwind compact schemes can be recommended for the direct computation of sound by DNS.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Adams N A, Shariﬀ K (1996) J Comput Phys 127:27–51 Andersson N, Eriksson L E, Davidson L (2005) Int J Heat Fluid Fl 26:393–410 Boersma B J (2005) J Comput Phys 208:675–690 Bogey C, Bailly C, Juv´e D (2000) AIAA J, 38(12):2210–2218 Bogey C, Bailly C (2004) J Comput Phys 194(1):194–214 Bogey C, Bailly C (2006) Comp Fluids 35:1344–1358 Bogey C, Bailly C (2006) Phys Fluids 18(6):065101 Colonius T, Lele S K, Moin P (1997) J Fluid Mech, 330:375–409 Fortun´e V, Lamballais E, Gervais Y (2004) Theor Comput Fluid Dyn 18:61–81 Freund J B (2001) J Fluid Mech 438:277–305 Honein A E, Moin P (2004) J Comput Phys 201(2):531–545 Kanevsky A, Carpenter M H, Hesthaven J S (2006) J Comput Phys (220) 41–58 Lele S K (1992) J Comput Phys 103:16–42 Mathew J et al. (2003) Phys. Fluids, 15(8):2279–2289 Moser C, Lamballais E, Gervais Y (2006) AIAAPaper 2006–2447 Nagarajan S, Lele S K , Ferziger J H (2003) J Comput Phys 191:392–419 Sandham N D, Li Q , Yee H C (2002) J Comput Phys 178:307–322 Sesterhenn J (2001) Comput Fluids 30:37–67 Tam C K W, Webb J C, Dong Z (1993) J Comput Acous, 1(1):1–30

DNS of Oriﬁce Flow with Turbulent Inﬂow Conditions George K. El Khoury1 , Mustafa Barri2, Helge I. Andersson2 , and Bjørnar Pettersen1 1

2

Department of Marine Technology, The Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, [email protected]; [email protected] Department of Energy and Process Engineering, The Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, [email protected]; [email protected]

Abstract Direct numerical simulation has been performed to study ﬂow through an oriﬁce at a bulk Reynolds number of 5700 and a blockage ratio of 1:2. In order to mimic an inﬁnitely long channel section upstream of the obstruction, realistic dynamic inﬂow conditions were provided by a promising technique proposed by Barri et al. (2009).

1 Introduction Understanding the characteristics of turbulent channel ﬂow with an oriﬁce is important in a variety of industrial applications as well as in the research area within the ﬁeld of ﬂow physics. The eﬀect of sudden expansion and contraction on the symmetry of the ﬂow ﬁeld, formation of large scale structures and Reynolds stresses are challenging issues associated with oriﬁce ﬂows. The ﬁrst and only direct numerical simulation (DNS) of ﬂow behind an oriﬁce was done by Makino et al. [1] with streamwise periodicity. Durst et al. [2] performed experimental investigation of the ﬂow through an axisymmetric constriction. In the present study, we perform DNS of a turbulent channel ﬂow with a rectangular oriﬁce where a realistic fully developed turbulent ﬂow is used as an inﬂow condition.

2 Computational approach A schematic view of the ﬂow domain is shown in Fig. 1. The oriﬁce is located at 12.4h with thickness t = 0.1h. The Reynolds number is 180, based on the channel half width h and wall friction velocity at the input uτ o. A total of about 25 × 106 grid points have been used. Stretching is employed in the V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 12, c Springer Science+Business Media B.V. 2010

83

84

G.K. El Khoury et al. Lx Lz

Ly Li y z

t x

Fig. 1. Oriﬁce ﬂow conﬁguration. Table 1. Computational parameters. Reτ o

Lx /h

Ly /h

Lz /h

Li /h

Nx × Ny × Nz

Δx+

Δ y+

Δz +

180

31

2

6.4

0.5

768 × 256 × 128

4.5 − 14

0.09 − 4.61

9

streamwise and wall-normal directions in order to adequately resolve the turbulence scales in the reattachment region and in the vicinity of the walls whereas a uniform mesh is used in the spanwise z-direction. The detailed computational parameters are provided in Table 1. The governing equations are the time-dependent, incompressible Navier– Stokes equations for a viscous ﬂuid in Cartesian coordinates non-dimensionalised by h and uτ o : ∇ · u = 0, (1) ∂u 1 + (u · ∇)u = −∇p + ∇2 u. (2) ∂t Reτ o The DNS code used to numerically solve the governing equations (1) and (2) is MGLET [3]. MGLET is a ﬁnite-volume code in which the Navier–Stokes equations are discretised on a staggered Cartesian mesh with non-equidistant grid-spacing. The discretisation is second-order accurate in space. A secondorder explicit Adams–Bashforth scheme is used for the time integration. The Poisson equation for the pressure is solved using a multi-grid algorithm. No-slip boundary conditions are used on the walls. The ﬂow in the spanwise direction is assumed to be statistically homogeneous and periodic boundary conditions are imposed. For the computational cells within the slits, the “blanking” technique is used where the velocity components are set to zero and the pressure is given a very large value. Dynamic inﬂow boundary conditions are provided by the cost-eﬀective method used by Barri et al. [4] in numerical simulation of plane channel ﬂow. They showed that an inﬂow condition for DNS of wall-bounded turbulent ﬂows is obtained by recycling a ﬁnite-length time series of the instantaneous velocity planes. At the exit, we solve the convective equation (3) to ensure a proper outﬂow condition. ∂u ∂u + Uc = 0. ∂t ∂x

(3)

DNS of Oriﬁce Flow with Turbulent Inﬂow Conditions

85

3 Results The streamline pattern of the mean ﬂow in Fig. 2 shows that the ﬂow is asymmetric with two large primary bubbles of separation. The reattachment lengths of the lower and upper primary bubbles are 8h and 2.31h, respectively. A tiny secondary bubble is observed at each slit corner embedded within each primary bubble. The lower and upper secondary bubbles extend about 0.89h. It is clearly noticed that the size of the separation zone on the lower wall exceeds the downstream length of the domain used by Makino et al. [1]. The pronounced asymmetry in the ﬂow is attributed to the “Coanda eﬀect”. The skin friction coeﬃcient Cf in Fig. 3 shows an alternating behaviour which conﬁrms the existence of the small separation regions at the slits corners. Figures 4 and 5 present the mean streamwise velocity proﬁles and turbulence intensities at four representative locations : inside the secondary bubbles, through the centers of the lower and upper primary bubbles and in the recovery region. The velocity proﬁles show a strong back-ﬂow beneath the cores of the primary bubbles. The turbulence intensities shown in Fig. 5 exhibit a high turbulence level immediately downstream the slit edges. This localized high-turbulence zone is obviously caused by the locally high mean-shear-rate in the mixing layer emanating from the slits. 2 y/h 1 0 10

15

20

25

x/h

Fig. 2. Streamlines of the mean ﬂow. 0.05 Lower wall Upper wall

0.03 0.01 Cf −0.01 −0.03 −0.05

15

18

21 x/h

24

27

30

Fig. 3. Skin friction coeﬃcient variation downstream the oriﬁce.

86

G.K. El Khoury et al. 2 x/h =12.7

13.7

16

30

1.5

y/h 1 0.5

0 −2

0

2

0

2 /Uo

0

2

0

2

Fig. 4. Mean streamwise velocity proﬁles. 2 x/h=12.7

13.7

16

30

1.5

y /h 1

0.5

0

0

0.5

0

0.5

Fig. 5. Turbulent intensities scaled with the maximum mean velocity at the input. —— streamwise direction; · · · · · · wall-normal direction; - - - - spanwise direction.

4 Conclusions DNS of turbulent ﬂow through an oriﬁce has been performed. Statistical results showed that the ﬂow is asymmetric due to the “Coanda eﬀect”. The streamline pattern of the mean ﬂow indicated that a long domain is required downstream of the oriﬁce in the case that periodic boundary conditions are to be used in the streamwise direction.

References 1. Makino S, Iwamoto K, Kawamura H (2008) Int. J. Heat Fluid Flow, 29, 602–611 2. Durst F, Founti M, Wang AB (1989) Turbulent Shear Flows 6, 338–350 3. Manhart M (2004) Computers and Fluids, 33, 435–461 4. Barri M, El Khoury GK, Andersson HI, Pettersen B (2008) Int. J. Numer. Meth. Fluids. 60, 227–235

The Mean Flow Proﬁle of Wall-Bounded Turbulence and Its Relation to Turbulent Flow Topology Vassilios Dallas1 , J. Christos Vassilicos2 , and Geoﬀrey F. Hewitt3 1

2

3

Institute for Mathematical Sciences & Department of Aeronautics, Imperial College London, SW7 2AZ, UK, [email protected] Institute for Mathematical Sciences & Department of Aeronautics, Imperial College London, SW7 2AZ, UK, [email protected] Department of Chemical Engineering and Chemical Technology, Imperial College London, SW7 2AZ, UK, [email protected]

1 Introduction The mean velocity proﬁle’s scaling in diﬀerent turbulent wall-bounded ﬂows and the so called von K´ arm´ an constant, κ in the case of a logarithmic velocity proﬁle, have been the source of controversy for the last decade. The classical log-law theory of von K´ arm´ an [11] and Prandtl [7] is questioned by some who propose power laws to describe the mean velocity proﬁle [1, 3]. Moreover, the most widely accepted value of κ used to be roughly 0.41 but recent estimates place this value as low as 1/e for channel ﬂows [6] and as high as 0.43 for pipe ﬂows [6]. It is also argued that for zero-pressure-gradient boundary layer ﬂows κ ≈ 0.38 [6]. It is therefore important to investigate the relation of the mean velocity proﬁle to the underlying turbulent ﬂow ﬁeld in order to gain an understanding of the key parameters that control this mean proﬁle. From there we will be able to explore the universality of the von K´ arm´ an constant, if it exists. This approach may also turn out to be instrumental in investigating drag reduction by dilute polymer solutions, in which case we know that the mean velocity proﬁle is modiﬁed dramatically [10]. In this paper, we summarise some of the arguments and results obtained by Dallas et al. [2] who present a phenomenology, based on DNS of various turbulent channel ﬂows, that relates the mean ﬂow proﬁle to the multiscale structure of stagnation points of the velocity turbulent ﬂuctuations.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 13, c Springer Science+Business Media B.V. 2010

87

88

V. Dallas et al.

2 Theory It has recently been shown under fairly reasonable assumptions that in homogeneous isotropic turbulence the Taylor length λ is proportional to the average distance s between stagnation points, i.e. points where the ﬂuctuating turbulent velocity is zero [4, 5]. The DNS of Dallas et al. [2] show that this relation remains approximately valid in an intermediate sublayer of turbulent channel ﬂows, i.e. λ(y) = B1 · s (y) (1) where y is the distance from the wall and B1 is a constant. For Reτ = uτ δ/ν 1, based on the friction velocity uτ , a large-scale δ characterising the geometry (i.e. channel, pipe, turbulent boundary layer) and the kinematic viscosity ν, we expect the production to balance dissipation in an intermediate sublayer δν (≡ ν/uτ ) y δ [8], i.e. P = B2 , with B2 → 1 as Reτ → ∞. Moreover, in this intermediate region one can deduce from the averaged streamwise momentum equation applied to channel/pipe ﬂows that −uv ∼ u2τ as δ/y → ∞ and Reτ → ∞. Hence, in this equilibrium region

du uτ u3τ if and only if κy dy κy

(2)

at least for turbulent channel/pipe ﬂows. We further deﬁne the following measure of ﬂow anisotropy, C = − 13 |u|2 /uv. According to classical results and claims [11], C ≈ 2 and κ ≈ 0.4 in the limit Reτ → ∞. Combining (1), (2), C = − 13 |u|2 /uv and the production–dissipation balance, the following functional dependence follows between the number density of stagnation points ns , which deﬁnes s , and the distance from the wall y+ (normalised with the wall unit δν ) for 1 y+ Reτ . ns =

Cs −1 y δν3 +

(3)

where Cs represents a number of turbulent velocity stagnation points at the upper edge of the buﬀer layer and is related to our other dimensionless quantities by B12 Cs = (4) κB2 C Dallas et al. [2] report that (3) is closely validated by their DNS of turbulent channel ﬂows with Cs about constant in the range δν y δ (δ being the half channel width) as Reτ increases beyond a few hundred. Incidentally, they ﬁnd that B1 is approximately constant in about the same y-range. Interestingly, though, the assumptions used to derive (3) with Cs constant are not supported by the DNS, since both 1/κ, (as calculated from the right-hand side equation in (2)) and C are far from well deﬁned constants throughout the ﬂow ﬁeld in the DNS of [2]. The phenomenology behind the new equations (1) and (3) of Dallas et al. [2] is discussed in the following section.

Mean Flow Proﬁle & Turbulent Flow Topology

89

3 Phenomenology The region where equation (1) is valid is the region where the eddy turnover time τ ≡ E/ ≡ 12 |u|2 / is proportional to 2 /ν, i.e. τ∼

2s ν

(5)

This follows directly from (1) and the deﬁnition of the Taylor length λ, = ν 2 2 3 |u| /λ where ≡ 2νsij sij and sij is the turbulence strain rate tensor. The scaling (5) implies that the eddy turnover time, i.e. the time the energy takes to cascade to the smallest scales, is at all locations where (5) holds and all large enough Reynolds numbers the same proportion of the time that viscous diﬀusion takes to spread over neighbouring stagnation points. Now, assuming that B1 and Cs are both constant in a particular range of y values, such as y δν at the very least, then equations (1) and (3) imply =

2 Euτ 3 κ∗ y

(6)

where the stagnation point K´ arm´ an constant is deﬁned as follows κ∗ ≡

B12 Cs

(7)

Hence, Cs constant means τ ∼ y/uτ , the constant of proportionality 32 κ∗ (τ = 3 2 κ∗ y/uτ ) being related to the stagnation point constants B1 and Cs . It must be stated that (6) is a priori diﬀerent from the usually accepted u3τ /κy unless 23 E/κ∗ u2τ /κ in the limit Reτ → ∞. In the equilibrium d d u as Reτ → ∞, (6) implies −uv dy u 23 E κu∗τy . layer where −uv dy 2 In turbulent channel/pipe ﬂows where −uv → uτ as Reτ → ∞, it then follows du uτ 2 = E+ (8) dy 3 κ∗ y where E+ ≡ E/u2τ . The DNS observations of Dallas et al. [2], namely Cs = const. and B1 = const. in a broad range of y-values larger than δν , imply a mean ﬂow proﬁle law (8) slightly diﬀerent from the log-law (2) in the equilibrium layer unless E+ tends to a constant in the limit Reτ → ∞. Even in this case, though, (8) may be expected to be a good approximation at values of Reτ where (2) is not.

4 Conclusion According to classical locality scalings, as Reτ → ∞, E ∼ u2τ independently of y in the equilibrium range δν y δ. If this is true, then the logd law will be recovered from dy u = 23 E+ κu∗τy but with a von K´ arm´ an constant

90

V. Dallas et al.

that is proportional to κ∗ which is inversely proportional to Cs , the number of stagnation points (number of “eddies”?) within a volume δν3 at the upper edge of the buﬀer layer. Why would anyone expect this number to be universal? If it is not, then the von K´ arm´ an constant might not be universal either. However, various DNS and experiments, as well as Townsend’s idea of inactive motions [9], seem to suggest that E does not scale as u2τ in the equilibrium region as Reτ → ∞. If this is the case, then there is, strictly speaking, no log-law (and power laws such as those of [1, 3] cannot be ruled out) and mean ﬂow data ﬁtted by a log-law may yield non-universal von K´ arm´ an “constants” as a result of κ∗ = B12 /Cs but also as a result of the small eﬀect that d inactive motions have on E and therefore on dy u = 23 E+ κu∗τy . The classical d intermediate asymptotics approach, which assumes independence of dy u on ν and δ where δν y δ, neglects the small eﬀect of inactive motions on d dy u. However, if the intermediate asymptotics argument is applied only to d τ yielding τ ∼ y/uτ , then the small eﬀects of inactive motions on dy u may 2 not be neglected. If E does not scale with uτ , then ∼ Euτ /y instead of ∼ u3τ /y. This approach is supported by the DNS observations that (1) and (3) are valid where δν y δ.

References 1. Barenblatt, G.I., Chorin, A.J. & Prostokishin, V.M. (1997), Scaling laws for fully developed ﬂows in pipes: discussion of experimental data, Proc. Natl Acad. Sci. 94, 773–776 2. Dallas, V., Vassilicos, J.C. & Hewitt, G.F. (2009), Stagnation point von K´ arm´ an constant, Phys. Rev. E 80, 046306 3. George, W.K. (2007), Is there a universal log law for turbulent wall-bounded ﬂows?, Phil. Trans. R. Soc. A 365, 789–806 4. Goto S. & Vassilicos, J.C. (2009), The dissipation rate coeﬃcient of turbulence is not universal and depends on the internal stagnation point structure, Phys. Fluids 21, 035104 5. Mazellier N. & Vassilicos, J.C. (2008), The turbulence dissipation constant is not niversal because of its universal dependence on large-scale ﬂow topology, Phys. Fluids 20, 015101 6. Nagib, H.M. & Chauhan, K.A. (2008), Variations of von K´ arm´ an coeﬃcient in canonical ﬂows, Phys. Fluids 20, 101518 7. Prandtl, L. (1925), Uber die ausgebildete Turbulenz, Z. Angew. Math. Mech. 5, 136 8. Townsend, A.A. (1961), Equilibrium layers and wall turbulence, J. Fluid Mech. 11, 97–120 9. Townsend, A.A. (1976), The Structure of Turbulent Shear Flow, Cambridge University Press 10. Virk, P.S. (1975), Drag reduction fundamentals, AIChE Journal 21, 625–656 ¨ 11. von K´ arm´ an, T. (1930), Mechanische Ahnlichkeit und Turbulenz, Proc. Third Int. Congr. Applied Mechanics, Stockholm, 85–105

Large Eddy Simulation of a Rectangular Turbulent Jet in Crossﬂow E.H. Kali1,2 , C. Brun1 , and O. M´etais1 1

2

MoST/LEGI, BP.53, 38041 Grenoble Cedex 09, France, [email protected]; [email protected] LMA/USTHB, BP.32 EL Alia 16111 Bab Ezzouar, Algeria, [email protected]

Abstract The eﬀect of Reynolds number and inlet ﬂow conditions on the topological structures of an incompressible jet in crossﬂow (JICF) with a passive scalar is investigated numerically using large-eddy simulation. The study is focused on two Reynolds number values of 2,000 and 20,000 and two kinds of inlet conditions, namely a randomly perturbed top-hat velocity proﬁle and a fully developed turbulent inﬂow issuing from a temporally-evolving duct. The jet to crossﬂow velocity ratio is 10. A passive scalar is also seeded in the jet. The ﬂow conﬁguration is a model of an experimental study performed at the CEA-Grenoble [1].

1 Numerical details The Trio U code [2] is used to solve incompressible Navier–Stokes equations including an advection–diﬀusion equation for the scalar, with an LES procedure designed for turbulent ﬂows. The computational domain (Fig. 1) extends on Lx = 15.9lj in the streamwise direction, Ly = 6.25lj in the spanwise direction and Lz = 7.5lj in the transverse direction where lj is the width of the rectangular jet section. The SubGrid-Scale (SGS) stress tensor and the SGS scalar ﬂux are respectively modeled by an eddy viscosity type selective structure function model [3] and an eddy diﬀusivity model with a constant SGS Prandtl number sets equal to 0.6 [4]. The discretisation used is a staggered ﬁnite volume method. Space derivatives are centered fourth order for the advection in the momentum equation and centered second order in the scalar equation. A third order Runge–Kutta scheme is used for time advancement. For pressure–velocity coupling, a Poisson equation is solved to ensure incompressibility at each step of the Rung–Kutta time advancing scheme and an SSOR conjugate solver is used for resolution.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 14, c Springer Science+Business Media B.V. 2010

91

92

E.H. Kali et al. Lx

Lz Z Uc,Tc

lx

Y

lj

X hj

Ly

Uj,Tj

Fig. 1. Geometry of the computational domain.

2 Boundary conditions 2.1 Jet ﬂow Two kinds of boundary conditions are used to deﬁne the inlet conditions for the jet namely a randomly perturbed top-hat proﬁle [5] for the velocity and a fully developed turbulent inﬂow issuing from a temporally-evolving duct. The general shape of the top-hat proﬁle is U(x0 , t) = Umean (x0 ) + Unoise (x0 , t)

(1)

where Umean (x0 )=(0, 0, W ) is the mean velocity given by a hyperbolictangent proﬁle with a slope R/θ0 = 6.25. θ0 is the momentum boundary layer thickness of the jet shear layer and R correspond to the rectangle half-width set to lj /2 and hj /2 in x and y directions respectively: W (x0 ) =

|x| 1 0.5lj (1 − tanh[6.25( )]) × − 4 0.5lj |x| |y| 0.3125lj (1 − tanh[6.25( )]) − 0.3125lj |y|

(2)

Unoise (x0 , t) is the inlet noise proﬁle which is deﬁned by Unoise (x0 , t) = An Ubase (x0 )f

(3)

An = 3%Ujb where Ujb the bulk jet velocity, is the maximum amplitude of the incoming noise. Ubase (x0 ) a weighting function that sets the noise location mainly in the shear layer gradients. f is a three component random noise. For the scalar, a similar hyperbolic tangent proﬁle is used, without superimposing a random noise. 2.2 Crossﬂow A steady uniform proﬁle is used for velocity and scalar at the channel inﬂow. It is a realistic model compared to the inﬂow issuing from a wind tunel nozzle.

Large Eddy Simulation of a Rectangular Turbulent Jet in Crossﬂow

93

A nonreﬂective [6] outﬂow advection condition is used at the channel outlet. No-slip conditions and adiabacity are imposed for velocity and scalar respectively on the lateral channel walls. Free slip boundary conditions for velocity and adiabatic boundary conditions for scalar are applied on the streamwise boundaries, so that no reﬁned resolution is necessary in the y direction at the boundary.

3 Results and discussion 3.1 Instantaneous ﬂow ﬁeld and vortex structures Visualizations analyzed by means of the positive Q-criterion are shown in Fig. 2 respectively for a Reynolds number (based on the bulk jet velocity Ujb and a rectangle width lj ) of 2,000 and 20,000. In the near ﬁeld ring vortex coherents structures are observed as in the case of free jets. The spreading of the jet and impingement is visible. The eﬀect of the Reynolds number is clear by mean of the more staggered structures pattern for the low Reynolds number case than for the higher one. The beginning of the transition process of the ﬂow close to the duct exit is traced by mean of the passive scalar isovalues set to θ = (T − Tc )/(Tj − Tc ) = 0.3 and is shown in Fig. 3. Tj , Tc are respectively the jet and crossﬂow temperatures. 3.2 Jet inﬂow conditions eﬀect Figure 4 shows the cross-sectional views of the ﬂow structures visualized by the scalar obtained by imposing the two type of jet inﬂow boundary conditions.

U 2

Fig. 2. Isosurfaces of the Q-criterion, Q = 0.1 l j2 . Left: Re = 2, 000. Right: Re = j 20,000.

Fig. 3. Instantaneous iso-scalar surface with a value of θ =

T −Tc Tj −Tc

= 0.3.

94

E.H. Kali et al.

Fig. 4. Near-ﬁeld structures visualized by a scalar. Left: top-hat. Right : periodic duct.

We can see that the core region is longer when deﬁning the inﬂow conditions from a periodic duct than in the top-hat one. Randomness of the shear layer vortices in the former case is obvious compared to those from the top-hat proﬁle which are more regular. This randomness is due to the uneven shear layer thickness caused by the complex interaction between the crossﬂow and the jet. With the top-hat proﬁle, the shear layer thickness is more uniform on exiting the jet nozzle, whereas with the proﬁle obtained from periodic duct conditions, the shear layer is thick to start with.

4 Conclusion A JICF problem was simulated with LES. The main conclusions are • • •

The jet development is quite sensitive to Reynolds number. The results show that the top-hat jet is more unstable and likely to shed shear layer vortices than parabolic jet. It is found that the jet issuing from a straight duct penetrate deeper into the ﬂow, but the large-scale structures appear to be less coherent than those issuing from the top-hat jet. This suggests that the characteristics of a JICF is a function of the velocity proﬁle and hence of the shear layer thickness, a well known property for spatially developing turbulent free jets [7, 8].

References 1. Fougairolle P, Moro J.P, Gagne Y Etude exp´erimentale du m´elange d’un jet traversier ` a l’aide d’une thermo-an´emom´etrie ´ a haute r´esolution. 18eme Congr`es Fran¸cais de M´ecanique Grenoble, 27-31 August :513–520, 2007. 2. Calvin C, Cueto P & Emonot P An object-oriented approach to the design of ﬂuid mechanics software. v1.5.5 M2AN., 36: 907–921, 2002. 3. Lesieur M, M´etais O New trends in Large Eddy Simulation of turbulence. Ann. Rev. Mech., 28: 45–82, 1996. 4. M´etais O, Lesieur M Spectral large-eddy simulation of isotropic and stably stratiﬁed turbulence. J. Fluid Mech., 239: 157–194, 1992.

Large Eddy Simulation of a Rectangular Turbulent Jet in Crossﬂow

95

5. Michalke A, Hermann G On the inviscid instability of a circular jet with external ﬂow. J. Fluid Mech., 114: 343–359, 1982. 6. Orlansky I A Simple boundary condition for unbounded hyperbolic ﬂows. J.Comp. Phys., 21:251–269, 1976. 7. G. Balarac, O. M´etais, and M. Lesieur. Mixing enhancement of coaxial jets through inﬂow forcing : a numerical study. Phys. Fluids., 19:1–17, 2007. 8. C. B. da Silva and O. M´etais. On the inﬂuence of coherent structures upon interscale interactions in turbulent plane jets. J. Fluid Mech., 473:103–145, 2002.

Numerical Simulation of a 2D Starting-Plume Cloud-Flow K. Aditya, S.M. Deshpande, K.R. Sreenivas, and R. Narasimha Engineering Mechanics Unit, JNCASR, Bangalore 560064, India, [email protected]; [email protected]; [email protected]; [email protected] Abstract This paper gives preliminary results of a numerical simulation of a 2D starting plume with oﬀ-source heating. A 2D plume would be relevant for ﬂow in line or row clouds. The eﬀect of heating on the shape and entrainment velocities in the plume ﬂow are discussed.

1 Introduction Cumulus clouds play an important role in weather prediction and climate research.Previous work has shown that the injection of volumetric heat into a jet or plume aﬀects ﬂow development dramatically [1–3]. It can be shown that the amount of heat injected can be made dynamically similar to the latent heat released by water vapour on condensation to liquid water in a cloud [1]. A temporal simulation using spectral methods was reported in [2]. The ﬂow has also been studied in [4] and a review of recent developments is presented by Narasimha and Bhat [5]. This paper describes preliminary results from direct numerical simulation of the 2D Boussinesq equations for a starting plume with and without heat addition. A starting plume is particularly relevant to clouds as the life cycle time of a single cloud is of the order of a few hours and is usually not long enough to generate completely self-similar ﬂows. A 2D plume would also be relevant for ﬂow in line or row clouds.

2 Simulation details A schematic of the ﬂow under consideration is shown in Fig. 1. The governing equations are the incompressible Navier–Stokes equations under the Boussinesq approximation [2],

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 15, c Springer Science+Business Media B.V. 2010

97

98

K. Aditya et al.

End of HIZ Heat-injection zone (HIZ) Besinning of HIZ

Tc dc

Fig. 1. Schematic of ﬂow under consideration.

∇.˜ u = 0,

(1)

∂u ˜ 1 + u˜. u ˜ = − p + ν 2 u ˜ − g˜αT, ∂t ρ

(2)

∂T J +u ˜ . T = κ 2 T + , ∂t ρcp

(3)

where u ˜, T, p, ρ, J, cp , α and κ are the velocity vector, change in temperature above ambient, pressure, density of the ﬂuid, heat added per unit volume per unit time, speciﬁc heat at constant pressure, coeﬃcient of thermal expansion and thermal diﬀusivity respectively. The acceleration due to gravity g˜ acts vertically downwards. The equations are non-dimensionalized using heat √ source diameter do , heat source temperature To and velocity scale Uo = gαTo do . Referring to the procedure adopted in [2], oﬀ-source heat is continuously added to the plume between the heights 15do and 35do from the source. The non-dimensional number corresponding to the oﬀ-source heating is G(x) = (J/ρCp )(do /Uo To ). The heat injected (J) varies across each plume section in a Gaussian distribution. A clustered cartesian grid has been used for the computation. The equations are solved using the projection method [6]. For integration in time, second order Adams–Bashforth and Crank–Nicolson methods are used for advective and viscous terms respectively. No-slip boundary condition is prescribed at the bottom surface. At the lateral and outﬂow boundaries, traction free and zero normal derivative conditions are used. The computational domain size is 80do in both horizontal and vertical directions. The present results are computed using a 256 × 256 grid at a Reynolds number Re = (Uo do )/ν of 3,000 and Prandtl number P r = 1.

3 Code validation As there are no comparable numerical simulations of starting plumes in the literature, the validation of the present code is based on conservation of buoyancy ﬂux and satisfaction of the divergence free condition (equation (1)).

Numerical Simulation of a 2D Starting-Plume Cloud-Flow

99

Fig. 2. Buoyancy ﬂux integral. (a) Laminar ﬂow, Re = 100. (b) Turbulent ﬂow, Re = 3,000.

Fig. 3. Vorticity contours. (a) Unheated, t = 300, (b) heated, t = 160, G = 0.01.

Figure 2 shows the buoyancy ﬂux integral (BFI) plotted along the streamwise coordinate at diﬀerent times. In laminar ﬂow, the peaks in the graphs correspond to the position of the plume head. After the plume head passes out of the domain, BFI remains nearly constant. In turbulent ﬂow, we can observe that BFI oscillates about a constant mean value. The divergence-free condition is satisﬁed to an order of 10−6 .

4 Results Figure 3a shows the vorticity contour plot of a classical turbulent starting plume. It can be seen that the plume spreads at a nearly constant rate and has developed a sinuous instability. One interesting observation in the turbulent ﬂow starting plume is that the plume head detaches from the main ﬂow and remains in an apparently laminar state with greater vertical velocity than the main ﬂow.

100

K. Aditya et al.

Fig. 4. Horizontal component of velocity (u) plotted along spanwise coordinate.

In case of a starting plume with oﬀ-source heating the spread rate declines in the HIZ (Fig. 3b). As the plume passes beyond the HIZ, the spread rate increases considerably. It is also seen that because of the oﬀ-source heating the entrainment velocity is signiﬁcantly lower (Fig. 4).

5 Conclusions The starting-plume model for clouds shows promise and gives results broadly similar to the model proposed in [1]. In the heat-injection zone the spread rate of the plume decreases. Immediately above the heat-injection zone instabilities begin to reappear and the spread rate tends to recover. More detailed numerical simulation of a 3D staring plume is now on hand.

References 1. 2. 3. 4.

Bhat G S, Narasimha R (1996) J Fluid Mech 325:303–330 Basu A J, Narasimha R (1999) J Fluid Mech 385:199–228 Venkatakrishnan L, Bhat G S, Narasimha R (1999) J Geo Res 104:14271–81 Agrawal A, Boersma B J, Prasad A K (2004) Flow, Turbulence and Combustion 73:277–305 5. Narasimha R, Bhat G S (2008) IUTAM symposium on computational physics and new perspectives in turbulence. In: Kaneda Y (eds). Springer 313–320 6. Guermond J L, Minev P, Shen J (2006) Comp methods App Mech Engg 195: 6011–6045

Part II

Methodologies and Modelling Techniques

Variational Multiscale Theory of LES Turbulence Modeling Y. Bazilevs1 , V.M. Calo2 , T.J.R. Hughes3 , and G. Scovazzi4 1 2

3

4

Department of Structural Engineering, UC San Diego, [email protected] Earth and Environmental Science and Engineering, KAUST, [email protected] Institute for Computational Engineering and Sciences, UT Austin, [email protected] 1431 Computational Shock and Multi-physics Department, Sandia National Labs, [email protected]

Abstract We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any ad hoc devices, such as eddy viscosities. We tested the formulation on a turbulent channel ﬂow. In the calculations, we employed quadratic and cubic B-Splines. The numerical results are very good and conﬁrm the viability of the theoretical framework. (This paper is excerpted from Bazilevs et al. [1], which is a much more comprehensive presentation of the theory, algorithms, implementation, and numerical studies. The reader is referred to it for further elaboration and many additional details.)

1 Variational multiscale formulation of the incompressible Navier–Stokes equations 1.1 Incompressible Navier–Stokes equations We consider a space–time domain Q = Ω×]0, T [⊂ R3 × R+ with lateral boundary P = Γ ×]0, T [, as illustrated in the left-hand side of Fig. 1. The initial/boundary-value problem consists of solving the following equations for u : Q → R3 , the velocity, and p : Q → R, the pressure (divided by the constant density), ∂u + ∇ · (u ⊗ u) + ∇p = νΔu + f ∂t ∇·u = 0 u=0 −

u(0 ) = u(0 ) +

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 16, c Springer Science+Business Media B.V. 2010

in Q

(1)

in Q

(2)

on P

(3)

on Ω

(4) 103

104

Y. Bazilevs et al. t

t

T =tN+1

T

tN tN−1

P

Q ⊂ R 3 × R+

tn+1 Qn

tn

Pn

t2 t1 0

0= t0

Ω ⊂ R3

Fig. 1. Space–time domain (left) and slicing into space–time slabs (right).

where f : Q → Rd is the given body force (per unit volume); ν is the kinematic viscosity, assumed positive and constant; u(0− ) : Ω → Rd is the given initial velocity; and ⊗ denotes the tensor product (e.g., in component notation, [u ⊗ v]ij = ui vj ). Equations (1)–(4) are, respectively, the linear momentum balance, the incompressibility constraint, the no-slip boundary condition and the initial condition. Global space–time variational formulation Let V = V(Q) denote both the trial solution and weighting function spaces, which are assumed to be identical. We assume U = {u, p} ∈ V implies u = 0 on P and Ω p(t) dΩ = 0 for all t ∈ ]0, T [. Let ( · , · )ω denote the L2 inner product with respect to the domain ω. The variational formulation is stated as follows: Find U ∈ V such that ∀W = {w, q} ∈ V: B(W , U ) = B1 (W , U ) + B2 (W , U , U ) = L(W ) with −

−

B1 (W , U ) = (w(T ), u(T ))Ω −

∂w ,u ∂t Q

(5)

Variational Multiscale Theory of LES Turbulence Modeling

+ (q, ∇ · u)Q − (∇ · w, p)Q + (∇s w, 2ν∇s u)Q B2 (W , U , V ) = − (∇w, u ⊗ v)Q

105

(6) (7)

L(W ) = (w, f )Q + (w(0+ ), u(0− ))Ω (8) where V = {v, · } and ∇s u = ∇u + (∇u)T /2. Note that B1 ( · , · ) is a bilinear form and B2 ( · , · , · ) is a trilinear form. Assuming suﬃcient regularity and integrating by parts, we obtain the Euler–Lagrange form of (5)–(8): ∂u s 0 = w, + ∇ · (u ⊗ u) + ∇p − ∇ · 2ν∇ u − f + (q, ∇ · u)Q ∂t Q +(w(0+ ), u(0+ ) − u(0− ))Ω

(9)

which reveals that the variational formulation implies satisfaction of the momentum equations, incompressibility constraint, and initial condition. The velocity boundary condition is built into the deﬁnition of the space V. In summary, the variational formulation is equivalent to (1)–(4). Sliced space–time variational formulation Consider a slicing of space–time obtained by replacing ]0, T [ by ]tn , tn+1 [, n = 0, 1, 2, . . . , N , and summing over the space–time slabs Qn (see Fig. 1). The counterparts of (5)–(9) for a typical slab are B(W , U )n = B1 (W , U )n + B2 (W , U , U )n = L(W )n ∂w − − ,u B1 (W , U )n = (w(tn+1 ), u(tn+1 ))Ω − ∂t Qn + (q, ∇ · u)Qn − (∇ · w, p)Qn + (∇s w, 2ν∇s u)Qn B2 (W , U , V )n = − (∇w, u ⊗ v)Qn − L(W )n = (w, f )Qn + (w(t+ n ), u(tn ))Ω ∂u s + ∇ · (u ⊗ u) + ∇p − ∇ · 2ν∇ u − f 0 = w, ∂t Qn + − + (q, ∇ · u)Qn + (w(t+ n ), u(tn ) − u(tn ))Ω

(10)

(11) (12) (13)

(14)

where, in (10)–(14), U = {u, p} and W = {w, q} belong to Vn = V(Qn ), the restriction of V to Qn . From the Euler–Lagrange form of the equation, (14), we see that the momentum equation and incompressibility constraint are satisﬁed on the slab, and the solution is continuous across slab interfaces. The formulation in terms of space–time slabs exploits the causal nature of the Navier–Stokes equations and reduces the overall problem to a succession of initial/boundary-value problems on the slabs. The solution is obtained by solving the variational equation on each slab successively, n = 0, 1, 2, . . . , N .

106

Y. Bazilevs et al.

We emphasize that this is an exact formulation, entirely equivalent to (5)–(9), and (1)–(4). However, it is a more suitable starting point for the development of numerical schemes. Remark In order to simplify notation, we will work with the global form of the variational equation subsequently. However, all results are equally applicable to the variational equations of the individual space–time slabs. 1.2 Scale separation We consider a direct-sum decomposition of V into “coarse-scale” and “ﬁne-scale” subspaces, V and V , respectively, V = V ⊕ V

(15)

V is assumed to be a ﬁnite-dimensional space and it will be identiﬁed later with the space of functions with which we actually compute. In order to make the decomposition well-deﬁned, we need to introduce a procedure for uniquely determining U ∈ V and U ∈ V from a given U ∈ V. This can be accomplished with the aid of a projector P : V → V. For example, P could be the L2 -projector, H 1 -projector, etc. There are inﬁnitely many possibilities.1 Once P is selected, we know how the coarse scales approximate all scales, viz., U = PU

(16)

U = U − PU = (I − P)U

(17)

where I is the identity operator. Likewise, we can decompose a weighting function into its coarse- and ﬁne-scale components: W = PW

(18)

W = W − PW = (I − P)W

(19)

With these, we may decompose the original variational equation into coupled coarse-scale and ﬁne-scale equations, viz., B(W , U + U ) = L(W )

B(W , U + U ) = L(W ) 1

(20) (21)

The way U is determined from U is a very important issue, and it has very signiﬁcant impact on the theory to be developed. An initiatory study of typical projectors is presented in Hughes and Sangalli [6]. Not only can one envision an inﬁnite number of possible projectors, but one can also envision an inﬁnite number of nonlinear optimization schemes that “ﬁt” U to U . In some applications nonlinear schemes will surely be important, an example being compressible turbulence with shocks where monotonicity is important. However, for incompressible turbulence, we feel linear projectors, such as the H 1 -projector, should suﬃce.

Variational Multiscale Theory of LES Turbulence Modeling

107

where B(W , U + U ) = B1 (W , U ) + B1 (W , U ) +B2 (W , U , U ) +B2 (W , U , U ) + B2 (W , U , U ) +B2 (W , U , U )

(22)

B(W , U + U ) = B1 (W , U ) + B1 (W , U ) +B2 (W , U , U ) +B2 (W , U , U ) + B2 (W , U , U ) +B2 (W , U , U )

(23)

In (22), B2 (W , U , U ) and B2 (W , U , U ) correspond to the cross-stress terms, and B2 (W , U , U ) corresponds to the Reynolds stress term. Equation (21) can be expressed as BU (W , U ) + B2 (W , U , U ) = W , Res(U )V ,V ∗

(24)

where BU (W , U ) = B1 (W , U ) +B2 (W , U , U ) + B2 (W , U , U ) W , Res(U )V ,V ∗ = L(W ) − B1 (W , U ) − B2 (W , U , U )

(25) (26)

in which Res(U ) is the coarse-scale residual “lifted” to the dual of the ﬁnescale space V ∗ , · , · V ,V ∗ is the duality pairing, and

BU ( · , U ) =

d B( · , U + εU ) dε ε=0

(27)

the linearization of B( · , U + U ) about U in the direction U . Note that the solution of (24) can be formally represented as a functional of U and Res(U ), namely, U = F (U , Res(U ))

(28)

The explicit dependence on U in the ﬁrst argument of F emanates from the dependence of the linearized operator BU on U . This expression can be inserted into (20) to “close” the ﬁnite-dimensional system for U , B W , U + F (U , Res(U )) = L(W )

(29)

(28) and (29) can be thought of in global terms or in terms of a sequence of space–time slabs. In both cases, they represent a procedure for solving the Navier–Stokes equations in terms of a scale decomposition of the solution.

108

Y. Bazilevs et al.

So far we have not discussed approximations or numerics. The solution U = U + U , where U is determined by solving (29) and U is determined from U through (28), is the exact solution of the original variational problem, (20)–(21), and (1)–(4), the Navier–Stokes initial/boundary-value problem. Our plan for turbulence modeling is to systematically approximate the functional F . This will provide us with a parameterization of the ﬁne scales in terms of the coarse scales, which can be substituted in the coarse-scale equation, “closing” it. The ﬁnite-dimensional coarse-scale equation can then be solved. In this way we obtain an approximate coarse-scale solution and an estimation of the ﬁne scales. In summary, our variational multiscale theory of turbulence modeling is encapsulated in the following equations: , Res(U )) (U = F U

(30)

+F , Res(U )) = L(W ) (U B(W , U

(31)

are the and U is an approximation of the exact functional F , and U where F approximations of U and U , respectively. We also note that (30) constitutes an a posteriori estimation of the error in the coarse-scale solution (see Hughes et al. [4, 7]). Remarks 1. (31) may be thought of as playing a similar role in the variational multiscale theory as the ﬁltered equations play in traditional turbulence modeling. Distinguishing features are (31) is ﬁnite-dimensional and closed, in contrast with the ﬁltered equations. 2. Intuitively, the “better” the ﬁne-scale approximation, the smaller the dimension of the coarse-scale space required, and consequently, the smaller the computational eﬀort. It is also possible to envision a hierarchy of approximations that produce variational multiscale analogues of traditional turbulence modeling concepts, such as large eddy simulation (LES), detached eddy simulation (DES), the Reynolds averaged Navier–Stokes (RANS) approach, etc. LES represents the turbulence modeling methodology requiring the greatest computational burden, but perhaps the least complex modeling. In the following sections we will endeavor to develop a variational multiscale analogue of LES within the theoretical framework of (30) and (31). 3. It is very important to emphasize that in practice we work directly with , (31), a ﬁnite-dimensional system, and we consider the solution of (31), U our approximation to U , and in turn our approximation to U . Recall, by design of P, U is an approximation to U . We do not need to solve for the ﬁne scales and because of this (30) is completely extraneous, unless we wish to use it to estimate the error in the coarse scales. That being +U as an alternative said, it may also be interesting to consider U

Variational Multiscale Theory of LES Turbulence Modeling

109

approximation to U . It will of course be necessary to assume that the coarse-scale space is suﬃciently large for the philosophy of LES to be appropriate. That is, if there is a well-deﬁned inertial sub-range, then we assume the cut-oﬀ between the coarse- and ﬁne-scale spaces resides somewhere within it. This assumption enables us to further assume that the energy content in the ﬁne scales is small compared with the coarse scales, an aspect of considerable importance in attempting to analytically determine the solution of the ﬁne-scale equations.

2 Turbulent channel ﬂow We consider an equilibrium turbulent channel ﬂow at Reynolds number 395 based on the friction velocity and the channel half width. The computational domain is a rectangular box of size 2π × 2 × 2/3π in the stream-wise, wallnormal, and span-wise directions, respectively. A no-slip Dirichlet boundary condition is set at the wall (y = ±1), while the stream-wise and the span-wise directions are assigned periodic boundary conditions. The no-slip condition is imposed strongly, that is, velocity degrees of freedom are explicitly set to zero at the wall. Alternatively, one may enforce the no-slip conditions weakly by augmenting the discrete formulation with terms that enforce Dirichlet conditions as Euler–Lagrange conditions (see Bazilevs and Hughes [2]). Although the weak boundary condition approach was shown to be superior to the strong imposition, we did not employ it in the computations reported in this paper. The ﬂow is driven by a constant pressure gradient, fx , acting in the streamwise direction. The values of the kinematic viscosity ν and the forcing fx are set to 1.47200 · 10−4 and 3.372040 · 10−3 , respectively. The computations were performed on meshes of 323 and 643 elements. For both meshes we employ C 1 -continuous quadratic, and C 2 -continuous cubic B-Splines. For both orders, in the stream-wise and the span-wise directions the number of basis functions is equal to the number of elements in these directions. On the other hand, the number of basis functions in the wallnormal direction is ny = nel + p, where nel is the number of elements in this direction and p is the polynomial order. Numerical results for this test case are reported in the form of statistics of the mean stream-wise velocity and root-mean-square velocity ﬂuctuations. Statistics are obtained by sampling the solution ﬁelds at the mesh knots and averaging in the stream-wise and span-wise directions as well as in time. Comparison of the statistical quantities of interest with the DNS data of Moser, Kim and Mansour [8] is made in order to assess the accuracy of the proposed turbulence modeling methodology. Note from Figs. 2 and 3 that for a mesh of 643 elements both quadratic and cubic solutions are almost identical to the DNS result. The results for 323 quadratic and cubic B-splines are even better than high-ﬁdelity spectral Galerkin LES results presented in Hughes et al. [5] and

110

Y. Bazilevs et al. 25 DNS Cubic

U+

20 15

64 10

32

5 0

100

10

1

0.1

u+

v+

w+

Mean stream-wise velocity 1.5 1 0.5 1.5 1 0.5 0 4 3.5 3 2.5 2 1.5 1 0.5

64

32

DNS

32

Cubic

DNS

64 0

100

200 y+

300

400

Velocity ﬂuctuations Fig. 2. Turbulent channel ﬂow at Reτ = 395 computed using quadratic B-splines.

Holmen et al. [3]. We note though that the formulation utilized in [3, 5] employed a ﬁne-scale eddy viscosity model and is quite diﬀerent from the one used here.

3 Conclusions We presented a general variational multiscale theory suitable for LES-type turbulence modeling. The theory is derived directly from the incompressible Navier–Stokes equations and does not involve any ad hoc mechanisms. In particular, it entirely avoids use of eddy viscosities. We feel that this theory of turbulence modeling is more fundamental and logically consistent than ones derived heretofore and it has signiﬁcant potential in practical engineering calculations. We might also mention that the turbulence modeling aspects remain unaltered when we consider laminar ﬂows. In this sense, our methodology may be viewed as an approach for solving the incompressible Navier–Stokes equations, whether the ﬂow under consideration is laminar or turbulent, or

Variational Multiscale Theory of LES Turbulence Modeling

111

25 DNS Cubic

U+

20 15

64 10

32

5 0 0.1

1

y+

10

100

u+

v+

w+

Mean stream-wise velocity 1.5 1 0.5 1.5 1 0.5 0 4 3.5 3 2.5 2 1.5 1 0.5 0

DNS

32

64

32

Cubic

DNS

64 100

200 y+

300

400

Velocity Fig. 3. Turbulent channel ﬂow at Reτ = 395 computed using cubic B-splines.

both. We also believe that this aspect separates our theory of turbulence modeling from predecessors. Given that the scale separation in the present methodology is performed with respect to the coarse-scale space actually used in the numerical computations, that is, the resolved scales, and that the ﬁne-scale approximation is rendered well-deﬁned by a projector used to make precise the direct sum decomposition into coarse and ﬁne scales, it is impossible to entirely separate modeling and numerical concepts. We accept this as a fact associated with correct LES-type modeling concepts, not a shortcoming. We also believe that our theory is more coherent mathematically than previous formulations and that it may be possible to use it as a basis of a statistical analysis of convergence and approximation. This would represent a very signiﬁcant step forward for the theory of turbulence modeling, but, admittedly, a very diﬃcult one to achieve. Nevertheless, we feel a door has been opened for the construction of a mathematical theory.

112

Y. Bazilevs et al.

In summary, we feel a new paradigm for turbulence modeling has been established. Initial results seem to indicate its accuracy per degree of freedom is superior or, at the least, equal to any procedure proposed heretofore. Its generality and geometric ﬂexibility also suggest it may provide a more powerful approach to turbulence calculations than previously existed.

Acknowledgements We wish to express our appreciation for support provided by the Oﬃce of Naval Research under Contract No. N00014-03-0263, Dr. Luise Couchman, contract monitor, and Sandia National Laboratories under Contract No. 114166.

References 1. Y. Bazilevs, V.M. Calo, J.A. Cottrel, T.J.R. Hughes, A. Reali, and G. Scovazzi. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible ﬂows. Computer Methods in Applied Mechanics and Engineering, 197:173–201, 2007. 2. Y. Bazilevs and T.J.R. Hughes. Weak imposition of Dirichlet boundary conditions in ﬂuid mechanics. Computers and Fluids, 36:12–26, 2007. 3. J. Holmen, T.J.R. Hughes, A.A. Oberai, and G.N. Wells. Sensitivity of the scale partition for variational multiscale LES of channel ﬂow. Physics of Fluids, 16:824–827, 2004. 4. T. J. R. Hughes, G. Feij´ oo., L. Mazzei, and J. B. Quincy. The variational multiscale method – A paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166:3–24, 1998. 5. T.J.R. Hughes, A.A. Oberai, and L. Mazzei. Large-eddy simulation of turbulent channel ﬂows by the variational multiscale method. Physics of Fluids, 13: 1784–1799, 2001. 6. T.J.R. Hughes and G. Sangalli. Variational multiscale analysis: the ﬁne-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM Journal of Numerical Analysis, 45:539–557, 2007. 7. T.J.R. Hughes, G. Scovazzi, and L.P. Franca. Multiscale and stabilized methods. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics, Vol. 3, Computational Fluid Dynamics, chapter 2. Wiley, 2004. 8. R. Moser, J. Kim, and R. Mansour. DNS of turbulent channel ﬂow up to Re = 590. Physics of Fluids, 11:943–945, 1999.

An Immersed Interface Method in the Framework of Implicit Large-Eddy Simulation M. Meyer, A. Devesa, S. Hickel, X.Y. Hu, and N.A. Adams Technische Universit¨ at M¨ unchen, Institute of Aerodynamics, 85747 Garching, Germany, [email protected]; [email protected]; [email protected]; [email protected]; [email protected]

1 Introduction Implicit Large Eddy Simulation (ILES) has shown considerable potential for the eﬃcient representation of physically complex ﬂows, see e.g. [4]. In ILES the truncation error of the discretization of the convective terms acts as a subgrid-scale model which is therefore implicit to the discretization. In this context, the Adaptive Local Deconvolution Method (ALDM) [5, 6] uses a discretization based on solution-adaptive deconvolution which allows to control the nonlinear truncation error. Deconvolution parameters are determined by an analysis of the spectral numerical viscosity. ALDM is incorporated into ﬁnite volume numerical solver for the incompressible Navier–Stokes equations based on ﬁnite volumes on a staggered grid. Applications to generic conﬁgurations, such as isotropic turbulence [5], plane channel ﬂow [7], and turbulent boundary layer separation [4], show excellent agreement with the corresponding results from theory, experiments, or direct numerical simulations. Given this performance the motivation is to use ALDM for the investigation of complex ﬂow conﬁgurations of practical relevance. Generally, the generation of good-quality body-ﬁtted grids can be timeconsuming and diﬃcult. Contradictory requirements, namely adequate local resolution and minimum number of grid points, can deteriorate the grid quality and therefore adversely aﬀect accuracy and numerical convergence properties. Alternatively, the discretization of the domain can be performed on a Cartesian mesh. Cartesian meshes also imply fewer computational operations per grid point. Bounding surfaces of the ﬂow can be accounted for by an Immersed Boundary (IB) approach. An overview on existing approaches is given in Reference [9], for which discrete mass and momentum conservation and high-order accuracy near the interfaces pose a challenge. Accurate representation of the ﬂow near the boundaries, however, is essential for LES of

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 17, c Springer Science+Business Media B.V. 2010

113

114

M. Meyer et al.

wall-bounded turbulence. The method proposed in this paper ensures mass and momentum conservation. In Section 2 a description of the diﬀerent steps for imposing the immersed boundary condition is given. Numerical examples, including validation results for laminar ﬂow around square and round cylinders at Re = 100 and turbulent ﬂow around a round cylinder at Re = 3, 900 are presented in Section 3, followed by concluding remarks in Section 4.

2 Conservative immersed interface method (CIIM) The immersion of an obstacle in the ﬂuid domain is characterized by a levelset ﬁeld Φ, i.e. a signed distance function for each point of the domain with respect to the immersed surface. The zero-levelset contour (Φ = 0) describes the interface between ﬂuid and the obstacle. The intersection of the obstacle with the Cartesian grid produces a set of cut cells, characterized by a solid and a ﬂuid part (see Fig. 1). By a special treatment of these cut-cells the immersed interface condition is imposed. 2.1 Cut-cell volume balance In terms of cut-cells the ﬁnite volume formulation only accounts for the ﬂuid part and the ﬂux is scaled with its corresponding face aperture. Depending on how the immersed boundary cuts the face of a computational cell, the face aperture A can have values between zero and one. Corresponding to the face apertures, volume fractions α for the ﬂuid part are determined. α can also have values between zero and one. The face apertures and volume fractions are determined by using the classiﬁcation of the Marching Square and Marching Cube approach by Lorensen [8]. The classiﬁcation allows for an eﬃcient calculation of the ﬂuid-related geometry portions and makes the approach suitable for treating ﬂuid–solid as well as for ﬂuid–ﬂuid boundaries.

Ai,j+1/2=1 Ai–1/2,j αi,j,k i,j,k

Fluid Ai+1/2,j

Immersed interface

(i,j,k) Solid

Fig. 1. Volume fraction αi,j,k , face apertures A and interface Γi,j,k for the cut cell i,j,k.

Immersed Interface Method for Implicit Large-Eddy Simulation

115

2.2 Friction force A ﬂuid moving along a solid boundary exerts a shear stress on that boundary due to the no-slip condition. For meshes aligned with the boundary, such as body-ﬁtted structured meshes, the imposition of the no-slip condition is straightforward. In order to account for the shear stress due to the immersed boundary on a Cartesian mesh, we add a friction force to the ﬂux balance of the cut cell. For a non-moving wall, this term can be expressed as ∂u utan − uinterface Fvisc = τwall dS = μ dS = μΓi,j,k · (1) ∂n hi,j,k Γ Γ (hi ni )2 + (hj nj )2 + (hk nk )2 (2) hi,j,k = 2 In accordance with the ﬁnite volume dicretization the shear stress τwall is integrated over the ﬂuid-solid interface Γ . ∂u∂n only considers the tangential velocity. μ represents the dynamic viscosity. In order to approximate the velocity diﬀerence, we use the diﬀerence of the tangential velocity and the velocity at the boundary. As geometric derivative the characteristic length of the cell hi,j,k is taken, where hi represents the cell size in i-direction and ni is the i-component of the normal vector. 2.3 Homogeneous Neumann condition for pressure For solving the elliptic Poisson equation we impose a homogeneous Neumann condition at the immersed interface. The discretized Poisson equation can be written as 1 ∇2 p = hj hk (Ai+1/2,j,k Fi+1/2,j,k − Ai−1/2,j,k Fi−1/2,j,k ) hi hj hk αi,j,k + hi hk (Ai,j+1/2,k Fi,j+1/2,k − Ai,j−1/2,k Fi,j−1/2,k ) + hi hj (Ai,j,k+1/2 Fi,j,k+1/2 − Ai,j,k−1/2 Fi,j,k−1/2 ) − Γi,j,k Fii,j,k (3) where F are the ﬂuxes of the cell faces and Fii,j,k is the interface condition for the pressure of the cut-cell i,j,k. The ﬂuxes on the diﬀerent computational cell faces are scaled with their corresponding face apertures of subsection (see Section 2.1). Setting the Fii,j,k to zero is equivalent to imposing ∂p∂n = 0 on the interface.

3 Numerical examples The computational domain for the following numerical examples is shown in Fig. 2. The upper and lower boundaries are characterized by periodic boundary conditions. At the inlet a uniform velocity and at the outlet a pressure

116

M. Meyer et al.

Fig. 2. Computational domain for numerical examples.

condition is deﬁned. On the boundary of the obstacle the immersed interface condition is imposed by CIIM (see Section 2). For the discretziation in space ALDM is used for the convective terms, and a second order accurate Central Diﬀerence Scheme for the diﬀusive terms. Time advancement is performed using a third order three step Runge–Kutta scheme. The Poisson equation is solved using Bi-Conjugate Gradient Stabilized (BiCGStab) iterative solver. All Computations are carried out with a CFL number of 0.5. 3.1 Square cylinder at Re = 100 At Re = 100, the ﬂow is 2D. Three diﬀerent grids with up to 520 ﬁnite volumes in the streamwise and 260 in the transverse direction equidistantly distributed on the computational domain were used (see Fig. 2). The ﬂowﬁeld of the calculation agrees well with the one described in literature. The ﬂow separation occurs at the rear edges and stays there. The wake shown (see Fig. 3) was found to be unsteady with vortex shedding at a Strouhal number St = 0.15, which agrees well with the experiments [10]. All examples are summarized in Table 1. 3.2 Round cylinder at Re = 100 Similar to the Square Cylinder at Re = 100, the ﬂow is 2D. The same three different resolutions as for the square cylinder were used. In good agreement with the literature, ﬂow separation occurs at rear part of the cylinder. The vortices are approximately half the size of the ones occurring in the square cylinder case, while reaching over the middle plane of the cylinder. The unsteady wake vortex shedding occurs at St = 0.17 which agrees well with the literature [1]. The convergence order study for the drag coeﬃcient shows second order convergence, while the drag coeﬃcient of the ﬁne grid perfectly corresponds with the literature (see [2]). All results are summarized in Table 2.

Immersed Interface Method for Implicit Large-Eddy Simulation

117

Fig. 3. U-velocity and streamlines of square cylinder at Re = 100. Table 1. Results of square cylinder at Re = 100. Case

St

Cd

Coarse (130 × 65) Medium (260 × 130) Fine (520 × 260) Franke et al. JoWiEng 1990 (CFD) Okajima JFM 1982 (Experiment)

0.143 0.158 0.155 0.154 0.14–0.15

0.86 0.98 1.38 1.61

Table 2. Results of round cylinder at Re = 100. Case

St

Cd

Coarse (130 × 65) Medium (260 × 130) Fine (520 × 260) Fey et al. PoF 1998 (Experiment) Franke et. al JoWiEng 1990 (CFD)

0.153 0.176 0.171 0.165

0.86 1.06 1.35 1.35

3.3 Round cylinder at Re = 3,900 The ﬂow over a round cylinder at Re = 3, 900 is three-dimensional and turbulent. The spanwise extension of the computational domain is 2π. Calculation were done on a hyperbolically stretched Cartesian mesh with 256 computational cells in streamwise, 128 is transverse, and 64 in spanwise direction. Good agreement is reached with the results of other calculations: the

118

M. Meyer et al.

Fig. 4. streamwise vorticity-isosurface ±1.5 s−1 of round cylinder at Re = 3, 900.

typical pressure tubes for the p-isosurfaces and the size recirculation zone corresponds with the results reported by [3]. The elongated structures for the x-vorticity shown in [11] can also be reproduced, see Fig. 4.

4 Concluding remarks In this paper a new Conservative Immersed Interface Method (CIIM) in the framework of ILES was introduced. According to the Cartesian ﬁnite volume approach for ALDM, CIIM operates on ﬂuxes of cut-cells only and therefore ensures mass and momentum conservation. The three major steps to impose the immersed interface condition were brieﬂy explained. As the presented method is constructed based on a standard Cartesian ﬁnite volume method and level set technique, it maintains the simplicity of GFM-like methods for implementation and handles topological changes naturally. A number of numerical examples were compared to experiments and Direct Numerical Simulations. The obtained results suggest that the method exhibits good accuracy and convergence properties. The eﬃcient classiﬁcation of cut-cells via a levelset function also allows the extension to moving boundaries and multiphase problems.

References 1. Fey U., K¨ onig M., and Eckelmann H. (1998) A new Strouhal-Reynolds-number relationship for the circular cylinder in the range 47 < Re < 2 · 105 . Physics of Fluids 10, 1547 2. Franke R., Rodi W., Sch¨ onung B. (1990) Numerical Calculation of Laminar Vortex Shedding Flow Past Cylinders. Journal of Wind Engineering and Industrial Aerodynamics 35, 237–257

Immersed Interface Method for Implicit Large-Eddy Simulation

119

3. Fr¨ ohlich J. (2006) Large Eddy Simulation turbulenter Strmungen. B.G. Teubner Verlag, Wiesbaden 2006 4. Hickel S., Adams N.A. (2008) Implicit LES applied to zero-pressure-gradient and adverse-pressure-gradient boundary-layer turbulence. International Journal of Heat and Fluid Flow 29, 626–639 5. Hickel S., Adams N.A., Domaradzki J.A. (2006) An adaptive local deconvolution method for implicit LES. Journal of Computational Physics 213, 413–436 6. Hickel S., Adams N.A. (2006) A proposed simpliﬁcation of the adaptive local deconvolution method. (Proceedings of CEMRACS 2005) ESAIM 16: 66–76 7. Hickel S., Adams N.A. (2007) On implicit subgrid-scale modeling in wallbounded ﬂows. Physics of Fluids 19: 105106 8. William E. Lorensen, Harvey E. Cline Marching Cubes: A high resolution 3D surface construction algorithm. In: Computer Graphics, Vol. 21, Nr. 4, Juli 1987, 163–169 9. Mittal R., Iaccarino G. (2005) Immersed boundary methods. Annual Review Fluid Mechanics 37, 239–261 10. Okajima A. (1982) Strouhal numbers of rectangular cylinders. Journal of Fluid Mechanics 123, 379–398 11. Kravchenko A. and Moin P. (2000) Numerical studies of ﬂow over a circular cylinder at ReD = 3900. Physics of Fluids 12, 403

Simulation of Gravity-Driven Flows Using an Iterative High-Order Accurate Navier–Stokes Solver R. Henniger and L. Kleiser Institute of Fluid Dynamics, ETH Zurich, 8092 Zurich, Switzerland, [email protected]; [email protected] Abstract We investigate by DNS the spatial growth of disturbances imposed at the inﬂow in 2D and 3D stratiﬁed mixing layer conﬁgurations. At the present moderate Reynolds, Schmidt and Richardson numbers the disturbances trigger Kelvin– Helmholtz waves which break down to small-scale ﬂuctuations. The results are found to be strongly inﬂuenced by the choice of the inﬂow setup, especially the disturbance type and magnitude.

1 Introduction and governing equations The disturbance growth in stratiﬁed mixing layers has been investigated by a number of authors in a temporal framework (see [3] for a review). In the present paper, we present direct numerical simulations (DNS) in 2D and 3D inﬂow-outﬂow conﬁgurations at moderate Reynolds, Schmidt and Richardson numbers to study the spatial disturbance growth and the basic mixing characteristics. The work targets at the future simulation of estuary mouth ﬂows by means of large eddy simulation (LES) where much higher Reynolds numbers are present. The density variations will then be caused by diﬀerent eﬀects like salinity and temperature gradients or suspended particles. We describe all density variations of the ﬂuid phase in an Eulerian manner by means of concentration ﬁelds which lead to additional volumetric forces on the carrier ﬂuid. The dimensionless transport equation for each of the concentration ﬁelds ck ∈ [0, Ck ] (k = 1, . . . , K) reads ∂ck 1 + u + usk eg · ∇ ck = Δck , ∂t Re Sc k

(1)

with Re as the Reynolds number, Sc k as the Schmidt number, u as the ﬂuid velocity, usk as a particle settling velocity and −eg as the unity vector in gravity

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 18, c Springer Science+Business Media B.V. 2010

121

122

R. Henniger and L. Kleiser

direction. Because the Boussinesq approximation can be applied at tolerable error, the dimensionless incompressible Navier–Stokes equations read ∇ · u = 0, ∂u 1 + (u · ∇) u = −∇p + Δu + eg Ri k ck , ∂t Re

(2a) (2b)

k

with the Richardson numbers Ri k . For equations (1) and (2) we employ either Dirichlet boundary conditions c = c0 ,

u = u0 ,

(3)

or convective and no-ﬂux boundary conditions, respectively, ∂c ∂c + Ung = 0 for Ung > 0, ∂t ∂n 1 ∂c = 0 for Ung ≤ 0, Ung c − ReSc ∂n ∂u ∂u + Un = 0, ∂t ∂n

(4a) (4b) (4c)

with the convection velocities Ung = n · (u + us eg ) and Un to be speciﬁed. n denotes the boundary-normal direction.

2 Numerical approach Equations (1) and (2) with boundary conditions equation (3) or (4) are discretized in time and space for a numerical solution. If the viscous time-step limit is more restrictive than the convective one, the diﬀusive terms are treated implicitly in time with the Crank–Nicolson scheme. Otherwise, a low-storage, third-order accurate Runge–Kutta integration scheme is employed as for all other terms in equations (1) and (2b). A new simulation code has been developed to be used on massively parallel computers [2]. Explicit, fourth-order accurate ﬁnite diﬀerences on staggered grids are chosen for the spatial discretization combined with a threedimensional domain-decomposition technique. This method requires ghost-cell updates using the Message Passing Interface (MPI) between adjacent processors before a spatial interpolation or diﬀerentiation of a quantity is performed. The discretized forms of equations (1) and (2) lead, in the implicit case, to linear systems of equations (LSE)

Hk ck = fk , k = 1, . . . , K f HG u = 0 D 0 p

(5) (6)

Simulation of Gravity-Driven Flows

123

for the unknown concentrations ck , velocity u and pressure p in each Runge–Kutta sub-time step. The matrices H, D and G are the discrete Helmholtz, divergence and gradient operators, respectively. The boundary conditions replace the conservation equations at the boundary in the respective rows, such that no boundary conditions for the pressure have to be formulated. The right-hand sides f comprehend all explicit terms of the discretized governing equations. The solution of equation (6) is achieved as follows. The Schur-complement problem DH−1 Gp = DH−1 f =: b for the pressure leads to a correspondingly smaller problem set, which formally replaces the lower matrix row in equation (6). Once the solution for the pressure is found, the velocity is determined by solving Hu = f − Gp. Because A := DH−1 G is a large and dense matrix, it is more eﬃcient to solve the equation in an iterative fashion instead to A is of a direct solution. Provided that an appropriate preconditioner A −1 r(l) with the residavailable, a simple Richardson iteration p(l+1) = p(l) + A ual r(l) = b−Ap(l) will converge quickly. We employ a so-called commutationbased preconditioner [1], such that the overall procedure is close to the well known SIMPLE algorithm but diﬀers fundamentally in the choice of A. Our preconditioner A involves two Poisson LSE, which can be solved with multigrid (MG) techniques very eﬃciently. The convergence rate is signiﬁcantly enhanced by employing the Krylov subspace method BiCGstab as primary solver with MG as preconditioner. A “Processor–Block Gauss– Seidel” smoother combined with line relaxation inside of MG leads to nearly parallelization-independent convergence rates. The Helmholtz problems in equations (5) and (6) are much better conditioned in practice than the pressure LSE, such that no multigrid preconditioning is necessary and the plain BiCGstab solver converges suﬃciently fast. A weak scaling test with up to 1.5 billion grid points on more than 1,700 processors has been performed [2] in which nearly no parallelization overhead was observed.

3 Flow conﬁguration The general ﬂow and simulation setup for a 2D conﬁguration is shown in Fig. 1. We consider only moderate Reynolds numbers Re = U h/ ν , Schmidt 2 numbers Sc and Richardson numbers Ri = g h/U in order to be able to capture some fundamental mixing processes of the considered stratiﬁed shear , ﬂows with DNS (U h = H/2, g are the reference velocity, reference length T

U z

I x

z0

H

fresh-water salt-water

O

g

B

Fig. 1. General setup in two dimensions at t = 0.

124

R. Henniger and L. Kleiser

and reduced gravitational acceleration, respectively. The heavier salt-water (index k = 1) is initially located in the lower part of the domain. Because the fresh-water is lifted above the salt-water in the vicinity of the estuary mouth before the mixing process takes place, we feed it into the domain at the upper part of the inﬂow ‘I’. The freshwater carries with it also suspended particles (k = 2). For the basic conﬁguration we choose the non-dimensional values H = 2, U = 1, C1 = C2 = 1, Re = 2, 000, Sc 1 = 1, Sc 2 = 2, Ri 1 = 0.5, Ri 2 = 0, us1 = 0 and us2 = −0.02. With a function f (y, z, t), the inﬂow proﬁle ‘I’ is given by u = {U f, 0, 0} for the velocity and c1 = C1 (1 − f ), c2 = C2 f for the concentration ﬁelds, cf. equation (3). In order to excite the instability of the shear ﬂow eﬃciently, we introduce small disturbances at the inﬂow plane by simply moving the z-position of the interface z i around an average position z0 = 0.75H. The outﬂow ‘O’ is modeled with boundary conditions equation (4) and Un = U . Since the boundary condition equation (4a) will wash the salt concentration successively out of the domain, equation (4b) is replaced by c1 = C1 for Ung ≤ 0 at the outﬂow. For the ﬂuid phase, the bottom ‘B’ is a no-slip and the top ‘T’ a non-deformable free-slip boundary. Accordingly, equations (4a) and (4b) are used in the vertical direction to model appropriate boundary conditions for the concentrations ﬁelds.

4 Results 4.1 Spatial growth of 2D and weak 3D disturbances Because we wish to study a sizable spatial growth in the downstream direction, we ﬁrst investigate the inﬂuence of a harmonic excitation z i (t) = z0 +z1 sin(ωt) for diﬀerent frequencies ω with z1 = 0.00025H√in two dimensions. For the inﬂow proﬁle we choose f (z, t)) = 0.5(1 + erf( π(z − z i (t))/m)) with the interface thickness m = 0.05H. The domain measures Lx × Lz = 12H × H and is discretized with Nx × Nz = 1537 × 129 grid points. A visualization of the obtained concentration ﬁeld is shown in Fig. 2 for ﬁve selected cases. The total kinetic energy in the computational domain is close to its minimum for ω = 3U/H, which is chosen in all successive simulations. To investigate the spatial growth of three-dimensional disturbances in the streamwise direction, we introduce the third, periodic, spatial dimension y with Ly = H and Ny = 129 grid points and excite the interface at the inﬂow additionally with a random disturbance, z i = z i (y, t). This is done usinga discrete Ornstein–Uhlenbeck process with reference time T = 1, 000 and var[f (y, t)] ≈ 0.0025H independently for each grid layer in the spanwise direction, such that the random disturbance on the average is an order of magnitude larger than the deterministic disturbance for the fundamental Kelvin–Helmholtz wave. Figure 3a reveals that mostly the smallest wavenumbers in spanwise direction are present beside the basic Kelvin–Helmholtz wave.

Simulation of Gravity-Driven Flows

125

z x

Fig. 2. Salt concentration (c1 = 0 (white), c1 = C1 (black)) at t = 150 for different excitation frequencies ω (2D simulations). From top to bottom: ωH/U = 1, 2, 3, 4, 5.

a z y x

Fig. 3. Salt concentration (c1 = 0 (white), c1 = C1 (black), iso-value c1 = 0.25C1 ) at t = 150 for a disturbed 2D inﬂow (ω = 3U/H) and additional weak 3D disturbances. (a) Random disturbances, (b) deterministic 3D disturbances.

This eﬀect can be strongly enhanced if two deterministic 3D disturbances with amplitude z2 = 0.00025H and wavenumbers k = ±2π/H are fed in instead, i.e. z i (y, t) = z0 + z1 sin(ωt) + z2 sin(ωt ± ky), as shown in Fig. 3b. 4.2 3D ﬂow conﬁguration and inﬂuence of suspended particles Now we investigate a genuinely 3D ﬂow conﬁguration by extending the physical domain to Ly = 6H and Ny = 769 grid points while the inﬂow extent is retained from the previous√2D conﬁguration. We√model this by redeﬁning f (y, z, t)) = 0.25(1 − erf( π(y − y i )/m)(1 + erf( π(z − z i (t))/m)) with m = 0.05H and the spanwise river half-extent y i . The ﬂow is assumed to be symmetric about y = 0. At y = Ly we introduce an additional convective outﬂow condition, cf. Section 3, in order to obtain an idealized estuary mouth conﬁguration. Based on the parameters in Section 3, four diﬀerent simulations are conducted with altered excitation frequency, inﬂow width and height and an additional particle concentration, cf. Fig. 4.

126

R. Henniger and L. Kleiser

a

z y x

Fig. 4. Salt concentration (c1 = 0 (white), c1 = C1 (black), iso-value c1 = 0.75C1 ) in an idealized estuary mouth at t = 150 for diﬀerent parameter variations. (a) Basic conﬁguration, (b) ω = 3.5U/H, (c) z0 = 0.666H, y i = 0.75H, (d) Ri 2 = 0.05.

Generally, the basic two-dimensional Kelvin–Helmholtz waves are still present downstream of the inﬂow up to x ≈ 6H. However, the breakdown of these structures is somewhat diﬀerent from the previous cases, especially the center of the vortices (“cat’s eye”) is pulled deeper under the water surface at this point. As a result, the increasingly unstable stratiﬁcation causes a quicker vortex breakdown and the mixing of the fresh-water with the ambient salt-water is correspondingly more eﬀective. This observation holds only for the central area of the estuary mouth near the symmetry plane. The mixing of fresh and salt-water in the other areas is obviously much less dominated by turbulence but rather inﬂuenced by diﬀusion.

5 Conclusions We investigated diﬀerent 2D and 3D stratiﬁed mixing layer conﬁgurations to study the spatial growth of disturbances imposed at the inﬂow. The disturbances are able to trigger Kelvin–Helmholtz waves at the present moderate Reynolds, Schmidt and Richardson numbers, accessible to DNS. The 2D disturbances grow much faster in the downstream direction than 3D ones, as expected from temporal simulations, even if strong 3D disturbances are imposed. However, our 2D and 3D simulations also reveal that the results are strongly inﬂuenced by the choice of the inﬂow setup. Especially the type and the magnitude of the disturbances have a large inﬂuence on the downstream ﬂow development. All computations were performed at the Swiss National Supercomputing Centre (CSCS).

Simulation of Gravity-Driven Flows

127

References 1. Elman H. (1999) Preconditioning for the steady-state Navier–Stokes equations with low viscosity. SIAM J. Sci. Comput. 20:1299–1316 2. Henniger R., Obrist D., Kleiser, L. (2007) High-order accurate iterative solution of the Navier–Stokes equations for incompressible ﬂows. Proc. Appl. Math. Mech. 7:4100009–4100010 3. Peltier W. R., Caulﬁeld C. P. (2003) Mixing eﬃciency in stratiﬁed shear ﬂows. Annu. Rev. Fluid Mech. 35:135–167

Compact Fourth-Order Finite-Volume Method for Numerical Solutions of Navier–Stokes Equations on Staggered Grids Arpiruk Hokpunna and Michael Manhart Fachgebiet Hydromechanik, Technische Universit¨ at M¨ unchen, Germany [email protected]; [email protected] Abstract A fourth-order ﬁnite-volume method for the three-dimensional Navier– Stokes equations is presented. It uses a novel divergence-free interpolation ensuring mass balance over the momentum cell. The fourth-order convergence rate is shown by the amplitude growth of the most unstable eigenmode in the plane channel ﬂow. Application of the method to a turbulent channel ﬂow demonstrates that the scheme is highly accurate and eﬃcient.

1 Introduction In ﬁnite-volume discretisation of the Navier–Stokes equations (NSE), the evolution of the cell-averaged value of the momentum is described by the summation of the ﬂuxes. Earlier developments of higher-order ﬁnite volume methods require surface and volume integrations which are expensive and had hindered the development for several decades until the work of Kobayashi in [1]. Soon after that, Pereira et al. [2] applied Kobayashi’s scheme to NSE on collocated grids using fourth-order accurate approximations. Piller and Stalio [3] presented fourth- and sixth-order schemes on staggered grids for the momentum term but retained a second-order approximation for the massconservation equation. There was a controversy whether the solution of pressure (mass-conservation and pressure gradient) must be fourth-order accurate to obtain fourth-order convergence for the velocities. The second-order solution of pressure is reported to be suﬃcient in [3] while it is found otherwise in [4]. In this paper we present a full fourth-order ﬁnite-volume method for the incompressible isothermal Navier–Stokes equations. The necessity of the fourth-order solution of pressure is investigated. The mass-conservation over the momentum term is given a special attention and a novel divergence-free interpolation for the convective velocity is developed.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 19, c Springer Science+Business Media B.V. 2010

129

130

A. Hokpunna and M. Manhart

z

p(i, j, k)

p(i+1, j, k)

y

u(is, j, k)

u(is+1, j, k)

x

w(i, j, ks)

w(i+1, j, ks)

Fig. 1. Arrangement of variables on a staggered grid. Solid line: p-cell, dotted line: u-cell, dash line: w-cell.

2 Cartesian grid system On Cartesian grids, a system of staggered grids can be set up by putting collocated grid points along a real line x using a strictly increasing function ξ(i), xi = ξ(i), i = 1, ..., nx. Staggered grid points are deﬁned by xis = 1 2 (xi + xi+1 ). The control volume Ωis of the momentum uis is deﬁned on the closed interval [xi , xi+1 ], likewise, the control volume of the pressure cells pi is deﬁned by Ωi = [xsi−1 , xsi ]. These deﬁnitions build up the staggered grid system (Fig. 1) considered in this work. It is important to make a distinction between the two averaged values, cell-averaged and surface-averaged. In this xyz paper we use [u]i,j,k to represent the cell-averaged u-momentum on (i, j, k)xy control-volume and [u]i,j,k is a surface-averaged one.

3 Numerical approximations We use the compact fourth-order interpolation and diﬀerentiation proposed in [1] to approximate the momentum and the diﬀusive ﬂuxes. In this section, we describe the fourth-order cell-centered interpolation, the novel divergence-free interpolation, and nonlinear corrections for staggered grids. 3.1 Cell-centered interpolation for the computation of mass ﬂuxes Mass conservation of the ﬂow ﬁeld is a crucial property. The common approach used to approximate the mass ﬂux at the surface of the pressure cell is the midpoint rule or the cell-centered interpolation. The second-order cell-centered interpolation is improved to fourth-order by using this explicit formula: xyz xyz xyz [u]yz is,j,k = β1 [u]is−1,j,k + β2 [u]is,j,k + β3 [u]is+1,j,k .

(1)

The interpolating coeﬃcients can be derived by matching the Taylor’s expansion.

Compact Fourth-Order FVM for Numerical Solution

131

3.2 Discretisation of Poisson equation In the projection methods, we need to solve a Poisson equation for the pressure ﬁeld p. The Laplacian operator in this method is given by the combination of the approximations of the divergence and the gradient operators. Upon solving the Poisson equation given by this Laplacian, the mass conservation can be ensured up to machine accuracy. Let Dx and Gx be the matrices responsible for the approximations of divergence and gradient in x-direction, respectively. Then the Laplacian operator in this direction is simply given by Lx = Dx Gx . 3.3 Divergence-free convective ﬂuxes In order to approximate the convection term for u-momentum, we make a distinction between the convective velocity and the momentum. The components of convective velocity are denoted as u, v and w. If the summation of these convective ﬂuxes is not zero but equals g then a source term Sis,j,k = xyz [g]xyz is,j,k [u]is,j,k will be added into the momentum equation. This source term directly aﬀects the local solution of momentum and degrades the global quality of the solution. In order to avoid this undesirable error we use the divergencefree interpolation to obtain the convective ﬂuxes. Divergence-free interpolation for convective ﬂuxes yz

A fourth-order Lagrange interpolation for [u]i,j,k from the mass ﬂuxes at the pressure cells is given by yz

yz

yz

yz

yz

[u]i,j,k = γis,1 [u]i− 3 ,j,k + γis,2 [u]i− 1 ,j,k + γis,3 [u]i+ 1 ,j,k + γis,4 [u]i+ 3 ,j,k . (2) 2

2

2

2

yz [u]i+1,j,k

is computed using the interpolating coeﬃcient γi+1 . The Similarly, coeﬃcients here are similar to a one-dimensional Lagarange interpolation. Suppose that the 2mth-order Lagrange formula is used for the above interpolation. The 2mth-order divergence-free interpolation for the convective velocity on the top surface of the u-momentum cell is then given by xy [w]is,j,ks

where, θis,l

1 = xis

xy θis,l [w]l− 1 ,j,ks ,

is+m

⎛ ⎞ 2m 2m =⎝ γis+1,j − γis,j ⎠ xl . j=l

(3)

2

l=is−(m−1)

(4)

j=l+1

Equation (3) is derived from a primitive value reconstruction via the second fundamental theorem of calculus. Due to the explicitness of the interpolation, the primitive values reconstruction, the interpolation and the conversion back to the cell-averaged values can be combined into a singlestep. The identical coeﬃcients are used to interpolate v. This divergence-free interpolation can be applied to every position in the domain.

132

A. Hokpunna and M. Manhart

3.4 Nonlinear correction The approximation of the nonlinear correction should be as local as possible. We approximate the nonlinear correction from the surface-averaged values. The nonlinear correction for [ui ui ] is given by 2 1 yz yz yz yz yz [u]is+ 1 ,j+1,k − [u]is+ 1 ,j−1,k [uu]is+ 1 ,j,k = [u]is+ 1 ,j,k [u]is+ 1 ,j,k + 2 2 2 2 2 48 2 (5) 1 yz yz [u]is+ 1 ,j,k+1 − [u]is+ 1 ,j,k−1 + 2 2 48 On the collocated direction(uj ui , j = i), it is more accurate to compute the correction term using the combination of the cell-averaged and the surfaceaveraged values because it delivers the shortest stencil.

4 Validation In the ﬁrst test case we evaluate the method using the instability of plane channel ﬂow. To this end , the parabolic proﬁle of the channel ﬂow is disturbed by the most unstable eigenfunction and the energy of this perturbation is expected to grow. The test case we are using here has the same settings as in [5]. The growth rate of the perturbation energy is measured at t = 50.29H/Ub where H is the channel half-width and Ub is the bulk ﬂow velocity. The convergence rates in Table 1 show that the proposed scheme is fourth-order accurate and the fourth-order solution of pressure is necessary. If the resolution in xdirection was reduced to 32, the fourth-order solution of pressure can still delivers a fourth-order accurate up to Nz = 512. However the convergence of the second-order pressure solution will stagnate already at Nz = 64, because the error in x-direction becomes dominant. In the second test case we perform direct numerical simulations of plane turbulent channel ﬂow. The target Reτ is 178.12 as obtained in [6]. The streamwise, spanwise and wall-normal directions are set to x, y and z accordingly. The periodic boundary conditions are applied in the streamwise Table 1. Error of the energy growth rate ε(G) of the ﬁrst mode and its convergence rate (C) in the instability of plane Poiseuille ﬂow using Nx = 64. The convergence with second-order (P 2) and fourth-order pressure (P 4) are shown in separate columns. Nz

ε(G) − P 2

ε(G) − P 4

C − P2

C − P4

32 64 128 256 512

5.3163e-03 1.9048e-03 1.9881e-04 1.8307e-05 5.7150e-06

5.1864e-03 1.8793e-03 1.8950e-04 1.0837e-05 7.3688e-07

— 1.48 3.26 3.44 1.68

— 1.46 3.31 4.13 3.88

Compact Fourth-Order FVM for Numerical Solution

133

Table 2. Speciﬁcations of numerical grids used in the grid dependency study. Grid

LX

LY

NX

NY

NZ

DX +

DY +

+ DZmin

+ DZmax

M F MKM1999, [6] KMM1987, [7]

12.57 12.57 12.57 12.57

4.20 4.20 4.19 6.28

96 128 128 192

80 128 129 129

96 128 128 160

23.6 17.7 17.7 11.9

9.4 5.9 5.9 7.14

1.1 0.72 0.054 0.054

5.8 4.4 4.4 4.4

20 18

3

Grid M KMM1987 MKM1999

16

2.5

+

u rms, MKM1999 + v rms, MKM1999 + w+rms, MKM1999 u rms, KMM1987 + v+rms, KMM1987 w rms, KMM1987

14 2

+

10

RMS

+

12

8

u++rms v rms w+rms

1.5 1

6 4

0.5

2 0 1

10

z+

100

0

20

40

60

80

100 +

120

140

160

180

200

z

Fig. 2. Mean streamwise velocity (left) and R.M.S. of the velocity (right) on grid M.

and spanwise directions. The top and the bottom walls are treated by no-slip boundary conditions. The ﬂow is driven by a constant pressure gradient which is added as a source term in the momentum equation of the streamwise velocity. The ﬂow is let to balance itself without mass ﬂow control. Due to the limited space we only report the results on two grids listed in Table 2 together with the grid spacings of the reference solutions by [6, 7]. The mean streamwise velocity and the R.M.S on grid M displayed in Fig. 2 are highly satisfactory. This solution of the fourth-order scheme using 0.7M grid point diﬀers from the two reference solutions not more than the diﬀerence between them, implying that the (very) small scales are the cause for this deviation. This is indeed conﬁrmed by the deviation in the higher-order statistics shown in Fig. 3. Increasing the resolution improved these statistics. Near the wall, the ﬁnest grid delivers an excellent agreement for the skewness but small deviations are seen from z = 0.4 until the center of the channel. This is because the grid spacing there is of a similar size to the spectral solutions and thus the small scales are not captured perfectly. The same reason explains the deviation of the ﬂatness factor near the wall. According to our study in the turbulent channel ﬂow, the fourth-order scheme is 2–3 times more expensive than the second-order scheme at the same number of grid cells. However, the second-order scheme requires eight times more grid cells to match the solution of the fourth-order and twice the number of time integration. Eﬀectively, the fourth-order scheme can be ten-times faster than the second-order one at a comparable accuracy.

134

A. Hokpunna and M. Manhart

1.5 1

F(u)-M F(w)-M F(u)-F F(w)-F F(u),MKM1999 F(w),MKM1999

25 20

0.5

Flatness

Skewness

30

S(u)-M S(w)-M S(u)-F S(w)-F S(u),MKM1999 S(w),MKM1999

0

15 10

– 0.5

5 0

–1 0

0.2

0.4

0.6 z

0.8

1

0

0.2

0.4

0.6

0.8

1

z

Fig. 3. Skewness (left) and Flatness factor (right) on grid M and F.

5 Conclusion The proposed scheme is highly accurate and eﬃcient. The engineering interesting scales of the turbulent channel ﬂow are well captured on 0.7M grid for the nominal Reτ = 180. It is the inevitable nature of non-spectral schemes that some information on the far-end of the spectrum will be lost. These eﬀects are observed on skewness and the ﬂatness factors.

References 1. M. H. Kobayashi, On a Class of Pade Finite Volume Methods, J. Comput. Phys. 156, 137 (1999). 2. J. M. C. Pereira, M. H. Kobayashi, and J. C. F. Pereira, A Fourth-OrderAccurate Finite Volume Compact Method for the Incompressible Navier-Stokes Solutions, J. Comput. Phys. 167, 217 (2001). 3. M. Piller and E. Stalio, Finite-volume compact schemes on staggered grids, Journal of Computational Physics 197, 299 (2004). 4. D. Ayodeji, W. Robert, V., and C. Mark, Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows, Part I: Theoretical Development, Numerical Heat Transfer, Part B 39, 207 (2001). 5. M. Malik, T. Zang, and M. Hussaini, A spectral collocation method for the NavierStokes equation, Journal of Computational Physics 61, 64 (1985). 6. Moser, Kim, and Mansour, Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590, Physics of Fluids 11, 943 (1999). 7. J. Kim, P. Moin, and R. D. Moser, Turbulence statistics in fully developed channel ﬂow at low Reynolds number, Journal of Fluid Mechanics 177, 133 (1987).

An Accurate Numerical Method for DNS of Turbulent Pipe Flow J.G.M. Kuerten Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, [email protected]

1 Introduction In contrast with channel ﬂow, there are only few numerical methods for DNS of turbulent pipe ﬂow [1]. The reason for this is the singularity at the pipe axis, which results from transformation of the governing equations to cylindrical coordinates. Most numerical methods which are presently being used can be divided in two classes. In the ﬁrst class [2–4] use is made of a ﬁnite volume approach on a staggered grid. In such an approach it is not so obvious how to ﬁnd suitable boundary conditions at the pipe axis. In the second class [5, 6] a spectral approach is chosen with a Fourier expansion in the two periodic (streamwise and azimuthal) directions and a Chebyshev collocation method in the radial direction. In order to avoid the clustering of collocation points near the pipe axis, where it is undesired, the radial direction is divided into several elements, which are coupled using continuity conditions for the velocity components and their radial derivatives. In such a method it is hardly possible to satisfy the continuity equation within machine accuracy. Moreover, the division of the radial direction into elements is somewhat arbitrary, reduces the global accuracy of the spectral method and makes it almost impossible to use the method for large-eddy simulation. Therefore, in the present work a novel spectral method for DNS of turbulent pipe ﬂow is developed, which circumvents the disadvantages of the spectral element method mentioned above. The method yields a fully divergence free velocity ﬁeld and satisﬁes all regularity conditions at the axis of the pipe.

2 Governing equations and numerical method The system of governing equations consists of the Navier–Stokes equation and the continuity equation, and reads in cylindrical coordinates (r, φ, z) and in non-dimensionalized form: V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 20, c Springer Science+Business Media B.V. 2010

135

136

J.G.M. Kuerten

⎧ ∂u 1 ∂ 2 u ∂ur ∂ 2 ur ur ∂P 2 1 ∂ 2 ∂uφ r r ⎪ r + +ω u −ω u + = + − − φ z z φ 2 2 2 2 2 ⎪ ∂t ∂r Re r ∂r ∂r r ∂φ ∂z r r ∂φ ⎪ ⎪ ⎪ 2 ⎪ ∂uφ ∂uφ ∂ 2 uφ uφ ∂P 2 1 ∂ 1 ∂ uφ 2 ∂ur ⎪ ⎪ r + +ω u −ω u + = + − + z r r z 2 2 2 2 2 ⎨ ∂t r∂φ Re r ∂r ∂r r ∂φ ∂z r r ∂φ ∂uz 1 ∂ 2 uz ∂ 2 uz ∂uz ∂P 2 1 ∂ − fz , ⎪ ∂t +ωr uφ −ωφ ur + ∂z = Re r ∂r r ∂r + r 2 ∂φ2 + ∂z 2 ⎪ ⎪ ⎪ ⎪ ∂u φ ∂u 1 ∂ ⎪ (rur ) + 1r ∂φ + ∂zz = 0 ⎪ ⎪ ⎩ r ∂r (1) where (ur , uφ , uz ) and (ωr , ωφ , ωz ) denote the cylindrical components of the velocity and vorticity vector, P is total pressure, t time and fz the mean axial pressure gradient. Finally, Re = Ub D/ν with D the pipe diameter, ν the kinematic viscosity and Ub the bulk velocity, is the Reynolds number. For the viscous ﬂow under consideration the boundary conditions are that all velocity components equal zero at the wall of the pipe. Moreover, we assume that the ﬂow is periodic over a length L in the axial direction. The numerical method is based on a Fourier–Galerkin expansion in the two periodic directions and a Chebyshev–Gauss–Lobatto collocation approach in the radial direction. This means that each velocity component and the pressure are written as u(r, φ, z, t) = u ˆkφ ,kz (r, t)ei(kφ φ+2πkz z/L) , (2) kφ ,kz

where kφ and kz are the azimuthal and axial wave number and u ˆkφ ,kz is the (complex) amplitude of the Fourier mode of quantity u. In order to avoid clustering of points near the pipe axis, the Chebyshev collocation points are chosen in the interval 1 ≥ r ≥ −1. Of course, the solution is not deﬁned for negative values of r, but all Fourier modes should satisfy symmetry relations: u ˆkφ ,kz (−r, t) = (−1)kφ +β u ˆkφ ,kz (r, t),

(3)

where β = 0 for the axial velocity component and pressure and β = 1 for the other velocity components. This property is used to deﬁne even and odd Chebyshev derivative matrices, so that radial derivatives of the solution can be calculated without storing the solution in negative points. By choosing the number of collocation points even, no collocation point is obtained in r = 0. This avoids special treatment of the terms in the Navier–Stokes equation at the pipe axis. Note, however, that all terms in (1) are regular in r = 0 and can be calculated for example by invoking l’Hˆ opital’s rule. Integration in time of the solution is performed by a combination of a compact-storage explicit three-stage Runge–Kutta method and the implicit Crank–Nicolson method. The nonlinear terms in the Navier–Stokes equation are handled in the explicit step, where fast-Fourier transforms are applied. Moreover, to prevent aliasing the 3/2-rule is adopted in the two periodic directions. This increases the stability of the method. The pressure and viscous

An Accurate Numerical Method for DNS of Turbulent Pipe Flow

137

terms are treated in the implicit step. In order to illustrate this method, the Navier–Stokes equation in (1) is schematically rewritten as ∂u 2 + N (u) = L(u) − ∇P, ∂t Re

(4)

where N (u) represent the nonlinear terms and L(u) the linear viscous terms. Then in every stage of the Runge–Kutta scheme a problem of the form 1 L(u(i+1) ) − γi u(i+1) − Re∇P (i+1) = 2 1 −γi u(i) + Re[βi N (u(i−1) ) + (1 − βi )N (u(i) )] − L(u(i) ) + Re∇P (i) 2 has to be solved. In this scheme γi = αRe , αi and βi are coeﬃcients of the i Δt Runge–Kutta method and the superscript refers to the stage. For channel ﬂow this problem is usually handled by ﬁrst solving a Poisson equation for the pressure at the new stage and subsequently use that to determine the new velocity ﬁeld. Since the equations for all components of the Fourier coeﬃcients are completely decoupled, this amounts to a set of one-dimensional linear systems of equations, one for every Fourier mode and velocity component. In cylindrical coordinates, the equations for the radial and azimuthal velocity components are coupled, as can be seen from (1). However, this coupling can be resolved by using instead of ur and uφ , u± = ur ± iuφ . Indeed, ∂u± 1 ∂ 2 u± ∂ 2 u± u± 2i ∂u± 1 ∂ r + 2 . (5) + − 2± 2 L(u± ) = r ∂r ∂r r ∂φ2 ∂z 2 r r ∂φ The velocity ﬁeld is made divergence free by application of a cylindrical variant of the inﬂuence matrix method, which was developed for channel ﬂow by Kleiser and Schumann [7]. This method diﬀers in two respects from the original one for channel ﬂow. On the one hand, the method for pipe ﬂow is simpler, as there is only one wall at r = 1. The solutions of the Helmholtz equations satisfy the correct regularity conditions at r = 0 by using the same symmetry properties as for the solution of the Navier–Stokes equation. On the other hand, the method for pipe ﬂow is more involved, since the viscous contributions to the radial and azimuthal components of the Navier–Stokes equations are coupled. In order to obtain the correct inﬂuence matrices, the Helmholtz equations needed for the inﬂuence matrix method also have to be coupled. The resulting velocity ﬁeld is divergence free up to machine accuracy.

3 Results Simulations are performed at a Reynolds number of 5,300 based on bulk velocity and pipe diameter. The length of the pipe was set equal to L = 5D. The simulations started from Poiseuille ﬂow onto which small divergence-free

138

J.G.M. Kuerten 0.1

0.05

0

–0.05

– 0.1

0

0.2

0.4

0.6

0.8

1

r/R

Fig. 1. Solid: amplitude of Fourier mode of radial velocity component for kφ = ur ) + (ˆ uφ ) for kφ = 1, kz = 2. 0, kz = 1; dashed: (ˆ

two- and three-dimensional perturbations were superposed. After some time a transition to a state of fully-developed turbulence takes place. Results shown are obtained with 128 Fourier modes in both periodic directions and 96 positive Chebyshev collocation points in the radial direction. The solution satisﬁes all regularity conditions near r = 0. These regularity conditions stem from the requirement that the Cartesian velocity components are continuous and have continuous derivatives at r = 0. As an example Fig. 1 shows the amplitude of the Fourier mode of the radial velocity component for kφ = 0 and kz = 1 which should be proportional to r close to r = 0. Also included in the ﬁgure is (ˆ ur ) + (ˆ uφ ) for kφ = 1 and kz = 2. Although no condition exists for the two contributions separately, their sum should be equal to zero at r = 0. The ﬁgure shows that these regularity conditions are accurately satisﬁed, in spite of the absence of a collocation point at r = 0. In Fig. 2 the mean axial velocity component is shown and compared with results from a spectral element method [5] at the same Reynolds number. Results of both methods are averaged over the two homogeneous directions and over more than 1,000 ﬁelds at diﬀerent times over a time span of approximately 1 × 104 in wall units. The results of both methods compare well. The slightly higher maximum axial velocity obtained in Ref. [6] can be attributed to the slightly higher value of the bulk Reynolds number in this simulation. Equally good agreement is obtained for the root-mean square of the three velocity components, which are shown in Fig. 3. Also the third and fourth order moments of the velocity components, i.e. the skewness and ﬂatness, agree well with results from Ref. [6]. The accuracy of the results can also be appreciated from the terms in the 2 2 equation for the turbulent kinetic energy. If Ek = 12 (u2 r + uφ + uz ) denotes

An Accurate Numerical Method for DNS of Turbulent Pipe Flow

139

20

15

10

5

0 −1 10

100

101 y+

102

Fig. 2. Mean axial velocity in wall units; solid: present result; dashed from Ref. [6]. 3 2.5

urms

2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1

r/R

Fig. 3. Root-mean square of velocity components in wall units; solid: present result; dashed from Ref. [6]; from top to bottom: axial, azimuthal and radial component.

the turbulent kinetic energy based on the ﬂuctuating parts of the velocity, the kinetic energy in cylindrical coordinates reads: 1 d 1 d 1 d d¯ uz + u σ + u σ ) + u u rur Ek + rur p − 2νr(ur σrr + = 0, z zr r z φ φr r dr r dr r dr dr (6) where a prime denotes the ﬂuctuating part of a quantity, σij is a component of the strain tensor rate in cylindrical coordinates and an overline denotes the mean part of a quantity. The ﬁrst term on the left-hand side is called the turbulent transport term T , the second term is the velocity-pressure gradient Π and the third term the viscous diﬀusion D. The fourth and ﬁfth terms are

140

J.G.M. Kuerten T Π D P ε sum

0.2

0.1

0

−0.1

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

Fig. 4. Terms in the balance equation for turbulent kinetic energy; T : turbulent transport term; Π: velocity–pressure gradient term; D: viscous diﬀusion; P : production; : dissipation; the sum of the terms is marked with +. 2 , with the production P and the dissipation , which we deﬁne as = 2νSij Sij the strain tensor in cylindrical coordinates. Figure 4 shows all terms in (6) scaled in wall units. Again the results are averaged over the homogeneous direction and time. It can be seen that apart from a region close the wall only the production and dissipation are important. The velocity–pressure gradient term is small everywhere. The sum of all terms in the equation is also shown in the ﬁgure. Its magnitude is smaller than 1 × 10−3 for all radial positions. An increase of the time averaging interval does not further reduce this. This shows that the time averaging interval is suﬃciently long to obtain converged statistical results and that the remaining error is caused by the accuracy of the spatial discretization.

References 1. Duggleby A, Ball KS, Paul MR, Fischer PF (2007) J Turbulence 8: Art. No. 43 2. Eggels JGM, Unger F, Weiss MH, Westerweel J, Adrian RJ, Friedrich R, Nieuwstadt FTM (1994) J Fluid Mech 268: 175–209 3. Wagner C, H¨ uttl TJ, Friedrich R (2001) Computers & Fluids 30: 581–590 4. Orlandi P, Fatica M (1997) J Fluid Mech 343: 43–72 5. Shan H, Ma B, Zhang Z, Nieuwstadt FTM (1999) J Fluid Mech 387: 39–60 6. Van Esch BPM, Kuerten JGM (2008) J Turbulence 9: 1–17 7. Kleiser L, Schumann U (1980) In: Hirschel EH, Proceedings of the 3rd GAMMConference on Numerical Methods in Fluid Mechanics, Vieweg, 165−173

Local Large Scale Forcing of Unsheared Turbulence Julien Bodart, Laurent Joly, and Jean-Bernard Cazalbou Universit´e de Toulouse, Institut Sup´erieur de l’A´eronautique et de l’Espace (ISAE) 10 av. Edouard Belin - BP 54032 - 31055 TOULOUSE Cedex 4, France, [email protected]; [email protected]; [email protected]

1 Introduction With the objective of studying the interaction between turbulence and a solid wall, a new way to generate a statistically steady turbulent state in a DNS setup is presented. Computation of ideal situations like turbulence diﬀusing from a plane/ point source requires to implement mechanisms of turbulent production that, (i) do not rely on the presence of mean-velocity gradients and, (ii) can be localized in space. A ﬁrst way to devise a shear-free turbulence-production mechanism has been proposed by Alvelius [1], it was based on the use of a random force ﬁeld, deﬁned in the spectral space and, consequently, not localized in the physical space. This method has been modiﬁed by Campagne et al. [2] in order to conﬁne the force ﬁeld in a ﬁnite-width, plane layer of ﬂuid and study the interaction between unsheared turbulence and a free-slip surface (Fig. 1). Since the Navier–Stokes solver was based on a pseudo-spectral method, the method of Alvelius was both a convenient and natural choice. In order to extend the work of Campagne et al. to the case of the interaction with a solid wall, we developed a mixed spectral/ﬁnite-diﬀerence Navier–Stokes solver. In this context, it is appealing to implement the forcing mechanism in the physical space. Such forcing methods have been recently proposed by Rosales and Meneveau [6] and Nagata et al. [5]. In the ﬁrst study the random force ﬁeld is linear (proportional to the instantaneous velocity ﬁeld) while the second study makes use of randomly-distributed elementary force ﬁelds (blobs). In either case, the forcing is statistically homogeneous in all the computational domain. In this paper, we present results obtained with a forcing method which is rather similar to that of Nagata et al. However, we use a diﬀerent kind of elementary force patterns, and conﬁne the forcing inside a plane layer of ﬂuid. The latter characteristic relies on the use of compact supports for the force patterns. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 21, c Springer Science+Business Media B.V. 2010

141

142

J. Bodart et al.

Fig. 1. Numerical analogy of the oscillating grid experiment, with turbulence selfdiﬀusion from a plane source.

2 Random force construction The random force fi appears as a source term in the incompressible Navier Stokes equation: ∂ui ∂uj 1 ∂P ∂ 2 ui + ui · =− +ν + fi (x, t) ∂t ∂xj ρ ∂xi ∂xj 2

(1)

The forcing is localized in the physical space, i.e., the vector ﬁeld is compactly supported, and excites the large scales of the ﬂow ﬁeld. In the context of projection methods, a divergence-free vector ﬁeld is preferable as it does not inﬂuence directly the pressure computation. We propose to synthetize such a vector ﬁeld using any function φ of class C1 and ψ of class C0 and the relations: ⎧ f = φ(x1 ) · φ(x2 ) · ψ(x3 ) →⎨ 1 − f2 = −φ(x1 ) · φ(x2 ) · ψ(x3 ) f ⎩ f3 = 0 To ensure both derivability and compact support, φ is chosen to be a second-order spline function, and ψ = φ for simplicity. Furthermore a low order spline function contains mainly low frequencies (Fig. 2). The above relations generate a two-component vector ﬁeld deﬁned in a 3D cubic box of size L. We generate three vector ﬁelds by means of permutation among the three coordinate indices. Each vector ﬁeld is characterized by its zerocomponent direction x1 , x2 or x3 (Fig. 3). Using three vector ﬁelds allows to excite the ﬂow ﬁeld in every directions. To build a plane source of turbulence (see Fig. 1), a 4 × 4 array of boxes divides the forced layer in the homogeneous directions. This corresponds to a turbulent input of energy at the wavenumber kL = 4 which is a good compromise between forcing at large length scales and keeping the forced layer

Local Large Scale Forcing of Unsheared Turbulence

143

Fig. 2. Second order spline function.

Fig. 3. Isomodule of the force, the three basis vector ﬁelds.

Fig. 4. Description of random shift in x3 , and x1 , x2 applied at each time step on the computed forcing ﬁeld.

relatively thin. At every time step, each box receives one of the basis vector ﬁelds randomly ampliﬁed, while keeping the time averaged input power constant. To avoid middle-plane symmetry of the domain, each box is slightly and randomly translated in the x3 direction. In the homogeneous directions random translations are applied as well to the resulting vector ﬁeld (Fig. 4). In the implementation point of view, the basis vector ﬁelds are computed at the preprocessing stage which keeps the simulation cost-eﬀective. The choice and the amplitude of the basis vector ﬁeld attributed to a given box at each time step as well as the amplitude of the diﬀerent translations are all governed by uniform distributions.

144

J. Bodart et al.

3 Results A MPI -based parallel solver has been developed, with an hybrid pseudospectral/ﬁnite diﬀerence approach. Fourier modes are used along directions x1 and x2 , while sixth order compact scheme have been implemented in x3 direction [4]. Time advancement uses a third order Runge–Kutta scheme for the advective terms and a second order Cranck–Nicholson for the viscous terms. The simulations are carried out using 216 fully idealized modes in the spectral directions and 256 grid points in the ﬁnite diﬀerence direction. In the presented case, free-slip boundary conditions are applied at the lower and upper side of the domain. After a transient, the power induced by the forcing ﬁeld is balanced by the dissipation rate in the whole domain and a statistically steady state is reached. Turbulence self-diﬀuses out of the forced layer, as presented in Fig. 5. Statistically steady state is necessary in every horizontal plane, and starting gathering statistics requires a particular care. Indeed, (i) the turbulence self-diﬀusion has to reach the wall (ii) the time scale is growing in the diﬀusive layer (Fig. 6), and so does the transient.

1

E /E

0.75 0.5 Whole domain Forced layer

0.25 0

0

5

10 15 20 Computational time

25

Fig. 5. Turbulent kinetic energy evolution in time.

Forced layer

z/L

10 7.5 5 2.5 0

0

0.5

1

1.5

1

τ

= ( /) 2

Fig. 6. Kolmogorov time scale evolution in the wall normal direction.

Local Large Scale Forcing of Unsheared Turbulence

Forced layer

10

Forced layer

10 7.5 z=L

7.5 z=L

145

5

5

Pure diﬀusion

Pure diﬀusion

2.5

2.5

0

0 0

0.25

0.5 0.75 I= w u

1

1.25

l=L∝−0.09·z=L 0

2

4 3 l=L=( k 2=²)=L

6

8

Fig. 7. Isotropy factor evolution an turbulent length scale evolution, averaged in x1 x2 planes.

10

St Sn

Forced layer

7.5 z=L

z=L

7.5 5

5

2.5 0 –2

Ft Fn

Forced layer

10

2.5

–1.5

–1

S˙ui =

–0.5 ∂ui3 ∂ui2 = ∂x ∂xi i

0

0.5

0

0

10

3 2

F˙ui =

20 ∂ui4 ∂ui2 = ∂x ∂xi i

30 2

40

Fig. 8. Skewness and ﬂatness factor of transverse and normal velocity derivatives evolution, averaged in x1 x2 planes.

This artiﬁcially generated turbulence remains physically consistent, as proved by the skewness and ﬂatness factors of the velocity derivatives (Fig. 8) in the forced layer. We ﬁnd Su −0.5 and Fu 5 in the middle plane, which is consistent with well known values of isotropic homogeneous turbulence. The resulting turbulent Reynolds number at the edge of the forced layer is Ret 150. Since the forced layer can be seen as the numerical analogue of the turbulence grid in oscillating-grid experiments, we shall examine our results with reference to the qualitative behaviour observed in these experiments. Characteristics of the pure-diﬀusion region are retrieved: a linear increase of the turbulent length scale and a nearly constant anisotropy ratio (Fig. 7). The values found are presented in Table 1 and compare very well with the experiments of De Silva and Fernando [3].

146

J. Bodart et al.

Table 1. Comparisons of the self-similarity characteristics in the pure diﬀusion layer.

I=

u3 2 u1 2

C=

∂k 2 /ε ∂x3

Experiment [3]

DNS

1.1–1.2

1.12–1.15

0.10

0.09

1 2

3

4 Conclusion We demonstrated that this conﬁned forcing is a well adapted tool to carry out Direct Numerical Simulation of self diﬀusion of unsheared turbulence. In our case, a plane source is built which fulﬁll all the requirements of our numerical experiment, i.e. to study kinematic-blocking eﬀect in the vicinity of a solid wall. The energy input can be adjusted, as well as the forcing ﬁeld anisotropy by changing the relative ampliﬁcation level of each basis vector ﬁeld. A pure diﬀusive state is recovered between the forced layer and the blocking layer and compares very well with oscillating grid experiments. Assuming the box size and the stroke length S of the grid in the experiments are comparable, this state is recovered much sooner. In our case less than 0.5L is needed while more than 4S in the experiment [3] which allows direct numerical simulations to be performed at a reasonable cost.

References 1. K. Alvelius. Random forcing of three-dimensional homogeneous turbulence. Physics of Fluids, 11:1880–1889, July 1999. 2. G. Campagne, J. B. Cazalbou, L. Joly, and P. Chassaing. Direct numerical simulation of the interaction between unsheared turbulence and a free-slip surface. In ECCOMAS CFD, 2006. 3. I. P. D. De Silva and H. J. S. Fernando. Oscillating grids as a source of nearly isotropic turbulence. Physics of Fluids, 6(7):2455–2464, 1994. 4. Sanjiva K. Lele. Compact ﬁnite diﬀerence schemes with spectral-like resolution. Journal of Computational Physics, 103(1):16–42, November 1992. 5. K. Nagata, P. A. Davidson, J. C. R. Hunt, Y. Sakai, and S. Komori. Direct numerical simulation of surface blocking eﬀects on isotropic and axisymmetric turbulence. In 5th International Symposium on Turbulence and Shear Flow Phenomena (TSFP-5 Conference), August 2007. 6. Carlos Rosales and Charles Meneveau. Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties. Physics of Fluids, 17(9), 2005.

Large-Eddy Simulations of a Turbulent Magnetohydrodynamic Channel Flow A. Vir´e1 , D. Krasnov2, B. Knaepen1 , and T. Boeck2 1

2

Physique Statistique et des Plasmas, Universit´e Libre de Bruxelles, Campus Plaine - CP231, Bd. du Triomphe, B-1050 Brussels, Belgium, [email protected]; [email protected] Fakult¨ at f¨ ur Maschinenbau, Technische Universit¨ at Ilmenau, P.O. Box 100565, 98684 Ilmenau, Germany, [email protected]; [email protected]

1 Introduction We consider the channel ﬂow of an electrically conducting ﬂuid subjected to a magnetic ﬁeld. In this framework, numerical predictions are particularly appealing because liquid metals are diﬃcult to study experimentally. In many industrial processes, the magnetic Reynolds number is low. Hence, the applied magnetic ﬁeld is not perturbed by the ﬂow (quasi-static approximation) and provides an additional force term in the equations of motion. Moreover, the Lorentz force acting on the ﬂow has globally dissipative and anisotropic eﬀects [1, 2]. In the case of low-intensity wall-normal magnetic ﬁelds, the turbulent ﬂuctuations tend to be suppressed at the center of the channel (ﬂattening eﬀect) and conﬁned in the near-wall region, where thin layers of high shear (the Hartmann layers) appear [3]. Direct Numerical Simulations (DNS) of magnetohydrodynamic (MHD) ﬂows require accurate numerical discretizations and are thus limited to moderate Reynolds number ﬂows and simple geometries. In Large-Eddy Simulations (LES), the dynamics of the unresolved scales is taken into account through a subgrid-scale model. The applicability of hydrodynamic models to MHD homogeneous turbulence has been proven successful [4]. Our aim is to evaluate the performances of under-resolved DNS of a MHD channel ﬂow (with and without subgrid model). Finite volume (FV) and spectral (PS) results are compared in terms of ﬁrst and second order statistics. A particular attention is also given to the contribution of the model to the kinetic energy budget.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 22, c Springer Science+Business Media B.V. 2010

147

148

A. Vir´e et al.

2 Equations of motion and subgrid modeling The relative importance of the nonlinear and the diﬀusive terms in the magnetic induction equation is quantiﬁed by the magnetic Reynolds number Rem = σμv U L, in which σ is the electric conductivity of the ﬂuid, μv is the vacuum magnetic permittivity and U , L are meaningful velocity and length scales, respectively. In the limit of low Rem , the perturbations of the magnetic ﬁeld induced by the ﬂow are negligible with respect to the applied magnetic ﬁeld. Hence, the Lorentz force becomes a linear function of the velocity and its eﬀects are taken into account through an additional term in the equations of motion. The Navier–Stokes equations become ∂t ui + ∂j (ui uj ) = −∂i p + ∂j (2νSij ) +

σ ijk jj Bk + Fi , ρ

(1)

together with the incompressibility condition, ∂j uj = 0. In equation (1), ρ, ν, σ are the ﬂuid density, kinematic viscosity and electrical conductivity, respectively; u, p, B are the velocity, kinematic pressure, external magnetic ﬁeld; and Sij = (∂j ui + ∂i uj )/2 is the strain-rate tensor. Repeated indices imply summation and ijk represents the permutation symbol. The forcing Fi keeps the mass ﬂow rate constant. In MHD, the induced electric current j is given by Ohm’s law, j = −∇Φ + u × B, assuming that the electric ﬁeld is the gradient of the electric potential Φ. The latter satisﬁes ∇2 Φ = ∇ · (u × B). Relevant non-dimensional parameters are introduced: the friction Reynolds number Reτ = uτ δ/ν, based on the frictionvelocity uτ and the channel halfwidth δ, the Hartmann number Ha = Bδ σ/ρν and the Reynolds number R = U0 d/ν based on the laminar centreline velocity U0 and the Hartmann layer thickness d = B −1 ρν/σ. In LES, the eﬀect of the unresolved scales is provided by subgrid-scale models. This work focusses on “viscosity-type” models (based on resolved scales), which consist in adding an eddy-viscosity νe to the kinematic viscosity. These models deﬁne τij = −2νe Sij + (τkk δij )/3 as the turbulent stress tensor and only its traceless part is modeled, the trace being taken into account into the pressure term. In the following, two models are discussed: Dynamic Smagorinsky (DSM) [5, 6] and Wall-Adapting Local Eddy-viscosity (WALE) [7]. No explicit ﬁltering is added to the classical formulations. In DSM, the ratio between the ﬁlter widths is set to 2 and the clipping procedure imposes ν + νe ≥ 0. In WALE, the model parameter C depends on the Reynolds number. The value C = 0.5 is chosen here, following [8].

3 Numerical methods DNS of MHD channel ﬂows have been investigated intensively in [3] using a pseudospectral (PS) code. Recently, the code has been extended to perform LES [9] and the results are used here as a benchmark. The ﬁnite volume (FV)

LES of a Turbulent Magnetohydrodynamic Channel Flow

149

computations are performed using the CDP code developed at the Center For Turbulence Research (Stanford/NASA-Ames). The details of the code are described in [10, 11]. A collocated discretization of the incompressible Navier–Stokes equations is used so that the velocity and pressure ﬁelds are both stored at the centroid of each control volume and the (independent) face normal velocities Uf are stored at the centroid of the volume’s faces. The discrete equations of motion are solved using the semi-implicit (Adams– Bashforth/Crank–Nicholson) fractional step method explained in [12]. In addition, a quantity known at two neighboring nodes is linearly interpolated to the mutual face and the gradients are evaluated using Gauss theorem and the summation-by-part (SBP) interpolation described in [11]. The face-centered gradient, which represents the largest contribution to the viscous stress, is treated more accurately using a central diﬀerence scheme. The resulting method is second-order accurate.

4 Results The channel ﬂow is bounded by parallel walls located at y = ±δ and its size is (2πδ) × (2δ) × (4πδ/5), in the streamwise x, wall-normal y and spanwise z directions. Periodic boundary conditions and a uniform mesh spacing are used in x, z. The no-slip condition is imposed at the walls and a cosine stretching function is adopted in y, following [13]. Moreover, the walls are insulating, i.e. (∂y Φ)walls = 0. Simulations are started from hydrodynamic δ ﬁelds at Reb = 2Ub δ/ν = 26, 600, Ub = −δ u(y)dy/2δ being the bulk velocity, and the forcing maintains a constant Reb . The wall-normal magnetic ﬁeld is uniform and such that Ha = 20, which corresponds to R = 700. The simulation parameters and results are reported in Table 1. As Reτ slightly varies S from one simulation to another, the reference value ReDN = 708.6 is used in τ the non-dimensionalizations. The mean streamwise velocity proﬁles are represented in Fig. 1 (left) as a function of the wall-normal coordinate in wall units. The proﬁles are ﬂattened at the center of the channel, where the magnetic ﬁeld tends to suppress the velocity perturbations [3]. This phenomenon is well captured by both solvers Table 1. Simulation parameters and results. The time interval Ta is the averaging time in convective units δ/U0 . Case

Nx × Ny × Nz

+ + Δx+ Δz + Δymin Δymax Model Method Reτ Ta

UDNS (FV) DSM (FV) WALE (FV) DSM (PS) DNS (PS)

64 × 64 × 64 64 × 64 × 64 64 × 64 × 64 64 × 64 × 64 256 × 256 × 256

70 70 70 70 17

28 28 28 28 7

0.85 0.85 0.85 0.85 0.05

35 35 35 35 9

None DSM WALE DSM None

FV FV FV PS PS

667 653 623 702 709

223 160 180 224 116

150

A. Vir´e et al. 25 20

u+

15

0.8

UDNS (FV) DSM (FV) WALE (FV) DSM (PS) DNS (PS)

UDNS (FV) DSM (FV) WALE (FV) DSM (PS) DNS (PS)

0.6 Rxy 0.4

10

0.2

5 0 10-1

100

101

y+

102

103

0

-1

10

10

0

1

10 y

2

10

3

10

+

Fig. 1. Non-dimensionalized mean velocity proﬁle (left) and Reynolds shear stress (right) as a function of the wall-normal coordinate, in wall units.

but the FV method slightly underestimates the velocity in the plateau region and in the buﬀer layer (5 ≤ y + ≤ 30). The FV proﬁles also exhibit a bump for 30 ≤ y + ≤ 200 in comparison with the spectral LES and DNS results. Finally, the UDNS run provides an accurate velocity proﬁle despite the absence of model. An important issue in under-resolved DNS of wall-bounded ﬂows is the accuracy of the Reynolds shear stress. A detailed study of the skin friction behaviour in hydrodynamic channel ﬂows has been discussed in [14] at different streamwise and spanwise resolutions and two Reynolds numbers. The authors highlighted that, for some set of parameters, the skin friction provided by UDNS might match the DNS value. In that case, the addition of a model worsens the results in terms of skin friction. Similar observations can be made here in MHD. Figure 1 (right) illustrates that, in the near-wall region, the Reynolds shear stresses obtained from the under-resolved cases are smaller than in DNS. This is conﬁrmed by the values of the friction Reynolds number S reported in Table 1. Without subgrid model, 94% of ReDN is obtained with τ the FV code. This percentage decreases to 92% (resp. 88%) when DSM (resp. S WALE) is used. Using the spectral method, 99% of ReDN is obtained using τ S DSM. At a resolution of 128 × 64 × 128 (not shown here), ReUDN = 709.9 in τ the FV case, which is slightly higher than the DNS value. It however decreases when a model is added, while keeping the same resolution, in fact DSM (resp. S WALE) gives 98% (resp. 96%) of ReDN . τ The proper comparison between DNS and LES Reynolds stress tensors has been discussed in [15]. In LES, a part of the small-scale motion is contained DN S LES • in the tensor τij , so that Rij = Rij + τij , where Rij = ui uj • and ui denotes the velocity ﬂuctuations. The bullet stands for the superscript either DNS or LES. The present eddy-viscosity models only approximate the traceless part of τij . Thus, only the traceless part of the tensors (denoted ∗DN S ∗LES by an asterisk) may be reconstructed, so that Rij ≈ Rij + τij∗ . In ∗ ∗DN S ∗LES the following, Rij stands for either Rij or (Rij + τij∗ ). Figure 2 (left) shows the diagonal component of the Reynolds stress tensor in the streamwise direction. In under-resolved FV and PS simulations, the absolute value of the

LES of a Turbulent Magnetohydrodynamic Channel Flow 6 5 4 R

xx

UDNS (FV) DSM (FV) WALE (FV) DSM (PS) DNS (PS)

0.4 0.2

3

0.06

CK (FV) CK (PS) Vν (FV) Vν (PS) MK (FV) MK (PS)

0.6

0.02 0

0 - 0.2

−0.02

1

- 0.4

−0.04

10

0

10

1

2

+

y

10

3

10

100

101

y+

102

Vsgs(FV) Vsgs(PS)

0.04

2

0

151

103

−0.06 100

101

y+

102

103

Fig. 2. Diagonal streamwise component of the Reynolds stress tensor (left). Resolved kinetic energy budget terms in DSM: CK , Vν and MK (center); Vsgs (right).

Reynolds stress is over-estimated in the near-wall region, when compared to the DNS. Moreover, the location of the peak value is shifted away from the walls using the FV method. Same observations are made in the wall-normal and spanwise directions (not shown here). Finally, we propose to evaluate the contribution of the subgrid model to the resolved kinetic energy budget. The balance is obtained by averaging (in the homogeneous directions) the scalar product of the velocity ﬁeld and the discrete equations of motion. Only the relevant contributions are presented here. They are written CK , Vν , Vsgs and MK for the convective, viscous, subgrid and magnetic contributions, respectively. For clarity, only the DSM results are presented. As shown in Fig. 2 (center), CK and Vν are the main components of the budget, and they almost cancel each other at each ylocation. The FV values are also systematically smaller than the PS ones. On the contrary, the MHD contribution is similar using the two methods but it is smaller than CK and Vν . It is also interesting to observe, in Fig. 2 (right), that Vsgs represents only between 1% and 8% (resp. 3% and 20%) of Vν using the FV (resp. PS) method, in the region 0 ≤ y + ≤ 30. It is recognized that discretization errors overwhelm the model contribution in wall-bounded ﬂows, for some set of parameters [16]. Our study also shows that the FV model has less impact on the kinetic energy balance than in the PS method.

5 Conclusions This work focussed on under-resolved DNS of a magnetohydrodynamic channel ﬂow, in a classical turbulent regime (Reb = 26, 600). Finite volume simulations without subgrid model (UDNS) and with two hydrodynamic models (DSM, WALE) were compared with spectral results in terms of ﬁrst and second order statistics. The study revealed that no signiﬁcant diﬀerence between UDNS and LES cases are observed in the FV velocity proﬁle and Reynolds stresses. In addition, the FV shear obtained from UDNS is closer to the DNS value than using DSM or WALE. An analysis of the kinetic energy budget also showed that the subgrid dissipation equals a small fraction of the

152

A. Vir´e et al.

viscous and convective contributions to the balance equation. This is due to a strong interaction between numerical errors and subgrid modeling. Additional investigations are needed to evaluate this interaction in both methods, at diﬀerent regimes and resolutions.

Acknowledgements A. V. is supported by the Fonds pour la Recherche dans l’Industrie et dans l’Agriculture (F.R.I.A - Belgium). B. K. acknowledges the ﬁnancial support of the award Modelling and simulation of turbulent conductive ﬂows in the limit of low magnetic Reynolds number made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) and funded by the Participating Organisations of EURYI and the EC Sixth Framework Programme. The support of FRS-FNRS Belgium is also gratefully acknowledged. T. B. and D. K. acknowledge ﬁnancial support from the DFG in the framework of the Emmy–Noether program (grant Bo1668/2-2) and computer resources provided by the Forschungszentrum Juelich.

References 1. Moreau R (1990) Magnetohydrodynamics. Kluwer Academic Publishers, Dordrecht 2. Davidson PA (2001) An introduction to magnetohydrodynamics. Cambridge University Press 3. Boeck T, Krasnov D, Zienicke E (2007) J Fluid Mech 572:179–188 4. Knaepen B, Moin P (2004) Phys Fluids 16:1255–1261 5. Germano M, Piomelli U, Moin P, Cabot WH (1991) Phys Fluids A 3:1760–1765 6. Lilly DK (1992) Phys Fluids 4:633–635 7. Nicoud F, Ducros F (1999) Flow, Turb and Comb 62:183–200 8. Bricteux L (2008) Simulation of turbulent aircraft wake vortex ﬂows and their impact on the signals returned by a coherent Doppler LIDAR system, PhD Thesis, Universit´e Catholique de Louvain 9. Vir´e A, Krasnov D, Knaepen B, Boeck T (2008) Parallel simulation of turbulent magneto-hydrodynamic ﬂows. In: Proc. of the Parallel Computing conference 2007. Parallel Computing: Architectures, Algorithms and Applications. IOS Press, The Netherlands 10. Ham F, Iaccarino G (2004) Energy conservation in collocated discretization schemes on unstructured meshes. In: Annual Research Briefs. Center for Turbulence Research, NASA Ames/Stanford University 11. Ham F, Mattsson K, Iaccarino G (2006) Accurate and stable ﬁnite volume operators for unstructured ﬂow solvers. In: Annual Research Briefs. Center for Turbulence Research, NASA Ames/Stanford University 12. Kim J, Moin P (1985) J Comput Phys 59:308–323 13. Kim J, Moin P, Moser R (1987) J Fluid Mech 177:133–166 14. Meyers J, Sagaut P (2007) Phys Fluids 19:048105 15. Winckelmans GS, Jeanmart H, Carati D (2002) Phys Fluids 14:1809–1811 16. Meyers J, Geurts BJ, Sagaut P (2007) J Comput Phys 227:156–173

Development of a DNS-FDF Approach to Inhomogeneous Non-Equilibrium Mixing for High Schmidt Number Flows Florian Schwertﬁrm and Michael Manhart Fachgebiet Hydromechanik, Technische Universit¨ at M¨ unchen, Germany, [email protected]; [email protected]

1 Introduction Turbulent mixing is of great importance in a wide range of engineering applications. In process engineering it is the rate determining step when the chemical time scales are small compared to the turbulent ones, and thus has a large impact on the product properties. As processes like precipitation take place in aqueous environments, such processes are challenging for numerical simulation. This is due to the wide range of involved scales, starting from the integral√ scale of the mixing apparatus LΦ down to the Batchelor scale ηB = ηK / Sc [1], which can be much smaller than the Kolmogorov scale ηK due to the high Schmidt numbers encountered in aqueous solutions. To correctly predict the reaction rates both, the large scale inhomogeneities as well as the molecular mixing on the Batchelor scale have to be predicted correctly. In the last two decades two useful trends could be observed. With the advent of increasing computer power Direct-Numerical Simulation (DNS) became available for the simulation of low Reynolds number ﬂows and was used intensively for analysis of such ﬂows. Nevertheless a DNS of very high Sc ﬂows is still out of reach. Also, probability-density function (PDF) approaches were developed for the statistical description of scalar ﬁelds in combination with Reynolds Averaged Navier Stokes (RANS) methods [2] in which the chemical source term is closed. The PDF transport equations can be solved with various methods like presumed PDFs or the DQMOM, but most accurately with the direct Monte-Carlo method, where a large number of stochastic particles follow a Fokker–Planck equation which produces the same statistics as the underlying PDF transport equation. This approach was adapted by Colucci et al. [3] to be used in conjunction with a Large Eddy Simulation (LES) of the ﬂow ﬁeld. In this context the scalar ﬁeld is described by the ﬁltered-density function (FDF) which has the same properties as the PDF but describes the local and instantaneous subgrid scale distribution of the V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 23, c Springer Science+Business Media B.V. 2010

153

154

F. Schwertﬁrm and M. Manhart

scalar ﬁeld. Due to this localness in space and time mixing models in the FDF transport equation perform better [4] and due to the LES of the ﬂow, large scale and time dependent inhomogeneities are generally predicted more accurately than in RANS-PDF approaches. Until now the LES-FDF approaches were focused on combustion simulations at low Sc ﬂows [3]. In this work we combine the above mentioned approaches, namely DNS of the ﬂow ﬁeld and a FDF simulation of the scalar ﬁeld with the focus on high Sc ﬂows.1 We formulate the FDF transport equation for the combination with a DNS of the ﬂow ﬁeld in which the convective term is closed and the only term which has to be modelled is the micro mixing term. With a good performance of simple mixing models in the LES-FDF context in mind we systematically test the LMSE model for the micro mixing term in the DNSFDF approach and develop a model for the prediction of the subﬁlter scalar dissipation rate used in the deﬁnition of the mixing frequency. To achieve this, we use DNS data of turbulent channel ﬂow and turbulent mixing with passive scalar transport up to Sc = 49 [5].

2 The DNS-FDF approach In the DNS-FDF approach a ﬁlter operation is applied to the ﬁeld variables with a ﬁlter width in the same order of magnitude as the Kolmogorov scale Δ ≈ ηK . Therefore the velocity ﬁelds are unchanged by the ﬁlter operation and for high Sc ﬂows only the small scale ﬂuctuations of the scalar ﬁeld are removed Φ = Φ + Φ . With this ﬁlter operation we deﬁne the FDF: +∞ [Ψ, Φ(xi , t)] G(xi − xi )dxi . (1) PL (Ψ ; xi , t) = −∞

Under the assumption that the velocity distribution is linear within the ﬁlter width the FDF transport equation results in "

# ∂2Φ ∂PL ∂ ∂PL ∂ 2 Φ + ui Γ 2 +Γ (2) =− |Ψ + ω(Ψ ) PL , ∂t ∂xi ∂Ψ ∂xi ∂x2i where the only unclosed term is the second term on the rhs that describes micro mixing.

1

As for Sc ≤ 1 a DNS of the ﬂow ﬁeld would imply that the scalar ﬁeld is also fully resolved.

DNS-FDF Approach for Mixing at High Schmidt Number Flows

155

3 The LMSE model for micro mixing To close this term we propose the linear mean square estimation model (LMSE). We performed a priori analysis of DNS data of mixing in turbulent channel ﬂow up to Sc = 49 and as can be seen in the scatterplot in Fig. 1 (left) the LMSE model Γ

∂ 2 Φ |Ψ = −ΩM Ψ − Φ ∂x2i

(3)

gives a good representation of the dynamics of the micro mixing term along Lagrangian paths, which gives a direct relation to the Monte-Carlo methods. The mixing frequency is deﬁned as

∂Φ ∂Φ ∂ 2 Φ2 1 ΩM = − 2 Γ (4) − 2Γ 2Φ ∂x2i ∂xi ∂xi and therefore the ﬁrst and second moments of equation (2) are transported exactly. Figure 1 (right) shows that the molecular transport of scalar variance in the deﬁnition of ΩM (4) can be neglected. In the limit of high Sc, the total dissipation of the scalar variance stems from the subgrid scalar dissipation rate Φ . As this is the key parameter in computing the micro mixing and as it can not be computed from the FDF it has to be modelled.

−ΩM Φ

2

Γ ∂∂xΦ2 i

−ΩM Φ

2

Γ ∂∂xΦ2 i

Fig. 1. Scatterplot of the diﬀusion by the subﬁlter ﬂuctuation and the corresponding LMSE model at Sc = 25; left: ΩM according to equation (4), right: diﬀusion of scalar variance neglected.

156

F. Schwertﬁrm and M. Manhart

4 Modelling the subgrid scalar dissipation rate As we aim for a model for non homogeneous, non equilibrium ﬂows, we analyse the transport equation of the subgrid scalar dissipation rate ∂Φ ∂ 2 Φ ∂Φ 2 ∂gk ∂gk =Γ − 4Γ + + uj ∂t ∂xj ∂x2j ∂xj ∂xj ∂ uj gk gk − uj gk gk −2Γ ∂xj ∂2 ∂uj + 4Γ gk uj Φ − uj Φ − 4Γ g g , ∂xk ∂xj ∂xk j k

(5)

with gi = ∂Φ/∂xi , along Lagrangian paths. As can be seen in Fig. 2, which shows the PDF of the terms sampled from a large number of Lagrangian paths in the DNS of the turbulent channel ﬂow at Sc = 3, the ﬁrst and the third term on the rhs, i.e. the molecular transport and the term originating from the convective term respectively, can be neglected. The second term D is the destruction of subgrid scalar dissipation rate and the last two terms, G and V , are production terms. For modelling equation (5) we follow Fox [6,7], who develops a formulation of the dissipation rate residing at the dissipative scales from a spectral point of view. In the model transport equation we relate the transfer of spectral scalar energy at the Kolmogorov scale τD (kK ) to a product of scalar variance computed at the Kolmogorov scale with the Kolmogorov time scale and get 1/2 DΦ Φ CD ∗2 = Φ + Cs Φ − Cd Φ Dt ln(Sc) ν ν Φ2

(6)

P DF

Fig. 2. PDF of the rhs of equation (5) at Sc = 25; M is the ﬁrst term, D the second, K the third, G the fourth and V the last term.

DNS-FDF Approach for Mixing at High Schmidt Number Flows

157

t uν2

τ

t uν2

τ

Fig. 3. Production and dissipation along a Lagrangian path from DNS (5) and model equation (6) at Sc = 3 and Sc = 25.

as a model for the transport equation of the subﬁlter scalar dissipation rate. The scalar variance at the Kolmogorov scale is approximated by applying the 2 relation Φ∗2 = Φ − Φ˜ where Φ˜ is the ﬁltered scalar ﬁeld with a ﬁlter width of 2Δ. To this end, production due to the ﬁltered scalar gradient is neglected. By comparing production and dissipation terms of equations (5) and (6) the model constants could be determined. Figure 3 shows the comparison of the production and dissipation terms between DNS data and model along selected Lagrangian paths at Sc = 3 and Sc = 25. Along the same paths model equation (6) was solved numerically in time. After initialisation of the scalar dissipation rate at t = 0 with the DNS value, equation (6) was integrated forward in time with the rhs taken from DNS data which would also be available in the DNS-FDF approach. As Fig. 4 shows, the model is capable of capturing both, the amplitude and the dynamics of the evolution of the subgrid scalar dissipation rate along Lagrangian paths quite accurately. Even if single events are not captured accurately, the dissipative character of the model equation leads to a stabilisation and return towards zero subgrid scalar dissipation.

5 Conclusions In this work, a combination of a DNS of the ﬂow ﬁeld with a FDF simulation of the scalar ﬁeld is proposed for the simulation of turbulent mixing of high Sc number ﬂows. It is shown that the simple LMSE model is adequate for

158

F. Schwertﬁrm and M. Manhart

t uν2

τ

t uν2

τ

Fig. 4. Time integration of model equation (6) compared with DNS data along a Lagrangian path at Sc = 3 and Sc = 25.

modelling the micro mixing term in the FDF transport equation and that the molecular transport of scalar variance in the deﬁnition of the mixing frequency ΩM can be neglected. As the subgrid scalar dissipation rate is unclosed in this formulation we introduce a model transport equation for this quantity. To model the individual terms we follow Fox [6] and determine the model constants by an a priori analysis of DNS data. A posteriori analysis of the subgrid scalar dissipation rate shows a good agreement between DNS and results from the modelled transport equation. As the subgrid scalar dissipation rate is represented dynamically and no assumptions for spectral equilibrium of production and dissipation of scalar variance are assumed we believe that this model is well suited for computing inhomogeneous and non equilibrium mixing of turbulent ﬂows at high Sc.

References 1. G. K. Batchelor, Small-scale variation of convected quantities like temperature in turbulent ﬂuid. Part 1. General discussion and the case of small conductivity, J. Fluid Mech. 5, 113 (1959). 2. S. B. Pope, PDF methods for turbulent reactive ﬂows, Prog. Energy Combust. Sci. 11, 119 (1985). 3. P. Colucci, F. A. Jaberi, P. Givi, and S. B. Pope, Filtered density function for large edddy simulation of turbulent reacting ﬂows, Phys. Fluids 10, 499 (1998). 4. S. Mitarai, J. Riley, and G. Kosaly, Testing of mixing models for Monte Carlo probability density function simulations, Phys. Fluids 17 (2005).

DNS-FDF Approach for Mixing at High Schmidt Number Flows

159

5. F. Schwertfirm and M. Manhart, DNS of passive scalar transport in turbulent channel ﬂow at high Schmidt numbers, Int. J. Heat and Fluid Flow 28, 1204 (2007). 6. R. O. Fox, The spectral relaxation model of the scalar dissipation rate in homogeneous turbulence, Phys. Fluids 7, 1082 (1995). 7. R. Fox, Computational Models for Turbulent Reacting Flows, Cambridge University Press, 2003.

Multi-Scale Simulation of Near-Wall Turbulent Flows Ayse Gul Gungor and Suresh Menon School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0150, USA, [email protected]; [email protected] Abstract A hybrid two-level large eddy simulation (TLS-LES) is used to study turbulent ﬂow in a converging-diverging channel, which represents a challenging test case due to the appearance of ﬂow separation and subsequent reattachment. While a conventional LES using a localized dynamic subgrid model is employed away from the wall, in the near-wall region a new two-scale approach called the twolevel simulation (TLS) approach is employed. A hybrid scale separating operator is used to couple the LES and TLS regimes, and the combined approach that does not contain any ad hoc adjustable parameters is used to simulate high Re wall-bounded ﬂows. The results show that the hybrid TLS-LES approach yields very reasonable predictions of most of the crucial ﬂow features in-spite of using a relatively coarse large-scale grid.

1 Introduction Large-eddy simulation (LES) equations, in general, are derived based on the assumption of commutativity of the ﬁltering operation with diﬀerentiation. For spatial derivatives, this condition is only satisﬁed for homogeneous ﬁlters. For wall-bounded ﬂows, this condition is barely satisﬁed, and also, the resolution requirement scales approximately as the square of the Reynolds number (Re). Consequently, LES of high Re wall-bounded ﬂows is too expensive unless particular methods are invoked to reduce the resolution requirement [1]. In conventional LES, the spatially ﬁltered (typically, using top-hat ﬁlter) equations are solved on a 3D grid and a subgrid model (typically, using an eddy viscosity closure) is used to include the subgrid stresses in the largescale equations. In contrast, the two-level simulation (TLS) approach [2, 3], decomposes the ﬂow ﬁeld variables into large-scale (LS) and small-scale (SS) components using a LS function (L ) that does not invoke spatial ﬁltering. Rather, as shown elsewhere [2], the LS is mathematically deﬁned as a local averaging operator uL (x, t) = L u(x, t), which in the simplest case is a sampling operator when the LS velocity is deﬁned at the nodes of the LS grid. The V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 24, c Springer Science+Business Media B.V. 2010

161

162

A.G. Gungor and S. Menon

SS is then obtained as uS = u − uL , where u is the fully resolved ﬁeld. More details of the application of this two-scale operation are in [2]. Using this twoscale decomposition, equations for both LS and SS are obtained formally in full 3D but for simulations, the SS equations are reduced to a one-dimensional domain (one in each Cartesian directions [2]) embedded within the LS grid. The resulting SS model is solved coupled to the LS 3D equations and contain in them “forcing” terms from the resolved LS. The TLS approach by itself has predicted accurately both isotropic and near-wall ﬂows at high Re [2,3]. Here, for a more practical generalized application the TLS model is re-formulated to work as a near-wall model and coupled with a conventional LES elsewhere. This hybrid approach is called TLS-LES, hereafter [4, 5]. Figure 1a illustrates the hybrid TLS-LES approach as implemented for a near-wall problem. To couple the LES spatially ﬁltered ﬁeld and the TLS LS near-wall ﬁeld a scale separation operator R is deﬁned as an additive operator that blends the LES ﬁltering operator F with the TLS scale separation operator L as: R = kF + (1 − k)L, where k is a transition function that relates the LES and the TLS domains. Both step and hyperbolic tangent blending functions have been investigated in this study. When this operator R is applied to the velocity components, the resolved ﬁeld (represented with F L superscript R) is obtained as uR i = Rui = kui + (1 − k)ui . The remaining un-resolved ﬁeld (identiﬁed with superscript U ) can be obtained from the decomposition as uU = u − uR . On applying this hybrid operator to the

b E

100

LS

(k)

10−2

t

−4 E(k) 10

E

SS

(k)

10−6 10−8 10−10

1

100

10 kz

c

25 20

d DNS, Moser et al. (256 x 193 x 192) LES (128 x 97 x 125) TLS-LES (64 x 46 x 64)

8 6

DNS LES TLS-LES

15

U+

4

10

<w 2 >

2 5 0

Reτ = 395 1

10 y+

100

2

0 1

10 y+

100

Fig. 1. (a) Illustration of wall-normal discretization for the TLS-LES model; (b) Evolution of SS energy spectra on a spanwise line in the separation region at three instants of time; (c) mean streamwise velocity and (d) rms velocity ﬂuctuation for channel ﬂow. LES and TLS-LES results are compared with DNS [8].

Multi-Scale Simulation of Near-Wall Turbulent Flows

163

Navier–Stokes equations and following the procedure discussed earlier [2] a coupled system of equations for resolved and un-resolved momentum equations are obtained as ∂uR ∂ ∂pR ∂ 2 uR i i U R U R + [(uR + u )(u + u )] + − ν =0 i j j ∂t ∂xj i ∂xi ∂x2j

(1)

∂pU ∂ 2 uU ∂ ∂uU U R U U i i [(uR −ν =0 + i + ui )(uj + uj )] + ∂t ∂xj ∂xi ∂x2j

(2)

For the hybrid TLS-LES approach used here the resolved equation are rewritten as ∂τijR ∂uR ∂ ∂pR ∂ 2 uR R R i i + [uR +ν − i uj ] = − 2 ∂t ∂xj ∂xi ∂xj ∂xj

(3)

R R F where τijR = (ui uj )R − (uR i uj ) is the unresolved term given as τij = kτij + F R R F L L R R L (1 − k)τijL + k[uF i uj − ui uj ] + (1 − k)[ui uj − ui uj ] , where the ﬁrst two terms are respectively, the LES and the TLS unresolved stresses while the others are hybrid terms that appear explicitly when the hybrid operator is used. Such hybrid terms appear even in hybrid RANS-LES approach [6] and become eﬀective in the transition region between the two domains (RANSLES or TLS-LES). They represent the un-resolved eﬀect that is not directly accounted by either approaches. For results reported here these hybrid terms are neglected, although new models are being pursued. The LES subgrid stress (τijF ) is closed by using an eddy viscosity model that employs the localized dynamic one equation model for the sub-grid kiS S L S S L netic energy [7]. In the TLS region, (τijL = [uL i uj + ui uj + ui uj ] ) is closed by solving a SS model on three one-dimensional lines embedded inside the LS grid. The resolution on the SS lines is ﬁne enough to resolve the smallest scales of interest. Details of the model, the underlying assumptions, and the justiﬁcations are given elsewhere [2, 3], and therefore avoided for brevity. Within each LS grid and at every LS (or LES) time step, for a given LS state, the SS ﬁeld evolves until the energy content in the largest scales of the SS approximately matches with the energy in the smallest scales of the LS, as shown in Fig. 1b. This evolution is on the 1D lines and in a parallel implementation is quite cost eﬀective. In the present study we apply the hybrid TLS-LES approach to turbulent channel ﬂow containing a convergent-divergent bump on the bottom wall. The Reynolds number is about 7900, based on the half width of the channel and the maximum streamwise velocity at the inlet. Turbulent ﬂow in such a channel permits examination of the ﬂow distortion due to the combined eﬀects of streamwise pressure gradient and surface curvature. For the current conﬁguration, the streamwise pressure gradient changes suddenly from strongly favorable to adverse due to the change in the lower wall shape change, which should show up in the turbulent stress proﬁle, and therefore, oﬀers a challenge to near-wall models.

164

A.G. Gungor and S. Menon

2 Results All numerical simulations are conducted using a spatially fourth-order accurate, ﬁnite-diﬀerence, parallel incompressible ﬂow solver. The solver uses a primitive variable formulation based on an artiﬁcial compressibility and a dual time stepping method. The SS equations are integrated eﬃciently in parallel, which reduces the overall cost of the TLS-LES approach. Although the computational cost of the hybrid TLS-LES model is signiﬁcant it is still considered substantially lower than a conventional LES model that employs a well wall-resolved grid. For high Re ﬂows, which is the focus of the current development, this advantage is beneﬁcial and critical. The inﬂow conditions for the current study are obtained from LES and TLS-LES of plane channel ﬂow at a Reynolds number (Reτ = 395), which is also used for ﬂow past the bump. These channel simulations are conducted in a domain of 2πδ × 2δ × πδ where δ is the inlet channel half height, using 128 × 97 × 128 points for LES and 64 × 46 × 64 points for TLS-LES. In the hybrid TLS-LES, the interface location is pre-deﬁned in advance and the SS region extends up to y + ≈ 30. The LES and TLS-LES results show good agreement with classical DNS data [8] (see Fig. 1c, d). The adverse pressure gradient (APG) ﬂow is created by a surface bump with concave and convex regions at the lower wall [9, 10]. The computational domain is chosen following [10] as 4πδ × 2δ × πδ, and a resolution of 128 × 97 × + 128 grid is used for LES (Δymin = 0.98) while 64 × 46 × 64 LS grid is used + for TLS-LES (Δymin = 5.4) with no stretching in streamwise and spanwise direction, and with a nominal stretching in the wall-normal direction. The resolution used for the hybrid TLS-LES study is considered very coarse (the DNS resolution is 1536 × 257 × 387 [10]) but is chosen in order to challenge the ability of the TLS-LES approach to deal with high Re ﬂows using very coarse grids. Thus, the near-wall turbulent ﬁeld is not expected to be captured properly in the LS resolved ﬁeld and the burden for the correct reconstruction of the near-wall ﬁeld is on the SS model. A uniform grid of 8 SS cells per LS cell is used in all directions, which gives a minimal resolution on wall-normal +SS lines of about Δymin = 0.68 (which is quite good, albeit in the 1D SS model). The near-wall SS region is within sixteen LS cells and therefore, extends up to y + ≈ 150 from both walls. Thus, most of the near-wall dynamics is expected to be resolved on the SS lines. The near-wall vorticity contours for LS and SS (Fig. 2a, b) show intense coherent structures near the separation region. The near-wall SS velocity ﬁeld exhibits strong streamwise vortical structures that are smaller but similar to those present in the LS ﬁeld suggesting that TLS-LES is capable of reconstructing physically correct ﬂow ﬁeld by combining the LS and SS ﬁelds in the near-wall region. A similar observation in channel ﬂow was shown by the TLS approach earlier [3]. 2 The DNS and TLS-LES pressure (Cp = Pw − Pw,ref / 21 Uref ) and skin 1 2 friction (Cf = τw / 2 Uref ) coeﬃcients along the lower wall (Fig. 2c, d) show

Multi-Scale Simulation of Near-Wall Turbulent Flows

c

0.5

d

DNS,Reτ=395(1536 x 257 x 384) LES,Reτ= 395(128 x 97 x 128)

0.0 Cp

0.03

165

0.02

TLS-LES,Reτ= 395(64 x 46 x 64)

−0.5 Cf 0.01

−1.0

Exp.,Reτ= 6500 DNS,Reτ= 395(1536 x 257 x 384)

−1.5 −2.0

LES, Reτ = 395(128 x 97 x 128)

0.00

TLS-LES, Reτ = 395(64 x 46 x 64)

0

4

8 x

12

-0.01

0

4

x

8

12

Fig. 2. Isosurfaces of the streamwise vorticity for (a) LS and (b) SS; Comparison of (c) pressure and (d) skin friction coeﬃcient predicted by LES and TLS-LES with the DNS [10]. Note the diﬀerence between the experimental and numerical Re.

that the present approach can capture the near-wall ﬁeld reasonably well. The minimum value of the pressure coeﬃcient at the summit of the bump is found to be 30% lower as compared to the experiment [9]. This diﬀerence between the numerical and the experiment studies is expected due to the large diﬀerence in Reynolds number and inﬂow conditions [10]. Both LES and TLSLES studies accurately predict the recovery of the pressure downstream of the bump summit. The two sudden jumps in Cf are found at the locations where the pressure gradient changes from adverse to favorable. In general, TLS-LES results show good agreement with DNS and LES data in spite of the very coarse LS grid employed. Prediction of Cf is better for ﬁner grid LES, as expected. Due to the very coarse LS grid in the near-wall TLS the separation region is not properly predicted, and this is expected given the coarse grid employed deliberately. Future studies will revisit this problem with more LS grid resolution in this region of ﬂow complexity. Wall-normal proﬁles of mean and rms ﬂuctuations of streamwise velocity are compared with DNS data [10] in Fig. 3a, b. The velocity proﬁles are normalized by the maximum mean velocity at the inﬂow. Both in the inlet and the downstream sections the TLS-LES predictions for mean and rms velocity show reasonable agreement with DNS and LES data, except in the region of separation region just downstream of the bump. Nevertheless, the overall trend is similar with an increase in the ﬂuctuation intensity at the onset of separation, which is also responsible for the sudden increase in Cf (see Fig. 2b) and an outward shift in the peak streamwise ﬂuctuation due to the strong adverse pressure gradient.

166

A.G. Gungor and S. Menon

a

2

y/δ 0

b

2

y/δ 0

0

2

4

6

x/δ

8

10

12

Fig. 3. Proﬁles of (a) mean streamwise velocity and (b) rms velocity ﬂuctuation. DNS (circles), LES (solid lines), TLS-LES (dashed lines).

3 Conclusion A new hybrid TLS-LES approach has been used to simulate ﬂow in a converging-diverging channel. This type of ﬂow represents a challenging test case, in particular due to the ﬂow separation and subsequent reattachment. The results obtained by TLS-LES have been compared with DNS and a more conventional LES. Results suggest that the TLS-LES approach has the potential for capturing the near-wall dynamics even when using very coarse grid. Some limitations of using very coarse grids in the near-wall region have been identiﬁed and will be addressed in the future. The authors are grateful to J.-P. Laval for providing the DNS data. This work is supported by the Oﬃce of Naval Research.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Piomelli U, Balaras E (2002) Annu Rev Fluid Mech 34:349–374. Kemenov K, Menon S (2006) J Comput Phys 220:290–311. Kemenov K, Menon S (2007) J Comput Phys 222:673–701. Gungor AG, Sanchez-Rocha M, Menon S (2007) In: Palma JMLM, Lopes S (eds) Advances in Turbulence XI. Springer, Berlin Heidelberg. Gungor AG, Menon S (2010) Prog Aerosp Sci 46(1):28–45. Sanchez-Rocha M, Menon S (2009) J Comput Phys 228:2037–2062. Kim W-W, Menon S (1999) Int J Numer Meth Fluids 31:983–1017. Moser R, Kim J, Mansour N (1999) Phys Fluids 11:943–945. Bernard A, Foucaut JM, Dupont P, Stanislas M (2003) AIAA J 41:248–255. Marquillie M, Laval J-P, Dolganov R (2008) J Turb 9:1–23.

Explicit Algebraic Subgrid Stress Models for Large Eddy Simulation L. Marstorp, G. Brethouwer, and A.V. Johansson Linn´e Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden, [email protected]; [email protected]; [email protected]

Abstract New explicit subgrid stress models have been developed which are based on the same methodology that leads to the EARSM formulation for RANS. The subgrid models involve the strain rate and rotation rate tensor, and can account for rotation in a natural way. A dynamic and a non-dynamic version are proposed. The non-dynamic version is a computationally cheap SGS model, whereas the dynamic version is more accurate. Large eddy simulations of rotating channel ﬂow have been carried out in order to test the models. Comparison with the standard and dynamic Smagorinsky models shows that the explicit dependence on the system rotation improves the description of the mean velocity proﬁles and the Reynolds stresses at high rotation rates. Furthermore, large eddy simulations with the new models are less sensitive to the grid resolution.

1 Introduction Subgrid scale stress (SGS) models based on the eddy viscosity assumption such as the Smagorinsky model are often used in large eddy simulations (LES). They are able to provide for a fairly correct amount of energy dissipation when a dynamic determination of the model constant is used, which is in many cases suﬃcient to obtain a good description of the resolved scales. However, mixed models, where the eddy viscosity term is accompanied with a second term which is not aligned with the resolved rate of strain, produce better results [5]. The models we develop depend explicitly on the resolved strain rate and rotation rate tensors and consist of two terms: an eddy viscosity term that provides for dissipation of energy, and a nonlinear term that accounts for subgrid stress anisotropy and improves the description of the individual SGS stresses. In that respect the new explicit models are mixed SGS models. Explicit algebraic Reynolds stress models (EARSM), whereby the Reynolds stress anisotropy is described in terms of the mean strain rate and the mean rotation rate tensors, have been shown to be superior to classical eddy viscosity based models, especially regarding the simulation of rotating ﬂows. A recent example of such a model is the EARSM by [6], which V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 25, c Springer Science+Business Media B.V. 2010

167

168

L. Marstorp et al.

is based on a modelled transport equation of the Reynolds stresses and on the assumption that the advection and diﬀusion of the Reynolds stress anisotropy are negligible. The aim of this study is to develop new models for the SGS stress by applying the same kind of methodology that leads to the EARSM for RANS. A dynamic model and a computationally more eﬃcient non-dynamic model are proposed. We think that these new models can improve the description of the SGS anisotropy compared to eddy viscosity models. Since the new models can include the eﬀect of system rotation in a natural way they have a particular potential for rotating ﬂows. To test the new models, LES of rotating and nonrotating channel ﬂow are carried out and compared to DNS data.

2 Model The transport equation for the SGS stress tensor τij = (u$ ˜i u˜j ), where i uj − u denotes a homogeneous ﬁlter operator, can be expressed in the same way as the transport equation for the Reynolds stress. In analogy with the Reynolds stress anisotropy tensor, the SGS stress anisotropy tensor is deﬁned as aij =

τij 2 − δij , KSGS 3

(1)

where KSGS = (u$ ˜i u ˜i )/2 is the SGS kinetic energy. In a non-rotating i ui − u frame the transport equation for aij reads

τij ∂Dijk Daij τij τij ∂DkKSGS − =− − (P − ) + Pij − ij + Πij , KSGS Dt ∂xk KSGS ∂xk KSGS (2) τij KSGS are the sum of the turbulent and molecular ﬂuxes where −Dijk and −Dk of the SGS stress and SGS kinetic energy, respectively. The production of the SGS stress, Pij , and the production of SGS kinetic energy, P = Pii /2, can be expressed in terms of τij and ﬁltered gradients, but the SGS pressure strain, Πij and the SGS dissipation rate tensor, ij , need to be modelled. We model these terms using the same modelling approach as in RANS, but in terms of ﬁltered quantities instead of averaged ones. We follow [6] and use a slightly modiﬁed version of the general linear model by [3] for Πij and an isotropic model for ij . The model for Πij diﬀers from the original RANS model in one of the coeﬃcients. We slightly reduce the coeﬃcient in order to increase the predicted SGS anisotropy. The parameter in the model for the slow part in Πij is ﬁxed using DNS data. The derivation of the EARSM involves the weak equilibrium assumption that the advection and diﬀusion of the Reynolds stress anisotropy can be neglected. We expect the weak equilibrium assumption to be a reasonable approximation for the subgrid scales as well, at least in the mean sense. Although

Explicit Algebraic Subgrid Stress Models for Large Eddy Simulation

169

the weak equilibrium assumption might be less motivated for LES than for RANS, it allows for an explicit SGS model that can account for rotation in a natural way and for a treatment of the subgrid anisotropy. By applying the weak equilibrium assumption, and following the steps by [6] using the same assumptions, we ﬁnally obtain a simpliﬁed explicit algebraic model for the SGS stress which reads 2 ∗˜ ∗2 ˜ ˜ ˜ ˜ τij = KSGS δij + β1 τ Sij + β4 τ (Sik Ωkj − Ωik Skj ) (3) 3 ˜ ij = where S˜ij = 0.5(∂ u˜i /∂xj + ∂ u˜j /∂xi ) is the resolved rate of strain, Ω ∗ 0.5(∂ u˜i /∂xj − ∂ u˜j /∂xi ) is the resolved rotation rate tensor, τ is the time ˜ik Ω ˜ki . In all versions of the present scale of the subgrid scales, IIΩ = τ ∗ 2 Ω SGS model we apply the approximation P/ = 1 which strongly simpliﬁes the expressions for β1 and β4 compared to the corresponding RANS model β1 = −

9c1 /4 33 , 20 [(9c1 /4)2 − 2IIΩ ]

β4 = −

1 33 . 2 20 [(9c1 /4) − 2IIΩ ]

(4)

where c1 is a model parameter. The approximation P/ = 1 has been validated using a` priori DNS tests. In the model system rotation is accounted for by ˜ij by replacing Ω ˜R = Ω ˜ij + 13 Ω s Ω (5) ij 4 ij s = jik Ωks is the system rotation tensor. where Ωij Two versions of the model have been developed: (i) A dynamic model with a dynamic determination of the subgrid kinetic energy KSGS = cΔ2 |S˜ij |2 ,

(6)

and τ ∗ , where Δ is the ﬁlter scale and c is determined using Germano’s indentity. (ii) A non-dynamic model with a non-dynamic determination of KSGS and τ ∗ . In [4] we present more details of the model derivation.

3 Results In order to test the developed subgrid models we have carried out LES of turbulent channel ﬂow rotating about the spanwise direction at wall friction Reynolds number Reτ = 180 and compared to DNS data of [2] at the same Reynolds number. Figure 1 shows the mean velocity proﬁle of a rotating channel ﬂow at rotation number Ro+ = 2Ω s /uτ = 37, where Ω s is the frame rotation and uτ the friction velocity at the wall. DNS results as well as LES results of the new explicit nondynamic model and the standard Smagorinsky model with wall damping are plotted. The non-dynamic model gives a much better prediction

170

L. Marstorp et al. 50 45 40 35

U

+

30 25 20 15 10 5 0 −1

−0.5

0

0.5

1

y/δ

Fig. 1. Mean velocity proﬁles in rotating channel ﬂow. DNS, solid line; non-dynamic model, dashed line; standard Smagorinsky model, dotted line. 55 50 45 Ub

+

40 35 30 25 20 15

0

0.5

1

1.5 Rob

2

2.5

3

Fig. 2. Mean bulk velocity Ub+ = Ub /uτ at various rotation speeds Rob = 2Ωs δ/Ub , where δ is the channel half width and Ub the mean bulk velocity. DNS, solid line; non-dynamic model, dashed line; standard Smagorinsky model, dotted line.

of the mean velocity proﬁles than the standard Smagorinsky model because the latter model has excessive SGS energy dissipation at the stabilised side of the channel. The non-dynamic model has a reduced SGS dissipation at the stabilised side of the channel, which improves the mean velocity proﬁle description. The new explicit dynamic model and the dynamic Smagorinsky model both predict the mean velocity proﬁle very well because the dynamic procedure reduces the SGS dissipation when laminarisation occurs [4]. Figure 2 shows the computed mean bulk velocity at various rotation numbers. All simulations have the same mean pressure drop. According to the DNS the mean bulk velocity increases with rotation and approaches the laminar value at high rotation speeds [2]. Because the mean velocity gradient is included in the standard Smagorinsky model it predicts a non-zero eddy viscosity even if the ﬂow is laminar. Consequently, this model cannot handle the laminarisation and strongly underpredicts the bulk mean velocity. The new non-dynamic model performs much better because it can predict the

Explicit Algebraic Subgrid Stress Models for Large Eddy Simulation

171

laminarisation at the destabilised side of the channel to some extent. At high rotation rates this model beneﬁts from the asymptotic behaviour β1 , β4 → 0. The diﬀerences between the results of the new dynamic and the dynamic Smagorinsky model simulations are small, see [4]. We have also carried out LES of nonrotating turbulent channel ﬂow at Reτ = 950 with the new dynamic model and the dynamic Smagorinsky and compared with DNS data by [1]. Three diﬀerent grid resolutions, 64 × 96 × 64, 96 × 96 × 96, 128 × 128 × 128, are used in order to determine the resolution sensitivity of the models. In wall units it corresponds to Δx+ = 187, 124, 93, Δz + = 93, 62, 47 and on average Δy + = 20, 20, 14 respectively. Figure 3 shows the computed streamwise mean velocity proﬁles. In the LES with a coarse resolution the streamwise velocity is overpredicted, but the dynamic Smagorinsky predicts a higher velocity than the explicit dynamic model at the same resolution. The streamwise velocity ﬂuctuations are also overpredicted by both models as shown in Fig. 4, in particular at low resolutions. However, 25 20 U+

15 10 5 0 100

101

y+

102

Fig. 3. Mean streamwise velocity proﬁle in channel ﬂow at Reτ = 950. DNS, solid line; non-dynamic model, dashed lines; standard Smagorinsky model, dotted lines. The arrow indicates the direction of increasing resolution. 1.5

16 14

1

10

+

+

12

8 6

0.5

4 2 0

0

50 100 150 200 250 300 350 400 y+

0

0

100 200 300 400 500 600 700 y+

Fig. 4. Streamwise (left ﬁgure) and wall normal (right ﬁgure) Reynolds stresses in channel ﬂow at Reτ = 950. DNS, solid line; non-dynamic model, dashed lines; standard Smagorinsky model, dotted lines. The arrow indicates the direction of increasing resolution.

172

L. Marstorp et al.

we can observe that the LES with the explicit model are less sensitive to the grid resolution and agree better with the DNS data than the LES with the dynamic Smagorinsky model. Figure 4 shows too that the wall normal velocity ﬂuctuations are severely underpredicted by the dynamic Smagorinsky model whereas the results of the explicit dynamic model agree quite well with the DNS data, even at rather low resolutions.

4 Conclusion Simulations show that our new non-dynamic model is a computationally eﬃcient SGS model which outperforms the standard wall-damped Smagorinsky model in case of LES of rotating channel ﬂow. The new explicit dynamic model is the most accurate and is an alternative to the dynamic Smagorinsky model. Both models’ explicit dependence on the system rotation improves the description of the mean velocity proﬁle as well as the Reynolds stresses and the subgrid scale stress anisotropy. Furthermore, simulations at relatively high Reynolds number show that the new dynamic model is less sensitive to grid resolutions than the dynamic Smagorinsky model. In [4] we present more results on the LES/DNS comparisons. Since the new subgrid models have strong similarities with the EARSM they might have promising features for detached eddy simulations.

Acknowledgement This study was ﬁnancially supported by the Swedish Research Council. Computational resources at PDC were made available by the Swedish National Infrastructure for Computing.

References 1. 2. 3. 4.

Del Alamo J C, Jimenez J, Zandonade P, Moser R D (2004) J Fluid Mech 500:135 Grundestam O, Wallin S, Johansson A V (2008) J Fluid Mech 598:177–199 Launder B E, Reece G J, Rodi W (1975) J Fluid Mech 41:537 Marstorp L, Brethouwer G, Grundestam O, Johansson A V (2009) submitted to J Fluid Mech 5. Vreman B, Geurts B, Kuerten H (1994) Phys Fluids 6:4057 6. Wallin S, Johansso A V (2000) J Fluid Mech 403:89

Scrutinizing the Leray-Alpha Regularization for LES in Turbulent Axisymmetric Free Jets F. Picano, C.M. Casciola, and K. Hanjali´c Dipartimento di Meccanica e Aeronautica, Sapienza Universit` a di Roma, via Eudossiana 18, 00184, Rome, Italy, [email protected]; [email protected]; [email protected]

1 Introduction The Leray-α regularization of the Navier–Stokes equations oﬀers a route to reduce the number of degrees of freedom in DNS [1]. This should make it possible to use coarser meshes as compared with standard DNS. However, the reports in the literature indicate that a sole regularization does not lead conclusively to satisfactory results. An alternative is to combine the regularization with the conventional LES using an sgs model. We report on testing the Leray-α regularization on its own and blended with LES eddy-viscosity models in simulating a turbulent axisymmetric jet over a broad range of Re numbers using diﬀerent meshes. This ﬂow was chosen because it is well documented by high-quality independent experimental data and complies with a well established similarity theory for the far-ﬁeld asymptotic regime. The absence of solid walls simpliﬁes the analysis, though, admittedly, limiting the conclusions to only free shear ﬂows. The computational code solves the N-S equations with a conservative second order ﬁnite diﬀerence scheme on a staggered grid. The time evolution is performed by a third order low-storage Runge–Kutta scheme [2]. Reference data were collected from a DNS [2] at ReD = 4,000 and from the experiments of Panchapakesan and Lumley [3] and of Hussein et al. [4] for higher ReD . All the models have been tested in the far-ﬁeld in the range 12 ÷ 32D (domain size up to 42D) of an annular axisymmetric jet Dout /Dinn = 1.6, 2 − D2 D = Dout inn for several Re numbers in the two-decades range, ReD = U0 D/ν = 4,000 ÷ 400,000 and in three diﬀerent meshes, details are given in Table 1. The inﬂow proﬁle is assigned by Dirichelet conditions. The axial velocity is given by a steady annular top-hat proﬁle, while the other components are null. The annular conﬁguration presents some beneﬁts for simulations as compared to the standard round jet. As an absolutely unstable ﬂow the transition to turbulence is independent of the details of the inﬂow. Moreover it develops faster to a self-similar asymptotic state (z ≥ 8 ÷ 10D) where we V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 26, c Springer Science+Business Media B.V. 2010

173

174

F. Picano et al.

1/3 to the Kolmogorov length Table 1. Ratio of the grid spacing Δ = (Δz Δr rΔθ) ηk = (ν 3 /ε)1/4 and the shear length Ls = ε/S 3 (S is the strain rate of the mean velocity ﬁeld) for diﬀerent meshes and Reynolds numbers at z = 20D and r = 1D. The meshes considered (Nθ × Nr × Nz ): 128 × 145 × 784 (∼15 Mill ), 32 × 37 × 196 (∼0.2 Mill) and 16 × 19 × 98 (∼30.000 points).

Mesh→

15 Mill

0.2 Mill

30.000

ReD

Δ/Ls

Δ/ηk

Δ/Ls

Δ/ηk

Δ/Ls

Δ/ηk

4,000 40,000 400,000

0.125 0.125 0.125

3 16 90

0.5 0.5 0.5

12 64 360

1 1 1

24 128 720

initially focus our statistical analysis [5]. The far-ﬁeld statistics of axisymmetric free jets maintain a weakly dependence on the inﬂow conditions. In fact, e.g. variations of the 10% are reported for the jet spreading rate when changing the details of the inlet: annular, round laminar or round turbulent; an extensive discussion on the dependence of the fully developed state statistics on the inﬂow conditions can be found in [2]. Nonetheless because errors of 10% order of magnitude can be accepted for simulations with LES models, reference data are collected both from round [3, 4] and annular [2] jets. The self-similarity fulﬁllment is measured by the root mean square of the diﬀerence between the self-similar solution and the simulated proﬁles at different axial distances. We tested classic and dynamic Smagorinsky models [6], the Leray-α one [7], and a blend of the two techniques. The test ﬁlter for the dynamic model was twice the grid implicit ﬁlter, while the α length of the Leray-α equations was assumed proportional to the grid spacing: α = a Δ.

2 Results and discussions The ﬁrst simulations focus on direct comparison of the two approaches with the reference (15M ) DNS data at ReD = 4, 000. The mesh adopted for the two methods contained 0.2M points and was generated by quadrupling the ﬁne DNS mesh spacing in each direction, see Table 1. A quantitative evaluation of the regularization method and the LES models considered is presented in Fig. 1, showing the far-ﬁeld self-similar proﬁles of the mean axial velocity and the streamwise turbulent stress. The dynamic and classic (Cs = 0.12) Smagorinsky and Leray-α (α = 0.3Δ = 0.15Ls) reproduce well the mean velocity proﬁle, demonstrating a remarkable self-similarity. The inﬂuence of the coeﬃcient value in the Leray-α approach is illustrated by the results obtained also with α = 0.5Δ. The eﬀects of the models are highlighted by a poor performance of the unresolved DNS on the same mesh, exhibiting very slow spreading (narrow velocity proﬁle). Clearly the width of the mean axial velocity proﬁle is strictly connected to the entrainment process

Leray-α Regularization for LES in Free Jets

175

Fig. 1. Far-ﬁeld (12 ÷ 32D) self-similar proﬁles of diﬀerent LES models at ReD = 4,000 on the 0.2M points mesh compared with the reference DNS data (15M ). Left: mean axial velocity; right: axial normal Reynolds stress (The numbers show the value of the constant adopted. Error-bars represent the self-similarity fulﬁlment).

Fig. 2. 2D-cut of the instantaneous axial velocity ﬁeld (0.2M grid points) for ReD = 40,000. Left: LES with dynamic model; middle: Leray-α, α = 0.3 Δ; right: mixed Leray-α-Smagorinsky, α = 0.3 Δ and Cs = 0.08.

and with the momentum preservation of the free jets dynamics [2, 4], hence it highlights how crucial is the correct estimate of the proﬁle width for jet simulations. Similar conclusions can be drawn for the second/moments, as illustrated by self-similar proﬁles of the axial normal Reynolds stress shown in Fig. 1 right. Hence, for a relative low Reynolds number and ratio Δ/ηk we ﬁnd that Smagorinsky models and Leray-α regularization provide accurate results on meshes coarser than those needed for a fully resolved DNS. Dramatically diﬀerent results are obtained when the models are applied to a much higher Reynolds number while using the same mesh, as illustrated in Fig. 2 by the snapshots of axial velocity for ReD = 40,000. The ratio between the typical mesh size and the dissipative length is increased to Δ/ηk 64, while the resolution of the large-scale motion deﬁned by Δ/Ls 0.5 remains ﬁxed, see Table 1 (Ls is the characteristic large scale at which kinetic energy production is active). For this situation the pure Leray-α model performs

176

F. Picano et al.

poorly for each value of α tested. In fact, as shown in the middle Fig. 2 the instantaneous axial velocity ﬁeld display a lot of spurious oscillations. For the same conditions, the dynamic Smagorinsky model reproduces apparently the correct behavior of the velocity ﬁeld, quite similar to that of the DNS at lower Reynolds number shown in Fig. 1. This behavior can be due to the absence of an extra-dissipation term in the Leray-α model. The reduced non-linearity of the regularized N-S equations tend to block the energy-cascade towards the small scales but without any dissipation that would drain energy to low scale motion [7]. So, by increasing the ratio Δ/ηk such a model can not provide the dissipation needed to control the growth of the velocity ﬂuctuations. In order to overcome the problem, we added an extra-dissipation term in the Smagorinsky form, but with a lower value of the constant Cs , namely using 0.07 ÷ 0.08. This mixed model gives a good overall impression for the instantaneous ﬁeld as shown in Fig. 2 right. The mean velocity proﬁles, displayed in Fig. 3 appear to conﬁrm this impression. In fact the pure Leray-α model exhibits a very narrow proﬁle similar to the unresolved DNS, independently of the α value. The simulations with larger values of α = a Δ failed. The Smagorinsky-based model and the blended model reproduced well the selfsimilar mean velocity proﬁles. Similar conclusions hold for the axial normal stresses displayed in Fig. 3 right. To approach the near-asymptotic conditions, the models are tested at a very large Reynolds number ReD = 400, 000 using still the same mesh of 0.2M points (Fig. 4). In this case the ratio Δ/ηk 360, while the ratio between the mesh size and the shear scale remain ﬁxed again at Δ/Ls 0.5. For this case the pure Leray-α model and the unresolved DNS fail. The Smagorinsky-based models and the blended Leray-α/Smagorinsky reproduce accurately the mean proﬁle and the axial Reynolds stress. The constant adopted for the mixed model are almost the same as used in the previous case. It seems that α = 0.3 Δ = 0.15 Ls and Cs = 0.08 are the asymptotic values for arbitrary large Reynolds numbers because they do not change when varying the Reynolds number from 40,000 to 400,000 while using the same mesh (the same Δ/Ls , but diﬀerent Δ/ηk ≤ 50).

Fig. 3. Far-ﬁeld self-similar proﬁles at ReD = 40,000 (0.2M point mesh). Symbols denote DNS [2] and experiments of PL [3] and HCG [4]. For legend refer to Fig. 1.

Leray-α Regularization for LES in Free Jets

177

Fig. 4. Far-ﬁeld (12 ÷ 32D) self-similar proﬁles of diﬀerent LES models at ReD = 400,000 on the 0.2M points mesh. For the caption refer to Figs. 1–3.

Fig. 5. Turbulent kinetic energy budget at Re = 400,000 (0.2M points). Lines show DNS data at Re = 4,000. Red denotes Advection, blue Production, orange Turbulent-Diﬀusion and green molecular dissipation. Left: LES with dynamic model; middle: mixed Leray-α-Smagorinsky, α = 0.3 Δ and Cs = 0.08. right: Leray-α, α = 0.3 Δ.

Figure 5 shows the turbulent kinetic energy budget in the far ﬁeld for the highest Reynolds number considered. The Smagorinsky models and the mixed one reproduce quite accurately the terms that originate from the large-scale interactions, i.e. advection, production and turbulent-diﬀusion. It is remarkable that the molecular dissipation computed by the simulations is negligible compared to the correct value, hence the models have to provide almost all the amount of the kinetic energy dissipation. By ﬁxing the mesh and varying the Reynolds number we maintained the constant ratio between the grid size and the typical length of the large scales Δ/Ls , while we alter the ratio between the grid resolution and the dissipative scale Δ/ηk . For such kind of tests we found that the Smagorinsky-based models and the mixed Leray-α/Smagorinsky perform well. Now we change Δ/Ls , while maintaining Δ/ηk in the range tested earlier. To this purpose we simulated the jet at ReD = 40,000 using a mesh obtained by doubling the mesh spacing in each direction, resulting in a very coarse mesh with total 30.000 mesh points and Δ/Ls 1, see Table 1. The results of these simulations are presented in Fig. 6. The best results for the mixed model have been obtained by ﬁxing the ratio α/Ls = 0.15 and not maintaining the constant α/Δ. The mean proﬁle in the self-similar variables

178

F. Picano et al.

Fig. 6. Far-ﬁeld (12 ÷ 32D) self-similar proﬁles of diﬀerent LES models at ReD = 40,000 on the 30,000 points mesh. For the caption refer to Figs. 1–3.

is predicted almost accurately by the models, while the ﬂuctuation levels are overestimated, although the self-similarity is well fulﬁlled. To conclude, we have found that for relative low Reynolds number Smagorinsky models and Leray-α give accurate results for free turbulent jets with ratio Δ/ηk ≤ 12. At a higher Reynolds number leading to a larger Δ/ηk and Δ/Ls ≤ 0.5 the Smagorinsky based models give good results, while the regularization method of Leray-α does not. At these conditions we have scrutinized a mixed model Leray-α plus a dissipative Smagorinsky model with a lower constant Cs (0.07 ÷ 0.08 vs 0.12) obtaining a good estimate of the mean and axial Reynolds stress proﬁles and their self-similarity achievement. By increasing Δ/Ls , we found that the limiting condition for the mesh coarsening is reached for the value of Δ/Ls ≈ 1 (though all models overestimate the ﬂuctuation level). Moreover the best results for the mixed model are obtained using α proportional to Ls , the optimum value being α = 0.15 Ls . The amount of extra dissipation needed, as measured by Cs , seems to be a function of Δ/ηk : in fact it is null for Δ/ηk ≤ 12, while Cs 0.08 appears the asymptotic value tested in the range Δ/ηk ∈ [64 : 360] for diﬀerent Δ/Ls . Based on the above ﬁndings, we conclude that the Leray-α regularization blended with the Smagorinsky model with a reduced coeﬃcient (ensuring a low-level extra dissipation) can be useful for simulating high Reynolds number free shear ﬂows.

References 1. A. Cheskidov, D. Holm, E. Olson, and E. Titi. On leray-alpha model of turbulence. Proc. Roy. Soc. Lond. A Mat., page 629, 2005. 2. F. Picano and C.M. Casciola. Small scale isotropy and universality of axisymmetric jets. Phys. Fluids, 19(11):118106, 2007.

Leray-α Regularization for LES in Free Jets

179

3. N. R. Panchapakesan and J. L. Lumley. Turbulence measurements in axisymmetric jets of air and helium. part 1. air jet. J. Fluid Mech., 246:197, 1993. 4. H. J. Hussein, S. P. Capp, and W. K. George. Velocity measurements in a highreynolds-number momentum-conserving, axisymmetric turbulent jet. J. Fluid Mech., 258:31, 1994. 5. N. W. M. Ko and W. T. Chan. Similarity in the initial region of annular jets: three conﬁgurations. J. Fluid Mech., 84:641, 1978. 6. M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A-Fluid, 3(7):1760, 1991. 7. M. van Reeuwijk, H. Jonker, and K. Hanjali´c. Incompressibility of the leray-alpha model for wall-bounded ﬂows. Phys. Fluids, 18:018103, 2006.

Localization of Unresolved Regions in the Selective Large-Eddy Simulation of Hypersonic Jets D. Tordella, M. Iovieno, S. Massaglia, and A. Mignone 1

2

Politecnico di Torino, Dip. Ing. Aeronautica e Spaziale, Torino, Italy, [email protected]; [email protected] Universit` a di Torino, Dipartimento di Fisica Generale, Torino, Italy, [email protected]; [email protected]

Abstract A method for the localization of the regions where the turbulent ﬂuctuations are unresolved is applied to the selective large-eddy simulation (LES) of a compressible turbulent jet of Mach number equal to 5. This method is based on the introduction of a scalar probe function f which represents the magnitude of the twisting-stretching term normalized with the enstrophy [1]. The statistical analysis shows that, for a fully developed turbulent ﬁeld of ﬂuctuations, the probability that f is larger than 2 is zero, while, for an unresolved ﬁeld, is ﬁnite. By computing f in each instantaneous realization of the simulation it is possible to locate the regions where the magnitude of the normalized stretching-twisting is anomalously high. This allows the identiﬁcation of the regions where the subgrid model should be introduced into the governing equations (selective ﬁltering). The results of the selective LES are compared with those of a standard LES, where the subgrid terms are used in the whole domain. The comparison is carried out by assuming as high order reference ﬁeld a higher resolution Euler simulation of the compressible jet. It is shown that the selective LES modiﬁes the dynamic properties of the ﬂow to a lesser extent with respect to the classical LES.

1 Small scale detection criterion The regions where the ﬂuctuations are unresolved are located by means of the scalar probe function [1, 2] f (u, ω) =

| (ω − ω) · ∇(u − u) | | ω − ω |2

(1)

where u is the velocity vector, ω = ∇ × u is the vorticity vector and the overbar indicates the statistical average. Function (1) is a normalized scalar form of the vortex-stretching term that represents the inertial generation of three dimensional vortical small scales. When the ﬂow is three dimensional and V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 27, c Springer Science+Business Media B.V. 2010

181

182

D. Tordella et al.

b 1024 512 256 128 64

0.8 p(f ≥ tω)

p(fLES ≥tω )−p(fDNS ≥tω )

1.0

0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

tω

2.0

2.5

0.3

9

512 256 128 64

8 7

0.2

6 5 4

0.1

3 2 0.0 0.0

0.5

0.1

0.5

2.0

1 2.5

p(fLES ≥tω ) / p(fDNS ≥tω )

a

tω

Fig. 1. (a) Distribution of the probability of having f for diﬀerent resolutions of incompressible homogeneous and isotropic turbulence; (b) comparison between the probability in the ﬁltered ﬁelds and in the unﬁltered 10243 DNS ﬁeld [2].

rich in small scales f is necessarily diﬀerent from zero. In a two-dimensional ﬂow it is instead equal to 0. The mean ﬂow is subtracted from the velocity and vorticity ﬁelds in order to consider the ﬂuctuating part only of the ﬁeld. The statistical distribution of f has been computed in a fully resolved turbulent ﬂuctuation ﬁeld (DNS of a homogenous and isotropic turbulent ﬂow (10243 , Reλ = 230, data from [3])) and in some unresolved instances obtained by ﬁltering this DNS ﬁeld on coarser grids (from 5123 to 643 ). Figure 1a shows that the probability that f assumes values larger than a given threshold tω is always higher in the ﬁltered ﬁelds and increases when the resolution is reduced. The diﬀerence between the probabilities in fully resolved and in ﬁltered turbulence is maximum when tω ∈ [0.4, 0.5] for all resolutions, see Fig. 1b. In such a range the probability p(f ≥ tω ) in the less resolved ﬁeld is about twice the probability in the DNS ﬁeld. Furthermore, beyond this range this probability, normalized over that of resolved DNS ﬁelds, is gradually increasing becoming inﬁnitely larger. This can lead to the introduction of a threshold tω of the values of f , such that, when f assumes larger values the ﬁeld could be considered locally unresolved and should beneﬁt from the local activation of the Large Eddy Simulation method (LES) by inserting a subgrid scale term in the motion equation. The values of this threshold can be chosen to be equal to that where the diﬀerence between the resolved and unresolved ﬁeld is maximum, that is tω ≈ 0.4. In order to investigate the presence of regions with anomalously high values of f , a set of a priori tests were carried out on existing Euler simulations of the temporal evolution of a perturbed, initially cylindrical, jet of initial M = 5 and density ratio between the ambient medium and the jet medium equal to 10, see Sect. 2 and [1, 2]. These tests positively compare with experimental results and show the convenience of the use of such a procedure [1, 2]. The regions where f assumes values larger than 0.4 have been ﬁlled

Selective Large-Eddy Simulation of Hypersonic Jets

183

Fig. 2. Contour plots of the small scale function f = 0.4 in longitudinal sections z = 0 at dimensionless time t = 20: (a) simulation with 1283 grid points, (b) simulation with 2563 grid points.

in black in the visualizations shown in Fig. 2. According to the present criterion, these black regions can be viewed as regions where small scales are present and unresolved and where subgrid-scale terms should be introduced into the governing equations. It should be noted that this statistical analysis is funded on the Morkovin hypothesis that the compressibility eﬀects do not have much inﬂuence on the turbulence dynamics, apart from varying the local ﬂuid properties [5].

2 Results We have studied numerically, and in Cartesian geometry, the temporal evolution of a 3D jet subject to periodicity conditions along the longitudinal direction. The ﬂow is governed by the ideal ﬂuid equations for mass, momentum, and energy conservation. In the astrophysical context, this formulation is usually considered to approximate the temporal evolution inside a spatial window of interstellar jets, which are highly compressible collimated jets characterized by Re numbers of the order 1013−15 . It is known that the numerical solution of a system of ideal conservation laws (such as the Euler equations) produces the exact solution of another modiﬁed system with additional diﬀusion terms. With the discretizations used in this study it possible to verify a posteriori that the normalized numerical viscosity is of the order of 1.3 · 10−3 which implies a Re of 7.5 · 102 . In such a situation it is clear that the addition of the diﬀusive-dissipative terms into the governing equations would be meaningless. The formulation used is thus the following: ∂ρ ∂ + ρui = 0 ∂t ∂xi ∂ ∂(ρuk ) ∂ SGS + ρui uk + pδik = H (fLES − tω ) τik ∂t ∂xi ∂xi & ∂E ∂ % ∂ + (E + p)ui = H(fLES − tω )qiSGS ∂t ∂xi ∂xi

(2) (3) (4)

184

D. Tordella et al.

where the ﬂuid variables p, ρ and ui and E are, as customary, the pressure, SGS density, velocity, and total energy respectively; and where τik and qiSGS are the subgrid stress tensor and total enthalpy ﬂow, respectively. H( · ) is the Heaviside step function, thus the subgrid scale ﬂuxes are applied only in the regions where f > tω . The threshold tω is taken equal to 0.4, which is the value for which the maximum diﬀerence between the probability density function p(f > tω ) between the ﬁltered and unﬁltered turbulence was observed (see Fig. 1). The initial ﬂow structure is a cylindrical jet in a parallelepiped domain, described by a cartesian coordinate system (x, y, z). The initial jet velocity is along the y-direction; its symmetry axis is deﬁned by (x = 0, z = 0). The initial jet velocity, at t = 0, is Vy (x, z) = V0 within the jet, i.e. for (x, z) ≤ R, where R is the initial jet radius, and Vy (x, z) = 0 elsewhere. The initial ﬂuid particle density is set to ρ = ρ0 within the jet and ρ = νρ0 outside, with ν the density ratio of the external medium to jet proper. Finally, we assume that the jet is in pressure equilibrium with its surroundings; for this reason, we assume an initial uniform pressure distribution p0 , see [4] for further details. In the following, we will express lengths in units of the initial jet radius R, times in units of the sound crossing time of the radius R/c0 (with c0 = Γ p0 /ρ0 ), velocities in units of c0 (thus coinciding with the initial Mach number), densities in units of ρ0 and pressures in units of p0 . The standard Smagorinsky model with Cs = 0.1 has been implemeted as subgrid model, the turbulent Prandtl number is taken equal to 1. Equations (2)–(4) have been solved using an evolution of the PLUTO code [6], which is a Godunov-type code that supplies a series of high-resolution shock-capturing schemes that are particularly suitable for the present application. In order to discretize the Euler equations, we chose a version of the Piecewise-ParabolicMethod (PPM), which is third order accurate in space and second order in time. The domain size is a 4π × 10π × 4π parallelepiped, with y along the initial jet velocity, covered by a uniform cartesian grid with 128×320×128 points (for the high resolution reference simulation the grid points were 256 × 640 × 256). We have adopted periodic boundary conditions in direction y and outﬂow conditions in the other directions. Two additional simulations were performed for comparison. A standard non selective LES where the subgrid model was introduced in the whole domain, which is obtained by putting H ≡ 1 in (2–4), and a higher resolution (640 × 2562 ) Euler simulation obtained by putting H ≡ 0. A visualization of the pressure ﬁeld in a longitudinal section at t = 36 can be seen in Figs. 3a–c for the three cases (selective LES, classical LES, high resolution Euler simulation). The comparison shows the higher smoothing and small scale suppression produced by the non selective use of the subgrid model. The time evolution of the enstrophy distribution at two time instants far from the initial one is shown in Fig. 4 as a function of the distance from the centre of the jet. While the agreement between the enstrophy distribution obtained with the selective LES simulation and with the reference high resolution Euler

Selective Large-Eddy Simulation of Hypersonic Jets

185

Fig. 3. Pressure distribution in a longitudinal section at t = 36: (a) selective LES, (b) standard LES, (c) higher resolution pseudo-DNS. The ﬁgures show the contour levels of log10 (p/p0 ), the mean ﬂow is from bottom to top. 35

ω⬘i ω⬘i

25 20 15

10

t⫽36

Euler High Res. Selective LES Standard LES

8 ω⬘i ω⬘i

t ⫽28 Euler High Res. Selective LES Standard LES

30

6 4

10 2

5 0

1

2

3 r/R

4

5

6

0

1

2

3 r/R

4

5

6

Fig. 4. Radial distribution of the enstrophy, r is the distance from the axis of the jet. All averages have been computed as space averages on cylinders.

simulation is very good, the non selective simulation damps out the vorticity magnitude in the center of the jet and in the outer part, and introduces a spurious accumulation in the intermediate radial region. As a results, the vorticity dynamics is highly modiﬁed. This is also evident when observing the spectrum of the turbulent kinetic energy. Figure 5 shows the kinetic energy spectrum in the intermittent region between the jet core and the surrounding ambient. In the non selective LES, for t = 28, there is a concentration of energy in the low wavenumber region, which becomes even more pronounced

186

D. Tordella et al. 100

100

t=28

Euler High Res. Selective LES Standard LES

10–2

10–2

10–3

10–3

–4

10

10–5

–6

10

10–6

10–7

10–7 10–8

–8

100

k

101

Euler High Res. Selective LES Standard LES

10–4

10–5

10

t =36

10–1

E(k)

E(k)

10–1

102

100

101

102

k

Fig. 5. Spectra of the turbulent kinetic energy at r = 2, computed as the Fourier transform of the two-point correlations of the ﬂuctuating kinetic energy ρu2i /2.

for t = 36. This is consistent with the higher level of enstrophy seen in Fig. 4 for the non selective LES at analogous distances from the centre of the jet. Thus, we can observe that the selective introduction of the subgrid model yields distributions much closer, with respect to the standard LES, to the distribution showns by the high resolution Euler simulation.

3 Concluding remarks In this work we showed that the selective LES, which is based on the use of a scalar probe function f – a function of the magnitude of the local stretchingtwisting operator – can be conveniently applied to the simulation of a compressible jet. The probe function f was coupled with the standard Smagorinsky subgrid model. However, it should be noted that the use of f can be coupled with any subgrid model because f simply acts as an independent switch for the introduction of a subgrid model. The comparison among the three kinds of simulation (selective LES, standard LES, high resolution reference) here carried out shows that this method can improve the dynamical properties of the simulated ﬁeld, in particular, the spectral and vorticity distributions.

References 1. D. Tordella, M. Iovieno, S. Massaglia. Comp. Phys. Comm. 176(8), 539–549 (2007), doi: 10.1016/j.cpc.2006.12.004. 2. D. Tordella, M. Iovieno, S. Massaglia. Comp. Phys. Comm. 178(8), 883–884 (2008), doi:10.1016/j.cpc.2008.02.005. 3. L. Biferale, G. Boﬀetta, A. Celani, A. Lanotte, F. Toschi, Phys. Fluids. 17(2), 021701/1-4 (2005).

Selective Large-Eddy Simulation of Hypersonic Jets

187

4. G. Bodo, P. Rossi, S. Massaglia, Astron. & Astrophys. 333, 1117–1129 (1998). 5. M.V. Morkovin, “Eﬀects of compressibility on turbulent ﬂows”, in M`ecanique de la turbulence, edited by A. Favre, 367, (1961). 6. A. Mignone, G. Bodo, S. Massaglia, T. Matsakos, O. Tesileanu, C. Zanni and A. Ferrari, Astr. J. Supplement Series 170(1), 228–242 (2007), and http:// plutocode.to.astro.it.

An ADM-Based Subgrid Scale Reconstruction Procedure for Large Eddy Simulation M. Terracol1 and B. Aupoix2 1

2

ONERA, CFD and Aeroacoustics Dept, BP 72 - 29 Av. de la division Leclerc, 92322 Chˆ atillon cedex, France, [email protected] ´ ONERA, Aerodynamics and Energetics Model Dept, BP 74025 - 2 Av. Edouard Belin, 31055 Toulouse cedex 4, France, [email protected]

1 Introduction Large Eddy Simulation relies on a scale separation which aims at resolving only the largest energy-carrying scales, while small scales are discarded. However, speciﬁc applications require to know explicitly the information carried by the smallest scales, e.g., in a coupled ﬂuid-particle simulation where the action of the smallest eddies on small particles can be signiﬁcant [1]. It therefore appears necessary to reconstruct the small unresolved scales. This study proposes a way to recover the unresolved information when explicit ﬁltering is introduced. Section 2 details the triple-scale decomposition introduced in this case, between resolved, sub-ﬁlter and subgrid scales. Section 3 investigates the use of the Approximate Deconvolution Method (ADM) to reconstruct the sub-ﬁlter ﬁeld. In Section 4, an original method using two deconvolution levels is proposed to get local estimates of the subgrid kinetic energy. The main conclusions and perspectives from this study are summarized in the last section of this paper.

2 Triple-scale decomposition We choose here to introduce explicitly the scale separation ﬁltering operator denoted by G. This operator acts as a low-pass ﬁlter in the wavenumber space, and is characterized by a cutoﬀ lengthscale Δ larger than the local grid space step Δx. Any variable φ can be split according to the following triple decomposition (see Fig. 1) as φ = φ + φsf + φsg

(1)

where φ = G ! φ is associated to the resolved scales (larger than Δ); φsf is associated to sub-ﬁlter scales, which are not directly resolved, but can be V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 28, c Springer Science+Business Media B.V. 2010

189

190

M. Terracol and B. Aupoix

Fig. 1. Triple-scale decomposition associated to explicit ﬁltering.

represented on the computational grid; φsg corresponds to unresolved scales which cannot be resolved on the grid (e.g., with a size smaller than Δx).

3 Use of ADM to reconstruct the sub-ﬁlter ﬁeld The present study is based upon the use of the Approximate Deconvolution Method (ADM), originally developed by Stolz and Adams [2]. The ADM technique proposes to reconstruct an approximation of the unﬁltered ﬁeld from the ﬁltered one, by approaching the inverse of the ﬁlter as:

N l G−1 QN = l=0 (Id − G) (2) Using this operator, the unﬁltered velocity ﬁeld components ui can be approximated as: ui = QN ! ui , and the sub-ﬁlter velocity ﬁeld components read: ui,sf (QN − Id)ui = ui − ui . (3) This procedure therefore provides a direct estimate of the sub-ﬁlter ﬁeld components, which only depend upon the deconvolution level N . Kuerten and Vreman [1] used that approach to evidence the importance of sub-ﬁlter ﬁeld transport on small particles. The performance of the method for sub-ﬁlter scales reconstruction has been investigated using an isotropic turbulence dataﬁeld on a 2563 grid. This ﬁeld has been obtained by DNS, introducing a numerical forcing in the lowest wavenumbers. The corresponding Reynolds number based upon Taylor’s micro-scale is Reλ = 140. A priori tests have been conducted using a dedicated spectral code, the LES ﬁeld being obtained by projecting the DNS ﬁeld on a coarser grid and by then applying explicitly the ﬁlter G. Figure 2 displays the energy spectra computed from the DNS ﬁeld and from the associated LES ﬁeld obtained by projecting this DNS ﬁeld on a 323 grid and applying a Gaussian ﬁlter to it. The spectra computed from the reconstructed ﬁeld using two and ﬁve deconvolution levels are also displayed. This ﬁgure evidences the potential of the approximate deconvolution to reconstruct the energy associated

An ADM-Based Subgrid Scale Reconstruction Procedure for LES

191

101 100

E(k)

10−1 10−2 10−3 10−4 10−5 10−6 0 10

DNS LES ADMN=2 ADMN=5 k−α 101 k

102

Fig. 2. Computed energy spectra from DNS, LES, and ADM reconstructions.

a

b

c

Fig. 3. Iso-contour maps of sub-ﬁlter kinetic energy in a x − y plane: (a) ADM reconstruction with N = 2; (b) ADM reconstruction with N = 5; (c) exact subﬁlter kinetic energy from DNS.

to sub-ﬁlter scales. The use of two deconvolution levels leads to the recovery of a signiﬁcant part of the sub-ﬁlter kinetic energy, while the use of ﬁve levels allows to retrieve almost perfectly this energy, up to the grid cut-oﬀ wavenumber. The same conclusion holds when looking at Fig. 3 which shows that the reconstruction procedure is local in space and is very eﬃcient to reconstruct all the sub-ﬁlter information at each point of the computational grid. Table 1 summarizes the performance of the method, for two diﬀerent ﬁlters (Gaussian and fourth-order accurate ﬁnite-diﬀerence) and several resolutions of the LES grid. For the Gaussian ﬁlter, ADM nearly restores all the sub-ﬁlter kinetic energy. The last two columns display the correlation between exact and reconstructed kinetic energy ﬁelds, for deconvolution levels N = 2 and N = 5 respectively. In all cases, the reconstructed sub-ﬁlter ﬁeld is observed to be highly correlated with the exact ﬁeld. Less good results are however obtained when the ﬁnite-diﬀerence ﬁlter is used. This can be explained by the fact that its transfer function falls to zero at the cutoﬀ, therefore reducing the eﬃciency of the approximate deconvolution (2) for the highest wavenumbers.

192

M. Terracol and B. Aupoix Table 1. Performance of sub-ﬁlter ﬁeld reconstruction by ADM. Filter

Grid

qsf 5 /qsf

Corr(qsf 2 , qsf )

Corr(qsf 5 , qsf )

Gaussian

643 323 163

0.956 0.929 0.928

0.975 0.972 0.974

0.996 0.995 0.996

FD4

643 323 163

0.803 0.715 0.706

0.837 0.804 0.808

0.893 0.863 0.871

4 Evaluation of the subgrid kinetic energy 4.1 The double deconvolution approach For the following developments, we shall assume that the energy spectrum obeys a power law of the form E(k) = Ck −α in the whole wavenumber range aﬀected by the ﬁlter, with C and α to be determined. Using a N -level deconvolution, the kinetic energy of the reconstructed sub-ﬁlter ﬁeld u is: kΔ % & 2 (k) − 1 G 2 (k) Ck −α dk qsf N = Q (4) N 0

where kΔ = π/Δx denotes the grid cutoﬀ wavenumber. The use of this relation for two diﬀerent deconvolution levels allows to estimate the two spectrum parameters C and α, and therefore the subgrid kinetic energy as: ∞ 1 1−α qsgs = CkΔ Ck −α dk = (5) α − 1 kΔ Applying relation (4) for two diﬀerent deconvolution levels N1 and N2 > N1 lead to a relation of the form: qsf N2 α = f G, (6) qsf N1 Thus, it appears that the ratio between the sub-ﬁlter kinetic energy computed for two distinct deconvolution levels only depends upon the ﬁlter G, and the spectrum slope α. This dependency, and the possibility to get an estimate of α has been investigated in detail for three diﬀerent ﬁlters (Gaussian, fourthorder accurate ﬁnite-diﬀerence, and Laplacian) using numerical integration of (4). In all cases, it was observed that relation (6) can be represented as: qsf N1 qsf N1 1 B log (7) = Aα + B or α = log − qsf N2 A qsf N2 A where the two constants A and B only depend upon the choice of the ﬁlter and the two deconvolution levels N1 and N2 . For each considered ﬁlter, a

An ADM-Based Subgrid Scale Reconstruction Procedure for LES

193

parametric study on N1 and N2 was carried out, and the associated values of A and B tabulated. From this study, the choice N1 = 2 and N2 = 5 was retained as optimal. Relation (7) provides an estimate of the spectrum slope α. From numerical integration of relations (4) and (5), the subgrid kinetic energy is ﬁnally represented as qsgs = aα−b qsf N2 (8) where a and b only depend upon the choice of the ﬁlter G and the deconvolution level N2 . These two constants were also computed for each ﬁlter. It must be pointed out that this approach only provides the subgrid kinetic energy. An extra procedure, e.g., a stochastic process, is required to generate a subgrid velocity ﬁeld consistent with this energy. 4.2 A priori tests on synthetic turbulent ﬁelds The previous approach relies upon statistical values. Its application to realistic turbulent ﬁelds however requires to introduce appropriate spatial averages. This section presents a priori numerical tests on synthetic turbulent ﬁelds obeying a k −α law on a 643 computational grid, for several values of the exponent α (1.25, 5/3, 2 and 3). Results are presented here for the fourthorder accurate ﬁnite-diﬀerence ﬁlter, with deconvolution levels N1 = 2 and N2 = 5. Plane averages yielded fair predictions of the exponent α in all cases. However, for more general applications, a local space averaging procedure is required. Spherical averages of diﬀerent radii were considered. Figure 4 presents the mean estimated values and variance of α as function of the radius of the averaging ball for two diﬀerent test cases. The average value of α is fair, although slightly underestimated, while its variance strongly decreases with the ball radius. A ﬁner analysis, considering α pdfs, shows that an optimal ball radius is about 4Δx, while representing a very small fraction (1/1,000) 10−1

1.6

3

102

2.98

101

2.96

100

2.94

10−1

2.92

10−2

2.9

10−3

1.59 10−3

<α>

< var α >

<α>

10−2

< var α >

1.595

1.585

1.58

2

4

Nb

6

8

10−4 10

2.88

2

4

6

8

10−4 10

Nb

Fig. 4. Mean and rms values of α using local averaging in a ball. Left: α = 5/3. Right: α = 3. “Nb” is the ratio of the ball radius to the grid space step.

194

M. Terracol and B. Aupoix

a

b

Fig. 5. Iso-contour maps of subgrid kinetic energy in a x−y plane: (a) reconstruction using the double deconvolution approach; (b) exact subgrid kinetic energy from DNS.

of the computational domain. This value is fully consistent with the spectral characteristics of the ﬁlter, which acts in the wavenumber range [kΔ /4; kΔ ]. 4.3 Tests on DNS ﬁelds The same DNS ﬁeld as in Section 3, projected on a 323 LES grid, was used to compare the exact subgrid kinetic energy to the one predicted with the above procedure. Figure 5 shows a poor prediction of the subgrid kinetic energy. On the one hand, the energy levels are overpredicted. On the other hand, exact and predicted energy contours appear to be poorly correlated. The ﬁrst failure can be blamed upon the assumption of a power law spectrum extending from the wavenumbers aﬀected by the ﬁlter to inﬁnity. Figure 2 shows the soestimated power law spectrum and evidences that this assumption strongly overestimates the subgrid energy spectrum. Cures to account for this low Reynolds number eﬀect are foreseen but not yet tested. The poor correlation between sub-ﬁlter and subgrid energies, which is a priori inconsistent with scale similarity, is also suspected to be a low Reynolds number eﬀect. Tests with higher Reynolds number DNS ﬁelds [3] are in progress.

5 Conclusions and future works The potential of ADM to reconstruct the unresolved ﬁeld in LES has been evidenced. ADM allows a nice recovery of the sub-ﬁlter ﬁeld. A new technique has been proposed to relate the subgrid energy to the sub-ﬁlter energy. It is local in space and has been assessed on synthetic ﬁelds. Its application to DNS ﬁelds has evidenced a need for low-Reynolds number corrections. These estimates of the sub-ﬁlter and subgrid ﬁelds will be used to better predict particle transport. They could also lead to improved subgrid scale models. This research was performed in the framework of ONERA federative project MOGADIR.

An ADM-Based Subgrid Scale Reconstruction Procedure for LES

195

References 1. Kuerten J.G.M, Vreman, A.W (2005) Can turbophoresis be predicted by LargeEddy Simulation?, Physics of Fluids 17:011701 2. Stolz S, Adams N.A (1999) An approximate deconvolution procedure for largeeddy simulation, Physics of Fluids 11(7):1699–1701 3. http://turbulence.pha.jhu.edu/

Large-Eddy Simulation of Turbulent Flow in a Plane Asymmetric Diﬀuser by the Spectral-Element Method Johan Ohlsson1 , Philipp Schlatter1 , Paul F. Fischer2 , and Dan S. Henningson1 1

2

Linn´e Flow Centre, KTH Mechanics, Stockholm, Sweden, [email protected]; [email protected]; [email protected] MCS, Argonne National Laboratory, Argonne, USA, [email protected]

Abstract LES and no-model LES (coarse-grid DNS) have been performed of turbulent ﬂow in a plane asymmetric diﬀuser by the Spectral-Element Method (SEM). Mean proﬁle and turbulent stresses compare well to LES results from Herbst et al., however the SEM generally predicts a later (i.e. further downstream) separation. It can be concluded that the use of a high-order method is advantageous for ﬂows featuring pressure-induced separation.

1 Introduction The Spectral-Element Method (SEM), introduced by Patera [10], is a highorder numerical method with the ability to accurately simulate ﬂuid ﬂows in complex geometries. SEM has opened the possibility to study – in great detail – ﬂuid phenomena known to be very sensitive to discretization errors, e.g. ﬂows exhibiting separation. Especially pressure-induced separation (as opposed to separation where the separation is induced by sharp edges or obstacles) is known to be particularly challenging, since the separation and reattachment points are hard to predict and are generally very sensitive to disturbances that could stem from an inaccurate numerical discretization. A typical engineering ﬂow where pressure-induced ﬂow separation is dominant is the turbulent diﬀuser ﬂow. The speciﬁc planar asymmetric diﬀuser considered here has been investigated experimentally by Obi et al. [9] and Buice and Eaton [2] and numerically by Kaltenbach et al. [7] (opening angle of 10◦ ). A similar geometry with an opening angle of 8.5◦ was studied by Herbst et al. [5], who performed LES at three diﬀerent Reynolds numbers (Reb = 4,500, 9,000, 20,000 based on bulk velocity and channel half-height) with a hybrid second-order ﬁnite-diﬀerence/spectral code. In the present work, a turbulent diﬀuser similar to Herbst et al. [5] is simulated by the SEM with large-eddy V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 29, c Springer Science+Business Media B.V. 2010

197

198

J. Ohlsson et al.

simulation (LES) and “no-model” LES at Reb = 4,500 and 9,000. For the LES, a version of the dynamic Smagorinsky model, speciﬁcally adapted for SEM, is used.

2 Numerical method and simulation setup The SEM code nek5000, developed by Paul F. Fischer, is used in the present study. It solves the three-dimensional, unsteady, incompressible Navier–Stokes equations by a Legendre-spectral element formulation and uses a rectangular structured grid with the ability to handle curved elements boundaries. It is massively parallelized and has shown a nearly linear speed-up for our computations up to 2048 cores. Two diﬀerent ways of stabilizing ﬂows at high Reynolds numbers are implemented in the code; overintegration (dealiasing) and polynomial ﬁltering (as proposed by Fischer and Mullen [3]). The combination of LES and SEM has only quite recently been explored, including the Rational LES (RLES) of channel ﬂow by Iliescu and Fischer [6], the dynamic Smagorinsky LES and the deconvolution based LES of the cubic cavity ﬂow by Bouﬀanais et al. [1]. Here, we use the dynamic Smagorinsky model, where the model coeﬃcient is computed according to the dynamic procedure proposed by Germano et al. [4]. In the framework of SEM, the deﬁnition of a test ﬁlter is implemented in Legendre space and constructed such that approximately half of the modes are aﬀected. To limit the ﬂuctuation of the model coeﬃcient, spatial averaging is used along the homogenous spanwise (z ) direction together with clipping of the model parameter (negative values are discarded).

3 Validation by turbulent channel ﬂow An extensive validation of the simulation setup was performed by means of DNS of turbulent channel ﬂow. The aim was to address the open questions about the stability of SEM at moderate to high Reynolds numbers. Fully turbulent periodic channel ﬂow was simulated at two diﬀerent Reynolds numbers, Reτ = 180 and 590. The results, shown in Fig. 1 for Reτ = 590, show that SEM is able to predict the mean velocity proﬁle and the Reynolds stresses with very good agreement to spectral DNS results at moderate Reynolds numbers if one employs either polynomial ﬁltering or overintegration as stabilization technique. If neither overintegration nor polynomial ﬁltering is used, the calculation will experience a numerical instability after a small number of time steps, even for a high resolution (also discussed by Fischer and Mullen [3]). Both ﬁltering and overintegration on their own are able to stabilize the calculation. This was also seen in transition simulations (K-type transition similar to Schlatter et al. [11]), where it turned out to be essential to employ a stabilization technique as soon as the turbulent state was reached.

LES of Turbulent Flow in a Plane Asymmetric Diﬀuser by SEM

199

8 7

20

2

6

u

+

/uτ

2

15 10

5 4

k

3 2

2 5

<w’ >

1

2

0 0 100

101

102

−1

y+

0

100

200

300 y+

400

500

600

Fig. 1. Turbulent channel ﬂow at Reτ = 590 (resolution: 288 × 192 × 288) showing mean velocity proﬁle and Reynolds stresses together with turbulent kinetic energy. Comparison to DNS results (resolution: 384 × 257 × 384) from Moser et al. [8] DNS data, Log Law, nek5000.

4 Diﬀuser 4.1 Geometry and parameter settings The geometry of the plane asymmetric diﬀuser is similar to Herbst et al. [5]. The geometry, with visible element boundaries, is sketched in Fig. 2. The simulations, including both LES and no-model LES, were performed at a Reynolds number of Reb = 4,500 and 9,000, corresponding to the lower and medium Reynolds number of Herbst et al. [5]. The cases are summarised in Table 1. The spanwise width of Lz = 8 is chosen in accordance with Kaltenbach et al. [7]. The inﬂow channel has a height of 2, which starts to expand at x = 0 with the diﬀuser wall inclined at a diverging angle of 8.5◦ . The diﬀuser reaches its ﬁnal height of 9.4 at x = 49.6. The edges where the inclined wall is attached, are rounded with a radius of 20.0. Spanwise (z ) periodicity is used for both cases. The aim was to have a fully developed turbulent ﬂow entering the diﬀuser expansion. This was achieved by an unsteady Dirichlet boundary condition, where random noise was superimposed on a turbulent mean velocity proﬁle. Having gone through transition to turbulence, the ﬂow was allowed to evolve for suﬃciently long distance upstream of the diﬀuser throat, so that an approximately fully developed turbulent state was reached just before the diﬀuser expansion. At the outﬂow, a sponge region is added, where the ﬂow is forced to a turbulent mean ﬂow proﬁle in order to damp out oscillations prior to reaching the zero-pressure outﬂow boundary. 4.2 Results Selected turbulent statistics, including the evolution of the mean streamwise velocity u and the Reynolds stress u v , are shown in Fig. 3 at seven streamwise positions at Reb = 4,500 and 9,000, respectively. The left column

200

J. Ohlsson et al. inflow

sponge

y

R = 20.0 2 0 −2 −4 −6 0

20

40

60

x

80

100

Fig. 2. Sketch of the diﬀuser geometry and the grid used for the Reb = 4,500 case. Note that only the element boundaries are visible. The edges marked by arrows are rounded with a radius of 20.0.

Table 1. Description of the diﬀuser cases Reb

Lx

Lz

Nx

Ny

Nz

Order

# Elements

# Grid points

4,500 9,000

121 131

8 8

260 346

43 61

64 91

7 15

1998 552

0.72 · 106 1.92 · 106

b

−2 −4 −6

y

y

a − 02 0

10

20

30

40

50

60

70

2 0 −2 −4 −6

x

d y

y

c

2 0 −2 −4 −6 0

10

20

30

40

50

60

0

10

20

30

0

10

20

30

70

x

f

2 0 −2 −4 −6 0

10

20

30

40

50

60

70

0

y

y

h 10

20

30

40 x

70

40

50

60

70

10

20

30

40

50

60

70

40

50

60

70

x

2 0 −2 −4 −6 0

60

2 0 −2 −4 −6

x

g

50

x

y

y

e

40

2 0 −2 −4 −6

50

60

70

2 0 −2 −4 −6 0

10

20

30 x

Fig. 3. Turbulent statistics for the diﬀuser at Reb = 4,500 (a − d ) and 9,000 (e − h): a & e 10 · u + x (LES), b & f 10 · u + x (no-model LES), c & g 500 · u v + x nek5000, Herbst et al. [5]. (LES), d & h 500 · u v +x (no-model LES),

(a - g) shows LES results and the right column (b - h) shows no-model LES results. The results are compared with LES data from Herbst et al. [5]. Good agreement is obtained for LES and no-model LES results at the two diﬀerent Reynolds numbers. Herbst et al. [5] performed simulations at three diﬀerent resolutions. The number of degrees of freedom in our simulations correspond (at both Reynolds numbers) to the lowest of the resolutions used in [5]. It should be pointed out, however, that the SEM results, although at the lowest resolution, compares better to the second highest resolution data from Herbst et al. [5], in particular in the separated region. This justiﬁes the idea that

LES of Turbulent Flow in a Plane Asymmetric Diﬀuser by SEM

b−220

2 0 −2 −4 −6

y

y

a

0

10

20

30

40

50

60

−4 −6 0

70

10

20

30

40

50

60

70

d 20

2 0 −2 −4 −6

y

y

c

201

0

10

20

30

40 x

50

60

70

−2 −4 −6 0

10

20

30

40

50

60

70

x

Fig. 4. Contours of the streamfunction at Reb = 4,500 (a − b) and 9,000 (c − d): a & c LES, b & d no-model LES. Thick white contour levels indicate the value 10−5 of the streamfunction, thin white contours range from −0.1 to 2.0 with spacing 0.2.

Fig. 5. Isosurface of −0.01 for the streamwise velocity of an instantaneous velocity ﬁeld at Reb = 4,500 (a − b) and 9,000 (c − d ), a & c LES, b & d no-model LES.

a high-order method makes a better use of the gridpoints than a low-order method. Contours of the streamfunction for a time and spanwise averaged ﬂow ﬁeld are shown in Fig. 4, with (a – c) and without (b – d) SGS model for Reb = 4,500 and 9,000, respectively. The trend of an increased separated region for higher Reynolds number is obvious in our no-model LES simulations (compare, e.g., Fig. 4b and 4d) and further conﬁrmed by numerical results by Herbst et al. [5] and experimental results by Obi et al. [9]. More speciﬁc, the position of the separated region (indicated by the mean diving streamline) almost coincides with data from Herbst et al. [5] (shown by arrows) for the lower Reynolds number (Fig. 4b). In the LES results, however, the separation bubble is not visible. For the higher Reynolds number, there is a general trend towards a later separation in our results compared to Herbst et al. [5]. The no-model LES results show a relatively early reattachment (Fig. 4d), whereas the LES results indicate a reattachment point that coincides with Herbst et al. [5] (Fig. 4c). Clearly, the presence of the SGS model aﬀects the size and the location of the separation bubble. In Fig. 5, the highly unsteady nature of pressure-induced separation is highlighted. The instantaneous separated

202

J. Ohlsson et al.

region, here visualized by an isosurface of −0.01 for the streamwise velocity, show diﬀerent separation and reattachment points compared to the averaged equivalences.

5 Conclusion and outlook LES and no-model LES have been performed of turbulent ﬂow in a plane asymmetric diﬀuser by the SEM. Turbulent statistics compare well to LES results by Herbst et al. [5]. It can be concluded that the use of a high-order method is advantageous, both in terms of the parallel eﬃciency of the method, but also the fact that less grid points were needed to predict a result with the same given accuracy compared to a low-order method. The inﬂuence of the SGS model is hardly noticeable in the statistics, except close to the diﬀuser throat where the SGS model might be slightly too dissipative (e.g. reduced turbulent activity in that region) and close to the outlet where the model seems to improve the results. Investigation of the location of the separation bubble reveals that the separation generally starts more downstream in the SEM results compared to Herbst et al. [5]. Improved methods of treating the turbulent inﬂow condition, e.g. the use of trip-forcing, will be considered and compared to the present method. The dynamic Smagorinsky model will be compared to other SGS models, in particular those based on high-pass ﬁltering. Further, as pointed out by Kaltenbach et al. [7], it may be desirable to increase the width of the domain in order decrease the presence of artiﬁcial coherent structures, which seems to delay the reattachment.

References 1. R. Bouﬀanais, M. Deville, P.F. Fischer, E. Leriche, and D. Weill. Large-eddy simulation of the lid-driven cubic cavity ﬂow by the spectral element method. J. Sci. Comput., 27, 2006. 2. C.U. Buice and J.K. Eaton. Experimental investigation of ﬂow through an asymmetric plane diﬀuser. J. Fluids Eng., 122(2):433–435, 2000. 3. P. Fischer and J. Mullen. Filter-based stabilization of spectral element methods. C.R. Acad. Sci. Paris, t. 332, Serie I:p. 265–270, 2001. 4. M. Germano, U. Piomelli, P. Moin, and W.H. Chabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids, 3(7):1760–1765, 1991. 5. A.H. Herbst, P. Schlatter, and D.S. Henningson. Simulations of turbulent ﬂow in a plane asymmetric diﬀuser. Flow Turbulence Combust, 79:275–306, 2007. 6. T. Iliescu and P.F. Fischer. Large eddy simulation of turbulent channel ﬂows by the rational large eddy simulation model. Phys. Fluids, 15(10):3036–3047, 2003. 7. H.-J. Kaltenbach, M. Fatica, R. Mittal, T.S. Lund, and P. Moin. Study of ﬂow in a planar asymmetric diﬀuser using large-eddy simulation. J. Fluid Mech., 390:151–185, 1999. 8. R.D. Moser, J. Kim, and N. Mansour. Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Phys. Fluids, 11(4), 1999.

LES of Turbulent Flow in a Plane Asymmetric Diﬀuser by SEM

203

9. S. Obi, H. Nikaido, and S. Masuda. Experimental and computational study of turbulent separating ﬂow in an asymmetric diﬀuser. Proceedings of the ninth symposium on turbulent shear ﬂows, 305.1–305.4, 1993. 10. A.T. Patera. A spectral element method for ﬂuid dynamics: Laminar ﬂow in a channel expansion. J. Comput. Phys., 54:468–488, 1984. 11. P. Schlatter, S. Stolz, and L. Kleiser. LES of transitional ﬂows using the approximate deconvolution model. Int. J. Heat Fluid Flow, 25(3), 2004.

h and p Reﬁnement with Wall Modelling in Spectral-Element LES S. Hulshoﬀ, E. Munts, and J. Labrujere Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, [email protected]; [email protected]; [email protected] Abstract The convergence behaviour of a variational multiscale method with mesh and polynomial reﬁnement is examined. The eﬀects of strong and weak boundary conditions on convergence are demonstrated.

1 Introduction Space–time spectral-element methods are an attractive base for adaptive techniques as they allow local reﬁnement of both mesh and interpolation order, and can exploit variational-multiscale concepts for SGS modelling. As a step towards developing adaptive strategies, we examine the convergence behaviour of space–time spectral-element computations of turbulent channel ﬂow as a function of h and p reﬁnement. We ﬁrst consider convergence of a low Reynolds number case using strong Dirichlet no-slip conditions, and then the convergence of a higher Reynolds number case using wall modelling via weak Dirichlet conditions.

2 Method The results have been computed using a space–time spectral-element method for the compressible Navier–Stokes equations. Denoting the vector of conservative variables with U, the source vector with S and the inviscid and viscous ﬂuxes with Fi (U) and Fvi (U), the corresponding variational formulation for a trial space Yn and test space Wn can be written as Find Y ∈ Yn such that ∀ W ∈ Wn ij Y,j − (W,t , U(Y))Qn − W,i , Fi (Y) − K + (W, (Fi (Y) −

Qn v Fi (Y)) ni )Pn

+ (W(tn+1 ), U(Y(tn+1 )))Ωn+1 − (W(tn ), U(Y(tn )))Ωn = (W, S)Qn . V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 30, c Springer Science+Business Media B.V. 2010

205

206

S. Hulshoﬀ et al.

Here (, )Ωn denotes the L2 -inner product over region Ωn , Ωn is the spatial domain at time tn , Qn is the portion of space–time domain between time levels tn and tn+1 , and Pn is the surface connecting the boundary of Ωn with ij are that of Ωn+1 . ni is the local space–time surface normal vector, and K v the diﬀusivity matrices deﬁned by Fi (U) = (Kij Y,j ),i . A space-continuous basis of Jacobi polynomials is used for both Yn and Wn (see [4]). The subgrid scale model is eddy-viscosity based and can selectively be applied to portions of the resolved scales as in the variational-multiscale approach of Hughes et al. [2]. The source term is held constant in space and time over the streamand spanwise-periodic domain (i.e. no bulk-velocity matching is used). Wall modelling is performed via weak Dirichlet conditions using an interior-penalty approach similar to that described in Bazilevs and Hughes [1]. In this case, however, the penalty parameter is held ﬁxed while a Schumannlike wall-stress model is used to determine the instantaneous state jump.

3 Error determination

0.35

0.08

0.3

0.07

rms u perturbation L2 error

rms u perturbation L2 error

The computed solutions are evaluated in terms of their L2 error relative to DNS mean and rms perturbation velocity proﬁles. For the latter, the results computed by Moser et al. [3] are used. As these are for incompressible ﬂow, modelling errors associated with compressibility are also present. The current results have been computed with a mean Mach number of 0.1. An investigation performed with mean Mach numbers ranging from 0.05 to 0.4 has shown that the errors due to compressibility are small relative to the remaining errors considered. Determining mean and rms proﬁles requires the selection of a sampling interval of appropriate length and initial time. Even with high-quality initial conditions, the results can be inﬂuenced by the transients arising due to modelling and discretisation eﬀects. The periodic channel geometry can also support relatively long-wavelength oscillations. To quantify uncertainty due to the choice of sampling interval, L2 errors have been computed using proﬁles obtained from a range of interval lengths and initial times. Figure 1 shows

0.25 0.2 0.15 0.1

T=5 T = 10 T = 20 T = 40

0.05 0 0

20

40 60

80 100 120 140 160 180 200

Sampling interval begin time

0.06 0.05 0.04 0.03 T=5 T = 10 T = 20 T = 40

0.02 0.01 0 0

50

100

150

200

250

Sampling interval begin time

Fig. 1. L2 errors for diﬀerent sampling interval lengths (T) and start times (x-axis). Coarse (left) and medium (right) levels of reﬁnement.

h and p Reﬁnement with Wall Modelling in Spectral-Element LES

207

L2 errors in the rms streamwise perturbation velocity proﬁle for coarse and medium levels of reﬁnement. As the reﬁnement is increased, the uncertainty in determining L2 errors becomes more signiﬁcant.

4 Results Figure 2 shows L2 errors of the mean and rms streamwise perturbation proﬁles for Reτ = 180 with increasing levels of reﬁnement and strong Dirichlet no-slip conditions. p1L and p2L indicate results from linear and quadratic interpolations where the SGS-model is applied to all resolved scales. p2V indicates quadratic results with the SGS model applied only to the smallest resolved scales. In these three cases, the number of degrees of freedom is increased by reﬁning the mesh (decreasing h). Also shown are results for an 83 -element discretisation where the number of degrees of freedom is increased by augmenting the interpolation order (h8V). For the latter the SGS model is applied only to the smallest resolved scales. The discretisations with higher-order interpolation show increased eﬃciency on a per degree-of-freedom basis. Signiﬁcant increases in accuracy are also obtained from restricting the subgrid-scale model to the smallest of the resolved scales. After p = 3, the computational cost per degree of freedom increases signiﬁcantly. Due to enlarged bandwidth, matrix storage can also become impractical in parallel environments when p > 3 and overlap is used. As the accuracy gains for p = 3 are not great, for this problem p = 2 is optimal. Figure 3 shows the L2 errors of the mean and rms streamwise perturbation proﬁles for Reτ = 395 with increasing levels of reﬁnement with and without wall modelling. Here p2V is the quadratic scheme with strong Dirichlet conditions, and p2Vw is the quadratic scheme with wall modelling via weak Dirichlet conditions. Both of these are reﬁned by decreasing h. h8Vw is 1

1

0.1 p1L p2L p2V h8V

0.01

0.001

1

10 100 Degrees of freedom per direction

rms u perturbation L2 error

mean u L2 error

10

0.1 p1L p2L p2V h8V 0.01

1

10 100 Degrees of freedom per direction

Fig. 2. L2 errors versus number of degrees of freedom for discretisations with hard Dirichlet no-slip conditions (Reτ = 180).

208

S. Hulshoﬀ et al. 1

p2V p2Vw h8Vw

rms u perturbation L2 error

mean u L2 error

1

0.1

0.01 10

100 Degrees of freedom per direction

0.1 p2V p2Vw h8Vw

0.01 10

100 Degrees of freedom per direction

Fig. 3. L2 errors versus number of degrees of freedom for discretisations using with and without wall modelling (Reτ = 395).

the 83 -element case with wall modelling via weak Dirichlet conditions, reﬁned in p. In all cases the reference location for computing the wall-model stress is 1/8 of the channel half height. The schemes converge over the observed range, with the weak Dirichlet condition solutions providing signiﬁcantly better accuracy for the mean velocity proﬁle. It can be expected, however, that at high levels of reﬁnement the weak Dirichlet condition results for the perturbations will fail to converge due to the inconsistency of the Schumann wall modelling procedure with the unsteady Navier–Stokes equations. p-reﬁnement appears to be more eﬀective than in the strong Dirichlet condition case. This may be due to reductions in interpolation error by eliminating the need to resolve a sharp near-wall layer.

5 Summary The results indicate that in spite of the roughness of the instantaneous LES solution, small increases in polynomial order can provide increased eﬃciency in the determination of low-order statistics. If at least quadratics are used, variational-multiscale concepts can also be exploited for signiﬁcant gains in accuracy. Initial results indicate that the eﬀectiveness of p-reﬁnement is greater when wall modelling via weak Dirichlet conditions is used.

References 1. Bazilevs Y. and Hughes, T. J. R. (2007) Computers & Fluids 36:12–26 2. Hughes, T. J. R., Mazzei, L., and Jansen, K. E. (2000) Comput. Visual. Sci. 3:47–59 3. Moser, R. D., Kim, J., and Mansour, N. N. (1999) Phys. Fluids 11:943 4. Munts, E. A., Hulshoﬀ, S. J., and de Borst, R. (2007) JCP 24:389–402

Error-Landscape Assessment of LES Accuracy Using Experimental Data Johan Meyers1 , Charles Meneveau2 , and Bernard J. Geurts3 1

2

3

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300A, B3001 Leuven, Belgium, [email protected] Department of Mechanical Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA, [email protected] Multiscale Modeling and Simulation, Faculty EEMCS, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, [email protected]

Abstract We present the setup of a database of large-eddy simulations of decaying homogeneous isotropic turbulence at high Reynolds numbers, corresponding to recent physical experiments in the tradition of Comte–Bellot and Corrsin. The errorlandscape approach is used for the evaluation of LES using the Smagorinsky model in a spectral and a second-order discretization method. The results identify an optimal combination of model parameter and resolution in a statistical robust fashion. Issues associated with dynamics at the largest length-scales are discussed.

1 Introduction Recently, error-behavior of large-eddy simulation (LES), and the assurance of quality in LES has gained considerable attention [1–3]. These studies aim to formulate a rigorous standard for the assessment of accuracy and reliability in LES. Most of these studies use direct numerical simulations (DNS) as reference case, since these provide very detailed information on statistical ﬂow properties. To employ these methods for the high-Reynolds-number applications in which LES is typically applied, one may wish to employ experimental data. The use of experimental data as reference for LES is non-trivial: less data is available, data are not always ﬁltered, uncertainty may exist about the boundary conditions, and measurement errors aﬀect the results. However, only experimental data are available since DNS are unfeasible at realistically elevated Reynolds numbers. The current work focusses on the use of experimental reference data in the multi-objective error-landscape approach developed in previous work [2, 4]. This approach provides a systematic framework to assess the errors of LES, characterizing both modelling and discretization eﬀects. In the current study, we use the decaying active-grid turbulence of Kang et al. [5], with initial V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 31, c Springer Science+Business Media B.V. 2010

209

210

J. Meyers et al.

Reynolds number Re λ = 716, as reference. This provides a much higher Reynolds-number reference than the DNS reference employed in previous studies [4]. We show that the evaluation of LES quality is best performed using a series of ﬂow properties that are sensitive to various scales of the ﬂow. As in Ref. [2], this leads to a characterization of a multi-objective optimal reﬁnement strategy. Secondly, for the current case, it is shown that the errorlandscape approach helps to identify inconsistencies in boundary conditions between the numerical set-up and the experimental set-up.

2 Methodology The active grid experiment of Kang et al. [5] provides spectra of decaying turbulence at four diﬀerent measurement stations in a windtunnel (x/M = 20, 30, 40, 48, where M = 5.08 cm is the mesh size of the grid). For the LES, experiments are translated into a time frame using the Taylor hypothesis and representing the ﬂow in a frame convecting downstream at the mean velocity. The ﬁrst measurement station (Re λ = 716) is used to generate the initial condition; data from the next measurement stations (Re λ = 676, 650, 626) are used as reference for the assessment of the numerical solution. For the setup of the LES, we follow the procedure set out in Ref. [5], with the diﬀerence that the experimentally obtained energy spectra are ﬁtted with new functional forms that provide better representations of the spectra [6]. Large-eddy simulations are performed using a standard Smagorinsky model. Two spatial discretization schemes are compared: (1) a pseudospectral method, and (2) a staggered second-order discretization mimicked by wavenumber modiﬁcations [7] in the same pseudo-spectral code. Equations are integrated in time with a four-stage fourth-order Runge–Kutta scheme. A cubical sharp cut-oﬀ ﬁlter is used, which corresponds to the cut-oﬀ of the pseudo-spectral discretization. Consequently, the ﬁlter width Δ = π/kc (with kc the grid cut oﬀ) used in the Smagorinsky model corresponds to B/N , with B the size of the computational box (B ≈ 108M ), and N 3 the grid size. To set up the LES database, a systematic variation of the grid size N 3 (163 –2563) and the model coeﬃcient Cs (0–0.284) is performed, such that the error behavior can be properly charted as a function of these parameters. In order to discuss the results in the next section, we brieﬂy introduce some concepts related to the multi-objective error-landscape approach [2]. Central to this approach is the use of a range of error measures, characterizing diﬀerent scales in the solution. First of all, we consider errors Dp (p = −1 to 2) which are based on weighted integrals of the energy spectrum, i.e., ⎡ ⎢ Dp (N, Cs ) = ⎣

* kc

k p (ELES (k) − EEXP (k)) dk x/M 0 * +2

kc p x/M 0 k EEXP (k) dk

+2 ⎤1/2 ⎥ ⎦

,

(1)

Error-Landscape Assessment of LES Accuracy Using Experimental Data

211

with kc = π/Δ the grid cut-oﬀ frequency, and where the summation is over results at the diﬀerent measurement stations. Depending on the values of p, this approach deﬁnes errors on large-scale properties (p = −1, 0) as well as resolved ﬁne-scale properties (p = 1, 2). For p = −1, 0, and 2, it is readily shown that Dp respectively represents the relative error on the integral length scale L , the resolved turbulent kinetic energy E, and the resolved enstrophy E (cf. Ref. [2] for details). s(p) (N ) is deﬁned Based on Dp (N, Cs ), an optimal reﬁnement trajectory C as the value Cs which provides a minimal error Dp at given resolution N . s(p) (N ) may depend on p, and hence, a multi-objective approach, Obviously, C where errors at diﬀerent p are considered simultaneously is called for. To this end, near optimal regions related to Dp (N, Cs ) are deﬁned as [2] " # / Dp (N, Cs ) + / Ωp (a) = N ∈ N; Cs ∈ R / ≤a , (2) s(p) (N )) Dp (N, C where we select a = 1.2 [2]. Further, we recall that a global weighted error is deﬁned as &

% (p) p Dp (N, Cs )/ Dp N, Cs (N ) (N, Cs ) = & , (3) D

% 0 (p) p 1 Dp N, Cs (N ) s (N ) is Based on D(N, Cs ), a multi-objective optimal reﬁnement trajectory C deﬁned. More information can be found in Ref. [2]. The error deﬁnition in equation (1) is based on a quadratic weighting of errors. However, there is no penalization for incorrect distributions of underlying energy spectra in wavenumber space. In Ref. [2], it was shown that this can lead to incorrect characterization of LES quality when only a few errors are included in the analysis. A more robust error deﬁnition, which penalizes incorrect spectral distributions corresponds to [2] 1 dp (N, Cs ) =

kc

2

k 2p (ELES (k) − EEXP (k)) dk kc

2p 2 x/M 0 k (EEXP (k)) dk

x/M

0

21/2 .

(4)

As for errors Dp , near-optimal regions can be identiﬁed with dp , and using a series of errors (e.g., p = −1, 0, 1, 2), a multi-objective optimal reﬁnement trajectory is deﬁned.

3 Results and discussion In this section, results from the analysis are presented and initial conclusions are drawn. Using a large set of LES at diﬀerent (N, Cs ) combinations, nearoptimal regions are readily identiﬁed in Fig. 1 for the Smagorinsky model.

212

J. Meyers et al.

Fig. 1. ‘Near optimal’ regions based on Dp (left) and dp (right) of Smagorinsky LES of Kang et al.’s (2003) experiment using pseudo-spectral discretization. The curves —, −−, − · , and · · · respectively mark the boundaries of Ωp (p = −1 to 2). Regions in between are ﬁlled using semitransparent gray, such that areas with overlapping near optimal regions appear with darker shades of gray. (bold dashed): multi-objective optimal reﬁnement trajectory.

We see that near-optimal regions for D0 , D1 , and D2 overlap; similarly, all near-optimal regions based on dp overlap. Both multi-objective optimal reﬁnes (N ) are properly located in these multi-objective overlap ment trajectories C appears to have a horizontal asymptote regions. For N > 60 (or L/Δ > 6), C with a value Cs = 0.13, which is very close to a theoretical high-Reynoldsnumber value of 0.135 for LES with a cubical sharp cut-oﬀ ﬁlter [8]. In Figure 1 (left), it is striking that the near-optimal region for D−1 (representing an error which is predominantly based on the large-scale region of the solution), has no signiﬁcant intersection with the other near-optimal regions. We believe that this points towards an inconsistency between the experimental and the numerical set-up. In the current case, the eﬀect of the side-walls in the windtunnel on the largest ﬂow scales are not properly represented in the numerical simulation, where periodic boundary conditions are employed. Since the box size B in our simulations is considerably larger then the integral length scale L (B/L ≈ 10), this eﬀect is seen to not strongly inﬂuence the smallerscale predictions (emphasized by p = 0 − 2). By evaluating the energy spectra in Fig. 2, it is appreciated that the evolution of the large-scale part of the spectrum is not predicted properly for any of the simulations, while results s (N ). for the rest of the spectrum are satisfactory for simulations near C When a second-order discretization is employed, we observe similar trends as for the spectral method in Fig. 3: multi-objective optimal reﬁnement trajectories are well deﬁned for both type of error measures Dp and dp , but the near-optimal region of D−1 has no signiﬁcant overlap with other regions. Finally, in Fig. 4 multi-objective optimal trajectories based on dp and Dp are shown for both methods, and trajectories for both error deﬁnitions correspond well. Further, we observe that the second order discretization requires

Error-Landscape Assessment of LES Accuracy Using Experimental Data

213

10−1

E(k)

E(k)

10−1

10−2

−2

10

100

101 kB=2=¼

102

10−3 100

101 kB=2=¼

102

Fig. 2. LES Spectra on multi-objective optimal trajectory. (Left) N = 128, at all measurement stations. (Right) N = 64 (· · · ), 128 (−−), 192 (− · ), 256 (—), at x/M = 48. Gray lines correspond to experimental spectra.

Fig. 3. ‘Near optimal’ regions based on Dp (left) and dp (right) of Smagorinsky LES of Kang et al. [5] experiment using second-order discretization. Lines as in Fig. 1.

higher values for the Smagorinsky coeﬃcient than the spectral discretization. This is related to the fact that the second-order discretization of the strain tensor in the Smagorinsky model has the eﬀect of reducing its magnitude.

4 Conclusions An error-landscape assessment of LES using a Smagorinsky model was presented. Instead of using direct numerical simulations as reference, experimental spectra of kinetic energy at high Reynolds numbers were used. A robust characterization of multi-objective optimal trajectories is obtained, independent of the error measure (dp or Dp ). Finally, it was found that near-optimal regions based on the integral length scale (D−1 ) did not overlap with other near-optimal regions. An argument was put forward that this

214

J. Meyers et al. 0.25

Cs

0.2

0.15

0.1

0.05

0

50

100

0

5

10

150 N

200

250

300

15

20

25

30

L /Δ

Fig. 4. Multi-objective optimal trajectories for the pseudo spectral method (), and the second-order method (). (—): Using dp ; (· · · ): using Dp .

deviation is related to diﬀerences that exist between boundary conditions in the simulations and the experiments, showing another application of the error-landscape approach.

References 1. Gualtieri P, Casciola CM, Benzi R, Piva R (2007) Preservation of statistical properties in large-eddy simulation of shear turbulence. J Fluid Mech 592: 471–494 2. Meyers J, Geurts BJ, Sagaut P (2007) A computational error assessment of central ﬁnite-volume discretizations in large-eddy simulation using a Smagorinsky model. J Comput Phys 227:156–173 3. Park N, Mahesh K (2007). Analysis of numerical errors in large eddy simulation using statistical closure theory. J Comput Phys 222:194–216 4. Meyers J, Geurts BJ, Baelmans M (2003) Database analysis of errors in largeeddy simulations. Phys Fluids 15:2740–2755 5. Kang HS, Chester S, Meneveau C (2003). Decaying turbulence in an activegrid-generated ﬂow and comparisons with large-eddy simulation. J Fluid Mech 480:129–160 6. Meyers J, Meneveau C (2008) A functional form for the energy spectrum parametrizing bottleneck and intermittency eﬀects. Phys Fluids 20:065109 7. Ferziger JH, Milovan P (1997) Computational Methods for Fluid Dynamics. Springer, Berlin Heidelberg New York 8. Meyers J, Sagaut P (2006) On the model coeﬃcients for the standard and the variational multi-scale Smagorinsky model. J Fluid Mech 569:287–319

The Role of Diﬀerent Errors in Classical LES and in Variational Multiscale LES on Unstructured Grids H. Ouvrard1, B. Koobus1 , A. Dervieux2 , S. Camarri3 , and M.V. Salvetti3 1

2 3

D´epartement de Math´ematiques, Universit´e de Montpellier II, Montpellier, France, [email protected]; [email protected] INRIA, Sophia Antipolis, France, [email protected] Dipartimento di Ingegneria Aerospaziale, Universit` a di Pisa, Pisa, Italy, [email protected]; [email protected]

Abstract The eﬀects of numerical viscosity, subgrid scale viscosity and grid resolution are investigated in LES and VMS-LES simulations of the ﬂow around a circular cylinder at Re = 3, 900 on unstructured grids.

1 Introduction The success of a large-eddy simulation depends on the combination and interaction of diﬀerent factors, viz. the numerical discretization, which also provides ﬁltering when no explicit one is applied, the grid reﬁnement and quality and the physical closure model. On the other hand, all these aspects can be seen as possible sources of error in LES. LES simulations oriented to industrial or engineering applications are often characterized by low-order non-conservative numerical schemes and coarse grid resolutions. In this context, the investigation of errors and of their interaction is an important issue in order to develop guidelines for veriﬁcation of the reliability and quality of LES results. The aim of the present study is to investigate the role of diﬀerent sources of error, namely numerical viscosity, SGS modeling and unstructured grid resolution both in classical LES and in Variational Multiscale (VMS) LES approaches and for an industrial numerical set-up. This industrial numerical set-up is based on a mixed ﬁnite-volume/ﬁniteelement discretization on unstructured grids, second-order accurate in space. The most critical point with this type of co-located schemes on unstructured grids is the need of numerical viscosity in order to obtain stable solutions, which could interact unfavorably with the subgrid scale model and signiﬁcantly deteriorate the results. Our proposition was to dedicate the subgrid modeling to a physics-based model and to use for numerics a second-order accurate MUSCL upwind scheme equipped with a tunable dissipation made V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 32, c Springer Science+Business Media B.V. 2010

215

216

H. Ouvrard et al.

of sixth-order [2] spatial derivatives of all ﬂow variables. Fourier analysis clearly shows that such a dissipation has a damping eﬀect which is much more localized on high frequencies than the one of stabilizations based on second-order derivatives. In this way we can reduce the interaction between numerical dissipation, which damps in priority the highest frequencies, and the SGS modeling. Moreover, a key coeﬃcient (γs ) permits to tune numerical dissipation to the smallest amount required to stabilize the simulation. As for SGS modeling, we investigate here diﬀerent eddy-viscosity models [6, 9, 10], ﬁrst used in a classical LES context, i.e. computed as a function of the whole resolved ﬂow ﬁeld. Furthermore, the same SGS models are considered within the VMS-LES approach [3]. The main idea of VMS-LES is to decompose, through Galerkin projection, the resolved scales into the largest and smallest ones and to add the SGS model only to the smallest ones. We remark that the role and the interaction/compensation of diﬀerent errors in the VMS-LES approach might be signiﬁcantly diﬀerent than in classical LES and, thus, the present paper also intends to give a contribution to investigate this issue. Our investigation is carried out for the ﬂow around a circular cylinder at Reynolds number, based on the freestream velocity and on the cylinder diameter, equal to 3,900. This ﬂow has been chosen since it is a classical and well documented benchmark (see, e.g., [7, 8] for experimental data and [1, 4, 5, 8] for numerical studies). Moreover, it contains all the features and all the diﬃculties encountered in the simulation of bluﬀ-body ﬂows also for more complex conﬁgurations and higher Reynolds numbers, at least for laminar boundary-layer separation.

2 Basic ingredients for numerics and modeling The governing equations for compressible ﬂows are discretized in space using a mixed ﬁnite-volume/ﬁnite-element method applied to unstructured tetrahedrizations. The adopted scheme is second-order accurate in space and introduces a numerical dissipation made of sixth-order space derivatives [2], and, thus, concentrated on a narrow-band of the highest resolved frequencies. Moreover, a parameter γs directly multiplies the upwind part of the numerical ﬂux, thus, permitting a direct control of the amount of introduced numerical viscosity. Finally, an implicit time marching algorithm is used, based on a second-order time-accurate backward diﬀerence scheme. More details on the numerical ingredients used in the present work can be found in [2]. In the Variational Multi-Scale approach, the ﬂow variables are decomposed as W = W + W , where W are the large resolved scales (LRS) and W are the small resolved scales (SRS). We follow here the VMS approach proposed in [3] for the simulation of compressible turbulent ﬂows through a ﬁnite volume/ﬁnite element discretization on unstructured tetrahedral grids. Let χl and φl be the N ﬁnite-volume and ﬁnite-element basis functions associated

Diﬀerent Errors in Classical LES and in VMS-LES

217

to the used grid. In order to obtain the VMS ﬂow decomposition, these can be expressed as: χl = χl + χl and φl = φl + φl , in which the overbar denotes the basis functions spanning the ﬁnite dimensional LRS spaces and the prime those spanning the SRS spaces. As in [3], the basis functions of the LRS space are deﬁned through a projector operator in the LRS space, based on spatial average on macro cells, which are obtained by an agglomeration process. Finally, a key feature of the VMS approach is that the SGS model is added only to the smallest resolved scales. Eddy-viscosity models are used here, and, hence, the SGS terms are discretized analogously to the viscous ﬂuxes. Thus, the Galerkin projection of the LES closure term is made on the space spanned by the φl basis functions. As previously mentioned, both for classical LES and for VMS-LES, three diﬀerent eddy-viscosity subgrid models have been considered, namely those proposed by Smagorinsky [9] and Vreman [10] and the so-called WALE model [6]. In classical LES, the eddy-viscosity introduced by these models can be generally expressed as μs = (CΔ)2 f (W ), where C is the model input parameter, f (W ) is a function of the velocity gradients, which depends on the speciﬁc model (see [6, 9, 10] for the relevant expressions), and Δ is the cubic root of the volume of each tetrahedron. In VMS-LES, the eddy-viscosity is computed as a function of the SRS variables, while the deﬁnition of Δ is the same as in classical LES. Finally, the model constant has been set equal to 0.1 for the Smagorinsky model, to 0.5 for WALE and to 0.158 for the Vreman one, and these values are used for both classical and VMS LES simulations. Thus, no adaptation of these free parameters to the VMS approach has been made.

3 Results and discussion The ﬂow over a circular cylinder at Reynolds number (based on the cylinder diameter and on the freestream velocity) equal to 3,900 is simulated. The computational domain is such that −10 ≤ x/D ≤ 25, −20 ≤ y/D ≤ 20 and −π/2 ≤ z/D ≤ π/2, where x, y and z denote the streamwise, transverse and spanwise directions respectively, the cylinder center being located at x = y = 0. Characteristic based conditions are used at the inﬂow and outﬂow as well as on the lateral surfaces. In the spanwise direction periodic boundary conditions are applied and on the cylinder surface no-slip is imposed. The ﬂow domain is discretized by two unstructured tetrahedral grids: the ﬁrst one (GR1) consists of approximately 2.9 × 105 nodes. The averaged distance of the nearest point to the cylinder boundary is 0.017D, which corresponds to y + ≈ 3.31. The second grid (GR2) is obtained from GR1 by reﬁning in a structured way, i.e. by dividing each tetrahedron in 4, resulting in approximately 1.46 × 106 nodes. A large number of simulations were carried out by varying diﬀerent parameters, as, for instance, the SGS model, the value of γs , the grid resolution or the cell agglomeration level for VMS-LES. We report here only the results obtained in some of these simulations. The

218

H. Ouvrard et al. Table 1. Main simulation parameters and ﬂow bulk coeﬃcients. SGS model

Grid

γs

Cd

St

lr

LES LES LES VMS-LES VMS-LES VMS-LES No model No model LES VMS-LES no model

Smagorinsky Vreman WALE Smagorinsky Vreman WALE – – WALE WALE –

GR1 GR1 GR1 GR1 GR1 GR1 GR1 GR1 GR2 GR2 GR2

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3

1.16 1.04 1.14 1.00 1.00 1.03 0.96 0.94 1.02 0.94 0.92

0.212 0.221 0.214 0.221 0.22 0.219 0.223 0.224 0.221 0.223 0.225

0.81 0.97 0.75 1.05 1.07 0.94 1.24 1.25 1.22 1.56 1.85

1.5 LES Smagorinsky

1

LES WALE

0.5 Cpm

b

LES Vreman

VMS − LES Smagorinsky VMS − LES Vreman

0

VMS − LES WALE

−0.5

Experiments

−1 −1.5 −2 0

20

40 60 80 100 120 140 160 180 Angle θ (0 at stagnation point)

Cpm

a

Turb. model

1 0.75 0.5 0.25 0 −0.25 −0.5 −0.75 −1 −1.25 −1.5

VMS−LES WALE LES WALE No model Experiment

0

20

40 60 80 100 120 140 160 180 Angle θ (0 at stagnation point)

Fig. 1. Mean pressure coeﬃcient distribution at the cylinder. (a) Simulations on GR1, (b) Simulations on GR2.

main parameters of the considered simulations are summarized in Table 1, together with some of the obtained ﬂow bulk parameters. The experimental reference value for the mean drag coeﬃcient, Cd , is 0.99 ± 0.05 from [7], which well agrees with those computed in well resolved LES in the literature [4, 5], while for the vortex-shedding Strouhal number, St, values in the range of [0.21, 0.22] are generally obtained. Finally, for the mean recirculation bubble length, a recent experimental and numerical study [8] seems to indicate a reference value of lr = 1.51 ± 10%. Figure 1a shows the mean pressure coeﬃcient distribution at the cylinder obtained on GR1 in LES and VMS-LES simulations, together with experimental data from [7]. From the discrepancy between numerical results and experimental data in the zone of the negative peak it is evident that in all cases the boundary layer evolution is not accurately captured in the simulations, due to the grid coarseness. Another symptom of a too coarse grid resolution (see the discussion in [4]) is the underestimation of the mean recirculation length lr in all the simulations

Diﬀerent Errors in Classical LES and in VMS-LES

a

2

1.5

10

1

9 8

0.5

b2 1.5

219 2 1.8

1

1.6

7

0.5

1.4

0

6 5

0

– 0.5

4

–0.5

0.8

–1

3

–1

0.6

–1.5 –2 –1 –0.5 0 0.5 1 1.5 2 2.5 3

2 1

1.2 1

0.4 –1.5

0.2

–2 –1 –0.5 0 0.5 1 1.5 2 2.5 3

Fig. 2. Instantaneous iso-contours of μs /μ. Simulations on GR1 with the Smagorinsky SGS model: (a) LES, (b) VMS-LES.

on GR1 (Table 1). However, some diﬀerences exist between the LES and the VMS-LES simulations. In particular, in LES the discrepancy observed in the negative peak of mean Cp is larger and the diﬀerences among the diﬀerent SGS models are more pronounced than in VMS-LES. This is due to the fact that the non-dynamic eddy-viscosity models here used, although mainly acting in the wake, also provide a signiﬁcant SGS viscosity in the laminar regions, as the boundary layer and the detaching shear layers (see, e.g., Fig. 2a). In the VMS-LES simulations the spatial distribution of the SGS viscosity is qualitatively similar to that obtained in LES, but the amount is signiﬁcantly reduced everywhere (compare the scales of Fig. 2a, b), and, thus, also in the laminar zones. Moreover, we recall that in the VMS-LES approach the SGS viscosity only acts on the smallest resolved scales. The diﬀerent distribution of SGS viscosity leads in LES to additional inaccuracies, besides those due to grid coarseness and previously discussed, which are not present in VMS-LES. For instance, Fig. 1a shows that the base pressure is inaccurately predicted in all LES simulations except for the Vreman model, leading to an inaccurate value of the mean drag coeﬃcient (Table 1) while for the VMS-LES ones the agreement with the experiments is fairly good. The pressure distribution obtained in the simulations without any SGS model (not shown) is very similar to the one obtained in the VMS-LES ones, as well as for low-order velocity statistics in the wake and for the bulk ﬂow parameters, except than for a signiﬁcantly higher lr given by the no-model simulations (Table 1). This is an a-posteriori conﬁrmation that the used MUSCL reconstruction indeed introduces a viscosity acting only on the highest resolved frequencies [2], as the SGS viscosity in the VMS approach and that this limits its negative eﬀects. Moreover, the results obtained with two diﬀerent(low) values of the parameter γs are also very similar (Table 1), consistently with our previous ﬁndings [2]. As for the results on the reﬁned grid GR2, as expected, in both LES and VMS-LES the agreement with the reference data is improved. However, in LES discrepancies

220

H. Ouvrard et al.

are still observed (see, e.g., Fig. 1b) due to the excessive introduced SGS viscosity, while with VMS-LES a general good agreement is obtained. Note that in this case, although for the pressure distribution, and thus for the drag coefﬁcient, the simulation without any model gives accurate results, the length of the mean recirculation bubble is largely overestimated, due indeed to the lack of SGS viscosity in the wake. Summarizing, our results conﬁrm that the idea of concentrating the SGS viscosity only on the smallest resolved scales actually permits to use simple eddy-viscosity SGS models and to obtain accurate results, comparable to those obtained in the literature with dynamic models. We recall that on unstructured grids this leads to signiﬁcantly more aﬀordable simulations, since the cost of the dynamic procedure may become prohibitive. In the same spirit, it has been conﬁrmed that the fact of also concentrating the numerical dissipation on the smallest resolved scales permits to limit its negative eﬀects.

References 1. M. Breuer. Numerical and modeling on large eddy simulation for theﬂow past a circular cylinder. Int. J. Heat Fluid Flow, 19:512–521, 1998. 2. S. Camarri, M. V. Salvetti, B. Koobus, and A. Dervieux. A low diﬀusion MUSCL scheme for LES on unstructured grids. Comp. Fluids, 33:1101–1129, 2004. 3. B. Koobus and C. Farhat. A variational multiscale method for the large eddy simulation of compressible turbulent ﬂows on unstructured meshes-application to vortex shedding. Comput. Methods Appl. Mech. Eng., 193:1367–1383, 2004. 4. A.G. Kravchenko and P. Moin. Numerical studies of ﬂow over a circular cylinder at re=3900. Phys. Fluids, 12(2):403–417, 1999. 5. J. Lee, N. Park, S. Lee, and H. Choi. A dynamical subgrid-scale eddy viscosity model with a global model coeﬃcient. Phys. Fluids, 18(12), 2006. 6. F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turb. Comb., 62(3):183–200, 1999. 7. C. Norberg. Eﬀects of reynolds number and low-intensity free-sream turbulence on the ﬂow around a circular cylinder. Publ. No. 87/2, Department of Applied Termosc. and Fluid Mech., Chalmer University of Technology, Gothenburg, Sweden, 1987. 8. P. Parneaudeau, J. Carlier, D. Heitz, and E. Lamballais. Experimental and numerical studies of the ﬂow over a circular cylinder at Reynolds number 3900. Phys. Fluids, 20(085101), 2008. 9. J. Smagorinsky. General circulation experiments with the primitive equations. Month. Weath. Rev., 91(3):99–164, 1963. 10. A.W. Vreman. An eddy-viscosity subgrid-scale model for turbulent shear ﬂow: algebraic theory and application. Phys. Fluids, 16:3670–3681, 2004.

Part III

LES Modelling Errors

Practical Quality Measures for Large-Eddy Simulation S.E. Gant Health and Safety Laboratory (HSL), Harpur Hill, Buxton, Derbyshire, UK [email protected]

Abstract Large-Eddy Simulation (LES), Detached-Eddy Simulation (DES) and Scale-Adaptive Simulation (SAS) models are increasingly being used as engineering tools to predict complex ﬂows where reference DNS or experimental data is not available. Frequently, the ﬂow has not been studied previously and the required grid resolution is unknown. Industrial users studying these ﬂows tend to be using commercial CFD codes and do not usually have access to high-performance computing facilities, making systematic grid-dependence studies unfeasible. There is a risk therefore that LES, DES and SAS simulations will be performed using overly coarse grids which may lead to unreliable predictions. The present work surveys a number of practical techniques that provide a means of assessing the quality of the grid resolution. To examine the usefulness of these techniques, a choked gas release in a ventilated room is examined using DES and SAS. The grid resolution measures indicate that overall the grids used are relatively coarse. Both DES and SAS models are found to be in poor agreement with experimental data and show greater grid sensitivity compared to RANS results using the SST model. The work highlights the need for grid-dependence studies and the dangers of using coarse grids.

1 Introduction There has been a steady increase in the use of LES for safety related studies in recent years. This has particularly been the case in the ﬁre safety industry, largely due to the increasing popularity of the Fire Dynamics Simulator freeware [1]. A recent report by the OECD [2] has also recommended that LES is used in assessing nuclear reactor safety. In view of the current and future use of LES in safety critical applications there is a need for practical advice for CFD users on quality and trust issues with LES.

c British Crown copyright (2008). V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 33, c Springer Science+Business Media B.V. 2010

223

224

S.E. Gant

2 Grid resolution measures A number of techniques have been suggested previously to assess the grid resolution in LES. Following Celik et al. [3], these can be classiﬁed into four groups: rules of thumb, techniques based on prior RANS results, single-grid estimators and multi-grid estimators. Assessing the grid resolution using a RANS simulation prior to running a full LES has signiﬁcant advantages since the computer runtime of a RANS simulation is typically an order of magnitude less than the equivalent LES. The RANS results can be used to examine the ratio of the ﬁlter width, Δ, to the integral turbulence length scale, lI = k 3/2 /ε. Recommended values are for the ratio lI /Δ to be above 12 [4, 5]. Prior RANS simulations can also be used to assess near-wall cell sizes in terms of wall units. Single-grid estimators include the ratio of the SGS to the laminar viscosity (νt /ν), the Relative Eﬀective Viscosity Index [3], the ratio of the cell size to the Taylor microscale [6], the Subgrid Activity Parameter [7], the ratio of resolved to the total turbulent kinetic energy, kres /ktot , and analysis of power spectra. Recommended values of these parameters and examples of their use in the literature are given by Gant [8]. Multi-grid estimators include the “Index of Quality”, LES IQk of Celik et al. [9] and the systematic grid and model variation approach of Klein [10]. The former is based on Richardson extrapolation on the resolved turbulent kinetic energy, while the latter involves running three LES calculations: a standard LES, a coarse-grid LES and an LES with the SGS model constant modiﬁed. Each of the above grid resolution measures has advantages and limitations. The RANS-based techniques are reliant upon the accuracy of the RANS model which may be a limiting factor in massively separated ﬂows, for example. If the ﬂow involves regions that are laminar, the turbulent length scale ratios from RANS simulations also become meaningless indicators. Celik et al. [9] noted that in most applied LES studies the turbulent viscosity is signiﬁcantly larger than the molecular viscosity (νt ν), and the subgrid activity parameter will therefore nearly always be close to unity. The fundamental problem with methods involving the resolved turbulent kinetic energy is that whereas kres might be expected to increase as the grid is reﬁned, it has been shown in mixing layers, jets and wakes [9, 10] that kres can actually decrease. This implies that the resolved turbulent kinetic energy is not a reliable indicator of grid resolution. A number of the above approaches do not account for the numerical dissipation which in many situations is of the same order or even larger than the modelled dissipation. Turbulence spectra cannot easily be plotted over the entire ﬂow ﬁeld to produce, for example, contour plots of grid resolution quality. Furthermore, the slope of the concentration or temperature spectra is modiﬁed in buoyant ﬂows, and in low Reynolds number ﬂows a distinct inertial subrange may not exist. Finally, Brandt [11] has shown that multi-grid approaches based on Richardson extrapolation can produce misleading results due to the grid resolution not being within the asymptotic range.

Practical Quality Measures for Large-Eddy Simulation

225

3 Application A continuous jet of methane gas released into a ventilated enclosure is modelled. The conﬁguration was previously examined by HSL to assess the implications of new hazardous area classiﬁcation legislation. The main interest in studying the ﬂow is to predict the size of the ﬂammable gas cloud. The room has internal dimensions 4 × 4 × 2.92 m with two ventilation inlets and two outlets. The ventilation rate is 12 ach and the gas is released in one corner of the room at a rate of 0.86 g/s. A description of the experiments undertaken for this work is given by Ivings et al. [12]. Four diﬀerent turbulence treatments are tested: steady RANS, unsteady RANS, SAS and DES. LES calculations could not be performed because of the nature of the high-speed jet ﬂow. The steady and unsteady RANS calculations used the SST model and the DES model was the SST-based version of Strelets [13]. Both the SAS and DES models used a spatial discretization scheme that switched from upwind biased second-order to central diﬀerencing in regions where ﬂow unsteadiness was resolved. All calculations were performed using the commercial code ANSYS-CFX11. Three unstructured computational grids were tested: coarse, medium and ﬁne, comprising 224,000, 412,000 and 660,000 nodes respectively. Further details of the CFD model are given by Gant [8]. Figure 1 shows the predicted streamlines from a steady RANS simulation. The gas jet is conﬁned within a narrow cavity in a corner of the room where the ﬂow recirculates, giving rise to locally high gas concentrations. The buoyant gas then rises as a plume towards the ceiling. Figure 2a shows the ratio of the integral turbulence length scale to the cell size, lI /Δ, for the ﬁne grid. The ratio is highest in the plume region with the ﬁne mesh where it has a maximum value of 10. Elsewhere in the room the ratio falls to around 5 and in the predominantly laminar regions a value of one or less. This implies that even the ﬁne grid is relatively coarse for LES.

Fig. 1. Streamlines coloured by mean velocity.

226

S.E. Gant b

a 10.0

1.0 0.9

8.0

0.8 0.7

6.0

0.6 0.5

4.0

0.4 0.3

2.0

0.2

0.0

0.0

0.1

Fig. 2. (a) Ratio of the integral length scale to the cell size calculated using steady RANS on the ﬁne grid (left); (b) ratio of resolved to total turbulent kinetic energy for the DES (right).

The ratio of the resolved to the total turbulent kinetic energy (kres /ktot ) is shown for the ﬁne-grid DES in Fig. 2b. The value of ktot is calculated from the sum of the resolved and the modelled components, kres and kmod . The results show that the DES model resolves 70 to 90% of the turbulence energy in the majority of the plume region which suggests that the grid resolution is reasonable. Broadly similar results were obtained for the SAS model. The ratio of the SGS to the laminar viscosity (νt /ν) for the DES model on the ﬁne grid is around 15 to 40 in the plume region. Given that the turbulent Reynolds number here is around 2000, these value are high compared to the target value of around 20 suggested by Celik et al. [9]. In the DES calculations, the subgrid activity parameter was close to one across the majority of the room and showed relatively little sensitivity to the grid resolution. The power spectral density based on the concentration ﬂuctuations in the plume for the DES results showed the spectra decaying at high frequencies faster than the −5/3 power law with no clearly discernible inertial subrange. This appeared to be related to the relatively small separation of turbulent length scales and the inﬂuence of buoyancy eﬀects. The “Index of Quality”, LES IQk , based on Richardson extrapolation of the turbulent kinetic energy between the medium and ﬁne grids produced values generally above 60% in the plume region although its value ﬂuctuated signiﬁcantly between neighbouring cells due to the non-smooth distribution of the ratio of cell sizes. At various points in the ﬂow the LES IQk value either exceeded a value of 100% or became negative. Mean gas concentrations at 14 points in the room were recorded in the experiments. Figure 3a shows the discrepancy between the CFD and the measured values averaged over all the measurement positions. For the ﬁne grid, the DES and SAS results have more than double the error of the URANS or RANS model predictions. The error diminishes as the grid is reﬁned in the RANS results, whereas with the DES and SAS models the reverse trend is produced with the greatest error on the ﬁnest grid.

Practical Quality Measures for Large-Eddy Simulation

0.4 0.2

ES oar se M ed D ium E SA S F in S e SA Coa rs S e M ed iu U SA m R AN S F in S U R C e AN oa rs S e M e U R diu AN m R AN S F S ine R AN Coa rs S e M e R diu AN m S Fi ne

0

C ES D

D

ES D

oa rs e M ed iu m D E SA S F in S SA Co e S ars M ed e iu U SA m R AN S F in S U e R AN Coa rs S e M e di U um R A R NS AN Fi n S C e R AN oa rs S e M e R diu AN m S Fi ne

0.6

ES

0.8

0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

D

1

b

Vz Volume (m 3)

1.2

C

Average Error in Predicted Gas Concentration (% vol/vol)

a

227

Fig. 3. (a) Average error between the CFD and experimental mean gas concentrations (left); (b) mean Vz gas cloud volumes and their 95% conﬁdence intervals (right).

The predicted gas cloud volumes deﬁned using the Vz criterion are shown in Fig. 3b. The statistical uncertainty in the mean values is indicated by error bars which show the 95% conﬁdence intervals. One of the claimed advantages of the SAS model is that it is less grid dependent than LES or DES since the model equations do not rely explicitly on the size of the grid cell. However, in the results shown in Fig. 3b, the SAS model exhibits the greatest grid sensitivity of the four models tested.

4 Discussion and conclusions Overall, the various grid quality measures tested here indicate that the spatial resolution was barely adequate for LES. Some of the grid quality measures could not provide useful information in regions of the ﬂow where the turbulence levels were very low. This was particularly an issue for the RANS-based approaches in laminar regions of the ﬂow and the power spectral density in the plume. The ratio of the modelled to the total turbulent kinetic energy appeared to show that the grid resolution was reasonable, but this may have been overly optimistic since it did not take account of any numerical dissipation. The LES “Index of Quality” did not provide useful information and produced values that were less than zero and greater than 100%. The relatively poor performance of the DES and SAS models is likely to have been a consequence of using overly coarse grids. It would have been useful to undertake simulations with ﬁner grids to establish that DES predictions could produce accurate results. However, the ﬁne-grid DES calculations took 134 CPU-days compared to around 8 CPU-days for the equivalent steady RANS calculation. The cost of undertaking well-resolved DES should not be underestimated, especially for industrial ﬂows. Although the SAS and DES models may be considered appropriate for modelling ﬂows exhibiting large-scale oscillatory behaviour, the present work has shown that on relatively coarse grids these models can produce less reliable predictions and results that are more grid-dependent than a standard RANS model.

228

S.E. Gant

References 1. A. Hamins, A. Maranghides, K. B. McGrattan, T. Ohlemiller, and R. Anleitner. Experiments and modeling of multiple workstations burning in a compartment. Technical report, NIST NCSTAR 1-5E, Federal Building and Fire Safety Investigation of the World Trade Center Disaster, September 2005. 2. J. Mahaﬀy, B. Chung, F. Dubois, F. Dubois, E. Graﬀard, M. Heitsch, M. Henriksson, E. Komen, F. Moretti, T. Morii, P. M¨ uhlbauer, U. Rohde, M. Scheuerer, B. L. Smith, C. Song, T. Watanabe, and G. Zigh. Best practice guidelines for the use of CFD in nuclear reactor safety applications. Technical report, NEA/CSNI/R(2007)5, OECD, 2007. 3. I. B. Celik, M. Klein, M. Freitag, and J. Janicka. Assessment measures for URANS/DES/LES: an overview with applications. J. Turbulence, 7(7):1–27, 2006. 4. Y. Addad, S. Benhamadouche, and D. Laurence. The negatively buoyant walljet: LES results. Int. J. Heat Fluid Flow, 25:795–808, 2004. 5. K. Van Maele and B. Merci. Application of RANS and LES ﬁeld simulations to predict the critical ventilation velocity in longitudinally ventilated horizontal tunnels. Fire Safety Journal, 43:598–609, 2008. 6. A. K. Kuczaj and E. M. J. Komen. Large-eddy simulation study of turbulent mixing in a T-junction. In Proc. Experiments and CFD Code Applications to Nuclear Reactor Safety, (XCFD4NRS), CEA, Grenoble, France, September 2008. 7. B. J. Geurts and J. Fr¨ ohlich. A framework for predicting accuracy limitations in large eddy simulation. Phys. Fluids, 14:41–44, 2002. 8. S. E. Gant. Quality and reliability issues with large-eddy simulation. Report RR656, Health and Safety Executive, UK, 2008. 9. I. B. Celik, Z. N. Cehreli, and I. Yavuz. Index of resolution quality for large eddy simulations. J. Fluids Eng., 127:949–958, 2005. 10. M. Klein. An attempt to assess the quality of large eddy simulations in the context of implicit ﬁltering. Flow, Turbulence and Combustion, 75:131–147, 2005. 11. T. Brandt. Study on numerical and modelling errors in LES using a priori and a posteriori testing. PhD thesis, Department of Mechanical Engineering, Helsinki University of Technology, Espoo, Finland, 2007. 12. M. J. Ivings, S. P. Clarke, S. E. Gant, B. Fletcher, A. Heater, D. J. Pocock, D. Pritchard, R. Santon, and C. J. Saunders. Area classiﬁcation for secondary releases from low pressure natural gas systems. Report RR630, Health and Safety Executive, UK, 2008. 13. M. Strelets. Detached eddy simulation of massively separated ﬂows. In Paper AIAA 2001-0879, Reno, Nevada, 2001. 39th Aerospace Science Meeting & Exhibit.

The Simplest LES M. Germano Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Torino, Italy, [email protected]

Abstract The simplest large eddy simulation of a turbulent ﬁeld is that produced by the two-point average, and it is intimately related to the two-point diﬀerence, basic ingredient of the small-scale turbulence theory, see Sreenivasan and Antonia for a recent review. This simple average is a useful tool to understand some peculiar aspects of LES and the duality between the two-point average and the two-point diﬀerence is important as regards the elementary interaction between large and small scales. In this paper the simplest LES is applied to the case of a passive scalar convected by a turbulent velocity ﬁeld and an equivalent form of the Yaglom inertial law is derived.

1 Introduction The motivations of this paper are twofold. The ﬁrst is related to the better knowledge of the elementary ingredients of the large eddy simulation, the subgrid ﬂuxes and the subgrid variance production. In this paper we will consider the simple two-point average applied to the passive scalar a a ¯(x, Δ, t) =

a(x + Δ, t) + a(x − Δ, t) 2

(1)

and we will see that the simple and signiﬁcative properties of this average are intimately related to the properties of the two-point diﬀerence a , deﬁned here for symmetrical reasons as a (x, Δ, t) =

a(x + Δ, t) − a(x − Δ, t) 2

(2)

and that is the second motivation of our study. Since the fundamental paper of Kolmogorov [5] the statistical properties of this quantity have been explored in great detail both in the case of a velocity ﬁeld and in the case of a passive scalar. The study of the inertial cascade, the study of the intermittency, the scaling theory of the structure functions are well known products V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 34, c Springer Science+Business Media B.V. 2010

229

230

M. Germano

of such approach. Strangely enough the study of the statistical properties of the simplest LES, the two-point sum, has not received a similar attention. One simple reason for that seems obvious : the idea that the inertial turbulence could be universal in terms of the increments remains the starting point of a well founded theory. It is more and more evident, however, that this universality cannot be so extended as previously postulated. In particular some inconsistencies and paradoxes related to the assumed independence of the two-point velocity diﬀerence and the two-point velocity sum have been recently evidenced by Frisch et al. [6], and Hosokawa [7]. It is clear that in order to better understand all that we need more information on the twopoint sum, the simplest LES, in particular as regards its relations with the two-point diﬀerence, and in this paper we try to consider them together, the elementary dual faces of the turbulence representation.

2 The two-point sum and diﬀerence operators Let us introduce the two-point sum operator S and the two-point diﬀerence operator D deﬁned as S[a(x, t)] = a ¯(x, Δ, t) =

a 1 + a2 a(x + Δ, t) + a(x − Δ, t) = 2 2

a 1 − a2 a(x + Δ, t) − a(x − Δ, t) = 2 2 where the overline and the apex stand for abbreviated writings. Important properties of these operators are the following D[a(x, t)] = a (x, Δ, t) =

a 1 b 1 + a2 b 2 =a ¯¯b + a b 2 a 1 b 1 − a2 b 2 =a ¯b + ¯ba (ab) = 2 a 1 b 1 c1 + a2 b 2 c2 =a ¯¯b¯ c + a b c¯ + b c a abc = ¯ + c a¯b 2 a 1 b 1 c1 − a2 b 2 c2 = a b c + a (abc) = ¯¯bc + ¯b¯ ca + c¯a ¯ b 2 ab =

(3)

and they commute with the time and space derivatives and with the statistical operator. Other important properties related to the derivatives in the physical space x and the scale space Δ are the following ∂¯ a ∂a ∂a ∂¯ a = ; = ∂xi ∂Δi ∂xi ∂Δi ∂2a ¯ ¯ ∂2a ∂ 2 a ∂ 2 a = ; = ∂xi ∂xj ∂Δi ∂Δj ∂xi ∂xj ∂Δi ∂Δj and in the particular case of homogeneous turbulence

(4)

The Simplest LES

a1 = a2 ;

a1 b1 = a2 b2 ;

231

a1 b1 c1 = a2 b2 c2

(5)

abc = abc

(6)

we have ¯ a = a ;

ab = ab ;

and a = 0 ab + ¯ba = 0 (ab) = ¯ a¯bc + ¯b¯ ca + ¯ ca ¯ b = 0 (abc) = a b c + ¯ (7) where the angled brackets stand for the ensemble average. It is worth noting that the two-point sum and the two-point diﬀerence are uncorrelated in the case of homogeneous turbulence ¯ a a = 0

(8)

but that does not mean that they are independent. We have a a a = −3¯ aa ¯ a

(9)

and as remarked by Hosokawa [7] that poses a lot of questions when applied to the longitudinal velocity diﬀerence in the inertial range as regards the reﬁned similarity hypothesis of Kolmogorov [8]. The same considerations can be extended to higher order moments, and here we will remark that in the case of homogeneous turbulence many other statistical relations between the two-point sum and the two point diﬀerence can be derived. We note that for incompressible ﬂows we have ∂un ∂un ∂un ∂u ¯n ∂u ¯n = = = = =0 ∂xn ∂xn ∂Δn ∂xn ∂Δn

(10)

and owing to the fact that in the case of homogeneous turbulence the statistical averages are space independent we also have ∂abun ∂(ab) un = =0 ∂Δn ∂Δn

(11)

In particular from the relation ∂ui ui un =0 ∂Δn

(12)

¯i un ∂ui ui un ∂¯ ui u =− ∂Δn ∂Δn

(13)

we can derive, see Germano [3]

232

M. Germano

3 The two-point LES of a passive scalar Let us now write the balance equations for a passive scalar a convected by an incompressible turbulent velocity ﬁeld ui ∂a ∂(aun ) ∂2a + =ν ∂t ∂xn ∂xn ∂xn

∂un =0 ∂xn

;

(14)

If we ﬁlter these equations we have ¯ au ¯n + τn ) ∂2a ∂¯ a ∂(¯ + =ν ∂t ∂xn ∂xn ∂xn

;

∂u ¯n =0 ∂xn

(15)

where τn is the subgrid ﬂux τn = aun − a ¯u ¯n

(16)

and we can also derive the following equation for the subgrid variance σ un + λn ) ∂2σ ∂σ ∂(σ¯ + =ν +P −ϑ ∂t ∂xn ∂xn ∂xn

(17)

σ = aa − a ¯a ¯

(18)

λn = aaun − aa¯ un − 2¯ aτn

(19)

where λn is the subgrid variance ﬂux

P is the subgrid variance production P = −2τn

∂¯ a ∂xn

(20)

ϑ is the subgrid variance dissipation ϑ = εa − 2ν

a ∂¯ a ∂¯ ∂xn ∂xn

(21)

and where εa is the molecular dissipation deﬁned as εa = 2ν

∂a ∂a ∂xn ∂xn

(22)

Simple calculations show that in the case of the simplest LES given by a ¯= we have

a(x + Δ) + a(x − Δ) 2

;

τn = a un

u¯n =

;

un (x + Δ) + un (x − Δ) 2

σ = a a

(23)

The Simplest LES

λn = 0 ϑ=

P =−

;

233

∂(a a un ) ∂Δn

εa ν ∂2σ ν ∂2σ + − 2 2 ∂xn ∂xn 2 ∂Δn ∂Δn

(24)

where a and un are the associated two point diﬀerences given by a =

a(x + Δ) − a(x − Δ) 2

;

un =

un (x + Δ) − un (x − Δ) 2

(25)

4 The simplest LES of a passive scalar in the case of a homogeneous turbulent ﬁeld Passive scalars have been largely studied in the case of homogeneous turbulent ﬁelds, and it is interesting to evaluate the variance σ captured by the simplest LES in this case. It is easy to see that in the case of homogeneous ﬂows we have aa = aa

;

εa = εa

aa Caa (2Δ) − 2 2 where the correlation Caa (2Δ) is given by σ = a a =

(26)

Caa (2Δ) = a(x + Δ)a(x − Δ)

(27)

We see that when Δ is larger than the integral length the variance captured by the two-point average goes statistically to the limiting value of aa/2, and as regards the balance equations we remark that in the homogeneous case we can write ∂a a un ∂¯ aa ¯un P =− = (28) ∂Δn ∂Δn so that we ﬁnally obtain ∂a a ∂a a un εa ν ∂ 2 a a + + =− ∂t ∂Δn 2 2 ∂Δn ∂Δn

(29)

We notice that for steady turbulence and in the inertial limit we recover both the well known relation of Yaglom [4] recently applied by Lindborg and Cho [9] to the determination of temperature and ozone variance in the lower stratosphere ∂a a un εa (30) =− ∂Δn 2

234

M. Germano

and the equivalent relation ∂¯ aa ¯un εa = ∂Δn 2

(31)

This last equation can be read as an equivalent form of the Yaglom inertial law for a passive scalar, and it is similar to the generalized inertial law for the turbulent velocity ﬁeld determined previously by the author [3].

5 Conclusions In this paper we have examined the simplest ﬁltering operator applied to the balance equation for a passive scalar : the two-point average in space, function both of the mean position x and the distance Δ between the two points selected. The properties of the two-point averages have been examined and have been related to the properties of the two-point diﬀerences. One important point is that in the case of homogeneous turbulence the two-point averages and the two-point diﬀerences are uncorrelated, but they are not independent. The two-point variance and the two-point subgrid ﬂux of a passive scalar are directly related to the two-point scalar and velocity diﬀerences, and the twopoint subgrid ﬂux of the variance of a passive scalar is zero. The two-point subgrid production of the variance of a passive scalar can be expressed as a ﬂux in the scale space and ﬁnally for homogeneous turbulence an equivalent form of the Yaglom inertial law has been derived. This last equation is similar to the generalized inertial Kolmogorov law for the turbulent velocity ﬁeld determined previously by the author [3].

References 1. Germano, M. (2007). The two point average and the related subgrid model. In: Turbulence and Shear Flow Phenomena TSFP-5. Conference Proceedings 1: 309–314 2. Sreenivasan, K. R., & Antonia, R. A. (1997). The phenomenology of small scale turbulence. Ann. Rev. Fluid Mech. 29, 435–472 3. Germano, M. (2007b). The elementary energy transfer between the two-point velocity mean and diﬀerence. Phys. Fluids 19, 085105 4. Yaglom, A. M. (1949). On the local structure of a temperature ﬁeld in a turbulent ﬂow. Dokl. Akad. Nauk SSSR 69, 743 5. Kolmogorov, A. N. (1941). Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 6. Frisch, U., Bec, J., & Aurell, E. (2005). Locally homogeneous turbulence: Is it an inconsistent framework ? Phys. Fluids 17, 081706 7. Hosokawa, I. (2007). A paradox concerning the reﬁned similarity hypothesis of Kolmorogov for isotropic turbulence. Prog. Theor. Phys. 118, 169–173

The Simplest LES

235

8. Kolmogorov, A. N. (1962). A reﬁnement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible ﬂuid at high Reynolds number. J. Fluid Mech. 13, 82–85 9. Lindborg, E., & Cho, J. Y. N. (2000). Determining the cascade of passive scalar variance in the lower stratosphere. Phys. Rev. Lett. 85, 5663–5666

Application of an Anisotropy Resolving Algebraic Reynolds Stress Model within a Hybrid LES-RANS Method M. Breuer, O. Aybay, and B. Jaﬀr´ezic Dept. of Fluid Mechanics, Institute of Mechanics, Helmut-Schmidt-University Hamburg, Holstenhofweg 85, D-22043 Hamburg, Germany, [email protected]; [email protected]; [email protected] Abstract In the present study a hybrid LES–RANS approach based on an explicit algebraic Reynolds stress model (EARSM) is suggested and evaluated. The model is applied in the RANS mode with the aim of accounting for the Reynolds stress anisotropy emerging especially in the near–wall region. Requiring solely the solution of one additional transport equation for the turbulent kinetic energy, the additional computational eﬀort is marginal. A dynamically predicted interface location avoids the predeﬁnition by the user and adjusts itself to the ﬂow situation. As a test case including ﬂow separation and reattachment, the ﬂow over a periodic arrangement of hills is used.

1 Introduction In principle, industry is willing to apply simulation methodologies which go beyond the capabilities of the well-known RANS approach. That is especially true for complex ﬂow phenomena for which RANS evidently suﬀers from reliability, e.g., large-scale separation/reattachment or vortex shedding. For such ﬂows LES is obviously the better choice. However, engineering applications are in most cases wall-bounded ﬂows, whereby wall-resolved LES requires very ﬁne near-wall grid resolutions. In order to avoid those at least in both wall-tangential directions, the idea of combining LES with RANS applied in the near-wall region was borne some years back. In the meantime, a variety of diﬀerent hybrid LES-RANS concepts were proposed with partially diﬀerent objectives but all aiming at a reduction of CPU costs compared to LES. What are the challenges for modeling the near-wall region appropriately? Close to a wall, large anisotropies of the Reynolds stresses are observed. Thus, a full Reynolds stress model (RSM) is obviously the best choice. However, RSM requires the solution of at least seven additional partial diﬀerential equations. Furthermore, these equations often turn out to be stiﬀ. Hence V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 35, c Springer Science+Business Media B.V. 2010

237

238

M. Breuer et al.

the computational eﬀort for RSM is too expensive to be used within a hybrid technique. A much cheaper alternative is found by an explicit algebraic RSM which avoids the solution of transport equations for each component. Instead algebraic relations are applied to account for the anisotropy emerging especially in the near-wall region. Such an anisotropy-resolving closure is implemented and evaluated in the present hybrid LES-URANS approach.

2 Hybrid LES-URANS methodology The hybrid method proposed relies on the idea to apply RANS or more speciﬁcally URANS in those regions, where statistical turbulence models in general perform properly. Moreover, LES is used in regions, where large unsteady vortical structures are present, which should be resolved directly. Thus URANS is used for the near-wall region, whereas LES is performed in the remaining computational domain. Overall this approach strongly reduces the resolution requirements since the application of RANS/URANS for the prediction of attached boundary layers allows to dramatically decrease the near-wall resolution with the exception of the wall-normal direction. This raises the hope that hybrid LES-URANS techniques can be used with acceptable eﬀort even for high-Re ﬂows encountered in technical applications. When setting up a hybrid approach, several questions have to be answered. The main ones are: (1) Which models should be used in the URANS and LES regions? (2) How should the LES-URANS interface be deﬁned? (3) How should both regions be coupled? In order to take the anisotropy of the Reynolds stresses in the near-wall region reasonably into account, the explicit algebraic Reynolds stress model (EARSM) proposed by Wallin and Johansson [11] is used here. It represents a compromise between the too expensive full RSMs and classical linear eddyviscosity models (LEVM) based on the Boussinesq approximation relating the shear stress component to the mean velocity gradient. The latter are not capable to account for the stress anisotropy. For the implementation into a CFD code, the EARSM can be formally expressed in terms of a non-linear eddy-viscosity model (NLEVM). Its extra computational eﬀort is small, still requiring solely the solution of one additional transport equation for the turbulent kinetic energy. The non-closed stress tensor in the Reynolds-averaged momentum equations is written as [11]: 2 (ex) eff δij − 2 Cμ S ij + aij (1) ui uj mod = kmod 3 Here, aij = f (S 2 , Ω 2 , S n Ω m , f1 , . . .) represents an extra tensor which takes the anisotropy of the stresses into account and is computed explicitly based on the mean strain S ij and rotation tensors Ω ij normalized by the turbulent time scale. As shown in [11] the model obtains the correct near-wall behavior. (ex)

Hybrid LES-RANS Method

239

The value of Cμ within the relation for the eddy viscosity νt = Cμ · vc · lc is not a constant but dynamically calculated, thus denoted Cμeff . However, the EARSM itself is not complete since the length scale lc and the velocity scale vc are not deﬁned. These have to be supplied by an additional scale-determining part. In principle, that can be achieved by a two-equation model such as a classical k- or k-ω model, which in the context of RANS is a natural choice since one transport equation is solved for the velocity scale and one for the length scale. However, for the present hybrid methodology a more or less unique modeling strategy in both regions has several advantages. Since in LES the length scale is naturally given by the ﬁlter width Δ, a one-equation model with a transport equation for the velocity scale is preferred. Thus the length scale in the URANS region has to be deﬁned by an algebraic relation. The resulting strategy consists of a single transport equation for the modeled turbulent kinetic energy kmod = kRAN S = vc2 in RANS mode and the subgridscale (SGS) turbulent kinetic energy kmod = kSGS in LES. This transport equation includes on the right-hand side the terms P, D, and , where P = −(ui uj )mod ∂U i /∂xj represents the production term closed by relation (1). Thus, in the non-linear model for the RANS zone P is determined on the basis of the consistent Reynolds stress formulation including the anisotropy term which compared to the originally applied linear model [4] improves the production term and subsequently kmod . Note that the extension to EARSM is actually not used in the LES mode. The unknown diﬀusion term D can be closed by a classical gradient-diﬀusion approach as done for the LES zone and previously also for the RANS zone [4]. However, for EARSM the enhanced representation of the Reynolds stresses can be introduced into D by applying the diﬀusion model of Daly and Harlow [7] which is preferred here. Finally, the dissipation rate needs to be deﬁned. For the LES zone the dissipation rate is 3/2 set to = Cd kSGS /Δ yielding the one-equation SGS model of Schumann [10]. For the URANS zone the formulation of can be overtaken from the near-wall one-equation model by Rodi et al. [9] or by Chen and Patel [6] both relying on algebraic relations for the length scales, where the former has the disadvantage that it is only applicable for y ∗ ≤ 60. Thus the latter is preferred. Based on previous experiences [4, 5] the predeﬁnition of LES and URANS regions is avoided in the present approach and a gradual transition between both methods is assured. The dynamic interface criterion [4] relies on the modeled turbulent kinetic energy and the wall distance leading to 1/2 y ∗ = kmod · y/ν ≤ Cswitch,y∗ . For y ∗ ≤ Cswitch,y∗ the method works in URANS mode, otherwise the LES mode is applied. In principle, the constant Cswitch,y∗ allows the user to vary the average interface position and to study its eﬀect. Typically, it should be set so that the interface is located in the logarithmic wall layer. However, it is worth to mention that the instantaneous interface position strongly varies in space and time depending on the local ﬂow ﬁeld close to the wall. Thus it relies upon physical quantities accounting for characteristic ﬂow properties and automatically reacts on dynamic ﬂow

240

M. Breuer et al.

ﬁeld variations. This interface criterion denoted version A partially provides no sharply delimited LES-URANS regions. Therefore, an enhanced version guaranteeing a sharp interface without RANS islands (denoted version Asi) was also taken into account, see, e.g., [4, 5]. This special interface treatment yields useful information whether the RANS islands found for version A inﬂuence the results or not. Presently, an interface treatment without synthetic or reconstructed turbulence is preferred in order to assess the method in its simplest form.

3 Numerical method and test case The LES code LESOCC used for the solution of the ﬁltered Navier-Stokes equations, is a 3-D ﬁnite-volume solver for arbitrary non-orthogonal and nonstaggered grids [1]. The spatial discretization of all ﬂuxes is based on a central scheme of second-order accuracy. A low-storage multi-stage Runge–Kutta method is applied for time-marching. The additional transport equation for kmod is discretized with the same scheme. Beside the hybrid approach diﬀerent SGS models for LES are implemented as well as DES. These features are used to provide reference data for comparison with the hybrid method. The emphasis of the present study is on separated ﬂows including largescale structures at which hybrid LES-URANS methods are mainly aiming at. For that purpose the ﬂow in a channel with streamwise periodic constrictions at Reb = 10, 595 (based on the bulk velocity Ub and the hill height h) [8] presents a geometrically simple but still challenging test case. The solution used to evaluate the hybrid simulations is a highly wall-resolved LES [2, 3] denoted WR-LES. In the present study, hybrid simulations are performed on two diﬀerent grids; grid A consists of 160 × 100 × 60 ≈ 1.0 × 106 CVs and grid B consists of 80 × 100 × 30 ≈ 0.25 × 106 CVs in streamwise, wall-normal (Δycrest /h = 5.0 · 10−3 , 1st CV height) and spanwise directions. Grid B is extracted from grid A by retaining every second grid point in streamwise and spanwise directions. The ratios of grid points in relation to the reference case WR-LES are about 13 and 52, respectively.

4 Results and conclusions In the following, four diﬀerent issues are addressed. I. Comparison of the hybrid technique based on LEVM vs. EARSM The evaluation relies on grid A, the interface criterion A with Cswitch,y∗ = 60, and the formulation of by Rodi et al. [9]. Figure 1 depicts results of the linear (denoted L) and the non-linear model (denoted nL). Regarding the proﬁles of the mean streamwise velocity U (Fig. 1a) a better agreement with the

Hybrid LES-RANS Method

a

b

3 2.5

L nL WR-LES

3 2.5 2

y/h

y/h

2 1.5

1.5

1

1

0.5

0.5

0

241

0 0

2

4

6

8

10

0

12

1

3.5 · U/Ub + x/h

2

3

4

5

6

7

8

9

25 · v v /Ub2 + x/h

d

c 3

0.012

2

0.01

τ wall

y/h

0.014 2.5

1.5

L nL WR-LES

0.008 0.006 0.004

1

0.002

0.5

0 0

–0.002 0

1

2

3

4

5

6

−25 · u v /Ub2 + x/h

7

8

9

0

1

2

3

4

5

6

7

8

9

τwall = f (x/h)

Fig. 1. Comparison of the hybrid technique based on LEVM vs. EARSM. (a) Mean streamwise velocity; (b) Wall–normal Reynolds stress; (c) Reynolds shear stress; (d) Wall shear stress at the lower wall.

reference data WR-LES is visible for nL, especially in the free shear layer and in the vicinity of the upper wall. Otherwise the results of both L and nL are in good accordance with WR-LES. Concerning the normal Reynolds stress u u (not shown here) a very good agreement between nL and WR-LES is found. Deviations between L and WR-LES are visible close to the upper wall and in the shear layer. The diﬀerences between L and nL are even more obvious for the second normal Reynolds stress v v (Fig. 1b), where L yields strange peaks in the vicinity of the upper wall. In the recirculation region both hybrid variants deliver an underprediction of v v with respect to WR-LES. The shear stress component u v (Fig. 1c) predicted by nL shows a very good agreement with WR-LES, whereas large deviations are observed for L in the shear layer. Finally, Fig. 1d depicts the wall shear stress distribution τw along the lower wall. For L a strong non-physical peak is found at the hill crest. This behavior was also noticed in pure RANS simulations and is caused by inaccurate predictions of the modeled Reynolds stresses. This disagreement

242

M. Breuer et al.

with WR-LES completely disappears for nL. Moreover, nL oﬀers a better prediction of τw along the entire wall. The separation and reattachment regions are reasonably predicted. Thus, the overall accuracy of the hybrid method is strongly enhanced by nL (EARSM) compared to L. II. Inﬂuence of the interface position The evaluation relies on grid A and the non-linear model based on the diﬀusion model by Daly and Harlow [7] and the dissipation formulation by Chen and Patel [6]. The interface criterion A is applied and the position is varied between Cswitch,y∗ = 60, 160, and 200. Figure 2a depicts the mean streamwise velocity U . Overall a good agreement between nL and WR-LES is observed independent of the interface position. That in general also applies for the normal Reynolds stresses (not shown here), where the results slightly improve for the larger Cswitch,y∗ values. The satisfactory agreement for all interface positions is also visible for the Reynolds shear stress component depicted in Fig. 2b. The same trend is found for the wall shear stress distribution (not shown here). Thus as desired, the results do not strongly depend on the interface position. III. Inﬂuence of the interface criterion As mentioned, the interface criterion A sometimes leads to RANS islands within the LES region. These islands appear irregularly and only survive a certain short time interval. Therefore, the question arises whether these islands are really critical or not. For that purpose the interface criterion Asi guaranteeing a sharp interface without RANS islands was installed [4,5]. The evaluation carried out on grid A using nL and the interface position Cswitch,y∗ = 200 yields nearly no diﬀerences between the results using the interface criteria A and Asi. Hence it can be concluded that the RANS islands do not disturb the overall solution signiﬁcantly. Therefore, it is not necessary to get rid of them by the time-consuming sorting algorithm applied within Asi.

a

b

3

3 2.5 2

y/h

y/h

y* = 60 2.5 y* = 160 y* = 200 2 WR–LES 1.5

1.5

1

1

0.5

0.5 0

0 0

2

4

6

8

3.5 · U/Ub + x/h

10

12

0

1

2

3

4

5

6

7

8

−25 · u v /Ub2 + x/h

Fig. 2. Comparison of the hybrid technique for diﬀerent interface positions.

9

Hybrid LES-RANS Method

243

IV. Inﬂuence of the grid resolution Finally, simulations based on nL (interface criterion A, Cswitch,y∗ = 200, D by Daly and Harlow [7], by Chen and Patel [6]) are carried out on grid A and B. The comparison with WR-LES shows that the results are nearly independent regarding the streamwise and spanwise resolution, which both diﬀer between grid A and B by a factor of 2. The ﬁne-grid results are marginally better than the coarse-grid predictions. However, overall a very good agreement with WRLES (e.g. for the separation and reattachment length) is found also on the coarse grid using only a quarter of a million grid points.

Acknowledgment ‘LES for Complex Flows’ (FOR 507), DFG BR 1847/8.

References 1. Breuer, M. (1998). Large-Eddy Simulation of the Sub-Critical Flow Past a Circular Cylinder: Numer. & Mod. Aspects, Int. J. Num. Meth. Fluids, 28, 1281–1302. 2. Breuer, M. (2005) New Reference Data for the Hill Flow Test Case, http://www.hy.bv.tum.de/DFG-CNRS/. 3. Breuer, M., Peller, N., Rapp, Ch., Manhart, M. (2008) Flow over Periodic Hills – Numer. & Exper. Study in a Wide Range of Re, Comp. & Fluids, 38, 433–457. 4. Breuer, M., Jaﬀr´ezic, B., Arora, K. (2008) Hybrid LES–RANS Technique Based on a One-Eq. Near-Wall Model, J. Theo. Comput. Fluid Dyn., 22, 157–187. 5. Jaﬀr´ezic, B., Breuer, M. (2008) Application of an Explicit Algeb. Reynolds Stress Model within an Hybrid LES–RANS Method, J. Flow Turb. Comb., 81, 415–448. 6. Chen, H.C., Patel, V.C. (1988). Near–Wall Turbulence Models for Complex Flows Including Separation, AIAA Journal, 26, 641–648. 7. Daly, B.J., Harlow, F.H. (1970) Transport Equations in Turbulence, Phys. Fluids, 13, 2634–2649. 8. Mellen, C.P, Fr¨ ohlich, J., Rodi, W. (2000) Large–Eddy Simulation of the Flow over Periodic Hills, 16th IMACS World Cong., Lausanne, Switzerland, 2000. 9. Rodi, W., Mansour, N.N., Michelassi, V. (1993) One–Eq. Near–Wall Turbulence Modeling with the Aid of Direct Simulation Data, J. Fluids Eng., 115, 196–205. 10. Schumann, U. (1975) Subgrid–Scale Model for Finite–Diﬀerence Simulations of Turb. Flows in Plane Channels and Annuli, J. Computat. Phys., 18, 376–404. 11. Wallin, S., Johansson, A.V. (2000) An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turb. Flows, J. Fluid Mech., 403, 89–132.

LES Meets FSI – Important Numerical and Modeling Aspects M. Breuer1,2 and M. M¨ unsch2 1

2

Dept. of Fluid Mechanics, Institute of Mechanics, Helmut-Schmidt-University Hamburg, Holstenhofweg 85, D-22043 Hamburg, Germany, [email protected] Institute of Fluid Mechanics, University of Erlangen-N¨ urnberg, Cauerstr. 4, D-1058 Erlangen, Germany, [email protected]

Abstract The paper is concerned with two main aspects, which should be considered when large–eddy simulation (LES) is married to ﬂuid–structure interaction (FSI). First, the inﬂuence of moving grids leading to temporally varying ﬁlter widths and thus additional commutation errors on the quality of the predicted results is thoroughly investigated. Second, a new partitioned coupling method based on the predictor–corrector scheme often used for LES is evaluated. A strongly coupled but nevertheless still explicit time–stepping algorithm results, which is very eﬃcient in the LES–FSI context. This new scheme is evaluated in detail based simulations around elastically supported cylindrical structures and a swiveling ﬂat plate.

1 Introduction Fluid–structure interaction (FSI) plays a dominant role in many technical applications such as suspension bridges, oﬀ-shore platforms or even vocal folds. Therefore, a strong need for appropriate numerical simulation tools exists for such coupled problems. In previous studies, FSI applications in the regime of laminar ﬂows as well as turbulent ﬂows using the RANS approach [5, 6] were numerically investigated. For that purpose, a partitioned fully implicit scheme was applied which coupled a three-dimensional ﬁnite-volume based multi-block ﬂow solver for incompressible ﬂuids with a ﬁnite-element code for the structural problem. This coupling scheme works eﬃciently for large time step sizes typically used for implicit time-stepping schemes within RANS predictions. However, ﬂow problems involving large-scale ﬂow structures such as vortex shedding or instantaneous separation and reattachment are often not reliably predicted by RANS and more advanced techniques such as largeeddy simulation (LES) are required. To resolve the turbulent ﬂow ﬁeld in time, LES uses small time steps. Thus, in general explicit time-marching schemes are favored, especially predictor–corrector schemes [1,2]. Furthermore, for FSI applications the solution domain changes in time due to the displacement of V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 36, c Springer Science+Business Media B.V. 2010

245

246

M. Breuer and M. M¨ unsch

the boundaries linked to the structure. Thus moving grids have to be used which has a direct inﬂuence on the ﬁltering approach in LES. Thus the paper addresses the aspects of additional errors introduced (e.g., commutation errors) and code coupling, which should be considered when LES is married to FSI.

2 Important steps for joining LES and FSI 2.1 LES on moving grids Within an FSI application the ﬂuid forces acting on the structure lead to the displacement or deformation of the structure. Thus the computational domain is no longer ﬁxed but changes in time. Besides other numerical techniques to account for these variations, the most popular one is the so-called Arbitrary Lagrangian–Eulerian (ALE) formulation. Here the conservation equations for mass, momentum (and energy) are re-formulated for a temporally varying domain. For a ﬁnite-volume scheme as applied in the present investigation, the grid and thus the size of the control volumes varies in time. In the context of LES relying on an implicit ﬁltering approach as used in many practically relevant LES predictions, the grid movement means that the ﬁlter width deﬁned as Δ = (Δx Δy Δz)1/3 now additionally varies in time, i.e., Δ = f (x, t). That leads to additional commutation errors depending on dΔ/dt and dΔ/dx [7]. In order to investigate the inﬂuence of the temporally varying ﬁlter width on the quality of the results, predictions of a plane channel ﬂow at Reτ = 590 on oscillating grids (domain: 2πδ × 2δ × πδ where δ denotes the channel halfwidth; grid: 128 × 128 × 128 CVs) were carried out applying the classical Smagorinsky model [9] with Smagorinsky constant Cs = 0.1 and Van Driest damping near solid walls as well as the dynamic model by Germano et al. [4] and Lilly [8]. The latter applies a test ﬁlter of ﬁlter width 2Δ taking the 27 neighboring control volumes into account. The sinusoidal grid movement was deﬁned by xmove /δ = A · sin(2π(t−t0 )/T ) · sin(x0 /δ). A variety of cases were computed (see Table 1) by using a normalized time step of Δt/(UB δ)=0.01 with UB denoting the bulk velocity. In one subset of cases, the amplitude A of the internal grid deformation was varied from 0.1 to 0.85 and the period T of the deformations was set to one ﬂow-through time (TF T ). Here, an amplitude of A = 0.5 leads to ±50% maximal reduction and expansion of the initial grid spacing during one cycle for example. In a second subset, computations for diﬀerent cycle periods T and a constant amplitude were performed. The results were compared to LES predictions on a ﬁxed grid (reference) by computing the root-mean-square deviations (rmsd) of the time and space averaged stresses u u , v v , w w , u v and of the mean ﬂow velocity U . As can be seen in Table 1 for a ﬁxed period T an increasing amplitude A leads to increasing deviations of the investigated ﬂow properties for the classical Smagorinsky model. Beside the shear stress quantities all other quantities reach their maximal deviations

LES Meets FSI – Important Numerical and Modeling Aspects

247

Table 1. Overview of moving grid computations performed showing amplitudes A and cycle periods T with the resulting root-mean-square deviations (rmsd) of u u , v v , w w , u v and U in [%]. Smagorinsky model: variation of amplitude A rmsd(u u ) rmsd(v v ) rmsd(w w ) rmsd(u v )

A

T/TFT

0.10 0.25 0.50 0.65 0.75 0.85

1 1 1 1 1 1

0.25 0.50

Dynamic model of Germano et al.: variation of amplitude A 1 1.70 4.37 2.46 33.69 1 2.71 6.79 4.28 20.02

0.50 0.50 0.50

1 0.2 0.1

2.63 2.98 2.55 2.63 3.34 4.19

0.81 1.11 2.68 3.71 5.13 6.35

Smagorinsky model: variation 2.55 2.68 4.98 10.58 10.15 26.57

0.85 1.16 2.36 2.53 3.63 3.92

8.93 5.22 9.36 5.74 13.73 12.06

of cycle period T 2.36 9.36 5.70 12.80 17.17 18.54

rmsd(U ) 0.16 0.21 0.48 0.67 0.81 1.06 0.41 0.68 0.48 1.40 3.04

at an amplitude of A = 0.85. The rmsd for u v reaches a maximum for an amplitude of A = 0.75. The mean velocity component U shows in general the smallest deviation from the reference test case. The results of the amplitude variation for the dynamic model of Germano et al. yield the same trend. Comparing the results of both models, the Smagorinsky model shows less deviations from the reference case than the dynamic model. The second test case, i.e., variation of the period T , shows a bigger impact on the rmsd-values than the ﬁrst test case. For decreasing period, i.e., increasing frequency of motion, the deviations from the reference case increase. A maximal rmsd-value of 26.57% for v v is obtained for T /TF T = 0.1. Here, again the mean ﬂow velocity U is the least sensitive variable with a deviation of 3.04%. Exemplarily, Fig. 1 depicts the results for the variation of the period T . In the ﬁrst case, one cycle of the grid deformation is equivalent to one ﬂowthrough time. In that case (T /TF T = 1) the deviations from the reference case are negligible. If the grid oscillates faster (i.e., 5 oscillations per ﬂow-through, T /TF T =0.2) ﬁrst deviations from the reference case appear. A further increase to 10 grid oscillations per ﬂow-through leads to strong deviations. In that case the errors introduced by the grid movement are obvious. Although diﬀerent eﬀects such as time-varying local resolution and the commutation error itself cannot clearly be distinguished, in summary, these additional errors due to moving grids signiﬁcantly aﬀect the LES results. The impact on the mean ﬂow properties is a minor one as long as the amplitudes and corresponding oscillations are not too high. The eﬀect on the secondary moments is much stronger leading to severe restrictions on tolerable amplitudes and cycle periods.

248

M. Breuer and M. M¨ unsch 1 1 0.8

0.6

v ´v ´/U B

U/UB

0.8 0.6 0.4

A = 0.5, T = 1TFT A = 0.5, T = 0.2TFT A = 0.5, T = 0.1TFT reference

0.2

– 0.5

0

Y/d

0.5

A=0.5, T = 1T F T A=0.5, T = 0.2T F T A=0.5, T = 0.1T F T reference

0.2

0 –1

0.4

1

0

–1

–0.5

0

Y/d

0.5

1

Fig. 1. LES prediction of plane channel ﬂow on temporally varying grids, A = 0.5, three diﬀerent grid cycle periods.

2.2 Partitioned coupled predictor–corrector scheme As mentioned above, LES often requires small time steps and thus predictor– corrector schemes relying on explicit time-marching are favored. Here the question arises how to appropriately couple such a scheme with the computation of the structure dynamics. Partitioned fully implicit schemes (e.g., [5, 6]) which guarantee a suﬃcient coupling between ﬂuid and structure, are in contradiction to the requirements. Therefore, the objective was to design a FSI coupling scheme, which on the one hand is appropriate for an explicit time-stepping scheme and on the other hand avoids instabilities known from loose coupling schemes. In the scheme suggested [3], each time step starts with an estimation of the structural displacement using a quadratic extrapolation of the three preceding time steps (see Fig. 2). Within the predictor step the momentum equations are solved by an explicit time integration scheme, namely a low-storage Runge–Kutta scheme. Here the ALE formulation is taken into account, which ensures that the space conservation law is fulﬁlled. In the corrector step a Poisson equation for the pressure correction variable is solved, which guarantees a divergence-free velocity ﬁeld and thus the fulﬁllment of the mass conservation. Whereas the predictor step is only done once per time step, the corrector step has to be repeated several times (5–10) until a predeﬁned convergence criterion is reached. The coupling and thus the exchange of ﬂuid forces in one direction and the resulting displacements in the other direction is conducted within the corrector step, which is repeated until a dynamic equilibrium between ﬂuid and structure is achieved. To reduce the number of subiteration the structural displacement ϕS is underrelaxated. Here, the structure itself can be described by an arbitrary approach such as equations of motion or ﬁnite-element-methods. Consequently, a strongly coupled but nevertheless still explicit time-stepping algorithm results. Within this study this FSI scheme was evaluated and compared with a partitioned fully implicit scheme by performing simulations around elastically

LES Meets FSI – Important Numerical and Modeling Aspects

249

Fig. 2. FSI coupling scheme, based on a predictor–corrector scheme with incorporated subiteration loop and underrelaxation for structural displacements.

Fig. 3. (a) LES prediction of an elastically supported cylindrical structure at Re=13,000, history of displacements for the present explicit scheme and the Implicit scheme [5, 6]. (b) Deﬂection angle and external momentum of the swiveling plate at Re = 64, 000.

supported cylindrical structures based on our in-house code FASTEST-3D. Exemplarily, Fig. 3a depicts the results for an elastically supported square cylinder with edge length b = 0.02 m at Re=U∞ b/ν=13,000 and with U∞ = 10 m/s for both coupling schemes, i.e., the present explicit one and the previously used implicit scheme [5, 6]. Here, the structural response on the acting ﬂuid forces is described by equations of motion, i.e., mx ¨(t) + d x(t) ˙ + c x(t) = Fx (t), m y¨(t) + d y(t) ˙ + c y(t) = Fy (t), with m = 0.005 kg, d = 0.1 Ns/m, and c = 5 N/m . After an initial excursion in x-direction, the cylinder oscillates in both directions owing to vortex shedding at the cylinder (see Fig. 3a). Furthermore, it is obvious that apart from the shedding frequency higher frequencies are involved, especially visible for the y-displacement. Although the momentum equations are only solved once

250

M. Breuer and M. M¨ unsch

per time step leading to a reduction of the computational eﬀort compared to the fully implicit scheme, the results obtained for both coupling schemes are in good agreement. Slight diﬀerences are attributed to diﬀerent starting time steps of the ﬂuid–structure simulation within the running LES and to the non-deterministic nature of turbulence. The maximal number of FSIsubiterations was very low, i.e., NF SI = 3, for the explicit time-stepping algorithm. Further testing of the coupling algorithm was performed by simulating the turbulent ﬂow around a swiveling plate at Re=U∞ L/ν=64,000 with plate length L = 0.064 m and U∞ =1 m/s. The LES is performed on a grid with 434,000 CVs using the classical Smagorinsky model with Cs = 0.1 and Van Driest damping. Here, the rotary movement of the rigid plate around an axis located at 0.3125×L from the leading edge is described by a spring-massmodel, i.e., I ϕ(t) ¨ + C sin(ϕ(t)) = Mz (t), with I=1.624 · 10−5 kg m2 and C=3.934 · 10−3 Nm. Again the mean value of required FSI-subiterations is low, i.e., N F SI = 5. In Fig. 3b the development of the forcing moment Mz (t) due to pressure and shear stresses acting on the plate and the resulting excursion angle ϕ(t) of the plate is plotted. Starting from an initial angle of ϕ = 0 maximal swiveling amplitudes of ±40◦ appear after about 2.5 s of transition time. Up to these amplitudes grid adaption can be tackled with transﬁnite interpolations performed in each subiteration step. Nevertheless, the need for more powerful grid adaption techniques becomes obvious when thinking about ﬂexible structures on the one hand. On the other hand, LES makes anyway high demands on the grid quality regarding smoothness and orthogonality, which have to be satisﬁed even for the special case of FSI application, i.e., the ALE formulation. Thus the eﬀect of elliptic grid smoothing techniques based on composite mapping as suggested by Spekreijse [10] and used by Yigit et al. [11] will be studied in the near future to maintain the grid quality within the coupled simulation.

Acknowledgement The project is ﬁnancially supported by the Deutsche Forschungsgemeinschaft within the research group ‘Fluid–Struktur–Wechselwirkung: Modellierung, Simulation, Optimierung’ (FOR 493) under contract number BR 1847/6.

References 1. Breuer, M.: Large–Eddy Simulation of the Sub–Critical Flow Past a Circular Cylinder: Numerical and Modeling Aspects, Int. J. for Num. Methods in Fluids, vol. 28, pp. 1281–1302, (1998).

LES Meets FSI – Important Numerical and Modeling Aspects

251

2. Breuer, M.: Direkte Numerische Simulation und Large–Eddy Simulation turbulenter Str¨ omungen auf Hochleistungsrechnern, Habilitationsschrift, Univ. Erlangen–N¨ urnberg, ISBN: 3–8265–9958–6, (2002). 3. Breuer, M., M¨ unsch, M.: Fluid–Structure Interaction Using LES – A Partitioned Coupled Predictor–Corrector Scheme, Proc. in Applied Mathematics and Mechanics, PAMM, vol. 8, no. 1, pp. 10515–10516, (2008). 4. Germano, M., Piomelli, U., Moin, P., Cabot, W.H.: A Dynamic Subgrid–Scale Eddy Viscosity Model, Phys. of Fluids A, vol. 3, pp. 1760–1765, (1991). 5. Gl¨ uck, M., Breuer, M., Durst, F., Halfmann, A., Rank, E.: Computation of Fluid–Structure Interaction on Lightweight Structures, Int. J. Wind Eng. & Indus. Aerodyn., vol. 89, pp. 1351–1368, (2001). 6. Gl¨ uck, M., Breuer, M., Durst, F., Halfmann, A., Rank, E.: Computation of Wind–Induced Vibrations of Flexible Shells and Membranous Structures, J. Fluids & Structures, vol. 17, pp. 739–765, (2003) 7. Leonard, S., Terracol, M., Sagaut, P.: Commutation Error in LES with Time– Dependent Filter Width, Computers & Fluids, vol. 36, pp. 513–519, (2007). 8. Lilly, D.K.: A Proposed Modiﬁcation of the Germano Subgrid–Scale Closure Method, Phys. of Fluids A, vol. 4, pp. 633–635, (1992). 9. Smagorinsky, J.: General Circulation Experiments with the Primitive Equations, I, The Basic Experiment, Mon. Weather Rev., vol. 91, pp. 99–165, (1963). 10. Spekreijse, S.P.: Elliptic Grid Generation Based on Laplace Equations and Algebraic Transformations, J. of Comput. Physics, vol. 118, pp. 38–61, (1995). 11. Yigit, S., Sch¨ afer, M., Heck, M.: Numerical Investigation of Structural Behavior During Fluid Excited Vibrations, REMN, vol. 16, pp. 491–519, (2007).

A New Multiscale Model with Proper Behaviour in Both Vortex Flows and Wall Bounded Flows L. Bricteux, M. Duponcheel, and G. Winckelmans Universit´e catholique de Louvain (UCL), Louvain School of Engineering (EPL), Mechanical Engineering Department, Division TERM, 1348 Louvain-la-Neuve, Belgium, [email protected]; [email protected]; [email protected]

1 Introduction A new subgrid-scale (SGS) model which has an adequate behaviour in both vortical ﬂows and wall-bounded ﬂows is proposed. In wall-bounded ﬂows computed using wall-resolved LES, the theory predicts that the SGS dissipation 3 should vanish with a y + behaviour near the wall. In the case of vortex ﬂows, one needs to have models which do not dissipate energy in the strongly vortical and essentially laminar part of the ﬂow, e.g. in vortex core regions. The model presented here aims at combining the strengths of two models as it is a regularized variational multiscale (RVM) model (as in Jeanmart and Winckelmans [3]), thus acting on the high pass ﬁltered LES ﬁeld, and for which the subgrid-scale viscosity is evaluated using the WALE (wall adapting local eddy viscosity) scaling of Nicoud and Ducros [5], itself computed using the high pass ﬁltered LES ﬁeld. Thus this model is only active when there is a signiﬁcant high wavenumber content in the ﬂow and it has a natural near wall damping behaviour. The ability of this model to simulate vortex and wall-bounded ﬂows is demonstrated on three test cases. The ﬁrst case is the turbulent channel ﬂow. The second case concerns a four-vortex system. The third case concerns a two-vortex system in ground eﬀect. It is shown that this new model allows to perform successfully LES of these ﬂows with the proper dissipative behaviour in both near wall and vortical regions. The LES are performed using a parallel fourth order code based on ﬁnite diﬀerences.

2 Presentation of the model Practical LES consists in a truncation to much less information than that required to numerically capture the complete ﬁeld, as in DNS. The truncation operator which consists in projecting a ﬁeld on a LES grid is written here as •. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 37, c Springer Science+Business Media B.V. 2010

253

254

L. Bricteux et al.

sgs Using this notation, one deﬁnes the subgrid-scale tensor as σ ij = u$ j − u$ iu i uj . This stress tensor has to be modeled since it cannot be expressed in terms of the LES ﬁeld. The model presented here is a regularized variational multiscale (RVM) version of the WALE model. The model acts on the small scale part (n) s . This small scale (high pass ﬁltered) LES ﬁeld of the LES velocity ﬁeld u , and the is computed from the diﬀerence between the LES velocity ﬁeld, u . As in [3], the low-pass ﬁltered ﬁeld is obtained using low-pass ﬁltered ﬁeld, u the multiple application of three-point discrete ﬁlters: % (n) n n n & = I − −δx2 /4 I − −δy2 /4 I − −δz2 /4 u (1) u

where I is the identity operator and δx2 fi,j,k = fi+1,j,k − 2fi,j,k + fi−1,j,k (on an uniform grid). The model SGS stress tensor is written: 1 s kk δij = 2 νsgs Sij ij − σ , τijM = σ 3

(2)

s Sij being the strain rate tensor computed using the small scale part of the ﬁeld. Thus this model acts mainly on the high pass ﬁltered LES ﬁeld. This ensures a good spectral behaviour, as highlighted in [2]. Furthermore, the SGS viscosity is here computed using the small scale ﬁeld and the scaling of the WALE model [5]:

νsgs = C Δ2

s d s d 3/2 (Sij Sij ) . sS s )5/2 + (Ss d Ss d )5/4 (Sij ij ij ij

(3)

sd s where Sij is deﬁned as the deviatoric part of Sij = 12 ∂k u si ∂j u sk + ∂k u sj ∂i u sk . The use of the WALE scaling for the evaluation of νsgs ensures a natural near wall damping behaviour. The fact that νsgs is evaluated on the high pass ﬁltered ﬁeld guarantees that the SGS model is active only when there are small scales in the ﬂow and that it is inactive in well-resolved zones. Indeed when using the WALE model on a non ﬁltered LES velocity ﬁeld, the produced SGS viscosity is too important in vortex cores, which constitutes a major drawback of this model for LES of vortex ﬂows.

3 LES of the turbulent channel ﬂow As the model is designed to have a proper near wall behaviour, it is important to test it on a ﬂow with walls. LES of the channel ﬂow at Reτ = uτνH = 395 is considered, the reference being the DNS of [4]. The computational domain is 2πH × 2H × πH in streamwise, normal, and spanwise directions. The LES grid is quite coarse: 64 × 48 × 48, thus Δx+ = 38.8 and Δz + = 25.8. The + grid stretching in the wall normal direction ensures that Δyw = 0.8 at the wall and Δyc+ = 45.4 at the center. The coeﬃcient used in the SGS model is

A New Multiscale Model for Vortical and Wall Bounded Flows

255

C = 0.56. The obtained mean velocity proﬁle is displayed in Fig. 1: it agrees very well with the reference DNS. The shear stress decomposition is shown in Fig. 2. It is important to note that the contribution of the SGS model to the mean total shear stress is much weaker than that of the classical WALE model. This is due to the fact that the model acts on small scales with a SGS M s viscosity evaluated on the small-scale ﬁeld: τ12 = 2νsgs (us )S12 . There is no reason to have an important time-averaged proﬁle on this quantity which is a major diﬀerence with the models acting on the complete ﬁeld, for which S12 has a non negligible time-averaged proﬁle. This also constitutes an interesting feature of this model; the time averaged LES solution − u v is able to recover the shear stress balance with only a minor average contribution of the SGS model. The dissipation proﬁle is displayed in Fig. 2:3one observes the proper 4 +3 sgs M y behaviour of the SGS model dissipation = τij Sij at the wall.

20

15

u+ 10 5

0 100

101

102

y+

Fig. 1. LES of channel ﬂow at Reτ = 395: mean velocity proﬁle. Reference DNS (solid) and LES (dash). 1

100

τxy

0.8

y+

uτ2

3

4 uτ

0.6 10–2

0.4 0.2 10–4

0 0

0.2

0.4

y H

0.6

0.8

1

100

101

102

y+

Fig. 2. LES 5of the Re6τ = 395: decomposition of the5 mean 6 channel ﬂow 5at visc 6 shear tot M (solid thick), τxy (solid thin), − u v (dash), stress (left). τxy τxy 3 4 (solidbullets). Dissipation proﬁles (right). Viscous dissipation ν = 2ν Sij Sij (dashthin), SGS dissipation sgs (solid-thin), total dissipation = ν + sgs (solid thick).

256

L. Bricteux et al. 1.1 1 0.9

E E0 0.8 0.7 0.6 0.5

0

0.5

1

1.5

2

τ

Fig. 3. Four-vortex system: kinetic energy evolution; 2D DNS (solid-thin) and LES (dash-dot) at Re = 20000; LES at very high Re (solid-thick).

4 LES of a counter-rotating four-vortex system The initial condition is made of a four-vortex system with two alternate pairs of counter-rotating vortices. The initial condition is the same as that used in Cocle et al. [1]. The Reynolds number is set to Re = 20, 000. We also investigate a case were the molecular viscosity ν is deliberately set to zero. Since the code is energy conserving, this allows to check that the model is inactive during the gentle phase of the ﬂow. The dimensionless time is τ , based on the descent velocity and initial spacing of the equivalent two-vortex system (i.e a two-vortex system with same momentum). The evolution of the kinetic energy is reported in Fig. 3. When the molecular viscosity is set to zero, it can be observed that the kinetic energy is strictly conserved in the ﬁrst phase. This is due to the fact that the model is inactive during the phase for which the ﬂow is not turbulent. Then a rapid decay phase occurs due to the growth of instabilities and the transition to turbulence. Concerning the case at Re = 20, 000, one observes that, in the ﬁrst part of the ﬂow development, the energy decay is solely due to the viscous dissipation of the laminar ﬂow: the curve indeed follows exactly the 2D spectral DNS reference (computed using a 1024 × 1024 grid). This means, again, that the model is inactive during this laminar part of the ﬂow. After, the rapid decay occurs and the model becomes active when the transition to turbulence takes place: when small scales develop, the model triggers and dissipates energy. It can be concluded that the model is suitable to simulate such vortical transitional ﬂows.

5 LES of a two-vortex system in ground eﬀect Further, wall resolved LES of a two-vortex system in ground eﬀect (IGE) is considered. The ﬂow consists in a pair of counter rotating vortices of circulation Γ0 evolving under their mutual inﬂuence and interacting with the ground

A New Multiscale Model for Vortical and Wall Bounded Flows

257

in a viscous way. The Reynolds number, based on the vortex circulation and the kinematic viscosity ν, is set to Re = Γν0 = 20, 000 which is indeed quite high for such wall-resolved LES. Deﬁning the initial distance between the vortex centers as b0 , the lengths of the computational domain are Lx = 4 b0 , Ly = 8 b0 and Lz = 3 b0 . The initial height of the vortex pair is set to h0 = b0 . 2 Here an algebraic proﬁle was used: ΓΓ(r) = r2r+r2 . The radius of maximum 0 c induced tangential velocity is set to rc = 0.05 b0. The domain is periodic in the x and y directions. A no-slip condition is set at the ground (z = 0) and a slip condition at the top of the domain (z = 3 b0 ). The number of grid points is 512 × 256 × 256. In order to capture the thin boundary layers developing at the ground, a clustering of the grid points close to the wall is used. The velocity scale for this problem is based on the descent velocity of the vortex Γ0 pair out of ground eﬀect: V0 = 2πb . This also deﬁnes the dimensionless time: 0 V0 τ = t b0 . A visualization of this ﬂow is provided in Fig. 4. The kinetic energy evolution is also provided in Fig. 5. Comparing the curve with that of a 2D

Fig. 4. Two-vortex system in ground eﬀect: visualization of the LES ﬂow ﬁeld using isosurfaces of ω b20 /Γ0 = 1 and 10 at τ = 2.0 and τ = 5.0. 1 0.8 E 0.6 E0 0.4 0.2 0

0

1

2

τ

3

4

5

Fig. 5. Two-vortex system in ground eﬀect: kinetic energy evolution: 2D DNS (solid-thin) and LES at Re = 20, 000 (solid-thick).

258

L. Bricteux et al.

DNS of reference (computed using a second order ﬁnite diﬀerences code with a very ﬁne grid: 2048×768), we observe that the vortex dynamics in the initial laminar phase is well preserved by the new model. The model is well adapted for LES of such transitional ﬂow as it does not dissipates in the well resolved and/or laminar regions and only triggers when small scales develop.

6 Conclusion The new RVMs -WALE model appears to be a good model to perform LES 3 of vortical ﬂows in presence of walls. This model oﬀers a y + damping at the wall, and it avoids excessive diﬀusion in vortex cores. This new model has been presented and tested on three challenging test cases. The ﬁrst case, aimed at assessing the quality of the model in wall bounded ﬂows, was the channel ﬂow at Reτ = 395. The agreement between the LES and the DNS results is very good. Moreover the contribution of this model in the shear stress balance is found to be negligible. The second ﬂow was a four-vortex system. It was shown that the model is inactive during the gentle, resolved phase of the ﬂow and only becomes active when there are turbulence ﬂuctuations. Finally, the model was assessed on a complex case which concerns a two-vortex system in ground eﬀect. This ﬂow includes both strong vortical structures and thin boundary layers.

References 1. Cocle R., Dufresne L. and Winckelmans G., Investigation of multiscale subgrid models for LES of instabilities and turbulence in wake vortex systems, Complex Eﬀects in LES: Springer, 56, 141-159, 2007. 2. Cocle R., Bricteux L., and G. Winckelmans “Spectral behavior of various subgridscale models in LES at very high Reynolds number” In: Quality and Reliability of Large-Eddy Simulations, ERCOFTAC Series, J. Meyers, B.J. Geurts and P. Sagaut ed(s), Springer, 2008, 12, p. 183-190. 3. Jeanmart H. and Winckelmans G., Investigation of eddy-viscosity models modiﬁed using discrete ﬁlters: a simpliﬁed “regularized variational multiscale model” and an “enhanced ﬁeld model”, Phys. Fluids, 19(5), 055110, 2007. 4. Moser R, Kim J, Mansour N (1999) Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590. Phys. Fluids11: 943–945. 5. Nicoud F. and Ducros F., Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow, turbulence and combustion, 62 183–200, 1999.

LES Based POD Analysis of Jet in Cross Flow ˇ c2 D. Cavar1 , K.E. Meyer1 , S. Jakirli´c2, and S. Sari´ 1

2

Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark, [email protected]; [email protected] Fachgebiet Str¨ omungslehre und Aerodynamik, Technische Universit¨ at Darmstadt, Darmstadt, Germany, [email protected]; [email protected]

Abstract The paper presents results of a POD investigation of the LES based numerical simulation of the jet-in-crossﬂow (JICF) ﬂowﬁeld. LES results are ﬁrstly compared to the pointwise LDA measurements. 2D POD analysis is then used as a comparison basis for PIV measurements and LES, and ﬁnally 3D POD analysis is conducted on the LES datasets, giving some clear depictions of interaction processes between dominant ﬂow structures pertinent to the JICF ﬂowﬁeld.

1 Introduction The Jet in Cross Flow (JICF) represents a basic ﬂow conﬁguration investigated quite extensively in the past, primarily due to its relevance to numerous engineering applications. The evolution of vortical ﬂow in and around JICF has been subjected to several experimental and numerical investigations aimed towards better understanding of a complex interaction mechanism between the jet and the cross-stream. Most of the experimental visualization studies, e.g., [2, 3] investigated laminar oncoming ﬂows (at least on the cross-stream side) making it possible to directly trace vorticity deformation and creation of vortical structures throughout the ﬂowﬁeld. However, in the most practical appliances where JICF occurs, both inﬂow conditions are turbulent. In such cases the inherited unsteadiness of the turbulent ﬂow makes the ﬂow patterns highly irregular, hence considerably more diﬃcult to visualize directly. In those situations methods like Proper Orthogonal Decomposition (POD) can be applied in order to investigate and better understand established complex ﬂow patterns.

2 Large Eddy simulation (LES) details The JICF ﬂow case has been studied numerically, by means of LES. The simulations have been performed utilizing the in-house ﬂow solver EllipSys [7], which is a multi-block ﬁnite volume solver for incompressible Navier–Stokes V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 38, c Springer Science+Business Media B.V. 2010

259

260

D. Cavar et al.

equations in general curvilinear coordinates. The code uses a collocated variable arrangement, where a revised Rhie/Chow interpolation is used to avoid odd/even pressure coupling. The pressure–velocity coupling is obtained by applying the well-known PISO algorithm. The solution is advanced in time using a second order iterative time stepping (dual-time stepping). The EllipSys code is parallelized with MPI message parsing library for execution on distributed/shared memory machines by a non-overlapping domain decomposition technique. Important ﬂow parameters for the considered case are Reynolds number based on the cross ﬂow velocity and the jet diameter (Re = 2, 400) and the jet-to-crossﬂow velocity ratio – R = 3.3. Figure 1 (left) outlines the geometrical layout and dimensions of the computational domain, while Fig. 1 (right) outlines the computational set-up and boundary conditions applied in the calculations. Two separate precursor computations have been performed in order to obtain suitable turbulent inlet boundary conditions – one simulating a fully developed pipe ﬂow and the other simulating spatially developing boundary layer ﬂow – see Fig. 1 (right). The latter ﬂow is simulated utilizing the method of Lund et al. [4]. The SGS stresses are modeled through the Eddy-viscosity assumption employing a Mixed Scale Eddy-Viscosity model of Sagaut [9]. The convective terms in the Navier–Stokes equations are discretized utilizing the QUICK scheme to avoid a wiggle contamination of the instantaneous ﬂow visualisations used in POD analysis. Actually 14 diﬀerent cases, where inﬂuence of various parameters (discretization schemes – CDS2, CDS4, QUICK and blended CDS4 and QUICK, domain inlet–outlet extensions, time step size, SGS eddy viscosity models – Smagorinsky (without explicit near wall damping), Dynamic Smagorinsky, Sagaut [9], etc.) have been investigated and

Lz

U∝ y

Lz

z

symmetry

Ly

x

D

D Lx 1

Ujet

Lx

outlet z

2

y

wall z inlet(s)

wall

Fig. 1. Flow conﬁguration, with domain dimensions – Lx1 = 5D, Lx2 = 12D, LzD = 3D, Ly = 12D, Lz = 10D (left). The computational setup shown as an instantaneous snapshot of the velocity magnitude visualized on inlet, outlet and symmetry – (y/D=0) planes. Domain boundaries, with boundary conditions applied are included in the Figure. Only periodical boundary conditions – applied in the spanwise (y) direction are not illustrated in the Figure (right).

LES Based POD Analysis of Jet in Cross Flow

261

grids of up to 14.8 million cells in size have been considered (for details see Cavar [1]) but none of them were able to better reproduce LDA measurements ¨ of Ozcan and Larsen [8] than the results presented in the following. The 2D part of POD analysis is conducted on a dataset consisting of 1, 000 instantaneous ﬂow snapshots extracted equidistantly in time during the total computational period of 510 D/U∞ , thereby accommodating comparison of PIV data of Meyer et al. [6] and LES results up to a similar level of detail. The 3D POD analysis is conducted in similar way – 1, 000 snapshots are extracted during the total time period of 1700 D/U∞ . The investigated near-ﬂow JICF region (−1 < x/D < 4,−1.5 < y/D < 1.5 and 0 < z/D < 4) is discretized by app. 500 000 mutually equidistantly displaced (Δ = 0.05 D) points. In both 2D and 3D cases, data points for LES based POD analysis were interpolated, using the second order inverse distance interpolation technique. The POD analysis itself has been conducted in similar way as described in Meyer et al. [5]. For further details regarding the present POD analyses the interested reader is referred to Cavar [1].

3 Results In the recent experimental studies of [6, 8] the turbulent jet exhausting into a relatively thick turbulent boundary layer has been investigated. General agreement between the computed mean streamwise velocity and the pointbased time averaged LDA measurements of [8] in the “symmetry” plane can be seen in Fig. 2. One can observe that the computations reproduce the measured ﬂow ﬁeld very well in the most ﬂow regions, but some diﬀerences can be observed at the position of the jet-to-crossﬂow interface, e.g., in −1 ≤ x/D ≤

a6 5

U/U∞, Exper. Δc=0.1

b6

1

1.1

0.9

1.1

0.9

0.7

0.8

4

0

0.5

0.3

z/D

3

3

0.5 0

0.7

0.3

0.7

2

2

−0.4

−0.4

1

1

0.5

0.5

0 −1

1

5

4

z/D

U/U∞,Comput. Δc=0.1

0

1

2

x/D

3

4

5

0 −1

0

1

2

3

4

5

x/D

Fig. 2. U/U∞ in the symmetry (y/D=0) plane : Experiment (a), LES (b). Curves depicted by the same color represent the same contour level. Dashed lines represent negative contour levels. Δc refers to diﬀerence between plotted contour levels. Diamonds depict jet trajectories, estimated as in Yuan et al. [10].

262

D. Cavar et al. 3.5

3.5

3

3 2.5

2

z/D

z/D

2.5

1.5 1

1

0.5 0

2 1.5

0.5

0

1

2 x/D

3

PIV 4

0

LES

0

1

2

3

4

x/D

Fig. 3. The symmetry plane, POD – mode 1: PIV (left), LES (right). Contour level scales are identical in both ﬁgures. Solid black line depicts the jet trajectory.

0, 0 ≤ z/D ≤ 3 region. The reason for those diﬀerences have been investigated in detail and reported in Cavar [1]. As the pointwise statistics showed reasonable agreement between LES and measurements, the LES analysis was extended by applying the POD technique in order to analyze the dynamic content of the ﬂow and identify large systematic ﬂow structures. Outcome of the POD analysis are so called “modes”, which represent systematic variations from the mean ﬂow. Figure 3 shows the dominant POD mode 1 in the plane shown in Fig. 2. The in-plane velocities are depicted as vectors, while the out-of-plane velocities are represented by contour plots. Velocities are all normalized by the freestream velocity U∞ . The results show that both PIV (Meyer et al. [6]) and LES based POD modes seem to have a very similar dominant out-of-plane pattern, which can be interpreted as “wake vortices” (see, e.g., Fric and Roshko [2]) parallel to the jet trajectory (seen as black solid line). The PIV based 2D POD analysis of Meyer et al. [5] showed that interaction between the standing (hanging) vortex of Yuan et al. [10] and wake vortices in the downstream region of the jet entry point can be identiﬁed by analyzing a POD based ﬂow reconstruction of the JICF ﬂow ﬁeld. Figure 4 depicts several aspects of a representative snapshot of the LES based JICF ﬂow ﬁeld at z/D = 0.55 position. Various vortical structures can be identiﬁed in Fig. 4a. A vortex positioned at (x/D, y/D) = (0.6, −0.1), another one at (x/D, y/D) = (1.25, 0.3) and the third one at (x/D, y/D) = (1.75, −0.2) can be clearly recognized indicating that a continuous vortex-shedding process takes place in the wake region. The corresponding POD based reconstruction, which is obtained by adding only the ﬁrst two POD modes to the mean ﬂow ﬁeld, – Fig. 4b, shows that only vortices in the vicinity of the jet can be identiﬁed. Signiﬁcant variations in position of the shedded vortices in the wake region (this can be visualized by inspection of several snapshots of the z/D = 0.55 plane – not shown here), indicate that the downstream transport of the wake vortices do not follow a regular pattern; hence each of the mentioned vortices singly is not capable of leaving a clear trace in the ﬁrst POD modes. Results of an application of the Q-criterion on the snapshot from Fig. 4a and its reconstruction from Fig. 4b are presented in Fig. 4c, d. As seen from Fig. 4c,

LES Based POD Analysis of Jet in Cross Flow

b

a

1.5

1

1

0.5

0.5 y/D

y/D

1.5

0

−0.5

0

−0.5

−1 −1.5 −1 −0.5

−1 0

0.5

1

1.5 x/D

2

2.5

3

3.5

−1.5 −1 −0.5

4

c

0

0.5

1

0

0.5

1

1.5 x/D

2

2.5

3

3.5

4

1.5

2

2.5

3

3.5

4

d 1.5

1.5

1

1

0.5

0.5

0

y/D

y/D

263

−0.5 −1 −1.5 −1 −0.5

0

−0.5 −1 0

0.5

1

1.5

x/D

2

2.5

3

3.5

4

−1.5 −1 −0.5

x/D

Fig. 4. Snapshot of the z/D = 0.55 plane (a) and its reconstruction (b). Q-criterion applied on the snapshot (c) and its reconstruction (d). The reconstruction is based on the ﬁrst two POD modes.

various vortices on the upstream side of the jet, actually not directly seen in the snapshot itself (traces of shear-layer vortices), as well as vortices in the wake region can be identiﬁed. However, the most interesting part here is represented by three vortices seen on both Fig. 4c and d positioned at (x/D, y/D) = (0.45, −0.5), (x/D, y/D) = (0.4, 0.5) and (x/D, y/D) = (0.65, −0.2). As hypothesized in Meyer et al. [5], the ﬁrst two vortices should represent traces of the standing (hanging) vortices of Yuan et al. [10], while the last one should represent traces of a wake vortex. Even though many relevant ﬂow features can be revealed by 2D POD analysis, a generally better understanding of diﬀerent ﬂow structures can be obtained by studying results of a corresponding 3D analysis. For that reason the present POD investigation of data attained by LES is continued in 3D (see Fig. 5) in order to better analyze vortical structures seen in Fig. 4c, d. Subﬁgures of Fig. 5 illustrate iso-surfaces, colored by ωz , of the same positive Q value, where Q-criterion has been applied on ﬂow reconstructions based on the ﬁrst two POD modes (the modes are added to the mean ﬂow ﬁeld). Moreover, subﬁgures of Fig. 5 actually follow the development of structures depicted in Fig. 5a throughout approximately half of the periodical cycle of dominant POD coeﬃcients A1 and A2 . It is shown in Cavar [1] that time variation of the ﬁrst two POD coeﬃcients – A1 and A2 closely follow a phase shifted pattern of sine and cosine functions.

264

D. Cavar et al.

a

b

c

e

d

f

z z x

x

y

x

y

x

z

z

z

z

y y

x

y y

x

Fig. 5. Iso-surfaces, colored by ωz , of the same positive Q value, with Q-criterion being applied on ﬂow reconstructions based on the ﬁrst two POD modes. Subﬁgures a–f follow the development of structures depicted in subﬁgure a throughout approximately half of the periodical cycle of coeﬃcients A1 and A2 .

Generally, in Fig. 5 three to four distinct vortical structures can be identiﬁed. Two of them, positioned on lateral edges of the jet, are visible in all subﬁgures. They slightly change their inclination angle with the z-axis (or jet trajectory), but always appear at approximately same domain position. According to the deﬁnition of Yuan et al. [10], they can be identiﬁed as hanging vortices. Inspecting Fig. 5a in a more detailed manner, two additional structures, both attached to the corresponding hanging vortex, can be identiﬁed. The process of continuous and alternating creation and destruction of these structures can be followed throughout subﬁgures a–f of Fig. 5. In Fig. 5a it is seen that one of the mentioned structures is attached to the left hanging vortex, while the other one is just about to be created in the close proximity of the right hanging vortex. Following the vortex dynamics further – Fig. 5b, one can observe that the vortex connected to the left hanging vortex become detached from it, while the vortex connected to the right hanging vortex grows in size, while it is convected downstream. The growth and downstream convection of the vortex attached to the right hanging vortex can be followed through subﬁgures c–e of Fig. 5 and Fig. 5f depicts a mirror image of Fig. 5a, indicating that a new analogous process of creation and destruction of the attached/detached vortices from the hanging vortices takes place.

4 Conclusions Present numerical study of the JICF ﬂow case shows a good agreement with LDA measurements with respect to the mean ﬂow statistics and PIV measurements regarding the ﬂow dynamics. Furthermore, the process of a dynamical interaction of wake vortices and corresponding hanging vortices in the JICF ﬂow is visualized based on the 3D POD analysis. From Fig. 5a, and 5f a clear indication of the signiﬁcant role the hanging vortex plays in the creation process of wake vortices is underlined. It is seen that the wake vortex practically originates from the hanging vortex and grows in size in a tornado vortex like manner by “sucking up” the boundary-layer ﬂuid.

LES Based POD Analysis of Jet in Cross Flow

265

References 1. D. Cavar. Large Eddy Simulation of Industrially relevant Flows. PhD thesis, Technical University of Denmark - MEK-DTU, 2006. 2. T. Fric and A. Roshko. Vortical structure in the wake of a transverse jet. J. Fluid Mech., 279:1–47, 1994. 3. R. Kelso, T. Lim, and A. Perry. An experimental study of round jets in crossﬂow. J. Fluid Mech., 306:111–144, 1996. 4. T. Lund, X. Wu, and K. Squires. Generation of turbulent inﬂow data for spatially-developing boundary layer simulations. J. Comp. Phys., 140(2): 233–258, 1998. ¨ 5. K. Meyer, J. Pedersen, and O. Ozcan. A turbulent jet in crossﬂow analysed with POD. J. Fluid Mech., 583:199–227, 2007. ¨ 6. K. E. Meyer, O. Ozcan, P. S. Larsen, and C. H. Westergaard. Stereoscopic PIV measurements in a jet in crossﬂow. In Proc. of 2. Intern. Sym. on Turb. and Shear Flow Phenomena, 27–29 June, Sweden, 2001. 7. J. A. Michelsen. Block structured Multigrid solution of 2D and 3D elliptic PDE’s. Technical Report AFM 94-06, Techn. Univer. of Denmark, 1994. ¨ 8. O. Ozcan and P. S. Larsen. An experimental study of a turbulent jet in crossﬂow by using LDA. Technical Report MEK-FM 2001–02, Technical University of Denmark - MEK-DTU, 2001. 9. P. Sagaut. Numerical simulations of separated ﬂows with subgrid models. Rech. Aero., pages 51–63, 1996. 10. L. L. Yuan, R. L. Street, and J. H. Ferziger. Large-Eddy Simulations of a round jet in crossﬂow. J. Fluid Mech., 379:71–104, 1999.

A Dissipative Scale-Similarity Model L. Davidson Division of Fluid Dynamics, Dept. of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, [email protected]

Abstract When Bardina in 1980 developed the scale-similarity model it was found that the gradient of the scale-similarity stress tensor was not suﬃciently dissipative. They had to add a Smagorinsky SGS model. The present paper presents a scalesimilarity model which is dissipative; this is achieved by using the fact that the viscous diﬀusion term is always dissipative. We include the gradient of the scalesimilarity stress tensor only when its sign agrees with that of the viscous diﬀusion term, otherwise it is set to zero.

1 The dissipative scale-similarity model When the ﬁrst scale-similarity model was proposed it was found that it is not suﬃciently dissipative [1]. An eddy-viscosity model has to be added to make the model suﬃciently dissipative; these models are called mixed models. The present work presents and evaluates a new dissipative scale-similarity model. The ﬁltered Navier–Stokes read ∂u ¯i ¯i ∂ 1 ∂ p¯ ∂2u ∂τik + (¯ ui u ¯k ) + =ν − , ∂t ∂xk ρ ∂xi ∂xk ∂xk ∂xk

∂u ¯i =0 ∂xi

(1)

where τik denotes the SGS stress tensor. In the scale-similarity model, the SGS stress tensor is given by ¯i u ¯k τik = u ¯i u ¯k − u

(2)

This model is not suﬃciently dissipative. Let us take a closer look at the equation for the resolved, turbulent kinetic energy, K = ui ui /2 (where ui = u ¯i − ¯ ui ), which reads

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 39, c Springer Science+Business Media B.V. 2010

267

268

L. Davidson

8 7 2 ∂K ∂ ui ∂p ui /ρ 1 ∂uk ui ui ∂ ui ∂¯ + (¯ ui K) + uk ui + + =ν u ∂t ∂xi ∂xk ∂xi 2 ∂xk ∂xk ∂xk i 8 8 7 2 8 7 8 7 7 ∂ ui ∂τik ∂τik ∂τik ui = ν − u − u = − ∂xk ∂xk ∂xk ∂xk i ∂xk i 8 8 7 7 ∂2K ∂ui ∂ui ∂τik −ν − u (3) ν ∂xk ∂xk ∂xk ∂xk ∂xk i ε

εSGS

The ﬁrst term on the last line is the viscous diﬀusion term and the second term, ε, is the viscous dissipation term which is always positive. The last term, εSGS , is a source term arising from the SGS stress tensor, which can be positive or negative. When it is positive, forward scattering takes place (i.e. it acts as a dissipation term); when it is negative, back scattering occurs. Figure 1 presents SGS dissipation, εSGS in equation (3), computed from ﬁltered channel ﬂow DNS data. The forward scatter, ε+ SGS , and back scatter, ε− , SGS dissipation are deﬁned as the sum of all instants when εSGS is SGS positive and negative, respectively. As can be seen, the scale-similarity model is slightly dissipative (i.e. εSGS > 0) , but the forward and back scatter dissipation are both much larger than εSGS . One way to make the SGS stress tensor strictly dissipative is to set the back scatter to zero, i.e. max(εSGS , 0). This could be achieved by setting ∂τik /∂xk = 0 when its sign is diﬀerent from that of ui (see the last term in equation (3)). This would work if we were solving for K. Usually we do not, and the equations that we do solve (the ﬁltered Navier–Stokes equations) are not directly aﬀected by the dissipation term, εSGS . Instead we have to modify the SGS stress tensor as it appears in the ﬁltered Navier–Stokes equations, equation (1). The second derivative on the right side is usually called a diﬀusion term because it acts like a diﬀusion transport term. When analyzing the stability properties of discretized equations to an imposed

100

50

0

−50 0

50

100

150

+

Fig. 1. Dissipation terms and production term from channel ﬂow DNS data. 963 mesh data ﬁltered onto a 483 mesh. Reτ = 500. : −ε+ : −ε− : SGS ; SGS ; D −εSGS (see equation (6)); ◦: −u v ∂U /∂y; +: −εSGS .

A Dissipative Scale-Similarity Model

269

disturbance, u , using Neumann analysis (see, for example, Chapter 8 in [2]), this term is referred to as a dissipation term. In stability analysis the concern is to dampen numerical oscillations; in connection with SGS models, the aim is to dampen turbulent resolved ﬂuctuations. It is shown in Neumann analysis that the diﬀusion term in the Navier–Stokes equations is dissipative, i.e. it dampens numerical oscillations. However, since it is the resolved turbulent ﬂuctuations, i.e. K in equation (3), that we want to dissipate, we must consider the ﬁltered Navier–Stokes equations for the ﬂuctuating velocity, ui . It is the diﬀusion term in this equation which appears in the ﬁrst term on the right side (ﬁrst line) in equation (3). To ensure that εSGS > 0, we set −∂τik /∂xk to zero when its sign is diﬀerent from that of the viscous diﬀusion term (cf. the two last terms on the second line in equation (3)). This is achieved by deﬁning a sign function ∂τik ∂ 2 ui Mik = sign − , no summation on i, k (4) ∂xk ∂xk ∂xk ¯i − ¯ ui ) until where Mik = ±1. The problem is that we do not know ui (= u the simulations have been carried out. Fortunately, the sign of the second derivative of the resolved velocity ﬂuctuation, ui , is mostly the same as that of the resolved velocity, u ¯i . Figure 2a presents a comparison of the two second derivatives using channel ﬂow DNS data from a 963 grid ﬁltered onto a 483 grid. As can be seen, the RMS of the second derivative of u is larger – or much larger – than that of ¯ u. Figure 2b shows the correlation of the signs of the two second derivatives. It can be seen that the correlation is larger than 95% for y + > 40. Hence equation (4) can be replaced by ¯i ∂τik ∂ 2 u , no summation on i, k (5) Mik = sign − ∂xk ∂xk ∂xk

a

b

x 104 2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0

0 0

50

y+

100

/ 2 / /∂ ¯ u/∂y 2 /; RM S(∂ 2 u /∂y 2 ). :

150

0

100

200

300

400

500

y+

:

Correlation coeﬃcient of ∂ 2 u /∂y 2 and ∂ 2 u ¯/∂y 2 .

Fig. 2. Channel ﬂow DNS data from a 963 grid ﬁltered to 483 . Reτ = 500.

270

L. Davidson

Each component of the divergence of SGS stress tensor in equation (1) is then ˜ ik = max(Mik , 0) i.e. simply multiplied by M D ∂τik ˜ ik ∂τik , no summation on i, k =M ∂xk ∂xk

(6)

D The SGS dissipation, εD SGS = (∂τik /∂xk )uk (cf. equation (3)), is shown in Fig. 1. ˜ ik operates on each cell rather It should be noted that, since the limiter M D than on each face, the SGS diﬀusive ﬂuxes, τik , are not conservative. However, this is unavoidable since we need to control the net force per unit volume, D D ∂τik /∂xk , rather than the stresses at the face, τik . It could also be mentioned D that ∂τik /∂xk is not coordinate invariant; however, this feature is shared by most bounded discretization schemes where numerical limiters are used for the convective ﬂuxes. It can be noted that by using equation (5) rather than equation (4) the model is no longer strictly dissipative in the K = ui ui /2 equation. It is now only 95% dissipative, see Fig. 2b. However, the model is – assuming that the diﬀusion term ν∂ 2 K/∂xj ∂xj in equation (3) is negligible – indeed strictly dissipative in the ¯ ui u ¯i /2 equation. In order to avoid that the sign function changes sign between two iterations within a time step, the second derivatives in equation (5) are evaluated using velocities at the old time step. ˜ ik we omit the back scatter caused by the SGS By using the limiter M stresses; another way to express it is that we exclude the part of the subgrid stress term that acts as counter-gradient diﬀusion.

2 Results 2.1 Decaying grid turbulence The domain is a cubic box of side 4π covered by 64 cells. Figure 3a presents the decay of the turbulent resolved ﬂuctuations versus time and Fig. 3b compares the predicted one-dimensional energy with experimental data. The pile-up of energy at the small scales exhibited by all models occurs because the smallest scales cannot be resolved by the grid. As can be seen, both the decay and the one-dimensional spectrum obtained with the dissipative scale-similarity model are very similar to those obtained with the Smagorinsky model. It can also be seen that the dissipative model is indeed much more dissipative than the original scale-similarity model. 2.2 Fully developed channel ﬂow The Reynolds number is 500 based on the half channel height and the friction velocity. The mesh has 64 × 80 × 64 (x, y, z) cells. The extent of the

A Dissipative Scale-Similarity Model

271

0.6 −2

10

0.4

E1(k)

u´u´ xyz

0.5

0.3

−3

10

0.2 0.1

−4

10

0

0.5

1

1.5

2

2.5

3

0

10

t

1

k

10

Fig. 3. Decaying isotropic grid turbulence. : Dissipative scale-similarity model; : scale-similarity model; : Smagorinsky model (CS = 0.14); ◦ experiments [3]. Left: Decaying of u u xyz versus time; right: Longitudinal one-dimensional spectra. 25 20

U+

15 10 5 0

100

101 y+

Fig. 4. Channel ﬂow. : Dynamic model; : no model; ◦: DNS [4].

102

dissipative scale-similarity model.

computational domain is 3.2 and 1.6 in the streamwise (x) and spanwise (z) directions, respectively. A grid stretching of 12% is used in the wall-normal direction. Figure 4 presents the velocity proﬁles obtained with the dissipative scalesimilarity model, the dynamic model and with no model. No converged results could be obtained with the standard scale-similarity model. As can be seen, no model gives perfect agreement with DNS and the log-law. Hence, this ﬂow is not a good test case for evaluating the accuracy of SGS models. Here it is used to analyze the dissipative scale-similarity model. The dynamic model gives slightly better agreement with DNS than the dissipative scale-similarity model. Figure 5a presents the momentum diﬀusion terms close to the wall. It can be seen that the SGS diﬀusion term evaluated using the standard scalesimilarity model is of opposite sign to that of the viscous diﬀusion. When introducing the sign function in equations (4) and (5), it can be seen that the SGS diﬀusion term takes the same sign as the viscous diﬀusion term

272

L. Davidson

b a

10 0

−10

20

−20 10

−30 −40

0

−50 −60

−10

−70 −20

−80 0

5

10

15

0

10

20

20

D : −5 · ∂τ12 /∂y; : ν∂ 2 ¯ u/∂y 2 ; : −∂u v /∂y; ◦: −5 · ∂τ12 /∂y.

:

y+

30

40

50

D −ui ∂τik /∂xk ; : : −2νsmag sij sij

−ui ∂τik /∂xk ; (Cs = 0.1).

Fig. 5. (a) Terms in the momentum equation. (b) Terms in the K equation.

for y + > 10. The fact that the two terms have opposite signs for y + < 10 simply means that the viscous diﬀusion is very large at instants when the SGS diﬀusion term is set to zero. The diﬀusion due to the resolved shear stress is included in the ﬁgure. It is, as can be seen, much larger (more than ﬁve times) than the SGS term. Figure 5b compares the SGS dissipation from the scale-similarity model with that from the dissipative scale-similarity model (recall that the simulation was carried out with the latter model). As can be seen, the SGS dissipation is indeed much larger with the dissipative model than with the standard model. For comparison, the SGS dissipation, εsmag , is also included.

3 Concluding comments In the proposed new scale-similarity model the back scatter generated by the model is omitted. An alternative way to modify the scale-similarity model is to omit the forward scatter, i.e. to include instants when the subgrid stresses act as counter-gradient diﬀusion. In hybrid LES-RANS, the stresses can then be used as forcing at the interface between URANS and LES. This new approach is the focus of [5].

Acknowledgments The ﬁnancial support of SNIC (Swedish National Infrastructure for Computing) for computer time at C3SE (Chalmers Center for Computational Science and Engineering) is gratefully acknowledged.

A Dissipative Scale-Similarity Model

273

References 1. J. Bardina, J.H. Ferziger, and W.C. Reynolds. Improved subgrid scale models for large eddy simulation. AIAA 80-1357, Snomass, Colorado, 1980. 2. C. Hirsch. Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretization, volume 1. John Wiley & Sons, Chichester, UK, 1988. 3. G. Comte-Bellot and S. Corrsin. Simple Eularian time correlation of full- and narrow-band velocity signals in grid-generated “isotropic” turbulence. Journal of Fluid Mechanics, 48(2):273–337, 1971. ´ 4. Juan C. del Alamo and Javier Jim´enez. Spectra of the very large anisotropic scales in turbulent channels. Physics of Fluids A, 15(6):L41–L44, 2003. 5. L. Davidson. Hybrid LES-RANS: back scatter from a scale-similarity model used as forcing. Phil. Trans. of the Royal Society A, 367(1899):2905–2915, 2009.

Optimization of Turbulent Mixing Restricted by Linear and Nonlinear Constraints Sara Delport, Martine Baelmans, and Johan Meyers Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300A, B3001 Leuven, Belgium, [email protected]; [email protected]; [email protected] Abstract The mixing in a temporal mixing layer after a selected simulation time horizon is maximized by optimizing the initial disturbances on the mean velocity ﬁeld. We concentrate on algorithms which impose the necessary energy and continuity constraints on the initial disturbances. A constrained optimization method is selected and applied to ﬁve diﬀerent cost functionals.

1 Introduction Thanks to the continuing increase of computational resources, it is at present possible to combine optimization techniques with direct numerical simulations or large-eddy simulations for the optimization and control of turbulent-ﬂow features [1, 8]. Important in any problem trying to ﬁnd an optimum to a desired cost functional, are the restrictions imposed by physical, and engineering constraints. In ﬂow optimization, a central relation is posed by the Navier– Stokes equations to which the ﬂow should strictly adhere. On top of that several constraints can enter the problem, and reduce the domain in which the parameter functions one wishes to optimize can freely change. When these constraints are nonlinear, and the parameter space has a large number of dimensions, such as encountered in turbulent-ﬂow optimization, they are not trivially imposed. In the current study, we focus on the enhancement of turbulent mixing in a temporal mixing layer by optimizing the spatial distribution of initial velocity perturbations φ imposed on a tangent-hyperbolic mean velocity proﬁle. Five diﬀerent cost functionals are considered based on the distribution of momentum at the ﬁnal time of a direct numerical simulation (DNS) with a time horizon T . The perturbations in this optimization problem are subject to two constraints: 1 1 ∇ · φ(x) = 0 and C(φ) ≡ φ · φ dx − E0 = 0. 2Ω Ω V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 40, c Springer Science+Business Media B.V. 2010

275

276

S. Delport et al.

The ﬁrst constraint ensures that the initial velocity ﬁeld complies with the incompressible continuity condition, imposing a linear constraint on the initial perturbations. The second constraint imposes the total energy of the perturbations, which is a nonlinear constraint. This constraint is essential for the current case, since a trivial improvement of the mixing at the end of the simulation can in general be obtained by increasing the energy of the initial perturbations.

2 Cost functionals In this work ﬁve diﬀerent cost functionals are used which are all based on the distribution of momentum in the domain Ω at the time horizon T : J = J(u(x, T ))dΩ Ω

with J a functional depending on the velocity ﬁeld at the time horizon u(x, T ) and u(x, T ) inﬂuenced by the initial disturbances φ. By optimizing φ the cost functional is minimized. One possibility to measure the mixing is to measure the width of the interaction zone between the two ﬂuid streams using the momentum thickness. Consequently, a cost functional JM which maximizes momentum thickness is a ﬁrst candidate for mixing optimization. A set of other cost functionals is based on energy considerations related to the mixing process. As a result of mixing, the total amount of kinetic energy in a volume of constant size surrounding the mixing region decreases, since it is converted to heat by viscous eﬀects. The turbulent mixing eﬃciency is related to the speed at which the kinetic energy of the mean ﬂow can be converted to small-scale motions, with high local velocity gradients, such that viscosity can play its role. Based on this mechanism, we formulate three cost functionals, which respectively minimize total kinetic energy JKE , minimize mean-ﬂow kinetic energy JMFE , and maximize turbulent kinetic energy JTKE . An alternative cost functional JENS , which measures the small-scale gradients in the ﬂow, can be based on the volume averaged enstrophy (ω, ω)T /(2Ω), with ω the vorticity (ω = ∇ × u).

3 Constrained optimization method To search the minimum of the cost functionals we employ the conjugate gradient method of Polak–Ribi`ere with a step search based on a variant of the Brent algorithm [2, 6]. To calculate the gradient of the cost functional, the continuous adjoint method (see, e.g. [1, 2]) is used. The linear constraint imposed by the continuity condition is explicitly enforced through parameter elimination [4] and reduces the parameters φ

Optimization of Turbulent Mixing Restricted by Constraints

277

to the parameters F. For the nonlinear energy constraint, two diﬀerent methodologies are investigated. A gradient-projection method [3] is used, which strictly enforces the constraint by projecting the search line on the parameter subspace compliant with the energy constraint. This is compared to an augmented Lagrangian method [4]. The latter adds a Lagrange term and a quadratic penalty term to the cost functional, with respectively Lagrange multipliers λ and μ. The augmented Lagrangian cost functional is repeatedly optimized, after convergence of each subproblem, the Lagrange multipliers are adapted such that the optima of the augmented Lagrangian cost functional gradually converge to an optimum of the constrained optimization problem.

4 Computational setup and discretization We consider a temporal mixing layer with a Reynolds number equal to 100 (based on half the velocity diﬀerence, and the initial vorticity thickness). The domain size is selected to be four times the most unstable wavelength following from linear stability theory. For the streamwise direction L1 = 4λ1 = 61.6 and for the spanwise direction L2 = 4λ2 = 0.6L1 [5]. Finally, in the normal direction, we take L3 = 60, which is large enough to exclude interactions between the boundary and the mixing region in the center of the domain. For the direct numerical simulations of the incompressible Navier–Stokes equations, a mixed pseudo-spectral ﬁnite-volume code is employed. In the two periodic directions a pseudo-spectral Fourier discretization is used. The normal direction is discretized using a fourth-order energy-conserving staggered ﬁnite-volume discretization [7]. The time integration is performed by a fourstage fourth-order accurate Runge-Kutta time integration with CFL number equal to 0.5. The resolution of the mesh is 32 × 32 × 64. By looking at spectra, we veriﬁed that this resolution is suﬃcient to capture the smallest scales in the current optimization cases. The parameters F optimized in this work represent initial perturbations on the mean velocity ﬁeld consisting of 16 Fourier-modes, i.e. with index (α, β) and α, β = 1, 2, 3, 4 (here the Fourier index (α, β) corresponds to wave number (α2π/L1 , β2π/L2 ) ). The imposed energy level per unit volume E0 is 0.1% of the mean-ﬁeld energy (ΔU )2 /8 and time horizon T = 40 is selected.

5 Results 5.1 Comparison of the augmented Lagrangian and gradient projection method The two methods to enforce the energy constraint are now evaluated and compared. The cost functional JTKE is used. In Fig. 1 the convergence history of optimization with the gradient projection method and the augmented Lagrangian method is shown. In Fig. 1a, the evolution of the cost functional

278

S. Delport et al.

a

b

0 −0.02

10

2

|C(φ)|/E0

100

JTKE

−0.04 −0.06 −0.08

10–2 10–4 10–6

−0.1

–8

−0.12 0

50

100

150

200

250

10

0

20

40

60

80

100

120

Iteration number

Iteration number

Cost functional

Relative error on the constraint

Fig. 1. Comparison of the gradient projection (— ◦), augmented Lagrangian method (− − ), gradient projection started from optimum attained by the augmented Lagrangian method (— •). (a): evolution of the cost functional; (b) evolution of the energy constraint (for the augmented Lagrangian).

is displayed; in Fig. 1b, the constraint violation is plotted. First of all, it is appreciated from this ﬁgure that optimization using gradient projection for the constraint, leads to a fast decrease in cost functional during the ﬁrst 40 iterations, afterwards the improvement slows down. An advantage of the gradient projection method is that the optimization cycle can be stopped at any time, since the constraint is strictly enforced. In this work the optimizations using this method were stopped after 200 line searches. In Fig. 1a results using the augmented Lagrangian method (based on the algorithm of Nocedal and Wright [4]) are also presented. The augmented Lagrangian converges after 119 iterations, and the energy constraint is satisﬁed up to 10−10 . However, the value of the cost functional is now higher than obtained using the gradient projection algorithm. Moreover, when the converged solution of the augmented Lagrangian is used as a starting point for the gradient-projection method (Fig. 1a), the cost functional converges further to the same level as the pure gradient-projection optimization. We conclude that the use of a gradient projection methodology to enforce the energy constraint is more robust in the context of our problem. For this reason the gradient projection method is used in the next section. 5.2 Optimization with diﬀerent cost functionals We will now focus on the results of mixing optimization using the gradient projection method for ﬁve diﬀerent cost functionals. Figure 2 displays the evolution as function of iteration number of the turbulent kinetic energy (Fig. 2a), enstrophy (Fig. 2b) evaluated at time t = T . The ﬁgure shows that the optimization converges along the same path when cost functionals JM , JTKE or JMFE are used. Further analyses shows that they converge to similar optima. As observed in Fig. 2, the optimization using cost functionals JENS and JKE follow diﬀerent trends.

Optimization of Turbulent Mixing Restricted by Constraints

b

0.1 0.09

0.035 0.03

0.08 Enstrophy (T)

Turbulent kinetic energy (T)

a

279

0.07 0.06 0.05

0.025 0.02 0.015

0.04 0.01

0.03 0.02

0

50

100 150 Iteration number

0.005

200

0

50

Turbulent kinetic energy

100 150 Iteration number

200

Enstrophy

Fig. 2. Evolution of mixing measures (turbulent kinetic energy (a) and enstrophy (b)) at T = 40 during optimization procedure. Optimization towards JM (◦), JTKE (♦), JMFE ( ), JKE (), and JENS (∇).

a

b

Unoptimized

J

c

TKE,

J

MFE,

J

M

d

J

KE

J

ENS

Fig. 3. λ2 visualization of the velocity-ﬁeld at the time horizon T = 40 colored with the distance from the mixing center line for unoptimized parameters (a), and for parameters optimized to the diﬀerent cost functionals (b), (c), and (d); proﬁle u1 (—).

Figure 2 shows that during optimization towards JM , JTKE , or JMFE , the level of turbulent kinetic energy at ﬁnal time T improves, but that the enstrophy decreases. This indicates that the turbulent kinetic energy for these solutions remains in large-scale modes with low vorticity. In Fig. 3 the ﬂow

280

S. Delport et al.

structures in the solution at T are visualized for the diﬀerent cost functionals. Here, it is observed that the cost functionals JM , JTKE and JMFE lead to large two-dimensional rollers. These solutions maximize the eﬀect of the mixing on mean momentum, but lack the small-scale structures needed for dissipation. In contrast, solutions optimized to JENS and JKE contain a lot of small-scale three-dimensional structures.

6 Conclusion We performed optimization of a turbulent mixing layer subject to a linear and a non-linear constraint. Two methods to impose the non-linear constraint were compared and optimization results for ﬁve cost functionals were presented. The gradient projection method is found to be a robust method to impose the non-linear constraint and is used to optimize mixing with ﬁve diﬀerent cost functionals. The ﬂow with parameters optimized to mean-ﬂow kinetic energy, turbulent kinetic energy or momentum thickness has large two-dimensional vortex structures while the kinetic energy and enstrophy cost functionals lead to complex three-dimensional vortex structures.

Acknowledgements We would like to acknowledge the ﬁnancial support of IWT-Vlaanderen and the FWO-Vlaanderen.

References 1. Bewley TR, Moin P, Temam R (2001), DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms, J. Fluid Mech. 447: 179–225. 2. Delport S, Baelmans M, Meyers J (2009), Constrained optimization of turbulent mixing-layer evolution, J. Turbul. 10 (N18). 3. Luenberger DG (2005), Linear and nonlinear programming, 2nd Ed., Kluwer Academic Publishers. 4. Nocedal J, Wright S (2006), Numerical optimization, 2nd Ed., Springer. 5. Pierrehumbert RT, Widnall SE (1982), The two- and three-dimensional instabilities of a spatially periodic shear layer, J. Fluid Mech. 114:59–82. 6. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996), Numerical recipes in FORTRAN 77: The art of scientiﬁc computing, 2nd Ed., Cambridge University press. 7. Verstappen R, Veldman AEP (2003), Symmetry-preserving discretization of turbulent ﬂow, J. Comput. Phys. 187 (1):343–368. 8. Wei MJ, Freund JB (2006), A noise-controlled free shear ﬂow, J. Fluid Mech. 546:123–152.

Stochastic Coherent Adaptive LES of Forced Isotropic Turbulence Giuliano De Stefano1 and Oleg V. Vasilyev2 1

2

Dipartimento di Ingegneria Aerospaziale e Meccanica, Seconda Universit` a di Napoli, I 81031 Aversa, Italy, [email protected] Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA, [email protected]

1 Introduction The stochastic coherent adaptive large eddy simulation (SCALES) method [3] exploits a wavelet thresholding ﬁlter-based dynamic grid adaptation strategy to solve for the energetic “coherent” eddies in a turbulent ﬂow ﬁeld. The eﬀect of the residual less energetic ﬂow structures is modeled by supplying the simulation with a suitable subgrid-scale (SGS) model. The SCALES approach was successfully applied to the simulation of decaying homogeneous turbulence (e.g., see [2, 10]). Here, the wavelet-based approach is applied to statistically steady turbulence by considering linearly forced homogeneous turbulence at moderate Reynolds-number. Due to the adaptive nature of the present approach, it is preferable to introduce forcing directly in physical space. For this reason, we adopt the linear forcing scheme proposed by Lundgren [5] and extensively studied by Rosales and Meneveau [7]. The continuity and Navier–Stokes equations for incompressible ﬂow can be written in the forced case as ∂ui = 0, ∂xi

(1)

∂ui ∂ui 1 ∂p ∂ 2 ui + uj =− +ν + Qui , ∂t ∂xj ρ ∂xi ∂xj ∂xj

(2)

where ρ and ν are the constant density and kinematic viscosity of the ﬂuid, while Q stands for the linear forcing coeﬃcient assumed constant. The latter can be easily expressed in terms of the turbulence parameters by considering the volume-averaged energy equation in the homogeneous case dK = −ε + 2QK , dt V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 41, c Springer Science+Business Media B.V. 2010

(3)

281

282

G. De Stefano and O.V. Vasilyev

where K stands for the mean kinetic-energy, ε for the turbulent dissipation, and angular brackets denote volume-averaging. Thus, in the equilibrium hyε pothesis for statistically steady turbulence, Q = 2K , and the characteristic time scale of the solution directly links to the forcing parameter – that is τeddy = (3Q)−1 .

2 Adaptive LES By applying the wavelet thresholding ﬁlter [3] to the continuity (1) and Navier–Stokes (2) equations, one gets the following governing equations for the adaptive LES of linearly forced isotropic turbulence ∂ui = 0, ∂xi

(4)

∂ui ∂ui 1 ∂p ∂ 2 ui ∂τij + uj =− +ν − + Qui , ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj

(5)

where ui stands for the wavelet-ﬁltered velocity ﬁeld, and τij = ui uj − ui uj represents the unknown SGS stress tensor. The corresponding evolution equation for the volume-averaged resolved energy becomes dKres = −εres − Π + 2QKres , dt

(6)

where Kres is the mean resolved kinetic-energy and εres the resolved turbulent dissipation. In the above equation the term Π stands for the SGS dissipation, which represents the energy transferred from the energetic resolved scales towards the unresolved background ﬂow. For a given forcing coeﬃcient, as the resolved kinetic-energy practically coincides with that one of the corresponding unﬁltered solution (Kres ∼ = K), the SGS model should provide the right amount of energy dissipation in order to have εres + Π ∼ = ε so to match results from a direct numerical simulation (DNS). In this study the localized dynamic kinetic-energy model (LDKM), proposed in [2], is exploited to close the ﬁltered momentum equation (5). The modeling procedure involves the numerical solution of an additional evolution equation for the SGS kinetic-energy. The corresponding dynamic procedure involves the deﬁnition of two parameters, one for the eddy-viscosity model, the other one for the SGS energy dissipation model. A Bardina-like approach is exploited here for the dynamical evaluation of both the parameters [2]. The ﬁltered momentum and the SGS energy equation are solved by means of the adaptive wavelet collocation methodology [4, 8, 9].

Stochastic Coherent Adaptive LES of Forced Isotropic Turbulence

283

3 Numerical experiments The numerical experiments were conducted starting with the initial velocity ﬁeld obtained by wavelet-ﬁltering of a pseudo-spectral solution at moderate Reynolds-number Reλ ∼ = 72 [1]. In order to build a reference DNS solution, a fully de-aliased pseudo-spectral simulation with 1923 Fourier modes was conducted with the same spectral code. In Fig. 1 the DNS kinetic-energy evolution for diﬀerent values of the forcing coeﬃcient is reported, along with the corresponding time-averaged energy spectra. Note that the actual turbulence Reynolds-number is too low to expect a clear inertial scaling in the energy spectra. The DNS solution is further analyzed by plotting in Fig. 2 the time evolutions of the eddy turnover time and the Taylor Reynolds-number. The diﬀerent numerical solutions show a characteristic time-scale that is in agreement with the theoretical values corresponding to the prescribed forcing coeﬃcients. For instance, it gives τeddy ∼ = 0.056 for Q = 6. 103

1200 Q=3 Q=5 Q=4 Q=6

energy spectrum at t = 4

102

energy

800

400

101 100 10–1 10–2

Q=3 Q=4 Q=5 Q=6

10–3 10–4

0 0

1

2

3

4

5

32

64

wave number

time

Fig. 1. DNS solution: energy evolution and averaged energy spectra for diﬀerent forcing coeﬃcients. 0.2

100

eddy turnover time

0.15

Taylor Reynolds number

Q = 3(0.111) Q = 4(0.083) Q = 5(0.067) Q = 6(0.056)

0.1

0.05

0 0

1

2

3 time

4

5

75

50

25

Q=3 Q=4 Q=5 Q=6

0 0

1

2

3

4

5

time

Fig. 2. DNS solution: eddy turnover time and Taylor Reynolds-number for diﬀerent forcing coeﬃcients.

284

G. De Stefano and O.V. Vasilyev

For the adaptive LES experiments the value Q = 6 was prescribed for the forcing parameter. The simulation was conducted for more than one hundred eddy-turnover times. The SCALES solution was obtained using a maximum resolution of 2563 grid points. However, owing to the choice of an appropriate threshold for the wavelet-ﬁltering procedure, only a very low fraction of these points was actually used during the simulation. The wavelet-based solution is able to reproduce the DNS energy level, as demonstrated in Fig. 3 (left side). More importantly, the energetic small-scale motions are resolved to some extent as it clearly appears by inspection of time-averaged energy spectra illustrated on the right side of the same ﬁgure. The energy content of the ﬂow is mostly captured by a limited number of wavelets. In fact, the SCALES solution uses in average only 3% of the available 2563 wavelets, as demonstrated in Fig. 4, where the grid compression (that is the number of ﬁltered-out wavelets with respect to total ones) is reported. It is worth 1200

103 DNS LDKM

102 101 energy spectrum

resolved & SGS energy

resolved SGS energy DNS 800

400

100 10–1 10–2 10–3 10–4

0

0

1

2

3 time

4

5

10–5

6

16 wave number

32 48 64

Fig. 3. Adaptive LES solution: energy evolution and averaged energy spectrum. 0.3

100

SGS-resolved energy ratio

% compression

98

96

94

92

90 0

1

2

3 time

4

5

6

0.2

0.1

0 0

1

2

3 time

4

5

6

Fig. 4. Adaptive LES solution: grid compression (percentage) and SGS to resolved energy ratio.

Stochastic Coherent Adaptive LES of Forced Isotropic Turbulence

285

100

8000

resolved & SGS dissipation

resolved SGS

% SGS dissipation

80

60

40

20 0 0

1

2

3 time

4

5

6

0 0

1

2

3 time

4

5

6

Fig. 5. Adaptive LES solution: SGS dissipation and percentage of SGS dissipation.

stressing that the actual grid resolution varies in space and in time following the location and size of energetic eddies. On the right side of Fig. 4, the ratio between SGS energy and resolved energy is plotted. This quantity represents a measure of the turbulent resolution achieved by the adaptive LES solution. The results show that the turbulence resolution is maintained during the simulation. Moreover, the SGS dissipation provided by the LDKM procedure is 50% of the total energy dissipation for the present moderate Reynolds-number ﬂow, as reported in Fig. 5. Finally, let us stress the distinctive feature of the SCALES method, namely the ability to locally reﬁne the computational mesh in ﬂow regions where the SGS model does not provide adequate dissipation, thus, resulting in a partial resolution of small dissipative scales. As a result, a direct comparison with DNS solution is more meaningful than for classical non-adaptive LES. However, the statistics that involve the level of energy dissipation must be “corrected” by expressing the deﬁnitions in terms of total dissipation rather than resolved one. For instance, since Kres practically coincides with K, the eddy turnover time is rewritten as τeddy =

(2/3)K , εres + Π

(7)

while, for the Taylor microscale deﬁnition (see, for instance, [6]), λ2 =

10νK . εres + Π

(8)

The evolutions of these two quantities are illustrated in Fig. 6. Note how the LES results, once corrected, match almost perfectly DNS data. In conclusion, the SCALES method supplied with the LDKM modeling procedure is able to reproduce DNS low-order statistics for statistically steady turbulence, with a high grid compression over long-time integrations.

286

G. De Stefano and O.V. Vasilyev 120

0.18

Taylor Reynolds number

eddy turnover time

LDKM (resolved) DNS LDKM (total)

0.12

0.06

90

60

30 LDKM (resolved) LDKM (total) DNS 0

0 0

1

2

3 time

4

5

6

0

1

2

3 time

4

5

6

Fig. 6. Adaptive LES solution: eddy turnover time and Taylor Reynolds-number.

The present approach appears very promising for the simulation of high Reynolds-number turbulent ﬂows with reduced computational cost with respect to classical non-adaptive methods.

Acknowledgments This work was supported by the US Department of Energy (DOE) under Grant No. DE-FG02-05ER25667 and the US National Science Foundation (NSF) under grant No. CBET-0756046. In addition G. De Stefano was partially supported by a research grant from Regione Campania (L.R.5).

References 1. De Stefano G, Goldstein DE, Vasilyev OV (2005) J Fluid Mech 525:263–274 2. De Stefano G, Vasilyev OV, Goldstein DE (2008) Phys Fluids 20: 045102.1–045102.14 3. Goldstein DE, Vasilyev OV (2004) Phys Fluids 16:2497–2513 4. Kevlahan NK-R, Vasilyev OV (2005) SIAM J Sc Comp 26:1894–1915 5. Lundgren TS (2003) Linearly forced isotropic turbulence. In: Annual Research Briefs:461–473. CTR, NASA Ames/Stanford University 6. Pope SB (2000) Turbulent ﬂows. Cambridge University Press 7. Rosales C, Meneveau C (2005) Phys Fluids. 17:1–8 8. Vasilyev OV (2003) Int J Comp Fluid Dyn, 17:151–168 9. Vasilyev OV, Bowman C (2000) J Comp Phys 165:660–693 10. Vasilyev OV, De Stefano G, Goldstein DE, Kevlahan NK-R (2008) J of Turbulence 9:1–14

An Improvement of Increment Model by Using Kolmogorov Equation of Filtered Velocity L. Fang1 , L. Shao1 , J.P. Bertoglio1, G.X. Cui2 , C.X. Xu2 , and Z.S. Zhang2 1

2

´ Laboratoire de M´ecanique des Fluides et d’Acoustique, Ecole centrale de Lyon, Ecully, France, [email protected]; [email protected]; [email protected] Department of Engineering Mechanics, Tsinghua University, Beijing, China, [email protected]; [email protected]; [email protected]

Abstract The velocity increment (VI) model of Brun et al. is improved by employing the Kolmogorov equation of ﬁltered velocity (KEF). The dynamic model coeﬃcient is derived. In high Reynolds number turbulence, it is simpliﬁed in a constant form. A priori and a posteriori tests are made to testify the model performance. The constant coeﬃcient form is low-cost and recommended.

1 Introduction In large-eddy simulation (LES), a way for subgrid modeling is to use the physical knowledge of the turbulence, i.e. the theory of multi-scale energy transfer. In physical space, the CZZS model [1] is an eddy viscosity model based on the two-point energy transfer equation, namely the Kolmogorov equation for ﬁltered velocity (KEF). C. Brun [2] introduced a velocity increment model (VI), which was inspired by M´etais and Lesieur’s structure function formulation [3]. Since VI model is a two-point model, we try to apply KEF to study the two-point energy transfer properties. A ﬁlter size is deﬁned as Δ = π/kc , where kc is the cut-oﬀ wave number in case of spectral cut-oﬀ ﬁlter. The superscripts •< and •> represent the gridscale and subgrid-scale parts respectively. The angle brackets • represent an ensemble average. The subscript l denotes a component in the direction of the two-point distance ξ, and the subscript n denotes the vertical component. KEF for homogeneous isotropic turbulence with high Reynolds number could be described as [4]: 4 < − f ξ = Dlll (ξ) − 6Tl,ll (ξ), (1) 5 where the two-point distance ξ is independent with the ﬁlter size Δ of LES. < The dissipation rate of subgrid scale turbulence f equals −τij< Sij , where V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 42, c Springer Science+Business Media B.V. 2010

287

288

L. Fang et al.

< τij = (ui uj )< − u< i uj . The dissipation term could be represented by using the 11 component in isotropic turbulence:

4 < < − f = 6τ11 S11 . 5

(2)

Deﬁne the velocity increment δu(x, ξ) = u(x + ξ) − u(x). On the right hand < < < (ξ) = δu< side of equation (1), Dlll 1 (ξ)δu1 (ξ)δu1 (ξ) is the third order longi< < tudinal structure function, and Tl,ll (ξ) = u1 (x1 )τ11 (x1 + ξ) represents the subgrid energy transfer. In this paper, we employ this method to improve the VI model. The criterion of selecting the suitable model scales is particularly discussed. To quantify the dissipative properties of VI or Improved VI (IVI) model, an equivalent eddy viscosity is introduced. The a priori results of channel Poiseuille ﬂow show that the IVI model presents correct near-wall behavior and predicts well both forward and backward energy transfer.

2 Improved increment model C. Brun suggested that the subgrid stresses tensor could be modelled by τij< (x, Δ) = Cf (Δ )Qij (x, Δ ),

(3)

where Cf is the dynamic model coeﬃcient, depending on another distance Δ . In this paper, the second order velocity increment tensor Qij with a distance Δ is written as (without summation convention): Qij (x, Δ ) =

< 1 < ui (x + Δ ei ) − u< uj (x + Δ ej ) − u< i (x) j (x) 2 < < + u< uj (x) − u< i (x) − ui (x − Δ ei ) j (x − Δ ej ) .

(4)

It is a little diﬀerent from Brun’s deﬁnition, but also satisfying the tensor symmetry Qij (x, Δ ) = Qij (x, −Δ ), Qij = Qji . From equations (2), (3), (4) and isotropy assumption, the subgrid dissipation could be denoted as / < 5 6 ∂Dlll (r) // 4 < − f ξ = 6Cf Q11 (Δ )S11 ξ = 2Cf ξ , (5) 5 ∂r /r=Δ < and the subgrid transfer term −6Tl,ll (ξ) = −6Cf u< 1 (x1 )Q11 (x1 + ξ, Δ ). Therefore, equation (1) could be written by using VI model, and the model coeﬃcient could be solved as

Cf = 2ξ

< ∂Dlll (r)

∂r

/ / /

r=Δ

< Dlll (ξ) < + 6 u< 1 (x1 )Q11 (x1 + ξ, Δ )

.

(6)

An Improvement of Increment Model

289

There are three diﬀerent scales implied in this result: the physical quantities are ﬁltered at a ﬁlter size Δ; the increment model supposes Cf and Q< ij at a distance Δ ; and the equation itself is based on a distance ξ. The Kolmogorov theory [5] predicts the scaling law of structure function n Fn (r) ∝ rn/3 , where Fn (r) = (u1 (r) − u1 (0)) is the nth order structure function of the non-ﬁltered velocity in longitude direction. Following Germano’s method [6], we could prove that when Δ r, the properties of structure functions are approximately the same between ﬁltered and nonﬁltered velocities. Therefore, the ﬁrst term of the denominator in equation (6) could be simpliﬁed when Δ Δ , both Δ and ξ are in inertial subrange and by using scaling law: / < ∂Dlll (r) // 2ξ < < 2ξ = Dlll (Δ ) = 2Dlll (ξ). (7) ∂r /r=Δ Δ The second term of the denominator in equation (6) corresponds to the transfer term of KEF. According to Meneveau’s analysis [4], it tends to zero when ξ is large. In fact, we could also easily evaluate the magnitude < < < < when Δ ξ, that u< 1 (x1 )Q11 (x1 + ξ, Δ ) ∼ δu1 (ξ)δu1 (Δ )Δu1 (Δ ) < Dlll (ξ). Thus the transfer term in equation (6) could be neglected, and the coeﬃcient value is D< (ξ) 1 Cf = lll< = . (8) 2Dlll (ξ) 2 The value does not depend on ξ, it agrees with the model assumption (3), where Cf is only a function of Δ and Δ . This result is satisﬁed only when the inertial subrange is wide enough, Δ and ξ are both in inertial subrange. In order to neglect the molecular terms in original Kolmogorov equation (Cui et al. [1]), the ﬁlter size of LES should be much larger than dissipation scale η. Finally we could write the multi-scale relation which should be satisﬁed: η Δ Δ ξ L,

(9)

where L is the integrated scale. However, this relation is diﬃcult to be satisﬁed in real LES practice. In fact, we apply the three scales as the same: Δ = Δ = ξ, and employ this constant coeﬃcient approximately in the next section.

3 A priori numerical veriﬁcations Two DNS cases of homogeneous isotropic turbulence with spectral method are used for a priori test. A deterministic forcing method is employed to simulate a statistically stationary turbulence. The computation domain has 2563 grid. The grid size is denoted as h. The two diﬀerent Reynolds numbers Reλ are 50

290

L. Fang et al. Reλ = 50 Reλ = 70 0

(E)kε

–3/2

K

5/3

10

10

–0

10

–2

10

–1

10

kη

0

Fig. 1. The compensate energy spectrum in homogeneous isotropic turbulence. Solid line: Reλ = 50; dashed line: Reλ = 70. Reλ = 50 Reλ = 70

1.5 1

<

<

<

< τ11 S11 >/< Q11S11 >

2

1/2

0.5 0

5

10

15 ξ/η

20

25

30

Fig. 2. The exact value of Cf against diﬀerent ﬁlter size and two-point distance Δ = ξ. Solid line with symbols: Reλ = 50; dashed line with symbols: Reλ = 70.

and 70. The compensate energy spectrums of DNS cases are shown in Fig. 1, where the plateaus represent the inertial subrange in spectral space. Although not obvious, the corresponding wave number is about 0.1 < kc η < 0.3. The corresponding ﬁlter size is 10 < Δ/η < 30. In addition, the integrated scale L ∼ 70η in the two cases, thus we could reasonably consider that η Δ = ξ L, and the error caused from scales is veriﬁed. In homogeneous isotropic turbulence, the exact subgrid stress is calculated with diﬀerent cut-oﬀ wave numbers. The model coeﬃcient is calculated as < < < Cf = τ11 S11 /Q11 S11 . Figure 2 shows the trend that the coeﬃcient value increases with Δ and ξ. And it is about 1/2 when 10 < ξ 18, which is a small part of inertial subrange. Therefore, in a practical LES case, if the ﬁlter size Δ and the two-point distance ξ are ﬁxed in this region, the simpliﬁed subgrid model (8) might be applied, to obtain the low cost in calculation. KEF is based on homogeneous isotropic turbulence, but in wall-bounded shear ﬂow, the local isotropy is also reasonably satisﬁed [1]. The slow parts are considered in shear ﬂow [7]. A DNS case of channel ﬂow is used for a priori test. The Reynolds number is ReH = 7,000, based on the bulk velocity Um and channel half-width H. The grid number is 128 × 128 × 64. The computation domain is 4πH, 2H and 2πH in streamwise, normal and spanwise directions.

An Improvement of Increment Model

a 3E-05

291

b 10

–5

+3

Y

ξ=h ξ = 2h ξ = 4h ξ = 6h

1E-05

v

t

vt

2E-05

10–6

ξ=h ξ = 2h ξ = 4h ξ = 6h

10–7

0 0

100

200 Y+

300

101

Y+

102

Fig. 3. A priori eﬀective νt in channel ﬂow, with two-point distance ξ = h, 2h, 4h, and 6h. The ﬁlter size Δ = ξ. (a) Original coordinate. (b) Log-log coordinate.

The pseudo-spectral method is used for computation. The DNS grid size is denoted by h. With the improved increment model, the eﬀective eddy viscosity < νt could be deﬁned as τij< = −2νt Sij . For comparison, it could be calculated 5 6 5 < <6 < as νt = −Cf Qij Sij /2 Sij Sij . The relation between the two-point distance and the model eﬀect is then analyzed. Figure 3a shows the eﬀective νt with ξ from h to 6h. The ﬁlter size Δ = ξ. Only when ξ ≤ 2h, the near-wall region (about 10 < Y + < 50) has a large eddy viscosity, which satisﬁes the empirical peak location. It means that the two-point energy transfer could only be satisﬁed in a proper two-point distance. The near-wall Y +3 law is satisﬁed well, which is shown in Fig. 3b. The further analysis is on the backward energy transfer property. The subgrid dissipation f represents the energy transfer between resolved and subgrid scales. The contributions of the components of subgrid dissipation were analyzed by Hartel et al. [8]. From his analysis, the subgrid energy is dissipated in streamwise and normal direction, and the backscatter is in spanwise direction. While Cf is always positive, the nondimensionalized subgrid dissipation tensor components + f ij caused by ﬂuctuations could be calculated 5 < 6 + < as f ij = − Qij Sij / rms(Qij )rms(Sij ) (without summation convention). The components of subgrid dissipation are shown in Fig. 4. A priori results are obtained from our DNS case of channel ﬂow. In the ﬁgures, positive value means subgrid energy dissipation, i.e. forward scatter; negative value represents the backscatter. Among the normal components, f 11 is the most important positive component which causes the dissipation, while the spanwise components f 33 has a strong energy backscatter. It shows nice agreement with Hartel’s analysis. The disagreement occurs in the Y + < 8 range, where the light energy backscatter of f 11 is not shown by our model. It might be caused by the strong inhomogeneity in this region. Among the deviatonic components, the most special component is f 12 , which has the backscatter in the region 10 < Y + < 20 and dissipates in the other ranges.

292

L. Fang et al.

a2.5

b ε11+ ε22+ ε33+

2 1.5

2.5

ε12+ ε13+

2

ε23+ 1.5

1

1

0.5

0.5

0

0 0

25

50

Y+

75

100

0

25

50 Y+

75

100

Fig. 4. The contributions of components for normalized subgrid energy dissipation. (a) Normal components. (b) Deviatonic components.

This result is in good agreement with Hartel’s analysis, including the peak locations. The model properties of backscatter could stem from the third order structure function. From equations (5) and (7), the energy dissipation is represented by the third order structure function, which could be diﬀerent values, either negative or positive, depending on diﬀerent directions and regions.

4 Conclusion The VI model is improved by introducing the Kolmogorov equation of ﬁltered velocity (KEF). The model coeﬃcient is solved dynamically, and is proved to be a constant in high Reynolds number turbulence. This constant value could represent the two-point energy transfer properties, and satisfy the classical scaling law. The later applications in shear ﬂow show the properties in nearwall region, respectably agreeing with the empirical and Germano dynamic method for wall correction. A priori results show the energy backscatter of diﬀerent components of subgrid dissipation, in good agreement with Hartel’s results. Therefore, we propose this improved increment model. It is clearly physical sounding. In particular, the constant coeﬃcient formulation (8) is extremely simple. It might be promising when applied in engineering projects.

References 1. G.X. Cui, H.B. Zhou, Z.S. Zhang, and L. Shao. A new dynamic subgrid eddy viscosity model with application to turbulent channel ﬂow. Physics of Fluids, 16(8):2835–2842, 2004. 2. C. Brun, R. Friedrich, and C.B. da Silva. A non-linear SGS model based on the spatial velocity increment. Theoretical and Computational Fluid Dynamics, 20:1, 2006. 3. O. M´etais and M. Lesieur. Spectral large-eddy simulation of isotropic and stably stratiﬁed turbulence. Joural of Fluid Mechanics, 256:157–194, 1992.

An Improvement of Increment Model

293

4. C. Meneveau. Statistics of turbulence subgrid-scale stresses: Necessary conditions and experimental tests. Physics of Fluids, 6(2):815–833, 1994. 5. A.N. Kolmogorov. The local structure of turbulence in incompressible viscous ﬂuid for very large Reynolds number. Proceedings: Mathematical and Physical Sciences, 30:299, 1941. 6. M. Germano. A direct relation between the ﬁltered subgrid stress and the second order structure function. Physics of Fluids, 19(3), 2007. 7. L. Shao, S. Sarkar, and C. Pantano. On the relationship between the mean ﬂow and subgrid stresses in large eddy simulation of turbulent shear ﬂows. Physics of Fluids, 11(5):1229–1248, 1999. 8. C. Hartel, L. Kleiser, F. Unger, and R. Friedrich. Subgrid-scale energy transfer in the near-wall region of turbulent ﬂows. Physics of Fluids, 6(9):3130–3143, 1994.

Symmetry-Preserving Regularization Modelling of a Turbulent Plane Impinging Jet O. Lehmkuhl1,2 , F.X. Trias1 , R. Borrell1,2 , and C.D. P´erez Segarra1 1

2

Centre Tecnol` ogic de Transfer`encia de Calor, Technical University of Catalonia, ETSEIAT, C/Colom 11, 08222 Terrassa, Spain, [email protected]; [email protected]; [email protected]; [email protected]; [email protected] Termo Fluids, S.L., Mag´ı Colet, 8, 08204 Sabadell (Barcelona), Spain

1 Introduction The Navier–Stokes and continuity equations can be written as ρ

∂u + C (u) u + Du + Gp = 0 ∂t Mu = 0

(1) (2)

where u ∈ Rm and p ∈ Rq are the velocities vectors and pressure, respectively. The matrices C (u), D ∈ Rm×m are the convective and diﬀusive operators, respectively. Note the u-dependence of the convective operator (non-linear operator). Finally, G ∈ Rm×q represents the gradient operator, and the matrix M ∈ Rq×m is the divergence operator. In the quest for a correct modelling of them, they can be ﬁltered spatially like in Large Eddy Simulation (LES). Doing so, the ﬁltered non-linear convective term must be modelled, ρ

∂u + C (u) u + Du + Gp = C (u) u − C (u) u ≈ model(u) ∂t

(3)

where the ﬁltered velocity is denoted by u. An appropriate model is hard to accomplish for several reasons. In practice, closure models are often based on phenomenological arguments that cannot be derived formally from the Navier Stokes equations. As a result, this methodology usually has severe numerical instabilities (these diﬃculties are more relevant under non-structured discretization), and is not the best alternative for turbulence modelling in engineering applications. Recently, relevant improvements on turbulence modelling based on symmetry preserving regularization models for the convective term (model(u) in equation (3)) have been developed. They basically alter the convective terms to reduce the production of small scales of motion by means of vortexstretching preserving all inviscid invariants exactly. To do so, symmetries and V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 43, c Springer Science+Business Media B.V. 2010

295

296

O. Lehmkuhl et al.

conservation properties of the convective terms are exactly preserved. Since now, they have been successfully tested for a channel ﬂow at diﬀerent Reynolds numbers [1] and a diﬀerentially heated cavity at high Rayleigh numbers [2] using staggered meshes. The aim of the present work is to test validity of these regularization approaches in more realistic conﬁgurations like a turbulent plane impinging jet and extend this modelling to collocated meshes.

2 C4 regularization modelling We restrict ourselves to the C4 approximation [1]: the convective term in the Navier–Stokes equations is then replaced by the following O(4k )-accurate smooth approximation C4 (u, v) given by C4 (u, v) = C(u, v) + C(u, v ) + C(u , v)

(4)

where a prime indicates the residual of the ﬁlter, e.g. u = u − u and ( · ) represents a symmetric linear ﬁlter with ﬁlter length k . An unstructured box ﬁlter is used in the study. This ﬁlter uses the vertex connectivity to do the agglomeration of the cell neighbours in order to construct the ﬁlter stencil. This ﬁlter is normalised conservative and also self-adjoint. The grid cutting length is evaluated using Deardorﬀ’s proposal [3]. This method is the most widely used today. It consists of evaluating the cutoﬀ length as the cube root of the volume Ω of the ﬁltering cell. Introducing the ratio e = k /i , where k is the ﬁlter cutoﬀ length and i is the grid cutting length, our ﬁnal turbulence model only contains one parameter (e). For further details about ﬁlters properties and how to construct them the reader is referred to [1, 4]. In general, equation (3) relates the approximation given by the regularization modelling (see equation (4)) one-to-one to LES-models if the ﬁlter is invertible. The approximation Cn may be seen in relationship to the approximate deconvolution method. If argument u of C(u, u) is replaced by the following approximate deconvolution of the ﬁltered velocity ﬁeld: u = F −1 u ≈ u, −1 where F approximates the inverse of the ﬁlter F u = u. The approximate of the inverse can be constructed by truncating the formal series expansion of the inverse. In our case, ρ

∂ u + F−1 F C( u, u ) + D u + G p=0 ∂t

(5)

if we take u = u, p = p, and Cn (u, u) = F −1 F C( u, u ), it is mathematically evident that any direct modiﬁcation of the convective term in the Navier– Stokes equation is simply related to an approximate deconvolution operator. From a physical point of view, however, not all, but only some modiﬁcations constitute a proper approximate model.

SP Regularization Modelling of a Impinging Jet

297

3 Numerical method: symmetry preserving discretization For the temporal discretization of the momentum equation (1), a second order backward diﬀerence scheme is used for the time derivative term, a fully explicit second-order one-leg scheme [5] for right-hand-side terms of the momentum equation except the pressure gradient (R (uc ) := Ω−1 (−C (uc ) uc − Duc )) and a ﬁrst-order backward Euler scheme for the pressure-gradient term. Incompressibility constraint is treated implicit. Thus, we obtain the fully-discretized Navier–Stokes equations Mun+1 =0 (6) c (β + 1/2) un+1 − 2βunc + (β − 1/2) un−1 c c −Gc pn+1 = R (1 + β) unc − βun−1 c c Δt (7) where the parameter β is computed each time-step to adapt the linear stability domain of the time-integration scheme to the instantaneous ﬂow conditions in order to use the maximum Δt possible [6]. The conservative nature of the Navier–Stokes equations is intimately tied up with the symmetries of the diﬀerential operators. In the following sections, we will see that retaining the symmetry properties of the continuous operators when discretizing equations is necessary in order to exactly conserve the inviscid invariants in a discrete sense. 3.1 Kinetic energy conservation The global discrete energy is deﬁned as ||uc ||2 ≡ u∗c Ωc uc

(8)

The evolution equation of ||uh ||2 can be obtained by left-multiplying equation (1) by u∗h and summing the resulting expression with its conjugate transpose ρΩc

d ||uc ||2 = −u∗c (Cc (us ) + Cc ∗ (uc )) uc − u∗c (D + D∗c ) uh dt −u∗c Ωc Gc pc − p∗c G∗c Ωc uc

(9)

In absence of diﬀusion, that is Dc = 0, the global kinetic energy ||uc ||2 is conserved if both the convective and pressure terms vanish (for any uc , Mc uc = 0c ) in the discrete energy equation, i.e., u∗c (Cc (us ) + Cc ∗ (uc )) uc = 0

(10)

u∗c Ωc Gc pc + p∗c G∗c Ω∗c uc = 0

(11)

These conservation properties are held if and only if the discrete convective operator is skew-symmetric (Cc (us ) = −Cc ∗ (uc )) and if the negative

298

O. Lehmkuhl et al.

conjugate transpose of the discrete gradient operator is exactly equal to the ∗ divergence operator (− (Ωc Gc ) = Mc ). Therefore, if the convective and gradient operators are properly chosen, the global kinetic energy equation (9) reduces to d ||uc ||2 = −u∗c (Dc + D∗c ) uc (12) dt Since the diﬀusive terms must be strictly dissipative (u∗c (Dc + D∗c ) uc ≥ 0) the diﬀusive operator, Dc , must be symmetric and positive-deﬁnite. 3.2 Solving the pressure–velocity coupling – Checkerboard problem To solve the velocity–pressure coupling, a classical fractional step projection method [7] is used, upc = un+1 + Gp˜c (13) c Δt/ (β + 1/2). Taking the divergence where the pseudo-pressure is p˜c = pn+1 c of (13) and applying the incompressibility condition yields a discrete Poisson equation for p˜c Mupc = Mun+1 + MGpn+1 −→ MGp˜c = Mups c c

(14)

Finally, using the deﬁnition of G previous equation becomes Lpc = Mupc

(15)

where discrete laplacian operator L ∈ Rq×q is, by construction, a symmetric positive deﬁnite matrix (L ≡ MΩ−1 M∗ ). Once the solution is obtained, un+1 c results from the correction = upc − Gp˜c un+1 c

(16)

Once the cell-center predictor velocity ﬁeld is calculated and using the gradient deﬁnition (i.e. G ≡ −Ω−1 M∗ ) , we can derive the following relations, = upc − Ω−1 M∗ L−1 Mupc un+1 c = I − Ω−1 P upc

(17) (18)

where matrix P ∈ R3q×3q is, by construction, symmetric positive deﬁnite. The origin of checkerboard problem is related with the unrealistic components of cell-centred velocity ﬁeld that projection matrix cannot eliminate, uCB = I − Ω−1 P uCB (19) c c that means that uCB lies in the kernel of P matrix (uCB ∈ Ker (P)). c c Hence, those velocity components that lie on the kernel of matrix P cannot be corrected. The most common way to tackle this problem is to slightly modify matrix P. Such modiﬁcations are generally introduced in the divergence

SP Regularization Modelling of a Impinging Jet

299

operator M as in the traditionally used Momentum Weighted Interpolation Method [8]. Our approach is to change the equation (16) with a Least Squares gradient reconstruction method. Doing so, the properties of matrix P are slightly changed and the system is conditioned without any modiﬁcation in the convective operator like in the Rhie and Chow technique [8].

4 Numerical results Impinging jets are used in heating, cooling or drying processes for the production of paper, glass, cooling of turbine blades, air curtains, etc. If turbulent impinging jets have been the subject of a lot of experimental research works, they have been less studied by numerical simulations like DNS or LES. The purpose of this study is to investigate the ability of the before presented regularization methodology to predict the overall ﬁeld quantities in the plane turbulent impinging jet of aspect ratio H/B = 4 (where H and B are the height and the nozzle width respectively) with a moderate Reynolds number of 2 · 104 using two diﬀerent experimental data from Ashforth–Frost et al. [9] and Zhe and Modi [10]. The computational domain is a rectangular box of dimensions Lx, H and Lz. The nozzle is centred on the upper boundary of the computational domain. The inlet vertical velocity is obtained from the Reynolds number value. No-slip condition is imposed at solid walls. At the exit boundary, a pressure outlet boundary condition is used. Periodic boundary condition is used in the spanwise direction. Two diﬀerent grids are used to evaluate the accuracy of the symmetry preserving formulation. The ﬁrst one is a coarse grid (m1) of 11.136 Control Volumes (CV) and the last one a medium sized grid (m2) of 94.080 CV. In order to solve the equation (14), a Fourier decomposition method is used. In Fig. 1a streamwise velocity at three diﬀerent locations in the impingement plate is plotted together with experimental data. Rather good agreement

a

0.7 0.6

0.3

exp [10] exp [9] C4 m1 C4 m2

0.25

X/B=7

0.2

0.4

y/B

y/B

X/B=4

X/B=2

0.5

b

exp [10] exp [9] C4 m1 C4 m2

X / B=2

X / B=4

X / B=8

0.15

0.3

0.1

0.2

0.05

0.1 0

0

0.5

1

1.5 U/Vref

2

2.5

3

0

0

0.2

0.4

0.6 0.8 1 Urms/Vref

1.2

1.4

Fig. 1. Streamwise velocity (a) and Reynolds stresses (b) at three diﬀerent locations in the impingement plate. Lines: numerical solutions using two meshes; symbols: experimental data from Ashforth–Frost et al. [9] and Zhe and Modi [10].

300

O. Lehmkuhl et al.

is obtained for the current results and the numerical data. Even for the simulation with the coarse mesh the results are reasonable good. The presented results all describe the mean ﬂow, which is steady in the time. However, the variation of the ﬂow is important and the Reynolds stresses are mostly used to quantify the ﬂuctuations of the ﬂow. In Fig. 1b some comparison of Reynolds stresses at three diﬀerent locations in the impingement plate between diﬀerent meshes and the experimental data is showed. There is a good agreement with experimental data for the ﬁne grid, but in this case the coarse grid does not perform well.

5 Conclusions In this paper a numerical approach for an accurate simulation of Navier– Stokes equations under turbulent complex ﬂows has been presented. As a ﬁrst step, the regularization turbulence modelling has been described taking special attention to the diﬀerent ﬁlters to be used. After this, the symmetry preserving discretization methodology has been extended to collocated meshes. Finally, using this formulation and modelling, some numerical simulations of a plane turbulent impinging jet with a moderate Reynolds number of 2 · 104 and H/B = 4 have been presented. A detailed validation using our numerical results together with experimental data from the literature have also been performed. As a conclusion, overall good agreement between regularization modelling using coarse meshes and reference data have been observed for both ﬁrst- and second-order statistics.

References 1. Roel Verstappen. On restraining the production of small scales of motion in a turbulent channel ﬂow. Computers and Fluids, Vol. 37(7):887–897, 2008. 2. F. X. Trias, R. Verstappen, M. Soria, and A. Oliva. Symmetry-preserving regularization modelling of a turbulent diﬀerentially heated cavity. In LES in Science and Technology, Poznan, Poland, April 2008. 3. P. Sagaut. Large Eddy Simulation for Incompressible Flows. Springer-Verlag, 2001. 4. A. W. Vreman. The adjoint ﬁlter operator in large-eddy simulation of turbulent ﬂow. Physics of Fluids, 16:2012–2022, 2004. 5. R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-Preserving Discretization of Turbulent Flow. Journal of Computational Physics, 187:343–368, May 2003. 6. F. X. Trias, M. Soria, A. Oliva, and C. D. P´erez-Segarra. Direct numerical simulations of two- and three-dimensional turbulent natural convection ﬂows in a diﬀerentially heated cavity of aspect ratio 4. Journal of Fluid Mechanics, 586:259–293, 2007. 7. N. N. Yanenko. The Method of Fractional Steps. Springer-Verlag, 1971.

SP Regularization Modelling of a Impinging Jet

301

8. C. M. Rhie and W. L. Chow. Numerical study of the turbulent ﬂow past an airfoil with trailing edge separation. AIAA Journal, 21:1525–1532, 1983. 9. S. Ashforth-Frost, K. Jambunathan, and C. F. Whitney. Velocity and Turbulence Characteristics of a Semiconﬁned Orthogonally Impinging. Exp. Thermal and Fluid Science, 14:60–67, 1997. 10. J. Zhe and V. Modi. Near Wall Measurements for a Turbulent Impinging Slot Jet. Journal of Fluids Enginnering, 123:112–120, 2001.

Progress in the Development of Stochastic Coherent Adaptive LES Methodology Oleg V. Vasilyev1 and Giuliano De Stefano2 1

2

Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA, [email protected] Dipartimento di Ingegneria Aerospaziale e Meccanica, Seconda Universit` a di Napoli, I 81031 Aversa, Italy, [email protected]

1 Introduction Stochastic Coherent Adaptive Large Eddy Simulation (SCALES) methodology, originally proposed by Goldstein and Vasilyev [1], is an extension of the Large Eddy Simulation (LES) approach that uses a wavelet ﬁlter-based dynamic grid adaptation strategy to solve for the most energetic coherent structures in a turbulent ﬂow ﬁeld, while modeling the eﬀect of the less energetic eddies. Despite some similarities, SCALES is drastically diﬀerent from classical LES, mainly in its ability to couple grid evolution and ﬂow physics. This direct coupling is achieved by resolving and “tracking” on a space–time adaptive mesh physically important ﬂow structures, whose evolution is inﬂuenced by a subgrid scale (SGS) model. This coupling provides a unique feedback mechanism that allows numerics to compensate for the inadequacies of the SGS model: mesh is automatically reﬁned in ﬂow regions, where the SGS model does not provide adequate dissipation, and coarsened in regions, where the model is over-dissipative.

2 SCALES methodology Let us very brieﬂy outline the main features of the wavelet thresholding ﬁlter used for the coherent vortex identiﬁcation and grid adaptation. More details can be found, for instance, in [2]. A velocity ﬁeld ui (x) can be represented in terms of wavelet basis functions as ui (x) =

l∈L0

+∞ 2 −1 n

c0l φ0l (x)

+

j=0

μ=1

μ,j dμ,j k ψk (x),

(1)

k∈Kμ,j

where φ0l and ψkμ,j are n-dimensional scaling functions and wavelets of diﬀerent families and levels of resolution, indexed with μ and j, respectively. One way V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 44, c Springer Science+Business Media B.V. 2010

303

304

O.V. Vasilyev and G. De Stefano

to think of a wavelet decomposition is as a multi-level or multi-resolution representation of ui , where each level of resolution j (except the coarsest one) consists of families of wavelets ψkμ,j having the same scale, but located at diﬀerent positions. Scaling function coeﬃcients represent the averaged values of the ﬁeld, while the wavelet coeﬃcients represent the details of the ﬁeld at diﬀerent scales. Wavelet ﬁltering is performed in the wavelet coeﬃcient space by discarding coeﬃcients below a given threshold ≥

ui (x) =

l∈L0

+∞ 2 −1 μ,j + dμ,j k ψk (x), n

c0l φ0l (x)

j=0

(2)

μ=1 k ∈ Kμ,j |dμ,j k | > u

where > 0 stands for the non-dimensional (relative) threshold parameter, u is the (absolute) dimensional velocity scale. Note that the wavelet thresholding ﬁlter (2) is uniquely deﬁned by the nondimensional threshold parameter and the velocity scale u. Depending on the threshold level, the eﬀect of the discarded (unresolved or subgrid scale) modes on the evolution of energetic (resolved) ﬂow structures can be insigniﬁcant or substantial. In the latter case, this eﬀect must be modeled. When the threshold is chosen simply to satisfy numerical accuracy and the eﬀect of the subgrid scales is negligible, we call this method Wavelet based Direct Numerical Simulation, or WDNS [3]. For larger values of the threshold parameter, it was shown that the unresolved SGS ﬁeld is nearly Gaussian white noise by [1, 4], which, due to its decorrelation with the resolved modes, results in practically no SGS dissipation. Therefore, simulations with no SGS model capture turbulent energy cascade and were shown to recover low order and some high order DNS statistics. This regime is called Coherent Vortex Simulation [5]. Further increase in the wavelet threshold parameter results in discarding of too many modes so that the energy cascade is no longer captured, which necessitates the use of a SGS model. This regime corresponds to the SCALES approach [1]. The ﬁrst step towards the construction of SGS models for SCALES was undertaken in [6], where a global dynamic Smagorinsky eddy viscosity model (GDM) based on the classical Germano procedure redeﬁned in terms of two wavelet thresholding ﬁlters was developed. The main drawback of this formulation is the use of a global (spatially non-variable) model coeﬃcient. In order to realize the full beneﬁt of SCALES approach for highly non-homogenous ﬂows in complex geometries two diﬀerent families of local dynamic SGS models have been developed: 1. Lagrangian Dynamic Model (LDM) with path-line/tube averaging [7] 2. Localized Dynamic Kinetic Energy based Models (LDKM) [8]

Progress in the Development of SCALES Methodology 1

b

Filtered DNS, eps = 0.43 Global Dynamic Model Lagrangian Dynamic Model Ksgsmodel (DSM) Ksgs model (LKM) LES

0.8

% compression (discarded points)

Resolved Kinetic Energy

a

0.6

0.4

0.2

0

100

99.9

99.8

99.7 Global Dynamic Model Lagrangian Dynamic Model Ksgs model (DSM) Ksgs model (LKM)

99.6

99.5 0

0.1

0.2

0.3

0

0.2

time

c

0.4

0.6

time

d

50 Global Dynamic Model Lagrangian Dynamic Model Ksgs model (DSM) Ksgs model (DSM)

102 101

Energy Spectrum

40 % SGS Dissipation

305

30

20

10

100 10−1

Global Dynamic Model Lagrangian Dynamic Model Ksgs model (DSM)

10−2

Ksgs model (LKM)

10−3

LES DNS, Reλ=72 DNS, Re =72, ε = 0.43 λ

0 0

0.2

0.4

0.6

10−4 0 10

101 wavenumber

time

102

e 10 Enstrophy (Dissipation) Spectrum

3

102 101 100 10−1 10−2 0 10

Global Dynamic Model Lagrangian Dynamic Model Ksgs model (DSM) Ksgs model (LKM) LES DNS, Reλ=72 DNS, Reλ=72, ε=0.43

101 wavenumber

102

Fig. 1. Energy decay (a); Field compression (b); Percentage of SGS (modeled) dissipation (c); Energy (d) and Enstrophy (e) density spectra at t = 0.08 for SCALES of decaying homogeneous turbulence at Reλ = 72 with Global Dynamic ( ), ), Kinetic Energy Dynamic Structure ( ), Local Lagrangian Dynamic ( ) models, reference LES with global and Localized Dynamic Kinetic Energy ( dynamic model ( ), DNS ( ), and wavelet ﬁltered DNS ( ◦ ).

306

O.V. Vasilyev and G. De Stefano

The LDM consists of Smagorinsky eddy-viscosity model with the dynamic procedure based on statistical moving ﬁltered averages over the trajectory of a ﬂuid particle, x = x (t): τ −t 1 t I i (x, t) = e T G (y − x (τ )) Ii (x (τ ) , τ ) dτ dy, T −∞ D where G (ξ, x) is the local low-pass ﬁlter moving together with ﬂuid particle and Ii (i = 1, 2) are instantaneous quantities used by the local dynamic model. The Localized Dynamic Kinetic Energy Models, of both eddy-viscosity [9] and non eddy-viscosity [10] types, involve the solution of an evolution equation for the additional ﬁeld variable that represents the kinetic energy associated with the unresolved motions. This way, the energy transfer between resolved and residual ﬂow structures is explicitly taken into account by the modeling procedure without an equilibrium assumption of the classical dynamic Smagorinsky approach. To demonstrate the eﬃciency and accuracy of the SCALES approach, the results of the simulation of incompressible isotropic freely decaying turbulence using diﬀerent SGS models are shown in Fig. 1. The initial velocity ﬁeld is a realization of a statistically stationary turbulent ﬂow at Reλ = 72 (λ being the Taylor microscale) as provided by a pseudo-spectral DNS [11]. The unique feature of the SCALES approach, namely the coupling of modeled SGS dissipation and the computational mesh, is illustrated in Fig. 1b, c: more grid points are used for models with lower levels of SGS dissipation. In other words, the SCALES approach compensates for inadequate SGS dissipation by increasing the local resolution and, hence, the level of resolved viscous dissipation. For example, the decrease of SGS dissipation in the Dynamic Structure Model (DSM) results in the decrease of grid compression and the increase of resolved energy dissipation. Another crucial strength of the SCALES approach is its ability to match the DNS energy and enstrophy density spectra (illustrated in Fig. 1d, e) up to the dissipative wavenumber range using very few degrees of freedom. It is important to emphasize that for all localized models the close match is achieved using less than 0.4% of the total non-adaptive nodes required for a DNS with the same wavelet solver (Fig. 1b). To highlight the signiﬁcance of such a close match, it is interesting to compare these results with those of an LES with the global dynamic Smagorinsky model. Despite the fact that LES uses almost four times the number of modes (1.56%), it fails to capture the small-scale features of the spectrum.

Acknowledgments This work was supported by the US Department of Energy (DOE) under Grant No. DE-FG02-05ER25667 and the US National Science Foundation (NSF) under grant No. CBET-0756046. In addition G. De Stefano was partially supported by a research grant from Regione Campania (L.R.5).

Progress in the Development of SCALES Methodology

307

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Goldstein DE, Vasilyev OV (2004) Phys Fluids 16:2497–2513 Daubechies I (1992) Ten Lectures on Wavelets, SIAM, Philadelphia Vasilyev OV (2003) Int J Comp Fluid Dyn 17:151–168 Farge M, Pellegrino G, Schneider K (2001) Phys Rev Lett 87:054501-1–054501-4 Farge M, Schneider K, Kevlahan N-KR (1999) Phys Fluids 11:2187–2201 Goldstein D, Vasilyev O, Kevlahan, N-KR (2005) J of Turbulence 6:1–20 Vasilyev OV, De Stefano G, Goldstein, DE, Kevlahan, N-KR (2008) J of Turbulence 9:1–14 De Stefano G, Vasilyev OV, Goldstein DE (2008) Phys Fluids 20:045102.1– 045102.14 Ghosal S, Lund TS, Moin P, Akselvoll K (1995) J Fluid Mech 286:229–255 Pomraning E, Rutland CJ (2002) AIAA Journal 40:689–701 De Stefano G, Goldstein DE, Vasilyev OV (2005) J Fluid Mech 525:263–274

Part IV

Scalars

LES of Heat Transfer in a Channel with a Staggered Pin Matrix G. Delibra, D. Borello, K. Hanjali´c, and F. Rispoli Dipartimento di Meccanica e Aeronautica, “Sapienza” University of Rome, Via Eudossiana 18, 00184 Roma, Italy, [email protected]; [email protected]; [email protected]; [email protected] Abstract Flow and heat transfer through a matrix of 8 × 8 cylindrical adiabatic rods in a staggered arrangement and connecting two non-isothermal walls of a plane channel have been studied with LES using an in-house unstructured ﬁnite-volume computational code. The results for the mean ﬂow properties agree well with the experimental data of Ames et al. The instantaneous velocity and temperature ﬁelds, vortical structures and their wall-signatures, unsteadiness and three-dimensionality – none of which is accessible to experiments - have been analyzed to gain a better insight into the ﬂow dynamics and its eﬀects on the local and averaged heat transfer.

1 Introduction Large-eddy simulations (LES) of technologically-relevant wall-bounded ﬂows and heat transfer at higher Reynolds number are still burdened with uncertainties related to the proper numerical resolution with an aﬀordable computational mesh, numerical issues in mesh quality, optimum subgrid-scale modelling, treating conjugate wall thermal conditions and others. We studied a ﬂow through a staggered pin matrix in a plain channel [1]. Unlike in heat exchangers, where tubes are usually heated/cooled by interior ﬂuid, here the lower endwall was heated with an imposed constant temperature. Thus, the main function of the thermally inactive pins is to enhance heat transfer between the two bounding non-isothermal walls by promoting vortex shedding and turbulence. The aim is to investigate the feasibility of applying LES to predict ﬂow and heat transfer in realistic conﬁgurations of gas-turbine blade cooling using an aﬀordable computational grid, but also to study vortical patterns and turbulence features around each pin and at the pin-endwall junctions, and eﬀects of ﬂow unsteadiness and three-dimensionality on the thermal ﬁeld and heat transfer.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 45, c Springer Science+Business Media B.V. 2010

311

312

G. Delibra et al.

2 Flow speciﬁcation and computational details The experimental pin ﬁn section consists of a matrix of 8 × 8 rods of diameter D = 2.54 cm, connecting the channel endwalls [1] . The rod spacing is 2.5D both in the streamwise and spanwise directions and the channel height (pin length) is h = 2D. In reality, the coolant ﬂowing through the channel removes heat from the walls through the pins, thus the main interest is in heat transfer on the pins and endwalls. Here, as in the experiment, the pins are treated as adiabatic and non-conducting, serving only to generate unsteady three-dimensional vortical structures and turbulence, all aimed at enhancing endwall heat transfer. The Reynolds number, based on pin diameter D and the averaged velocity between the minimum passage area between the pins, Vmax , was set to 10,000 to correspond to one of the experimentally investigated cases. The heated surface extends from 2.75D downstream from the inlet to 2.75D upstream from the outlet, Fig. 1. The solution domain encompasses one streamwise row of rods with periodic boundary conditions imposed on the domain sides. Two unstructured meshes were used containing 2 and 5.5 million cells, clustered in the vicinity of rods and the endwalls. For the ﬁner mesh y + of the wall-nearest cell centers was less than 1 over the whole domain, Fig. 1. The center of the ﬁrst pin is located 7.5D downstream from the nozzle, thus allowing for some development of boundary layers on endwalls. The channel exit is located 7.5D downstream from the center of the eighth pin. One of the uncertainties in mimicking experiment is the proper speciﬁcation of inﬂow conditions. In order to diminish as much as possible the eﬀect of the inﬂow and outﬂow conditions, we considered in all computations the complete experimental channel length. In the experiment the ﬂuid enters the channel through a nozzle with a free stream turbulence of 1.4%. In compliance with this we imposed a constant streamwise bulk velocity of 0.625Vmax and the same turbulence level. No-slip conditions were imposed on solid walls for velocity, and zero ﬂux conditions for heat transfer, except for heated bottom endwall. The temperature of the incoming ﬂuid is 291 K and the heating was imposed by setting a constant temperature of 316 K on the heated portion of the lower endwall, Fig. 1. The temperature ﬁeld is treated as a nondimensional passive scalar.

Fig. 1. Solution domain (left); grid discretisation around the rod (right).

LES of Heat Transfer in a Pinned Channel

313

The simulation was performed using the in-house unstructured collocated control-volume-based Navier–Stokes solver T-FlowS [2]. The subgrid scale (sgs) deviatoric stresses were modelled with the dynamic Smagorinsky model. A second-order CDS was used for discretizing the convective terms for all variables. The iterative pressure correction algorithm (SIMPLE) was used for the pressure–velocity coupling, and the diagonally preconditioned conjugate gradient (CG) method was used to solve the linearised system of equations. The time marching was performed using a fully-implicit three-level time scheme, with the time step chosen to ensure a typical CFL number always lower than 0.8. We performed 50,000 timesteps (corresponding to more than two ﬂowthrough times). For the simulation we used a CentOs LINUX cluster featuring up to 32 XEON processors for a total CPU time equal to 30,000 h.

3 Discussion of results It is noted that ﬂow around the ﬁrst rod is predominantly laminar except for some weak turbulence generated in the wake, and LES on both grids (as well as URANS and hybrid models) returned very good agreement with experiments. The performance of LES is better illustrated in subsequent rods, and as an illustration we shows in Fig. 2 the averaged velocity proﬁles and pressure coeﬃcient along lines A and B for the second rod, compared with the experimental results. The ﬁner-grid LES returns a credible representation of the velocity proﬁles between the rods in the same row (Fig. 2b). Similar quality of agreement with experiments was found for other rods downstream. The pressure coeﬃcient Cp around all rods complies also well with the experimental data, Fig. 2c. We proceed by presenting other results only for the ﬁner grid, considered to be more credible. Snapshots of the instantaneous streamlines in two horizontal planes, Fig. 3, at midheight z/h = 0.5 (up) and close to the bottom wall at z/h = 0.05 (below) give an impression of the ﬂow unsteadiness. In the near-wall plane, the pattern is somewhat less vigorous due to the wall and viscous suppression (y + varies from ≈10 to 40 along the ﬂow), but it is 1.4

0

1.2

−0.5

LES 2M LES 5M exp

−1

0.8 row #2

0.6

LES 2M LES 5M exp

0.4 0.2 0 0

0.5 1 1.5 distance from pin (cm)

pin #2

Cp

U/ Vmax

1

−1.5 −2 −2.5

0

50 100 theta [deg]

150

Fig. 2. Streamwise velocity proﬁles and pressure coeﬃcient near the second pin along reference lines A and B, respectively.

314

G. Delibra et al.

Fig. 3. Snapshots of streamlines patterns in the midplane at z/h = 0.5 (up) and in a near-all plane at z/h = 0.05 (y + ≈ 10 − 40) (below); gray shades denote the temperature ﬁeld.

Fig. 4. Instantaneous temperature ﬁeld in a near-wall plane at z/h = 0.05, where y + varies from ≈ 10 behind the 1st pin row to ≈ 40 behind the last pin.

still markedly unsteady in the whole domain. The LES realisation highlights a spanwise ﬂapping of the wakes after the 4th row and a visible interaction of diﬀerent wakes leading to ﬂuid meandering from one wake to another. It is worth noting that U-RANS and hybrid simulations [3] (not shown here) do not show the same intensity of direct wake interactions. The temperature ﬁeld, indicated by gray shades in Fig. 3, shows the development of temperature plumes sustained by the complex streamlines pattern. A better view of the thermal activity is presented in Fig. 4 showing a snapshot of the instantaneous temperature in the near-wall plane z/h = 0.05. In view of the close proximity of the heated wall kept at constant temperature, the instantaneous ﬂuid temperature in Fig. 4 shows a remarkable nonuniformity with hot and cold spots permeating over the whole area. A familiar horse-shoe pattern of cold ﬂuid (dark shades) is visible only around the ﬁrst two rod rows reﬂecting intensive cooling by the necklace vortices. It is expected that the temperature ﬁeld in the rest of the passage should also bear signature of the vortical pattern, but its interpretation can be envisaged only by a blowup analysis of diﬀerent regions.

LES of Heat Transfer in a Pinned Channel

315

Fig. 5. Streamlines and temperature shades in ﬂow-normal planes between neighbouring pins (second, fourth, sixth and eighth row) and near the outlet.

Figure 5 illustrates further the complexity of the three-dimensional vortical structure by showing the instantaneous streamline patterns with superimposed shades of the temperature ﬁeld in four ﬂow-normal cross-sectional planes between the successive rows along the ﬂow. Each of these planes extends between the two rods in the same row. The last picture refers to a section placed 2.5D downstream from the last pin. Each pattern is very different from one another, and so are the diﬀerent time realisations in each plane, indicating vigorous time dynamics and spatial evolution. A remarkable coincidence of the streamlines patterns and the contours of thermal plumes (identiﬁed by isolines/shades of constant temperature), especially the closed loops, conﬁrms that the vortical structures engulf cold or hot ﬂuid and convect it while moving and expanding, and this mechanism seems to be crucial in enhancing heat transfer as compared with common wall-attached boundary layers. Moving downstream, the vortex structures expand while weakening in strength, and the mixing of the cold and hot ﬂuid leads gradually to more uniform streamline patterns and temperature ﬁeld. Downstream from the last pin the extent of the ﬂow three-dimensionality diminishes and consequently the temperature ﬁeld returns towards a two-dimensional stratiﬁcation. The time averaged Nusselt number, normalised with the bulk value, Nu/Nuave , is shown in Fig. 6 separately for the fore and aft four pins with different gray-scale bars to match the experimental records. The Nu distribution in both segments agrees reasonably well with experiments, though the bulk averaged value Nu = 44 is notably lower than the experimental value of 54. This is surprising in view of good agrement of other mean ﬂow properties and could be attributed to a mismatch of the inﬂow turbulence level and scales, or to an eﬀect of radiation in the experiment, which was not accounted for in the simulations. The frequency spectrum of velocity signals at a point placed in the wake region (P in Fig. 1b) shows a peak at Strouhal number of 0.24 for all the pins, but with diﬀerent amplitudes, which indicates a diﬀerent distribution of energy among the modes. For comparison with URANS and Hybrid simulations, see Table 1.

316

G. Delibra et al.

Fig. 6. Averaged Nusselt along the heated wall for the fore (left) and aft four rods (right). Top: experiments; bottom: present LES. Table 1. Velocity ﬂuctuations within the wake. Parameter

LES

Hybrid

URANS

Max excursion (m/s) Strouhal number Mean value (m/s)

2.6 0.24 −0.69

0.6 0.23 0.532

0.004 0.18 0.513

4 Conclusions An LES of ﬂow and heat transfer in a staggered wall-bounded rod array provided an insight into the dynamics of shed vortical structures and their role in local heat removal. The close correlation between the streamline pattern and temperature contours shows that heat is transported by organized vortical systems convecting away from the wall the engulfed temperature ﬁeld. The analysis of unsteadiness showed that LES analysis outperforms URANS and hybrid models in predicting the unsteady velocity ﬁeld and its ﬂuctuations.

References 1. Ames F E, Nordquist C A, Dvorak L A (2007) Endwall heat transfer measurements in a staggered pin-ﬁn array with an adiabatic pin. In: ASME Turbo Expo, GT2007-27432 2. Niceno B, Hanjali´c K, (2005) Unstructured large-eddy- and conjugate heat transfer simulations of wall-bounded ﬂows. In Modeling and Simulation of Turbulent Heat Transfer (Chapter 2), (Developments in Heat Transfer Series), Editors: M. Faghri and B. Sunden, WIT Press, pp. 35–73 3. Delibra G, Borello D, Hanjali´c K and Rispoli F (2008) U-RANS of ﬂow in pinned passages relevant to gas-turbine blade cooling, ETMM7, selected for publications on Int J Heat and Fluid Flows

Turbulent Channel Flow with Λ-Shape Turbulators on One Wall Jaime A. Toro Medina, Benjamin Cruz Perez, and S. Leonardi Department of Mechanical Engineering, University of Puerto Rico at Mayaguez, Puerto Rico, USA, [email protected]; [email protected]; [email protected] Abstract The cooling system of the gas turbine blades plays a critical role in increasing the thermal eﬃciency and power output of advanced gas turbine. In fact, by increasing the heat transfer, the turbine blade can resist to an impinging ﬂuid with higher temperature. Direct Numerical Simulations of Λ-shape turbulators have been performed in order to enhance the heat transfer in the channel of the turbine blades. Several geometrical conﬁgurations have been study in order to obtain conﬁguration that produces the maximum heat transfer.

1 Introduction Gas turbine engines are designed to operate with high turbine inlet temperature. Sophisticated cooling techniques are necessary to maintain a reasonable blade life. One of them consists in the passage of cool air in internal ducts of a blade. To increase the heat transfer the walls are roughened. The heat transfer over rough surfaces have not been completely understood. Kim and Moin [7], Kasagi et al. [5], Kawamura et al. [6] studied the transport of a passive scalar in a turbulent channel ﬂow with smooth walls. Miyake et al. [10] considered the transport of passive scalar over a rough wall made of square bars with w/k = 6 (w is the spacing between roughness elements and k is the roughness height) on the temperature ﬁeld. Leonardi et al. [8] carried out DNSs of turbulent channel ﬂow over square bar roughness varying the pitch to height ratio. They showed how the form drag inﬂuences the frictional and form drag and their correlation with the roughness function. Recently, for the same geometry, Leonardi et al. [9] have studied the dependence of the heat transfer on w/k. Similarly to what happens for the roughness function, they found that w/k = 7 is the geometry which maximizes the heat transfer. The heat and mass transfer over 3D roughness has been studied mostly experimentally [3], and numerically by RANS [4]. Results have shown that 3D

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 46, c Springer Science+Business Media B.V. 2010

317

318

J.A. Toro Medina et al.

A

A

B

B

C

C

k Sec.AA

Sec.BB

Sec.CC w ¸

Fig. 1. Top: velocity vectors superimposed to color contours of spanwise velocity in plane view. Bottom: velocity vectors superimposed to color contours of wall normal velocity for w/k = 3. Deﬁnitions of λ, k, x, y, z are included.

roughness are more eﬀective in enhancing the heat transfer. The present paper represents the ﬁrst attempt to perform Direct Numerical Simulations (DNS) of the turbulent ﬂow in a square channel with Λ-shape turbulators on both the walls. Square rib turbulators are placed on the lower wall (see Fig. 1). Several pitch to height ratio have been considered, w/k = 3, 7, 9, 14 where k is the height of the elements, w the width of the cavity (w = λ − k) for k/h = 0.1 (h is the channel half-height). The angle of inclination of the ribs is 45 degrees. The temperature on the lower and side walls is T = 1 while on the upper wall is T = −1. Therefore heat is transported away from the lower wall and dissipated on the upper wall.

Turbulent Channel Flow with Λ-Shape Turbulators on One Wall

319

2 Numerical procedure The non-dimensional Navier–Stokes and continuity equations for incompressible ﬂows are ∂Ui ∂P 1 ∂ 2 Ui ∂Ui Uj =− + + Πδi1 , ∇·U = 0 (1) + ∂t ∂xj ∂xi Re ∂x2j Π is the pressure gradient required to maintain a constant ﬂow rate, δij is the Kronecker delta, Ui is the component of the velocity vector in the i direction and P is the pressure. The energy equation is ∂T ∂T Uj 1 ∂2T + = , ∂t ∂xj Re P r ∂x2j

(2)

where T is the temperature and P r = ν/α is the Prandtl number, with α the thermal diﬀusivity. The Navier–Stokes and energy equations have been discretized in an orthogonal coordinate system using the staggered central second-order ﬁnite-diﬀerence approximation. Here, only the main features are recalled since details of the numerical method can be found in Orlandi [11]. The roughness is treated by the eﬃcient immersed boundary technique described in detail by Orlandi and Leonardi [12]. The Reynolds number Re = Ub h/ν is 7000, while the Prandtl number P r is set equal to 1; here, Ub is the bulk velocity and ν is the kinematic viscosity, h the channel half height. The computational box is 6h × 2h × πh in x (streamwise), y (wall-normal) and z (spanwise direction) respectively (Fig. 1). Periodic boundary conditions apply in the streamwise and spanwise direction.

3 Results Vectors of the mean velocity (averaged in x, t) are shown in Fig. 1 in plane and side view. Near the side boundaries of the channel, Sec. AA, a separation occurs at the leading edge of the element, and the ﬂow is driven inward, in the cavity between two ribs. Two streams form and follow, to a good approximation, the direction of the Λ-shape ribs. Therefore, the ﬂuid entering in the cavity does not impinge on the following rib, as in 2D transverse ribs [8], but is driven towards the center of the channel. Near the trailing edge of the rib, the ﬂuid has low momentum and in the center of the cavity a weak vortex can be observed. Moving towards the center, ejections can be observed (Sec. BB) near the trailing edge of the rib. The ejection is however limited to a small region of the cavity and the intensity is rather weak so that most of the ﬂuid remains trapped within the cavity. At the center of the channel (Sec. CC), two streams coming from the side merge and for impermeability condition ∂w/∂z is large and it has to be balanced by ∂v/∂y. This causes, all over the roughness layer, a strong ejection. The ejected ﬂow travels all the way to the other

320

J.A. Toro Medina et al.

w/k = 3

w/k = 14

Fig. 2. Velocity vectors superimposed to color contours of wall-normal velocity. Averages in time and streamwise direction.

wall (ﬂat). As a consequence, 2 large scale vortices are formed (Fig. 2). The wall normal velocity intensity is proportional to the intensity of the spanwise velocity, which has a maximum for w/k = 7 and decreases by increasing the distance between the ribs. The larger the spanwise velocity the stronger is the ejection and the penetration into the outer ﬂow. With respect to transverse square bars, the eﬀect of the roughness into the outer layer is much higher and this question the general validity of the Taylor similarity hypothesis. The total heat transfer q is the sum of the molecular conduction 1 ∂T ( P rRe ∂y ) and of the turbulent heat ﬂux T v ([1], page 164). Angular brackets denote averaging with respect to time as well as streamwise and spanwise directions, T is the instantaneous temperature and v is the wallnormal velocity. Since the walls are isothermal, the heat ﬂux has to be constant within the channel, i.e. all the heat transported from the hot walls has to exit from the upper wall. In fact the volume averaged temperature is constant in time (not shown here for lack of space). The recirculations generated by the Λ-shape turbulators aﬀect the turbulent heat ﬂux. Figure 3 shows color contours of the turbulent heat ﬂux averaged in time, in a plane section at half the rib height. Near the kink of the Λ-shaped turbulators, hot ﬂuid is convected outward the wall (T v is positive). Near the side boundaries, cold ﬂuid is convected towards the wall and then T v is negative. Therefore, heat transfer is larger near the center of the duct and at the trailing edge of the ribs turbulators, it is small near the leading edge of the ribs and at the side boundaries of the duct. As a consequence, large temperature gradients occur on the wall, and this may be a concern in structural analysis of turbine blades. The intensity of T v increases with w/k up to a maximum for the case w/k = 7. For larger cavities T v decreases by increasing w/k. In fact, by increasing the distance between the elements the motion of ﬂuid towards the center of the duct and then the ejections become less intense. To quantify the overall heat ﬂux on each surface, the mean value T v + 1 dT ReP r dy at the crests plane of the turbulators has been calculated (an overline denotes averages in x, z, t) for all the cases here considered, varying the pitch

Turbulent Channel Flow with Λ-Shape Turbulators on One Wall

w/k = 3

w/k = 7

w/k = 9

w/k = 14

321

Fig. 3. Color contours of the time averaged turbulent heat ﬂux in a horizontal plane at y = 0.8k from the bottom wall (within the roughness elements). 0.005 0.0045

Tv

+

1 dT ReP r dy

0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 0

2

4

6

8

10

12

14

w/k Fig. 4. Dependence of the total heat ﬂux on w/k.

to height ratio and the height of the turbulators. The maximum heat ﬂux is observed for w/k = 7 and it decreases by increasing the distance between the ribs (Fig. 4). The maximum eﬀect of 2D square or circular rods [2, 8] on the overlying ﬂow was observed for a similar pitch to height ratio.

322

J.A. Toro Medina et al.

4 Conclusions Direct Numerical Simulations of the ﬂow in a square channel with Λ-shape turbulators on both walls have been performed varying the pitch to height ratio. The present simulations clearly demonstrate that Λ-shape roughness can be very eﬀective in enhancing the turbulent (and total) heat transfer. The maximum heat ﬂux occurs when w/k is equal to 7. Turbulent heat transfer distribution showed a highly inhomogeneous heat transfer which cause high temperature gradients on the walls. This may be a concern for turbine blades durability.

Acknowledgments This research was supported in part by the National Science Foundation through TeraGrid resources provided by Texas Advanced Computer Center.

References 1. Davidson P.A. (2004). Turbulence and introduction for scientists and engineers. Oxford University Press. 2. Furuya, Y., Miyata, M. & Fujita, H. (1976). Turbulent boundary layer and ﬂow resistance on plates roughened by wires. J. Fluids Engng. 98, 635−644. 3. Han, J.C., Dutta, S. & Ekkad S. (2000). Gas Turbine Heat Transfer and Cooling Technology. Taylor & Francis. 4. Jang Y., Chen. & Han J. (2001). Flow and Heat Transfer in a Rotating Square Channel with 45 deg Angled Ribs by Reynolds Stress Turbulence Moled. J. Turbomach., 123, 124. 5. Kasagi, N., Tomita, Y. & Kuroda, A. (1992). Direct numerical simulation of passive scalar ﬁeld in a turbulent channel ﬂow. Trans. ASME 114 598–606. 6. Kawamura, H., Abe, H. and Matsuo, Y., (1999). DNS of turbulent heat transfer in channel ﬂow with respect to Reynolds and Prandtl number eﬀect. Internat. J. Heat Fluid Flow 20 196–207. 7. Kim J. & Moin P. (1989). Transport of passive scalars in a turbulent channel ﬂow. Turbulent Shear Flows 6. Springer Verlag, Berlin 85–96. 8. Leonardi, S., Orlandi, P., Smalley, R.J., Djenidi, L. & Antonia, R.A. (2003). Direct numerical simulations of turbulent channel ﬂow with transverse square bars on one wall. J. Fluid Mech. 491, 229–238. 9. Leonardi, S., Orlandi, P. & Antonia, R.A. (2007). Heat Transfer in a Turbulent Channel Flow with Roughness, 5th International Symposium on Turbulence and Shear Flow Phenomena TU Munich 2729 August 2007. 10. Miyake Y., Tsujimoto K. & Nakaji M. (2001). Direct numerical simulation of a rough–wall heat transfer in a turbulent channel ﬂow. J. Heat Fluid Flow 22:237–244. 11. Orlandi, P. (2000). Fluid ﬂow phenomena, a numerical toolkit. Kluwer Academic Publishers. 12. Orlandi P. & Leonardi S. (2006). DNS of turbulent channel ﬂows with two- and three-dimensional roughness. Journal of Turbulence, 7.

Implicit Large-Eddy Simulation of Passive-Scalar Mixing in a Conﬁned Rectangular-Jet Reactor Antoine Devesa1 , Stefan Hickel2 , and Nikolaus A. Adams2 1 2

FluiDyna GmbH, 85748 Garching, Germany, [email protected] Technische Universit¨ at M¨ unchen, Institute of Aerodynamics, 85748 Garching, Germany, [email protected]; [email protected]

1 Introduction The subgrid-scale (SGS) modeling environment provided by the Adaptive Local Deconvolution Method (ALDM) has been recently extended to LargeEddy Simulations (LES) of passive-scalar transport. The resulting adaptive advection algorithm has been described and discussed with respect to its numerical and turbulence-theoretical background in Ref. [1]. Results demonstrate that this method allows for reliable predictions of the turbulent transport of passive-scalars in isotropic turbulence and in turbulent channel ﬂow from small to moderate Schmidt numbers. Due to the strong inﬂuence of the Schmidt number Sc on the Batchelor characteristic scales, diﬃculties in the modeling of the passive-scalar transport arise when the diﬀusive structures are several order of magnitude smaller than the viscous scales (Sc 1). In this paper, we present the results from implicit LES of the ﬂow in a conﬁned rectangular-jet reactor and an analysis of the mixing of a high Schmidt number scalar: Sc = 1, 250. The numerical study is carried out in collaboration with experimentalists from Iowa State University.

2 Experimental conﬁguration The mixing of passive-scalars at very high Schmidt numbers was recently studied at the Iowa State University [2], where measurements of velocity by Particle Image Velocimetry (PIV) and scalar concentration by Planar LaserInduced Fluorescence (PLIF) were carried out in a conﬁned rectangular-jet reactor. The conﬁguration (Fig. 1) was designed to provide a shear ﬂow with a Reynolds number based on the channel hydraulic diameter d = 20 mm of Re = 50, 000. The Schmidt number of the scalar in the liquid-phase ﬂow, injected at the center of the reactor inlet, is Sc = 1, 250. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 47, c Springer Science+Business Media B.V. 2010

323

324

A. Devesa et al.

Fig. 1. Sketch of the experimental setup.

Observations are undertaken in a rectangular Plexiglas cross-section measuring 60 mm (height) by 100 mm (width) and with an overall length of 1 m, at ﬁve sections S1 to S5 located at x/d = 1.0, 4.5, 7.5, 12.0, and 15.0. The height of each of the inlet channels, separated by two splitter plates, is equal to d and the aspect ratio of the rectangular-jet is 5 : 1. Free-stream velocities are 0.5, 1.0, and 0.5 m/s in the top, center and bottom inlet channels respectively.

3 Numerical method The ﬂow is described by the incompressible Navier–Stokes equations, that are discretized on a staggered Cartesian mesh. For time advancement, the fractional step method with an explicit third-order Runge–Kutta scheme is used. The time-step is dynamically adapted to satisfy a Courant–Friedrichs–Lewy condition with CF L = 1.0. The pressure–Poisson equation and diﬀusive terms are discretized by second-order centered diﬀerences. The convective terms of the momentum and passive-scalar transport equations are discretized by the ALDM, which also provides a SGS model. The Poisson solver employs the stabilized bi-conjugate gradient (BiCGstab) method. The presented results are obtained by the Simpliﬁed Adaptive Local Deconvolution algorithm [1] which is an implementation of the implicit LES ALDM, with improved computational eﬃciency [3].

4 Implicit subgrid-scale modeling for passive-scalar transport We consider the turbulent transport of passive-scalars, which do not measurably aﬀect the velocity ﬁeld. This case represents a one-way coupling of the scalar to the ﬂuid. Hence, the closure problem is restricted to the scalar transport equation, where the ﬂux function F is formally linear in c: ∂c + ∇.F(u, c) = 0 ∂t

with:

F(u, c) = uc −

1 ∇c ReSc

(1)

Implicit LES of a Conﬁned Rectangular-Jet Reactor

325

Fig. 2. Left: Low Schmidt number regime. Right: High Schmidt number regime at moderate Reynolds number. Solid line: scalar variance; dashed line: kinetic energy; dotted line: numerical cutoﬀ wavenumber.

Turbulence modeling and discretization for the momentum equations remain unchanged [3]. However, the evolution of a non-uniform scalar ﬁeld is subject to the velocity dynamics. Small-scale ﬂuctuations of velocity and scalar are correlated in the presence of a scalar-concentration gradient. The projection of equation 1 on a grid with ﬁnite resolution results in the modiﬁed equation: ∂cN + ∇.FN (uN , cN ) = τSGS ∂t

(2)

The subgrid tensor: τSGS = F(u, c) − FN (uN , cN ) originates from the grid projection of advective terms and represents the eﬀect of the action of subgrid scales and has to be approximated by a SGS model. The various regimes (Fig. 2) that exist for the variance spectrum of passivescalars at diﬀerent Schmidt numbers [4] have to be recovered by the SGS model. In the ALDM framework, implicit SGS modeling is accomplished by calibrating free discretization parameters. An analysis of the typical wave numbers, for low Schmidt-number and high Schmidt-number scalars, revealed that diﬀerent parameters are required for each of these regimes. This approach for passive-scalar mixing has already been validated for several canonical ﬂows [1]. The computation of the experimental setup presented here will then assess the high Schmidt number model in a complex conﬁguration.

5 Computational details and numerical results The computational domain is divided in two parts. The inlet, located upstream from the measurement domain, is composed of three channels, one in the center and two lateral ones, where the bulk velocity is twice lower. The three channels are periodic in stream- and spanwise directions, while bounded by solid walls in the transverse direction. The measurement section of the domain is conﬁned by side walls in spanwise and transverse directions, while inﬂow conditions are taken from the inlet channels. The computational domain has a total extent of 60d × 3d × 5d and is discretized by 14.4 × 106 ﬁnite volumes.

326

A. Devesa et al.

Fig. 3. Snapshot of the streamwise velocity ﬁeld.

Fig. 4. Snapshot of the scalar concentration ﬁeld. 3 2.5 2 1.5 1 0.5 0

S1

S2

3 2.5 2

1.5 1

1.5

–1

0 3 2.5 2 1.5 1 0.5 0

1 0.5 0

1

–1

S4

–1

0

S3

3 2.5 2

1

0

0.5 0

1 3 2.5 2 1.5 1 0.5 0

0

–1

1

S5

–1

0

1

Fig. 5. Mean streamwise velocity proﬁles at the 5 measurement stations. Symbols: experiment; solid line: ILES.

The inlet and reactor parts of the computational domain can be identiﬁed on the snapshots Figs. 3 and 4, showing velocity and scalar concentration ﬁelds respectively. The jet expansion close to the splitter plates and its further development downstream towards a ﬂat channel ﬂow can be observed as well. Figures 5 and 6 show a ﬁrst comparison between experimental data and numerical results for the temporal averaged streamwise velocity and scalar concentration respectively. The ﬁve observation stations S1 to S5 are depicted.

Implicit LES of a Conﬁned Rectangular-Jet Reactor S1

1

S2

1 0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

–1

0

1

–1

0

S4

1

0.8 0.6

0.4

0.4

0.2

0.2 0

0

1

–1

0

1

S5

1

0.6

–1

0

1

0.8

0

S3

1

0.8

327

–1

0

1

Fig. 6. Mean scalar concentration proﬁles at the 5 measurement stations. Symbols: experiment; solid line: ILES. S1

0.6

S2

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

–0.2

–0.2

–0.2

–0.4

–1

0

–0.4

1

–1

S4

0.6

0

1

0.4

0.2

0.2

0

0

–0.2

–0.2 –1

0

1

–0.4

–0.4

–1

0

1

S5

0.6

0.4

–0.4

S3

0.6

–1

0

1

Fig. 7. Streamwise velocity/scalar concentration cross correlation proﬁles at the 5 measurement stations. Symbols: experiment; solid line: ILES.

The ﬂow topology and expansion of the jet are found to be in agreement with the experiment. Results concerning the scalar variance, that are not displayed here, show the high mixing regions accordingly to the scalar variance peaks from the experimental data. Even though the second-order moments are after more than 1, 800 samples not fully converged, they show a good agreement with experimental data. An equivalent trend is observed for the velocity/scalar crossed-correlations < u c > and < v c >, that were obtained experimentally thanks to simultaneous PIV and PLIF measurements. The comparison with the numerical results is shown on Figs. 7 and 8 and demonstrate a satisfactory agreement.

328

A. Devesa et al. S1

S2

0.6 0.4 0.2 0 –0.2 –0.4 –0.6

S3

0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –1

0

1

0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –1

0

1

–1

S4

0

1

S5

0.6 0.4 0.2 0 –0.2 –0.4 –0.6

0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –1

0

1

–1

0

1

Fig. 8. Cross-stream velocity/scalar concentration cross correlation proﬁles at the 5 measurement stations. Symbols: experiment; solid line: ILES.

References 1. S. Hickel, N.A. Adams, and N.N. Mansour. Implicit subgrid-scale modeling for large-eddy simulation of passive-scalar mixing. Phys. Fluids, 19:095102, 2007. 2. H. Feng, M.G. Olsen, Y. Liu, R.O. Fox, and J.C. Hill. Investigation of turbulent mixing in a conﬁned planar-jet reactor. AIChE J., 51:2649–2664, 2005. 3. S. Hickel, N.A. Adams, and J.A. Domaradzki. An adaptive local deconvolution method for implicit LES. J. Comput. Phys., 213:413–436, 2006. 4. G.K. Batchelor. Small-scale variation of convected quantities like temperature in turbulent ﬂuid. part 1. general discussion and the case of small conductivity. J. Fluid Mech., 5:113–133, 1959.

Direct Numerical Simulation of a Turbulent Boundary Layer with Passive Scalar Transport Qiang Li, Philipp Schlatter, Luca Brandt, and Dan S. Henningson Linn´e Flow Centre, KTH Mechanics, Stockholm, Sweden, [email protected]; [email protected]; [email protected]; [email protected]

Abstract A fully-resolved direct numerical simulation (DNS) of a spatially developing turbulent boundary layer with passive scalars over a ﬂat plate under zero pressure gradient (ZPG) has been carried out using a spectral method with about 40M grid points. The highest Reynolds number based on the momentum thickness and free-stream velocity is Reθ = 850 and the molecular Prandtl numbers for the scalars range from 0.2 to 2.0. The intermittent region near the boundary-layer edge was identiﬁed by investigating the high-order moments and PDF. In addition, it was found that the streamwise velocity is similar to the scalar distribution at P r = 0.71 with isoscalar wall boundary condition. Far away from the wall, the two quantities become less correlated.

1 Introduction Direct numerical simulation (DNS) of turbulent ﬂows, especially in channel geometry, has matured to an important research tool. However, for ﬂat-plate boundary layers with zero pressure gradient (ZPG), which is a relevant canonical ﬂow case for theoretical, numerical as well as experimental studies, the progress has been slower. The simulation by Spalart [7] using an innovative spatio-temporal approach provided valuable data at Reθ = 300, 670, 1, 410 with the Reynolds number based on the momentum thickness θ and the freestream velocity U0 . Komminaho and Skote [5] performed a true spatial DNS up to Reθ = 700. Concerning boundary-layer simulations with passive scalars, to our knowledge the only DNS was performed by Kong et al. [6] at a low Reynolds number of about Reθ = 420. Accurate and reliable simulation data of spatially developing turbulent boundary layer involving passive scalars at medium to high Reynolds numbers are thus clearly needed. Therefore, in the present study, a DNS of spatially developing turbulent boundary layer under ZPG has been performed reaching up to Reθ = 850. Several passive scalar ﬁelds are computed with molecular Prandtl numbers between 0.2 and 2 with either isoscalar or isoﬂux wall boundary conditions. The present DNS V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 48, c Springer Science+Business Media B.V. 2010

329

330

Q. Li et al.

Fig. 1. Instantaneous contour plot of the scalar ﬁeld in a x − y plane, starting from Reθ = 175 to Reθ = 850. Note that the fringe region is not shown in the ﬁgure. The box height is enlarged by a factor of 2.

is aiming at providing the research community with useful data at a medium Reynolds number in order to improve our understanding of the physics of these boundary-layer ﬂows, especially in the outer region.

2 Numerical approach The three-dimensional, time-dependent, incompressible Navier–Stokes equations are solved using a pseudo-spectral method (see Chevalier et al. [2]) within a computational box of 750 × 40 × 34 (non-dimensionalised by the displacement thickness of the Blasius solution at the inlet δ0∗ ) and 1024 × 289 × 128 grid points in the streamwise, wall-normal and spanwise directions, respectively. Standard Fourier/Chebyshev spectral discretisations are used with a mixed Runge–Kutta/Crank–Nicolson scheme for time advancement. At the downstream end of the domain, a “fringe region”, in which the outﬂow is forced by a volume force to the laminar Blasius inﬂow, is added to fulﬁl the periodic boundary condition in the streamwise direction. Additionally, a trip forcing is located a short distance downstream of the inlet to cause a rapid (natural) laminar-turbulent transition. Figure 1 provides a snapshot of the computational domain with contours of the scalar evolution. An exhaustive amount of low and high-order statistics are collected and averaged during the simulation together with two-point correlation and time series at selected positions. The sampling time for the statistics is about δ∗ 5,000 ( U00 ).

3 Results Mean-ﬂow quantities and ﬂuctuations from both the velocity and scalar ﬁelds are shown in Figs. 2 and 3. The agreement with Komminaho and Skote [5] and other literature data sets is excellent, however the small deviation from the simulation by Spalart [7] is most probably due to the temporal approach used for the latter simulation. The ﬁve scalars are summarised in Table 1. The diﬀerent scalar boundary conditions only have small eﬀects on the mean proﬁles. As expected, the ﬂuctuations of the scalar variable show diﬀerences in the near-wall region owing to the diﬀerent boundary conditions.

DNS of Turbulent Boundary Layer with Passive Scalar Transport

331

3

25

2.5

20 + urms

+

2 15

1.5

10

1

5 0 100

0.5 0 101

y+

102

103

0

100

200

y+

300

400

500

Fig. 2. Streamwise velocity proﬁle u+ and ﬂuctuations u+ rms at Reθ = 670. present DNS, u+ = κ1 ln y + + 5.2 with κ = 0.41, ◦ Spalart [7] at Reθ = 670, Komminaho and Skote [5] at Reθ = 670. 5

30

P r = 2.1

<θ>+

20

P r = 0.71

15 10

10

1

2

y+

10

3 2

P r = 0.71

1

P r = 0.2

5 0 0 10

P r = 2.0

4

θ+rms

25

10

3

0

10−1

P r = 0.2 100

101

102

103

y+

+ Fig. 3. Scalar proﬁles θ+ and ﬂuctuations θrms at Reθ = 830. + θ1 , θ2 , θ3 , θ4 , θ5 . Isoscalar wall: θ1 , θ2 and θ4 . Isoﬂux wall: θ3 and θ5 .

Table 1. Parameters for the scalars. Scalar no. θ1 θ2 θ3 θ4 θ5 Wall boundary condition isoscalar isoscalar isoﬂux isoscalar isoﬂux Pr 0.2 0.71 0.71 2.0 2.0

The full Reynolds-stress and scalar-ﬂux budgets are obtained from the DNS. All the terms appearing in the budget equations are explicitly evaluated including the pressure terms. The residual is at most O(10−3 ) in viscous scaling. Figure 4 compares the budgets of the streamwise and wall-normal scalar-ﬂux ﬂuctuations with the channel data from Kawamura et al. [3]. The near-wall behaviour is similar to that observed in channel ﬂow; the exception is the mean convection terms which are non-zero in the boundary layer albeit comparably small. As also noted in Kawamura et al. [3], the dissipation term seems unchanged with increasing Reynolds number whereas the other dominant terms increase for higher Reynolds number in the budget for v θ . The skewness and ﬂatness factors of the scalars are shown in Fig. 5. For the simulations with isoﬂux boundary condition in the vicinity of the wall, the skewness and ﬂatness factors are almost constant and close to the Gaussian values, respectively. In the outer region, the skewness and ﬂatness factors for

Q. Li et al.

budget for

0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3

0.05

budget for

332

0

−0.05

0

10

20

y+

30

40

50

−0.1

0

10

20

y+

30

40

50

Fig. 4. Budgets of scalar-ﬂux of θ2 with P r = 0.71 and isoscalar wall at diﬀerent Reynolds numbers in viscous units. at Reθ = 390, at Reθ = 620, at Reθ = 830, ◦ Production, Dissipation, Scalar pressure diﬀusion, Molecular diﬀusion, Turbulent diﬀusion, ∗ Mean convection. Symbols are from Kawamura et al. [3] at Reτ = 395 with P r = 0.71. 10

2

8 F (θ)

S (θ)

0 −2 −4 −6

6 4 2

0

50 100 150 200 250 300 350 y+

0

0

50 100 150 200 250 300 350 y+

Fig. 5. Skewness and ﬂatness factor distributions of scalars at Reθ = 830. + θ1 , θ2 , θ3 , θ4 , θ5 .

both wall boundary conditions increase rapidly which indicates the existence of a region of intermittency. Outside the boundary layer, both skewness and ﬂatness factors decrease rapidly towards the Gaussian values of 0 and 3, respectively. A diﬀerent perspective on the characteristics of the ﬂuctuations is provided by the probability density function (PDF). The PDF distributions of the scalar ﬂuctuation at various wall-normal positions, ranging from y + ≈ 5 in the conductive sub-layer to y + ≈ 300 near the boundary-layer edge, were collected (see Fig. 6). A highly positively skewed distribution for the isoscalar wall is observed in the near-wall region whereas the one for the isoﬂux wall is more symmetric. From y + ≈ 30 until the boundary-layer edge, P (θ ) is nearly symmetric and the inﬂuence from the diﬀerent wall boundary conditions is not noticeable. The distribution of θ2 compares very well with Abe et al. [1] in the near-wall region. Note that near the boundary layer edge, P (θ ) is highly negatively skewed which indicates the intermittent region. Outside the

DNS of Turbulent Boundary Layer with Passive Scalar Transport

a

b

0.5

P (θ’/ θrms)

P (θ’/θrms )

0.4 0.3 0.2

333

0.4 0.3 0.2 0.1

0.1 0 −2

c

0

2

θ’/ θrms

4

0 −4

6

0

4

2

θ’/ θrms

d

0.4

−2

2.5

P (θ’/ θrms)

P (θ’/ θrms)

2 0.3 0.2 0.1 0 −4

1.5 1 0.5

−2

0

θ’/θrms

2

4

0 −5

−4

−3

−2

θ’/ θrms

−1

0

1

Fig. 6. PDF of the scalar ﬂuctuation θ of P r = 0.71 at Reθ = 830. • present DNS, Gaussian distribution. (a) θ2 at y + ≈ 5, (b) θ3 at y + ≈ 5, (c) θ2 at y + ≈ 100, (d) θ2 at y + ≈ 300.

boundary layer, the PDF distribution is close to the Gaussian distribution again. It is noted that for the present Prandtl number range, the inﬂuence of P r on the PDF distributions is negligible. The correlation between the velocity and scalar is also examined through the joint probability density function (JPDF). A strong positive correlation between u and θ and a mild negative correlation between v and θ are observed. For θ2 , i.e. P r = 0.71 with isoscalar boundary condition, the distribution of P (u , θ2 ) and P (v , θ2 ) are essentially the same as reported by Kim and Moin [4] in a heated channel. For θ4 and θ5 (P r = 2.0), P (u , θ ) and P (v , θ ) are shown in Fig. 7 at y + ≈ 5. The lower correlations between u and θ5 in Fig. 7b are due to the isoﬂux boundary condition. The strong correlation between u and θ becomes milder when increasing the wall distance.

4 Conclusion A spectral DNS of a spatially developing turbulent boundary layer has been performed and validated with other numerical and experimental results. All the terms in the budget equations, including pressure, are explicitly evaluated. Exhaustive single and two-point statistics have been collected to provide insight in the ﬂow physics and serve as a data set for modelling purposes. The

Q. Li et al.

a

b

θ’/ θrms

3

3

2

2

1

1

θ’/ θrms

334

0

−1 −2 −3 −3

0 −1 −2

−2

−1

0

1

2

−3 −3

3

5

d

5

0

v’/vrms

v’/vrms

c

0

−5 −5

0

θ’/ θrms

−2

−1

0

1

2

3

u’/urms

u’/urms

5

−5 −5

0

θ’/ θrms

5

Fig. 7. JPDF of the scalar ﬂuctuation θ of P r = 2.0 at y + ≈ 5 and Reθ = 830. (a) P (u , θ4 ), (b) P (u , θ5 ), (c) P (v , θ4 ), (d) P (v , θ5 ). Contour levels are from 0.016 to 0.32 by increments of 0.016 for (a), 0.025 to 0.5 by increments of 0.025 for (b), 0.012 to 0.24 by increments of 0.012 for (c) and 0.01 to 0.2 by increments of 0.01 for (d).

inﬂuences from the wall boundary conditions for the scalars are conﬁned to the near-wall region. Additionally, the results in the near-wall region for the boundary layer are quantitatively and qualitatively similar to those obtained from channel-ﬂow simulations. However, diﬀerences appear in the outer part of the boundary layer. The main focus of this contribution is on the presentation of the averaged turbulence scalar statistics pertaining to the outer region of the boundary layer. Very high peaks are observed in skewness and ﬂatness of all the scalars in the outer region and the magnitudes are much higher than those of the velocity ﬁeld. The intermittent outer region is also identiﬁed by the PDF distribution. Moving away from the wall, less correlated relations between streamwise velocity and scalars are observed in JPDF of P (u , θ ).

References 1. H. Abe, H. Kawamura, and Y. Matsuo. Direct numerical simulation of a fully developed turbulent channel ﬂow with respect to the Reynolds number dependence. J. Fluids Engng., 123(2):382–393, 2001. 2. M. Chevalier, P. Schlatter, A. Lundbladh, and D. S. Henningson. A pseudospectral solver for incompressible boundary layer ﬂows. Technical Report TRITAMEK 2007:07, Royal Institute of Technology, Stockholm, 2007.

DNS of Turbulent Boundary Layer with Passive Scalar Transport

335

3. H. Kawamura, H. Abe, and Y. Matsuo. DNS of turbulent heat transfer in channel ﬂow with respect to reynolds and prandtl number eﬀects. Int. J. Heat Fluid Flow, 20(3):196–207, 1999. 4. J. Kim and P. Moin. Transport of passive scalars in a turbulent channel ﬂow. In J.-C. Andr´e, J. Cousteix, F. Durst, B. E. Launder, and F. W. Schmidt, editors, Turbulent Shear Flows 6, pages 85–96. Springer-Verlag, Berlin, 1989. 5. J. Komminaho and M. Skote. Reynolds stress budgets in Couette and boundary layer ﬂows. Flow, Turbulence Combust., 68(2):167–192, 2002. 6. H. Kong, H. Choi, and J. S. Lee. Direct numerical simulation of turbulent thermal boundary layers. Phys. Fluids, 12(10):2555–2568, 2000. 7. P. R. Spalart. Direct simulation of a turbulent boundary layer up to Reθ = 1410. J. Fluid Mech., 187:61–98, 1988.

Part V

Active Scalars

Numerical Experiments on Turbulent Thermal Convection Roberto Verzicco Dipartimento di Ingegneria Meccanica, Universit` a degli studi di Roma ‘Tor Vergata’, Roma, Italy, [email protected]

Abstract Numerical simulations of Rayleigh–B´enard convection have become a valid complement to laboratory experiments to disentangle the complex ﬂow dynamics. In this paper we describe some of the state-of-the-art experiments and discuss how numerical simulations help in elucidating some phenomena. In particular we show that in the high-end of the Rayleigh number there is an unavoidable deviation of the real ﬂow from its ideal counterpart thus leading to diﬃculties in the interpretation of the results and in the comparison among diﬀerent experiments. In this respect numerical simulation can be a very helpful tool since each eﬀect can be tested separately.

1 Introduction The ﬂuid motion generated by temperature induced buoyancy is called thermal convection and it is being investigated for more than a century, since B´enard [1] and Rayleigh [2], owing to its relevance in Nature and technology. The basic ﬂow consists of a ﬂuid layer vertically bounded by ﬂat plates at different temperatures (the lower being hotter than the upper) with horizontal dimensions which are inﬁnite in a truly Rayleigh–B´enard conﬁguration but always limited in practical realizations. This problem is the base for many theoretical analyses whose results are used as a guideline for the interpretation of experimental ﬁndings and observations. The main control parameters are the Rayleigh (Ra), and Prandtl (P r) numbers deﬁned as Ra =

gαΔh3 , νk

Pr =

ν , k

(1)

where Δ and h denote the temperature diﬀerence between the plates and their separation distance, respectively, and g is the acceleration of gravity. The ﬂuid properties are α, ν and k, respectively, the thermal expansion coeﬃcient, the kinematic viscosity and the thermal diﬀusivity. The outputs of the ﬂow are the heat transfer between the plates φ that, when normalized by the purely V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 49, c Springer Science+Business Media B.V. 2010

339

340

R. Verzicco

conductive value yields the Nusselt number (N u) and the strength of the ﬂow quantiﬁed by the Reynolds number (Re). The practical realization of the ﬂow has to cope with several technological limitations that make the real ﬂow and the ideal problem similar only to a certain degree. More in detail, since the Rayleigh number depends on h3 a modest temperature diﬀerence (Δ) of few Kelvins in a big system is enough to yield a large Rayleigh number (O(1015 −1020 )). On the other hand in a compact experimental setup, cost and controllability considerations suggest to limit h to O(1)m and large Δ are used to make up for the small h. The temperature diﬀerence Δ, however, cannot be increased beyond a certain limit (which will be better speciﬁed later) in order to maintain the validity of the Boussinesq approximation in which the only variation of the ﬂow properties is the density in the buoyancy term. The temperature boundary conditions in an experimental setup are another cause of concern because in the ideal Rayleigh–B´enard problem the temperature is strictly constant over the heated and cooled plates while the sidewall is adiabatic. The practical realization of these conditions involve, respectively, materials of very high and very poor thermal conductivity and deviations from the ideal intended problems are unavoidable. In particular, the parasite heat currents through the side walls are important at low Rayleigh numbers while the ﬁnite thermal conductivity of the horizontal plates becomes relevant for high Rayleigh numbers. Since the aim of most of the nowadays experiments is to hit the high end of Ra the limits of each experimental setup are often stressed thus posing the question of the reliability of the results or of the proper comparison among diﬀerent experiments. In the last decade impressive advances of the CPU architectures and parallel computing have made the direct numerical simulation (DNS) of turbulent thermal convection an attractive option since they can avoid the above technological limitations just discussed and obtain ﬂow details that are inaccessible, despite considerable gains, in experiments. On the other hand simulations must solve appropriately the smallest scales in the bulk of the ﬂow as well as viscous and thermal boundary layers. In addition the time advancement of the solution must cover several of the largest time scales with steps of the same order as the smallest time scales of the ﬂow. Both, space- and time-resolution requirements imply an increase of the computational cost with Ra that make the simulations very challenging for the Rayleigh and Prandtl number of typical experiments. In the present paper we discuss the results of extensive simulations of Rayleigh–B´enard convection that we have carried out with the aim of covering the widest range of Rayleigh numbers with constant Prandtl number, strict Boussinesq conditions, constant temperature for bottom and top plates, and no side-wall conduction. We also discuss the deviations of the experiments from the ideal conditions by taking into account the ﬁnite conductivity of the top and bottom plates.

Numerical Experiments on Turbulent Thermal Convection

341

2 The problem The Rayleigh–B´enard ﬂow is a classical and well deﬁned problem that can be summarized as follows: A ﬂuid layer of thickness h is heated from below by a smooth ﬂat horizontal plate at constant and uniform temperature Th and cooled from above by an identical plate but at temperature Tc . Because of thermal expansion the warmer ﬂuid becomes lighter and tends to rise while the colder more dense ﬂuids tends to sink thus generating a motion. The ﬂow dynamics is governed by the Navier–Stokes equations with the Boussinesq approximation that read: Du = −∇p + θˆ x+ Dt

Pr Ra

12

∇2 u,

1 Dθ 2 = 1 ∇ θ, Dt (P rRa) 2

∇ · u = 0,

(2) (3)

being u, p and θ, respectively, velocity vector, pressure and temperature while x ˆ the unity vector pointing in the opposite√ direction with respect to gravity. In equations (2–3) the main scales are U = gαΔh for the velocity, Δ for the temperature and h for lengths. It is worth noting that the thermal forcing can induce a ﬂow only when the buoyancy overcomes the viscous drag on a ﬂuid parcel and this condition is quantiﬁed by the Rayleigh number that must exceed the threshold Rac = 1,704 to generate the ﬂuid motion. Rac is independent of the Prandtl number but it strongly depends on the velocity boundary conditions over the solid surfaces (no-slip or stress-free) and on the aspect ratio of the container [3, 4]. When the Rayleigh number is just above Rac the ﬂow is steady because the viscous and diﬀusive terms exceed the convective ones; as the Rayleigh number is increased nonlinear terms start to dominate thus leading to unsteady periodic motions followed by multi-periodic, chaotic regimes and eventually to turbulence. The transition to each regime is marked by a critical Rayleigh number that, diﬀerently from Rac , is extremely dependent on the details of the set-up and on the Prandtl number. The dependence on the latter parameter is easily understood looking at equations (2–3) and noting that P r acts in opposite directions in the viscous and diﬀusive terms of the momentum and energy equations. This implies that the nonlinear terms can dominate earlier the momentum or temperature depending on the particular combination of Ra and P r thus yielding a variety of possible scenarios. Among the various ﬂow regimes the turbulence is the most investigated, either for the interest of applications and because of its rich dynamics. In this case, however, the problem is extremely sensitive also to the experimental set-up details [5] thus making possible that apparently similar experiments diﬀer form each other and that numerical simulations are can be compared to laboratory ﬂows only to some extent: Clarifying this issue is the focus of this paper.

342

R. Verzicco

3 Experimental setups We refer to the sketch of Fig. 1 to describe the arrangement of typical experimental setups. It is immediately evident that although the classic Rayleigh–B´enard problem has an inﬁnite lateral extension its practical realization must be laterally bounded by sidewalls. From one hand this introduces the aspect ratio of the cell Γ = d/h as an additional control parameter and, on the other hand, it poses the question of how to estimate the spurious heat ﬂux through the lateral boundaries. Up to a decade ago the latter was accounted for by measuring the heat current when the cell was completely empty and by subtracting that value from the heat transfer in presence of ﬂuid. It has become evident, however, that the temperature proﬁle within the sidewall is linear in the case of pure conduction but it copies the temperature distribution in the ﬂuid when the cell is ﬁlled with ﬂuid, thus completely changing the parasite heat current. To this aim diﬀerent correction were derived by ‘ad hoc’ experiments [6, 7] with diﬀerent side-walls while dedicated numerical simulations [8] not only proposed another correction but also showed that the mean ﬂow structure was modiﬁed by the presence of a conducting side-wall. Nevertheless this eﬀect was observed to be relevant only for reduced aspect-ratio containers (Γ ≤ 1) and decreased in magnitude for increasing Ra, thus becoming negligible in the high-end of the Rayleigh numbers (Ra ≥ 109 − 1010 ). The ﬁnite conductivity of the horizontal plates, in contrast, has an eﬀect on the heat transfer which increases with Ra and depends on the particular coupling of the plate material/ﬂuid in addition to the plate thickness and mounting details (like ﬂanges, thermal links, coupling with the side-wall, etc.). In particular each plate produces a static temperature drop that can be easily accounted for but it has also a dynamic eﬀect on the shedding of plumes which is more subtle and less easy to correct. In addition the large majority

termostatic bath Tc

e

Th

e

h

heater d

Fig. 1. Sketch of a typical experimental apparatus.

Numerical Experiments on Turbulent Thermal Convection

343

of experiments have the heat source on the hot plate that provides the ﬂow with constant heat ﬂux thus allowing for substantial temperature ﬂuctuations. The upper cold plate is indeed maintained at constant temperature but this is true only for the side of the plate directly in contact with the thermostatic bath.

4 Numerical simulations As mentioned in the Introduction, numerical simulations do not have the problems previously discussed even if they must assure an appropriate spatial and temporal resolution of the smallest and fastest ﬂow scales in the bulk of the ﬂow as well as close to the wall. In particular, assuming that far from the boundaries the ﬂow is homogeneous and isotropic, the smallest scales in the bulk can be estimated as the Kolmogorov η and Batchelor ηT scales [9] thus yielding η/h π[P r2 /(RaN u)](1/4) for P r ≤ 1 with η ≤ ηT and ηT /h π[1/(P rRaN u)](1/4) for P r > 1 with ηT ≤ η. Close to the horizontal plates there are viscous and thermal boundary√layers √ whose thickness, δu and δT respectively, can be estimated as δu /h P r/(5 Ra) and δT /h 1/(2N u). The above estimates can be used to compute the appropriate grid spacing only once a N u = N u(Ra, P r) correlation is available; although this correlation is often one of the main objects of each investigation to this aim it is enough a relation of the form N u = ARaβ (with values A = 0.124 and β = 0.31 for P r = O(1)). Concerning the time discretization, indicating by T the large scale ﬂow that quantiﬁes the time taken by a ﬂuid particle to revolve inside the cell, we can write for the fastest times tη in the ﬂow tη /T P r/Ra thus indicating that the time step size Δt ≤ tη must decrease for increasing Ra. In addition to the physical constrain on the time step there is also its numerical counterpart Δt ≤ CΔx/u (being C the stability parameter which is scheme dependent, and Δx/u is the maximum over the whole ﬁeld of the ratio of grid spacing and velocity) that is needed in order to maintain the time integration of the Navier–Stokes equations stable; this constrain is usually more stringent than the previous one and the most computationally expensive. Finally each numerical simulation has to be carried out for a number of large scale times T big enough to compute converged statistics; this number increases with Ra while both time step size and grid spacing decrease. This implies that accurate direct numerical simulations become computationally more expensive as Ra increases or P r deviates signiﬁcantly from O(1). As a last comment we wish to point out that large aspect ratios have to be traded with the Rayleigh number: In fact, since Ra is deﬁned with the height of the cell, big values of Γ imply large ﬂuid volumes that in turn require equally large computational grids.

344

R. Verzicco

5 Results In this ﬁnal section we describe some examples in which numerical simulations provided results that were complementary to laboratory experiments and that helped in shedding light on complex phenomena. One example is given by the ﬁnite conductivity of the plates that were found to limit the maximum Nusselt number or, in other words, the non dimensional heat transfer between the plates. Denoting by λw and λf the thermal conductivity of the plates and of the ﬂuid and by e the plate thickness it is possible to deﬁne the parameter X = λw e/(λf hN u) (which is in fact the inverse of the Biot number) that quantiﬁes how far the experimental conﬁguration is far from the ideal conditions in which the horizontal plates have a temperature distribution exactly constant and uniform and can provide the ﬂuid with as much heat as is requested by the ﬂow. Within this ideal condition we indicate as N u∞ the non dimensional heat transfer and the ratio N u/N u∞ as a measure of the non perfect ﬂow conditions. The results of Fig. 2a show that there is a good correlation between the two parameters and an empirical correction can be derived in the form N u/N u∞ = 1 − exp[−(X/4)1/3 ] that can be used to correct experimental data. As an application we show the results by Nikolaenko and Ahlers [11] that were obtained using aluminum plates with a thermal conductivity which is about half of a similar plate made with copper (Fig. 2b). Once the raw data are corrected for the ﬁnite conductivity of the plates they agree very well with the model by Grossmann and Lohse [12] that is the present standard guideline for the interpretation of the results. Another important advantage of numerical simulations is the possibility of testing models and theories. Among many, one of the simplest and dated

a

b

1.1 1

0.14

Nu Ra−0.3

0.136 Nu Nu∞

0.132 0.128 0.124

0.4 10

λw h λ fNu e

100

0.12 1e+08

1e+09

1e+10

1e+11

Ra

Fig. 2. (a) N u/N u∞ vs λw h/(N uλf e) ﬁt of the data: N u/N u∞ = 1 − exp[−(X/4)1/3 ]. The various symbols indicate simulations at diﬀerent Ra, P r, e/h, λw /λf and ρw Cw /(ρf Cpf ). For more details see [10]. (b) Compensated Nusselt numbers versus Rayleigh numbers: × acetone data without side wall correction, ◦ acetone data with side wall correction, acetone data with side wall and plate data for water corrections, • data for water (side wall correction is not needed) with plate correction; all data from [11]. theoretical prediction by [12].

Numerical Experiments on Turbulent Thermal Convection

345

is that by Malkus [13] and Priestley [14] according to which if the Rayleigh number is high enough upper and lower thermal boundary layer become uncorrelated and the non dimensional heat transfer scales as N u ∼ Ra1/3 . Despite the simplicity of the theory its experimental veriﬁcation proved to be diﬃcult since nowadays high-Ra experiments are mostly performed in cylindrical cells of small aspect-ratio [15,16] and the thermal boundary layer remain connected by a persistent large-scale recirculation. The problem is further complicated by the Prandtl number variation with Ra that in turn inﬂuences strength and topology of the mean ﬂow [17]. Within this scenario numerical simulations turned out to be the only viable possibility to test the theory of Malkus that indeed proved to be correct. In particular, it has been observed that in that geometry the mean ﬂow weakens with Ra owing to the random shedding of strong thermal plumes from the plates and eventually vanishes (Fig. 4b). The latter condition was evidenced by investigating the probability density function of the vertical velocity product over diametrically opposite positions (Fig. 3) whose deviation from symmetry and negative tails were a measure of the strength of the mean ﬂow [18]. Figure 4a shows that indeed the numerical simulations are the only ‘experiments’ showing a 1/3 power law over an extended range of Ra although there are quantitative discrepancies of the Nusselt number that are still waiting for a fully satisfactory explanation.

a

b pdf

100

10

1 −0.03

−0.015 0 vx(θ)vx(θ+π)

0.015

Fig. 3. (a) Instantaneous snapshot of temperature showing one possible mean ﬂow conﬁguration with a single recirculation at Ra = 2 × 1010 , and P r = 0.7. Only the temperature range 0.475 ≤ θ ≤ 0.525 is represented. Light gray indicates warm ﬂuid while the dark gray the cold ﬂuid. The symbols • are in the same position as the numerical probes. (b) Probability density function of the product vx (θ)vx (θ + π) the Ra = 2 × 1014 , velocities being sampled at the positions of the • in Fig. 3a: 13 12 11 Ra = 2 × 10 Ra = 2 × 10 , Ra = 2 × 10 long Ra = Ra = 2 × 109 ; P r = 0.7. 2 × 1010 and long

346

R. Verzicco

a

b

Nu Ra−1/3

0.1

100

0.08

1R

10 Pr

0.06 1 0.04

NMF

2R 6

10

1010 Ra

1014

106

1010

1014 Ra

Fig. 4. (a) Compensated Nusselt number versus the Rayleigh number. (b) P r vs Ra phase diagram with the mean ﬂow structure according to the model by [17]. Experimental data: ◦ , Chavanne et al. [15]; ×, Niemela et al. [16]. Numerical simulations: •, Amati et al. [18].

References 1. B´enard, H. (1900) Revue Gen. des Sci. pures et appl. 11:1261–1271 2. Rayleigh, Lord (1916) Phil. Mag. 32:529–546. 3. Chandrasekar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Oxford University Press. 4. Charlson, G.S. & Sani, R.L. (1971) Int. J. of Heat and Mass Transfer. 14: 2157–2160. 5. Daya, Z. A. & Ecke, R. E. (2001) Phys. Rev. Lett. 89:4501, U81–U83. 6. Ahlers, G. (2001) Phys. Rev. E. 6302(2):5303 U7–U9. 7. Roche, P.E., Castaing, B., Chabaud, B., Hebral, B. & Sommeria, J. (2001) Eur. Phys. J. B. 24:405–408. 8. Verzicco, R. (2002) J. of Fluid Mech. 473:201–210. 9. Gr¨ otzbach, G. (1983) J. Comput. Phys. 49:241–264. 10. Verzicco, R. (2004) Phys. of Fluids. 16(6):1965–1979. 11. Nikolaenko, A. & Ahlers, G. (2003) Phys. Rev. Letters. 91:084501–1/4. 12. Grossmann, S. & Lohse, D. (2000) J. Fluid Mech. 407:27–56. 13. Malkus, M.V.R. (1954) Proc. Roy. Soc. Lond. A225:196. 14. Priestley, C.H.B. (1959) Turbulent Transfer in the Lower Atmosphere, U. Chicago Press. 15. Chavanne, X., Chill` a, F., Chabaud, B., Castaing, B. & Hebral, B. (2001) Phys. Fluids. 20:1300–1320. 16. Niemela, J. J., Skrbek., L., Sreenivasan, R. R. & Donnely, R. J. (2000) Nature 404:837–841. 17. Stringano, G. and Verzicco, R. (2006) J. Fluid Mech. 548:1–16. 18. Amati, G., Koal, K., Massaioli, F., Sreenivasan, K.R. and Verzicco, R. (2005) Phys. Fluids. 17:121701.

Direct Numerical Simulation of Turbulent Reacting and Inert Mixing Layers Laden with Evaporating Droplets J. Xia1 and K.H. Luo2 1

2

Energy Technology Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected] Energy Technology Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected]

Abstract The diﬀerence of droplet dynamics in reacting and non-reacting turbulent mixing layers has been investigated using direct numerical simulation. The multiphase reacting ﬂow is described by the hybrid Eulerian-Lagrangian approach. In the reacting mixing layer, the combustion-induced evaporation causes droplet size to signiﬁcantly decrease in comparison with the non-reacting case, where evaporation is inconsiderable and the size of droplets remains. The droplet size PDF and instantaneous droplet distribution in the reacting case reveals a polydisperse system, whereas the monodisperse droplets of St0 = 1 in the non-reacting case preferentially concentrate in high-strain low-vorticity regions of the turbulent ﬂow.

1 Introduction Multiphase combustion is encountered in numerous engineering applications including diesel engines and pulverized coal combustors. Combustion eﬃciency and emissions in such devices are determined by the complex interactions among turbulence, combustion, evaporation and particles/droplets dynamics. For example, evaporation aﬀects the mixing and micromixing between the fuel and the oxidizer, which in turn inﬂuence the combustion process. In the meantime, heat release from combustion causes evaporation in a turbulent environment, with turbulence-enhanced mass and heat exchange. Such interactions are not only local, but also change with time. Direct Numerical Simulation (DNS) has been widely used over the last two decades to investigate interacting phenomena in order of increasing complexity: ﬁrst the dispersion of solid particles in turbulent ﬂow [1–4], then evaporating droplets [5–7] and ﬁnally multiphase combustion [8–10]. Various studies have been devoted to investigation of the preferential segregation of dispersed particles/droplets in turbulent ﬂow [1, 3, 7, 11–13].

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 50, c Springer Science+Business Media B.V. 2010

347

348

J. Xia and K.H. Luo

In this paper, the dynamics of evaporating droplets in reacting and nonreacting turbulent ﬂow are investigated using DNS. The focus is on how combustion, through its eﬀect on evaporation, aﬀects the distribution of droplets.

2 Direct numerical simulation The governing equations for the gas phase comprise the compressible timedependent Navier–Stokes equations and transport equations for the fuel, oxidizer and water vapour [14]. A global one-step ﬁnite-rate chemical reaction is simulated using the Arrhenius law for the reaction rate. Using the Lagrangian approach, every droplet has its own governing equations for mass, momentum and energy [6, 14]. The droplet evaporation is described by the ClausiusClapeyron law. In the droplet momentum equation, only the Stokes drag force between the phases is included. The two-way coupling between the continuum and the dispersed phases is represented by the extra source terms in the carrier gas-phase equations. In order to perform DNS of multiphase reacting ﬂow, a temporal mixing layer has been chosen as the computational conﬁguration due to its lower computational cost in comparison with a spatial mixing layer or jet. The fuel and oxidizer streams are the upper and lower streams, respectively and move in opposite directions with equal streamwise speed U0 . Note the letters x, y, z or numbers 1, 2, 3 refer to the streamwise, cross-stream and spanwise directions, respectively. All the variables and quantities presented herein are nondimensionalized using standard reference quantities for temporal mixing layer [9, 10]. Error functions are prescribed for the initial proﬁles of the gas streamwise velocity ug,1 , mass fractions of fuel Yf and oxidizer Yo , and droplet number density nd . The initial laminar mixing layer is excited by spanwise and streamwise vorticity perturbations. To attain fully turbulent ﬂow, a large initial Reynolds number based on the vorticity thickness Re = 1, 000, strong disturbance amplitudes F2D = 0.1 and F3D = 0.0875, measured as the ratio of the respective disturbance circulation to the mean circulation [6], has been deployed [9]. At t = 110, a fully turbulent mixing layer is reached after the roll-up, paring and transition stages. The reaction and/or droplet eﬀects are then switched on at t = 110 and simulations are continued until t = 250, by which the self-similar state of the mixing layers has been well established. The liquid droplets are initially randomly seeded in the gas oxidizer stream according to the speciﬁed number density proﬁle, with initial droplet velocity vd,0 and temperature Td,0 identical to the local gas velocity ug,0 and ambient temperature Tg,0 , respectively. Table 1 presents the key parameters. Mc is the convective Mach number. The Damk¨ ohler number Da, Zel’dovich number Ze and non-dimensional heat release parameter Qh were chosen to represent a ﬁnite-rate slow diﬀusion ﬂame with a typical peak temperature (Tf = 4), without explicit inclusion

DNS of Droplet-Laden Reacting/Inert Mixing Layers

349

Table 1. Simulation parameters. Re

Mc

Pr

Sc

1000 0.5 0.697 0.697

Da Ze Qh 5

3

hfg

Lx

Ly

Lz

λx

λz

nx × ny × nz

7.5 19.16 2λx 2Lx 2λz 1.16(2π)δω,0 0.6λx 192 × 384 × 116

Cases

St0

M LR0

Nd,0

Reacting

D4 D4n

1 1

0.4 0.4

16,925,496 16,925,496

Yes No

of radiation calculations. The latent heat of vaporization of water hfg was obtained by setting the normal boiling temperature of water as the reference temperature. The viscosity μ is kept constant in order to avoid the complication of ﬂow re-laminarisation due to heat release and to avoid further complexity introduced in the Stokes number, which is solely dependent on the droplet diameter in this paper. St0 , MLR0 and Nd,0 denote the initial Stokes number ρ D2

N

m

d d,0 d,0 of droplets, St0 = 18μ/Re , initial mass loading ratio, M LR0 = ρg,0 Ld,0 x Ly Lz /2 (ratio of the initial droplet mass to the mass of the carrier stream – oxidizer), and initial droplet number in the domain, respectively. The Stokes number St0 = 1 is chosen due to its prototype value and its special eﬀect on preferential distribution of droplets. The computational domain contains two disturbance wavelengths, denoted as λx and λz , in both the streamwise and spanwise directions. Ly = 2Lx , which allows the mixing layer to develop well after the self-similar state of the mixing layer has been established. The grid spacing is uniform in each direction. To fully resolve the thin reaction zones, the grid resolution is chosen to be on the order of the Kolmogorov scale η of the fully developed turbulent ﬂow under investigation. For both gas and droplet phases, periodic boundary conditions are set in the streamwise x and spanwise z directions, while adiabatic slip walls imposed in the cross-stream direction y. High-order compact ﬁnite diﬀerence schemes are employed to calculate the spatial derivatives. Time advancement for the gaseous phase is achieved by a 3rd-order Runge–Kutta method, while for the droplet phase, a semi-analytic approach [5] is employed to save the huge cost of droplet marching. A droplet is assumed to be completely evaporated and removed from the simulation when its Stokes number reaches 0.075, i.e., Stdel = 0.075, to avoid extremely small time steps. Further details on the mathematical, physical models and numerical procedures can be found in [9, 14]. All the simulations were run on the UK national high-end computing facility HECToR using 192 CPUs. The number of droplets traced is up to the order of 107 , which is more than that in most previous studies by DNS.

350

J. Xia and K.H. Luo

3 Results and discussion Presented in Fig. 1a are contours of the mass fraction of water vapour in the central plane in the spanwise direction zcp = Lz /2 at t = 210 (100 time units after the initiation of reaction and droplet eﬀects) for the reacting case D4. The location of the stoichiometric mixture fraction Zst is also shown in the ﬁgure together with the reaction rate contours. Intense reaction takes place in zones centered around the Zst lines. Vapour concentration is high in the broad central region of the mixing layer, where droplets interact with the reaction zones. Reaction may have been quenched by the evaporating droplets in some areas, as evidenced by the lack of reaction along some segments of the Zst line. It is noted that as ﬁnite-rate chemistry is used, the interactions among mixing, reaction, evaporation and ﬂame quench have diﬀerent time scales. Shown in Fig. 1b is the Probability Density Function (PDF) of the droplet size compared to the initial value for Case D4 at the same time. The PDF is calculated using all the droplet samples in the region (Ly − δω )/2 < y < (Ly + δω )/2, sorted into 100 discrete sample bins. Figure 1b shows that the droplet size spans the range from the initial diameter to the smallest diameter (Stdel /St0 = 0.075/1 ⇒ Dd,del/Dd,0 ≈ 0.27) below which the droplet is considered to be completely evaporated and removed. The PDF of the droplet size is quite broad, although peaks appear at Dd,0 /Dd = 1 and Dd,0 /Dd ≈ 0.75. Figure 2 shows the instantaneous distribution of droplets overlapped with the contours of the second invariant of the deformation tensor Πd , which is

b

a

2 25 1.5 PDF

y

20 15

1

10 0.5

5 0

0

5

x

10

0

0

0.2

0.4 0.6 Dd/Dd,0

0.8

1

Fig. 1. (a) Filled contours of the mass fraction of vapour at t = 210 for the reacting case D4. The bold line denotes the location of the stoichiometric mixture fraction and the thin contour lines show the reaction rate ﬁeld. (b) PDF of the droplet size for the reacting case D4 at t = 210.

DNS of Droplet-Laden Reacting/Inert Mixing Layers

351

Fig. 2. The droplet distribution superimposed by contours of the second invariant of the deformation tensor Πd [1] in the central plane in the spanwise direction zcp = Lz /2 for (a) the reacting case D4 and (b) the non-reacting case D4n at t = 210. Only the central mixing region (Ly − δω )/2 < y < (Ly + δω )/2 is shown for clarity. Note δω is diﬀerent for D4 and D4n. The solid contour lines indicate positive Πd and dashed lines negative Πd , respectively. The plotted droplets are proportional to their sizes.

deﬁned as Πd = −0.5(S 2 − 0.25ωj ωj ) [1], in the central plane in the spanwise direction zcp = Lz /2 at t = 210 for both the reacting case D4 and nonreacting case D4n. Droplets located within the half-grid space surrounding the cut zcp in the spanwise direction, i.e., |zd − zcp | ≤ z/2, have been selected and shown. For clarity, only the region (Ly − δω )/2 < y < (Ly + δω )/2 is shown in the ﬁgure. As indicated by the deﬁnition, high-strain-rate regions are designated by negative Πd values, while high-vorticity regions by positive Πd values. In non-reacting ﬂow, the St = 1 particles/droplets were known to preferentially accumulate in the high-strain-rate regions [1, 6, 7, 10–14], as can be found in Fig. 2b. A clear tendency can be seen that droplets cluster in the regions where Πd ≤ 0. For the combusting case D4, on the other hand, droplets are widely seen in Πd > 0 regions in Fig. 2a. In general, droplets do not accumulate preferentially in the Πd ≤ 0 regions. The changed dynamics of droplets can be explained as follows. In a reacting ﬂow, the predominant eﬀect of heat release is to signiﬁcantly increase the evaporation rate of individual droplets. As the droplet size decreases, its Stokes number also decreases. Therefore, the majority of droplets, especially in the central region of the mixing layer where reaction rate is high (see Fig. 1), have Stokes number of less than unity. As the study of Hu et al. [4] showed, particles of non-unity

352

J. Xia and K.H. Luo

Stokes numbers do not show the same preferential distribution of St = 1 particles/droplets. In the present reacting case, there is a range of Stokes numbers due to diﬀerent evaporation rates of individual droplets. As a consequence, the dynamics of droplets in Case D4 is much more complex than studies assuming monodisperse particles/droplets. With such a broad PDF shown in Fig. 1b, the droplet dynamics is closer to a polydisperse droplet system than to that of the monodisperse droplets speciﬁed initially. As the reaction rate changes both spatially and temporally, droplet size and correspondingly its dynamics in a reacting ﬂow will vary in space and time. The diﬀerent dynamics of droplets in reacting and non-reacting ﬂow have implications on micromixing as measured by the scalar dissipation rate. As discussed in [10], the production term for the scalar dissipation rate χ is closely related to the ﬂow strain, and enhanced by the concentration of droplets in the high-strain-rate regions. As evaporation due to combustion changes the Stokes number and dynamics of individual droplets, it will modify droplet eﬀects on micromixing. To isolate each eﬀect in a reacting turbulent ﬂow with evaporating droplets is a complex task that merits further study in the area.

4 Conclusions DNS has been performed to investigate the dynamics of evaporating droplets in turbulent reacting and non-reacting mixing layers. Due to combustioninduced local evaporation, the initially monodisperse droplet system changes to a polydisperse system, whose size PDF is broad and time-dependent. Therefore, evaporating droplets in the reacting ﬂow do not show the typical preferential distribution of monodisperse particles/droplets in a non-reacting ﬂow. Such a combustion-induced change in droplet dynamics is only one aspect of a complex interacting loop linking turbulence, chemical reaction, evaporation and droplet dynamics. The isolation of each eﬀect will form future work.

Acknowledgements The work was funded by the UK EPSRC under the grant No. EP/E011640/1. The computing resources were supported by the UK Consortium on Computational Combustion for Engineering Applications (COCCFEA) under the EPSRC Grant No. EP/D080223/1.

References 1. Squires KD, Eaton JK (1991) Phys Fluids 3:1169–1178. 2. Elgobashi S, Truesdell GC (1992) J Fluid Mech 242:655–700. 3. Wang LP, Maxey MR (1993) J Fluid Mech 256:27–68.

DNS of Droplet-Laden Reacting/Inert Mixing Layers 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

353

Hu ZW, Luo XY, Luo KH (2002) Theoret Comput Fluid Dynamics 15:403–420. Ling W, Chung JN, Troutt TR, Crowe CT (1998) J Fluid Mech 358:61–85. Miller RS, Bellan J (1999) J Fluid Mech 384:293–338. R´eveillon J, Demoulin FX (2007) J Fluid Mech 583:273–302. Mashayek F (2000) J Fluid Mech 405:1–36. Xia J, Luo KH (2009) Proc Combust Inst 32:2267–2274. Xia J, Luo KH (2010) Flow Turbulence Combust. DOI:10.1007/s10494-0099238-7. Simonin O, Fevrier P, Lavi´eville J (1993) J Turbulence 118:97–118. Fessler JR, Kulick JD, Eaton JK (1994) Phys Fluids 6:3742–3749. Aliseda A, Cartellier A, Hainaux F, Lasheras JC (2002) J Fluid Mech 468: 77–105. Xia J, Luo KH, Kumar S (2008) Flow Turbulence Combust 80:133–153.

Large Eddy Simulation of a Two-Phase Reacting Flow in an Experimental Burner M. Sanjos´e1, E. Riber1 , L. Gicquel1 , B. Cuenot1 , and T. Poinsot2 1

2

CERFACS – CFD Team, 42 av. G. Coriolis, 31057 Toulouse, France, [email protected]; [email protected]; [email protected]; [email protected] IMFT, UMR CNRS-INPT-UPS 5502, Toulouse, France, [email protected]

Abstract Large eddy simulations are performed on the experimental burner MERCATO operated at ONERA fed with air and liquid kerosene. The purpose of the present study is to assess the reliability of the Euler-Euler approach for two-phase ﬂow with and without combustion. The mesoscopic Eulerian quantities for the liquid dispersed phase and the gaseous variables are advanced in space and time with a third-order accurate numerical scheme. Mean and ﬂuctuating velocity components for both phases extracted from the numerical results are compared to existing LDA measurements. The gaseous swirling ﬂow is accurately predicted by the compressible solver. In the non-reacting case, the velocity proﬁles of the liquid dispersed phase show good agreements with experimental data. In the reacting case, the simulation accurately localizes the two premixed and rich burning zones identiﬁed in the experiments.

1 Introduction Large-eddy simulation (LES) has demonstrated its ability to predict turbulent and reactive gaseous ﬂows in complex geometries. It is now applied in real gas turbine geometries as a standard tool to study mean and unsteady ﬂows [7,14]. As most industrial burners are fuelled with liquid hydrocarbons, the extension of LES to two-phase ﬂows is necessary to investigate spray combustion in such geometries [1]. The Euler–Euler approach for two-phase ﬂow provides a good parallel eﬃciency [3], which has been an advantage to perform the LES of the two-phase reacting ﬂow in an experimental burner. This present work is a joint eﬀort between CERFACS, ONERA and IMFT to perform LES of an experimental two-phase combustor operated at ONERA, Toulouse. The experimental rig MERCATO (Fig. 1) is a swirled combustor fed with air and Jet-A liquid fuel.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 51, c Springer Science+Business Media B.V. 2010

355

356

M. Sanjos´e et al.

Fig. 1. The MERCATO conﬁguration (ONERA Toulouse).

Table 1. Summary of test-rig MERCATO ﬂow regimes. Case I: gaseous ﬂow II: gaseous ﬂow + droplets III: reacting two-phase ﬂow

Pressure Air Liquid Air Fuel Equivalence (atm) temperature (K) ﬂow rate (g/s) ratio 1 1

463 463

− 300

15 15

− 1

− 1.0

1

285

285

26

2.9

1.6

This conﬁguration has been computed with the LES code AVBP of CERFACS and results have been compared to experimental data for three regimes, presented in Table 1: (I) purely gaseous non-reacting ﬂow (II) gaseous non-reacting ﬂow with evaporating droplets and (III) reacting ﬂow with droplets. Available experimental data for regimes I and II include velocity ﬁelds for the gas and for the droplets measured by LDA. For case III, highspeed camera visualizations of the reacting ﬂow are available.

2 Numerical conﬁguration and simulations 2.1 Calculation domain and mesh The conﬁguration sketched in Fig. 1 has been entirely meshed, as well as the exit geometry: for cases I and II the ﬂow exits in the atmosphere, while for case III an additional exhaust pipe is added after the combustion chamber. The unstructured meshes contain 3.5 million cells and 650, 000 nodes. 2.2 Description of the Euler–Euler (EE) solver The AVBP code solves the compressible Navier–Stokes equations with a third-order schemes for both spatial diﬀerencing and time advancement [5].

LES of Two-Phase Reacting Flows

357

The Wale model is used to model the sub-grid scales tensor [9]. Boundary conditions are handled with the NSCBC (Navier–Stokes Characteristics Boundary Conditions) formulation [10]. The treatment of the dispersed phase is based on a monodisperse Eulerian mesoscopic approach, which has been previously used in several studies [2, 3, 11]. Conservation equations are solved for droplet number density, liquid volume fraction, droplet mesoscopic velocity components and droplet enthalpy. The present EE method takes into account neither local poly-dispersion nor trajectory crossing of droplets. If two packets of droplets, having diﬀerent mesoscopic velocities, temperatures and mean droplet diameters, intersect, they collapse into one packet having the averaged properties of the preceding two. 2.3 Description of the evaporation and combustion models In the experiments, the injected fuel is a commercial aviation kerosene Jet-A, which is a mixture of a large number of hydrocarbons and additives. In the LES calculations, kerosene is modeled by a single meta-species built as an average of the thermodynamic properties of a kerosene multi-component surrogate. The surrogate is deﬁned by the following molar repartition: 74% n-decane, 15% n-propylbenzene and 11% n-propylcyclohexane. The boiling temperature at atmospheric pressure is 445.1 K. A Stokes’ law with a Reynolds correction is employed for a two-way coupling of drag-force. A classical model is employed for evaporation, which assumes inﬁnite thermal conductivity of the liquid: the evaporation rate is driven by the thermal and species diﬀusion from the droplet surface into the gas phase. A Reynolds correction is applied on Nusselt and Sherwood numbers. One-third to two third rule is employed for constant reference mass fractions and temperature. The reaction rates of the combustion of the kerosene surrogate are calculated by an Arrhenius law. The chemical scheme is a two-step mechanism, with an adaptation of the preexponential factor, built to ﬁt the laminar ﬂame speed values as a function of local equivalence ratio evaluated with the detailed mechanism of [8]. The interaction between turbulence and combustion is modeled using the DTF (dynamic thickened ﬂame) model [4]. 2.4 Description of the kerosene injection Modeling the liquid kerosene spray produced by the atomizer is a crucial step for reacting stationary ﬂows simulations. The ﬂame is indeed highly sensitive to the fuel vapor ﬁeld. The primary and secondary liquid jet breakups are not taken into account in the simulation. The pressure atomizer is modeled by a Dirichlet boundary condition, where functional proﬁles for droplet number density, mean diameter and droplet velocity components reproduced a hollow cone spray based on the empirical correlations of [12] and the air entrainment model of [6].

358

M. Sanjos´e et al.

3 Results 3.1 Gas ﬂow without droplets (case I) Before considering dispersion, evaporation and combustion of droplets, the precision of the results for the carrier phase in the MERCATO conﬁguration was evaluated in [13]. The numerical results of the AVBP gas-phase simulation (case I) were compared to numerical results obtained with the LES code CDP from Stanford and to experimental data provided by ONERA. The AVBP simulation captures the ﬂow correctly in terms of mean ﬂow and unsteady structures. Mean and RMS AVBP results agree well with the experimental proﬁles, with a comparable precision to the CDP simulation. 3.2 Gas ﬂow with evaporating droplets (case II) Figure 2 (left) shows an instantaneous view of droplet distribution in the combustion chamber. The swirled air ﬂow produces droplet preferential concentrations which are commonly observed in turbulent particle-laden ﬂows. The central recirculation zone has a low droplet density. Dense pockets of droplets can be seen in the shear layer of the swirled air ﬂow. Droplets are trapped in a single recirculation zone in the corners of the chamber. Such a droplet distribution leads to fuel vapor inhomogeneities through the evaporation. The RMS ﬁeld of kerosene vapor is shown in Fig. 2 (right). The velocity proﬁles of droplets are compared with LDA experimental data provided by ONERA in Figs. 3 and 4. The experimental values, represented by square symbols, correspond to the averaged velocity of the droplets locally recorded in the laser beam, no matter what the droplet size might be. Such an average value can be directly compared to the resolved Eulerian mesoscopic velocity from the simulation. To delay the spray impingement on the visualization windows, the experimentalists have increased the air ﬂow rate to perform the LDA measurement at the z = 56 mm station. Thus the comparisons at this location must be taken with care. The averaging time in the simulation is of the order of 80 ms corresponding to approximately 2 ﬂow-through times.

Droplet density number.

RMS kerosene mass fraction.

vapor

Fig. 2. Instantaneous droplet distribution and consequences on fuel vapor ﬁeld (case II).

LES of Two-Phase Reacting Flows 0 mm

26 mm

6 mm

56 mm

0 mm Distance to axis (mm)

Distance to axis (mm)

60 40 20

0 –20 –40

6 mm

359

56 mm

26 mm

60 40 20 0 – 20 – 40 – 60

–60 0

5

10 15 20 0

10

20 –12

0

12

24 – 6 0

6

12 18

0

4

Droplet mean axial velocity (m/s)

8 0

2

4

6

8 0

3

6

9

3

0

6

Fig. 3. Left: droplet mean axial velocity. Right: droplet RMS axial velocity. ONERA, − AVBP-EE. (case II). 6 mm

26 mm

56 mm

0 mm

60

6 mm

26 mm

LDA-

56 mm

60

Distance to axis (mm)

Distance to axis (mm)

0 mm

9

Droplet RMS axial velocity (m/s)

40 20 0 – 20 – 40 – 60 –12 – 6 0 6 12 –10 0 10 –14 –7 0 7 14 –6 0

6

40 20 0 –20 –40 –60 0

Droplet mean tangential velocity (m/s)

3

6 0 3 6 0 3 6 90 2 Droplet RMS tangential velocity (m/s)

4

6

Fig. 4. Left: droplet mean tangential velocity. Right: droplet RMS tangential velocity. LDA-ONERA, − AVBP-EE. (case II).

The agreement between the LES results and the experimental proﬁles is good for the mean values. This shows that the injection procedure used in this study, summarized at Section 2.4, is suitable for such ﬂows, where two-phase eﬀects and droplet drag are signiﬁcant. The LES results for the droplet RMS velocity agree with experimental data reasonably. The axial RMS numerical values show more discrepancies from experimental data, especially at the ﬁrst measurement location z = 6 mm. This highlights that the injection procedure does not take into account the ﬂuctuations at the discharge oriﬁce of the simplex atomizer. 3.3 Reacting two-phase ﬂow (case III) The ﬂame structure obtained at steady state is complex: the zones where the droplets evaporate overlap the reaction zones. Due to the low inlet gas temperature, the droplet life time is large compared to the ﬂame characteristic time-scale. Droplets disperse, through the ﬂame front, far downstream in the chamber, as it can be seen on both numerical and experimental instantaneous visualizations in Fig. 5. The two combustion regimes which are highlighted by the color of the ﬂame light emissions, are captured in the simulation. Diﬀusion ﬂames (yellow) are formed between oxygen injected in the air stream and kerosene evaporated

360

M. Sanjos´e et al.

LES instantaneous ﬁeld of droplet distribution (grayscale) and gaseous temperature isolines.

High-speed visualizations. Left: droplets enlightened in a laser plane. Right: ﬂame light emissions.

Fig. 5. Visualization of the two-phase ﬂame structure. (a) Numerical results, (b) experimental views. (case III).

in the burnt gases. A partially premixed ﬂame (blue) occurs in the central recirculation zone as the droplets number is lower and hot gases are stabilized.

4 Conclusions The precision of the Euler–Euler approach for the LES of two-phase ﬂows has been evaluated on a non-reacting but evaporating case. The numerical results obtained in this study show reasonably good agreements with the experimental measurements of ONERA. The two-phase reacting simulation well captures two-phase and ﬂame structures which can be observed on the experimental visualizations. Further results evaluations are needed and require highly detailed experimental measurements.

Acknowledgments The support of Turbomeca, of the French DGA are gratefully acknowledged. Part of this work received funding from the European Community through the project TIMECOP-AE (project #AST-CT-2006-030828). It reﬂects only the authors’ views and the Community is not liable for any use that may be made of the information contain therein.

References 1. S. V. Apte, K. Mahesh, P. Moin, and J. C. Oefelein. Large-eddy simulation of swirling particle-laden ﬂows in a coaxial-jet combustor. Int. J. Multiphase Flow, 29(8):1311–1331, 2003. 2. M. Boileau, S. Pascaud, E. Riber, B. Cuenot, L.Y.M. Gicquel, T. Poinsot, and M. Cazalens. Investigation of two-ﬂuid methods for Large Eddy Simulation of spray combustion in Gas Turbines. Flow, Turb. and Combustion, 80(3): 291–321, 2008.

LES of Two-Phase Reacting Flows

361

3. M. Boileau, G. Staﬀelbach, B. Cuenot, T. Poinsot, and C. B´erat. LES of an ignition sequence in a gas turbine engine. Combust. Flame, 154(1-2):2–22, 2008. 4. O. Colin, F. Ducros, D. Veynante, and T. Poinsot. A thickened ﬂame model for large eddy simulations of turbulent premixed combustion. Phys. Fluids, 12(7):1843–1863, 2000. 5. O. Colin and M. Rudgyard. Development of high-order taylor-galerkin schemes for unsteady calculations. J. Comput. Phys., 162(2):338–371, 2000. 6. G.E. Cossali. An integral model for gas entrainment into full cone sprays. J. Fluid Mech., 439:353–366, 2001. 7. W. W. Kim, S. Menon, and H. C. Mongia. Large-eddy simulation of a gas turbine combustor. Combust. sci. technol., 143(1-6):25–62, 1999. 8. J. Luche. Elaboration of reduced kinetic models of combustion. Application to a kerosene mechanism. PhD thesis, LCSR Orleans, 2003. 9. F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient. Flow, Turb. and Combustion, 62(3):183–200, 1999. 10. T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible viscous ﬂows. J. Comput. Phys., 101(1):104–129, 1992. 11. E. Riber, M. Garc´ıa., V. Moureau, H. Pitsch, O. Simonin, and T. Poinsot. Evaluation of numerical strategies for LES of two-phase reacting ﬂows. J. Comput. Phys., page under review, 2008. 12. N.K. Rizk and A.H. Lefebvre. Internal ﬂow characteristics of simplex atomizer. J. Prop. Power, 1(3):193–199, may-june 1985. 13. M. Sanjos´e, T. Lederlin, L. Gicquel, B. Cuenot, H. Pitsch, N. Garc´ıa-Rosa, R. Lecourt, and T. Poinsot. LES of two-phase reacting ﬂows. In Proc. of the Summer Program. Center for Turbulence Research, NASA Ames/Stanford Univ., 2008. 14. L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.-U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, and D. Veynante. Compressible large-eddy simulation of turbulent combustion in complex geometry on unstructured meshes. Combust. Flame, 137(4):489–505, 2004.

Hybrid LES/CAA Simulation of a Turbulent Non-Premixed Jet Flame C. Klewer, F. Hahn, C. Olbricht, and J. Janicka Institute for Energy and Powerplant Technology, Darmstadt University of Technology, Petersentr. 30 D-64287 Darmstadt, Germany, [email protected]; [email protected]; [email protected]; [email protected] Abstract In this work, a numerical study of combustion induced noise is performed. For this purpose, a hybrid LES/CAA approach is applied to the simulation of a turbulent jet ﬂame. The approach is based on a low-mach number LES and linearized acoustic equations. Both LES and CAA computations are compared to experimental investigations.

1 Introduction Knowledge on combustion generated noise and especially on combustion instability is of great importance in the design process of stationary gas turbine combustors and aircraft engine combustors. In the presence of walls and therefore enclosed ﬂames, a feedback of the acoustics onto the ﬂame is possible, if the ﬂame drives a resonant mode of the combustion chamber. The combustion instability phenomenon is known to appear prevalent in the context of lean, premixed combustion since there are many diﬀerent mechanisms exciting resonant modes, e.g. equivalence ratio oscillations [1]. As a ﬁrst step towards the simulation of the combustion instability phenomenon, the noise emission by an open non-premixed jet ﬂame is studied in this work employing a hybrid approach combining large eddy simulation (LES) with computational aeroacoustics (CAA). In order to simulate combustion induced noise, diﬀerent hybrid approaches were proposed in the close past [2, 3]. Questions to be addressed towards the LES/CAA simulation of conﬁned cases are matching computational grids and appropriate boundary conditions. The current study addresses these issues by using the same computational grid for both the LES and the CAA computations within the source region together with an adequate formulation of the governing equations.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 52, c Springer Science+Business Media B.V. 2010

363

364

C. Klewer et al.

2 LES/CAA methodology In order to describe the reacting ﬂow ﬁeld of a non-premixed ﬂame, the mixture fraction approach is used. The relation of the mixture fraction and the density is determined from detailed, one dimensional chemistry computations in a pre-processing step and then stored in lookup tables. A detailed description of the modelling and the numerical procedure can be found in [4]. The governing equations are solved within the LES code FLOWSI, which has shown to predict jet ﬂames with promising results before, e.g. [4]. The acoustic wave propagation is described using linearized acoustic equations. These are obtained by adding an acoustic perturbation to the mean ﬂow: p = p1 + p 0 ; ρ = ρ 1 + ρ0 ; ui = u1,i + u0,i (1) In the case of combustion, the mean quantities are usually a function of space. Substituting p and u1,i back into the governing equations of mass and momentum assuming negligible viscous forces and conserving only ﬁrst order terms gives [5]: ∂ρ1 ∂u1,j + ρ0 =0 (2) ∂t ∂xj ρ0

∂u1,i ∂p1,j + =0 ∂t ∂xj

(3)

In the case of combustion, the major acoustic source generating sound can be expressed through the ﬂuctuating heat release [5]. Using the energy equation for reacting ﬂows including transport eﬀects [6], an expression relating the material time derivative of the density to the heat release, which is included in the second term of the right hand side of equation (4), can be obtained. This formalism was applied by Bui et al. [3]. ⎛ ⎞⎤ ⎡ / N Dρ ⎣ 1 Dp DYn ∂ui ⎠⎦ α ⎝ ∂h // ∂qj ·ρ − τij (4) = 2 + + / Dt c Dt cp n=1 ∂Yn / Dt ∂xj ∂xj ρ,p,Ym

For the derivation of an appropriate combustion noise source term in conjunction with a reactive low-mach number LES, the only source term taken into account is the material derivative of the density containing the heat release as proposed by Bui et al. [3]. According to this simpliﬁcation, the following pressure density relation is used and applied to the governing equations: ∂ρ1 1 ∂p1 ρ0 Dρ = 2 + ∂t c0 ∂t ρ Dt

(5)

∂p1 ∂u1,j ρ0 Dρ + c0 2 ρ 0 = −c0 2 ∂t ∂xj ρ Dt

(6)

Hybrid LES/CAA Simulation of a Turbulent Non-Premixed Jet Flame

ρ0

∂u1,i ∂p1,j + =0 ∂t ∂xj

365

(7)

The acoustic wave propagation is solved using the CLAWPACK [7] software. Since this solver is a general purpose solver for ﬁrst order hyperbolic problems, its use is predestined for the current study. The solver applies a high-resolution Godunov scheme in a ﬁnite volume context. In order to avoid spurious oscillations a van Leer limiter is used [7]. For the current study, the solver has been extended accounting for cylindrical grids. Furthermore, the governing equations are changed compared to previous work [8]. Here, the acoustic variables were transformed in order to solve the second order wave equation. This requires a mathematical reconstruction of the acoustic pressure ﬂuctuation as a post processing step and the treatment of solid surfaces is questionable.

3 Experimental test case and numerical setup Throughout this work the so called DLR-A ﬂame has been investigated. Detailed measurements on the reactive ﬂow ﬁeld as well as measurements of sound pressure levels exist allowing for a comparison of the results obtained by the current LES/CAA approach. The DLR-A ﬂame consists of a central fuel jet, which is fed coaxially through a round nozzle into a constant coﬂow of ambient air. The basic parameters of the conﬁguration are given in Table 1. The numerical grid used for the LES computations consists of 752 × 80 × 32 ≈ 1.9 · 106 cells in axial, radial and circumferential direction covering a region of 0.75 × 0.2 m × 2π, while the computational grid used for the CAA computations consists of 752 × 120 × 32 ≈ 2.9 · 106 cells and covers a region of 0.75 × 0.45 m × 2π. Within the radius of the LES domain, the two grids are identical. Outside this radius, only the acoustic wave propagation is solved.

4 Results An instantaneous cross-sectional view of the mixture fraction, the local speed of sound as well as the OH mass fraction is provided in Fig. 1 giving an impression of the ﬂow ﬁeld and the ﬂame position. A statistical evaluation of the ﬂowﬁeld is given in Fig. 2 for three distinct axial heights setting x/D = 5, 40, 80. The position close to the nozzle Table 1. Parameters of the DLR-A ﬂame. F uel N ozzle Re

22.1/33.2/44.7 D = 0.008 15200

vol% CH4 /H2 /N2 m −

366

C. Klewer et al. 90

90 0.9 [−] 0.8

80 70

850 [m/s] 800

80 70

0.7 60

x 10−3

90

750

80

3 [−]

70

2.5

60

700

60

50

650

50

40

600

2

0.4 30

0

30

1

0

0 10 r/D [−]

20

450

10

0.1 10

1.5

40

500 20

0.2

10

550

30

0.3

20

x / D [−]

0.5 40

x/D [−]

x/D [−]

0.6 50

10

0 10 r/D [−]

400

10

350

0

0.5

10

0 10 r/D [−]

[m/s] 1/2

12 9 6 3 0 20 15 10 5 0 48 36 24 12 0 0.00

[m/s]

Fig. 1. Snapshots of the LES computations.

0.02

0.04 r [m]

0.06

0.08

3.2 2.4 1.6 0.8 0.0 6.0 4.5 3.0 1.5 0.0 12 9 6 3 0 0.00

LES Exp

0.02

0.04 r [m]

0.06

0.08

Fig. 2. Statistical proﬁles of the LES computations.

(x/D = 5) reveals, that the jet break up point is slightly underestimated, since the simulated proﬁle of the mean axial velocity is somewhat steeper than the experimental one. This is consistent with the velocity ﬂuctuation at this point, since its peak position is shifted towards a smaller radius compared to the experimental investigation. One possibility to overcome this eﬀect is to employ turbulent inﬂow data for the LES simulation. This was not applied for the present case to avoid the emerging of artiﬁcial distinct frequencies in the acoustic spectra. Nevertheless, the simulated ﬂow ﬁeld matches the experimental results very well. In Fig. 3, two snapshots of the simulated acoustic pressure are given providing a qualitative insight into the acoustic ﬁeld. One can clearly observe wave patterns emerging from the ﬂame. Regarding the two dimensional view, the acoustic waves generated near the nozzle show a rather uniform behavior, while the ﬁeld more downstream exhibits random-like structures. This observation is emphasized by the three dimensional view. The propagation of the acoustic waves near the nozzle appears to be almost spherically. The most active regions of the noise generation are connected to the location of the ﬂame, where the unsteady heat release is generated.

Hybrid LES/CAA Simulation of a Turbulent Non-Premixed Jet Flame

367

Fig. 3. Instantaneous view of the acoustic ﬁeld.

60 LES/CAA EXP

40 20

Lp [dB/Hz]

Lp [dB/Hz]

60

x/D=12 r/D= 50

0 100

300

1000 f [Hz]

2000

5000

LES/CAA EXP

Lp [dB/Hz]

Lp [dB/Hz]

20

x/D= 0 r/D= 50

0 100

300

1000 f [Hz]

20

x/D= 37 r/D= 50 0 100

300

1000 f [Hz]

60

60 40

LES/CAA EXP

40

2000

5000

2000

5000

LES/CAA EXP

40 20 x/D = 25 r/D = 50 0 100

300

1000 f [Hz]

2000

5000

Fig. 4. Comparison of simulated spectra with experimental investigations.

A spectral analysis of the sound pressure level at four diﬀerent axial positions computed by means of a FFT is presented in Fig. 4. The simulated spectra are obtained via averaging eight individual spectra along the circumference. Within a range from approximately 100 to 2, 000 Hz, promising agreement between simulation and experiment can be observed. Beyond this frequency, the simulated spectra overestimate the experimental ﬁndings. This is especially visible at x/D = 37. The deviations in the high frequency range might be due to the numerical scheme being second order accurate. Hence high wave number components may be inﬂuenced by numerical dissipation and dispersion.

368

C. Klewer et al.

5 Conclusions and outlook A hybrid LES/CAA approach has been proposed and was applied to the simulation of a turbulent non-premixed jet ﬂame. The LES calculations were validated brieﬂy by ﬁrst and second order statistics. A spectral comparison of the sound pressure levels to measurements revealed promising agreement within the low frequency range. In contrast to previous work, the governing equations are changed and the acoustic solver was extended to handle cylindrical computational grids. This enables the treatment of solid surfaces or reﬂecting outlets. Within conﬁned conﬁgurations, a ﬂame/acoustic coupling occurs and therefore, this has to be incorporated by an appropriate model into the simulation procedure. One promising way to achieve this is to modulate the ﬂow mass rate at the inlet of such a system according to the acoustic pressure waves. Another approach was mentioned by Duwig et al. [9]. They propose to couple back the acoustic pressure to the aerodynamic pressure within the Navier–Stokes equations. These issues can be addressed in future work with the extended source code.

Acknowledgements The authors gratefully acknowledge the ﬁnancial support by the following institutions: • • •

The German Research Council (DFG) through the Research Unit FOR 486 and the Collaborative Research Center SFB 568. The German Ministry for Economy and Technology (BMWi) through the joint project COORETEC-turbo (grant ID 0327725B). Rolls-Royce Germany through the joint project COORETEC-turbo (grant ID 0327725B).

References 1. T. Lieuwen and B. T. Zinn. The role of equivalence ratio oscillations in driving combustion instabilities in low NOx gas turbines. Proc. Combust. Inst., 27: 1809–1816, 1998. 2. M. Ihme, D. J. Bodony, and H. Pitsch. Prediction of combustion-generated noise in non-premixed turbulent jet ﬂames using large-eddy simulation. In 13th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, USA, May 2006. AIAA-2006-2614. 3. T. Ph. Bui, W. Schr¨ oder, and M. Meinke. Acoustic perturbation equations for reacting ﬂows to compute combustion noise. Int. Journal of Aeroacoustics, 6: 335–355, 2007. 4. A. Kempf, F. Flemming, and J. Janicka. Investigation of lengthscales, scalar dissipation, and ﬂame orientation in a piloted diﬀusion ﬂame by LES. Proc. Combust. Inst., 30:557–565, 2005.

Hybrid LES/CAA Simulation of a Turbulent Non-Premixed Jet Flame

369

5. T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. R.T. Edwards, Flourtown, USA, 2001. 6. F.A. Williams. Combustion Theory. The Fundamental Theory of Chemically Reacting Flow Systems. Addison-Wesley, 1964. 7. R. J. LeVeque. Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Physics, 131:327–353, 1997. 8. F. Flemming, A. Sadiki, and J. Janicka. Investigation of combustion noise using a LES/CAA hybrid approach. Proc. Combust. Inst., 31:3189–3196, 2007. 9. B. Duwig, C. Gherman, M. Mihaescu, M. Salewski, and L. Fuchs. Numerical study on thermo-acoustic waves generation by a swirling ﬂame using a new approach based on large-eddy simulation. In ASME Turbo Expo 2005, Reno, NV, USA, 2005. GT2005-68136.

LES/CMC of Forced Ignition of a Bluﬀ-Body Stabilised Non-Premixed Methane Flame A. Triantafyllidis1 , E. Mastorakos1, and R.L.G.M. Eggels2 1

2

Department of Engineering, University of Cambridge, UK, [email protected]; [email protected] Rolls Royce Deutschland, Blankefelde-Mahlow, Germany, [email protected]

Abstract A Large Eddy Simulation/Conditional Moment Closure calculation of forced ignition of a turbulent bluﬀ-body stabilised non-premixed methane ﬂame is performed and the results are qualitatively investigated to assess the suitability of this model for highly transient combustion phenomena. The qualitative features of the ﬂame expansion process agree with experimental ﬁndings. Quantitatively, the time for the establishment of the whole ﬂame is under-predicted, which could be improved by using a ﬁner CMC grid.

1 Introduction Predicting the success or failure of a spark event is important from a fundamental point of view, but also for practical engineering applications, such as industrial gas turbines and jet engines. The propagation speed of a ﬂame and its eﬀect on the ﬂow ﬁeld can inﬂuence the design of a combustor, the size of which is often determined by its ignition performance. Large Eddy Simulation (LES) promise to oﬀer accurate predictions of transient phenomena. The larger energy containing scales are resolved, while the eﬀect of the smaller (sub-grid) scales is modelled. Since most chemical reactions take place on a sub-grid scale level, a combustion model is needed. The Conditional Moment Closure (CMC) approach [1] has been used in Reynolds Averaged Navier Stokes simulations to investigate diﬀerent turbulent combustion problems, such as attached ﬂames [2], lifted ﬂames [3], bluﬀ-body stabilised ﬂames [4], opposed jet ﬂames [5] and autoignition [6]. Recently, the CMC was formulated for LES [7] and used in simulations of statistically steady turbulent ﬂames [7]. In the CMC, equations are solved for the conditionally ﬁltered reactive scalars, the conditioning being on the mixture fraction. Here, the CMC is used in LES of forced ignition of a bluﬀ-body stabilised non-premixed methane ﬂame. The purpose of this work is to investigate whether it is possible to capture the qualitative behaviour of an igniting V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 53, c Springer Science+Business Media B.V. 2010

371

372

A. Triantafyllidis et al.

non-premixed ﬂame. In the following section, the LES and CMC equations are described, followed by the presentation of the results and discussion before the conclusions are drawn.

2 Modelling Applying a ﬁltering operation to the Navier–Stokes equations gives the LES equations: ∂ρ ∂(ρ ui ) =0 + ∂t ∂xi

(1)

∂(ρτijr ) ui ) (ρ ui u j ) ∂(ρ ∂p ∂ τij + =− + − ∂t ∂xj ∂xi ∂xj ∂xj

(2)

where ρ, u and p are the density, velocity and pressure respectively, while ∂ u ∂ ui + ∂xji ). The term τijr = u$ i u j is the sub-grid scale stress τij = μ( ∂x i uj − u j tensor. The equation for the ﬁltered mixture fraction ξ is

ui ) ∂(ρξ) (ρξ ∂ ∂ ∂ ξ + − = ρD ρ(ui ξ − u i ξ) (3) ∂t ∂xi ∂xi ∂xi ∂xi An equation for ξ2 is solved:

∂ ∂ ξ2 ui ξ2 ) ∂(ρξ2 ) ∂(ρ 2 − 1 ρ(u$ − 2ρN = ρD i ξ2 ) + iξ − u ∂t ∂xi ∂xi ∂xi ∂xi

(4)

2 = D ∂ξ where N is one half of the scalar dissipation rate. The sub-grid ∂xi 2 = ξ2 − ξ2 . scale variance can then be calculated as ξ The sub-grid scale stress tensor is modelled using the standard Smagorin2 sky model, through a turbulent viscosity νt = (CS Δ) S, where the parameter CS is calculated dynamically based on a scale similarity approach [8] and S is the characteristic ﬁltered rate-of-strain tensor. The sub-grid scale ﬂuxes of ξ and ξ2 are modelled using gradient approaches with a turbulent diﬀusivity Dt = νt /Sct , where Sct = 0.7 is the turbulent Schmidt number. The scalar 2 /Δ2 , based on a characteristic = Cξ (ν + νt ) ξ dissipation is modelled as N timescale [9]. In the context of LES, the density-weighted conditionally ﬁltered average of a random variable f is deﬁned as [10]: +∞ ρ(x , t)f (x , t)δ[ξ(x , t) − η]G(x − x ) dx f |η(η, x , t) = −∞ (5) +∞ , t)Δ[ξ(x , t) − η]G(x − x ) dx ρ(x −∞

Stabilised Non-Premixed Methane Flame

373

where G(x −x ) is a ﬁltering function of speciﬁed width Δ, δ[ξ(x , t)−η] is the x , t) = +∞ ρ(x , t)δ[ξ(x , t)−η]G(x −x ) dx /ρ ﬁne-grained density and P(η; −∞ is the density-weighted Filtered probability Density Function (FDF). The ﬁl1 tered value of the variable f can be obtained as f = 0 f |η P(η) dη. The CMC equations can be derived by ﬁltering the transport equations for the reactive scalars Yα [7] and can be written as & ∂ % ∂Qα $ ∂Qα ∂ 2 Qα 1 $ $ + ui |η =N |η + ω |η− ρ P(η) u Y |η − u |ηQ α i α i α ∂t ∂xi ∂η 2 ∂xi ρ P(η) (6) $ |η is the conditionally ﬁltered reactive scalar, u |η is the conwhere Qα = Y$ α i $ ditionally ﬁltered velocity, N |η is the conditionally ﬁltered scalar dissipation rate, ω α |η is the conditionally ﬁltered reaction rate, while the last term on the left-hand-side is the sub-grid scale conditional ﬂux. Several terms in this equation require modelling. The conditional velocity is considered to be equal to the unconditional, while the Amplitude Map$ ping Closure [12] is used to model N |η. The sub-grid scale conditional ﬂux $ is modelled as ui Yα |η − ui |ηQα = −Dt ∂Qα /∂xi , where Dt is the turbulent diﬀusivity. A single-step chemical mechanism where the activation energy and heat release rate are functions of the local equivalence ratio is used to provide the chemical source terms [13]. This tuning was made to achieve good agreement with experimental data of the laminar ﬂame speed. After this tuning was made, it was also found that the extinction strain rate of a non-premixed ﬂame could also be reproduced [13]. The spark is modelled by assuming that initially in a region of the ﬂow (spark location), the corresponding ﬂamelet (Qα (η)) would be that of a steady ﬂame.

3 Results and discussion The case investigated here is shown in Fig. 1, used previously in spark ignition experiments [14]. The diameter of the bluﬀ body is 25 mm, while the burner is enclosed in a cylindrical quartz tube of diameter 70 mm and length 80 mm. The bulk velocity of the air stream is Vb = 10 m/s, while the fuel (pure methane) is injected radially through a 0.7 mm slit situated 2 mm before the exit of the bluﬀ body. This burner consists of regions with diﬀerent levels of mixing. This creates diﬃcult circumstances for the CMC model and makes it a good application for investigating whether the correct behaviour of the ﬂame expansion can be captured in regions which are partially premixed. The ﬁnite volume, low Mach number CFD code PRECISE was used to perform the LES [15], while an in-house CMC code was used to solve the CMC equations [5, 6, 11]. Three grids with diﬀerent resolutions were used for the inert LES; a coarse grid with approximately 800 K nodes, a normal grid

374

A. Triantafyllidis et al.

Spark location

Fuel

Air

Fuel

Air

Fig. 1. Schematic of the bluﬀ-body burner showing the inlet of the fuel and air, the recirculation zones behind the bluﬀ body and the location of the spark in the ignition case.

with 1.8 M nodes and a ﬁne grid 4.9 M nodes. The ignition simulations were performed on the ﬁne grid. The CMC equations were solved on a grid with 15.5 K nodes in physical space, while 51 nodes were used to discretise η-space. The mean and the Root Mean Square (RMS) of the axial velocity and the mixture fraction are compared with experimental data (obtained with LDA and acetone PLIF) in Fig. 2. For the velocity, there is excellent agreement of the LES with the experimental data using both grids, which implies that the velocity ﬁeld is not sensitive to the resolution of the grid. The recirculation zone is approximately 1.0Db long and 0.5Db wide, in agreement with experimental studies of ﬂows behind bluﬀ bodies [16]. For the mixture fraction distribution, when the ﬁne grid is used, ξ is predicted accurately. Because of the high resolution, the break-up of the jet before the exit of the bluﬀ body is captured and the value of ξ is predicted accurately in all axial positions. Using the normal grid leads to over-prediction of ξ, especially at the exit of the bluﬀ body. This demonstrates the high sensitivity of the mixture fraction distribution to the grid resolution and that attention must be given to the very early fuel jet break-up phase. For the ignition simulation, the location of the spark is shown in Fig. 1; it is a rectangular spark covering 5.5 mm (two CMC cells), 5 mm (two CMC cells) and 7.5 mm (three CMC cells) in the x, y and z-direction respectively. Isosurfaces of stoichiometric mixture fraction, coloured by temperature at diﬀerent times from the moment of ignition are shown in Fig. 3. Initially, under the strong inﬂuence of convection, the ﬂame expands in the axial direction while at the same time turbulent diﬀusion causes the ﬂame to expand in the azimuthal and radial direction. Heat is transferred from the location of the spark to the neighbouring locations and after approximately 10 ms the ﬂame has expanded in all directions and a three dimensional bluﬀ-body stabilised turbulent ﬂame has been established. This demonstrates that the correct dynamics of the

Stabilised Non-Premixed Methane Flame 1 0

y/Db=2.0

b 0.25

y/Db=2.0

1 0

y/Db=1.4

0.25

y/Db=1.4

1 0

y/Db=1.0

0.5 0.25

y/Db=1.0

1 0

y/Db=0.62

0.5 0.25

y/Db=0.62

1 0 –1

y/Db=0.22

0.25

y/Db=0.22

0

1

0.5

Vrms/Vb

V/Vb

a

0

1.5

0

0.5

r/Db

c

d y/Db=1.4

0 0.1

y/Db=1.0

ξRMS

0.2 ξ

1.5

r/Db

0.1

y/Db=0.6

0.1 0.2 0.1 0.2 0.1 0

1

375

y/Db=0.4

0.05

y/Db=1.4

0.05

y/Db=1.0

0.05

y/Db=0.6

0.05

y/Db=0.4

0.1

y/Db=0.2 0

1

0.5 r/Db

y/Db=0.2

0.05 0 1.5

0

0.5

1

1.5

r/Db

Fig. 2. Radial proﬁles of (a) the mean, (b) RMS of the axial velocity, (c) the mean and (d) the RMS of the mixture fraction at diﬀerent distances from the bluﬀ body. Symbols: experimental data [14], black lines: coarse grid, red lines: normal grid, blue lines: ﬁne grid.

Fig. 3. Instantaneous isosurfaces of the stoichiometric mixture fraction at diﬀerent times from the moment of ignition, coloured by temperature [K].

ﬂow are captured by this simulation, since this behaviour is in agreement with experimental ﬁndings. The time for the ﬂame to fully expand, however, is under-estimated. The fact that the correct dynamics are captured

376

A. Triantafyllidis et al.

demonstrates that this method is promising and further improvement on the accuracy of the results may be achieved by modifying the resolution of the CMC grid and the complexity of the chemical mechanism.

4 Conclusions Large Eddy simulations of ignition of a turbulent non-premixed bluﬀ-body stabilised methane ﬂame have been performed. The Conditional Moment Closure was used to model the turbulence-chemistry interaction, while a singlestep chemistry model was used to provide the chemical source term. The inert ﬂow was captured accurately, with increased resolution required to predict the distribution of the mixture fraction. This is due to the fact that the fuel is injected before the exit of the bluﬀ body and it is crucial to capture that initial break-up of the jet. The correct dynamics of the ﬂame propagation process were captured, with the LES/CMC predicting the correct behaviour of the ﬂame expansion. The time for the ﬂame to fully expand was under-estimated, which may be due to the simpliﬁcations made in the application of the CMC in the LES (single-step chemistry, coarser CMC grid, modelling of spark).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Klimenko A Y, Bilger, R W (1999) Prog. Energy Combst. Sci. 25:595–687 Roomina M R, Bilger R W (2001) Combust. Flame 125:1176–1195 Kim I S, Mastorakos E (2005) Proc. Combust. Inst. 30:911–918 Fairweather M, Wooley R M (2007) Combust. Flame 151:397–411 Kim I S, Mastorakos E (2006) Flow Turb. Combust. 76:133–162 Wright Y M, De Paola G, Boulouchos K, Mastorakos E (2005) Combust. Flame 143:402–419 Navarro-Martinez S, Kronenburg A, Di Mare F (2005) Flow Turbulence Combust. 75:245–274 Germano M, Piomelli U, Moin P, Cabot W H (1991) Phys. Fluids 3:1960–1965 Pera C, R´eveillon J, and Vervisch L, and Domingo P (2006) Combust. Flame 146:635–648 Colucci R J, Jaberi F A, Givi P, Pope S B (1998) Phys. Fluids 10:499–515 De Paola G, Mastorakos E, Wright Y M, Boulouchos K (2008) Combust. Sci. Tech. 180:883–899 O’Brien E E, Jiang T-L (1991) Phys. Fluids 3:3121–3123 Fern´ andez-Tarrazo E, and S´ anchez A L, and Lin˜ an A, and Williams F A (2006) Combust. Flame 147:32–38 Ahmed S F, Balachandran R, Marchione T, Mastorakos E (2007) Combust. Flame 151:366–385 James S, Zhu J, Anand M S (2006) AIAA J. 44:674–686 Taylor A M K P, Whitelaw J H (1984) J. Fluid Mech. 139:1079–1090

Large Eddy Simulation of a High Reynolds Number Swirling Flow in a Conical Diﬀuser C´edric Duprat, Olivier M´etais, and Guillaume Balarac L.E.G.I, B.P 53, 38041 Grenoble Cedex 09, France, [email protected]

1 Introduction The ﬂow downstream an hydraulic turbine is a swirling ﬂow which presents a strong unsteady vortex core within the draft tube. It may trigger instabilities whose development has serious impacts on the eﬃciency of the system. The draft tube is also sensitive to ﬂow separation due to the presence of a strong pressure gradient. It is still a great challenge to numerically reproduce the ﬂow dynamics and to predict the associated instabilities. The purpose of the present study is a ﬁrst step towards the numerical simulation of a full draft tube conﬁguration: we here consider a simpliﬁed draft tube consisting of a straight conical diﬀuser (see Fig. 1). This ﬂow exhibits most of the complex features of a draft tube: presence of swirl generated by the inﬂow conditions, high Reynolds number and adverse pressure gradient which may eventually lead to ﬂow separation. The code validation is based on an ERCOFTAC data base which provides accurate experimental data even close to the wall [1]. The experiments reproduce the essential features of the complex ﬂow and are here used to test the numerical procedure and the modeling assumptions. Since high Reynolds numbers are here involved, the goal of the present study is to test the wall function procedure originally proposed by Manhart et al. [2] which deals with the presence of local pressure gradient and is therefore suitable to reproduce ﬂow separation. A crucial point to correctly reproduce the ﬂow within a diﬀuser is an adequate representation of the inﬂow conditions.

2 Physical and numerical modelling The simulations are performed with the newly distributed open source CFD code based on the Field Operation and Manipulation C++ class library for continuum mechanics (OpenFOAM, http://www.opencfd.co.uk). It solves the Navier–Stokes equations for turbulent incompressible ﬂow with a polyhedral ﬁnite-volume approach. The numerical scheme is second order accurate in V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 54, c Springer Science+Business Media B.V. 2010

377

378

C. Duprat et al.

−25

60

175

200

260 mm 10 degree

400 mm

100 mm

510 mm

Fig. 1. Experimental geometry, location of the probes.

both time and space. The preconditioned Conjugate Gradient [3], an Algebraic Multigrid solvers and parallel computers in the domain decomposition mode have been used in order to reduce the computational time by preserving the accuracy. This open code has been already tested for turbulent ﬂow in water turbines and the results are similar with other CFD codes in terms of accuracy [4]. The C++ code library of classes includes a well-tested and validated largeeddy simulation (LES) capability [5]. A one-equation type model is used in this work. This choice is motivated by several factors, as it has been demonstrated in the comparative SGS model study done by Fureby [5] 2.1 Axisymetric diﬀuser The ﬂow conﬁguration is the following: The swirling component is deﬁned by uθ max /Ub = 0.59 where Ub is the average axial velocity at the inlet and uθ max is the maximum circumferential velocity in the inﬂow; the angle of the conical diﬀuser was chosen to be 20◦ . The swirl parameter used is deﬁned at the inlet of the experimental geometry x = −25 mm as R ux uθ r2 dr Sw = 0 R (1) R 0 ux 2 r dr Where ux and uθ are the axial and azimuthal velocities respectively. Its value is Sw = 0.295. The block-structured mesh consists in about 2 million hexahedral cells with 126 cells in azimuthal direction. The grid is stretched in both the axial and radial direction in order to reﬁne close to the wall and in the diﬀuser. The ﬁrst node close to the wall is placed at a maximum value of 11 wall units which is the upper limit of the wall model deﬁned later (Section 2.2). In the experiment (see Fig. 1), the swirl is generated using an honneycomb located 500 mm before the diﬀuser. In experiments, outlet of the diﬀuser is discharged to the atmosphere. To reproduce the outlet numerically, a large dump is added to the computational domain as displayed in Fig. 2. To avoid recirculation behind the end of the diﬀuser, we impose a swirling coﬂow at the annular side of the dump. Appropriate outﬂow boundary conditions should ensure that vorticies can approach

High Reynolds Number Swirling Flow in a Conical Diﬀuser

379

Fig. 2. Computational geometry.

and cross the outﬂow boundary without signiﬁcant disturbances. For this purpose we use a convective boundary condition of the type ∂u ∂u + Uconv =0 ∂t ∂χ

(2)

where Uconv is a convective velocity ﬁxed by velocity value at the previous cell, and χ the main ﬂow direction [6]. In the present work, we want to perform LES at high Reynolds number for industrial applications: the Reynolds number of the ﬂow based on the bulk velocity of the inlet and the inlet diameter of the diﬀuser is Re = 202, 000. We also wish to develop a tool for engineering applications. This implies to keep a relatively coarse mesh with an adapted wall function to take into account the wall informations. Moreover, a way to economically specify the inﬂow conditions needs to be designed. 2.2 Wall modelling To use a coarse mesh, the near wall behaviour should be modelled through a wall model. Because of the geometry, we need to take into account both wall shear stress and pressure gradient. The pressure gradient is indeed far from being negligible in a draft tube and they can lead to ﬂow separation. The present wall function has previously been tested in turbulent ﬂow with boundary layer separation successfully and yield accurate results [2]. In addition to the standard friction quantities uτ , the authors determined a pressure / / / ∂p /1/3 based characteristic velocity up = / ρμ2 ∂x / , and deﬁnes combined veloci9 ties uτ p = u2p + u2τ . The near-wall scaling for velocity is derived involving pressure gradient eﬀects : ∗

∗

U = sign(τw )αy + sign

∂p ∂x

(1 − α) 2

3/2

y∗

2

(3)

380

C. Duprat et al.

where U ∗ = U/uτ p and y ∗ = ρyuτ p /μ. The ratio α = uτ /uτ p quantify the friction, if α = 0 there is ﬂow separation and if α = 1 there is no pressure gradient. 2.3 Generation of realistic inlow The numerical representation of an hydraulic turbine, in one case, and of a honneycomb, in an other one, are very demanding because of the large number of cells and the high Reynolds number. An alternative methods was chosen to generate a turbulent inﬂow conditions by performing a precursor simulation on a simpler geometry. A streamwise periodic LES of developed swirling ﬂow in an annular pipe was performed, using body forces as described by Pierce [7]. The body force proposed by Landman in a DNS calculation [8] is used in this study: F fθ (r) = (4) 2 2 1 + (r/r0 ) where r0 is the radius of the inlet and F the magnitude of the force fθ (r) needed to correct the swirl number. In the current work, we introduced an axial body force to preserve the massﬂow average, and a tangential body forces to keep the same swirl number as the experimental setup one. To avoid performing a separate calculation for the inﬂow, a mapped inlet method was used. This employees a simple scaled mapping of the velocity (and other data such as the subgrid scale kinetic energy k as appropriate) from some plane in the interior of the main domain to specify the inlet velocity and turbulence properties. The mapping plane must be far enough upstream of signiﬁcant features to prevent corruption of the inlet by downstream eﬀects. Validation of this method was done with a periodic swirling pipe. Initial ﬁeld was a zero-swirl number laminar proﬁle, perturbed close to the wall. Figure 3 displays time signals of the swirl number inside the pipe. The feedback control algorithm check the azimuthal body force generate the desired 1.2

Sw/Sw_expe

1 0.8 0.6 0.4 0.2 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Times

Fig. 3. Time signals of Sw .

High Reynolds Number Swirling Flow in a Conical Diﬀuser

381

1.2 1

mean axial velocity mean orthoradial velocity

U/Uo

0.8 0.6 0.4 0.2 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 radius/ro

1

Fig. 4. Mean velocity proﬁle.

swirl number. After a few time, the swirl become stable and the value is the right swirl number. Streamwise proﬁle and azimuthal proﬁle can be seen in Fig. 4, despite we initialize the ﬁeld with a laminar one, a turbulent proﬁle is obtained. The azimuthal body force generate an azimuthal velocity which is null at the center of the pipe (r = 0) and at the wall (r = r0 ).

3 Results 3.1 Proﬁles of mean velocity Because the swirl number represent an integrated quantity, one swirl number may have completely diﬀerent velocity distributions. Although, the swirl number provides an incomplete description of the properties of a swirling ﬂow, it is commonly used in the literature and therefore also employed in the present study. Inﬂow proﬁles have not been corrected. Even if this choice lead to different mean proﬁle, in comparison to the experiment, it include more physics in the calculation. Figure 5 shows the computed time-averaged proﬁles in the azimuthal direction for diﬀerent axial position. Streamwise velocity proﬁles are represented in Fig. 6 and shows less inﬂuence of the swirl number. However, because the inlet azimuthal velocity is smaller than expected close to the wall, the streamwise velocity is overpredict in the center of the diﬀuser. However, despite the diﬀerence of the velocity proﬁles, similar behaviors with the Clausen’ experiments are found. The given swirl number value allows indeed to prevent boundary layer separation but the value is insuﬃcient to cause recirculation in the core ﬂow. 3.2 Instantaneous ﬂow In this section, the instantaneous ﬂow is discussed with special emphasis on coherent structures and scales present in the ﬂow. Figure 7 shows two isosurfaces of instantaneous pressure. It is the swirl eﬀect that creates the conical

382

C. Duprat et al. 0.6 Measured W-025 W-025 Measured W060 W060 Measured W175 W175

0.5

U/(U_0)

0.4 0.3 0.2 0.1 0 0

20

40

60 80 100 120 140 160 180 proble location (mm)

Fig. 5. Mean azimuthal velocity proﬁle.

1.4

Measured U-025 U-025 Measured U060 U060 Measured U175 U175

1.2

Ux/(U_0)

1 0.8 0.6 0.4 0.2 0

0

20

40

60 80 100 120 140 160 180 probe location

Fig. 6. Mean streamwise velocity proﬁle.

Fig. 7. Coherent structures visualized using two diﬀerent isosurfaces of instantaneous pressure.

shape of the iso-pressure contour. A large-scale coherent structure, rotating, is also detected in the diﬀuser. Due to the pressure gradient in the diﬀuser, a part of the vortex becomes an helical structure, rotating at a constant rate. Pressure spectra (not shown here) exhibit pics in frequency, corresponding to this helical structure. Positive isosurface of Q criterion, second invariant of velocity-gradient tensor, characterises coherent vorticies for incompressible

High Reynolds Number Swirling Flow in a Conical Diﬀuser

383

Fig. 8. Coherent structures visualized using two isosurfaces of Q = 0.9 Ub2 /r02 .

ﬂows. Hence, small structures can be seen in Fig. 8, but because of the Reynolds number, there are a huge number of structures which lead to a more complex analysis. The eﬀect of the swirl can be seen by a tube of the isosurface in the cylindrical part.

4 Conclusion Large Eddy Simulation are carried out in a conical diﬀuser at high Reynolds number. Speciﬁc methods had been employed to reach this goal. To reach high Reynolds number at low cost, an extended inner scaling was chosen to model the inﬂuence of both wall shear stress and streamwise pressure gradient close to the wall, in the viscous part. The choice was done to keep proﬁle free, by imposing only two parameters; the mass ﬂow average and the swirl number. This choice lead to diﬀerent velocity proﬁle in comparison to experiments but the same physics as Clausen’ observations.

Acknowledgements The authors acknowledge the ADEME (French Environment and Energy Management Agency) and Alstom Hydro France for their support.

References 1. P.D. Clausen, S.G. Koh, and D.H. Wood. Measurements of a swirling turbulent boundary layer developing in a conical diﬀuser. Experimental Thermal and Fluid Science, 6:39–48, 1993. 2. M. Manhart, N. Peller, and C. Brun. Near-wall scaling for turbulent boundary layers with adverse pressure gradient. Theor. Comput. Fluid Dyn., 22:243–260, 2008. 3. H. Jasak, H.G. Weller, and N. Nordin. In cylinder cfd simulation using a c++ object-oriented toolkit. Technical Report 2004-01-0110, SAE Technical papers, 2004.

384

C. Duprat et al.

4. H. Nilsson. Evaluation of openfoam for cfd of turbulent ﬂow in water turbines. In 23 IARH Symposium on Hydraulic Machinery and Systems, 2006. 5. C. Fureby, G. Tabor, H.G. Weller, and A.D. Gosman. A comparative study of subgrid scale models in homogeneous isotropic turbulence. Phys. Fluid, 9: 1416–1429, 1997. 6. I. Orlanski. A simple boundary condition for unbounded hyperbolic ﬂows. Journal of Computational Physics, 21(3):251–269, july 1976. 7. C.D. Pierce and P. Moin. Method for generating equilibrium swirling inﬂow conditions. AIAA, 36(7):1325–1327, 1998. 8. M.J. Landman. On the generation of helical waves in circular pipe ﬂow. Phys. Fluids, 2(5):738–747, 1990.

Direct Numerical Simulation of Hot and Highly Pulsated Turbulent Jet Flows V. Clauzon and T. Dubois Laboratoire de Math´ematiques, Universit´e Blaise Pascal and CNRS (UMR 6620), 63177 Aubi`ere, France, [email protected]; [email protected] Abstract In this paper, we present numerical simulations of highly pulsated jet ﬂows at 12, 000 K developing in a colder environment. Such ﬂows are used to model plasma jets generated by direct current plasma torch. Plasma spraying is used to deposit thick coatings on a substrate. We focus here on situations where plasma jets are randomly forced by an electric arc resulting in a highly turbulent ﬂow at large Reynolds number. DNS were performed for diﬀerent ambient temperatures T∞ ≥ 6, 000 K. For T∞ = 3, 000 K, the use of a high-order explicit ﬁlter was necessary in order to remove aliasing oscillations. The eﬀects of T∞ on the ﬂow properties are discussed.

1 Introduction Plasma spraying is a materials processing technique for producing coatings using a plasma jet. Deposits can be produced from metals or ceramics which are introduced into the plasma jet. The jet temperature is typically of the order of 10, 000 K so that the material is melted and propelled towards a substrate. The resulting materials are used for engineering applications including automotive, biomedical or aerospace areas. In direct current (dc) plasma torch, the plasma forming gas, here Ar–H2 , enters the torch with a speed of 30 m · s−1 and a temperature of 1, 000 K. While passing between the two concentric electrodes, the gas is heated by Joule eﬀects reaching 12, 000 K. The mean axial velocity of the plasma jet is above 1, 000 m · s−1 at the torch exit. Flows induced by plasma jets develop in a chamber ﬁlled with air at 300 K. When plasma torches are operated in restrike mode (see [1, 2]), plasma jets are forced by the chaotic movement of the arc. The induced ﬂow is highly turbulent. Due to the large diﬀerence between the plasma and the ambient gas temperatures, the Reynolds number based on environment characteristics is of the order of 55, 000. DNS at this Reynolds number is not accessible with the computational resources which are available. To overcome this diﬃculty, V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 55, c Springer Science+Business Media B.V. 2010

385

386

V. Clauzon and T. Dubois

we focus in this paper on hot and highly pulsated jet ﬂows developing in a cold environment with ambient temperature above 3, 000 K, which corresponds to Reynolds numbers smaller than 3, 500. In order to model the chaotic behavior of the arc, perturbations up to 30% of the signal magnitude are imposed to the jet inﬂow. DNS, for ambient temperatures T∞ decreasing from 12, 000 K to 6, 000 K, are presented and discussed. Finally, a simulation was performed with the ambient temperature set to T∞ = 3, 000 K. In this case, an explicit ﬁltering is used to remove aliasing oscillations.

2 Formulation of the problem 2.1 Modeling assumptions and governing equations We assume that plasma and ambient gas are the same, that is Ar–H2 . Also, as in [1,2], the ﬂow is assumed to be in local thermodynamic equilibrium (LTE). Therefore, the plasma is considered as a compressible, perfect gas. Thermodynamic and transport properties are evaluated by using tabulated values for the viscosity and the thermal conductivity (see Fig. 1). This is essential in order to capture changes due to gas ionization and dissociation occurring at 3, 500 K. As mathematical model, we use the compressible Navier–Stokes equations supplemented with the equation of state for ideal gas. The ﬂow variables are made non-dimensional by using as reference values the torch radius R0 = 5.25 mm, the temperature of the plasma jet at the torch exit, T0 = 12, 000 K, and the speed of sound at temperature T0 , that is c0 = 1, 641 m · s−1 . Therefore, the density ρ, the velocity ﬁeld u and the total energy E satisfy the following equations 3

–4

3×10

2

µ (T )

K (T )

2×10–4

1

1×10–4

0

0 0

4000 8000 Temperature T

12000

0

4000

8000

Temperature T

12000

Fig. 1. Tabulated values of the kinematic viscosity μ (T ) (left) and the thermal conductivity κ (T ) (right).

DNS of Hot and Highly Pulsated Jet Flows

387

∂ρ ∂ρuj + = 0, ∂t xj ∂ρui ∂ρui uj ∂p ∂τij + + = , ∂t ∂xj ∂xi ∂xj ∂E ∂(E + p)uj ∂ ∂T ∂τij ui cp (T0 ) + κ(T ) , = − ∂t ∂xj ∂xj γR Re Pr ∂xj ∂xj where τij =

(1)

μ(T ) ∂ui ∂uj 2 ∂uk + − δij Re ∂xj ∂xi 3 ∂xk

is the viscous stress tensor, μ(T ) = μ (T )/μ (T0 ) is the kinematic viscosity and κ(T ) = κ (T )/κ (T0 ) the thermal conductivity. The constants R = 187 J · kg−1 · K−1 and γ = 1.2 have been computed from tabulated values of the speed of sound for the plasma gas Ar–H2 . The values of the Reynolds, Prandtl and Mach numbers are respectively Re = 840, Pr = 0.5 and Ma = u0 /c0 = 0.7. The jet ﬂow in its expansion region is characterized by the Reynolds number ρ∞ c∞ R0 Re∞ = μ (T∞ ) where T∞ (≤ T0 ) is the ambient gas temperature; Re∞ is related to Re by Re∞ =

T0 T∞

12

μ (T0 ) Re. μ (T∞ )

For plasma torches, we have T∞ = 300 K so that Re∞ = 54, 600. DNS at this Reynolds number can not be performed with the available computational resources. 2.2 Numerical procedure The computational domain is a rectangular box discretized by a cartesian grid. In the lateral directions, the grid is stretched near the jet axis. In the streamwise direction, the mesh size is constant for x ≤ 20R0 and increases linearly until the outﬂow boundaries are reached. Spatial derivatives are approximated by using a sixth-order compact scheme (see [3]). For time advancement, the explicit third-order Runge–Kutta scheme is used. A sponge zone acts on 10% (resp. 20%) of the domain size on the lateral (resp. outﬂow) boundaries. Non-reﬂecting boundary conditions (see [4]) are applied at the inﬂow and outﬂow planes. A weak condition smoothly forces the pressure to adjust to the freestream pressure at the outﬂow. The jet temperature and velocity are prescribed at inﬂow. Classical hyperbolic tangent proﬁles are used for the mean streamwise velocity and temperature, namely

388

V. Clauzon and T. Dubois

r − 1 Ma 1 − tanh , vjet = wjet = 0, ujet (r) = 2 δ r − 1 T∞ T∞ 1 Tjet (r) = 1− , 1 − tanh + T0 2 T0 δ

(2) (3)

where δ = 0.2 is the shear layer thickness parameter. Meshes in the transverse directions are chosen so that approximately 10 grid points are located in the jet shear layer. In order to model the eﬀects of the chaotic movement of the arc inside the plasma torch on the jet ﬂow, axial and helical ﬂuctuations are imposed on the inlet streamwise velocity, that is % y & u(0, y, z, t) = ujet (r) 1 + εa sin (Sta t) + εh sin Sth t + arccos r (4) r where r = y 2 + z 2 . The parameters εa and εh deﬁning the amplitude of the axial and helical perturbations were set to 0.3; note that Boersma and Danaila in [5] used two times weaker perturbations in their study of bifurcating incompressible jet ﬂows. The Strouhal numbers were set to Sta = 0.36 and Sth = 0.45 which respectively correspond to frequencies of arc ﬂuctuations of 40 khz and 50 khz. These values are above the critical Strouhal number for jet ﬂows so that turbulence rapidly develops. Note that experimental values for dc plasma torch (see [2]) are a factor of 2–4 smaller.

3 Numerical results Numerical simulations in the computational domain (0, 60R0 ) × (0, 40R0 ) × (0, 40R0 ) were performed for ambient gas temperature T∞ varying in the range 3, 000 K to 12, 000 K(= T0 ). The computational parameters are reported in Table 1. For T0 /T∞ ≤ 4/3 (Re∞ < 900), a grid with 320×161×161 mesh points is ﬁne enough to provide an eﬃcient resolution of the ﬂow variables as well as their derivatives. When T∞ is decreased so that T0 /T∞ = 2 (Re∞ = 1, 441), grid oscillations appear close to the jet inlet, that is for x/R0 ≤ 5. However, the resolution is suﬃcient to avoid accumulations of aliasing oscillations. For T0 /T∞ = 4, while the grid was reﬁned to 500 × 241 × 241 points, the use of a high-order compact ﬁlter, as in [6], was necessary in order to avoid the development of numerical instabilities. The time integration was performed on 100 time periods τa = 2π/Sta of the jet perturbations for T0 /T∞ ≤ 2 Table 1. Computational parameters. T0 /T∞

Nx

Ny = Nz

Re∞

Outﬂow sponge

Spatial ﬁltering

1 4/3 2 4

320 320 320 500

161 161 161 241

840 882 1, 441 3, 442

Yes Yes Yes No

No No No Sixth-order

DNS of Hot and Highly Pulsated Jet Flows

389

and on only 50 time periods for T0 /T∞ = 4. In the former case, statistics are accumulated on 60 time periods so that convergence, at least for the ﬁrst order statistics, is obtained. In the latter case, only 30 time periods could be used. For T0 /T∞ < 2, the use of a sponge zone at the outﬂow boundary is necessary. For smaller values of T∞ , ﬂow ﬂuctuations are rapidly damped due to the diﬀerence between the jet and ambient densities. Indeed, for T0 /T∞ ≥ 2 and x/R0 ≥ 12 (see Fig. 2) the mean axial velocity is reduced by approximately 30% compared with the values obtained for larger ambient temperatures: this results in weaker advection of ﬂow perturbations in the axial direction. In the case T0 /T∞ = 4, the non-reﬂecting boundary conditions adjust the values of the ﬂow variables at the outﬂow without requiring the use of a sponge zone. The mean axial velocity (see Fig. 2) has a rapid decay in the transverse directions so that only small ﬂow ﬂuctuations reach the sponge zone (see Fig. 3). Reﬂections on the lateral boundaries are weak and their eﬀects are negligible. 0.5

Mean streamwise velocity

Mean streamwise velocity

0.8

0.6

0.4

0.2

0 0

10

20

30 x/R0

40

50

60

0.4 0.3 0.2 0.1 0 –4

–3

–2

–1

0 y/R0

1

2

3

4

Fig. 2. Mean streamwise velocity along the jet axis y = z = 0 (left) and at x = 10 R0 (right). T∞ = 12, 000 K (dashed line), T∞ = 9, 000 K (dotted-dashed line), T∞ = 6, 000 K (double dotted-dashed line), T∞ = 3, 000 K (solid line). 10

y/R0

y/R0

5

0

−5

−10 x/R0

0

5

10 x/R0

15

20

Fig. 3. Instantaneous temperature (left) and magnitude of the instantaneous vorticity (right) for the simulation with T∞ = 3, 000 K on the grid with 500 × 241 × 241 mesh points.

390

V. Clauzon and T. Dubois

The centerline velocity (Fig. 2) starts to drop along the jet axis at a location such that x/R0 ≤ 6. This is much earlier than in weakly perturbed, heated jet ﬂows which are generally studied (see [7] and the references therein). Here, the pulsations (4) imposed on the jet inﬂow represent 30% of the mean inﬂow velocity. Also in Fig. 2, we observe that the decay starts earlier when the ambient temperature decreases. For classical round jets, the mean axial velocity is nearly constant close to the jet oriﬁce. Further downstream from the inﬂow, the centerline velocity decreases: the decay rate of round jets is proportional to 1/(x − xc ) where xc is the virtual origin of the jet. Here, the mean axial velocity proﬁles (Fig. 2) exhibit three regions with diﬀerent behaviors. In the ﬁrst zone, that is for x/R0 ≤ 6, the centerline velocity is almost constant for T0 /T∞ ≤ 4/3. For smaller T∞ , a peak develops close to the jet oriﬁce, that is for x/R0 ≈ 5: it looks like a time persistent oscillation induced by the strong pulsations imposed to the jet inﬂow. A transition zone with a superlinear decay follows: its extent is of the order of 14R0 for T0 /T∞ ≤ 4/3 and 9R0 for T0 /T∞ ≥ 2. Finally, the decay is linear until the outﬂow plane is reached. In the isothermal case T0 /T∞ = 1, the slope of the decay is of the order of 0.14 which is close to the expected value (see [7]). For T0 /T∞ = 2, the predicted decay rate, that is 0.3, compares well with the value found in [7] for similar temperatures and Mach numbers. Note that in [7], the Reynolds number is much larger (Re = 200, 000). For T0 /T∞ = 4, a faster decay, with a slope of the order of 0.48, is found. Therefore, in the area where the turbulence develops, the decay of the hot jets studied in this paper depends on the ambient temperature. As expected (see [7]), a faster decay is found when the ambient gas is colder. Finally, Fig. 3 representing the temperature and the magnitude of the vorticity shows that, in the case T0 /T∞ = 4, most of the turbulence activity takes place in the region corresponding to 7 ≤ x/R0 ≤ 20 and −5 ≤ y/R0 ≤ 5. Further in the domain, that is for x ≥ 20R0 and |y| > 5R0 , the ﬂow ﬂuctuations are rapidly damped indicating that the turbulence is decaying.

Acknowledgments The numerical simulations presented in this paper were performed on the cluster of 10 vectorial supercomputers NEC-SX8 of the Supercomputing Center IDRIS of CNRS (Orsay, France, http://www.idris.fr).

References 1. Baudry C (2003), Contribution a ` la Mod´elisation Instationnaire et Tridimensionnelle du Comportement Dynamique de l’Arc Dans une Torche de Projection Plasma, Ph.D. thesis, Universit´e de Limoges, France. 2. Trelles P, Heberlein JVR (2006), Simulation Results of Arc Behavior in Diﬀerent Plasma Spray Torches, J. Thermal Spray Tech. 15:563-569.

DNS of Hot and Highly Pulsated Jet Flows

391

3. Lele SK (1992), Compact ﬁnite diﬀerence schemes with spectral-like resolution, J. Comput. Phys. 103:16-42. 4. Poinsot TJ, Lele SK (1991), Boundary Conditions for direct Simulations of Compressible Viscous Flows, J. Comput. Phys. 101:104-129. 5. Danaila I, Boersma BJ (2000), Direct numerical simulation of bifurcating jets, Phys. Fluids 12:1255-1257. 6. Bogey C, Bailly C (2006), Computation of a high Reynolds number jet and its radiated noise using large eddy simulation based on explicit ﬁltering, Computers and Fluids 35(10):1344-1358. 7. Lew P, Blaisdell GA, Lyrintzis AS (2005), Recent Progress on Hot Jet Aeroacoustics Using 3-D Large Eddy Simulation, AIAA paper 2005-3084:1-24.

DNS of Convective Heat Transfer in a Rotating Cylinder R.P.J. Kunnen1,2 , B.J. Geurts1,3 , and H.J.H. Clercx1,3 1

2

3

Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]; [email protected] Current address: RWTH Aachen University, Institute of Aerodynamics, W¨ ullnerstraße 5a, 52062 Aachen, Germany, [email protected] Department of Applied Mathematics and J.M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Abstract The eﬀects of rotation on the convective heat transfer and ﬂow structuring in a cylindrical volume of ﬂuid is investigated with direct numerical simulation (DNS). A formulation of the discrete equations of motion in cylindrical coordinates is solved with ﬁnite-diﬀerence approximations. At constant Rayleigh (Ra; dimensionless temperature gradient) and Prandtl (σ; characterises diﬀusive properties of the ﬂuid) numbers, the Rossby number Ro, the ratio of buoyancy and Coriolis forces, is varied between runs with 0.045 ≤ Ro ≤ ∞. For Ro 1.2 we ﬁnd the so-called large-scale circulation. At Ro 1.2 slender columnar vortical plumes are found. In a range of Ro heat transfer is increased by Ekman pumping in the vortical plumes.

1 Introduction Large-scale geophysical ﬂows, as found in the atmosphere and in the oceans, are often driven by buoyancy, and subsequently shaped by the rotation of our Earth through the Coriolis force. Similar conditions can be found in the convective layer in the Sun and inside the giant planets. Next to fundamental interest, rotating convection is a relevant topic in meteorology, climatology, oceanography and astrophysics. We approximate these complicated ﬂows by a simple model. A cylindrical volume of ﬂuid is heated from below, cooled from above, and subjected to a rotation with orientation parallel to the cylinder axis and counter to gravity. The destabilising temperature gradient is designated in dimensionless form by the Rayleigh number Ra. The rotation rate is speciﬁed by the Taylor number T a. The diﬀusive properties of the ﬂuid are characterised by the Prandtl number σ. These numbers are deﬁned as V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 56, c Springer Science+Business Media B.V. 2010

393

394

R.P.J. Kunnen et al.

Ra ≡

gαΔT H 3 , νκ

Ta ≡

2ΩH 2 ν

2 ,

σ≡

ν , κ

(1)

with H the vertical separation of the plates and ΔT the temperature diﬀerence between them, g the gravitational acceleration, Ω the rotation rate, and ν, κ and α the kinematic viscosity, thermal diﬀusivity and thermal expansion coeﬃcient of the ﬂuid, respectively. The cylindrical domain adds a fourth parameter, the diameter-to-height aspect ratio Γ , which in this study is set to one. Another useful deﬁnition is the Rossby number Ro = Ra/(σ T a) as a direct indication of the ratio of buoyancy over Coriolis forces. An important result in convection, the heat transfer, is commonly expressed in dimensionless form as the Nusselt number N u, the convective heat transfer normalised by the conductive part of the heat ﬂux. A cylindrical geometry is the default choice for experimental investigations. It also adds a phenomenon of fundamental interest: the organisation of the motion into a domain-ﬁlling large-scale circulation (LSC). In non-rotating cylindrical convection the LSC has received considerable attention (see, e.g., Refs. [1,2]). It is the starting point of theories [3] that describe the convective heat transfer as a function of Ra and σ. Rotation adds interesting temporal dynamics in the LSC [4]. It is found that at a certain rotation rate the LSC is no longer present; instead an array of strongly localised vertical vortices is found, transporting ﬂuid and heat vertically [5]. We study rotating convection in a cylinder with direct numerical simulation (DNS) at constant Ra and σ, where the rotation rate is varied between runs. The following topics will be addressed: (i) the persistence of the LSC under rotation [4], (ii) the organisation of the ﬂow in the vortical state at higher rotation rates [5], and (iii) the convective heat ﬂux (N u) as a function of Ro [6,7]. These results reinforce some of our recent experimental ﬁndings [8] found using stereoscopic particle image velocimetry in turbulent rotating convection.

2 Numerical procedure The governing equations are the incompressible Navier–Stokes and temperature equations, written in the Boussinesq approximation [9] with the Coriolis term added: : ∂u σ 2 1 ˆ + (u · ∇)u + z × u = −∇p + ∇ u + T ˆz , ∂t Ro Ra ∂T 1 + (u · ∇)T = √ ∇2 T , ∂t σRa ∇·u = 0, (2) where u is the (co-rotating) velocity vector, p the reduced pressure, and T the temperature relative to the cold top wall. Velocities are scaled with

DNS of Convective Heat Transfer in a Rotating Cylinder

395

√ the so-called free-fall velocity U ≡ gαΔT H [10], lengths are scaled with height H, and the temperature with ΔT . The boundary conditions for velocity are no-slip on all walls. The sidewall is thermally insulating (zero temperature gradient) while the bottom and top walls are isothermal: T = 0 at the top and T = 1 at the bottom wall. The solution to the Navier–Stokes equations in cylindrical coordinates provides some challenges, like the treatment of the equations on the cylinder axis r = 0. The discretisation proposed by Verzicco and Orlandi [11] treats the singularity by writing the equations in terms of uφ , qr = r ur and uz , with uφ , ur and uz the azimuthal, radial and axial velocity component, respectively. The ﬁnite-diﬀerence discretisation on a staggered mesh is second-order accurate. Time advancement is done using a third-order Runge–Kutta scheme with an adaptive time step. For further details on the numerical procedure we refer to Verzicco and Camussi [10] and Verzicco and Orlandi [11]. The simulations are carried out at constant Ra = 1 × 109 and σ = 6.4, corresponding to strongly turbulent convection in water. The Rossby number takes values between ∞ (no rotation) and 0.045 (strong rotation). The grid resolution is set to 385 × 193 × 385 for the azimuthal, radial and axial directions, respectively. Near the bottom and top walls, as near the sidewall, the grid spacing decreases to cluster more grid points in the boundary layers. In the most demanding case, Ro = 0.045, there are 8 grid points in the sidewall boundary layer and 10 in the layers near bottom and top. Averaging was done for at least 150 large-eddy turnover times, tested separately and found to yield accurate results for approximating the long-time averages. Furthermore, 64 numerical probes placed at half-height are distributed evenly over a circle at r = 0.45 close to the sidewall (r = 0.5). At these points, local time histories of velocity, temperature, and other quantities are recorded, to be used momentarily in the analysis of the LSC.

3 Coherent structures in the ﬂow The variations in ﬂow structuring due to rotation are shown in Fig. 1. Three cases are included: (a) no rotation Ro = ∞, (b) medium rotation Ro = 0.72, and (c) strong rotation Ro = 0.045. In plot (a) at Ro = ∞ isosurfaces of vertical velocity are plotted: dark (light) colouring corresponds to upward (downward) ﬂow. A spatial separation of upward and downward ﬂow is observed, a ﬁngerprint of the LSC. In (b,c) isosurfaces of the so-called Q criterion for the detection of vortices are plotted [12]: ; ; ; 1 ; ;∇u − (∇u)T ;2 − ;∇u + (∇u)T ;2 . Q≡ 2 Q is positive when the antisymmetric (rotational) part of ∇u dominates over the symmetric (strain) part, i.e., inside a vortex. At Ro = 0.72 (b) there is no LSC; a few coherent vortices can be found. At Ro = 0.045 (c) many slender

396

R.P.J. Kunnen et al.

Fig. 1. (a) Vertical-velocity isosurfaces at Ro = ∞: dark (light) colouring is for upward (downward) ﬂow. (b) Q isosurfaces at Ro = 0.72, indicating vortical regions. (c) Q isosurfaces at Ro = 0.045.

a

b

0.2 0.15

1 Ro = 1.2

Ro = 2.5

0.8

0.1 0.6 En / E

uz

0.05 0

0.4

−0.05 −0.1

0.2

−0.15 −0.2

0 0

0.2

0.4

φ/ 2π

0.6

0.8

1

10−1

100 Ro

101

Fig. 2. (a) Snapshot of the azimuthal proﬁle uz (φ) (circles) at Ro = 5.76, with the corresponding n = 1 Fourier mode (solid line). (b) Averaged relative energy content En /E of Fourier modes: n = 0 (crosses), n = 1 (circles), n = 2 (squares), n = 3 (triangles) and the sum of modes n ≥ 4 (pluses). The vertical dashed lines indicate Ro = 1.2 and 2.5.

columnar vortices are found, stretching vertically across the domain. A quasitwo-dimensional bulk state is achieved, with boundary layers to connect to the solid walls. The vertical transport of ﬂuid and heat is conﬁned within the vortical tubes. Statistics concerning the number and size of the vortices are presented elsewhere [8]. While such visualisations may aid in the detection of an LSC or vortices, we also deﬁne a quantitative criterion for presence of an LSC. A procedure used by Brown and Ahlers [1] is followed. We ﬁt sinusoidal functions to the axial velocity as recorded by the numerical probes using a Fourier transform in the azimuthal direction. The average energy content En of each Fourier mode n can then be determined, where the n = 1 mode can be interpreted as the ‘LSC mode.’ In Fig. 2a an example ﬁt of the n = 1 mode is shown. Figure 2b gives the relative energy content of modes n = 0, 1, 2, 3 and the

DNS of Convective Heat Transfer in a Rotating Cylinder

397

sum of modes n ≥ 4. Our criterion for existence of an LSC demands that the n = 1 mode is the most energetic mode, i.e., E1 is the largest contribution to the total energy E. We ﬁnd that for Ro 1.2 indeed E1 is the largest contribution and there must be an LSC, while at Ro 1.2 more and more of the energy is found in the higher-n modes, corresponding to smaller and smaller ﬂow structures (the narrow vortical tubes). There is a rather narrow transition region 1.2 Ro 2.5 in which the strength of the n = 1 mode (the LSC) is smaller than at larger Ro, but still the largest individual mode.

4 Heat transfer The dependence of N u on Ro is shown in Fig. 3, depicted as N u(Ro) normalised by its non-rotating value N u(Ro = ∞). Results from other numerical [13] and experimental [6, 14] studies are also included. Remarkable is the 15% increase of N u at Ro ≈ 0.2 in the current results (squares). Thus, despite the increased stability under rotation [9] the convective heat transfer is larger for a range of Ro. All other included results indicate that heat transfer can be increased by rotation, but the peak heat ﬂux is not at a uniform Ro value. A considerable dependence on the parameters (Ra, σ, Γ ) is expected. In the current work, the increased heat transfer is found only when the LSC is absent according to the criterion mentioned before. The vortical state is thus able to transport more heat. The so-called Ekman pumping [15], occurring when a vortex connects to a solid wall in rotating ﬂows, provides a means of entraining ﬂuid from very close to the walls into the vortices, that subsequently transport this ﬂuid to the other side. This mechanism is considered responsible for the increase of N u observed in Fig. 3. At Ro < 0.2, however, the stabilising eﬀect of rotation takes precedence and N u is reduced.

1.4

Nu(Ro) / Nu(Ro = ∞)

1.2 1 0.8 0.6 0.4 0.2 0 10−2

10−1

100 Ro

101

102

Fig. 3. Normalised heat transfer N u(Ro)/N u(Ro = ∞) as a function of Ro. Squares: current results (Ra = 109 , σ = 6.4, Γ = 1). Circles: Ra = 2 × 108 , σ = 0.7, Γ = 0.5 [13]. Triangles: Ra = 109 , σ = 6.3, Γ = 0.78 [6]. Dashed line: Ra = 2.5×106 , σ = 6.8, Γ > 6 [14].

398

R.P.J. Kunnen et al.

5 Concluding remarks Convection in a rotating cylinder displays various types of ﬂow-structuring, dependent on the rotation rate. The LSC is found for zero and small rotation, while at higher rotation rates many vortical structures can be observed. The organisation of the ﬂow into coherent structures has important consequences for the heat transfer. There is a peak heat transfer at a Rossby number for which no LSC is formed. The vortical plumes, rather than the LSC, are involved in the increased heat transfer through Ekman pumping. The stabilising eﬀect of rotation is felt at the lowest Rossby number used in this work. At even lower Rossby numbers the rapid reduction of the heat transfer is expected to continue. The boundary-layer regions are currently under investigation. We wish to quantitatively describe the Ekman pumping, to improve our understanding of the changes in the heat ﬂux under rotation. The analysis will also shed more light on what triggers the ‘destruction’ of the LSC.

Acknowledgements RPJK wishes to thank the Foundation for Fundamental Research on Matter (FOM) for ﬁnancial support. This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with ﬁnancial support from the Netherlands Organisation for Scientiﬁc Research (NWO).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Brown E, Ahlers G (2006) J Fluid Mech 568:351–386 Xi HD, Zhou Q, Xia K-Q (2006) Phys Rev E 73:056312 Grossmann S, Lohse D (2000) J Fluid Mech 407:27–56 Hart JE, Kittelman S, Ohlsen DR (2002) Phys Fluids 14:955–962 Sakai S (1997) J Fluid Mech 333:85–95 Liu Y, Ecke RE (1997) Phys Rev Lett 79:2257–2260 Kunnen RPJ, Clercx HJH, Geurts BJ (2006) Phys Rev E 74:056306 Kunnen RPJ (2008) Turbulent rotating convection. PhD Thesis, Eindhoven University of Technology, Eindhoven Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford Verzicco R, Camussi R (2003) J Fluid Mech 477:19–49 Verzicco R, Orlandi P (1996) J Comput Phys 123:402–414 Haller G (2005) J Fluid Mech 525:1–26 Oresta P, Stringano G, Verzicco R (2007) Eur J Mech B/Fluids 26:1–14 Rossby HT (1969) J Fluid Mech 36:309–335 Gill AE (1982) Atmosphere-Ocean Dynamics. Academic Press, London

Numerical Simulations of Thermal Convection at High Prandtl Numbers G. Silano1 , K.R. Sreenivasan2 , and R. Verzicco3 1 2 3

Universit` a degli Studi di Trieste, Trieste, Italy, [email protected] International Centre for Theoretical Physics, Trieste, Italy, [email protected] DIM, Universit` a degli Studi di Roma ‘Tor Vegata’, Roma, Italy, [email protected]

Abstract In this paper we present some results of an extensive campaign of direct numerical simulations of Rayleigh–B´enard convection at high Prandtl numbers (100 ≤ P r ≤ 104 ) and moderate Rayleigh numbers (105 ≤ Ra ≤ 109 ). In particular, we examine the Nusselt and the Reynolds number dependences on Ra and P r. A short discussion on the characteristic ﬂow velocity is also presented.

1 Introduction The paradigm of thermal convection is the ﬂow between inﬁnitely conducting parallel plates heated from below and cooled from above. This model, called the Rayleigh–B´enard convection, is governed by two non-dimensional parameters: the Rayleigh number Ra = gαΔT h3 /(νκ) and the Prandtl number P r = ν/κ, where g is the acceleration of gravity, h is the ﬂuid layer depth, ΔT is the temperature diﬀerence and the ﬂuid properties α, ν and κ are, respectively, the thermal expansion coeﬃcient, kinematic viscosity and thermal diﬀusivity. The Rayleigh–B´enard model is generally based on the Boussinesq approximation, in which the ﬂuid properties are assumed to be constant despite the temperature gradient across the ﬂuid depth, and the only eﬀect of the temperature in the momentum equation is to modify the buoyancy term. The inﬂuence of the Prandtl number on thermal convection dynamics is diﬃcult to investigate experimentally, because P r can be substantially changed only by changing the ﬂuid. Examples of studies following this approach can be found in [1, 2]. A diﬀerent strategy for varying P r was adopted in [3–5]. It consists of working close to the critical point of compressed gas. This technique enables the exploration of the inﬂuence of P r variations at quite high Ra numbers. However, in both strategies complications arise from the great diﬃculty of maintaining constant properties across the ﬂuid depth, with consequent violation of the Boussinesq approximation, especially at high P r numbers. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 57, c Springer Science+Business Media B.V. 2010

399

400

a

G. Silano et al. 105

b

104

104

103 102 Pr

103 109x18x70 161x26x101 101x25x101 129x31x136 121x45x201 201x45x201 217x55x217 217x75x301 385x81x321

RaC

101 100 RaU

10–1 10–2 4 10

105

105

106

107 Ra

108

109

1010

102 Pr 101 100 10–1 10–2

0

0.002 0.004 ηΒ /h

Fig. 1. (a) Map of the performed simulations in a cylindrical cell of aspect ratio Γ = 1/2. The legend shows the corresponding number of grid points in the azimuthal, radial and vertical directions. The vertical solid line represents the critical Ra for the onset of convection [11]. The dashed line represents the threshold Ra to pass from steady to unsteady ﬂow. (b) Batchelor scale ηB /h = 1/4 1/(Re3 ε) · P r −1/2 as a function of Pr. Squares: Ra = 108 , upright triangles: 9 Ra = 10 ; ﬁlled symbols: ηB /h based on εV,t (average in space and time); unﬁlled symbols: ηB /h based on max εt (average only in time); vertical solid and dashed lines: respectively minimum and maximum grid size Δ = (2πrΔθΔrΔz)1/3 for the biggest grid (385 × 81 × 321). It is worth noting that the Batchelor scale is quite independent of P r.

Numerical simulations can completely overcome these issues, even if they face other kinds of diﬃculties such as adequate spatial resolution and the integration of long enough time evolutions. Some numerical studies deal with varying P r numbers [6–10]. Only few of them [7, 10] mimic a set-up similar to experiments, thus allowing a proper comparison of the results. A smaller number of studies concerns very high-P r regimes. On the other hand, in [10] the discussion is mainly focused at lower P r regimes (2.2 × 10−3 ≤ P r ≤ 15). In [7] a wider range of P r numbers is explored (10−3 ≤ P r ≤ 102 ), but only at ﬁxed Ra = 106 . We propose a numerical study of thermal convection at high-P r regimes and moderate Ra numbers for a wide range of P r and Ra (see Fig. 1a). Numerical simulations at high P r are very challenging. They require highly reﬁned grids to solve the smaller temperature scales of the ﬂow and large time windows to attain the steady state for mean quantities and to correctly represent the statistics of the slow dynamics of the ﬂow.

2 Physical and numerical setup The ﬂow considered here develops in a cylindrical cell of aspect ratio (diameter to cell height) Γ = 1/2. Cold and hot temperatures are imposed, respectively,

Numerical Simulations of Thermal Convection at High Prandtl Numbers

401

on the top and bottom plates. The sidewall is adiabatic and all the boundaries satisfy the no-slip condition. The ﬂow is solved by numerically integrating the three-dimensional unsteady Navier–Stokes equations with the Boussinesq approximation. To solve these equations a code [12] based on second-order accurate ﬁnite-diﬀerence schemes on a staggered mesh in cylindrical coordinates and an optimized fractional-step method have been used. The grid size has been chosen to solve the Batchelor scale [13] in the bulk (see Fig. 1b as an example) and 7 − 15 points have been placed inside the thermal boundary layers. Reﬁnement analysis has been extensively applied (see Fig. 1a) in order to assess the grid independence of the solutions. In particular, it has been checked that all the quantities considered in this paper, when computed on most reﬁned grids, diﬀered from those of coarser grids by less than 4%.1 Where possible, the results from the most reﬁned grids have been used. The simulations have been run for long enough time windows to obtain statistically converged quantities; in particular, after the initial transient was exhausted each simulation was continued for further 1,500 time units based on the large-scale ﬂow. This time unit tc is deﬁned as the time that a ﬂuid particle needs to cross the cell depth with a typical velocity Uc of the large scale structures: tc = h/Uc. The precise deﬁnition of Uc , as well as some discussion of it, will be given in Section 3.2.

3 Results 3.1 Nusselt number The Nusselts number (N u) is the non-dimensional measure of the heat ﬂux crossing the layer of ﬂuid and is one of the most important quantities to calculate. Figure 2a shows N u divided by Raα as a function of Ra at P r = 103 . The exponent α = 1/3 corresponds to the best power-law ﬁt in the interval Ra = 105 − 108 . This exponent, however, appears to be over-estimated as Ra increases. Including a provisional result at Ra = 109 and excluding the simulation at Ra = 105 , the best exponent is equal to 0.307. The exponent α = 2/7 corresponds to the best ﬁt in the interval Ra = 2 · 106 − 107 . It is worth noting that for all the three exponents, the data depart at most by about 11% with respect to perfect straight lines, except for α = 2/7 and Ra = 105 where the departure exceeds 20%. This scatter is quite small compared to the data usually available in literature. However it is larger than the error bars which are less than 3.5% (hidden by symbols in both the plots of Fig. 2). In the case of the dependence of N u on P r at ﬁxed values of Ra (Fig. 2b), the deviations from constant values are also around 10%, even if for P r ≥ 100 1

The same result has been obtained also for second order moments of temperature and velocities ﬁelds (average in space and time).

402

G. Silano et al.

b

a α = 0.307

Nu

Nu/Raα

α = 2/7

0.10 α = 1/3

101

105

106

107 Ra

108

109

100

10

1

2

10 Pr

3

10

4

10

Fig. 2. (a) The Nusselt number, N u, divided by Raα versus Ra at P r = 103 , where α assumes diﬀerent values. The results at Ra = 109 are provisional. (b) N u number versus P r at Ra = 2 · 106 (circle), Ra = 107 (square) and Ra = 108 (up triangle). The error bars are smaller than the symbols size. The error bar corresponds to the maximum diﬀerence of the values obtained computing the N u number through its relationships with the viscous and the temperature dissipation rate, and averaging the non-dimensional heat ﬂux at each horizontal section of the cell and in the whole volume [6].

they fall below about 2%. At Ra = 2 · 106 and Ra = 107 we ﬁnd a trend that decreases slightly with P r, while at Ra = 108 the behavior is more scattered. The decreasing behavior is in line with the presence of an overshoot in the N u versus P r trend, while passing from low to high P r regimes [15]. This trend is conﬁrmed from a running simulation at Ra = 107 and P r = 10−1 . The scattered behavior at Ra = 108 was not expected. However, a possible explanation lies in diﬀerences of the large-scale ﬂow structures found in our simulations. This work is in progress and the results are encouraging. 3.2 Characteristic velocity and Reynolds number In addition to the Nusselt number, the Reynolds number is an output of the problem reﬂecting the strength of the ﬂow. For P r of the order of unity, the typical √ large-scale velocities of the ﬂow are scaled well by the free fall velocity U = gαΔT h which results from the balance between the inertial term and the buoyancy term of the momentum equation. At high Prandtl numbers, however, the momentum tends to be very diﬀusive and inertial forces become small; accordingly, the present simulations at high P r showed that U does not represent the typical velocity of the large scale structures, which is much smaller. By scaling the results from several simulations we have found that the √ large-scale characteristic velocity is Uc = U/ P r (note that U and Uc coincide for P r = O(1)) √ which yields the estimate of the Reynolds number to be Re = Uc h/ν √ = Ra/P r. As a consequence, the P´eclet number becomes P e = Uc h/κ = Ra independent of P r. To verify the reliability of this estimates we show the results from two velocities that can be assumed as typical of large scale structures. One is the maximum value of the root-mean-square (rms) of

Numerical Simulations of Thermal Convection at High Prandtl Numbers

a

101

b 10 4

Urms Wmax

Re /Pr–1

+0.676

Re

100

10

+0.581

–1

10–2

10 3

10 2 107

106

105

10 0

108

10 1

10 2 Pr

Ra

c

403

d

10 0

10 3

10 4

102

U rms

101 –1/2

100

10 –1 –0.589

–0.501

δ|rms

δT|rms

W max

10–1 10–2

10 –2 1 10

10 2

10 3 Pe

10 4

10–3 –2 10

100

102

104

Re

Fig. 3. Filled symbols show Re and P e based on the rms horizontal velocity Urms and unﬁlled symbols Re and P e are based on maximum vertical velocity Wmax . Diamonds represent trends at ﬁxed P r = 103 , circles those at ﬁxed Ra = 2 · 106 , squares those at ﬁxed Ra = 107 and upright triangles those ﬁxed Ra = 108 .

5 6 the horizontal velocity proﬁle Urms = max[ u2h A,t ]1/2 , averaged in time and over horizontal surfaces. The other is the time-averaged peak vertical velocity Wmax = max[uz ]t . Figure 3a shows the behavior of Re with respect to Ra at P r = 103 . The power-law exponent signiﬁcantly diﬀers for the two velocities and both of them are diﬀerent from 1/2. However, it is possible that the exponents closer may be to 1/2 after the transition from the so-called soft to hard turbulence, as shown in [1]. In the range of Ra numbers explored up to now, no transition is visible. A simulation at higher Ra is presently running. Figure 3b shows the behavior of Re with respect to P r at diﬀerent values of Ra. The Reynolds number is divided by P r−1 to better show the trends. For all Rayleigh numbers and for both choices of typical velocities, the data tend to approach constant values as P r increases. At P r ≥ 102 the deviation from a constant value is less than 3.6%. At lower P r the deviations can exceed 50%, strongly depending from Ra and on the kind of velocity considered. Finally, Fig. 3c, d shows the behavior of thermal (δT ) and viscous (δ) boundary layer thickness, respectively, as functions of P e and of Re. Both of them are obtained considering the distance from the wall corresponding to

404

G. Silano et al.

the maximum values of rms proﬁles (of temperature for δT and of horizontal velocity for δ). The boundary layers have laminar-like behavior, and the viscous boundary layer saturates to a constant value as Re decreases.

4 Conclusions In the convective regimes considered here, the Nusselt number seems to follow a power-law of the kind N u ∼ Ra1/3 P r0 . However, a lower value exponent for Ra dependency better represents the data trend at higher Ra numbers. Instead, the tendency of N u to become independent of P r for large P r seems to be unambiguous. √ As a main result, we have found a 1/ P r correction to apply to the free fall velocity, obtaining a more appropriate representation of the large scale velocity at high P r. Using this new characteristic velocity, it is possible to derive a new non-dimensional form of the Boussinesq equations and a rough estimation of the Re and P e numbers that are more suitable at high-P r analysis than those which derive from the characteristic velocities commonly used to make the equations non-dimensional. Considering N u ∼ Ra1/3 P r0 and the mean peak vertical velocity Wmax as the characteristic velocity, the results qualitatively agree with aspects of the Grossmann–Lohse theory [15] concerning very high-P r regimes (III∞ ). We thank CASPUR (www.caspur.it) for providing the computational resources.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Lam S, Shang XD, Zhou SQ, Xia KQ (2002) Phys. Rev. E 65:066306 Ahlers G, Xu X (2001) Phys. Rev. Lett. 86:3320–3323 Niemela JJ, Skrbek L, Sreenivasan KR, Donnelly RJ (2000) Nature 404:837–840 Roche PE, Castaing B, Chabaud B, H´ebral B (2002) Europhys. Lett. 58: 693–698 Ashkenazi S, Steinberg V (1999) Phys. Rev. Lett. 83:3641–3644 Calzavarini E, Lohse D, Toschi F, Tripiccione R (2005) Phys. Fluids 17:055107 Breuer M, Wessling S, Schmalzl J, Hansen U (2004) Phys. Rev. E 69:026302 Schmalzl J, Breuer M, Hansen U (2002) Goephys. Astrophys. Fluid Dynamics 96:381–403 Kerr RM, Herring JR (2000) J. Fluid Mech. 419:325–344 Verzicco R, Camussi R (1999) J. Fluid Mech. 383:55–73 Oresta P, Stringano G, Verzicco R (2007) Euro. Mech. Flu. 26:1:14 Verzicco R, Orlandi P (1996) J. Comput. Phys. 123:402–414 Batchelor GK (1959) J. Fluid Mech. 5:113–133 Akselvoll K, Moin P (1996) J. Comput. Phys. 125:454 Grossmann S, Lohse D (2001) Phys. Rev. Lett 86:3316-3319

Inﬂuence of the Lateral Walls on the Thermal Plumes in Turbulent Rayleigh–B´ enard Convection in Rectangular Containers M. Kaczorowski1 and C. Wagner2 1

2

DLR G¨ ottingen, Institute of Aerodynamics and Flow Technology, G¨ ottingen, Germany, [email protected] DLR G¨ ottingen, Institute of Aerodynamics and Flow Technology, G¨ ottingen, Germany, [email protected]

1 Introduction Turbulent Rayleigh–B´enard (RB) convection is one of the classical problems in ﬂuid mechanics, where ﬂuid with a Prandtl number Pr = ν/κ is exposed to a vertical temperature gradient between a hot lower and a cold upper surface. The Rayleigh number Ra = αgH 3 ΔT /(νκ) is a non-dimensional parameter to specify the ratio between the buoyancy and viscous forces, where α, ν and κ denote the thermal expansion coeﬃcient, kinematic viscosity and thermal diﬀusivity, respectively. ΔT is the vertical temperature gradient between the two bounding surfaces, H the height of the ﬂuid layer and g the gravitational acceleration. For high enough Ra the thermal plumes that are rising and falling from the respective hot and cold surfaces become increasingly irregular. In recent years many fundamental studies dealt with the Rayleigh–B´enard problem in order to gain deeper knowledge of the underling mechanisms driving the convective ﬂow and the heat transport. Verzicco and Sreenivasan [10] investigated the inﬂuence of an isothermal and a constant heat ﬂux bottom plate in a cylindrical geometry on the heat transport for Rayleigh numbers up to 1013 . They concluded that the unsteady ejection of thermal plumes with a constant heat ﬂux boundary is closer to experimental conditions. The thermal plumes are, however, a subject area that attracted a lot of attention in the ﬁeld of thermal convection. Shishkina and Wagner [9] also performed direct numerical simulations (DNS) in a cylindrical geometry and analysed the geometrical properties of sheet-like thermal plumes in horizontal slices through the bulk ﬂow. Zhou and Xia [11] conducted experiments showing that the skewness of the temperature signal can be taken to separate the ﬂow ﬁeld into three regions: bulk, boundary and mixing layers. Daya and Ecke [2] investigated the inﬂuence of the shape of the container on the statistics of the convective ﬂow. They found that global values like the V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 58, c Springer Science+Business Media B.V. 2010

405

406

M. Kaczorowski and C. Wagner

Nusselt number Nu are not aﬀected, but signiﬁcant diﬀerences were observed in local properties of the turbulent ﬂow. In the present study we investigate two geometries, a rectangular cell and a cube, for which we compare the mean heat transport at diﬀerent Ra and the probability density functions (PDFs) of the thermal dissipation rates in order to determine the impact of the additional lateral wall on the behaviour of the large coherent structures.

2 Computational setup In the present numerical simulations the non-dimensionalisation of the governing equations is carried out using xi = xˆi /W , ui = uˆi /(αg ΔT H)1/2 , θ = (T − T0 )/ΔT , p = pˆ/(ˆ ραgH ΔT ) and t = tˆ(αg ΔT H)1/2 /W . Together with the Boussinesq approximation we solve the incompressible Navier–Stokes equations given by ∂ui =0 ∂xi ∂ui ∂ 2 ui ∂ui ∂p + uj + = Pr /(Γ 3 Ra))1/2 +θδ1i ∂t ∂xj ∂xi ∂x2j ∂θ ∂2θ ∂θ + ui = 1/(Γ 3 RaPr )1/2 2 ∂t ∂xi ∂xi

(1) (2) (3)

where ui are the velocity components in i = x, y, z direction, p and θ are the pressure and temperature, respectively. The volume balance procedure by Schumann et al. [7] is used for the integration over the ﬂuid cells and the solution is evolved in time by means of the explicit Euler–Leapfrog scheme. Spatial derivatives and cell face velocities are approximated by piecewise integrated fourth order accurate polynomials where the velocity components are stored on staggered grids which are described in detail by Shishkina and Wagner [8]. The velocity pressure coupling is carried out through the projection method by Chorin [1] which requires the solution of a Poisson equation. Simulations are conducted in two diﬀerent geometries, a cube and a long rectangular cell with a square cross section and periodic boundaries in the longitudinal direction, where the period L of the cell is L = 5H. All vertical walls are adiabatic and the isothermal bottom and top walls have non-dimensional temperatures θ1 = 0.5 and θ2 = −0.5, respectively. No-slip and impermeability conditions are applied to all walls and the grid spacing is gradually reduced towards the rigid walls, so that the computational domain is discretised using non-equidistant meshes. In the periodic cell a fast Fourier transform is used to decouple the Poisson equation in the longitudinal direction, whereas the separation of variables method (see Kaczorowski et al. [4]) is used for the cube.

Inﬂuence of the Lateral Walls in Turbulent Rayleigh–B´enard Convection

407

The ﬂuid has Pr = 0.7 and simulations are performed for Ra = 3.5 × 105 , 3.5 × 106 , 3.5 × 107 and Ra = 2.31 × 108 . The grid sizes for the cube are 643 , 963 , 1523 and 2563, so that at least 10 grid points are within the thermal boundary layers. The grid sizes for the simulations of the respective Rayleigh numbers in the periodic cell are 64 × 128 × 64, 96 × 256 × 96, 152 × 512 × 152 and 190 × 512 × 190 with 12 to 8 gridpoints within the thermal boundary layer. However, the bulk resolution for Ra = 2.3 × 108 is approximately as large as the Kolmogorov length scales, so that the tensor-diﬀusivity model by Leonard and Winkelmanns [6] is used to model the subgrid scales. For more details on the resolution of the periodic domain we refer to Kaczorowski and Wagner [5].

3 Results Instantaneous ﬂow ﬁelds represented by isothermal surfaces are given in Fig. 1 for two Ra and for both geometries. It illustrates the development of increasingly smaller plumes rising and falling from the bottom and top walls as Ra increases as well as the diﬀerent topological structure of the ﬂow in both geometries. In the periodic domain two pairs of convection rolls are observed with rotational axes orthogonal to the longitudinal direction. In the cubic domain, however, the thermal plumes are predominantly rising and falling in the corners of the cube leading to a large scale circulation that is oriented along one of the diagonals of the container.

Fig. 1. Snapshots of the ﬂow represented by 18 instantaneous isothermal surfaces with |θ| ≥ 0.1; Ra = 3.5 × 106 (top) and Ra = 2.31 × 108 (bottom) for a cubic and a rectangular domain.

408

M. Kaczorowski and C. Wagner 0.2 periodic box cube

NuRa−0.29

0.18

0.14

0.12

0.1 104

105

106

108

109

Ra

Fig. 2. Nu normalised with Ra β and β = 0.29 following the theoretical prediction by [3].

The analysis of the mean heat transport represented by the Nusselt number Nu shows, however, that the diﬀerences between the two geometries are relatively small, once the ﬂow has become turbulent. Figure 2 shows that Nu diﬀers about 10% for a laminar ﬂow case with Ra = 3.5 × 104. When the ﬂow becomes weakly turbulent at Ra = 3.5 × 105 the diﬀerences in Nu decrease to about 2% with a little lower mean heat transport in the cubic domain. With increasing Ra the diﬀerences decrease further, so that it is concluded that in the limit of Ra → ∞ the mean heat transport is not aﬀected by the geometry. This is considered to be an eﬀect of the size of the plumes that at relatively low Ra are of the same order of magnitude as the dimensions of the cube, so that their interaction reduces the heat transfer. Kaczorowski and Wagner [5] have shown that the inﬂection points of the thermal dissipation rate PDFs extracted from the long rectangular cell give a good indication of the boundaries between the conductive sublayers, mixing layers and the bulk ﬂow, respectively. These so deﬁned regions are referred to as III, II and I, respectively. Integration of the thermal dissipation rates in the limits of I, II and III hence yields an estimate for the respective contributions of these features of the ﬂow. For the periodic cell it was found that the thermal dissipation rate contributions θ I from the bulk are approximately constant. Contributions from the plumes and mixing layers (region II) on the other hand increase, while those from the conductive sublayer decrease with Ra. This behaviour is depicted in Fig. 3, together with the contributions of these regions obtained in the cube. It is observed that the contributions of regions II and III in both geometries are similar at high Ra, but at the smallest Ra the results of the cube show a signiﬁcantly diﬀerent behaviour. Here the plume and mixing layer dominated region (II) contributes signiﬁcantly more to the volume averaged mean thermal dissipation rate. This might be due to the fact that the thermal plumes preferably rise and fall along the side

Inﬂuence of the Lateral Walls in Turbulent Rayleigh–B´enard Convection

409

1 θi , i=II,III θV

0.1 B2

0.8

II

0.05

periodic box cube

0.1

0.4

0 0.2 0 105

cube periodic box

0.05

III 106

Ra

108

109

Δt =10

0

Fig. 3. Left: Comparison of the estimated contributions of region II and III for the cube and the periodic cell from the onset of turbulence until a fully turbulent ﬂow is achieved. Right: Comparison of the time series of the least-squares coeﬃcient B2 obtained for Ra = 3.5 × 106 .

walls as stated by Zhou and Xia [11]. Therefore the cubic geometry might enhance the formation of plumes at the onset of a 3D ﬂow, whereas at higher Ra turbulence dominates the ﬂow. The evaluation of the PDFs of the thermal dissipation rates shows that the scatter of the tails in the cubic domain is signiﬁcantly larger than in the periodic domain. This is also reﬂected by the time-series of the least-squares coeﬃcient B2 of the PDF’s right tail which represents the distribution within the conductive sublayer. Further analyses of the diﬀerences between the cube and the periodic cell are conducted evaluating the temporally averaged temperature and velocity ﬁelds in horizontal and vertical slices through the cells. This shows that the global ﬂow structure in the periodic cell remains unchanged for the Rayleigh numbers simulated, whereas a transition from a roll with its rotational axis parallel to one side wall (Ra = 3.5 × 105 ) to a roll with its orientation in the diagonal direction (Ra ≥ 3.5 × 106 ) is found in the cube. We therefore conclude that the transition to fully developed turbulence in both geometries shows signiﬁcantly diﬀerent behaviours. However, the diﬀerences vanish at suﬃciently high Rayleigh numbers.

4 Conclusions DNS of Rayleigh–B´enard convection have been performed in a cube for Ra up to 2.31×108 and compared to simulations conducted in a rectangular periodic cell [5]. Comparison of the mean heat transfer shows only small diﬀerences after the onset of turbulence, which appear to vanish for Ra → ∞. Evaluation of the contributions of the regions associated with the bulk (I), plumes and mixing layers (II) and the conductive sublayer (III) reveal that the respective contributions are similar for Ra ≥ 3.5 × 106 , whereas at the lowest Ra region

410

M. Kaczorowski and C. Wagner

II dissipation rates contribute signiﬁcantly more to the mean heat transfer in the cube. Signiﬁcant diﬀerences are also observed in the temporal behaviour of the least-squares ﬁt coeﬃcient B2 of the conductive sublayer for Ra up to 3.5 × 106 . This diﬀerent ﬂow behaviour in the cube is also reﬂected by the mean temperature and velocity ﬁelds that reveal a transition from a roll parallel to the side wall similar to laminar convection to a roll structure in the diagonal direction for Ra ≥ 106 . We therefore conclude that the ratio of the size of the thermal plumes to the width of the cell determines the dynamic behaviour of the large-scale ﬂow, which explains the large diﬀerences of the ﬂow behaviour at the onset of turbulence.

References 1. A. Chorin. Numerical solution of the Navier-Stokes equations. Mathematics of Computations, 22:745–762, 1968. 2. Z. A. Daya and R. E. Ecke. Does turbulent convection feel the shape of the container? Phys. Rev. Lett., 87(18):184501, 2001. 3. S. Grossmann and D. Lohse. Fluctuations in turbulent Rayleigh-B´enard convection: The role of plumes. Phys. Fluids, 16(12):4462–4472, Dezember 2004. 4. M. Kaczorowski, A. Shishkin, O. Shishkina, and C. Wagner. New Results in Numerical and Experimental Fluid Mechanics VI, volume 96 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, chapter Developement of a Numerical Procedure for Direct Simulations of Turbulent Convection in a Closed Rectangular Cell, pages 381–388. Springer, 2008. 5. M. Kaczorowski and C. Wagner. Analysis of the thermal plumes in turbulent Rayleigh-B´enard convection based on well-resolved numerical simulations. J. Fluid Mech., 618:89–112, 2009. 6. A. Leonard and G. S. Winkelmans. A tensor-diﬀusivity subgrid model for LargeEddy Simulation. Technical Report Tech. Rep. 043, Caltech ASCI, 1999. 7. U. Schumann, G. Gr¨ otzbach, and L. Kleiser. Direct numerical simulations of turbulence. In Prediction methods for turbulent ﬂows, number 2 in VKI-lecture series 1979. Von K´ arm´ an Institute for Fluid Dynamics, Brussels, 1979. 8. O. Shishkina and C. Wagner. Boundary and interior layers in turbulent thermal convection in cylindrical containers. Int. J. Sci. Comp. Math., 1(2/3/4):360–373, 2007. 9. O. Shishkina and C. Wagner. Analysis of sheetlike thermal plumes in turbulent Rayleigh-B´enard convection. J. Fluid Mech., 599:383–404, 2008. 10. R. Verzicco and K. R. Sreenivasan. A comparison of turbulent thermal convection between conditions of constant temperature and constant heat ﬂux. J. Fluid Mech., 595:203–219, 2008. 11. S.-Q. Zhou and K.-Q. Xia. Plumes statistics in thermal turbulence: Mixing of an active scalar. Phys. Rev. Lett., 89(18):184502, 2002.

DNS of Mixed Convection in Enclosed 3D-Domains with Interior Boundaries Olga Shishkina, Andrei Shishkin, and Claus Wagner DLR-Institute of Aerodynamics and Flow Technology, G¨ ottingen, Germany, [email protected]; [email protected]; [email protected]

Abstract Engineering problems of climate control in buildings, cars or aircrafts, where the temperature must be regulated to maintain comfortable and healthy conditions, can be formulated as mixed convection problems, in which the ﬂows are determined both by buoyancy and by inertia forces, while neither of these forces dominate. The objective of the present study is to investigate by means of Direct Numerical Simulations (DNS) instantaneous and statistical characteristics of turbulent mixed convection ﬂows around heated obstacles which take place in indoor ventilation problems for Grashof number up to 1.0e11 and Reynolds numbers based on the height of the domain and the inlet velocity up to 1.0e5. The chosen computational domain, which is a parallelepiped with four parallelepiped obstacles inside, can be assumed as a generic room in indoor ventilation problems. The DNS of turbulent convective ﬂows are carried out with a fourth order accurate ﬁnite volume code solving the three-dimensional incompressible Navier–Stokes equations in Boussinesq approximation.

1 Introduction The air-conditioning of buildings, cars and aircrafts, where the temperature must be regulated to maintain healthy and comfortable conditions, can be formulated as a mixed convection problem in enclosed three-dimensional (3D) domains with interior boundaries. This type of convection is determined both by the buoyancy force like in natural convection and by inertia forces like in forced convection, while neither of these forces dominates. Mixed convection is characterized by Archimedes number Ar = Gr/Re2 , which is neither small nor large in the considered case, i.e. Ar ∼ 1, where ˆ 3 ΔTˆ/ν 2 denotes Grashof number and Re = Dˆ ˆ uinlet /ν is Reynolds Gr = αg D number. α is the thermal expansion coeﬃcient, ν the kinematic viscosity, g the gravitational acceleration, ΔTˆ the temperature diﬀerence between the warm ˆ the width of interior boundaries of the container and the cold inlet ﬂow, D the container and u ˆinlet is the mean velocity of the inlet ﬂow.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 59, c Springer Science+Business Media B.V. 2010

411

412

O. Shishkina et al.

The aim of our study is to investigate numerically by means of Direct Numerical Simulations (DNS) instantaneous and statistical characteristics of the turbulent ﬂows past heated parallelepiped obstacles, which develop owing to mixed convection inside cuboid domains, depending on Gr and Ar.

2 Problem deﬁnition In our test problem we consider mixed convection of air in a box, inside which four parallelepiped heated obstacles are situated (see Fig. 1). The container has four long slits close to the top and the bottom. Through two thin upper slits cold ﬂows enter the box. The outlet slits are located close to the bottom. The temperatures of the obstacles and inlet ﬂows are ﬁxed to certain values. In the outlet ﬂows and at the outer rigid adiabatic walls ∂ Tˆ/∂n = 0, where n is the normal vector. The mean velocities at the inlet are speciﬁed as u ˆinlet and at all solid walls the velocity ﬁeld vanishes according to impermeability and ˆ /∂n = 0. All the ﬂuid properties are set to no-slip conditions. In the outlet ∂ u those of air at the temperature +25◦ C. For diﬀerent values of two parameters, namely uˆinlet and ΔTˆ , we investigate the ﬂows inside the container. We use the following reference constants for non-dimensionalization: 1/2 ˆ for distance, Tˆref = ΔTˆ for temperature, u ˆ ΔTˆ x ˆref = D ˆref = αg D for velocity, tˆref = xˆref /ˆ uref for time and pˆref = u ˆ2ref ρ for pressure. Thus the system of the governing dimensionless equations in Boussinesq approximation reads ut + u · ∇u + ∇p = Gr −1/2 Δu + T ex , ∇ · u = 0,

(1) (2)

Tt + u · ∇T = Gr −1/2 Pr−1 ΔT.

a inlet

−→

outlet

←−

(3)

b inlet

←−

outlet

−→

–0.5

0.5

Fig. 1. (a) Sketch of the computational domain and the boundary conditions. Warm interior boundaries and cold inlet ﬂows are shown in red and blue, respectively. (b) Instantaneous temperature distribution with superimposed velocity vectors in a central vertical cross-section for Gr = 1010 , Pr = 0.71 and Ar = 4.

DNS of Mixed Convection in 3D-Domains with Interior Boundaries

413

Here u is the velocity vector-function, T the temperature, ut and Tt their time derivatives, p the pressure, Pr = ν/κ Prandtl number, κ the thermal diﬀusivity, ρ the density and ex the unit vector in the vertical direction. The boundary conditions are taken in accordance with those for the dimensional problem. T varies between +0.5 at the obstacles and −0.5 at the inlets.

3 Numerical method To simulate turbulent ﬂows in complicated 3D-domains, a fast high-order ﬁnite-volume method [5] which uses the Chorin ansatz [1] was developed. Owing to the continuity equation (2), the convective terms u · ∇u and u · ∇T in equations (1) and (3), respectively, are reduced as follows u · ∇uβ = ∇ · (uβ u) − uβ ∇ · u = ∇ · (uβ u),

u · ∇T = ∇ · (T u).

(4)

To discretize equation (1) in time, the Leapfrog scheme for the convective components ∇ · (uβ u), β = x, y, z, and the Euler scheme for the diﬀusive terms Δuβ , β = x, y, z, are employed. Thus, for (1), (4) one obtains the following explicit time integration scheme un+1 − un−1 β β 2Δt

+ ∇ · unβ un + ∇pn · eβ = μ Δun−1 + T δβz , β = x, y, z, (5) β

where Δt denotes the time step, n the number of the considered time step and δβα is the Kronecker symbol, μ = Gr −1/2 . To solve equation (5) we use Chorin’s scheme [1] based on the Helmholtz decomposition. The algorithm for solving equation (5) consists of three steps: •

•

Compute an auxiliary function u∗ from /(2Δt) + ∇ · unβ un = μ Δun−1 + T δβz , u∗β − un−1 β β Solve the following Poisson equation for pn Δpn = ∇ · u∗ /(2 Δt),

•

β = x, y, z.

(n · ∇pn )|∂Υ = 0.

(6)

Calculate un+1 = u∗ − 2 Δt ∇pn .

To discretize equation (5) in space, the ﬁnite-volume approach is used employing hexahedral staggered grids for the velocity components. We average (5) over each ﬁnite volume V and represent the averaged convective and diﬀusive components, applying the Gauß–Ostrogradsky theorem, as follows < 5 6 1 n n ∇ · (uβ u ) V = n · unβ un dS, |V | S < 4 4 3 3 μ n−1 μΔun−1 = μ ∇ · ∇u = n · ∇un−1 dS, β β β |V | S V V

414

O. Shishkina et al.

where < · >V denotes averaging over V , |V | is the volume and S the surface ∂ of V . The surface-averaged quantities like uα S and ∂β uα S are calculated with fourth order accuracy using linear combinations of the volume-averaged quantities uα Vi−1 ,. . . , uα Vi+2 , where S is the adjacent surface of the ﬁnite volumes Vi and Vi+1 . For details on the construction of such approximation schemes we refer to [7]. Further, to compute the convective term (4) we need to approximate the averaged product of two quantities like uα uγ S using averaged single quantities like uα S and uγ S . Following [3], we get the following Fourier series for uα uγ S in the case of the top-hat ﬁltering and hexahedral ﬁnite volumes: ⎛ ⎞ (Δβ)2 ∂ uα ∂ uγ S S + O ⎝ (Δβ)4 ⎠ . uα uγ S = uα S uγ S + 12 ∂β ∂β β

β

To obtain the second- or the fourth-order approximations of uα uγ S we use, respectively, the ﬁrst or the ﬁrst two terms of the above expansion. In our simulations ﬁne enough meshes are used, according to the requirements by Gr¨ otzbach [2] for the resolution in DNS. The adaptive time step Δt is chosen small enough to provide the numerical von Neumann stability of the Leapfrog–Euler scheme (5) according to [8]. In the case of the fourth-order spatial approximation schemes, the time step is restricted by −1 (Δt) > max 1.5 | uβ V | (Δβ)−1 + (16/3)μ max{1, Pr−1 }(Δβ)−2 . V

β

4 Poisson solver In order to solve the Poisson equation (6) using computational meshes, which are regular in all three directions and equidistant in one direction, one can apply Fast Fourier Transform (FFT) in the equidistant direction and further utilize any fast solver for structured matrices of relatively low order in other directions. This approach was used in diﬀerent DNS [6, 9] of turbulent Rayleigh–B´enard convection. If the mesh is regular but non-equidistant in all three directions, the FFT is not applicable, although separation of variables in the regular directions is still possible. In order to solve equation (6) for the pressure in complicated domains using a computational mesh with at least one regular direction (z), we apply separation of variables [5] with respect to the regular direction and use the capacitance matrix technique [4] in the irregular directions. To separate the variables, we assume that the computational mesh is regular in the z-direction and consists of N = Nz × Nxy hexahedral ﬁnite volumes.

DNS of Mixed Convection in 3D-Domains with Interior Boundaries

415

Thus, the Poisson equation (6) averaged over a ﬁnite volume Vi,j,k with the center (xi , yj , zk ) can be represented in the form Dxy φi + Dz φi = fi , φi = pn Vi , fi = ∇ · u∗ Vi /(2 Δt), i ≡ (i, j, k)

(7)

and the operators Dz and Dxy act, respectively, in the z-direction and in the directions, orthogonal to z. If the eigenvalues λl and the corresponding eigenvectors vl , l = 1, . . . , Nz , of the operator Dz are calculated and stored before the simulations are started, the algorithm to solve the discretized Poisson equation (7) can be formulated as follows: •

l Calculate fˆi,j , l = 1, . . . , Nz , from

l fˆi,j =

Nz 1 Δz k fi,j,k vkl , vl 2Δz

v2Δz ≡ (v, v)Δz =

k=1

•

Δz k (vk )2 .

k=1

Solve linear systems (each with Nxy unknowns φˆli,j ) l , Dxy φˆli,j + λl φˆli,j = fˆi,j

•

Nz

Calculate φi,j,k =

l = 1, . . . , Nz .

(8)

Nz ˆl l l=1 φi,j vk .

To solve systems like (8) on non-regular meshes, we use the capacitance matrix technique, the idea of which can be summarised as follows. Suppose that the system with positive deﬁned matrix Al of the order Nx × Ny reads Al φˆl = ˆf l ,

T Al = Al1 , Al2 ,

T ˆf l = ˆf l , ˆf l , 1 2

(9)

with the matrix Al1 of the order n× (Nx × Ny ), n Nx × Ny . We assume that there are no available fast solvers for (9), but a fast solver exists for linear systems with the matrix Bl , which diﬀers from the matrix Al only in a few T n ﬁrst lines, i.e. Bl = Bl1 , Al2 . Then the solution of the system (9) can be obtained in three steps: the ﬁrst and the last of them require solving the systems of linear equations with the matrix Bl , for which the fast solver exists, and the second step requires the solution of a system of linear equations with a dense matrix of low order n. Formally, the algorithm reads • •

Solve the system of linear equations Bl xl = ˆf l for the vector xl with the matrix Bl of the order Nx × Ny . Solve the system Cl ω l = ˆf1l − Al1 xl for the vector ω l with the matrix −1 T (I, 0) Cl = Al1 Bl

•

(10)

of the order n. Here I is the unit matrix of the order n and 0 is (Nx × Ny − n) × n matrix with zero components. T Solve the system of linear equations Bl φˆl = ˆf l + ω l , 0 for the vector φˆl .

416

O. Shishkina et al.

The algorithm is used to solve eﬀectively systems (8) in 2D-domains with interior boundaries if the mesh, which divides the original computational domain into Nxy cells, can be extended into an ambient rectangular domain to a regular mesh with Nx × Ny cells. In this case n denotes the number of the cells of the original domain, which are in contact with the interior boundary, n Nx ×Ny , and Nxy ≤ Nx ×Ny . Then system (8) for a ﬁxed l can be written in a form (9) in the ambient rectangular domain, while the components of the ˆ which correspond to the cells inside the ambient obtained solution vector φ, domain but outside the original one, are ignored. To solve eﬀectively equations (8), we compute the matrices Cl (10) and make their LU -decompositions for all l = 1, . . . , Nz once before the simulations are started. The procedure of the LU -decomposition is time-consuming but it is used only once for each l = 1, . . . , Nz applicably to the matrix Cl of low order n, n Nx × Ny .

5 Results We investigate mixed convection in air in the above discussed domain, for diﬀerent Gr up to 1011 and Ar of order 1. The simulations are started with quiescent ﬂow ﬁeld. As time goes by, warm thermal plumes develop at the upper corners of the heated obstacles and cold plumes develop close to the inlet slits. Warm plumes rise up, while cold plumes fall down. Further the thermal plumes initiate a large-scale ﬂow inside the container, similar to those observed in Rayleigh–B´enard convection in cells of the aspect ratio of order one and for similarly high Gr. In Fig. 1b the instantaneous temperature distribution in a vertical crosssection is presented for the case Ar = 4, Pr = 0.71 and Gr = 1010 to illustrate the obtained ﬁelds. First results of DNS of mixed convection show that for Ar 1 the inﬂuence of the inlet ﬂows is negligible, i.e. in this case natural convection dominates over forced convection. In room ventilation problem with Ar = 1 neither buoyancy nor inertia forces dominate. Further, with growing Gr the ﬂow becomes stronger and its structures – more complicated.

6 Conclusion The ﬁnite-volume fourth order numerical method and corresponding computational code were developed to conduct DNS of turbulent convection in enclosed 3D-domains with and without inlet/outlet ﬂows, with and without obstacles. It allows to investigate ﬂow structures and mean ﬂow characteristics in mixed convection for diﬀerent Gr and Ar. In the future the numerical predictions will be compared with experiments conducted at the RWTH Aachen, E. ON ERC, Institute for Rational Use of Energy in Buildings.

DNS of Mixed Convection in 3D-Domains with Interior Boundaries

417

Acknowledgements The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG) for supporting this work.

References 1. Chorin A. J. & Marsden J. E. (1993) A Mathematical Introduction to Fluid Mechanics. Springer Series: Texts in Applied Mathematics, 4, Springer-Verlag. 2. Gr¨ otzbach G. (1983) Spatial resolution requirements for direct numerical simulation of Rayleigh–B´enard convection, J. Comput. Phys. 49: 241–264. 3. Leonard A. & Winckelmans G. S. (1999) A tensor-diﬀusivity subgrid model for Large-Eddy Simulation, Caltech ASCI technical report, cit-asci-tr043, 043. 4. Schumann U. & Sweet R. A. (1976) A direct method for the solution of Poisson’s equation with Neumann boundary conditions on a staggered grid of arbitrary size, J. Comput. Phys. 20: 171–182. 5. Shishkina O., Shishkin A. & Wagner C. (2009) Simulation of turbulent thermal convection in complicated domains, J. Comput. Appl. Maths 226, 336–344. 6. Shishkina O. & Wagner C. (2008) Analysis of sheet-like thermal plumes in turbulent Rayleigh–B´enard convection, J. Fluid Mech. 599: 383–404. 7. Shishkina O. & Wagner C. (2007) A fourth order ﬁnite volume scheme for turbulent ﬂow simulations in cylindrical domains, Comput. Fluids 36: 484–497. 8. Shishkina O. V. (2007) The Neumann stability of high-order symmetric schemes for convection–diﬀusion problems, Siberian Mathematical J. 48: 1141–1146. 9. Verzicco R. & Camussi R. (2003) Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell, J. Fluid Mech. 477: 19–49.

LES and Hybrid RANS/LES of Turbulent Flow in Fuel Rod Bundle Arranged with a Triangular Array Stefano Rolfo1 , J.C. Uribe1 , and D. Laurence1,2 1

2

School of MACE, The University of Manchester, Manchester M60 1QD, UK, [email protected]; [email protected]; [email protected] EDF R&D, 6 Quai Waitier, Chatou, 78420, France

Abstract Turbulent ﬂow parallel to fuel rod bundles arranged in a triangular array are computed using LES and hybrid RANS/LES. Inner-channel ﬂow pulsations are captured and the dominant frequency is in agreement with the experiments. The hybrid method shows a lack of accuracy in the near wall region probably due to the formulation of the blending function.

1 Introduction Rod bundles are a typical constitutive element of a very wide range of nuclear reactor design. They are composed of a set of rod elements, containing the nuclear fuel, arranged usually in square or triangular conﬁgurations. Those elements are cooled by a ﬂuid ﬂowing parallel to the rods. Experimental studies [7,10] found that the distribution of the turbulent intensities are quite diﬀerent from those in pipes and channels. For long time the explanation was the presence of a secondary ﬂow [15], as in corners of rectangular or triangular ducks. The intensity of this secondary ﬂows was measured directly for the ﬁrst time by Vonka [15], ﬁnding a value about 0.1 % of the mean bulk velocity. In nuclear reactors a high value of burn-up1 is desiderable, which can be achieved with small value of the pitch-over-diameter ratio (P/D). In those conﬁgurations an energetic and almost periodic azimuthal ﬂow pulsation is present in the gap region between two sub-channels. This phenomena was already observed by Rowe [11] and it was deﬁnitely conﬁrmed and measured by Hooper [6]. Krauss and Meyer [7] demonstrated that those ﬂuctuation are the reason for higher mixing between sub-channels. These ﬂow pulsations can be described as coherent large-scale structure that are ﬂowing in the stream-wise direction, causing an unsteady meandering oﬀ the mean ﬂow. Usual steady state 1

Burn-up: measure of the neutron irradiation of the fuel.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 60, c Springer Science+Business Media B.V. 2010

419

420

S. Rolfo et al.

CFD techniques cannot suitable describe this type of ﬂow, in particular when the geometry is very tight. A ﬁrst attempt to use unsteady RANS approach (URANS) was undertakes by Chang and Tavoularis [4, 5]. In the second article they solved the isothermal ﬂow in a sector of 60o in a 37-rod bundle and reported time average velocity and ﬂuctuations as well as high correlations between turbulent structures in the entire geometry. Another URANS simulation was presented in Merzari et al. [9], where the authors use an ad hoc turbulence model presented in [1]. The results are compared with the experiments of Krauss and Mayer [7] reporting good agreement in term of wall shear stress and turbulent kinetic energy, but with an underestimation of the dominant frequency. The authors reports also the coherence pattern for the azimuthal velocity between two reference gaps, showing a coherence peak in accordance with the experiment. This proved that even when the simulation domain cross section is limited to two sub-channels the main features of the ﬂow could be captured. Consequently the conclusion of Chang and Tavoularis [5] that large cross sections, with a high number of sub-channels needed, might be considered too conservative. Another interesting conclusion of Baglietto’s work is that the wavelength and dominant frequency are strongly dependent from the length of the computational domain in the streamwise direction. Finally a LES and DNS study is reported by Baglietto et al in [2] for P/D equal 1.06 until 1.2 Reynolds numbers from 6,000 up to 24,000. The cross section of the computational domain is composed by only four elementary units, arranged in a pattern that is covering the gap region between the centres of two adjacent sub-channels. In this case ﬂow pulsation are not reported, probably because of the short length used in the stream-wise direction, rather than the limited cross section employed, as will be demonstrated in this work.

2 Hybrid RANS/LES The Hybrid RANS/LES can be classiﬁed as a zonal approach with a smooth transition realized through a blending function Fb . The model follows Schumann’s decomposition [12] into a locally isotropic and inhomogeneous parts. The subgrid stress tensor and the subgrid heat ﬂuxes have the following formulation: ⎧ Locally Isotropic Inhomogeneous ⎪ ⎪ 5 6 5 6 ⎨ τijr = −2νr Fb Sij − Sij − 2νa (1 − Fb ) Sij (1) ⎪ 5 6 ⎪ ⎩ σ r = −Fb κr ∂ T − T − (1 − Fb ) κa ∂ T j

∂xj

∂xj

for the isotropic turbulent viscosity νr a Smagorinsky model [13] based on the ﬂuctuating rate of strain is employed, and an elliptic relaxation model v 2 /k − f [8] is used for the RANS contribution νa . The blending function is parameterized by the ratio between the RANS turbulent length scale Lt = √ v 2 k/ and the ﬁlter width Δ obtaining:

LES and Hybrid RAN/LES in Rod Bundle

421

n Lt Fb = tanh Cl Δ

(2)

A more exhaustive description of the model is given in [14]. The eddy diﬀusivity modelling is using a turbulent Prandtl number analogy as κ = ν/P rt with constant P rt .

3 Case description The conﬁguration of a fuel rod bundle arranged with a triangular array is presented in Fig.1. The geometry is fully parameterized using only one geoP metrical parameter the pitch over diameter ratio or D . Two diﬀerent ratio P P are taken into consideration: D = 1.06 and D = 1.15. In order to reduce the computational cost a cross section composed by four elementary units is used. Periodic boundary conditions are deﬁned in the streamwise direction. Rotational periodicity is also employed in the cross section. For both conﬁgurations a LES with near wall resolution is performed at Re = 6, 000. The mesh details are summarized in Table 1. A key point of the computational domain is the length in the streamwise direction, which needs to be long enough to permit ﬂow pulsations to develop.

Open Region

Temp 2.980e+02

Gap Region

P

2.965e+02

Mid Region

2.950e+02

Point A

PointB

D

Computational Domain

2.935e+02 2.920e+02 x z

Fig. 1. Sketch, on the left, of the rod bundle geometry,where D is the diameter of a fuel pin and P is the distance between the center of two adjacent fuel elements (pitch). The colored region is the computational domain used. On the left P instantaneous temperature ﬁeld for LES, D = 1.06 @ Re = 6, 000. Table 1. Test case mesh deﬁnition. Case LES6K LESWPD HYB6K HYB39K Exp [7]

Model P/D LES LES Hyb Hyb Exp

Re

r+

r + Δθ

x+

Mesh Size Heat Tran

1.06 6,000 0.7–1.06 7.5–11 16–22.5 1.6 × 106 1.15 6,000 0.8–1.1 6.5–10 16–22.5 1.4 × 106 1.06 6,000 0.8–1.3 15–20 40–60 0.36 × 106 1.06 39,000 0.8–1.2 20–25 50–70 0.9 × 106 1.06 39,000 / / / /

Yes No Yes No Yes

422

S. Rolfo et al.

4 Results Flow ﬂuctuations can be seen in Fig. 1, where an instantaneous temperature ﬁeld is plotted. The ﬁgure clearly shows a meandering behavior of the temperature in the gap region. Krauss and Meyer [7] deﬁne a Strouhal number based on the average velocity in the narrow gap and for P/D = 1.06 and Re number in the range 30 k to 50 k found it to be constant: St =

fD Umean,gap

= 0.93

(3)

with f being the dominant frequency. Figure 2 shows the spectra of the different cases and Table 2 the correspondent Strouhal number. For all cases computed at lower P/D the ﬁrst dominant frequency is in relatively good agreement with the experimental value. However a second dominant frequency is also visible at higher value. This could be a consequence of the small cross section employed for the computational domain, limiting the amplitude of the pulsations. For the case HYB39K few samples for the spectra calculation are available. Higher number of samples might lead to a merging of the two dominant frequencies. Table 2 reports also the adimensionalized total wall shear. The comparison between case LES6K and HYB6K, and between HYB39K and the experiments shows the tendency of the hybrid model to overestimate the total wall shear stress. The eﬀects on the ﬂow ﬂuctuations on the average 100

LES P/D=1.06 Re = 6000 LES P/D=1.15 Re= 6000

−1

10

10−1 10−2 Φw [m2 / s]

Φw [m2 / s]

10−2 10−3 10−4

10−3 10−4

10−5

10−5

10−6 101

10−6

102

103

Hy P/D= 1.06 Re=6000 HY P/D=1.06 Re= 39000

10

f [Hz]

1

102

103

104

f [Hz]

Fig. 2. Power spectra of velocity ﬂuctuations in the angular direction. Table 2. Summary of the mean wall shear and dominant frequency for the diﬀerent cases. CASE τw,m /(0.5ρUb2 ) St Umean,gap /Ub

LES6K LESWPD HYB6K HYB39K Exp [7] 0.0086 0.98–1.96 0.75

0.0025 / 0.89

0.0106 0.0104 0.0057 0.91–1.85 0.92–1.4 0.93 0.77 0.83 0.78

LES and Hybrid RAN/LES in Rod Bundle

<ww>

0.21

0.93

1.66

2.38

0.11

3.10

0.48

0.39

0.90

1.40

1.91

√

2.42

uu/u2τ

0.20

and

√

ww/u2τ

0.53

0.62

1.23

1.60

0.85

1.18

1.50

for case LES6K (top) and HYB39K

Tmean 0.31

0.86

<ww>

Fig. 3. Reynolds stresses (bottom) (Table 1).

0.00

423

t2 0.93

1.24

Fig. 4. Adimesionalized mean temperature √ θθ/Tτ for case LES6K.

1.60

Tw,mean −T Tw,mean −Tb

2.04

2.47

2.91

3.35

on the left and temperature

ﬂuctuations

√ ﬁelds is very clear on Fig. 3. In the gap region the maximum of ww/u2τ is located in the middle of the geometry and not close to wall, as in classical pipe or channel ﬂows. This correspond to the oscillating structure seen in Fig. 1. For HYB39K the coherent structure motion captured a ww less prominent against the classical turbulence level. As expected the proﬁle of the non-dimensional mean temperature is very similar to that one of the nondimensional mean streamwise velocity (Fig. √ 4). The same analogy is no more valid for the temperature ﬂuctuations θθ/Tτ2 and the streamwise ﬂuctuations. In the ﬁrst case the maximum is located close to the gap region, when, instead for the second-one is placed in the open region.

5 Conclusion Flow parallel to a fuel rod bundle arranged in a triangular array was investigated using LES and Hybrid RANS/LES coupling. Flow pulsations in the gap region were observed and captured for the narrow-gap geometry with all the turbulence models employed. The ﬁrst dominant frequency is in agreement with the experimental value, but a second dominant frequency, not detected by the experiments was also found which could be an artiﬁcial eﬀects due to the small cross section of the computational domain. Calculations employing a larger cross section composed by two complete sub-channels will be carried out. Results of the hybrid method cannot be fully trusted in the near wall

424

S. Rolfo et al.

region, because unfortunately the switch-over from the RANS to LES takes place around y + = 10 to 20, i.e. the buﬀer layer, which is probably the most challenging (i.e. worst?) location to switch approaches. In eﬀects the RANS model was only driving the simulations for the ﬁrst three cells near the wall. The function in equation (2) is based on the classical lengthscale k 3/2 / which erroneously goes to zero at the wall (whereas the turbulent structure very near to the wall scale with the Kolmorov scales). Meanwhile the “elliptic relaxation near the wall RANS” community of modelers (Manceau, Hanjali´c, Billard et al. [3]) is currently developing a more general elliptic blending function, α, which clearly distinguish the buﬀer layer from the fully developed turbulent layer, without reference to modelled integral lengthscale or y + . This α function could thus advantageously replace or complement equation (2) in the hybrid RANS/LES context.

Acknowledgements This work was carried out as part of the TSEC programme KNOO and as such we are grateful to the EPSRC for funding under grant EP/C549465/1.The authors would like also to acknowledge the use of the UK National Grid Service in carrying out this work and Dr E. Baglietto for his helpful comments.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Baglietto E, Ninokata H (2005) Nuc. Eng. Des. 235: 773–784. Baglietto E, Ninokata H, Misawa T (2006) Nuc. Eng. Des. 236: 1503–1510. Billard F, Uribe J C, Laurence D (2008) TSFP7 Vol 1: 89–94. Chang D, Tavoularis S (2005) J. Fluid Eng.: 458–466. Chang D, Tavoularis S (2007) Nuc. Eng. Des. 237: 575–590. Hooper J D , Rehme K (1984) J. Fluid Mech. 145: 305–337. Krauss T, Meyer L (1998) Nuc. Eng. Des. 180: 185–206. Laurence D, Uribe J C, Utyuzhnikov S (2004) Flow, Turbulence and Combustion 73: 169–185. Merzari E, Ninokata N, Baglietto E (2008) Nuc. Eng. Des. 238: 1703–1719. Rehme K (1989) Exp. Thermal and Fluid Science 2: 341–349. Rowe D S, Johnson B M, Knudsen J G (1974) Int. J. Heat Mass Transfer 17: 407–419. Schumann U (1975) J. of Computational Phy. 18: 376–404. Smagorinsky J (1963) Month. Weath. Rev. 91(3): 99–165. Uribe J C, Jarrin N, Prosser R, Laurence D (2007) TSFP5 Vol 2: 701–706. Vonka V (1988) Nuc. Eng. Des. 106: 191–207 and 209–220.

Large-Scale Patterns in a Rectangular Rayleigh–B´ enard Cell A. Sergent1 and P. Le Qu´er´e2 1

2

LIMSI CNRS, Campus de l’Universit´e Paris-Sud, 91403 Orsay, France Universit´e Pierre et Marie Curie, 75005 Paris, France, [email protected] LIMSI CNRS, Campus de l’Universit´e Paris-Sud, 91403 Orsay, France, [email protected]

1 Introduction Rayleigh–B´enard convection is known to present a large variety of coherent ﬂow, which can persist in turbulent regime [1]. The mean large-scale ﬂow generated by the convection can feature metastable ﬂow structures over hundreds or thousands of large eddy turnover times between which spontaneous sudden reversals or changes of direction occur [2, 3]. For example, in a cylindrical unity-aspect-ratio cell, a single circulation ﬁlls the whole volume. But in a cell with aspect ratio 1/2, the ﬂow structure evolves with the Rayleigh number (Ra) from a single roll ﬂow to a ﬂow with two super-imposed counter-rotating rolls [4, 5], and both kinds of ﬂow structures can coexist in a particular Ra range. A model of prediction of the mean ﬂow structure has been proposed for the particular case of 1/2 aspect ratio cylinder [4]. Most of the previously cited experimental or numerical studies have been performed in small aspect ratio containers. But it has been shown numerically in a periodic ﬂuid layer [1] that up to Ra = 107 the selected size of circulation cells in turbulent convection increases with Ra in the case of moderate Prandtl numbers (P r). The aim of the present work is to study by means of Large Eddy Simulations, the inﬂuence of the conﬁnement on the time evolution of the network of quasi-stationary rolls in a rectangular container.

2 Numerical set-up Let us consider the air-ﬂow (Pr = 0.71) developing in a rectangular cell of moderate aspect ratios (lengths over height, Ay = 1 and Az = 5) for Ra ranging from 107 to 1010 . The ﬂuid is heated from below and cooled from above by two isothermal walls. The four vertical walls are perfectly adiabatic. No slipping condition is applied on the six walls. This geometry corresponds to the experimental rectangular cell studied in [6]. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 61, c Springer Science+Business Media B.V. 2010

425

426

A. Sergent and P. Le Qu´er´e

The LES equations for convective ﬂows are obtained from the 3D ﬁltered incompressible Navier–Stokes equations with the Boussinesq approximation. The equations are discretized on a staggered mesh by ﬁnite volume approximations. All the terms involved in the conservation equations are evaluated with a second-order accurate centered scheme, except the nonlinear terms of the momentum equations for which a QUICK scheme is used. The choice of the QUICK scheme is made to improve numerical stability. It is also substantiated by a priori test which were carried out in the case of the 2D diﬀerentially heated cavity at Rayleigh number Ra = 5.1010 [7]. It has been shown that the numerical dissipation of the QUICK scheme is able to correctly reproduce the energy transfer between resolved and unresolved scales of the ﬂow. To summarize, the Mixed Scale Diﬀusivity Model [7] is used in the energy equation, while no explicit subgrid viscosity model is introduced for the momentum equation. The code has been validated for the diﬀerentially heated cavity [7] and Rayleigh–B´enard convection in a periodic ﬂuid layer [8]. In particular, a 2/7 scaling behavior of the heat transfer (N u = 0.161Ra0.286, N u: Nusselt number) has been reproduced over a large Ra range and a regime transition towards a 1/3 turbulent regime has been observed at Ra = 2.108 (Fig. 1). The resolution requirement for solving a natural convection ﬂow can be speciﬁed from an estimate of the thermal boundary layer thickness and the dissipative scale in the bulk, as mentioned by [9]. The spatial resolution used in the present work is N x × N y × N z = 50 × 34 × 130 unequally spaced in the vertical direction,with at least 5 grid points in the thermal boundary layer and a ﬁrst point at wall located at x+ ≈ 0.5. The maximum horizontal mesh size in the bulk is 20 times the Kolmogorov scale for the highest Ra.Consequently, it is not necessary to model the near-wall dynamics (nor the heat transfer). Indeed, a DNS like criteria is applied to the grid distribution in the boundary layer, and a Mixed Scale approach is used, which adapts the SGS modelling to the local ﬂow. This way, the ﬂow structure has the capacity to re-arrange by itself.

NuRaβ

1

100

β = –2/7 β = –0.309 β = –1/3

0.1

Nu

104 105 106 107 108 109 1010 Ra

10 DNS - Kerr(1996) Niemela et al. (2000) Chavanne et al. (2001) Fleischer & Goldstein (2002) present study

104

105

106

107

108

109

1010

Ra

Fig. 1. N u versus Ra. Inset: Compensated Nusselt number N u × Raβ versus Ra concerning the present results.

Large-Scale Patterns in a Rectangular Rayleigh–B´enard Cell

427

3 Results Numerical simulations have been performed for Ra = 107 , 6.108, 1010 . All computations have been started from the conductive temperature proﬁle with random noise as initial conditions. For each Rayleigh number, a sudden transition occurs without any external intervention between two mean ﬂows, which are stable on very long time period (∼ 104 convective time units). Only a single large-scale reorganization has been observed in our computations. But due to the long characteristic times, we cannot conclude if the second observed mean ﬂow is the unique stable state. Similar transitions have been observed in two studies: (i) experimentally at Ra = 4.1012 in a water ﬁlled cylindrical cell with aspect ratio 1/2 [5] with a transition characteristic time in the order of 300 h, (ii) in a numerical study with a similar geometry [4], where two diﬀerent ﬂow structures have been observed. In the present work, the ﬁrst observed mean ﬂow is the same for the three Ra. It corresponds to the steady-state solution with 4 counter-rotating rolls placed side by side, on which is superimposed the background turbulence (Fig. 2). The fact that the ﬁrst ﬂow is independent on Ra demonstrates that the 4-roll ﬂow structure is only transitional before a physical solution is established. The large coherent ﬂow structures of the second ﬂow correspond to 3 or 2 rolls placed side by side depending on Ra. Their length scale can be determined by computing the spectral distribution of the vertical heat transfer at mid-height, where the wavelength of the maximum spectral density characterizes the typical size of the largest ﬂow structures [1]. In agreement with results in [1, 10], it is observed that the spontaneous selected size of the circulation cells increases with Ra (Fig. 3) and the gap between large and small scales becomes larger with increasing Ra. But at Ra = 6.108 , the two roll ﬂow structure (Fig. 2) has been obtained by decreasing Ra from the Ra = 1010 solution ﬁeld. This third ﬂow remains stable during at least 104 convective times units. But due to the long time period over which the large coherent ﬂow structures can persist with this set of physical parameters, it is not clear whether the 2- and 3-roll ﬂow structures are two possible solution ﬁelds, or if the ﬂow will converge eventually towards one of these mean ﬂow conﬁgurations.

Fig. 2. Ra = 6. 108 . Three diﬀerent patterns of time-averaged iso-surfaces of y-component of vorticity Ωy = ±0.3. (top), and the relative iso-surfaces of instantaneaous vertical heat ﬂux (∼ 2N u), coloured by the temperature ﬁeld (bottom).

428

A. Sergent and P. Le Qu´er´e 103 DNS-Hartlep et al., 03 DNS-Shishkina & Wagner, 06 present work

0.8

103 k|Uxθ|

λmax / Lref

1

0.6

103

Ra = 107 Ra = 6.108 Ra = 6.108 Ra = 1010

0.4 105

106

107

108 Ra

109

1010

103 1

10 k

= Fig. 3. Left: wavelength λmax /Az of the maximum spectral density |U x θ| at mid= height versus Ra; right: k|U θ| as a function of the wavenumber k. x

Acknowledgements Numerical simulations were performed on a NEC-SX8 supercomputer at the computing center IDRIS-CNRS under research project i20080210326. This work was supported by the French National Research Agency (ANR).

References 1. T. Hartlep, A. Tilgner, and F.H. Busse. Transition to turbulent convection in a ﬂuid layer heated from below at moderate aspect ratio. J. Fluid Mech., 554: 309–322, 2005. 2. E. Brown, A. Nikolaenko, and G. Ahlers. Reorientation of the large-scale circulation in turbulent Rayleigh-B´enard convection. Phys. Rev. Lett., 95:084503, 2005. 3. F. F. Araujo, S. Grossmann, and D. Lohse. Wind reversals in turbulent RayleighB´enard convection. Phys. Rev. Lett., 95(8):084502, 2005. 4. G. Stringano and R. Verzicco. Mean ﬂow structure in thermal convection in a cylindrical cell of aspect ratio one half. J. Fluid Mech., 548:1–16, 2006. 5. F. Chill` a, M. Rastello, S. Chaumat, and B. Castaing. Long relaxation times and tilt sensitivity in Rayleigh-B´enard turbulence. Eur. Phys. J. B, 40: 223–227, 2004. 6. A. Ebert, C. Resagk, and A. Thess. Experimental study of temperature distribution and local heat ﬂux for turbulent Rayleigh-B´enard convection of air in a long rectangular enclosure. Int. J. Heat Mass Transfer, 51:4238–4248, 2008. 7. A. Sergent, P. Joubert, and P. Le Qu´er´e. Development of a local subgrid diﬀusivity model for large eddy simulation of buoyancy driven ﬂows: application to a square diﬀerentially heated cavity. Num. Heat Transf. A, 44:789–810, 2003. 8. A. Sergent, P. Joubert, and P. Le Qu´er´e. Large eddy simulation of RayleighB´enard convection in an inﬁnite plane channel using a mixed scale diﬀusivity model. P. Comp. Fluid Dyn., 6:40–49, 2006. 9. R. Verzicco and R. Camussi. Numerical experiments on strongly turbulent thermal convection in slender cylinder cell. J. Fluid Mech., 477:19–49, 2003. 10. O. Shishkina and C. Wagner. Analysis of thermal dissipation rates in turbulent Rayleigh-B´enard convection. J. Fluid Mech., 546:51–60, 2006.

LES and Laser Measurements of Dynamic Flame/Vortex Interactions V. Di Sarli1 , A. Di Benedetto1 , G. Russo2 , E.J. Long3 , and G.K. Hargrave3 1

2

3

Istituto di Ricerche sulla Combustione - CNR, Via Diocleziano 328, 80124, Napoli, IT, [email protected]; [email protected] Dipartimento di Ingegneria Chimica, Universit` a degli Studi di Napoli Federico II, P.le Tecchio 80, 80125, Napoli, IT, [email protected] Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK, [email protected]; [email protected]

1 Introduction The dynamic interaction that occurs between a propagating ﬂame front and a rotating vortex structure is one of the key mechanisms within turbulent combustion [1]. As such, it is found within many applications such as industrial combustors and burners, internal combustion engines, and obstacle-induced explosions. The exact manner in which the ﬂame propagates through a turbulent ﬁeld is strictly dependent on the turbulent structures encountered by the ﬂame. However, the ﬂuid structures generated in practical conﬁgurations are highly complex, providing a diﬃcult environment in which to examine interactions between ﬂame propagation and ﬂow ﬁeld. This research was aimed at studying dynamic interactions between premixed ﬂame fronts and simple toroidal vortex structures within a controlled environment. Large Eddy Simulation (LES) and experimental optical diagnostics were jointly applied to investigate the impact of these vortices on the structure of the ﬂame front and the progression of the ﬂame through the vortices themselves. Numerical and experimental results provide characterisation of the toroidal vortex/ﬂame front interaction for a range of vortex strengths.

2 Experimental work In this work, numerical results and experimental data, acquired using the test rig schematised in Fig. 1a, are compared. Within this rig, a quiescent premixed charge of stoichiometric methane and air was ignited inside a small cylindrical pre-chamber (35 mm in height and 70 mm in diameter), linked V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 62, c Springer Science+Business Media B.V. 2010

429

430

V. Di Sarli et al.

a

b

Fig. 1. Schematics of the test rig (a) and the ﬂame/vortex interaction (b).

to the main chamber (150 × 150 × 150 mm) via a small oriﬁce. The main chamber was constructed from polycarbonate to provide optical access. The combustion chamber was sealed using a PVC membrane, which ruptured soon after ignition allowing the exhaust gases to escape. After ignition, the ﬂame propagated through the pre-chamber, pushing unburned charge ahead of the ﬂame front through the oriﬁce. This motion of the reactants through the constriction resulted in a toroidal vortex being shed into the main chamber. As the ﬂame continued to propagate through the charge, it interacted with the vortex structure (Fig. 1b), distorting the ﬂame and altering its burning velocity. The nature of the vortices produced in the main chamber is strongly dependent on the oriﬁce diameter. Three diﬀerent oriﬁce diameters were tested during this investigation: 20, 30, and 40 mm. All three oriﬁces had a constriction length of 25 mm, with sharp edges at both the pre-chamber and main chamber sides. To image the ﬂame propagation, High-Speed Laser-Sheet Flow Visualisation (HSLSFV) was employed. Application of this technique, which consisted of an Oxford Lasers Copper Vapour Laser LS20-40 synchronised to a Photron APX-RS CMOS high-speed camera, has been detailed in a previous paper [2]. The image recording was initiated on ignition of the charge, with a recording rate of 4,500 frames per second at a resolution of 896 × 784 pixels. An image area of 35 × 30 mm was obtained, a region size large enough to capture the interaction between the ﬂame front and one side of the toroidal vortex.

3 Large Eddy simulation (LES) model The LES model equations were obtained by Favre-ﬁltering the governing equations for unsteady compressible ﬂows with premixed combustion, i.e., the Navier–Stokes equations for conservation of mass, momentum, energy and species, joined to the constitutive and state equations. The species transport equation was recast in terms of a transport equation for the reaction progress variable, c (c = 0 within fresh reactants, c = 1 within burned products) [3].

LES and Laser Measurements of Dynamic Flame/Vortex Interactions

431

The closure of the sub-grid scale (sgs) stress tensor was achieved with the dynamic Smagorinsky–Lilly model [4]. The sgs ﬂuxes of heat and reaction progress variable were modelled through the gradient hypothesis [3]. To handle the ﬂame/turbulence interaction at the sgs level, the ﬂame wrinkling model by Charlette et al. [5] was implemented in the LES model in the context of the ﬂame surface density formalism [3]. The model equations were discretised using a ﬁnite volume formulation on a non-uniform structured grid (minimum cell size = 0.5 mm; maximum cell size = 1 mm). For the spatial discretisation, second order bounded central schemes were employed. The time integration was performed using the second order implicit Crank–Nicholson scheme. Adiabatic and no-slip wall boundary conditions were applied at the solid interfaces. Outside the combustion chamber, the computational domain was extended to allow for a more realistic reproduction of the exit of the expanding gas from the chamber into the atmosphere. A condition of ﬁxed static pressure was assigned at the boundaries of this additional domain. Initial conditions had velocity components, energy and reaction progress variable set to zero everywhere. Ignition was obtained by means of a hemispherical patch, with a radius equal to 1 mm, of hot combustion products at the centre of the bottom end of the pre-chamber. Computations were performed by means of the segregated solver of the Fluent code (version 6.3.26) (see http://www.ﬂuent.com/).

4 Results and discussion The time sequences of the HSLSFV images and the LES reaction progress variable maps are presented in Fig. 2 for all three oriﬁce diameters investigated. This ﬁgure shows the ﬂame as it propagates inside the main chamber at diﬀerent time instants after ignition/initialisation. The LES model produces good prediction of the ﬂame shape and the arrival time for each oriﬁce. From both the experimental and LES results of Fig. 2 it can be seen that, as the ﬂame ﬁrst exits the oriﬁce into the main chamber, it is almost ﬂat. This behaviour occurs for all oriﬁce sizes. However, from this point on, three distinct modes of ﬂame/vortex interaction can be observed. For the 40 mm oriﬁce, the ﬂame propagates around the vortex, pushing the vortex ahead of the ﬂame front and slowly consuming the mixture in the vortex. With the 30 mm oriﬁce, the ﬂame initially curves into the vortex and then consumes the vortex more rapidly than in the 40 mm case. Finally, the ﬂame exiting the 20 mm oriﬁce immediately consumes the mixture in the vortex, via a thin ﬂame sheet, without rolling up in the vortex. Figure 2 also shows that, with both the 40 and 30 mm oriﬁces, the ﬂame front is aﬀected by the vortex only in its surface area, while its structure remains undisturbed. More precisely, the ﬂame/vortex interaction wrinkles the ﬂame surface. The degree of ﬂame front wrinkling increases the further

432

V. Di Sarli et al.

Fig. 2. Time sequences of the HSLSFV images and the LES reaction progress variable maps at diﬀerent oriﬁce diameters.

the ﬂame propagates through the vortex. It is more intense with the 30 mm oriﬁce, thus leading to faster ﬂame propagation than for the 40 mm oriﬁce. Diﬀerently from the 40 and 30 mm cases, the ﬂame structure produced from the 20 mm oriﬁce appears to be strongly aﬀected by the vortex: the ﬂame/vortex interaction disrupts the continuity of the ﬂame front, giving rise to the formation of ﬂame pockets which leave the main front. This diﬀerent ﬂame structure signiﬁcantly increases the ﬂame propagation rate, as the ﬂame burns through both the main front and the pockets themselves. The variation in ﬂame/vortex interaction can be linked to the recirculation velocity of the vortex. As the oriﬁce size reduces, the velocity of the ﬂow exiting the oriﬁce increases, thereby increasing the vorticity of the rotation and thus the rate at which the ﬂame propagates through the vortex.

References 1. Renard P-H, Th´evenin D, Rolon JC, Candel S (2000) Prog Energy Combust Sci 26:225–282 2. Long EJ, Hargrave GK, Jarvis S, Justham T, Halliwell N (2006) J Phys Conf Ser 45:104–111 3. Poinsot T, Veynante D (2005) Theoretical and numerical combustion (second ed). R.T. Edwards, Philadelphia 4. Lilly DK (1992) Phys Fluids A 4:633–635 5. Charlette F, Meneveau C, Veynante D (2002) Combust Flame 131:159–180

3D Direct Simulation of a Nonpremixed Hydrogen Flame with Detailed Models G. Fru1 , D. Th´evenin1 , C. Zistl1 , G. Janiga1 , L. Gouarin2 , and A. Laverdant2 1

2

Lab. of Fluid Dynamics and Technical Flows, University of Magdeburg “Otto von Guericke”, Universit¨ atsplatz 2, D-39106 Magdeburg, Germany, [email protected]; [email protected]; [email protected]; [email protected] ONERA, Avenue de la Division Leclerc, BP 72, F-92322 Chˆ atillon, France, [email protected]; [email protected]

Abstract We present a three-dimensional direct simulation of a turbulent nonpremixed H2/air ﬂame described with a complete chemical scheme and a multicomponent transport model. When analyzing for example the local ﬂame structure with a ﬂame index, various burning regimes (nonpremixed, partially-premixed, premixed) are shown to coexist and to contribute to fuel oxidation. Nevertheless, nonpremixed combustion globally dominates the process, as expected for such a conﬁguration. The distribution of mixture fraction can be quite accurately reconstructed using a β-function approximation based on mean and variance. A γ-function approximation leads always to a higher error level. Furthermore, the evolution of scalar dissipation rate versus mixture fraction is investigated, revealing a highly non-symmetrical distribution.

1 Introduction Nonpremixed ﬂames correspond to situations where the reactants are not initially mixed and originate from two separate streams. Mixing, which can be modeled by an appropriate description parametrized by the mixture fraction Z, must then bring them together for combustion to take place. The understanding of turbulent combustion relies on strongly coupled and highly complex physical phenomena that involve a large number of species and elementary reactions. As a consequence of this complexity, small-scale turbulence/chemistry interactions are diﬃcult to measure experimentally, highlighting the importance of complementary analyses based on Direct Numerical Simulations (DNS) [1, 2]. DNS are becoming increasingly useful for combustion applications and constitute a natural complement to experiments, in particular to investigate V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 63, c Springer Science+Business Media B.V. 2010

433

434

G. Fru et al.

in detail complex physical processes in simple geometries. Nevertheless, this is only true if the employed physical models are close enough to reality. When considering some essential issues like pollutant formation or ignition/extinction processes, detailed models must thus be employed to describe the chemical and transport phenomena with suﬃcient accuracy. Despite the huge computational cost of the resulting simulations, progress in computer technology now allow DNS of turbulent ﬂames while employing complete reaction schemes, at least for some simple conﬁgurations [2,3]. In the present work DNS results are presented for a three-dimensional, turbulent nonpremixed H2 /air ﬂame described with a complete chemical scheme and a multicomponent transport model. The aim is to examine the coupling of an initially homogeneous isotropic turbulence with the ﬂame and to deliver further important information concerning ﬂame structure, impacting turbulent modeling.

2 Physical and computational model The compressible 3D reactive Navier–Stokes equations are solved using a 6thorder central scheme and an explicit 4th-order Runge–Kutta scheme for time integration. Figure 1 illustrates the employed conﬁguration and boundary conditions. Further details concerning the code can be found for instance in [4]. A nonpremixed diluted H2 –air ﬂame is ignited using numerical catalysis [5]. A 3-D ﬁeld of synthetic turbulence (generated assuming a turbulent kinetic energy distribution given by a von K´ arm´ an spectrum with Pao correction [4]) is imposed on top of it. H2 oxidation is described with 9 species and 19 elementary reactions [6]. The initial conditions correspond to a global mixture ratio φ = 0.8, with Zs = 0.56. The computational domain is a 1.0 cm cube with a uniform grid spacing of 50 µm. The initial turbulence corresponds to an integral scale lt = 1.55 mm (mixture viscosity ν = 1.55 10−5 m2 /s), velocity ﬂuctuation u = 2.05 m/s, turbulent Reynolds number Ret = 205 and eddy

b

a

Non reflective boundary

Non reflective boundary Y_H2 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002

Periodic

Fig. 1. (a) General conﬁguration of the nonpremixed ﬂame, showing instantaneous iso-surfaces of the mass fraction of H2 at the end of the simulation. The mixing zone is clearly visible, extending around the center of the domain. (b) Employed boundary conditions.

3D Direct Simulation of a Nonpremixed Hydrogen Flame

435

turn-over time τ = 0.76 ms. 125 processors from a parallel cluster HP SC-2 are employed, requiring approximately 12, 500 h of CPU time. In what follows, all results are presented for time t = 0.94τ .

3 Results Since these DNS simulations are 3-D in space, unsteady, and involve nine diﬀerent chemical species beyond the classical ﬂow variables, post-processing issues become essential. A dedicated open-source post-processing MATLABbased library has been developed [7] in our group and has been used for the following analysis. 3.1 Turbulent ﬂame structure We deﬁne a normalized ﬂame index (see [8] and included references for details) as ξ = (∇YF /|∇YF |) · (∇YO /|∇YO |) with ξ −1: purely nonpremixed, ξ 1: purely premixed and −1 ξ 1: partially premixed. YF and YO are the fuel and oxidizer mass fractions, respectively. This index conditioned by Z is shown in Fig. 2a, excluding the regions with Z ≤ 0.05 (pure oxidizer) and Z ≥ 0.95 (pure fuel). A large region in Z-space around Z = Zs is dominated by pure nonpremixed conditions, as expected for this globally nonpremixed conﬁguration. A non-negligible amount of partially-premixed zones are nevertheless visible on the fuel side (around Z 0.81) and, to a lesser extent, on the oxidizer side (around Z 0.13). For these same conditions the peak of conditional ﬂame index lies systematically around 1, showing locally the presence of fully premixed conditions. These two regions, just outside of the main reaction zone, correspond to the burning conditions most strongly disturbed by interaction with the turbulence. Figure 2b shows the maximum values of temperature and heat release conditioned by ξ. As expected, most of the heat

Fig. 2. (a) Maximum (−−), mean (—) and minimum (−.) values of the ﬂame index ξ conditioned by the mixture fraction. The solid vertical line corresponds to Zs . (b) Maximum temperature (—, left scale) and heat release (−−, right scale) conditioned by the ﬂame index.

436

G. Fru et al.

Fig. 3. PDF of Z in the sub-region where the worst (a) and the best (b) agreement is observed. (—): DNS; (−−): β-function and (−.): γ-function.

release takes place in a nonpremixed mode, though the peak of conditional heat release and temperature is found for a value of ξ slightly higher than −1. For ξ > −0.5 a slowly decreasing plateau is observed, followed by a sharp drop when approaching ξ = 1. 3.2 Reconstructing the PDF of mixture fraction Nonpremixed ﬂames are classically modeled using a description based on Z, allowing to decouple mixing and reaction processes [9, 10]. A presumed PDF using either β- or γ-functions is often used [11]. For this analysis the central part of the numerical domain in x-direction (ﬂame zone) is separated in nine identical cubic sub-regions with a side length of 1/3 cm (roughly twice the integral scale of turbulence). In each sub-region the Favre-averaged mean Z $ 2 and variance Z are computed by post-processing the DNS data (considering the y- and z-directions as homogeneous), and are afterwards used to reconstruct the PDF of Z. Figure 3 shows two of these reconstructed PDF together with the exact PDF obtained from the DNS data within the corresponding sub-region. The quantitative agreement is measured via their correlation coeﬃcients (not shown).Without any exception, the correlation associated with the β-function is considerably higher than with the γ-function, as observed for example in Fig. 3b. Nevertheless, for a few cases where the exact PDF exhibits three peaks, both correlations may be poor (Fig. 3a).

4 Conclusions The coupling of an initially homogeneous isotropic turbulence with a nonpremixed hydrogen–air ﬂame has been examined with DNS, using detailed models. The coexistence of diﬀerent burning regimes have been quantiﬁed by using the ﬂame index. Two thirds of the heat release take place in the

3D Direct Simulation of a Nonpremixed Hydrogen Flame

437

nonpremixed mode, as expected for this globally nonpremixed conﬁguration. Typical correlation values above 0.9 are obtained for a β-function PDF reconstruction of Z for 90% of the cases. The γ-function leads to a much poorer approximation.

References 1. L. Vervisch. Proc. Combust. Inst., 28:11–24, (2000). 2. E. Hawkes, R. Sankaran, J.C. Sutherland, and J.H. Chen. Proc. Combust. Inst., 31:1633–1640, (2007). 3. J. Bell, M. Day, J. Grcar, M. Lijewski, J. Driscoll, and S. Filatyev. Proc. Combust. Inst., 31:1299–1307, (2007). 4. R. Hilbert and D. Th´evenin. Combust. Flame., 128(1-2):22–37, (2002). 5. R. Hilbert and D. Th´evenin. Combust. Flame., 138:175–187, (2004). 6. R. Yetter, F. Dryer, and H.J. Rabitz. Combust. Sci. Tech, 79:91–128, (1991). 7. C. Zistl, R. Hilbert, G. Janiga, and D. Th´evenin. Increasing the eﬃciency of post-processing for turbulent reacting ﬂows. Comput. Vis. Sci., (2008) in press. 8. R. Hilbert, F. Tap, H. El-Rabii, and D. Th´evenin. Prog. Energy Combust. Sci., 30:61–117, (2004). 9. D. Veynante and L. Vervisch. Prog. Energy Combust. Sci., 28(3):193–266, (2002). 10. T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. Edwards Publishing, PA, (2006). 11. N. Peters. Turbulent Combustion. Cambridge University Press, (2000).

Part VI

Environmental and Multiphase Flows

Large Eddy Simulation of Pollen Dispersion in the Atmosphere Marcelo Chamecki1 , Charles Meneveau2 , and Marc B. Parlange3 1

2 3

Johns Hopkins University, Baltimore, USA, [email protected] Present aﬃliation: The Pennsylvania State University, University Park, USA Johns Hopkins University, Baltimore, USA, [email protected] ´ Ecole Polytechnique F´ed´erale de Lausanne EPFL, Switzerland, [email protected]

1 Introduction Pollen dispersion by the wind is a subject of interest in many ﬁelds which has gained renewed attention in the recent years due to the increasing level of landscape fragmentation and the introduction of genetically modiﬁed crops. Accurate quantitative estimates of the three-dimensional plume of airborne pollen grains and consequent ground deposition from an emitting ﬁeld are usually required. A large number of ﬁeld experiments has been conducted to investigate the problem. However, the results are expected to depend upon several factors, such as aerodynamic properties of the pollen grains, geometric properties of the ﬁeld, and local vegetation and atmospheric conditions, limiting the validity of experimental results to speciﬁc ﬁeld conditions. Up to date, simulations of pollen dispersal were restricted to the use of Lagrangian stochastic models [6] and two-dimensional RANS models [4]. In this work we describe tools and frameworks to simulate pollen dispersal using large eddy simulation (LES).

2 Model description In this work the wind ﬁeld is represented by the incompressible ﬁltered threedimensional momentum equations in rotational form. Neutral atmospheric stability is assumed. The subgrid scale (SGS) momentum ﬂuxes are modeled using the dynamic Smagorinsky model [5], following the Lagrangian scale-dependent implementation [1]. Boundary conditions for the momentum equations are imposed using a wall layer model based on test-ﬁltered ﬁelds [1]. A continuous pollen concentration ﬁeld C(x; t) is deﬁned and represented on an Eulerian grid. The evolution of the concentration ﬁeld is obtained from V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 64, c Springer Science+Business Media B.V. 2010

441

442

M. Chamecki et al.

the conservation of pollen mass, represented by the ﬁltered advection–diﬀusion equation with an additional term representing the gravitational settling: ∂C = −∇ · π C + Qsrc , + ( u − ws e3 ) · ∇C ∂t

(1)

where a tilde represents a ﬁltered ﬁeld, ws is the settling velocity of pollen grains (assumed to be constant and equal to the terminal velocity in a still ﬂuid), Qsrc is a pollen source term, and π C is the subgrid scale pollen ﬂux modeled as: νsgs πC ∇C. (2) model = − Scsgs Here νsgs is the SGS viscosity obtained using the Lagrangian scale-dependent dynamic model and Scsgs is a prescribed constant SGS Schmidt number (the value adopted here is Scsgs = 0.4). In the proposed approach the pollen emission and ground deposition are modeled through the lower boundary condition for pollen concentration. For this purpose, the equilibrium proﬁle for particles settling in the atmospheric boundary layer [3] is used as a wall model: −α C Φ z − do Φ = +1 − . (3) z − d Cr C r ws C r ws r o Here an overline represents an (ensemble) averaged ﬁeld, C r is the pollen concentration at the reference level zr , do is the displacement height, and Φ is T ws the pollen surface ﬂux. The parameter α = Scκu represents the ratio between ∗ settling velocity and a turbulent diﬀusive velocity at the boundary (u∗ is the friction velocity). Note that in principle the turbulent Schmidt number ScT does not have to be equal to the SGS Schmidt number Scsgs . In fact, we use ScT = 0.95, which yields the generally accepted form of Monin–Obukhov similarity function for scalars if the limit of ws → 0 is adopted. Equation (3) can be rearranged and expressed in terms of resolved quantities to yield the required expression for the SGS pollen ﬂux at the surface (Φsgs ): −α z1 − do (x, y) C(x, y, z1 ) − Cr zr (x, y) − do (x, y) Φsgs (x, y) = −ws . (4) −α z1 − do (x, y) 1− zr (x, y) − do (x, y) r at The SGS pollen ﬂux can then be determined once the reference value C zr is speciﬁed. The proposed approach is to model the pollen source by using r above the source ﬁeld. Ground deposition experimental data to specify C r = 0 at zr = zo , where zo is the surface is modeled by simply specifying C roughness. In this case, equation (4) reduces to z1α y, z1 ) Φdep. . (5) (x, y) = −w C(x, s sgs z1α − zoα

Large Eddy Simulation of Pollen Dispersion in the Atmosphere

443

3 Numerical discretization A common approach for LES of the atmospheric boundary layer is to combine pseudospectral discretization in the horizontal directions and second-order centered ﬁnite-diﬀerences in the vertical non-homogeneous direction [7]. Here this approach is used to discretize the velocity ﬁeld. However, pollen sources are fairly localized in space, generating strongly nonhomogeneous concentration ﬁelds. For this reason, the pseudospectral approach is not a good option for the pollen concentration ﬁeld. It can generate unphysical behavior such as spatial oscillations and negative values of pollen concentration. Therefore, we discretize the pollen conservation equation using a ﬁnite-volume approach. The advection term is discretized using the bounded third-order upwind interpolation scheme SMART. The usual second order centered scheme is used for the turbulent diﬀusion term. Coupling the two discretization schemes requires interpolation of the velocity ﬁeld from the spectral nodes to the ﬁnite-volume surfaces. Unique to this problem is the fact that on the original grid the divergence-free condition is enforced based on the nonlocal spectral derivatives while on the new target grid the interpolated velocity is required to satisfy the same constraint in a local discretization. We use a new interpolation scheme that produces a locally divergence-free velocity ﬁeld by construction. The main idea is to use the spectral derivatives (instead of the spectral velocity ﬁeld) to obtain the velocity ﬁeld on the required grid. The derivatives are integrated using a scheme consistent with the ﬁnite-volume discretization. The method takes advantage of speciﬁc characteristics of both grids, yielding a simple and eﬃcient algorithm [2].

4 Validation The numerical model is validated against the classical experimental data of point source release of glass beads reported by Walker (1965). Glass beads were released at a constant rate of Q = 0.11875 g s−1 during 60 min from a height h = 15 m above the ground. The average particle diameter was d = 107 µm and the settling velocity in still ﬂuid ws = 0.58 m s−1. Experimental conditions correspond to near neutral atmospheric stability with u∗ = 0.44 m s−1 . Three simulations are carried out with progressive mesh reﬁnement. The main grid (intermediate resolution) has 64 × 32 × 48 points with horizontal grid spacing dx = dy = 10 m and vertical grid spacing dz = 2 m. A high resolution grid with half of the grid spacing and a coarse resolution grid with twice the grid spacing are also used. = An instantaneous snapshot of the three-dimensional isosurfaces for C = 0.5 g m−3 is shown in Fig. 1. The high concentration 1, 500 g m−3 and C plume clearly illustrates the eﬀect of the large settling velocity for the glass

444

M. Chamecki et al.

AWID/Qt [m-1]

Fig. 1. Particle concentration plume for the point-source release. Isosurface for = 0.5 g m−3 (lighter surface). Also shown = 1, 500 g m−3 (darker surface) and C C the ﬁnal ground deposition (plane is oﬀset for better visualization). 0.01 0.001 0.0001 1e-05 1e-06 1e-07 1e-08 10

100 r [m]

1000

Fig. 2. Normalized AWID for the point-source release. Experimental data (circles) and LES results: high (solid line), intermediate (dashed line) and coarse (dotted line) resolutions.

beads, as the plume has a strong inclination towards the ground surface. However, as can be seen in both plumes, strong enough turbulent ﬂuctuations carry particles farther downwind. The two-dimensional time-integrated ground deposition is also shown, the region of larger deposition corresponding to the region where the high concentration plume touches the ground. Experimental data of arcwise integrated deposition (AWID) is compared to the simulation in Fig. 2. The general agreement is very good for the intermediate and high resolution simulations, but quite poor for the coarser one. The peak in deposition is correctly predicted (both its position and intensity) and the width is also well represented. Very good agreement is obtained for the near ﬁeld in the high resolution case. It is clear that even the coarser resolution is capable of predicting the far ﬁeld deposition with good accuracy. The cross-arc standard deviation of the deposition is fairly well reproduced by the simulations, which tend to underestimate the spread in the far ﬁeld (not shown).

Large Eddy Simulation of Pollen Dispersion in the Atmosphere

445

5 Pollen dispersion from a ragweed ﬁeld The numerical code is applied to study pollen emission and deposition from a natural ragweed ﬁeld. Experimental data was obtained during the months of September and October of 2006 in a fairly ﬂat ﬁeld near Upper Marlboro, MD. The ﬁeld measured 245 × 245 m and was occupied homogeneously by common ragweed (average height h = 0.69 m with an estimated density of 89 plants/m2 ). The ﬁeld experiment consisted of simultaneous measurements of meteorological and turbulent quantities and pollen release, airborne concentration and ground deposition. The ﬁeld was mowed, leaving only a north– south stripe ragweed patch of 48 × 245 m. A 6 m tower with meteorologic instrumentation was placed upwind of the ragweed patch as shown in the sketch presented in Fig. 3. Two sonic anemometers were used to measure turbulent ﬂuxes of momentum and heat. Measurements of airborne pollen concentrations were performed using six Rotorod rotating impact samplers. Five samplers were placed at the downwind edge of the ﬁeld to characterize the pollen source: one at the southern station, three at the center station and one at the northern station. The last sampler was placed upwind to characterize pollen background concentration. Finally two lines of greased microscope slides were positioned downwind to measure ground pollen deposition as a function of distance from the ﬁeld. The simulation domain chosen is larger than the experimental site (see Fig. 3). A grid with 100 × 50 × 50 points is used resulting in a resolution dx × dy × dz = 12 × 12 × 3 m). Ragweed pollen has a small settling velocity and is dispersed far from the source. If the main purpose is to simulate relevant length scales for ragweed pollen dispersion, coarse resolutions are unavoidable. The simulation spans one diurnal pollen cycle (i.e. from 5 am to 8 pm, when measured concentrations where signiﬁcantly diﬀerent from zero) with a timestep dt = 0.1 s. An unsteady mean pressure gradient drives the simulation and is adjusted to reproduce the diurnal variation of the friction velocity. Measured pollen concentration above the canopy is used to specify Real Field

Simulation Domain N

θ = – 30° Pollen Sampler

W

E

240m

Mean Wind

S

64m

Deposition Lines

48m

y

52m x Met Tower

1200m

Fig. 3. Sketch of the ragweed ﬁeld simulations.

600m

446

M. Chamecki et al.

a

b

D [grains / m2]

200000 150000 100000 50000 0

0

10

20

30 40 x [m]

50

0

10

20

30

40

50

60

x [m]

Fig. 4. Ground deposition downwind of the ﬁeld along (a) north line and (b) south line. Experimental data (circles) and LES simulation (line).

Fig. 5. Pollen deposition form ragweed ﬁeld.

the pollen source. Surface roughness and displacement height were adjusted over the ragweed patch to represent the drag exerted by the canopy. There is fairly good agreement between ground deposition measurements and simulation results (Fig. 4). The pollen deposition inside the entire computational domain is shown in Fig. 5. Eﬀects of the mean wind direction and heterogeneity of the pollen source have a strong eﬀect on the deposition pattern. From the total emission ﬂuxes it is possible to estimate the total number of pollen grains emitted from the ﬁeld during the entire day. The result obtained is 62.61 billion grains, which gives an average emission of approximately 60 thousand grains per plant. However, 37.79 billion grains (about 60%) are deposited within the ragweed patch and only the remaining 24.82 billion grains are available for cross-pollination. Time series of instantaneous deposition ﬂuxes Φdep sgs were stored at the ten points indicated in Fig. 5 to study the occurrence of sporadic (but signiﬁcant) deposition events far from the source ﬁeld. Statistics were calculated using a running window of 1 h. The time evolution of the ﬂuctuation of the deposition ﬂuxes at points 1 and 8 is shown in Fig. 6 for part of the day. Far from the source ﬁeld, the instantaneous deposition is clearly more intermittent and

Large Eddy Simulation of Pollen Dispersion in the Atmosphere 200 150 100 50 0 –50 –100 –150 –200

b Φ–<Φ> [grains/m2s]

Φ–<Φ> [grains/m2s]

a

447

9

10 11 time [hours EDT]

0.2 0.1 0 –0.1 –0.2

12

9

10 11 time [hours EDT]

12

Fig. 6. Time series of deposition ﬂux ﬂuctuations for points (a) 1 and (b) 8.

a

5

b

1000 100

3

ΚΦ

σΦ /<Φ>

4

2

10

1 0

7

9 11 13 15 17 19 time [hours EDT]

1

0

100 200 300 400 500 x [m]

Fig. 7. (a) Time evolution of ﬂuctuation intensity at point 1 inside the ragweed patch (dotted line) and at point 8 outside (solid line). (b) Average kurtosis as a function of distance form the trailing edge of the ﬁeld (solid line) and Gaussian value for reference (dashed line).

skewed. The time evolution of the intensity of the ﬂux ﬂuctuations (i.e. the rms normalized by the mean deposition ﬂux) is shown for the same points in Fig. 7a. The higher value far from the ﬁeld conﬁrms the relative importance of the isolated strong events. Fig. 7b shows the average kurtosis for the ten points selected. The behavior follows the Gaussian value (3.0) up to 250 m from the edge and then the kurtosis display a fast increase, indicating that infrequent high peaks in deposition become more relevant.

6 Conclusions A new approach to simulate pollen dispersal using large eddy simulation has been developed. Validation against classical data of a point source release in the atmospheric boundary layer conﬁrms the model capability in predicting ground deposition. The approach is applied to study pollen dispersal from a natural ragweed ﬁeld. Measured pollen concentration above the canopy and the use of the theoretical equilibrium proﬁle allow a simple and consistent approach to specify the pollen sources and sinks using the lower boundary condition. Fairly good agreement with measured deposition is obtained even for this complex case. The model provides an estimate of the total pollen

448

M. Chamecki et al.

emission and its deposition partition inside the ﬁeld and downwind. The large amount of pollen deposited inside the ragweed ﬁeld (about 60%) is a good strategy to ensure successful pollination, since there is a large probability that pollen grains in this region will actually end up on female ﬂowers. The pollen remaining airborne at the edge of the ﬁeld (40%) is critical to ensure cross-pollination and increases biodiversity. Analysis of the deposition ﬂux also indicates the importance of sporadic intense events far from the source ﬁeld.

References 1. Bou-Zeid E, Meneveau C, Parlange MB (2005) Phys Fluids 17:025105 2. Chamecki M, Meneveau C, Parlange MB (2008) Bound-Layer Meteor 128: 473–484 3. Chamecki M, van Hout R, Meneveau C, Parlange MB (2007) Bound-Layer Meteor 125:25–38 4. Dupont S, Brunet Y, Jarosz N (2006) Agric For Meteor 141:82–104 5. Germano M, Piomelli U, Moin P, Cabot WH (1991) Phys Fluids A 3:1760–1765 6. Jarosz N, Loubet B, Huber L (2004) Atmos Environ 38:5555–5566 7. Moeng CH (1984) J Atmos Sci 41:2052–2062

Internal Wave Breaking in Stratiﬁed Flows Past Obstacles Sergey N. Yakovenko, T. Glyn Thomas, and Ian P. Castro Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, Highﬁeld, Southampton SO17 1BJ, UK, [email protected]; [email protected]; [email protected]

1 Introduction Turbulent patches occurring in environmental ﬂows can arise from breaking internal waves generated topographically. The convective overturning of waves leads to shear instability and then to turbulence which develops into the fully mixed region in the place of initial wave breaking. The objective of the present studies is to explore the steady internal wave breaking observed in stably stratiﬁed ﬂows past obstacles [1]. We use a numerical approach based on well-resolved Navier–Stokes DNS/LES methods using parallel multi-block architecture [2] with the Boussinesq approximation and sponge layer treatment to avoid wave reﬂection from upstream/downstream boundaries.

2 Results As a test of the code, which had to be modiﬁed by insertion of the density equation, good agreement was obtained with data from DNS/experiments of other authors for the velocity and scalar ﬁeld characteristics in fully-developed turbulent channel ﬂows with zero, passive and active scalars, for prescribed values of either the scalar or its ﬂux at boundaries. In particular, a ﬂow with active scalar (temperature T ), i.e. combined forced and natural convection between two vertical parallel plates kept at diﬀerent constant temperatures Tc (cold) and Th (hot), was considered. The results for velocity and scalar ﬁelds (Fig. 1) show clear eﬀects of stratiﬁcation parameterised by the Grashof number Gr on distributions at horizontal channel cross-sections near both the heated and cooled walls, and agree closely with the previous DNS results [3]. To generate breaking internal waves, a slightly transparent (emulated) obstacle was inserted in the ﬂow, of a shape and size which, given a linear upstream density variation and constant inﬂow velocity U , would be expected to produce such waves [1] at suﬃciently low values of the Froude number Fh V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 65, c Springer Science+Business Media B.V. 2010

449

450

S.N. Yakovenko et al. 20 40 Gr = 0 ---> 30

f+

U+

present DNS, case 2 Kasagi, Nishimura (1997), case 2b present DNS, case 2b

<--- Gr = 960,000

15

Kasagi, Iida (1999), case 2

10

Gr = 0 ---> 20

<--- Gr = 960, 000

Kuroda, Kasagi (1990), cases 1 and 2 5

present DNS, cases 1 and 2

10

Kasagi, Nishimura (1997), case 2b present DNS, case 2b 0

100

+

z

200

300

0 100

z+

200

300

Fig. 1. Mean velocity and scalar proﬁles in plane-channel ﬂow, Reτ = 150.

based on the obstacle height h. With increase of the spanwise aspect ratio W/L of the obstacle, the internal wave breaking occurs at 0.2 < Fh < 0.8 [1]. This phenomenon can be captured in numerical simulations by the modiﬁed code and is rather sensitive to various geometrical and physical parameters, e.g. the Reynolds number Re = U h/ν, grid resolution, boundary conditions and obstacle parameterisation accuracy. After numerous runs with diﬀerent parameters varied, it turned out that at Re > 2, 000 there is turbulence generation in the breaking zone similar to that in experiments [1, 4] and DNS [5]. We present here (Figs. 2–7) data for the 2D case (since the breaking zone intensity increases with W/L) at Re = 4, 000 and the Froude number Fh = 0.6; the hill length L/h = 3.56 is tuned to be two times larger than in [1] and approximately the same as in [5]. All boundaries have no-ﬂux conditions for scalar except the upstream one. Initial conditions correspond to ’impulsive start’, i.e. sudden obstacle insertion into the ﬂow with the velocity and scalar ﬁelds assigned at t = 0 to be the same as at inﬂow. To enhance the modiﬁed code eﬃciency, a special sponge-layer procedure with extra forcing terms in both velocity and scalar equations has been incorporated to prevent reﬂection from inﬂow/outﬂow boundaries. This allows us to reach longer times with relatively short domains, as well as to reﬁne the grid and produce accurate results. Initially, the computation domain was extended in the horizontal direction from x/h = 0 up to x/h = 200 with the obstacle located in the middle (at x/h = 100), but the return time of the fastest reﬂected waves was tr U/h = 39 only. The sponge layers, with carefully selected widths/locations and window function shapes/coeﬃcients, allowed us to reduce the domain size by a factor of two (over the range x/h = 50 to x/h = 150) and use run times up to tU/h = 250 without signiﬁcant differences with corresponding long-domain computations. (These comparisons used coarse grids and were at relatively low Reynolds number.) The ﬁnest applied grid comprised about 170 million nodes (2, 560 × 256 × 256) and had a resolution of Δx = Δy = Δz = h/25.6.

Internal Wave Breaking in Stratiﬁed Flows Past Obstacles 5

4

4

3

3 z/h

z/h

5

451

2

2

1

1

0 100

102

104

106

108

0 100

110

102

104

x/h

106

108

110

106

108

110

x/h 5

4

4

3

3

z/h

z/h

5

2

2

1

1

0 100

102

104

106

108

0 100

110

102

104 x/h

x/h

Fig. 2. Streamlines of velocities averaged in the time interval 5h/U , and along y. 10

10

t = 30U/h t = 50U/h t = 70U/h t = 100U/h t = 190U/h

8

t = 30U/h t = 50U/h t = 70U/h t = 100U/h t = 190U/h

8 6

z/h

z/h

6 4

4

2

2

0

0

1 /U

2

3

0

0

5 <ρ>

10

Fig. 3. Proﬁles of velocity and density (at x = 103h) averaged as in Fig. 2.

The time evolution visualised by pathlines of ﬂuid particles shows the well resolved recirculation arising from internal wave overturning and breaking (Figs. 2 and 4) which replicates the results of previous studies. This breaking region, generating strong turbulence activity, destroys the recirculation structure as seen from the averaged data, but maintains a quasi-steady fully mixed zone which moves slowly upwards together with the corresponding peaks of turbulent stresses and t.k.e. production terms (Fig. 7). The maintenance of the vertical streamlines in the mixed zones is conditioned by the fast growth of instability for negative slopes of streamlines (and density contours – heavy

Internal Wave Breaking in Stratiﬁed Flows Past Obstacles

References 1. 2. 3. 4. 5.

Castro IP, Snyder WH (1993) J Fluid Mech 255:195–211 Thomas TG, Williams JJR (1997) J Wind Eng Ind Aero 67&68:155–167 Kasagi N, Nishimura M (1997) Int J Heat Fluid Flow 18:88–99 Castro IP, Snyder WH, Marsh GL (1983) J Fluid Mech 135:261–282 Gheusi F, Stein J, Eiﬀ OF (2000) J Fluid Mech 410:67–99

455

452

S.N. Yakovenko et al. t = 30h/u

t = 40h/u 5

4

4

3

3 z/h

z/ h

5

2

2

1

1

0 100

102

104

106

108

0 100

110

102

104

106

108

110

108

110

x/h

x/h

t = 50h/u

t = 190h/u 5

4

4

3

3 z/h

z/h

5

2

2

1

1

0 100

102

104

106

108

110

0 100

102

104

106 x/h

x/h

Fig. 4. Instantaneous ﬂuid particle pathlines near 2D bump in the central plane. 1.0 u /U

/U = 0.295

0.0

50

60

70

80

tU/h

Fig. 5. Time history of velocity values at x/h = 107.5, y/h = 0.0, z/h = 2.0.

ﬂuid overlying lighter ﬂuid) and by tendency of the resulting turbulent ﬂuxes to reduce them due to turbulent diﬀusion. Both time and spatial spectra produced from the computation data in some points of the mixed zones contain clear parts of the ‘−5/3’ inertial ranges (Fig. 6). This is in contrast with the data of [5], which showed significantly greater powers – −2.4 to −3.5. The obvious reason is the enhanced viscous dissipation inherent to the case of the low Reynolds number Re = 200 studied in [5] rather than a slope of −3 corresponding to the buoyancy spectrum. In the mixed zone the density proﬁle is ﬂat (Fig. 3) so we do not expect stably stratiﬁed behaviour of the turbulence in this particular region (having

Internal Wave Breaking in Stratiﬁed Flows Past Obstacles 100

10−1 −5/3

Eu = 0.014(fh/U)

K

Eu

10−1 10−2

computed spectrum K=0.7n-5/3 K=1.0n-5/3

100

fh /U = 3.8

453

10−2

10−3

10−3

10−4 10−1

100 fh/U

10−4

101

100

101

102

0

10

fh/U = 3.8

100

10−1 −5/3

y Ed

Ed

10−1 10−2

computed spectrum Eyd = 0.001 n-0.37 Eyd = 0.02 n-5/3 Eyd = 0.01 n-5/3

Ed = 0.014(fh/U)

10−3

10−2 10−3

10−4 10−1

100 fh/U

10−4

101

100

101 n

102

Fig. 6. Longitudinal velocity and density (time) spectra at x/h = 107.5, y/h = 0.0, z/h = 2.0 averaged over four intervals within the period 47.6 < tU/h < 87.5 (left); t.k.e. and density (spatial) spectra in the cyclic y-direction at tU/h = 42.5, averaged for the box 102.8 < x/h < 103.3, 1.4 < z/h < 1.9 (right) where the wavenumber n = Ly /λ [5] is the number of periods over the domain width Ly . 4

4

t = 30U/h t = 50U/h t = 70U/h t = 100U/h t = 190U/h

3 z/h

z/h

3

t = 30U/h t = 50U/h t = 70U/h t = 100U/h t = 190U/h

2

1

2

0

1

0.2

1/2

0

/U 4

3

3

< u’w’>/U

2

0.01

z/h

z/h

4

2

1

2

0

0.01 Ph/U

3

0.02

1

−0.01

0 3 Gh/U

0.01

Fig. 7. Proﬁles of Reynolds stresses, t.k.e. mean-shear (P ) and buoyancy (G) production terms (at x = 103h and various times) averaged as in Fig. 2.

454

S.N. Yakovenko et al.

high turbulence activity). Moreover, the velocity spectra show no unrealistic features (such as peaks at the end of the inertial region which would suggest the need for a subgrid-scale model). Thus, at least for these ﬁne-grid computations at the moderate Reynolds number Re = 4, 000, the use of the DNS approach is reasonable and suﬃcient. However, one can see a reduced-slope section in the spatial spectrum of density and this is almost certainly a consequence of under-resolution of the scalar ﬁeld, because of the high Sc number. Note that the cut-oﬀ frequency (Fig. 6) representing the smallest scale resolved on the grid is fc =< u > /(2Δx) = 3.8U/h with < u >= 0.295U estimated from velocity series (Fig. 5).

3 Conclusions DNS studies on generation, development and breaking of internal waves past the cosine-shaped obstacle have been performed. Turbulence generation takes place at Re > 2, 000 within the wave breaking regions, and the recirculation zone forming after wave overturning/breaking is destroyed by turbulence activity which maintains constant values of mean density and (near-zero) longitudinal velocity. The quasi-steady mixed region moves gradually upwards together with the peaks of Reynolds stresses and energy production terms. Both velocity spatial and time spectra demonstrate the inertial range interval typical for neutral ﬂows (without any peaks requiring subgrid-scale models) whereas the density time spectrum indicates under-resolution. Further studies of interest are on the turbulence kinetic energy balance (dissipation), LES model eﬀects at higher Re, grid independence and spatial structure in the breaking regions. Preliminary studies of 3D visualisation by vorticity invariants performed on relatively coarse grids did not reveal clear structures for this high-Reynolds-number case.

Acknowledgements This work was supported by DSTL and EPSRC (EP/C008561/1), making use of both the Iridis2 computing resources at University of Southampton and the facilities of HPCx, the UK’s national high-performance computing service, which is provided by EPCC at the University of Edinburgh and by STFC Daresbury Laboratory, and funded by the Department for Innovation, Universities and Skills through EPSRC’s High End Computing Programme.

DNS of a Gravity Current Propagating over a Free-Slip Boundary Alberto Scotti Department of Marine Sciences, UNC, Chapel Hill, NC 27599-3300, USA, [email protected]

1 Introduction The frontal structure of gravity currents depends on the nature of the boundary condition applied to the surface along which the current propagates. Experiments and numerical calculations have focused on the no-slip case, such as when heavy ﬂuid spreads underneath a lighter ﬂuid over a solid surface. Soon after the commencement of the ﬂow, a three-dimensional instability develops (lobe-and-cleft instability, see, e.g. [3]) along the foot of the front. This instability, which appears as lobes protruding from the frontal region and racing each other, is due to the unstable stratiﬁcation that develops as the foremost point of the gravity current (the nose) is lifted up from the no-slip boundary. Under the lifted nose lighter ﬂuid remains trapped and becomes convectively unstable. Along a free-slip boundary the tip of the gravity current remains attached to the boundary. Under inviscid conditions, Benjamin [1] derived an analytical solution which predicts that the stagnation streamline (in a frame of reference stationary relative to the front) issuing from the front slants upward at a 600 angle, and argued that the theoretical solution is subject to Kelvin–Helmholtz instabilities. This view has become the standard explanation for the onset of turbulent entrainment in the lee of free-slip gravity currents. Alas, it is not easy to perform free-slip experiments at laboratory scales, since surface tension and impurities turn a free surface into a no-slip surface unless the E¨otv¨ os number is large [2]. Recent experiments have circumvented this problem by releasing ﬂuid of density (ρ1 + ρ2 )/2 along the interface separating ﬂuids of equal depth and density ρ1 and ρ2 [6]. The resulting intrusion is equivalent to two gravity currents, one being the mirror image of the other. In this case, the proﬁle of the head agrees very well with Benjamin’s theoretical prediction from the tip of the front to about 1.5 channel height downstream, well beyond the point where accurate numerical two-dimensional simulations [4] show Kelvin–Helmholtz billows start developing. Further, Lowe et al. [6] report that the head of the intrusion is ﬁlled with stagnant ﬂow (when observed in a frame of reference moving with the front), V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 66, c Springer Science+Business Media B.V. 2010

457

458

A. Scotti

and that most of the ﬂuid approaching the frontal region from behind does not reach the head. In this note, we show that the spanwise direction (absent in two-dimensional simulations) is necessary to recover the experimental results, and thus prove that the ﬂow is inherently three-dimensional.

2 Approach Without lack of generality, we consider a ﬂuid of density ρ0 propagating underneath a ﬂuid of density ρ1 < ρ0 . The Early experiments of Keulegan [5] demonstrated that lock-exchange generated gravity currents approach a statistically steady state after traveling 50 channel heights or more. This makes three-dimensional simulations problematic, since very long channels would be needed. For this reason, we study the problem in a frame of reference moving with the speed of the front, limiting the streamwise extent of the domain to ﬁve channel heights (one ahead of the front and four behind, see Fig. 1). The challenge of this setup is that the vertical boundaries ahead and behind the front are open, and appropriate boundary conditions have to be imposed. Ahead of the front a uniform velocity equal to the velocity at which the front propagates in a ﬁxed frame of reference has to be set; behind the front we need to both supply heavy ﬂuid and remove mixed ﬂuid at a rate set by the entrainment taking place within the domain. This is accomplished by a control

m

sid

e

Downstream side

z1

Ou

tle

t

Up

str

ea

z2

0

b

1

Inl

et

U1 h3

H

h4 H

LY

U4

0 LX

Fig. 1. Sketch of the computational domain and notation used in the paper. The shaded strip on the downstream side separates the inlet where heavy ﬂuid is injected from the outlet where mixed ﬂuid is allowed to escape. Conditions are adjusted so that the front remains one channel height away from the upstream boundary (except during the sensitivity test in Section 3.4). The origin of the frame of reference is set at the front. h3 (the thickness of the mixing layer) and h4 (the thickness of the stem) are deﬁned after Britter and Simpson [2], namely b(h4 ) = 0.95binlet and b(h3 + h4 ) = 0.05binlet .

DNS of a Gravity Current Propagating over a Free-Slip Boundary

459

algorithm that adjusts the boundary conditions in real time to maintain the system stationary. After an initial transient, the ﬂow selects its own Froude √ number U1 / g H and entrainment ratio U4 /U1 (for deﬁnitions see below), which agree well with available experimental data. 2.1 Implementation We solve the Navier–Stokes equations in the Boussinesq approximation for a ﬂuid of variable buoyancy Du = −∇p − bk + ν∇2 u, Dt Db = κ∇2 b, Dt ∇ · u = 0,

(1)

where b = g(ρ−ρ0 )/ρ0 is the buoyancy anomaly relative to a reference density ρ0 , The equations are solved in a channel of depth H, length Lx and width Ly (Fig. 1). For details of the implementation see Scotti [7]. Of importance here is the fact that free-slip conditions are applied along the upper and lower walls. are made nondimensional with the buoyancy velocity Uref = √ The equations g H, where g ≡ (ρ0 − ρ1 )/ρ) is the reduced gravity of the ﬂuid and H, the height of the channel. Expressed in these units, the Reynolds and Prandtl number are Re = Uref H/ν, Pr = Re−1 Uref H/κ.

(2) (3)

Pr ranges from 7 to 700 in water depending on whether the stratifying agent is temperature or salinity.1 In our experiments, we set P r ≤ 10 and considered values of Re ranging from 103 to 104 . This range overlaps with the values reported in Lowe et al. [6] and Britter and Simpson [2], while Pr is smaller, since salt is commonly used in the lab as a stratifying agent. Unfortunately, numerical constraint limit the maximum value of Pr attainable. The same resolution (dx × dy × dz = 0.0049 × 0.012 × 0.0039 H) is employed in all runs. Most runs were also repeated in two-dimensions (with identical streamwise and vertical resolution).

3 Results The analysis shows striking diﬀerences between two- and three-dimensional simulations. Figure 2 shows simulated shadowgraphs (essentially, shaded plots 1

In the latter case, it is customary to refer to the diﬀusivity ratios as the Schmidt number.

460

A. Scotti

Fig. 2. Shadowgraphs of three-dimensional (left column) and two-dimensional (right column) gravity currents with diﬀerent values of Reynolds and Prandtl number. In each plot, the dashed line shows the location of the inviscid stagnation streamline.

of spanwise-averaged ∇2 b) of gravity currents with diﬀerent values of Reynolds and Prandtl number. The three-dimensional currents follow the theoretical prediction one or more channel heights behind the location of the front (left side), with little evidence of Kelvin–Helmholtz roll-ups forming in the wake. This is consistent with the experimental ﬁnding of Lowe et al. [6]. The twodimensional simulations are characterized by Kelvin–Helmholtz roll-ups starting 0.5 height downstream of the tip of the current, similarly to what H¨ artel et al. [4] ﬁnd. Thus, two-dimensional and three-dimensional simulations show qualitative as well as quantitative diﬀerences. Entrainment is substantially reduced in these three-dimensional experiments (Fig. 3) relative to laboratory experiments of stationary gravity current [2] which show the characteristic billows. Interestingly, it is possible to excite Kelvin–Helmholtz billows in the three-dimensional simulations if turbulence is added to the medium in which the current propagates (Fig. 4). In the present set up, this is achieved by adding noise to the upstream boundary conditions. In the latter case, entrainment falls in line with the experimental values (solid squares in Fig. 3). The amount of turbulence needed to trigger Kelvin–Helmholtz billows was found to be small,and decrease with increasing Reynolds number. This may explain why the experiments of Britter and Simpson [2] found billows (the reported value of rms velocity ﬂuctuations in the arresting stream was reported to be 0.5% of the mean velocity, suﬃcient, based on our calculations, to trigger the billows). On the other hand, the currents in Lowe et al. [6] propagate in still water and thus do not develop billows. The mean velocity within the head diﬀers between the billow and nobillow case. Without billows, the ﬂow in the head is virtually quiescent (when see in a frame of reference moving with the front). The ﬂuid approaching

DNS of a Gravity Current Propagating over a Free-Slip Boundary

461

0.35 Britter and Simpson (1978) 2D 3D with billows 3D, no billows

0.3

U4 /U1

0.25 0.2 0.15 0.1 0.05 0

0

0.05

0.1

0.15

0.2

0.25

h4 / H

Fig. 3. Bulk entrainment properties of the ﬂows simulated in our experiments vs. experimental results. Refer to Fig. 1 for a deﬁnition of U1 , U4 , h3 and h4 . All numerical experiments reported in the plot have P r = 10, except for the data surrounded with a gray circle, for which P r = 2. Solid diamonds show results from three-dimensional experiments with low-level of entrainment associated with the absence of well deﬁned Kelvin–Helmholtz billows. Solid squares are from experiments in which the addition of upstream turbulence triggers the development of Kelvin– Helmholtz billows, resulting in higher level of entrainment.

Fig. 4. Currents without upstream turbulence (left panel) and with upstream turbulence (right) panel. Billows (arrows) are present when turbulence is added. In birth cases, P r = 10, Re = 6 × 103 .

from behind mixes behind the head (two channel heights or more behind the front). With billows, only the very tip of the front is quiescent, and ﬂuid reaches well into the head, where a signiﬁcant portion of mixing occurs. This is consistent with the structure reported by Britter and Simpson [2][ﬁgure 10], and also with the results from two-dimensional simulations. Analysis of the vorticity ﬁeld in the three-dimensional case shows that in the absence of turbulence, streamwise vortices appear slightly downstream of the tip of the front on the light-ﬂuid side, and intensify downstream. Analysis of the distribution of momentum across the curved streamlines shows a centrifugally unstable region located where the vortices initially appear. Even though the unstable region is conﬁned to a narrow region in the proximity of the front, the shear ﬁeld further downstream provides positive vortex stretching along

462

A. Scotti

the streamwise direction, allowing streamwise vorticity to grow. We speculate that these vortices act as “stiﬀeners” for the interface, restricting the growth of shear billows in a manner similar to the stabilizing action of surface tension [7]. Such vortices are obviously absent in two-dimensional simulations. Also, we have veriﬁed that the addition of even modest amount of turbulence upstream impairs the development of these streamwise vortices, allowing K-H billows to develop unimpeded. In the latter case, the large-scale ﬂow follows the familiar two-dimensional script until the billows succumb to secondary instabilities. To summarize we have shown that even in the case of free-slip conditions, the ﬂow within the frontal region is inherently three-dimensional, due to centrifugal instabilities along the curved streamlines of the light ﬂuid ﬂowing over the head. This instability in turn exerts a strong control on the development of shear instabilities and entrainment in the wake of the current. To make further progress, it is necessary to develop a theoretical framework to study these instabilities, ﬁrst in isolation, and then as they aﬀect each other.

Acknowledgements This work was supported by NSF, grants 000-83-976 and OCE-0726475 and ONR, grant N00014-05-1-036.

References 1. T. B. Benjamin. Gravity currents and related phenomena. J. Fluid Mech., 31, 1968. 2. R. E. Britter and J. E. Simpson. Experiments on the dynamics of a gravity current head. J. Fluid Mech., 88:223–240, 1978. 3. C. H¨ artel, F. Carlsson, and M. Thunblom. Analysis and direct numerical simulation of the ﬂow at a gravity-current head. Part 2. The lobe-and-cleft instability. J. Fluid Mech., 418:213–229, 2000. 4. C. H¨ artel, E. Meiburg, and F. Necker. Analysis and direct numerical simulation of the ﬂow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries. J. Fluid Mech., 418:189–212, 2000. 5. G. H. Keulegan. Twelfth progress report on model laws for density currents. The motion of saline fronts in still water. Technical Report 5831, Nat. Bur. Standards, 1958. 6. R. J. Lowe, P. F. Linden, and J. W. Rottman. A laboratory study of the velocity structure in an intrusive gravity current. J. Fluid Mech., 456:33–48, 2002. 7. A. Scotti. A numerical study of gravity currents propagating on a free-slip boundary. Theor. Comp. Fluid Dyn., 22:383–402, 2008.

Large Eddy Simulation of Turbulent Mixing in an Estuary Region F. Roman1 , V. Armenio1 , R. Inghilesi2 , and S. Corsini2 1

2

DICA, Universit´ a degli studi di Trieste, Trieste, Italy, [email protected]; [email protected] ISPRA, istituto superiore per la protezione e la ricerca ambientale, Roma, Italy, [email protected]; [email protected]

Abstract A Large Eddy Simulation (LES) of mixing in the estuarine area of the Tiber river is performed. Sea coastal domains are characterized by a strong grid anisotropy due to the diﬀerent length scales on the horizontal and vertical direction. We propose to model the sub-grid stress by means of a directional eddy viscosity model. The density anomaly, due to salinity diﬀerence between the river fresh water and the sea basin, is treated as an active scalar in the momentum equation. Besides the complex geometry due to bathymetry, coast line and human structures is treated with a combination of curvilinear grid and an Immersed Boundary Method (IBM). The results show that the ﬂow feels the Coriolis force tending to the north direction and that the fresh light water tends to rise and to spread over the sea water that is heavier. In general the model seems to be able to catch the sea coastal dynamics.

1 Introduction One of the most important aspects, when facing with sea coastal domain, is that it is characterized by two length scales, when using 3D model, one for the horizontal plane (x, z or 1, 3) of order of kilometers and one for the vertical direction (y or 2) of orders of some meters. This has lead previous literature, in approaching such problems, to use 2D shallow water approximation. On the other hand many coastal problems are characterized by buoyancy eﬀects and relevant 3D complex geometry making the 2D approximation unfeasible. This diﬀerence by two order of magnitude in the length scales gives diﬃculties when a characteristic length scale is required. In fact once the domain is discretized it is usual to have pancake−type anisotropic cells, with an aspect ratio of 20 : 1. This can be a problem in LES to reproduce the sub-grid stress (SGS). In LES it is common to take, for cells with unequal sides, the Deardroﬀ equivalent length scale Δeq = (Δ1 Δ2 Δ3 )1/3 , where Δ1,2,3 represents the cell side. This is a good choice for weakly anisotropic grid. But in the case of cigar cells or sheet cells, as in the present case, determines anisotropic ﬁltering on isotropic turbulence. Consequently there is a deviation from main statistical quantities, V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 67, c Springer Science+Business Media B.V. 2010

463

464

F. Roman et al.

the rms values can diﬀer of ten per cent respect to the use of isotropic ﬁlter [2]. Scotti et al. [8] investigate on Smagorinsky model with anisotropic grid in the case of inhomogeneous ﬂow. They deﬁne a single length scale obtaining a model close to that for isotropic case. Also model with diﬀerent length scales to face this type of problem were proposed [9]. To work in sea coastal area with strong anisotropic grid, we propose a Smagorinsky model using diﬀerent length scales. These types of model are justiﬁed only for highly anisotropic ﬁltering cells, where the use of an unique characteristic length is no longer pertinent [7]. An other important aspect is the complex geometry involved. This is mainly due to the bathymetry, the coast line, the presence of human structures like jetty and wave breakers. We use an immersed boundary method in conjunction with curvilinear grid as in [6]. Due to the complex geometry and to the physics involved there is the development of fully three dimensional phenomena like upwelling and downwelling close to the coast line. The main forcing comes from tides, sea current and wind action on the sea surface, also Coriolis eﬀect plays an important role. Finally density anomalies, due to temperature gradient or salt concentration, are very important. In the present study a river provides an inﬂow of fresh water in a salty environment with the consequent buoyancy eﬀects.

2 The mathematical model We use the Navier–Stokes equations under the Boussinesq approximations. In fact the density anomaly related to the salinity diﬀerences between a river stream and sea salt water is small and its eﬀect is not negligible just in the vertical momentum equation. For the SGS we use a mixed model composed of a scale similar part and of an eddy-viscosity one. Literature studies have shown that the dynamic model is not suited for large-scale ﬂows [1]. This is attributed to lacking of scale invariance of the SGS and subtest stresses in applicative large scale ﬂows. Moreover, the explicit ﬁltering operation required by the dynamic procedure can be problematic when working with IBM. An alternative is to move back to the Smagorinsky model, which works well in conjunction with wall layer models and immersed boundaries. The SGS is expressed as the sum of an eddy viscosity part and a scale similar one as follows: 1 ∂ui ∂uj τSGS,ij = −2νt S ij + ui uj − ui uj with S ij = (1) + 2 ∂xj ∂xi where Sij is the strain rate tensor, xi is the spatial coordinates in direction i = 1, 2, 3, ui is the velocity component, νt is the eddy viscosity, the · denotes ﬁltered quantities. The scale similar part in equation (1) accounts for local backscatter and anisotropy, whereas the Smagorisnky part of the model

Large Eddy Simulation of Turbulent Mixing in an Estuary Region

465

supplies most of SGS dissipation. The original model is isotropic, based on the assumption that the small scales tend to isotropy; this is not true in large scale coastal dynamics, where the SGS part of the spectrum contains a wide range of anisotropic structures. The eddy viscosity is evaluated as the product of a length scale CΔ, proportional to the grid size, and a velocity scale CΔ|S|, with C a constant. The requirement of a single length scale in strongly anisotropic grid is a problem, which can be overcome considering different eddy viscosities, for the vertical and for the horizontal direction. This approach is a standard technique used in large-scale ocean models, however as far as we know it has never been applied in the LES context. Two eddy viscosity νt,h and νt,v are commonly used in geophysical ﬂuid dynamics [5], with index h for the horizontal and v for the vertical. Then the diﬀusive term for the Navier–Stokes equations reads as Fi =

∂ u¯i ∂ u¯i ∂ u¯i ∂ ∂ ∂ νt,h + νt,v + νt,h ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3

(2)

Although widely in use, this formulation is not mathematically consistent. It takes into account just deformation and not rotation to represent the stress. This can be done only assuming a linear proportionality with S¯ij , that is not true if we introduce directional eddy viscosities. A correct tensorial analysis brings to three coeﬃcients for the eddy viscosity [3, 4]: ν11 = ν13 = ν33 , ν12 = ν23 and ν22 with νij = νji . Using a Smagorinsky model: ν11 = (CLh )2 |S h |

ν12 = (CLv )2 |S v |

ν22 = (CLv )2 |S r |

(3)

with Lh and Lv proper length scales for horizontal and vertical direction, while the strain rate tensor is decomposed as follows: 9 9 9 2 2 2 2 2 2 |S h | = 2(S 11 + S 33 + 2S 13 ) |S v | = 4S 12 + 4S 23 |S r | = 2S 22 (4) If we consider ν11 = νt,h and ν12 = νt,v the diﬀusive term for the horizontal plane can be written as in equation (2). While for the vertical direction 2 we have F2 =

∂ u¯2 ∂ u¯2 ∂ u¯2 ∂ ∂ ∂ νt,v + νt,r + νt,v ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3

(5)

where νt,r = ν11 − 2ν12 + 2ν22 . A dimensional analysis shows that νt,r is of the same order of νt,h . The coeﬃcients of the model need calibration and this is still an open issue for lacking of proper test cases. The same approach is used for the subgrid diﬀusivity, considering directional eddy diﬀusivities. The algorithm integrates the equations using the curvilinear-grid, fractional-step method of [10]. All terms are treated explicitly through secondorder Adams–Bashforth technique except the diagonal diﬀusive terms which

466

F. Roman et al.

are treated implicitly. Spatial derivatives are treated using central diﬀerences but the advective terms discretized using a 3rd-order accurate QUICK scheme. The pressure equation is solved using a line-SOR with line solution in the vertical direction and point iteration in the horizontal ones in conjunction with a Multigrid technique to speed up the convergence.

3 Application to an estuarine ﬂow: results and discussion Using this mathematical model we study the mixing eﬀects due to the incoming ﬂow from Tiber river in the sea. For the subgrid model we use a constant C = 0.085 in the horizontal direction and C = 0.035 in the vertical one. A sketch of the area is in Fig. 1. The domain covers an area of 5 × 6 km in the horizontal plane and it is discretized with 385 grid points both for x and z direction, while 33 points are taken in the vertical y. This determines cells of about ten meters in the horizontal and 0.4 meters in the vertical direction. Figure 2a shows a horizontal grid plane, for a coarser grid while Fig. 2b shows the grid bottom constructed from bathymetric data. Part of the coast line

Fig. 1. Bathymetry, grid edges and physical domain for Tiber river area.

a

Z Y

X

Fig. 2. (a) Horizontal plane for a coarse grid. (b) Bottom of the grid from bathymetry.

Large Eddy Simulation of Turbulent Mixing in an Estuary Region

467

and of the bathymetry is treated using the IBM. The boundary conditions are chosen in the following way. On the sea bottom and on solid walls we use a wall function with a Werner Wengle exponential law. In the vertical boundary SE-SO a longshore sea current of 0.02 m/s is considered. This is a typical value for the region examined. On the Boundary NE-SE in the correspondence of the Tiber river we consider an inﬂow condition. This inﬂow comes from a pre simulation of a turbulent channel ﬂow with the characteristics in geometry and mass transport of the river. The average velocity is about 0.3 m/s while the ﬂow rate is of about 300 m3 /s. This value is typical for Tiber in winter time. Based on the river mean velocity and on the vertical length close to the river the Reynolds number is about 4.5 × 106. The channel ﬂow is simulated considering the Coriolis eﬀect, so its proﬁle is not symmetric, and it provides the turbulent kinetic energy for the simulation. On the sea surface we consider a free-slip condition, while on the remaining boundaries an Orlanski condition is enforced. The incoming ﬂow from the river has the density of fresh water, this light ﬂuid goes into a salt heavier environment. Stratiﬁcation is only due to this salt concentration diﬀerence, then we work under Reynolds analogy. We can consider that the rate of transport of a scalar like concentration is the same of momentum. In the advection–diﬀusion equation for salinity concentration we consider a turbulent Schmidt number Sct = 0.5 (where Sct = νt /kt , with kt the eddy diﬀusivity). Besides the domain is so large and the characteristic velocities are so small that Coriolis eﬀect cannot be neglected. Figure 3a shows the contour plot of the horizontal velocity component u. The river ﬂow once in the sea tends to turn north due to the presence of the meridional sea current and because of the Coriolis eﬀect. It remains far from the coastline and a large recirculation area is trapped between the main ﬂow and the coastline. In particular it can be seen the asymmetric proﬁle of the river inﬂow, because of the Coriolis force there are higher velocity close to the north bank of the river’s mouth. A mixing layer is observable where the river ﬂow meets the meridional sea current. Figure 3b shows a vertical velocity plot. As can be seen from the contour legend the vertical velocity magnitude is of one order less than the horizontal velocity components, this

Fig. 3. (a) u velocity at 1m below the surface. (b) v velocity at 1m below the surface.

468

F. Roman et al.

Fig. 4. (a) ρ /ρ0 at 5m from the sea surface. (b) ρ /ρ0 at the sea surface.

diﬀerence in the magnitude is typical for sea ﬂow. These two plots refer to an horizontal layer just below the sea surface. The same mean behavior is observed at diﬀerent vertical levels. In Fig. 4a, b can be seen a density contour plot at diﬀerent levels on vertical direction. The light water coming from the river, because of buoyancy tends to rise over the salt water and to spread in the horizontal direction. This is due to the fact that density is treated as an active scalar in momentum equations. Then the fresh water is transported by the mean current and so turns north without reaching the coast far from the river’s mouth. In the present study we have proposed an approach to face sea coastal domain with LES. The results show that the model is able to catch the expected behavior for a sea coastal area. Also real observations conﬁrm this.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bou-Zeid E., Meneveau C., Parlange M.B., (2005), Physics of Fluids, vol. 17. Kaltenbach H.J., (1997), J. Comput. Phys., 136: 399–410. Kamenkovich V. M., (1977), Fundamentals of ocean dynamics, Elsevier. Mills J., (1994), Journal of Physical Oceanogrphy, 24: 1077–1079. Pedlosky J., (1987), Geophysical ﬂuid dynamics, Springer. Roman F., Napoli E., Milici B., Armenio V., (2009), Computer & Fluids, 38:1510–1527. Sagaut P., (1998), Large Eddy Simulations for incompressible ﬂows, Springer Verlag. Scotti A., Meneveau C., Lilly D.K., (1993), Physics of Fluids, 5(9): 2306–2308. Zahrai S., Bark F.H., Karlsson R.I., (1995), Eur. J. of Mech. B/Fluids, 14: 459–486. Zang Y., Street R.L., Koseﬀ J., (1994), J. Comput. Phys., 114:18–33.

Dispersion of (Light) Inertial Particles in Stratiﬁed Turbulence M. van Aartrijk and H.J.H. Clercx Fluid Dynamics Laboratory, Department of Physics, Eindhoven University of Technology, The Netherlands, [email protected]; [email protected] Abstract We present a brief overview of a numerical study of the dispersion of particles in stably stratiﬁed turbulence. Three types of particles are examined: ﬂuid particles, light inertial particles (ρp /ρf = O(1)) and heavy inertial particles (ρp /ρf 1). Stratiﬁcation suppresses the vertical dispersion of all three types of particles compared to isotropic turbulence. The horizontal dispersion, on the other hand, is enhanced. The importance of the forces that act on inertial particles are examined and the results are shown for particles with ρp /ρf = 10. The inertia of the particles results in a nonuniform particle distribution over the domain. This preferential concentration eﬀect is presented here for heavy particles, and it is found that the eﬀect is stronger in isotropic turbulence than in stratiﬁed turbulence.

1 Introduction The turbulent dispersion of inertial particles plays an important role in stratiﬁed oceanic and atmospheric ﬂows, such as estuaries, coastal areas or the nocturnal atmospheric boundary layer. The ﬁnal applications that we have in mind are oceanographic environments, where the density of the particles (plankton, algae, sand) is of the same order as that of the surrounding ﬂuid. In stably stratiﬁed turbulent ﬂows a negative vertical density gradient is present, which suppresses vertical ﬂuid motions. The ﬂow typically displays large horizontal vortical structures and thin, sheared layers in vertical direction [6]. Previous dispersion studies in these stratiﬁed ﬂows mainly focused on ﬂuid particles, see for example, Refs. [2, 7, 8, 10].

2 Numerical approach We study a model system by means of direct numerical simulations (DNS): the dispersion of particles in statistically stationary homogeneous stably stratiﬁed turbulent ﬂows. The ﬂow ﬁeld is solved using the Eulerian approach. The V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 68, c Springer Science+Business Media B.V. 2010

469

470

M. van Aartrijk and H.J.H. Clercx

Navier–Stokes equations with Boussinesq approximation are solved on a tripleperiodic domain [2, 13]. A linear stable background density stratiﬁcation is imposed which is kept constant throughout a simulation. Density ﬂuctuations are present on top of the linear proﬁle, and the total density is given by ρf = ρ0 +ρ(z)+ρ (x, y, z, t) (ρ0 a reference value, ρ the background proﬁle, ρ density ﬂuctuations). From the density gradient the buoyancy frequency N 2 = − ρg0 ∂ρ ∂z can be computed, with g the gravitational acceleration. Large-scale forcing is applied in the horizontal direction to obtain a statistically stationary state. A description of the resulting ﬂows can be found in Refs. [1, 2]. The Lagrangian approach is applied to study the particle trajectories. Three types of particles are tracked in the ﬂows: ﬂuid particles, light particles (ρp /ρf = O(1)) and heavy particles (ρp ρf ), with ρp the particle density. dx Particle trajectories are obtained from dtp = up , with xp the particle position and up its velocity. For ﬂuid particles their velocities are derived from cubic spline interpolation of the velocity ﬁeld at the particle position [2]. The velocities of the inertial particles (light and heavy) are obtained by solving the Maxey–Riley equation [9]: 1 2 2 dup Du mp = 6πaμ u − up + a ∇ u + mf dt 6 Dt 1 2d 2 Du dup 1 − + a ∇ u + (mp − mf ) g + mf 2 Dt dt 10 dt t du/dτ − dup /dτ + 16 a2 d∇2 u/dτ dτ . (1) + 6πa2 μ 1/2 0 [πν(t − τ )] The particle mass is given by mp , a is the radius of the particle and mf is the mass of a ﬂuid element with a volume equal to that of the particle. The ﬂuid velocity is denoted by u, ν is the kinematic viscosity and μ = νρf is the dynamic viscosity. The forces on the right-hand side of this equation are viscous drag, a local pressure gradient in the undisturbed ﬂuid, gravitational forces, added mass and the Basset history force, successively. For the added mass term the form described by Auton et al. [4] is used. du Equation (1) reduces to dtp = τ1p (u − up ) + g in the limit of small heavy particles. For the results presented in this paper gravity is set to zero (g = 0). d2 ρp /ρ

A measure of the particle inertia is the particle response time τp = p 18ν f , with dp = 2a the particle diameter. The particle inertia will be expressed in the following using the Stokes number St = τp /τK (τK the Kolmogorov time). Fluid particles can be seen as particles in the limit of ρp = ρf and St → 0. The particles are released when the ﬂow has reached a stationary state and velocity and position time series of O(104 ) − O(106 ) particles are collected for about 40 eddy turnover times. It has been tested how many previous data points are needed for the Basset force to reproduce this force accurately. Since the particles are small, the

Dispersion of (Light) Inertial Particles in Stratiﬁed Turbulence

471

smallest scales of the ﬂow are the most important for the strength of the forces that act on a particle. It is found that the history term has to be calculated over a time interval of at least one Kolmogorov time. For the runs presented here a history of about 2τK (about 800 time steps) is chosen; increasing this time does not signiﬁcantly change any of the forces acting on a particle. The computation of especially the Basset force is highly memory-demanding. This makes the study of light particle behavior in turbulent ﬂows a computational challenge.

3 Results The results presented in this paper are obtained in either a moderately (case N10, N = 0.31 s−1 ) or a strongly (case N100, N = 0.98 s−1 ) stably stratiﬁed ﬂow [1,2]. For several cases also the results for isotropic turbulence are shown for comparison. Several quantities are studied; the main interest here is on single-particle dispersion, on the eﬀect of preferential concentration and on the relative importance of the diﬀerent forces that are acting on the particles. 3.1 Single-particle dispersion The horizontal and vertical dispersion (mean-squared displacement, δi2 = 2

[xp,i − xp,i (0)] ) of both ﬂuid particles and inertial particles in isotropic and stratiﬁed turbulence (case N10) are shown in Fig. 1. For isotropic turbulence the classical initial ballistic t2 -regime and long-time linear diﬀusion limit are retrieved [12]. For stratiﬁed turbulence the vertical dispersion is clearly

a

b

104 10

t

St1+ 3 St5 St6

2

2

fluid iso

103 10

St1=0.30 St2=0.55 St3=0.80 St4=1.38 St5=1.81 St6=6.75 fluid iso

2

t 101

100 10

−2

t2

St6 St5 St3 St1

St2 St4

t

10

2

0

t

2

10−1

10−4 10−2

10−1

100

101

102

10−2

10−1

100

101

102

Fig. 1. Horizontal (a) and vertical (b) single-particle dispersion as a function of time for case N10. Results for ﬂuid particles, light particles (St2 and St4) and heavy parti2 cles. The axes are scaled using the horizontal (u2 h ) and vertical (w ) rms-velocities, the Lagrangian time scale TL and the buoyancy frequency N . For reference, also the result for isotropic turbulence is added, in the right plot this graph is shifted and the axes are scaled using w2 and TL .

472

M. van Aartrijk and H.J.H. Clercx

suppressed. Fluid particles that are displaced from their original equilibrium height tend to return to that equilibrium height. After the t2 -regime a plateau is found that scales as N−2 [7]. For long times again a transition towards a diﬀusive t-regime can be seen, which is caused by molecular diﬀusion of the density [2, 11]. In the horizontal direction it can be seen (Fig. 1a) that the dispersion of ﬂuid particles is enhanced in stratiﬁed turbulence. This is a result of strong local shear between large-scale horizontal vortical structures in diﬀerent vertical layers [2]. When looking at the transport of more realistic particles, inertial eﬀects have to be considered. Inertial particles do not exactly follow the ﬂow like ﬂuid particles, as will be seen in the next section, and they follow biased trajectories. The eﬀect of inertia, expressed using the Stokes number, can be seen for six diﬀerent St in Fig. 1. Particles denoted with St2 and St4 have density ratios of 10 and 25, the others are heavy particles with ρp /ρf = O(104 ). Under the assumption that the particles are small (diameter not exceeding O(η)), particles wit ρp /ρf = O(1) have Stokes numbers smaller than about 0.1 and it is found that their results resemble the results obtained for ﬂuid particles. The horizontal long-time dispersion of heavy particles is slightly smaller than that of ﬂuid particles, but the slope remains of order t2 . Results for St2 and St4 are omitted in Fig. 1a for clarity, but they show similar behavior. In the vertical direction a clear diﬀerence is found for the dispersion of ﬂuid particles and that of inertial particles. With increasing St the long-time dispersion is enhanced. The typical plateau becomes less pronounced and the transition to a ﬁnal linear diﬀusion limit sets in at earlier times. Here, the Stokes number is the determining factor. The density ratio only has a minor eﬀect on the vertical dispersion in stably stratiﬁed turbulence, as becomes clear from the strong resemblance between the light (St2 and St4) and heavy particle results. 3.2 Preferential concentration The distribution of the particles over the ﬂow domain can give an impression of the mixing properties of the ﬂow. Several previous studies (for example, Bec et al. [5]), looking at heavy particles in isotropic turbulence, observe the socalled eﬀect of preferential concentration. The particle distribution is found to be highly nonuniform, and the particles collect in regions of high strain rate and low vorticity. Also for stratiﬁed turbulence we obtain this nonuniform distribution of particles [1]. Strong local heavy particle accumulation is shown in Fig. 2 for isotropic turbulence and for strongly stratiﬁed turbulence. The particle distribution in stratiﬁed turbulence diﬀers from that obtained in isotropic turbulence, and a clear diﬀerence can be seen between the horizontal and the vertical direction. The particle distribution reﬂects the anisotropy of the ﬂow. In the horizontal direction particles cluster on larger scales than in isotropic turbulence, whereas in the vertical direction thin, sheared layers are observed. The correlation dimension D2 as used by Bec et al. [5] is applied to

Dispersion of (Light) Inertial Particles in Stratiﬁed Turbulence

a

b

473

c

Fig. 2. Particle distributions in horizontal (b) and vertical (a,c) cross-sections of the ﬂow domain. Results are shown for isotropic turbulence (a) and for stably stratiﬁed turbulence (b,c) (case N100), from runs with 106 particles and Stokes number around the optimum for clustering. No gravitational forces are acting on the particles.

quantify the preferential concentration. This leads to the conclusion that in stably stratiﬁed turbulence the eﬀect of preferential concentration decreases with increasing stratiﬁcation [1]. 3.3 Forces acting on the particles Elaborate studies on the importance of the diﬀerent forces that are acting on inertial particles are scarce in the literature. The topic is studied for a turbulent channel ﬂow by Armenio and Fiorotto [3]. It is mainly of interest for light particles. All forces in equation (1) are taken into account, except for the gravitational force (to avoid a mean drift velocity). The forces act at the smallest scales of the ﬂow. At these scales the ﬂow is more or less isotropic, also for the moderately stratiﬁed case N10. Therefore, a strong resemblance is obtained between the results for isotropic turbulence and for stratiﬁed turbulence. For particles with density ratios of order one, all forces are relevant except for the added mass Fax`en correction term. With increasing ρp /ρf the importance of the diﬀerent forces decreases. For particles with ρp /ρf = 10 and St = 0.55 in a case N10 ﬂow the probability density functions (PDFs) of the ratio of the diﬀerent forces and the Stokes drag are shown in Fig. 3. The dispersion results for these particles are shown previously in Fig. 1. It can be seen that for this combination of ρp /ρf and St, apart from the Stokes drag, only the pressure gradient and the Basset force play a signiﬁcant role. In a separate simulation with the same St but ρp /ρf = 144 these two forces are found to be smaller by a factor of ten and two, respectively. It can therefore be concluded that the density ratio is an important parameter in the study of the diﬀerent forces.

474

M. van Aartrijk and H.J.H. Clercx 1

103

pdf(Fi/FSD)

2 102 10

3 4

1

5 6

100

10−1 10−5

10−4

10−3

10−2

10−1

100

Fi /FSD Fig. 3. PDF of the ratio of diﬀerent forces Fi and the Stokes drag FSD ; F1 = added mass Fax`en correction, F2 = Basset force Fax`en correction, F3 = Stokes drag Fax`en correction, F4 = added mass, F5 = pressure gradient, F6 = Basset force. Particles with St = 0.55 and ρp /ρf = 10 are tracked in a case N10 ﬂow.

4 Concluding remarks In this work we showed the behavior of light and heavy particles in stably stratiﬁed turbulence. Three topics are discussed; single-particle dispersion, the eﬀect of preferential concentration and the importance of the diﬀerent forces that are acting on the particles. In future work we will study in more detail the inﬂuence of the diﬀerent forces on the dispersion of (light) inertial particles in stably stratiﬁed turbulence. Furthermore, we will examine the inﬂuence of gravitational forces that are acting on the particles on the presented results.

Acknowledgments This programme is funded by the Netherlands Organisation for Scientiﬁc Research (NWO) and Technology Foundation (STW) under the Innovational Research Incentives Scheme grant ESF.6239. This work is sponsored by the Stichting Nationale Computerfaciliteiten (NCF, NWO) for the use of supercomputer facilities.

References 1. 2. 3. 4. 5. 6.

van Aartrijk M, Clercx HJH (2008) Phys Rev Lett 100:254501 van Aartrijk M, Clercx HJH, Winters KB (2008) Phys Fluids 20:025104 Armenio V, Fiorotto V (2001) Phys Fluids 13(8):2437–2440 Auton TR, Hunt JCR, Prud’homme M (1988) J Fluid Mech 197:241–257 Bec J, et al. (2007) Phys Rev Lett 98:084502 Brethouwer G, et al. (2007) J Fluid Mech 585:343–368

Dispersion of (Light) Inertial Particles in Stratiﬁed Turbulence 7. 8. 9. 10. 11. 12. 13.

475

Kimura Y, Herring JR (1996) J Fluid Mech 328:253–269 Liechtenstein L, Godeferd FS, Cambon C (2005) J Turbul 6(24):1–18 Maxey MR, Riley JJ (1983) Phys Fluids 26(4):883–889 Nicolleau F, Vassilicos JC (2000) J Fluid Mech 410:123–146 Pearson HJ, Puttock JS, Hunt JCR (1983) J Fluid Mech 129:219–249 Taylor GI (1922) Proc Lond Math Soc Series II 20:196–212 Winters KB, MacKinnon JA, Mills B (2004) J Atmos Ocean Tech 21(1):69–94

The Inﬂuence of Magnetic Fields on the Rise of Gas Bubbles in Electrically Conductive Liquids Daniel Gaudlitz and Nikolaus A. Adams Technische Universit¨ at M¨ unchen, Lehrstuhl f¨ ur Aerodynamik, Boltzmannstr. 15, 85748 Garching, Germany, [email protected]; [email protected] Abstract The inﬂuence of magnetic ﬁelds on single gas bubbles rising in electrically conductive liquids is investigated by direct numerical simulation. For the description of moving and deformable gas-liquid interfaces the hybrid particle-level-set method is employed. As low magnetic Reynolds numbers are considered, the induced magnetic ﬁeld can be neglected and a simpliﬁed magnetohydrodynamic approach can be used. In case magnetic ﬁelds are applied, distinct diﬀerences with respect to the bubble wakes have been found, which in turn lead to signiﬁcant changes regarding bubble shapes, rising paths and velocities.

1 Introduction Two-phase ﬂows play an important role in current technology. In metallurgy injection of gas bubbles into a melt is used for stirring and alloying processes. Another possibility to enhance mixing is to apply magnetic ﬁelds. In electrically conductive ﬂuids Lorentz forces are generated, which induce ﬂuid velocities for stirring the melt. In order to optimize technical processes knowledge of the mixing characteristics of bubbly ﬂows inﬂuenced by magnetic ﬁelds is a prerequisite. The rise of gas bubbles in liquids has been subject of numerous experimental and numerical investigations [7]. By performing numerical simulations of single gas bubbles rising on linear or zig-zagging paths we have obtained detailed information on the bubble shape and on characteristic turbulent structures in the bubble wake [4]. In experimental investigations of bubbles in opaque liquid metals the established optical measurement techniques can not be applied, which makes the evaluation of the ﬂow ﬁeld diﬃcult. Therefore we investigate the inﬂuence of magnetic ﬁelds on single gas bubbles in an electrically conductive liquid by numerical simulation. The observed changes of the rising characteristics of the bubbles are analyzed and compared with available data for bubbles, which are not exposed to magnetic ﬁelds. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 69, c Springer Science+Business Media B.V. 2010

477

478

D. Gaudlitz and N.A. Adams

2 Numerical method For the description of moving interfaces we employ the hybrid particle-levelset method (HPLS) proposed by Enright et al. [3]. This method combines the eﬃcient reconstruction of interfaces provided by the level-set method with the good mass-conservation properties of marker-particle schemes. We have evaluated this method for two-phase-ﬂow computations and a modiﬁed version for improved accuracy for these applications has been developed [5]. The evolution of incompressible, non-reacting two-phase ﬂows is governed by the Navier–Stokes equations 1 ∇p + ∇ · (2 μ D) + ρ Reref ρ ey ρb 1 κδn+ + 1− 2 − We ρ F rref ref ρ 1 (j × B), + Nref ρ ∇ · u = 0.

ut + u · ∇u = −

(1)

(2) √ Herein Reref = ρb U L/μb is the reference Reynolds number, F rref = U/ g L is the reference Froude number, W eref = ρb U 2 L/σsurf is the reference Weber number and Nref = σelectr (B0 )2 L/(ρb U ) is the reference Stuart number. The volume-equivalent diameter of the bubble has been chosen as√the reference length L = dB , and the reference velocity is deﬁned by U = g L. Further quantities used for non-dimensionalization are the density ρb and viscosity μb of the bulk phase, the surface-tension coeﬃcient σsurf , the electrical conductivity of the liquid σelectr , and the magnetic ﬂux density B 0 . Buoyancy eﬀects have been included in the Navier–Stokes equations by subtracting a reference state, in which the density is ρ(x) = ρb and the pressure gradient is ∇p = ρb g. The fourth term on the right-hand side (RHS) of equation (1) gives the surface-tension force, whose components are determined by the local interface curvature κ and the unit-normal vector n. By multiplying with a molliﬁed Delta function δ this source term is active only within a narrow region at the interface. The last term on the RHS of equation (1) gives the Lorentz force acting on electrically conductive ﬂuid elements due to applied magnetic ﬁelds. As we consider only low magnetic Reynolds numbers (Rem = μmagn σelectr U L 1, with μmagn being the magnetic permeability) and magnetic ﬁelds of constant strength, the induced magnetic ﬁeld can be neglected and the so called simpliﬁed magnetohydrodynamic approach can be used [6]. Therefore, the electric current is given by j = −∇ψ + u × B 0 .

(3)

The electric potential ψ is determined by solving a Poisson equation ψ = ∇ · (u × B 0 ) = B 0 · ω.

(4)

Rising Gas Bubbles Inﬂuenced by Magnetic Fields

479

For the discretization of the above equations we use ﬁnite diﬀerences on an equidistant staggered grid. At the domain boundaries periodic boundary conditions are applied. When solving equation (4) the electric potential is set to zero inside the non-conductive gas phase and at the bubble surface a Neumann boundary condition is enforced. Incompressibility is satisﬁed by a fractional step projection method. The resulting pressure-Poisson equation is solved by a conjugate-gradient method. For time integration a third-order Runge–Kutta method is used. Spatial derivatives within the HPLS-method are discretized using a ﬁfth-order WENO-scheme.

3 Results 3.1 Gas bubbles rising on linear paths First, bubbles rising on a linear path and trailing a closed laminar wake are considered. In the following we investigate the eﬀect of a vertically oriented magnetic ﬁeld on this two-phase-ﬂow conﬁguration. In order to compare the numerical results with experimental data for the case without magnetic ﬁeld, we have chosen the parameters as in the experiments of Bhaga and Weber [1] (see Fig. 3f of their paper). The density ratio is ρgas /ρliquid = 1/1, 048 and the viscosity ratio is μgas /μliquid = 1/15, 822, the non-dimensional parameters are Reref = 62.03, W eref = 116.0, and F rref = 1.0. If magnetic ﬁelds are present, we assume that the electrical and magnetic properties of the liquid are equal to the ones of mercury. For the highest Stuart number of Nref = 1.0 this requires a magnetic ﬂux density of B = 0.16 T, which is within the range of industrial applications. We applied static magnetic ﬁelds as well as unsteady ﬁelds, characterized by square pulses of diﬀerent frequencies. The initially spherical bubble of diameter d = 1 was centered in a computational domain of the size Lx × Ly × Lz = 4 × 8 × 4 and a resolution of Nx × Ny × Nz = 240 × 480 × 240 grid cells was used. The bubble accelerates from rest until a steady terminal rising velocity UT is reached. The bubble shape changes from spherical to a cap-type form. Characteristic shape parameters of the bubble and its wake are deﬁned in Fig. 1 wb hb hs hw

ww

Fig. 1. Geometry parameters of the bubble and the wake.

480

D. Gaudlitz and N.A. Adams Table 1. Bubbles rising on linear paths: global characteristics of the ﬂows.

UT cD wb hb ww hw hs

Exp. [1]

Nref = 0.0

Nref = 0.5

Nref = 1.0

Nref = 1.0 fpuls = 25 Hz

Nref = 1.0 fpuls = 75 Hz

0.68 2.89 1.50 0.45 1.70 1.70 0.50

0.65 3.13 1.53 0.50 1.63 1.65 0.52

0.88 1.72 1.34 0.63 1.28 1.15 0.38

0.93 1.54 1.27 0.69 1.20 0.97 0.32

0.79 2.14 1.39 0.60 1.33 1.18 0.36

0.79 2.14 1.39 0.60 1.33 1.17 0.35

a

b

Nref = 0.0.

Nref = 1.0.

Fig. 2. Flow ﬁeld of a gas bubble rising on a linear path. (a) Without magnetic ﬁeld, and (b) with magnetic ﬁeld applied.

and their values for the diﬀerent cases considered are given in Table 1. For the bubble rise without magnetic ﬁeld good agreement of experimental and numerical results has been found. When the magnetic ﬁeld is applied, the size of the bubble wake is reduced signiﬁcantly, compare Fig. 2a and b. This is due to a damping of velocity gradients in direction of the magnetic ﬁeld, which has also been found in MHDsimulations of isotropic turbulence [9] and turbulent channel ﬂows [6]. With increasing Stuart number the reduction of the wake size becomes stronger. If the magnetic ﬁeld of strength Nref = 1.0 is pulsed, the obtained ﬂow characteristics are almost identical for both frequencies. Obviously, for these frequencies only the time-averaged magnetic forcing is of major importance for the ﬂow evolution. Due to a damping of the velocities inside the wake, the bubble experiences less shear forces at its lower side and hence is less ﬂattened than in the simulation without magnetic ﬁeld. The reduced wake size and the more spherical-like bubble shape lead to a lower drag and an increase in rising speed.

Rising Gas Bubbles Inﬂuenced by Magnetic Fields

481

3.2 Gas bubbles on unsteady paths We now focus on MHD-simulations of more complex bubble ascents with open turbulent wakes. The numerical investigation of an air bubble rising on a zigzag path in water [4] revealed a very good agreement with recent experimental results [2, 8]. We now apply a constant external magnetic ﬁeld to this ﬂow conﬁguration and investigate changes in the rising behavior of the bubble. Again we assume that the liquid has electrical and magnetic properties like mercury. The parameters of the simulation are the following: ρgas /ρliquid = 1/775, μgas /μliquid = 1/54, Reref = 600.0, W eref = 3.66, F rref = 1.0, and Nref = 0.5. The bubble of diameter d = 1 is placed in a domain of the size Lx × Ly × Lz = 5 × 12 × 5, which has been discretized using Nx × Ny × Nz = 120 × 288 × 120 grid cells. In the simulation without the magnetic ﬁeld the bubble accelerates from rest on a linear path and after development of a path instability the bubble ﬁnally rises on a regular zigzag path with a pathoscillation frequency of fpath = 4.3 Hz. The bubble reaches a rising velocity of UT = 0.226 m/s and its ﬁnal shape is an ellipsoid with the axes {a, b, c} = {1.47, 1.40, 0.60}. The lateral motion of the bubble is caused by the formation and periodic shedding of cascades of hairpin vortices in the bubble wake, see Fig. 3a. Such a vortex system consists of up to four interconnected hairpin vortices with closed vortex loops. After detachment of the vortex cascade from the lower side of the bubble these vortex loops persist as far as seven diameters below the bubble. When the magnetic ﬁeld is active the path characteristics change substantially. After acceleration from rest also wake instabilities develop, but the bubble rises on an elongated spiral path. The zigzag mode, which often is the precursor of a spiraling ascent, is entirely suppressed. However,

Nref = 0.0.

Nref = 0.5.

Fig. 3. Wake structures (iso-surfaces λ2 = −1.2, colored with vertical vorticity) of a gas bubble rising on an unsteady path. (a) Without magnetic ﬁeld, and (b) with magnetic ﬁeld applied.

482

D. Gaudlitz and N.A. Adams

the bubble changes its direction of the lateral motion with a frequency of fpath = 4.0 Hz, which is comparable to the simulation without magnetic ﬁeld. The ellipticity of the bubble shape increases, the axes have been determined as {a, b, c} = {1.50, 1.29, 0.58}. Also the terminal rising velocity, UT = 0.244 m/s, is slightly higher. The damping eﬀect of the Lorentz forces changes the wake structures signiﬁcantly, see Fig. 3b. Fewer hairpin vortices can be observed and the vortex loops are broken up already in a short distances below the bubble. However, the vertical vortex legs are less aﬀected by the magnetic ﬁeld and are approximately of the same strength as in the case without magnetic ﬁeld. The hairpin vortices are elongated in vertical direction and also a distinct twisting of these vortices around the vertical axis is apparent. The latter possibly causes the spiraling motion of the bubble.

4 Conclusions A numerical investigation of gas bubbles rising in electrically conductive liquids and being inﬂuenced by vertically oriented magnetic ﬁelds has been performed. For bubbles rising on a linear path a strong reduction of the wake size due to the damping eﬀect of the Lorentz forces has been found. This in turn leads to more spherical bubble shapes and higher rising velocities. For the considered bubble rising on an unsteady path, the application of a magnetic ﬁeld causes a change of the path characteristics to a spiraling mode without an intermediate zigzagging motion. Fewer hairpin vortices are found in the bubble wake and these vortices are elongated in direction of the magnetic ﬁeld.

Acknowledgements This work is supported by the German Research Council (DFG). The simulations were performed on the national super computer NEC SX-8 at the High Performance Computing Center Stuttgart (HLRS).

References 1. D. Bhaga and M.E. Weber. Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech., 105:61–85, 1981. 2. C. Br¨ ucker. Structure and dynamics of the wake of bubbles and its relevance for bubble interaction. Phys. Fluids, 11:1781–1796, 1999. 3. D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle level set method for improved interface capturing. J. Comput. Phys., 183:83–116, 2002. 4. D. Gaudlitz. Numerische Untersuchung des Aufstiegsverhaltens von Gasblasen in Fl¨ ussigkeiten. PhD thesis, Technische Universit¨ at M¨ unchen, 2008.

Rising Gas Bubbles Inﬂuenced by Magnetic Fields

483

5. D. Gaudlitz and N. A. Adams. On improving mass-conservation properties of the hybrid particle-level-set method. Computers and Fluids, 37:1320–1331, 2008. 6. D. Lee and H. Choi. Magnetohydrodynamic turbulent ﬂow in a channel at low magnetic Reynolds number. J. Fluid Mech., 439:367–394, 2001. 7. J. Magnaudet and I. Eames. The motion of high-Reynolds-number bubbles in inhomogeneous ﬂows. Annu. Rev. Fluid Mech., 32:659–708, 2000. 8. C. Veldhuis, A. Biesheuvel, and van Wijngaarden L. Shape oscillations on bubbles rising in clean and in tap water. Phys. Fluids, 20, 2008. 9. O. Zikanov and A. Thess. Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number. J. Fluid Mech., 358:299–333, 1998.

Large Eddy Simulation of a Turbulent Droplet Laden Mixing Layer W.P. Jones, S. Lyra, and A.J. Marquis Department of Mechanical Engineering, Imperial College London, London, UK, [email protected]; [email protected]; [email protected] Abstract The present study investigates, by the use of Large Eddy Simulation (LES), a turbulent, droplet laden mixing layer and focuses on the preferential concentration of the dispersed phase. A Lagrangian formulation is adopted for the dispersed phase coupled with an Eulerian description for the carrier phase. A stochastic model has been used for the representation of the sub-grid scales on the particle motion. Sensitivity analysis on the eﬀect of the dispersion constants was conducted. Results showed good agreement with measurements, demonstrating that the characteristics of the ﬂow are well captured.

1 Introduction Two-phase ﬂows are encountered in several practical systems arising in industrial, environmental and biomedical applications. Thus, extensive studies have been conducted to determine the response of particles in shear ﬂows in a number of diﬀerent geometrical conﬁgurations such as turbulent jets and mixing layers [1–3]. Mixing layers in particular, are characterised by the presence of large-scale stream and spanwise vortical structures that determine the preferential concentration of the dispersed phase [4]. LES is a promising tool for the prediction of two-phase turbulent ﬂows [5]. For the continuous phase ﬁltered source terms arise, representing the inﬂuence of the dispersed phase. In the present LES study of a droplet laden mixing layer, the dispersed phase is composed of discrete spherical droplets having a relatively small volume fraction; eﬀects due to droplet collisions, breakup and coalescence are negligible. The structure of this paper is as follows. The mathematical formulation is presented in the next section. The experimental apparatus, the computational details and the results are presented and discussed in the Section 3. The conclusions of the study are summarised in the ﬁnal section.

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 70, c Springer Science+Business Media B.V. 2010

485

486

W.P. Jones et al.

2 Mathematical modelling 2.1 Filtered Navier–Stokes equations The density weighted ﬁltered Continuity and Navier–Stokes equations can be written: ∂ ρ¯ ∂(¯ ρui ) + = S¯mass . (1) ∂t ∂xi ∂(¯ ρui ) ∂(¯ ρui uj ) ∂ p¯ ∂σ ¯ij + =− + + τij + ρ¯gi + S¯mom,i ∂t ∂xj ∂xi ∂xj

(2)

The Smagorinsky model [6] is used for the sub-grid scale tensor: τij = ρ¯ (u$ i uj ) with μsgs = ρ¯ Cs Δ2 ||Sij ||, where Cs is the Smagorinsky i uj − u 9 constant equal to 0.1 and ||Sij || is the Frobenius norm ||Sij || = 2Sij Sij of the ﬁltered strain tensor. The source terms account for contribution of

the n the dispersed phase and can be evaluated as S¯ = Δ13 α=1 S (α) where the summation is deﬁned over the number of the droplets present in the cell volume under consideration. The mass and momentum source terms for the αth (α) (α) d α α droplet are S¯mass = − dm and S¯mom,i = − dt (mui ) . dt 2.2 PDF modelling of fuel sprays The dispersed phase is described in terms of a set of macroscopic variables, the droplet radius, r the droplet number n, the droplet velocity u and the droplet temperature θ. In the present case the Weber number may be presumed small, (W e < 20). The equation for the evolution of the corresponding ﬁltered joint pdf P¯ (r, n, v, θ, x, t) [7] is ∂ P¯ ∂ ¯ ∂ ˙¯ ∂ ˙¯ ∂ ˙ ¯ + (N P ) + (RP ) + (Θ P ) = 0 aj P + ∂t ∂vj ∂n ∂r ∂θ

(3)

where aj is the droplet acceleration, N˙ = dn/dt is the rate of change of the droplet number through the process of droplet breakup and coalescence, R˙ = dr/dt the rate of change of the droplet size through evaporation and Θ˙ = dθ/dt the rate of change of droplet temperature caused by heat trans˙ N, ˙ Θ˙ are zero. fer from the surrounding gas phase. In the present case R, Only viscous drag and gravitational forces are considered and a stochastic Markov model is used to represent the eﬀects of unresolved ﬂuctuations on the droplets [8]: > ksgs (i) −1 (i) dV = τp Co dWt + gdt U (t) − V dt + (4) τt is the ﬁltered gas velocwhere V(i) is the velocity of the ith particle, U ity at the particle position, ksgs is the unresolved kinetic energy of the gas

Large Eddy Simulation of a Turbulent Droplet Laden Mixing Layer

487

phase, Co is a model constant, dWt represents the increment of the Wiener process, g is the gravitational acceleration, τt is a sub-grid timescale which aﬀects the rate of interaction between the particle and turbulence dynamics, deﬁned as: τt =

τp2α 2α−1 .

√Δ

The particle relaxation time, τp is given by:

ksgs

τp−1 =

3 ρf CD 8 ρp R |U − V|."Thedrag

coeﬃcient CD is obtained from the Yuen and 1+ : 0 < Re < 1, 000 Chen law [9]: CD = where Re is the 0.44 : Re > 1, 000 Reynolds number based on the droplet diameter and the relative velocity of the droplet with respect to the gas phase. 24 Re

Re2/3 6

3 Results and discussion The case considered corresponds to the planar mixing layer studied experimentally by [10], formed between two ambient temperature air streams with a velocity ratio of 0.28. The test section is shown in Fig. 1. The fastest air stream was seeded with water droplets with a Sauter Mean Diameter, SM D = 35 µm. The resulting spray ﬂow is characterised as dilute with a volumetric void fraction of 5.5 × 10−6 . The LES code BOFFIN was used for the computations conducted. The computational domain, consisted of 150 × 80 × 60 nodes in the x, y, z directions respectively. At the inﬂow plane the ﬂow is fully developed thus a turbulent generator, [11] was used. The measured droplet diameter pdf at the inlet plane is well represented by the Nukiyama–Tanasawa [12] distribug tion function p (D) = aDp e−(bD) with a suitable selection of the parameters p, b, g, Fig. 2. Figure 3 shows computed and experimental time averaged gas and droplet velocity cross stream proﬁles at selected downstream stations, located in the near-ﬁeld of the mixing layer. Following the experimental procedure of [10], the proﬁles are presented in similarity coordinates, scaled with the momentum thickness θu and the velocity diﬀerence ΔU across the layer. The predicted proﬁles are in good agreement with the measurements and are independent of the downstream distance, preserving the self-similarity inflow boundary mesh screen

outflow boundary

y

test section spliter plate

Ua x, g

Ud

atomizers free slip walls

200mm

Fig. 1. Experimental set up. The test section corresponds to the region 200 × 150 × 150.

488

W.P. Jones et al. 0.1

N-T exp

0.08

1

0.06

CDF

PDF

N-T exp

1.2

b=1.2, p=2, q=0.83

0.04

0.8 0.6 0.4

0.02 0

0.2 0

10

20 30 40 Diameter [ μm]

50

0

60

0

20 40 Diameter [μm]

60

Fig. 2. Droplet size distribution in the droplet laden stream.

(U-Umin)/ΔU

0.6

(Ud -Ud,min)/ΔUd

x=1mm x=5mm x=10mm x=15mm exp

1 0.8

0.4 0.2 0 -5

-3

-1 1 (y-y0.5) /θυ

3

0.6 0.4 0.2 0

5

x=1mm x=5mm x=10mm x=15mm exp

1 0.8

-5

-3

-1 1 (y-y 0.5 ) /θυ

3

5

Fig. 3. Normalised stream wise velocity proﬁles for the gas (left) and the dispersed (right) phase: experimental data (symbols) and predicted proﬁles (solid lines).

a

50

30 20 10 0

10

calc exp

8 θu [mm]

40 δ08 [mm]

b

calc exp

6 4 2

0

5

10 x [cm]

15

20

0

0

5

10

15

20

x [cm]

Fig. 4. Longitudinal evolution of the a) velocity level thickness and b) momentum thickness: experimental data (symbols) and computed results (solid lines).

of the ﬂow. Predicted and measured proﬁles of the 0.1–0.9 layer thickness δ08 (y) = y (U = 0.1U∞ ) − y (U = 0.9U∞ ) and momentum thickness θu are presented in Fig. 4a, b, respectively. The calculations are in good agreement with the measurements conﬁrming the linear growth of the layer. Figure 5 compares measured and predicted droplet size distributions in the fast stream of the mixing layer (y = −3 mm) in the middle of the shear layer (y = 0 mm) and in the low speed stream (y = 2.5 mm) at three downstream locations. Parametric investigation were conducted on the inﬂuence of the constants Co and a on the particle diﬀusion (equation (4)) and results for four diﬀerent computations are presented. The maximum deviation of the particle distribution

pdf %

Large Eddy Simulation of a Turbulent Droplet Laden Mixing Layer 60 y=-3.0mm x=1cm 40

y=0.0mm x=10cm

y=2.5mm x=15cm

α=0.5, α=0.8, α=0.8, α=0.8, exp

20 0

0

10 20 D [ μm]

0

10 20 D [ μm]

0

10 20 D [ μm]

Co Co Co Co

= = = =

489

1 1 2 0.5

30

Fig. 5. Pdf of the droplet diameters in the fast stream (left), at the edge of the layer (middle) and in the slow stream (right). 30

30

pdf (D|y) %

25 20

Stokes number

y=-0.05:-0.03 y=-0.03:-0.01 y=-0.01:0.0 y=0.0:0.01

15 10

2-10 11-19 20-28 29-37 38-46

25 20 15

calc μm μm μm μm μm

10 5

5

0

0 0

10

20 30 D [μm]

40

50

0

5

10 x [cm]

15

20

Fig. 6. Pdf of the droplet diameters at x = 15.0cm and at diﬀerent y intervals (left), Longitudinal evolution of the Stokes number as a function of the droplets’ diameter (right).

in the fast and slow speed region was found to be 3%. At the edge (y = 0 mm) of the mixing layer though, an increase of the dispersion constant Co from 1 to 2 (for the same value of a = 0.8), is found to shift the distribution towards smaller particle sizes suggesting that the contribution of the stochastic term becomes more dominant for smaller particles. Figure 6a shows the evolution of the diameter pdf at x = 150 mm and at various positions across the mixing layer. In moving from the fast stream region towards the edge of the layer and the slow stream zone the accumulation of small particles increases by around 5%. For each droplet class the Stokes number, deﬁned as the ratio of the viscous droplet relaxation time to the characteristic timescale of the ρd d2 ΔU energy containing eddies, St = was computed and is compared with 18μδ08 the experimental data, Fig. 6b. Predicted Stokes numbers are found to agree well with measured values and to increase with increasing droplet size at a given location x. The decrease in the Stokes number, is due to the continuous increase of the layer thickness δ08 that characterises the spreading of the mixing layer.

490

W.P. Jones et al.

4 Conclusions A numerical investigation on the spreading behaviour of liquid droplets in a turbulent mixing layer using LES was conducted. A stochastic model was incorporated to simulate the eﬀects of unresolved scales on particle dispersion and the model reproduces well the observed size-selective phenomena in shear ﬂows. The calculations presented are relevant to spray combustion dynamics, since they focus on the dispersion eﬀects in the near-ﬁeld of the mixing layer, a region important in combustion processes related to ignition and ﬂame stabilisation.

Acknowledgements This work received funding from the European Community through the project TIMECOP-AE (Project No. AST5-CT-2006-030828). It reﬂects only the authors views and the Community is not liable for any use that may be made of the information contained therein.

References 1. J. N. Chung, T. R. Troutt, J Fluid Mech. 186 (1988) 199–222. 2. R. Chein, J. N. Chung, International Journal of Multiphase Flow 6 (1987) 785–802. 3. B. J. Lazaro, J. C. Lasheras, J Fluid Mech. 235 (1992a) 135–178. 4. G. Brown, A. Roshko, J Fluid Mech. 64 (1974) 775–816. 5. Q. Wang, K. Squires, Phys. Fluids 8 (1996) 1207–1223. 6. J. Smagorinsky, Monthly Weather Review 91 (1963) 99–164. 7. M. Bini, W. P. Jones, J Fluid Mech. to appear. 8. M. Bini, W. P. Jones, Phys. Fluids 19 (2007) 035104. 9. M. C. Yuen, L. W. Chen, Comb. Sci. & Tech. 14 (1976) 147–154. 10. M. S. Tageldin, B. M. Cetegen, Comb. Sci. & Tech. 30 (1997) 131–169. 11. L. D. Mare, M. Klein, W. Jones, J. Janicka, Phys. Fluids 18 (2006) 025107. 12. A. H. Lefevre, Atomization and Sprays, Hemisphere Publishing Corp., 1989.

The Diﬀuse Interface Method with Korteweg Approach for Isothermal, Two-Phase Flow of a Van der Waals Fluid A. Pecenko and J.G.M. Kuerten Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]

1 Introduction Direct numerical simulation of multiphase ﬂows represents a challenging task, since the interfacial layer separating the bulk phases experiences complex topological changes. In our work, we consider the two-phase ﬂow of a singlecomponent, viscous, Newtonian ﬂuid undergoing isothermal phase transitions and/or phase boundary deformations and we propose a method for stable numerical simulation of the ﬂow dynamics based on a single system of governing equations for the whole ﬂow domain. Our formulation of the problem does not require any special treatment for the interfacial zone, since we make use of a diﬀuse interface concept [1], and of the Korteweg capillary tensor in the Navier–Stokes equations [2]. This approach allows to overcome some typical disadvantages of sharp interface model-based methods in which the interfacial layer separating the bulk phases is captured or rather tracked, provided that the assumption of zero interface thickness is made. A review of these methods can be found for example in [3]. Their reliability depends on the way the front is marked during its motion, and on the type of grid that is used. The main disadvantage of the sharp interface concept, however, lies in discontinuities which ﬂuid properties exhibit over each phase boundary. Moreover, the so-called interface source terms, primarily caused by surface tension, must be included in some way in the system of equations. With the approach that we follow, the functions representing ﬂuid properties are continuous everywhere, and interfacial forces are incorporated into the model in a consistent way. The notion of phase boundary as a small but ﬁnite region of the ﬂow domain, where density, pressure and other quantities vary rapidly but smoothly, can be related to the interfacial Helmholtz free energy, which can be expressed (see [4, 5]) by means of the local density gradient. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 71, c Springer Science+Business Media B.V. 2010

491

492

A. Pecenko and J.G.M. Kuerten

This gradient theory allows a continuum mechanical description of the interface between two phases of the same substance, as the capillary forces, and therefore surface tension, can be explicitly written in terms of the density gradients by minimizing the Helmholtz energy expression. The result is the aforementioned Korteweg tensor [1] 1 T = −p + Kρ∇2 ρ + K | ∇ρ |2 I − K∇ρ ⊗ ∇ρ (1) 2 where K, often called energy-gradient coeﬃcient, plays the role of a capillary coeﬃcient related to surface tension. In general, K is a function of density and temperature, but in the present study we ignore the dependence on temperature, since we only consider isothermal situations. We then refer to the following system of governing equations in Eulerian coordinates and in conservative form ρt + ∇ · (ρu) = 0 (ρu)t + ∇ · (ρuu) = ∇ · (d + T)

(2) (3)

ρ being mass density, u velocity. Here, d denotes the usual viscous stress tensor with linear stress–strain relation. In order to close the previous system, an equation of state relating pressure to density must be assigned. If temperature is chosen slightly below its critical value, the Van der Waals equation of state provides a suﬃciently accurate description in both vapour and liquid phases of the ﬂuid, and reads p(ρ, T ) =

RT a 2 ρ ρ− M − bρ M2

(4)

R being the gas constant, M the molar mass and a and b two constant coeﬃcients which depend on the substance considered and are usually determined experimentally. Below the critical temperature, the relation between p and ρ at constant T has the form shown in Fig. 1. The non-monotonic shape of the isotherm brings intrinsic instability into the system of equations, since the mathematical nature of the problem becomes hyperbolic–elliptic. In addition, the Korteweg tensor itself may introduce a severe dispersion eﬀect onto the numerical solution, if no adequate countermeasure is taken. In Section 2, we present the outlines of the numerical method that we have devised in order to obtain stable solutions.

2 The numerical method In [6] a numerical scheme is presented for one-dimensional, isothermal propagation of phase boundaries in solids when a non-monotonic constitutive relation for pressure is taken. Solutions are obtained for the Riemann problem

Diﬀuse Interface Method for Two-Phase Flow

493

x 106 8 7 6 5

psat

p(Pa) 4 3 2 1 0

VS 0

VM

PM

100

150

50

LM LS 200

ρ(kg / m3)

250

300

Fig. 1. A Van der Waals isotherm below the critical point in the (p, ρ) plane. The horizontal solid line represents the saturation pressure psat at the assigned temperature. The vertical lines mark the diﬀerent regions of the solution domain according to equation of state (4). VS= vapour stable, VM= vapour metastable, PM= phase mixture, LM= liquid metastable, LS= liquid stable. The region of instability for the system of equations (2)–(3) corresponds to the phase mixture.

in a Lagrangian frame, by adding both a viscous and a capillary term to the momentum equation and for vanishing viscosity and capillary coeﬃcients, the usual ”entropy condition” being not suﬃcient for a hyperbolic–elliptic system of equations. In our study, we consider a single-component ﬂuid as a substance endowed with a given viscosity coeﬃcient μ(ρ) and a given capillary coeﬃcient K(ρ). Moreover, even apart from the nonlinear relation (4) for p(ρ), the Eulerian system of equations (2)–(3) is aﬀected by further nonlinearities due to convective, diﬀusive and capillary terms. Therefore, we have generalized the method in [6] to provide stable solutions for liquid/vapour one-dimensional and multidimensional isothermal ﬂow problems. The core of our formulation is a transformation of variables in the equations, which makes it possible to control the magnitude of the capillary term in the discretized equations: ρ = ρˆ

ρu = ρ? u − ν0 (ˆ ρ)∇ˆ ρ.

(5)

The main source of dispersion is cancelled out of the transformed equations if ν0 is chosen in such a way that the diagonal term of the matrix of transformed capillary coeﬃcients K0 (ˆ ρ) = K(ˆ ρ) −

ρ) 4 ν0 (ˆ ρ) − ν0 (ˆ ( μ(ˆ ρ)ˆ ρ) 2 ρˆ 3

is identically equal to zero. Space discretization uses a central ﬁnite diﬀerence scheme that is globally accurate at the second order. A TVD–Runge–Kutta third-order accurate time integration scheme is adopted for the control of the oscillations produced in the unstable, elliptic region of the solution domain.

494

A. Pecenko and J.G.M. Kuerten

We have developed the method starting from the one-dimensional formulation in order to assess the consistency of the numerical results with the mixed mathematical nature of the problem, by choosing diﬀerent sets of initial conditions and comparing our results with relevant examples in literature. Next, we have extended the method to two and three dimensions and we have tested it on classic benchmark cases, namely the retraction of a non-circular drop surrounded by vapour in quiescent state, and the head-on collision of two circular drops. These results are presented in Section 3.

3 Benchmark simulations 3.1 Drop retraction The retraction of a drop in a quiescent medium is a simple but eﬀective way to observe the action of surface tension. It is well known that the equilibrium condition for a static drop, in the absence of other forces, is represented by a spherical shape. Hence, if another shape is prescribed as initial state, the nonuniform capillary forces will drive the system towards the state of mechanical equilibrium. Figure 2 shows the two-dimensional case. 3.2 Two-drop collision The time evolution of the coalescence process of two circular drops is shown in Fig. 3. The initial impulse that pushes the drops towards each other along the collision trajectory is given by a divergence-free velocity ﬁeld.

tfinal

t=0

1.6

220

1.4

200 180

1.2

160

y 1

1.6

220

1.4

200 180

1.2

160

y 1

140 0.8

140 0.8

120 0.6

120 0.6

100 0.4 0.5

1 x

1.5

100 0.4 0.5

1 x

1.5

Fig. 2. Retraction of an elliptical drop surrounded by quiescent vapour, under the eﬀect of capillary forces: initial state (left) and ﬁnal equilibrium state (right). The interface is represented by means of density isolines. Time t is expressed in arbitrary units.

Diﬀuse Interface Method for Two-Phase Flow t=6

t=0 2

220

2

220

1.8

1.8 200

1.6 1.4

200

1.6 1.4

180

180

1.2

1.2

y 1

160

y

1

160

0.8

0.8 140

0.6 0.4

140

0.6 0.4

120

120

0.2

0.2 0

0

0.4

0.8

1.2

1.6

2

0

100

0

0.4

0.8

x

220

1.2

1.6

2

2

1.8

100

220

1.8 200

1.6 1.4

200

1.6 1.4

180

1.2

180

1.2

1

160

0.8

y

160

1 0.8

140

0.6 0.4

120

0.2 0

x

t=99

t=8 2

y

495

140

0.6 0.4

120

0.2 0

0.4

0.8

x

1.2

1.6

2

100

0

0

0.4

0.8

x

1.2

1.6

2

100

Fig. 3. Time evolution of the collision of two circular drops surrounded by the vapour phase and endowed with the same density distribution across their diﬀuse interfaces represented by isolines. The picture at time t = 99 shows the ﬁnal equilibrium state.

When drops collide, the eﬀect of short-ranged attractive forces between the molecules on the interface causes the coalescence into a single drop of larger size. Then, after a transient, the drop that has originated from this process acquires the circular, equilibrium shape. 3.3 Determination of surface tension As further validation of the numerical results, the surface tension coeﬃcient γ can be determined from the radius R of the drop and the pressure diﬀerence Δp = cγ/R between inside and outside the drop, according to the Laplace equation where c equals 1 in two dimensions and 2 in three dimensions. In Fig. 4 the theoretical value of γ for the prescribed isothermal situation and for various drop sizes is compared with the numerical value, showing that the latter is accurate, within a reasonable uncertainty, and remains almost constant as the size of the drop is increased. This result is achieved in both two- and three-dimensional simulations. The picture on the right-hand side shows a study of grid convergence.

496

A. Pecenko and J.G.M. Kuerten 1.1

1.05 γ / γth 1

0.95

2D 3D 0

0.5

1

1.5 R

2

2.5 3 x 10−7

Fig. 4. (left) Comparison between numerical and theoretical value of surface tension coeﬃcient for diﬀerent radii of a static drop. The error bars represent the uncertainty in the drop radius. Abscissa is expressed in meters. (right) Grid reﬁnements are shown for a portion of the pressure proﬁle as a function of x, along a line passing through the center of the drop.

4 Concluding remarks In our work, we have used a single system of governing equations for the whole computational domain of a two-phase ﬂow, where the interface is represented by a stress tensor in terms of density gradients. This formulation allows to follow the evolution in shape and position of the interface without need for any additional speciﬁc relation for the interfacial layer. Furthermore, surface tension can be easily computed by using the data acquired from the numerical simulation. Later, we intend to include the equation of conservation of energy, so that non-isothermal situations and related heat exchange can be studied.

Acknowledgements This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Aﬀairs.

References 1. Anderson DM, McFadden GB, Wheeler AA (1998) Annu Rev Fluid Mech 30: 139–165 2. Korteweg DJ (1901) Arch N´eerl Sci Exactes Nat s´er II 6:1–24 3. Tryggvason G, Bunner B, Esmaeeli A (2001) J Comput Phys 169:708–759 4. Cahn JW, Hilliard JE (1958) J Chem Phys 28,2:258–267 5. Cahn JW (1959) J Chem Phys 30,5:1121–1124 6. Cockburn B, Gau H (1996) SIAM J Sci Comput 17,5:1092–1121

Numerical Simulation of Air Flows in Street Canyons Using Mesh-Adaptive LES Dimitrios Pavlidis1, Elsa Aristodemou2 , Jeﬀerson L.M.A. Gomes1 , Christopher C. Pain1 , and Helen ApSimon3 1

2

3

Dept. Earth Science & Engineering, Imperial College London, London, UK, [email protected]; [email protected]; [email protected] School of Science & Technology, Nottingham Trent University, Nottingham, UK, [email protected]; [email protected] Centre for Environmental Policy, Imperial College London, London, UK, [email protected]

Abstract In this study a novel approach for modelling urban atmospheric ﬂows is presented. It uses a modiﬁed general purpose CFD model with anisotropic mesh adaptivity. The eﬀect of traﬃc induced turbulence is modelled through a two-ﬂuid approach giving the ability to explicitly model the movement of individual vehicles. Results are presented from the application of the model to a real urban area.

1 Methodology Eﬀective air quality management and response to air quality emergencies necessitates the implementation of micro-scale models that are able to capture adequate spatial and temporal variability of urban pollution dispersion patterns [1]. Although Direct Numerical Simulation (DNS) can produce results that are indistinguishable from measurements [2], the computational demands tend to impose severe restrictions on the complexity that can be considered. Thus, to model realistic urban ﬂows, the best compromise between steady-state ﬂows and DNS is large eddy simulation (LES), especially when an adaptive-mesh is employed [3]. We implement in this study the LES methodology developed by [4], which combines a Smagorinsky-type sub-grid-scale turbulence model, with a fully adaptive unstructured mesh that optimises the mesh throughout the ﬂow. Transport of pollutant concentrations is determined by a high resolution method, which is globally high order accurate in space and time. This is suﬃciently robust to be used with multiphase ﬂow problems [5]. The model employs a world-leading anisotropic mesh adaptivity method based on mathematical optimisation as described in [5]. The methodology has been validated against wind tunnel data [6]. Traﬃc modelling is based on the two-ﬂuid approach, with air representing the ﬁrst ﬂuid, whilst traﬃc being the V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 72, c Springer Science+Business Media B.V. 2010

497

498

D. Pavlidis et al.

second one. Individual vehicle shapes are represented by simple shapes which are moved around the ﬁrst ﬂuid (air) by solving a set of equations associated with the second ﬂuid phase (traﬃc). This extremely ﬂexible approach enables resolution of individual vehicles shapes (given adequate numerical resolution due to mesh-adaptivity) or parametrisation of a vehicle or ﬂow of traﬃc with an eﬀective drag force. 1.1 Inlet boundary conditions The inlet boundary conditions for this study are based on the turbulent coherent structures approach of [7]. Coherent structures are introduced around the inlet plane, and are deﬁned by a time-varying and spatially-varying shape function. The ﬁnal velocity boundary condition is reconstructed according to: ¯i + aij u , where U ¯i is the mean velocity represented by a logarithUi = U j mic function, αij is an amplitude tensor and is correlated to the prescribed Reynolds stress tensor via: ⎛ √ ⎞ R11 0 0 ⎠ R22 − α221 0 αij = ⎝ R21 /α11 (1) 2 2 R31 /α11 (R32 − α21 α31 )/α22 R33 − α31 − α32 and uj is the normalised ﬂuctuating component calculated using: N 1 ui (x, t) = √ εij fi (x − xj (t)) N j=1

(2)

where, N is the number of coherent structures, εij is the sign of structure j on component i and is randomly drawn to be +1, or −1 and f is a Gaussian function that evaluates the eﬀect of each structure on any given inlet plane node based on the structure’s length-scale. 1.2 Traﬃc induced turbulence In order to model traﬃc induced turbulence individual moving vehicles were included. They were modelled using a two ﬂuid approach, where the ﬁrst ﬂuid is the air and the second is the vehicle. Assuming in-compressibility, for any phase f , the continuity equation becomes: ∂αf + ∇ · (αf Uf ) = 0, ∂t

(3)

where αf is the volume fraction of phase f , Uf is the vector of velocity of the phase f . The phase f refers either to the air (subscript: g), or the vehicle volume fraction (subscript: t). The equations for the conservation of momentum for the gas phase and the traﬃc can be appropriately combined, given that the velocity of the second phase (traﬃc) is deﬁned. This allows the momentum equation for the gas phase to be written as:

Air Flows in Street Canyons Using Mesh-Adaptive LES

αg ρg

∂Ug + (Ug − UG ) · ∇Ug ∂t

499

= −∇p + ∇ · τg + FD ,

(4)

where in UG is the ﬁnite-element grid velocity, Ut is the traﬃc velocity and FD is the total drag force caused by both friction and pressure diﬀerences and exerted on the dispersed phase and is given by: FD =

1 CD AI ρg |Ut − Ug | (Ut − Ug ), 8

(5)

where AI is the interface area density exposed to the ﬂow and CD the drag coeﬃcient. Appropriate expressions for AI need to be speciﬁed, depending on the traﬃc volume i.e. αt < 0.25. Similarly, an appropriate expression for the drag coeﬃcient is implemented, based on the Reynolds number of the second phase (traﬃc).

2 Results and conclusions Numerical LES results for air ﬂows in the Marylebone area, central London [8] are presented in this study. Of primary importance in these simulations, is the eﬀect of traﬃc turbulence. The computational domain with the building conﬁguration is shown in Fig. 1a. Figure 1b shows the eﬀect of the vehicle movement on the adaptive mesh in a horizontal plane, whilst Fig. 2a shows the eddies formed above the vehicle in a vertical cross-section. It is clear that the motion of vehicles disturbs the airﬂow patterns in the street, and it is of prime importance to be able to model these eﬀects, particularly in relation to pollution studies. A comparison of the diﬀerence in the velocity values, for all three velocity components, for the cases when vehicles are present and not, is shown in Fig. 2b. The study shows the implementation of an adaptive-mesh LES methodology, in a complex street canyon environment for

Fig. 1. (a) The 48 building computational domain within the Marylebone area, and (b) the adaptive mesh in a horizontal plane showing eﬀect of motion of 10 identical vehicles travelling at a constant speed in the main street at low ambient wind speeds. There are 10 vehicles in total in this simulation.

500

D. Pavlidis et al.

a

Velocity timeseries at middle of intersection 0 0.08

0.8 0.7 0.6

V (m/sec)

U (m/sec)

0.06

−0.2

0.5 0.4 0.3 0.2

−0.3 −0.4

0

10 20 time (sec)

30

−0.7

0.02

−0.02

−0.6

0

0.04

0

−0.5

0.1

−0.1

incl. traffic no traffic

−0.1

W (m/sec)

b

0

10 20 time (sec)

30

−0.04

0

10 20 time (sec)

30

Fig. 2. (a) A vertical cross-section, showing the eddies produced around the vehicles, and (b) comparison of velocity components at the central intersection with and without traﬃc: blue line – with traﬃc; green line – without.

pollution studies. Vehicle movement is taken into account, and the advantage of the adaptive LES approach is clear, as it is able to resolve ﬂow movement around the vehicle, thus capturing the eﬀects necessary for pollution studies. Testing and validation of the results from wind tunnel data is in progress.

References 1. Air Quality Expert Group: Particulate Matter in the UK. Department of the Environment, Food, and Rural Aﬀairs. (2005) 2. Coceal, O., Dobre, A., Thomas, T.G. and Belcher, S.E.: Structure of turbulent ﬂow over regular arrays of cubical roughness. J. Fluid Mech. 589 (2007) 375–409 3. Pope, S.B.: Turbulent Flows. Cambridge University Press. (2000) 4. Bentham, T.: PhD thesis. Imperial College London, London, UK. (2004)

Air Flows in Street Canyons Using Mesh-Adaptive LES

501

5. Pain, C.C., Umpleby, A.P., de Oliveira, C.R.E., Goddard, A.J.H.: Tetrahedral mesh optimisation and adaptivity for steady-state and transient ﬁnite element calculations. Comp. Meth. in Appl. Mech. and Eng. 190 (2001) 3771–3796 6. Aristodemou, E., Bentham, T., Pain, C.C., Robins, A.: A comparison of meshadaptive LES with wind tunnel data for ﬂow past buildings: mean ﬂows and velocity ﬂuctuations. Atm. Env. Reviewed (2009) 7. Jarrin, N., Benhamadouche, S., Laurence, D., Prosser, R.: A synthetic-eddymethod for generating inﬂow conditions for large-eddy simulations. Heat and Fluid Flow. 27 (2006) 585–593 8. Arnold, S., ApSimon, H., et al.: Dispersion of Air Pollution and Penetration into the Local Environment - DAPPLE. Sci. of the Total Env. 332 (2004) 139–153

Part VII

Aerodynamics and Wakes

LES of the Flow Around a Two-Dimensional Vehicle Model with Active Flow Control S. Krajnovi´c and J. Fernandes Chalmers University of Technology, Department of Applied Mechanics, Gothenburg, Sweden, [email protected]

Abstract The technique of large-eddy simulation (LES) was used to study the inﬂuence and the resulting ﬂow mechanisms of the active ﬂow control applied to a two-dimensional vehicle geometry. Duplication of a previous experimental study performed with LES was used to verify the numerical approach. This was followed with an exploration of the inﬂuence of ﬂow actuation on the near-wake ﬂow and resulting aerodynamic forces. Not only was good agreement with the previous experimental study obtained, but new knowledge was gained in the form of complex interaction of the actuation with the coherent ﬂow structures. The resulting time-averaged ﬂow shows a strong inﬂuence of the extension of the actuation slits and the lateral solid walls on the near-wake ﬂow structures and thereby on the resulting drag.

1 Introduction Energy-eﬃcient ground vehicles require careful geometrical styling to decrease drag primarily due to the formation of wake region. However, commercial vehicles such as trucks and buses do not permit any great change of the rear part of the vehicle. Instead of changing their geometry attempts can be made to inﬂuence the ﬂow using ﬂow control, which can be either active or passive. Both strategies suﬀer from the ability to adjust to the ﬂow conditions if there is no feedback information from the actual ﬂow. The development of an eﬃcient strategy for closed-loop active ﬂow control requires an increased understanding of open-loop active ﬂow control and the ﬂow mechanisms that are likely to produce an increase in the base pressure of ground vehicles. The goal of the present paper is twofold: to investigate the applicability of the LES technique for the purpose of ﬂow control and to increase our understanding of the ﬂow mechanisms acting in an active ﬂow control process. A two dimensional bluﬀ body with a lateral shape similar to a so called Ahmed body [1] used in the experimental study by Pastoor et al. [4] was

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 73, c Springer Science+Business Media B.V. 2010

505

506

S. Krajnovi´c and J. Fernandes

b

a

8

U

W H

c

L

Fig. 1. (a) Geometry of the 2D Ahmed body. (b) Zoom of the trip tape. (c) Zoom of the actuation slots.

employed in the present study. The interaction of the upper and lower shear layers after the leading edges of the 2D Ahmed body (Fig. 1) results in von Karman-like instabilities. Such instabilities early produce two large 2D vortices in alternating order. As the vortices are formed very early, the near-wake separation bubble (the dead water) is short producing low base pressure and large drag. An increase of the base pressure can be achieved by elongation of the near wake region and suppressing or delaying the shear layer interaction. To achieve this objective, the present work applies the strategy used in [4] to force symmetric vortex shedding and thereby delay the wake instabilities.

2 Description of the model and numerical set-up The geometry of the body is shown in Fig. 1. It has a cross section from a side of a simpliﬁed bus with chord length L = 262 mm, height H = 72 mm and spanwise width W = 550 mm. The front of the body has a radius of 25 mm. Two trip tapes of height 0.8 mm and length 5 mm were placed on the upper and the lower face of the body 30 mm downstream of the front. Two pairs of slots extending in the spanwise direction are used for actuation of the ﬂow. The slots of width 1 mm and spanwise length of 250 mm were placed on the upper and the lower face (one per face) and on the rear face of the model (two slots). This is the only diﬀerence in the set-up conﬁguration from the experiments, where only two slots were placed at the upper and lower trailing edges with an angle of 45◦ relative to the streamwise direction. The model was centered in the computational domain with a height of 7.7H. The distance from the model to the inlet and the outlet was 10.25H and 20.83H, respectively.

LES of Active Flow Control on a 2D Vehicle

507

2.1 Boundary conditions and the actuation The Reynolds number of the ﬂow was ReH = 2 × 104 based on the inlet velocity and the height of the model. A constant velocity proﬁle was applied at the inlet, while the homogeneous Neumann boundary condition was used at the outlet. A no-slip boundary condition was applied on the surface of the body, and wall functions were used on the upper, lower and lateral walls. Harmonic actuation was applied to each slot by imposing a velocity on the boundary equal to us = uA sin(ωA t)(cos(φ)i + sin(φ)j), where i and j are the unit vectors in the x and y directions, respectively, and the actuation angle, φ, was 45◦ , in agreement with the experiments. The actuation amplitude, uA , can be derived from the expression for the momentum coeﬃcient, Cμ = 4su2A /HU∞ . The actuation frequency and the moment coeﬃcients in the present study were StA = 0.17 and Cμ = 0.015, respectively. 2.2 Numerical simulations Large eddy simulations using the standard Smagorinsky model with the constant CS = 0.1 were made with a commercial ﬁnite volume solver, STAR-CD. A blend of a second-order central diﬀerence and second order upwind scheme was used for the spatial discretization of the convective terms. An implicit second-order scheme was used for the temporal discretization. A structured computational grid containing a total of 5.5 × 106 nodes was used for both the natural and the controlled ﬂow simulations. This resulted in a resolution expressed in the wall units of approximately y + = 1, Δx+ = 14 and Δz + = 30 in the wall-normal, streamwise and spanwise directions, respectively. A non-dimensional time step, t∗ = tU∞ /H = 0.0058, resulted in the maximal CFD number of approximately 0.3 and 8.5 in the natural and controlled ﬂow simulations, respectively. However, the CFL number is smaller than one in 96.5% of the computational cells during the entire simulation. Both simulations were ﬁrst run until the values of the aerodynamic coeﬃcients stabilized before the averaging was initiated.

3 Results Figure 2a shows that the actuation dominates the near-wake ﬂow resulting in a drag signal that is almost aligned with the signal of the actuation (Fig. 2b). Figure 2b shows that blowing and suction produce maximum and minimum drag, respectively. The reason for this is found in the 2D coherent structures observed in the natural ﬂow that are also formed during the blowing sequence of the actuation but are destroyed in the course of the suction cycle. The LES of the natural and the controlled ﬂows resulted in mean CD values of 1.02 and 0.92, i.e. a decrease of about 11% using ﬂow actuation. This can be compared with 0.98 and 0.83 obtained in the experimental study at

508

S. Krajnovi´c and J. Fernandes

a

b 1.15 CD

1.1 1.05 CD

0.95 0.9 0.85 0.8 0.75

A

C B

D

1 0.95 0.06

0.9 0.85

CL 0.04

0.8

0.02 50

100

150

200 8

t*=t*U /H

250

300

A

274

280

E 286

F 291

297

303

309

t* = t * U∞/H

Fig. 2. (a) Time history of the drag coeﬃcient for the controlled ﬂow. (b) Partial time history of drag and lift coeﬃcients for the controlled ﬂow. Positions A and C are locations of maximal blowing, while positions B and D are locations of maximal suction. E and F are intermediate points between blowing and suction.

Fig. 3. Velocity vector ﬁeld and vorticity ωz (left ﬁgure) and an isosurface of p = −12, colored with ωz for natural ﬂow (right ﬁgure).

slightly higher Reynolds number, Re = 2.3 × 104 [2, 4], for the natural and the controlled ﬂow, respectively. The source of the relatively large diﬀerence (≈ 10%) in drag coeﬃcient for the controlled case between the experiment and the LES is not clear. There are two possible sources of error: the difference in the set-up conﬁguration of slots between the experiment and the LES and the usage of a blend of central diﬀerence and second-order upwind scheme for the convective terms, which might be too dissipative for an LES. However, with small diﬀerences in velocity proﬁles between experiments and the LES (not shown here), it is diﬃcult to explain the large diﬀerence in the drag coeﬃcient. A comparison of Figs. 3, 4a, c shows that blowing postpones the rollover of the shear layer and formation of strong 2D alternating vortices. On the other hand, the suction leads to strong perturbations of the 2D structures and their breakdown into small coherent structures (Fig. 4b, d). Note, however, that even these structures are predominantly 2D. Figures 3 and 4d indicate

LES of Active Flow Control on a 2D Vehicle

509

Fig. 4. Velocity vector ﬁeld and vorticity ωz (upper ﬁgures) and isosurface of p = −12 colored with ωz for maximal (a and c) and minimal (b and d) values of drag coeﬃcient cd in Fig. 2. 2.5

x 10

−3

1

x 10

−3

0.9 0.8

2

PSD

PSD

0.7

1.5

0.6 0.5 0.4

1

0.3 0.2

0.5

0.1

0 0.1

0.15

0.2

0.25

0.3

0.35

St = fH/U∞

0.4

0.45

0.5

0 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

St = fH/U∞

Fig. 5. Power spectral density of the drag signal for (a) natural ﬂow and (b) controlled ﬂow.

that the actuation strategy used in the present work is eﬃcient in postponing the vortex formation. For example, it is obvious that, during blowing, the length of the free shear layer in Fig. 4a is longer than length in the natural ﬂow case in Fig. 3. On the other hand, all large coherent structures are eﬃciently destroyed during the suction phase (Fig. 4b, d). Thus both cycles desynchronize shear layers and thereby inﬂuence wake dynamics in a way that the interaction of vortices rolling up from the two shear layers is decreased or delayed. Both result in an increase of the base pressure and a reduction of the drag. Power density spectra of both drag and lift signals were computed for the natural and the controlled ﬂows. Figure 5 shows the more interesting spectral picture of the drag signal. A very clear dominant frequency was found at Stnat = 0.31 in the natural ﬂow case. This can be compared with several dominant frequencies of StEXP nat = 0.23 − 0.25 found in the results of Pastoor et al. [4]. However, in the controlled ﬂow, the dominant frequency was found at St = 0.17, and St = 0.31 was the second peak frequency. This result

510

S. Krajnovi´c and J. Fernandes

Fig. 6. Time-averaged ﬂows for natural ((a)–(c)) and controlled ﬂows ((d)–(f )). Figures (a) and (d) show a view from above the body and vortex cores and particle traces on the body surface. The body is colored with the surface pressure. Figures (b), (c), (e) and (f ) show streamlines projected on the plane z = 0 ((b) and (e)) and z/H = 3.47 ((c) and (f )).

is interesting as it shows that the actuation (at Stact = 0.17) dominates the response and that the natural ﬂow shedding frequency remains superimposed onto the drag signal (note that the frequency of the natural ﬂow, Stnat = 0.31, is identical to the second peak frequency in the controlled ﬂow). The controlled ﬂow signal transformed into the frequency space also shows a considerably larger number of distinct frequencies (between 0.1 < St < 0.3) which is in line with the break-up of large alternating vortices into a large number of smaller coherent structures with diﬀerent characteristic frequencies observed and discussed above. Figure 6 shows the vortex cores together with the streamlines projected onto the symmetry plane and a plane in the vicinity of one lateral wall of the domain. As seen in this ﬁgure, the actuation has a larger impact on the spanwise part of the near wake where the actuation was not applied, i.e. close to the lateral walls. Streamlines in Fig. 6b, e show that there is an elongation of the near-wake separation bubble in the center plane. However, this eﬀect is much stronger close to the lateral wall (Fig. 6c, f). This is an unexpected result and could be a consequence of the presence of solid lateral walls. This raises questions about the eﬃciency of the proposed actuation strategy in the case of a 3D Ahmed body [3]. Besides, it suggests an elongation of the spanwise length of the computational domain. Such an elongation of the domain would reduce the inﬂuence of the boundary layer and the coherent structures (horse-shoe vortices formed close to the lateral walls in the interaction of the boundary layer with the body) on the actuated near wake.

Acknowledgments The work presented here was supported by Banverket (the Swedish National Rail Administration). CD-adapco contributed licenses of the solver, STAR-CD. Computer time at SNIC (the Swedish National Infrastructure for

LES of Active Flow Control on a 2D Vehicle

511

Computing) at the Center for Scientiﬁc Computing at Chalmers (C3SE) is gratefully acknowledged. The help of Dr. Lars Henning at the Measurement and Control Group at TU Berlin in providing the experimental data and details about the experiments is gratefully acknowledged.

References 1. S. R. Ahmed, G. Ramm, and G. Faltin. Some salient features of the time averaged ground vehicle wake. SAE Paper 840300, 1984. 2. L. Henning. Private communication. TU Berlin, 2008. 3. S. Krajnovi´c and L. Davidson. Flow around a simpliﬁed car, part 1: Large eddy simulation. ASME: Journal of Fluids Engineering, 127:907–918, 2005. 4. M. Pastoor, L. Henning, B. R. Noack, R. King, and G. Tadmor. Feedback shear layer control for bluﬀ body drag reduction. Journal of Fluid Mechanics, 608: 161–196, 2008.

Wake-Vortex Decay in External Turbulence Bernard J. Geurts1 and Arkadiusz K. Kuczaj2 1

2

Multiscale Modeling and Simulation, Faculty EEMCS, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands and Anisotropic Turbulence, Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands, [email protected] Computational Fluid Dynamics, Nuclear Research and Consultancy Group (NRG), P.O. Box 25, 1755 ZG Petten, The Netherlands, [email protected]

1 Introduction Wake vortices that form behind a moving aircraft represent a safety concern for other aircrafts that follow. These tornado-like wake structures may persist for several minutes, extending for many kilometers across the sky. This safety issue is particularly important close to major airports where successive aircrafts follow approximately the same approach or departure paths at relatively high frequency. For this reason, safety regulations require aircrafts to be well separated in ﬂight. This obviously limits the airport’s throughput capacity, even to the point of leading to congestion and delays. Current aircraft separation rules have proven to be very eﬀective in preventing wake vortex encounters, but there is a consensus that these rules are often overly conservative under most meteorological conditions. A timely challenge is to relax current separation rules by applying dynamic aircraft spacing to increase airport capacity and to reduce air traﬃc delays. This requires accurate wake-vortex prediction systems to enable an appropriate response to weather conditions, particularly where atmospheric turbulence is concerned. We investigate the decay of a model wake-vortex system, and focus on the inﬂuence of external turbulence on the decay-rate. It may be expected that the ‘entrainment’ of external turbulence into the vortex cores will enhance viscous dissipation and thereby speed-up the decay. Moreover, turbulent ﬂuctuations will bring together regions of the ﬂow with opposite vorticity more rapidly, thereby also enhancing the overall reduction of circulation. A qualitative illustration may be seen in Fig. 1. The ﬂow problem is simulated using direct numerical simulation (DNS), which provides a point of reference for assessing the Leray and NS-α regularization models. In Section 2 we introduce the computational setting in which we model the vortex system. Then, in Section 3 we present results of DNS, illustrating the quantitative diﬀerences in the vortex decay-rate due to diﬀerences in V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 74, c Springer Science+Business Media B.V. 2010

513

514

B.J. Geurts and A.K. Kuczaj

Fig. 1. Eﬀect of external turbulence on the decay of trailing vortices [1]. Left: decay against a quiescent background, Middle: decay in a weakly turbulent ﬁeld. Right: decay in a strong turbulent ﬁeld. Both ‘sinusoidal’ patterns (left) and localized ‘bursts’ (middle and right) can occur. Vertical axis denotes time.

the strength of the external turbulence. In Section 4 we assess regularization modeling. Finally, concluding remarks are collected in Section 5.

2 Computational setting and regularization modeling The decay of a vortex system in an external turbulent ﬁeld can be systematically investigated in a simple temporal setting in which periodic boundary conditions are adopted. In such a temporal setting the initial state of the vortex system and a realization of the turbulent background are all that is required to simulate the corresponding decay. In this paper we will restrict ourselves to external turbulence that arises from decaying a ‘random’ initial condition in a pre-cursor simulation of homogeneous, isotropic conditions [3]. We adopt a parallelized, fully de-aliased pseudo-spectral method to simulate the ﬂow in a computational box endowed with periodic boundary conditions [2, 3]. The equations for individual Fourier coeﬃcients u are: 1 1 2 2 ∂t + k u(k, t) = W(k, t) − ıkF p(x, t) + |u(x, t)| , k (1) Re 2 where F (a(x, t), k) denotes the Fourier-coeﬃcient of the function a(x, t) corresponding to wave-vector k. This deﬁnes W(k, t) corresponding to the nonlinearity W(x, t) = u(x, t) × ω(x, t) with ω the vorticity. LES using regularization models for the nonlinear term [4,5] implies modiﬁcations W(k, t) of W(k, t). The Leray [6] and NS-α [7] models are: Leray : W(k, t) = −F (u(x, t) · ∇)u(x, t), k (2) NS−α : W(k, t) = −F (u(x, t) · ∇)u(x, t), k 3 −F j=1 uj (x, t)∇uj (x, t), k

Wake-Vortex Decay in External Turbulence

515

where the overbar denotes spatial ﬁltering. This is readily implemented in the pseudo-spectral framework [4]. A faithful model for the dominant ﬂow-features in the far wake of an aircraft can be obtained by adopting a system of so-called strained Batchelor vortices. These are formed by extending 2D Lamb–Oseen vortices and allow the vortex core to be a general space-curve. Denoting the velocity by (u, v, w) in the streamwise, the spanwise and the vertical directions a single strained Batchelor vortex with circulation Γ is given by u(x, y, z) = u∞ + U e−(r/R) 2 Γ z − zc (x) v(x, y, z) = (1 − e−(r/R) ) 2π r2 2 Γ y − yc (x) w(x, y, z) = (1 − e−(r/R) ) 2π r2 2

(3)

where R is the vortex-width, r2 = (y − yc (x))2 + (z − zc (x))2 and (yc (x), zc (x)) denotes the axis of the vortex as function of the downstream distance. We will consider initial vortex cores that are periodic functions of x with amplitude 0.1. In addition, (u∞ , 0, 0) is the velocity in the far ﬁeld and U the velocity along the axis in the x direction. Finally, Γ denotes the circulation of the vortex. Extensions to multiple vortices are readily generated by superposition. A background turbulent ﬁeld obtained from a homogeneous turbulence simulation is added to such a vortex system, to study its decay. The resulting turbulent ﬁeld is rescaled to a desired input energy level and added to the strained Batchelor vortices: u(x, y, z, 0) = aU(x, y, z) + (1 − a)V(x, y, z)

(4)

where U is the initial vortex pair and V a realization of the turbulent background [3]. The parameter a measures the strength of the vortical and turbulent ﬁelds.

3 Direct numerical simulation In this section, we present some DNS results of the decaying wake-vortex ﬂow. Figure 2 illustrates the evolution of the magnitude of the vorticity |ω|. The pure vortical state in (a) starts from a pair of counter-rotating vortices tilted at an angle of 45◦ . At t = 0.1 we notice additional vortex ﬁlaments to emerge. When adding a small amount of external turbulence we notice that the largescale features of the decay process remain largely unchanged, but additional small scales and asymmetries arise very quickly. Adding very strong external turbulence destroys the general vortical impression and yields a ﬂow ﬁeld that is dominated by small-scale turbulence from the onset. The total kinetic energy E(t) = u · u is shown in Fig. 3. A small amount of added turbulence seems to marginally slow down the decay. At more pronounced external turbulence levels we observe that the decay of the kinetic

516

B.J. Geurts and A.K. Kuczaj

t=0

a

b

c

d

1 0.8

1 0.8

1 0.8

1 0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2 0 1 0.8 0.6 0.4 0.2 0 0

0.2 0 1 0.8 0.6 0.4 0.2 0 0

0.2 0 1 0.8 0.6 0.4 0.2 0

0.2 0 1 0.8 0.6 0.4 0.2 0

t = 0.1

0.6 0.8 0.2 0.4

1

0.6 0.8 0.2 0.4

1

1 0.6 0.8 0.2 0.4

0

1 0.8

1 0.8

1 0.8

1 0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2 0 1 0.8 0.6 0.4 0.2 0

0.2 0 1 0.8 0.6 0.4 0.2 0

0

1 0.6 0.8 0.2 0.4

0

0.2 0 1 0.8 0.6 0.4 1 0.2 0.6 0.8 0 0.2 0.4 0

0.6 0.8 0.2 0.4

1

0.2 0 1 0.8 0.6 0.4 0.2 0

e

1 0.8 0.6 0.4

0

0.2 0 1 0.8 0.6 0.4 1 0.2 0.6 0.8 0 0.2 0.4 0

0.6 0.8 0.2 0.4

1

1 0.8 0.6 0.4

0

1 0.6 0.8 0.2 0.4

0.2 0 1 0.8 0.6 0.4 0.2 0

0

0.6 0.8 0.2 0.4

1

Fig. 2. Magnitude of vorticity |ω| at t = 0 and t = 0.1 in case a = 1 (a), a = 0.9 (b), a = 0.8 (c), a = 0.7 (d) and a = 0.5 (e); a = 1 denotes the pure vortical state.

a

b 1.2

0.5

1

0.45

E

S

0.4

0.8

0.35

0.6

0.3 0.4 0.25 0.2 0.2 0

0.15 0.1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

1

−0.2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

Fig. 3. Decay of total kinetic energy E (a) and evolution of skewness S (b) in case a = 1 (solid), a = 0.9 (dash), a = 0.8 (dash–dot), a = 0.7 (triangle), a = 0.6 (circle) and a = 0.5 (diamond); a = 1 denotes the pure vortical state.

energy is strongly enhanced. A purely vortical initial condition ﬁrst needs to go through a transition to small scales before it can achieve strong decay rates. The eﬀect of external turbulence implies a signiﬁcant energy decay already from the onset. The evolution of the ﬂow is further illustrated by looking at the skewness S. While an initial case with strong external turbulence yields values close to S = 0.5 early on, indicative of homogeneous, isotropic turbulence, the purely vortical initial condition is seen to go through transition much more slowly and even at t = 1 achieves values of skewness about S = 0.8, indicating a ﬂow that is still very heterogeneous and anisotropic. The spectrum of kinetic energy showed that a signiﬁcant amount of external turbulence results in a strong depletion of the energy in the larger scales. The smaller scales were quite independent of the initial level of turbulence that was introduced.

Wake-Vortex Decay in External Turbulence

517

b

a 0.5

0.5

0.45 E

0.45

E

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

t

Fig. 4. Decay of total kinetic energy E at a = 0.3 in case Δ = /16 (a) and Δ = /32 (b). Lines are labeled: DNS (solid), ﬁltered DNS (dash), Leray (dot at N = 64, ◦ at N = 128), NS-α (triangle at N = 64, diamond at N = 128).

4 Regularization modeling of vortex decay We consider regularization modeling comparing predictions with ﬁltered DNS at a ﬁlter-width Δ = /16 and Δ = /32 where denotes the length of a side of the cubical ﬂow domain. The decay of kinetic energy is displayed in Fig. 4. Both the Leray and NS-α models provide insuﬃcient decay of kinetic energy. The NS-α model requires higher resolutions to approximate grid-independence [4]. The NS-α model was found to yield rather poor predictions for the skewness; too high values of about 0.75 were observed compared to 0.6 obtained in ﬁltered DNS at Δ = /32. The Leray model slightly underpredicts skewness at values around 0.55. The prediction of the Taylor Reynolds number Reλ is shown in Fig. 5. The exaggerated small-scale variability of the solution obtained with the NSα model expresses itself through a close agreement with the unﬁltered DNS results. Theoretically, the LES solution should approximate the ﬁltered DNS result. This is achieved to a higher degree on the basis of the Leray model.

5 Concluding remarks In this paper we considered the role of external turbulence in the decay of two counter-rotating vortices. At suﬃciently high turbulence levels the decay of kinetic energy was found to be strongly increased. Particularly, the large scales are considerably reduced, hence constituting an interesting option for

518

a

B.J. Geurts and A.K. Kuczaj

b

300

240 220

Reλ 250

Reλ

200 180

200

160 140

150

120 100

100

80 60

50

40

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

t

Fig. 5. Taylor Reynolds number Reλ at a = 0.3 in case Δ = /16 (a) and Δ = /32 (b). Lines are labeled: DNS (solid), ﬁltered DNS (dash), Leray (dot at N = 64, ◦ at N = 128), NS-α (triangle at N = 64, diamond at N = 128).

future aircraft separation under suﬃciently lively meteorological conditions. The prediction of the vortical decay with LES based on regularization modeling was shown to yield accurate capturing of the velocity skewness and the Taylor Reynolds number. The NS-α model was shown to exaggerate the small scale variability of the LES solution, even to the degree that correspondence with unﬁltered DNS was closer than with ﬁltered DNS data. The parametric study documenting the precise inﬂuence of external turbulence on vortical decay at realistic Reynolds numbers is subject of ongoing research and will be published elsewhere.

References 1. Liu, H.-T.: 1992. Eﬀects of ambient turbulence on the decay of a trailing vortex wake. J. Aircraft 29, 255–263 2. McComb, W.D.: 1990. The physics of ﬂuid turbulence. Oxford University Press. 3. Kuczaj, A.K., Geurts, B.J.: 2007. Mixing in manipulated turbulence. J. of Turbulence 7, 1–28, DOI: 10.1080/14685240600827534 4. Geurts, B.J., Kuczaj, A.K, Titi, E.S.: 2008. Regularization modeling for largeeddy simulation of homogeneous isotropic decaying turbulence, Journal of Physics A: Math. Theor. 41, 344008. 5. Geurts, B.J., Holm, D.D.: 2003. Regularization modeling for large eddy simulation. Phys. of Fluids, 15, L13 6. Leray, J.: 1934 Sur les movements d’un ﬂuide visqueux remplaissant l’espace. Acta Mathematica, 63, 193 7. Foias, C., Holm, D.D., Titi, E.S.: 2001. The Navier-Stokes-alpha model of ﬂuid turbulence. Physica D, 152, 505

DNS of Aircraft Wake Vortices: The Eﬀect of Stable Stratiﬁcation on the Development of the Crow Instability G.N. Coleman, R. Johnstone, C.P. Yorke, and I.P. Castro School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected]; [email protected]; [email protected]; [email protected] Abstract A numerical experiment is performed to determine the likelihood that the Crow instability will mitigate the potentially hazardous eﬀects of the buoyancyinduced rebound of initially parallel line vortices. Parameters are chosen to correspond to wake vortices downstream of an elliptically loaded wing with large span (typical of the A380) landing in very stable conditions. The DNS is initiated by perturbing the vortex pair into the shape of the linearly most unstable Crow eigenmode, with a maximum displacement of 1% of the initial distance between the pair. Under these conditions, the Crow instability progresses fast enough to break the two dimensionality of the vortex system before it returns to its original elevation. This suggests that in many cases the Crow instability will prevent the rebounding vorticity from being a serious danger to following aircraft. Whether or not this will always happen in practice is an open question, requiring further investigation.

1 Introduction Stable buoyancy can have a profound impact upon the behaviour of aircraft wake vortices. Spalart [7] found in a numerical study of a pair of plane/line vortices that when it is strong enough, stable stratiﬁcation can even change the propagation direction of the vortex pair, such that the vortices ‘rebound’ and return to their original elevation. The potential consequences of this phenomenon, from an air-traﬃc-control (ATC) point of view, are obvious. The critical parameter in this case is the ‘stratiﬁcation number’ (an inverse Froude number), which can be deﬁned as N0∗ = 2πN b20 /Γ0 , where N is the Brunt– V¨ais¨ al¨a frequency of the stratiﬁed ﬂuid in which the line vortices are embedded, b0 the initial distance between the pair, and Γ0 their vortex strength. Thus, b0 is proportional to1 the span b, and Γ0 equal to the root circulation, of the lifting wing from which the vortex pair emanates, where W = ρ U Γ0 b0 1

Recall that for an elliptic loading, b0 = πb/4 [10].

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 75, c Springer Science+Business Media B.V. 2010

519

520

G.N. Coleman et al.

is the aircraft weight and U its ﬂight velocity. The other relevant parameters are Reynolds number and the manner in which the vorticity is concentrated around the axis of the vortices (see below). Spalart found that the rebounding behaviour occurs when N0∗ is of order one. Although not common, this magnitude of N0∗ is not inconceivable; one could imagine situations in which a very large aircraft, such as the A380 (large b0 ), was landing at the end of a long ﬂight (relatively low Γ0 ) in strongly stratiﬁed conditions (large N ), producing N0∗ ≥ 1. On the other hand, the computations that revealed the rebounding behaviour were purely two dimensional, and assumed a quiescent background [7]. It is therefore an open question whether this phenomenon could ever occur in practice, or is instead mitigated by real-world eﬀects such as background turbulence, or overwhelmed by the Crow instability which becomes active once the third dimension is included. The purpose of this study is to consider the eﬀect of the latter, by investigating how the Crow instability alters the evolution of the stratiﬁed vortex pair. It is thus similar to that of Garten et al. [3], who performed DNS of stably stratiﬁed vortex pairs in 3D domains, focussing on the vortex reconnection processes involved in the breakdown of the Crow instability. (The LES study of Holz¨ apfel et al. [4] is also relevant.) In the present work, the emphasis is upon the time required for the Crow breakdown to occur, and its eﬀectiveness in preventing the rebound mechanism described above.

2 Approach DNS is used to solve the incompressible Navier–Stokes equations, invoking the Boussinesq approximation to account for buoyancy eﬀects. A Fourier representation is assumed in all three spatial directions. The viscous terms are time advanced analytically, via an integration factor [6], while the advective and buoyancy terms employ a compact third-order Runga–Kutta scheme [8]. The code has been previously applied to study stably stratiﬁed vortex rings and stably stratiﬁed turbulent wakes [5]. The present version makes use of Corral and Jim´enez’ [1] boundary condition treatment to allow one of the inﬁnite directions to be accurately represented with a compact ﬁnite domain (resulting in signiﬁcant computational savings). As a result, the inﬁnite periodic array of vortex pairs that would otherwise be present does not exist in the lateral (y) direction. (The periodicity remains in the other two directions, which requires the axial and especially vertical domains to be large enough to avoid any unphysical behaviour associated with the ‘vertical stacking’ of the wake vortices.) The strategy employed here is to determine how likely it is that the stratiﬁcation-induced rebound will occur in practice: we consider worse-case conditions and perform a three-dimensional numerical experiment involving a purely quiescent background, applying a small perturbation solely to the fastest growing (long-wave) linearly unstable Crow mode, choosing N0∗ = 1.2

DNS of Stably Stratiﬁed Aircraft Wake Vortices

521

Fig. 1. Vortex pair initialization, with most unstable Crow eigenfunction [3] perturbed by (a) 0% and (b) 30% of b0 .

as a large but plausible value. Initial conditions consist of a vortex pair whose nominally parallel axes are ‘warped’ such that they trace the shape of the most unstable Crow eigenfunction (i.e. are displaced into a sinusoid of amplitude δ and wavelength 8.6 b0 in planes at 48◦ above the horizontal [2, 3]; see Fig. 1). The results shown here are for δ = 0.01b0. We follow Spalart [9] and specify at t = 0 an axisymmetric radial vorticity distribution ωx (r) approximately corresponding to that produced by an elliptically loaded wing, with the azimuthal velocity component given by Γ0 r uθ (r) = min 1, 1.27 + 14 ln 2πr b0 for r/b0 > 0.0103 and by solid-body rotation for r/b0 < 0.0103. The Reynolds number is Re = Γ0 /ν = 105 . The size of the domain is 8.6 b0 , 4πb0 and 8πb0 respectively in the axial x, lateral y and vertical z directions. The number of corresponding grid/collocation points is (nx , ny , nz ) = (128, 720, 1440). (A two-dimensional ∂/∂x ≡ 0 run is also performed, using the same uθ (r) initial condition, Reynolds number, and lateral/vertical domain and grid, to establish the non-Crow benchmark.) The question is then whether or not the Crow instability is able to progress suﬃciently under these (stably stratiﬁed) conditions to break the two-dimensional coherence of the vortex pair before it returns to its original elevation.

3 Results The study was begun by reproducing the two-dimensional results of Spalart [7], in order to verify the Corral and Jim`enez correction [1]. Figure 2 shows a comparison of the vertical trajectories of the Re = 2 × 104 Gaussianωx (‘Type I’) thick-core (radius/span ratio rc /b0 = 0.25) vortices for N0∗ = 1.

522

G.N. Coleman et al.

(z−z0)/b0

0

–1

–2

0

1

2

3

4

5

Nt

Fig. 2. History of elevation z of stably stratiﬁed ‘Type I’ [7] Gaussian-core-vorticity two-dimensional wake-vortex pair with rc /b0 = 0.25 and Re = 2 × 104 for N0∗ = 1: , Spalart [7]; , present 3D DNS with third direction inactive. 0.2 0

(z−z0)/b0

–0.2 –0.4 –0.6 –0.8 –1 –1.2

0

1

2 Nt

3

4

Fig. 3. History of elevation z of stably stratiﬁed two-dimensional elliptic-loading wake-vortex pair with Re = 105 for N0∗ = 1.2: ◦, present 3D DNS with third direction inactive.

The agreement is close enough to provide conﬁdence in the current results. (The diﬀerences between the present and previous cases are primarily due to the use of diﬀerent vortex-tracking algorithms, and the use by Spalart of a slightly too small lateral domain.) Turning now to the aerodynamically more relevant case with Re = 105 and the thinner elliptic-loading cores (rc /b0 ∼ 0.01), we ﬁnd the tendency for the vortex pair to return to its original elevation when N0∗ ≈ 1 is also present when the core is thin (Fig. 3). Although the distance between the vortices is slightly larger after the rebound (Fig. 4), the two-dimensional benchmark

DNS of Stably Stratiﬁed Aircraft Wake Vortices

a

523

b

Fig. 4. Vorticity countours from two-dimensional results for N0∗ = 1.2 at (a) N t = 0 and (b) N t = 4.1. 0.2 0

(z−z0)/b0

–0.2 –0.4 –0.6 –0.8 –1 –1.2

0

1

2

3

4

Nt

Fig. 5. History of elevation z of wake-vortex pair for N0∗ = 1.2: ◦, 2D results; •, 3D results at ﬁxed x for δ/b0 = 0.01.

reveals that the strength and concentration of the vorticity is, from an ATC perspective, still strong enough to cause concern. The role of the Crow instability for N0∗ = 1.2 in breaking the twodimensionality of the vortex pair – and thus its danger for aircraft operations – during its buoyancy-induced descent and rise, can be inferred from Figs. 5 and 6. The fact that a relatively small displacement (1% of b0 ) causes the line vortices to pinch oﬀ into loops before the ascent phase begins (near N t ≈ 1.9) suggests that for N0∗ ≈ 1 the velocities induced by the rebounding vortex structure may develop enough streamwise variation to prevent them from being a serious threat to aircraft following the same ﬂight/landing path. This conclusion needs to be conﬁrmed by examining the state of the ﬂow at times after the vortex merging process begins (which will require ﬁner resolution than that used here), to determine how fragmented and three dimensional the

524

G.N. Coleman et al.

Fig. 6. Vorticity contours from three-dimensional results for N0∗ = 1.2 with δ/b0 = 0.01: vorticity contours at (a) N t = 0.39 and (b) N t = 1.86.

vortices become as they approach their original elevation. We will also need to consider initial perturbations more representative of actual worse-case atmospheric conditions found when N0∗ ≥ 1. Since these very stable (large N ) conditions could include both smaller initial displacements (perhaps limited to the horizontal plane) and wavelengths other than the ones used here, the possibility that they might lead to unexpected and dangerous conditions behind a very large and lightly loaded long-haul landing aircraft cannot yet be ruled out. We hope to clarify the situation in the near future.

Acknowledgement This work was done as part of the UK Turbulence Consortium, sponsored by the Engineering and Physical Sciences Research Council (Grant EP/D044073/1). The computations were done on the EPSRC HPCx and HECToR clusters. We are grateful to Dr Philippe Spalart for his contribution to this study.

References 1. Corral R, Jim´enez J (1995) Fourier/Chebyshev methods for the incompressible Navier-Stokes equations in inﬁnite domains. J Comp Phys 121:261–270 2. Crow SC (1970) Stability theory for a pair of trailing vortices. AIAA J 8: 2172–2179 3. Garten JF, Werne J, Fritts DC, Arendt S (2001) Direct numerical simulations of the Crow instability and subsequent vortex reconnection in a stratiﬁed ﬂuid. J Fluid Mech 426:1–45 4. Holz¨ apfel F, Gerz T, Baumann R (2001) The turbulent decay of trailing vortex pairs in stably stratiﬁed environments. Aerosp Sci Tech 5:95–108

DNS of Stably Stratiﬁed Aircraft Wake Vortices

525

5. Redford JA, Coleman GN (2007) Numerical study of turbulent wakes in background turbulence. Proc 5th Int Sym Turbulence and Shear Flow Phenomena/TSFP-5: 561–566 6. Rogallo RS (1981) Numerical experiments in homogeneous turbulence. NASA Tech Memo 81315 7. Spalart PR (1996) On the motion of laminar wing wakes in a stratiﬁed ﬂuid. J Fluid Mech 327:139–160 8. Spalart PR, Moser RD, Rogers MM (1991) Spectral methods for the NavierStokes equations with one inﬁnite and two periodic directions. J Comp Phys 96:297–324 9. Spalart PR (2008) On the far wake and induced drag of aircraft. J Fluid Mech 603:413–430 10. Spreiter JR, Sacks AH (1951) The rolling up of the trailing vortex sheet and its eﬀect on the downwash behind wings. J Aero Sci 18:21–32

On the Download Alleviation for the XV-15 Wing by Active Flow Control Using Large-Eddy Simulation M. El-Alti, P. Kjellgren, and L. Davidson Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, [email protected]; per [email protected]; [email protected] Abstract The ﬂow around a XV-15 wing with and without active ﬂow control (AFC) is investigated using large-eddy simulation (LES). Results show that a drag reduction of 34% is achieved with AFC.

1 Introduction A part of the current ﬂow control research at Chalmers University is directed towards vehicle aerodynamics optimization. The focus is on drag reduction using periodic excitation. As a ﬁrst step in our research, we investigated the tilt-rotor XV-15 wing, which is a continuation of previous research [1–3]. The wing has a deﬂected ﬂap at the trailing edge. The optimal angle is 70◦ , where the ﬂow reattaches. If the deﬂection angle is increased, the ﬂow separates and hence the download increases. With active ﬂow control (AFC), the ﬂow reattaches, the wake becomes narrower and the download is alleviated. In AFC the ﬂow is controlled by supplying energy to the system. The energy input is provided in this study by an actuator that can blow in or suck out ﬂow. The use of periodic excitation was shown to be more eﬀective than steady blowing or suction [3]. Periodic excitation depends on many parameters that must be optimized. We have used the optimal values found in [1,2] to analyze the download reduction process.

2 Numerical method The ﬂow is computed by commercial ﬁnite-volume code, STAR-CD ver 4.02, with large-eddy simulation for solving the turbulent ﬂow and the Smagorinsky SGS model. The Reynolds number is 130, 000 based on the chord. The V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 76, c Springer Science+Business Media B.V. 2010

527

528

M. El-Alti et al.

y

x z

Fig. 1. The computational domain. Inlet at x = −6c; outlet at x = 8c; slip sidewalls at y = −6c and y = 7c.

temporal discretization is a three-level, second-order scheme. Three diﬀerent schemes are investigated for the spatial discretization: a pure central diﬀerence scheme (CD), a blended central diﬀerence with upwind and ﬁnally a monotone advection and restruction scheme (MARS). The conﬁguration includes the wing with ﬂap without rotors. At take-oﬀ, the wing is exposed by the ﬂow from the rotors that yields the high download. The two-dimensional domain is span-wise extruded so that the span of the wing is 30% of its chord. The domain is shown in Fig. 1. The forcing is modeled as a transient velocity inlet, and the governing variables are the slot width, the velocity (both magnitude and direction) and the frequency. The RMS momentum from the slot is deﬁned as Jrms = ρu2rms dh = ρu2rms Δh (1) We have a constant spatial velocity proﬁle at the slot. Δh is the eﬀective slot width. Further, we deﬁne the momentum-coeﬃcient as Cμ =

Jrms u2 Δh = rms 1 2 c 2 ρu∞ c 12 u2∞

(2)

where c is the length of the chord. Thus the velocity out from the slot is given by : √ Cμ c u2∞ , u(t) = 2urms sin(2πF t) urms = (3) 2Δh where F is the frequency. Finally, we deﬁne the non-dimensional forcing frequency as F · XT E F+ = (4) U∞ where XT E is the distance from the slot to the trailing edge of the ﬂap.

On the Download Alleviation for the XV-15 Wing by AFC Using LES

529

The mesh consists of around 700, 000 unstructured quadrilateral cells in the xy−plane and structured hexahedral cells in the yz−plane. The mesh quality for the forced case at the ﬂap is: n+ = 0.8 − 11, Δs+ = 6 − 65 and Δz + = 40 − 400. Where n+ and s+ denote normal and tangential due to the wall, respectively.

3 Results Simulations were made with and without AFC. The following values of the diﬀerent parameters were chosen. The free-stream velocity is U∞ = 7 m/s, F + = 0.7, and the slot is located at 10% of the ﬂap (see Fig. 1), α = 35◦ denotes the forcing angle due to the ﬂap surface, Δh = 2 mm which gives Cμ = 1.5%. The spatial scheme dependence was investigated. For the blended scheme, the blending factor chosen was 0.95, i.e. 95% CD and 5% UD. 3.1 Drag and lift The predicted drag for the unforced case is in good agreement with experimental results and previous FEM computations [1,2]. The normalized drag for the unforced case is 0.99. The experimental value is 1.03 [1]. This is the case for the pure CD scheme; however, this scheme showed wiggles in the region upstream of the wing. The MARS scheme over-predicted the drag to be 1.52, and the blended CD scheme also over-predicted to be 1.27. These two latter schemes however removed the wiggles. The predicted drag for the forced case is also in good agreement with the experimental results and previous FEM computations. The normalized drag is 0.76 for the pure CD scheme, compared with the experimental value of 0.73 [1]. We did not run the forced case with the MARS-scheme due to the high over-prediction. However, for the blended scheme, the normalized drag is 0.84. The results below were all obtained with the blended scheme. In the drag and lift history plots in Fig. 2, we can see that there are large ﬂuctuations for the unforced case and that the peak to peak value is reduced with forcing. This is conﬁrmed by calculating the RMS of the drag coeﬃcient. The RMS of CD is 0.29 and 0.14 for the unforced and forced case, respectively. We can also see that the drag history is modulated for the forced case. This is the actuation frequency, which is higher than the vortex shedding frequency. We can also see that the lift increases with forcing. 3.2 Pressure coeﬃcient distribution and its RMS Figure 3a plots the non dimensional pressure coeﬃcient Cp,x obtained by using the projection of the pressure load in the x-direction for both the upper and lower wing surfaces. Cp,x is investigated because it is the quantity that

530

a

M. El-Alti et al.

b

2

0.6 0.4

1.5

0.2 0 −0.2

1

−0.4 −0.6

0.5

190

200

210

220

230

−0.8

240

190

Normalized drag history.

210

220

230

240

Normalized lift history.

Fig. 2. Time histories of drag and lift,

a

200

2

b

: AFC OFF;

: AFC ON.

6 4

0

2

−2

0 −4 −2 −6 0

0.2

0.4

0.6

0.8

1

Mean Cp,x along the wing. Fig. 3. Upper surface: OFF; + : AFC ON.

: AFC OFF;

0

0.2

0.4

0.6

0.8

1

RMS of Cp,x along the wing. : AFC ON, Lower surface:

o : AFC

contributes to the drag, CD . High values of Cp,x are found at the upper surface where the ﬂow impinges on the wing. Cp,x however decreases a great deal at the curvature connecting the ﬂap and becomes negative. The comparison of the pressure coeﬃcient distribution with the unforced case shows that a strong reduction is obtained at the ﬂap surface. The reduction of pressure is a result of the ﬂow reattachment mechanism due to AFC. We also observe that the peak values are reduced in the RMS plot in Fig. 3b, which is the reduction of peak to peak value of drag history for the AFC case, see Fig. 2a. At the lower surface, the pressure coeﬃcient is higher with AFC than without. This conﬁrms that, in the mean, the base pressure has increased with forcing. The high base pressure with AFC covers the entire lower surface, and is probably a side-eﬀect of the reduction done on the ﬂap surface. We observe from the RMS of Cp,x in Fig. 3b that the main reduction in peak to peak in CD is caused by the lower surface.

On the Download Alleviation for the XV-15 Wing by AFC Using LES

Mean U-velocity, AFC OFF.

Mean U-velocity, AFC ON.

Resolved KE, AFC OFF

Resolved KE, AFC ON

531

Fig. 4. Mean U-velocity with streamlines (top) and resolved turbulent kinetic energy (bottom).

3.3 Mean velocity, pressure and resolved kinetic energy Figures 4a and b show contour plots of the time average of the x-component of velocity together with streamlines. For the unforced case, we observe a large separation region along the ﬂap and a large wake region. We also observe that there is a large region with negative velocities (dark blue) behind the wing in the unforced case. With AFC, the large separation on the ﬂap region is removed (i.e. the time-averaged ﬂow is reattached), the wake size narrows a great deal and the region with negative velocities behind the wing is smaller. The wake structures become smaller. The resolved turbulent kinetic energy (KE) is shown in Figs. 4c and d as contour plots. There are three main achievements with forcing in this state. We observe a high KE region on the ﬂap, creating low pressure and thus sucking the ﬂow onto the surface, hence reattaching the ﬂow. We also observe the huge decrease of KE in the wake due to forcing. The vortex shedding has

532

M. El-Alti et al.

been destroyed so that the wake is less intense. Further, the wake is narrower and smaller.

4 Conclusion An XV-15 wing with a ﬂap was investigated with and without AFC. Results show that a drag reduction by 34% is achieved with AFC. The reduction is due to ﬂow reattachment on the ﬂap surface, which gave rise to a less intensive wake with smaller wake structures and a narrower wake size.

Acknowledgements This work is supported by the Swedish Agency of Innovation Systems (VINNOVA), Volvo 3P, SKAB and CD-ADAPCO. Financial support by SNIC (the Swedish National Infrastructure for Computing) for computer time at C3SE (Chalmers Centre for Computational Science and Engineering) is gratefully acknowledged.

References 1. P. Kjellgren, A. Hassan, J. Sivasubramanian, L. Cullen, D. Cerchie, and I. Wyganski. Download alleviation for the XV-15: computations and experiments of ﬂows around the wing. In Biennial International Powered Lift Conference and Exhibit, Williamsburg, Virginia, Nov 5–7 2002. 2. P. Kjellgren, D. Cerchie, L. Cullen, and I. Wyganski. Active ﬂow control on bluﬀ bodies with distinct separation locations. In 1st Flow Control Conference, St. Louis, Missouri, June 24–26 2002. 3. P. Kjellgren, N. Anderberg, and I. Wyganski. Download alleviation by periodic excitation on a typical tilt-rotor conﬁguration - computation and experiment. In Fluids 2000 Conference and Exhibit, Denver, CO, June 19–22 2000.

Turbulent Flow Simulations Around an Airfoil At High Incidences Using URANS, DES and ILES Approaches Bowen Zhong, Satish K. Yadav, and Dimitris Drikakis Department of Aerospace Sciences, Cranﬁeld University, Bedfordshire MK43 0AL, UK, [email protected]; [email protected]

1 Introduction Flows around airfoils at high incidences are highly unsteady and characterized with complex ﬂow phenomena like boundary layer separation, separation reattachment, unsteady vortex shedding, etc. [1, 2]. These phenomena have direct impact on the performance of airfoils such as lift and drag characteristics and hence are required to be accurately predicted or captured. Most of the well known turbulence models are unable to predict the ﬂow correctly in near stall and post stall regimes. This is due to the presence of massive separated ﬂows. As a result one tends to employ high level simulation such as Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) for better accuracy. However, due to the ﬁne grid demands, DNS is not feasible for most practical engineering problems [3]. Also LES still requires very large grids. In order to overcome the limitations of the traditional LES approach for the compressible ﬂow simulation, Detached Eddy Simulations(DES) [4] and implicit LES (ILES) [5] have been introduced. The aim of the present work is to perform turbulent ﬂow simulations around the NACA 0012 airfoil at high angles of incidence using URANS, DES and ILES approaches. It is intended to investigate the capabilities of the DES and ILES approaches for capturing ﬂow phenomena such as separation and vortex shedding over the airfoil.

2 Methodologies Limitation of LES and DNS to relatively simple ﬂows has given rise to a new class of methods, the hybrid LES/RANS methods [6]. One of the most popular of these methods is called Detached-Eddy Simulation (DES) proposed by Spalart [4]. Here near the wall the ﬂow is modeled using RANS and away V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 77, c Springer Science+Business Media B.V. 2010

533

534

B. Zhong et al.

from the wall LES is employed. In order to improve the simulation in the compressible turbulent ﬂow, Implicit Large Eddy Simulation (ILES) approach was also introduced [5]. In ILES, the small scales are not explicitly modeled, instead the numerical dissipation of the numerical scheme is used to represent the eﬀect of the small eddies. DES (or hybrid LES/RANS approach) and ILES can be regarded as modiﬁcations of the traditional LES. The above approaches result in diﬀerent formulations of the governing equations for turbulent ﬂow. The FLOWer code developed by German Aerospace Center (DLR) is used for this study. The integral forms of the governing equations formulated above are discretized using a ﬁnite-volume method. The Jameson’s central diﬀerence scheme with artiﬁcial viscosity and implicit residual smoothing technique is used in the original code. The HLLC approximate Riemann solver [7], in conjunction with various ﬂux limiters [8], has been implemented into FLOWer as a part of the work for the GOAHEAD project [9]. High-order accuracy is achieved using the MUSCL scheme. Dual time step techniques are used for the time integration.

3 Simulation conditions and setup The Mach number for the simulation is ﬁxed at 0.3 and the Reynolds number is ﬁxed at 9×105. To study three-dimensional ﬂows around NACA0012 the 2-D airfoil shape is extruded on spanwise direction. It is assumed that by keeping thickness as 10% of chord length the complex three-dimensional eﬀect of the ﬂow can be eﬀectively captured. The angles of attack for simulations are chosen in order to capture the eﬀect of separation and stall. On NACA0012 the stall is reported to occur at angles between 14◦ and 15◦ . Thus the obvious choice of angles of attack for simulation is a range from 12◦ to 16◦ . However some simulations are also carried out at smaller and larger angles of attack to study the eﬀect of separation. The current calculations for URANS, DES and ILES are carried out on a mesh with grid size of 413 × 85 × 45. Figure 1 is a close-up view of the grid which shows the grid on the airfoil surfaces and over the side boundary. Figure 2 shows the boundaries and the interfaces between diﬀerent blocks. The length of the ﬁrst cell from the wall is kept at 10−4 wall units. A ﬁne mesh with grid size of 601 × 111 × 85 is also used for ILES calculations in order to examine the grid sensitivity. The boundary condition on the wall is treated as no-slip. The boundary condition at the domain boundaries is given as far ﬁeld. On both sides of the span, periodic boundary conditions are applied.

Simulations Around an Airfoil Using URANS, DES and ILES

535

Fig. 1. The surface mesh and the grid on the periodic boundary.

Fig. 2. Grid block interfaces and boundaries.

4 Numerical results and discussions URANS calculations have been performed at the following angles of attack: 5◦ , 10◦ , 12◦ , 14◦ , 16◦ , 20◦ and 25◦ . For URANS, Jameson’s central diﬀerence scheme with 5 stage Runge–Kutta time integration and k − ω SST turbulence model are employed. For DES, calculations have been carried out at the incidences of 5◦ , 12◦ , ◦ 14 , 16◦ , 20◦ and 25◦ . ILES calculations have been performed only at the airfoil post stall region, i.e. at the angles of attack of 16◦ , 20◦ and 25◦ . The RANS and LES regions in the DES simulations are speciﬁed through a switch. Figures 3 and 4 show the interface between RANS and LES regions. It can be seen that the position of the interface is grid dependent. Computational tests show that the numerical results match very well with the measurements for both URANS and DES at an angle of attack of 12◦ . However, at the angle of 16◦ , the pressure distribution from URANS is very diﬀerent from experiment; see Fig. 5. At an angle of attack of 16◦ , experiments have reported sudden drop in lift and surge in drag values. The comparison of Cp with experiments reveals that URANS is unable to predict Cp distribution accurately. The time histories of lift coeﬃcient Cl and drag coeﬃcient Cd

536

B. Zhong et al.

Fig. 3. Interface between the RANS and LES regions.

Fig. 4. Close-up view of the division of the RANS and LES regions. –9 EXPERIMENT NASA URANS DES ILES

–8 –7 –6

Cp

–5 –4 –3 –2 x –1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1

Fig. 5. Comparison of pressure distributions between URANS, DES and ILES.

shown in Fig. 6 do not exhibit oscillatory behaviour, which also indicates the absence of vortex shedding. For the case of angle 16◦ where URANS fails to give correct pressure coeﬃcients, the averaged pressure distributions obtained from DES and ILES approaches compared amazingly well with the experimental data. This is very encouraging. The time histories of Cl and Cd in DES and ILES also exhibit

Simulations Around an Airfoil Using URANS, DES and ILES

537

Fig. 6. Time history of lift and drag coeﬃcients for URANS.

Fig. 7. Time history of lift and drag coeﬃcients for DES.

diﬀerent behaviours from URANS, see Figs. 7 and 8. The unsteadiness in Cl and Cd of DES and ILES can clearly be identiﬁed. Figures 9 and 10 give the instantaneous vorticity iso-surface contours obtained with DES and ILES respectively. These ﬁgures show the vortex shedding of the ﬂow. It also shows that DES and ILES have the capabilities in capturing small scale turbulent ﬂow motions. The overall performances of DES, ILES and URANS in predicting the Cl and Cd characteristics are summarized in Figs. 11 and 12. The experimental data for the comparison is taken from McCroskey [10]. Uncertainties in the measurements are depicted with a grey region.

538

B. Zhong et al.

Fig. 8. Time history of lift and drag coeﬃcients for ILES.

Fig. 9. The instantaneous vorticity iso-surfaces of DES.

Fig. 10. The instantaneous vorticity iso-surfaces of ILES.

Simulations Around an Airfoil Using URANS, DES and ILES

539

Fig. 11. Comparison of lift coeﬃcients between numerical results and experiments. 0.8 0.7 EXPERIMENT Mc Croskey EXPERIMENT NASA URANS DES ILES

0.6

Cd

0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12 14 16 Angle of attack

18

20

22

24

26

28

Fig. 12. Comparison of drag coeﬃcients between numerical results and experiments.

5 Conclusions It is observed that URANS have been very eﬀective in predicting ﬂow behavior at small (pre-stall) angles of attack. For ﬂows at high angle of incidences, it has been found that URANS over predicts values of CL. The failure of URANS model at post stall angles of attack is due to presence of large separation. The simulation results from DES indicate an impressive overall performance. Numerical tests show that the DES and ILES give good results in the post stall ﬂow regimes whilst URANS results are completely oﬀ the target. Results also demonstrate the capability of ILES and DES approaches in capturing small scale turbulent ﬂow motions. Comparative studies show that ILES is less dissipative than the DES approach used.

Acknowledgements The authors acknowledge that DLR gives them access to the FLOWer code. The assistance and help of Dr. Jochen Raddatz, Dr. Thorsten Schwarz and Dr. Walid Khier of DLR, Dr. Frederic Le Chuiton of Eurocopter Deutschland in

540

B. Zhong et al.

using the FLOWer code is greatly appreciated. The support of Dr. Klausdieter Pahlke and Prof. Norbert Kroll of DLR is also greatly appreciated. The research is partially funded by the European 6th Framework GOAHEAD project, for which the authors are very thankful.

References 1. Mary I, Sagaut P (2002), Large eddy simulation of ﬂow around an airfoil near stall, AIAA J. 40(6):1139–1145 2. Strets M (2001), Detached eddy simulation of massively separated ﬂows. AIAA 2001-0879, 39th AIAA aerospace sciences meeting and exhibit 3. Piomelli U (1999), Large-eddy simulation: achievements and challenges, Prog. in Aero. Sci. 35:335–362 4. Spalart PR, Jou WH, Strelets M, Allmaras SR (1997) Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. First AFOSR international conference on DNS and LES. In: Advance in DNS/LES 5. Fureby C, Grinstein FF (1999), Monotonically integrated large eddy simulation of free shear ﬂows, AIAA J., 37:544–556 6. Zhong B, Tucker PG (2004), k-l Based Hybrid LES/RANS Approach and Its Applications to Heat Transfer Simulation, Int. J. for Num. Meth. in Fluids 46:983-1005 7. Toro EF (1999) Riemann solvers and numerical methods for ﬂuid dynamics, Springer-Verlag, Berlin Heidelberg 8. Drikakis D, Rider WJ (2005) High-resolution methods for incompressible and low-speed ﬂows. Springer. Berlin Heidelberg 9. Pahlke K, (2007) The GOAHEAD project. Proceedings of the 33rd european rotorcraft forum. Russia 10. McCroskey WJ, (1987) A critical assessment of wind tunnel results for the NACA 0012 airfoil. NASA TM 100019

Large Eddy Simulation of Flow Around an Airfoil Near Stall Jaber H. Almutairi, Lloyd E. Jones, and Neil D. Sandham School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected]; [email protected]; [email protected]

Abstract Large-eddy simulation is used to study the stalling behaviour of a NACA0012 airfoil at low Reynolds number. The method is validated against direct numerical simulation for an airfoil test case with a transitional separation bubble. The role of the sub grid model is considered together with explicit ﬁltering and it is found that results are improved when both methods are used in combination. Large ﬂuctuations in lift coeﬃcient are observed during the initial stages of calculations near stall and it is found that these are sensitive to the spanwise extent of the computational domain.

1 Introduction Airfoil stall can be classiﬁed into three main types: leading-edge stall, thin-airfoil stall and trailing-edge stall. Early studies such as Gaster [1] and Horton [2], made great contributions to understanding the behaviour of laminar separation bubbles (LSB), which play a role in the ﬁrst two of these types of stall. Laminar separation bubbles are classically described as being either “short” or “long” (see e.g. [1]). As the incidence is increased a short bubble may burst, i.e. fail to reattach. This can lead to fully stalled ﬂow, or else the separated shear layer may reattach and form a long bubble. Unsteady separation bubbles exhibiting bubble bursting and low frequency oscillation during stall have been observed recently, for example Bragg et al. [3] and Broeren and Bragg [4]. They found that as the incidence is increased the Strouhal number (deﬁned as St = f c sin α/U where f is the frequency of the ﬂow oscillation, c is the chord length, α is the incidence and U is the free stream velocity) tends to increase. Strouhal numbers associated with the oscillation lay in the range between 0.017 to 0.03, which is around ten times lower than the Strouhal number of bluﬀ body vortex shedding. More recently Rinoie and Takemura [5] performed an experimental study of a NACA-0012 airfoil at Reynolds number 1.3 × 105 and α = 10◦ that provided insight into

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 78, c Springer Science+Business Media B.V. 2010

541

542

J.H. Almutairi et al.

the mechanism behind the low-frequency oscillation. Sandham [6] showed that a viscous-inviscid modelling method reproduces some of the characteristics of the low-frequency oscillation, but such models need further development and calibration. The increasing capacity and performance of computers allows the study of turbulent ﬂow phenomena by high accuracy numerical methods, such as direct numerical simulation (DNS) and large eddy simulation (LES). DNS has been successfully implemented by Alam and Sandham [7] for a short LSB in both two and three dimensional ﬂows. Numerical simulations of the ﬂow around an airfoil at or near stall may lead to improved understanding of the mechanism behind the low-frequency oscillations. In the current study, LES has been applied to ﬂow with low Reynolds number around NACA-0012 airfoil to study the behaviour of an LSB near stall. The numerical method is essentially the same as that used by Jones et al. [8], originally written and validated by Sandham et al. [9] to solve the Navier–Stokes equations for a compressible ﬂow using a fourth order central ﬁnite diﬀerence scheme in space, and a fourth order Runge–Kutta method in time. LES is performed by ﬁltering the conservative variables and adding a sub-grid scale (SGS) model. Filtering is applied using a fourth order tridiagonal compact scheme [10].

2 Results The numerical code has been validated by performing LES with Mach number M∞ = 0.4 at an incidence of 5◦ and a Reynolds number of 50,000. A grid of 637 × 375 was generated and data were compared with the DNS of Jones [11] with a grid of 2570 × 692. The 2D simulation was then extended to a 3D simulation with 32 grid points in the spanwise direction and compared with DNS results on a 2570×692×96 grid. The 3D computations were conducted in two ways; ﬁrstly by applying the LES without using any subgrid-scale (SGS) model, i.e. relying on ﬁltering alone, and secondly by adding a Mixed-TimeScale (MTS) model. Statistical results such as the pressure coeﬃcient and skin friction, as well as the ﬂow behaviour like the vorticity, exhibit good agreement with the DNS of Jones [11] and Jones et al. [8]. Figure 1 shows that in the 2D case the lift coeﬃcient oscillates periodically around a mean value of 0.495 which agrees with the DNS results. The frequency is obtained as f = 3.32, which matches reasonably the DNS value of f = 3.37. In the 3D case, the pressure distribution over the airfoil obtained by LES clearly matches the DNS without any signiﬁcant error, as illustrated in Fig. 2a. However the skin friction proﬁle, which is depicted in Fig. 2b, appears more sensitive, and noticeable error is present. It is also clear to see that the LES with the SGS model is closer to the DNS than the LES with ﬁltering alone. An LES near stall was performed for a ﬂow at Re = 130,000 and α = 11.5◦ . The transient low-frequency behaviour can be clearly seen from Fig. 3a but

Large Eddy Simulation of Flow Around an Airfoil Near Stall

543

DNS LES

0.54 0.52 CL

0.5 0.48 0.46 0

1

2 t

3

4

Fig. 1. Comparison of time dependence lift coeﬃcient for two dimensional simulations at Re = 50,000 with incidence of 5◦ by DNS [11] and the current study (LES).

a

b

DNS LES (without SGS) LES (MTSM)

1.5

0.02

1 0.5

Cf

–Cp

LES (MTS) LES (without SGS) DNS

0.01

0

0

– 0.5 –1 0.2

0.4

x/c

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

x /c

Fig. 2. Comparison between (a) pressure and (b) skin friction coeﬃcients for DNS, LES without SGS model and LES using MTS model.

a

b

0.9

0.8

0.8 0.7

CL

CL

0.9

0.7

0.6

0.6

0.5

0.5

0

5

10

15 t

20

25

25

30

35 t

Fig. 3. Time variation of lift coeﬃcient at Re = 130,000 and incidence of 11.5◦ , showing the airfoil is (a) initially switching between stalling and non-stalling and (b) fully developed stalled ﬂow.

the ﬂow eventually becomes fully stalled (Fig. 3b). A possible reason behind not detecting the low-frequency ﬂow oscillation is the inﬂuence of the small spanwise computational domain on the results where some vortical structures may be longer than the spanwise domain used in the simulation. In order to in-

544

J.H. Almutairi et al. 0.9

LES3 (L Z = 0.2 c) LES4 (L Z = 0.5 c)

0.8

CL

0.7 Attached flow 0.6

0.5 stalled flow 0

10

30

20

40

t

Fig. 4. Eﬀect of increasing the spanwise length of the computational domain on the LES time dependence lift coeﬃcient.

vestigate the eﬀect of the spanwise domain, two further simulations near stalls were performed at Re = 50,000 and α = 9.25◦ . The grid for these simulations were 637 × 320 × 32 and 637 × 320 × 80 for narrow (Lz = 0.2c) and wide domain (Lz = 0.5c), respectively. As can be seen from Fig. 4, the ﬂow behaviour is signiﬁcantly changed when the domain width has been increased and the ﬂow becomes fully attached instead of fully stalled in the narrow domain. This implies that in order to achieve accurate ﬂow prediction near stall, the domain width should be at least 50% of the chord length, and possibly significantly more than this.

3 Conclusions Large eddy simulation has been performed to study the behaviour of laminar separation bubbles near stall. For a validation test case the LES results compare favourably with the DNS data. It was found that a method incorporating a sub-grid scale model in addition to explicit ﬁltering gave more accurate results than a method with ﬁltering alone. Transient low-frequency ﬂow behaviour has been found at Re = 130,000 and α = 11.5◦ but eventually the ﬂow became fully stalled. Increasing the computational domain width was found to change the ﬂow behaviour during stall at Re = 50,000 and α = 9.25◦ .

References 1. Gaster, M. (1967). The structure and behaviour of laminar separation bubbles. ARC R&M 3595. 2. Horton, H.P.(1967). A semi-empirical theory for the growth and bursting of laminar separation bubbles. ARC CP 1073.

Large Eddy Simulation of Flow Around an Airfoil Near Stall

545

3. Bragg, M. B., Heinrich, D. C., Balow, F. A. & Zaman, K. B. M. Q.(1996). Flow oscillation over an airfoil near stall. AIAA 34(1), 199–201. 4. Broeren, A. P. & Bragg, M. B. (1998). Flowﬁeld measurements over an airfoil during natural low-frequency oscillations near stall. AIAA 37(1), 130–132. 5. Rinoie, K. & Takemura, N. (2004). Oscillating behaviour of laminar separation bubble on an aerofoil near stall. The Aeronautical Journal 108, 153–163. 6. Sandham, N. D. (2008). Transitional separation bubbles and unsteady aspects of aerofoil stall. The Aeronautical Journal 112, 395–404. 7. Alam, M. & Sandham, N.D. (2000). Direct numerical simulation of “short” laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 403, 223–250. 8. Jones, L. E., Sandberg, R. D., Sandham, N. D. (2008). Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. J. Fluid Mech. 602, 175 –207. 9. Sandham, N. D., Li, Q. & Yee, H. C. (2002). Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comp. Phys. 178, 307–322. 10. Lele, S.K. (1992). Compact ﬁnite-diﬀerence schemes with spectral-like resolution. J. Comp. Phys. 103, 16–42. 11. Jones, L. E., (2007). Numerical studies of the ﬂow around an airfoil at low Reynolds number. PhD thesis, Southampton University, UK

Large Eddy Simulation of Turbulent Flows Around a Rotor Blade Segment Using a Spectral Element Method A. Shishkin1 and C. Wagner2 1

2

Institute for Aerodynamics and Flow Technology, DLR – German Aerospace Center, G¨ ottingen, Germany, [email protected] Institute for Aerodynamics and Flow Technology, DLR – German Aerospace Center, G¨ ottingen, Germany, [email protected]

Abstract Large Eddy Simulations (LES) of turbulent ﬂows around a segment of the FX-79-W151 airfoil have been performed for diﬀerent Reynolds numbers between Re = 5 · 103 and Re = 105 and the angle of attack 12◦ using a Spectral Element Method (SEM). The results of the LES for Re = 5 · 103 and Re = 5 · 104 are compared. The turbulence statistics obtained in the LES reveal laminar and turbulent ﬂow separation regions for the lower and higher Reynolds numbers, respectively.

1 Numerical method and computational parameters Numerical methods based on Reynolds-Averaged Navier–Stokes (RANS) equations are successfully applied for a wide range of industrial applications. Nevertheless, many unsteady ﬂow problems dealing with complex threedimensional domains can not be accurately simulated in the framework of statistical turbulence modeling due to unsteady nature of the problem. In this respect, the Large Eddy Simulation (LES) has proven to be a promising technique. Here we present the results obtained in LES of the turbulent ﬂow around a segment of the FX-79-W151 proﬁle with an angle of attack α = 12◦ . The early results of the simulation for Re = 5 · 103 and some related questions were discussed in [4]. We use the spectral/hp element method based on the high order 2D polynomial representation of the solution combined with the Fourier extension in homogenous spanwise direction. This method was developed by Karniadakis and Sherwin [1] and co-workers and implemented in the N εκT αr code. The LES is governed by the dimensionless incompressible ﬁltered Navier– Stokes equations together with the Smagorinsky subgrid scale model adapted to high order SEM by Karamanos [2]. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 79, c Springer Science+Business Media B.V. 2010

547

548

A. Shishkin and C. Wagner

20 CL 12° CL

20 CL 1 CL

5 CL

Fig. 1. Schematic view of the computational domain (left) and unstructured 2D mesh (right) consisting of 2,116 elements.

The computational domain is schematically shown in Fig. 1 (left). The size of the computational domain is 25cl × 20cl × 1cl (cl denotes the chord length) in streamwise, vertical and spanwise directions, respectively. We use time-independent laminar inﬂow boundary conditions, Neumann boundary conditions at the outﬂow boundary and periodic conditions in spanwise direction. We implemented the sponge zone in order to damp the turbulent vortical structures at the outﬂow boundary. The LES are carried out for the Reynolds numbers based on the chord length and the freestream velocity Re = 5 · 103 , Re = 5 · 104 and Re = 105 on the hybrid structured/unstructured mesh consisting of 2,116 2D-elements (Fig. 1, right) with 64 Fourier planes in spanwise direction. The polynomial order P = 9 yields about 5.5 · 106 degrees of freedom. The simulations are performed on 64-bit Linux Cluster with 1.7 GHz AMD Opteron processors. The memory usage is 20G and one time step takes approximately 5.5 CPU seconds on 32 processors. The computational expenses of the SEM in this “2D+Fourier”case are much lower than those in “full 3D” SEM considered in [3].

2 Results of large eddy simulations The ﬂow ﬁelds predicted in the LES are averaged in time and over the spanwise length to get the turbulence statistics. Some of the results are depicted in Figs. 2–3. The mean streamwise velocity components (Fig. 2) reveal the ﬂow acceleration over the leading edge and the backﬂow regions (outlined with white curves) for both Reynolds numbers. One can observe that the backﬂow region is closer to the leading edge for the higher Reynolds number. The turbulent kinetic energy (TKE) (Fig. 3) reveals an essential diﬀerence between the two studied cases. The zone of higher TKE is located downstream

Large Eddy Simulation of Turbulent Flows Around a Rotor Blade

549

Fig. 2. The mean streamwise velocity components as obtained in the LES for Re = 5 · 103 (left) and for Re = 5 · 104 (right). The regions of negative mean velocity (backﬂow) are outlined.

Fig. 3. Distributions of the turbulent kinetic energy obtained in the LES for Re = 5 · 103 (left) and Re = 5 · 104 (right).

the trailing edge for Re = 5 · 103 . In the case of the higher Reynolds number this zone is located in the suction side area close to the trailing edge. Considering both, the distributions TKE and the mean streamwise velocity components, one concludes that ﬂow separation is observed for both Reynolds numbers, but a turbulent separation is predicted for the higher Reynolds number while the separation is laminar for the lower one. The further analysis of the spatial energy spectra at three locations over the suction side of the rotor blade marked in Fig. 3 (right) conﬁrms the conclusions. The locations have been chosen as follows: position one is in a region where high TKE values are observed for both Reynolds numbers (right point in Fig. 3); at position two (middle point in Fig. 3) large TKE values are obtained only for the higher Reynolds number; ﬁnally, position three (left point in Fig. 3) is located in a low TKE region for both cases. Energy spectras taken in the higher TKE region (Fig. 4, middle and right top and right bottom) reﬂect a decay which partly agrees with the Kolmogorov −5/3 law of turbulence, while the spectras taken at the other locations are characterized by low energy values and faster decay for all components.

550

A. Shishkin and C. Wagner

0.01 0.001

0.1

1

0.01

0.1

1e-04 1e-05

0.001

1e-06 1e-07

1e-04

1 1 0.01 1e-04 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1

0.01

Kz–5/3

0.001 1e-04

1e-05

1e-08 1e-09

Eu Eu Eu

10

100

1e-06

1e-05 1

10

100

100

1

0.01 1e-04

0.001

1e-06

1e-04

1e-08

1e-05 1e-06

1e-10

10

1e-06

1e-12 1e-14

10

100

10

100

0.1 0.01

1

1e-07 1

10

100

1e-08

1

Fig. 4. Energy spectra components for Re = 5 · 104 (top) and Re = 5 · 103 (bottom) at the three positions above the trailing edge in Fig. 3. The ordering from left to right corresponds to that on Fig. 3.

3 Outlook and future work The above presented results are obtained in LES of turbulent separated ﬂows around the rotor blade segment for Reynolds numbers Re = 5 · 103 and Re = 5 · 104 using SEM. The study of the streamwise velocity components, the distributions of the TKE and the energy spectras at some positions over the suction side reveals a turbulent separation for the higher Reynolds number, while the laminar separation region is observed for the lower Reynolds number. The LES for Re = 105 is currently running. The results for Re = 5 · 104 and Re = 105 will be compared with experiments in the near future.

References 1. Karniadakis, G.E. , Sherwin, S. (1999). Spectral/HP Element Methods for CFD.Oxford University Press, Oxford. 2. Karamanos, G.-S.(1999) Large Eddy Simulation Using Unstructured Spectral/hp Elements. PhD Thesis, Imperial College. 3. Shishkin, A., Wagner. C. (2006). Direct Numerical Simulation of a turbulent ﬂow using a spectral/hp element method.Notes on Numer. Fluid Mech., v.92, 405-412, Springer, Germany. 4. Shishkin, A., Wagner. C. (2006). Large eddy simulation of the ﬂow around a wind turbine blade. In Wesseling, P., Onate, E., Periaux, J., eds., European Conference on Computational Fluid Dynamics.

Part VIII

Compressible Flows

DNS of Compressible Turbulent Flows Rainer Friedrich and Somnath Ghosh Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany, [email protected]; [email protected]

Abstract In the ﬁrst part of this paper direct numerical simulation (DNS) is used to explore similarities and diﬀerences between fully developed supersonic turbulent plane channel and axisymmetric pipe ﬂow bounded by isothermal walls. The comparison is based on equal friction Mach and Reynolds numbers. In the second part a comparison between supersonic turbulent nozzle and diﬀuser ﬂow is undertaken based on DNS, with the aim to better understand the basic physics of these nonequilibrium ﬂows which develop from upstream fully developed supersonic pipe ﬂow. We focus on eﬀects of mean extra rate of strain and mean dilatation on the axial Reynolds stress, the turbulence production and redistribution mechanisms.

1 Introduction Compressible wall-bounded turbulent ﬂows are an important element of highspeed ﬂight. Although they have attracted researchers since the ﬁfties of the last century, they are not completely understood, even today. For recent reviews we refer to Lele [7] and Smits and Dussauge [10]. It was the DNS of supersonic fully developed turbulent channel ﬂow by Coleman et al. [3] and the companion work of Huang et al. [5] on data analysis and modelling issues which contributed to a better understanding of ‘compressibility’ eﬀects in the form of mean property variations in shear ﬂows bounded by isothermal walls. Nearly a decade later Foysi et al. [4] gave an explanation for the near-wall reduction of pressure–strain correlations in supersonic channel ﬂow by linking it to the sharp wall-normal density variations in the framework of a Green-function-based analysis of the pressure ﬁeld. Accelerated/decelerated compressible wall-bounded turbulent shear ﬂows provide further challenges for understanding the ﬂow physics and modelling. In his pioneering work, Bradshaw [2] studied eﬀects of mean dilatation on the turbulence structure in wall-bounded ﬂows using engineering calculation methods and found that mean dilatation eﬀects have a greater impact than would be expected from the extra production terms in the Reynolds stress transport equations. V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 80, c Springer Science+Business Media B.V. 2010

553

554

R. Friedrich and S. Ghosh

It is our objective in this paper to identify similarities and diﬀerences between fully developed compressible channel and pipe ﬂow up to secondorder turbulence statistics and to provide explanations for the corresponding behaviour. The results on supersonic nozzle and diﬀuser ﬂow will improve our understanding of ﬂow acceleration and retardation, will underline the lack of importance of pressure–dilatation correlation and compressible dissipation rate, but highlight the need for proper modelling of pressure–strain correlations.

2 Comparison of supersonic channel and pipe ﬂow The compressible Navier–Stokes equations are solved in direct simulations using the pressure–velocity–entropy formulation of Sesterhenn [9]. Speciﬁc heats are assumed constant at a ratio γ of 1.4 for air. The Prandtl number is P r = 0.71 and the viscosity exponent, n, is 0.7 (μ ∝ T n ). The remaining ﬂow parameters are the Mach and Reynolds numbers. We use the channel half-width h and the pipe radius R to deﬁne the Reynolds number and prefer the friction Reynolds and Mach numbers as characteristic ﬂow parameters, viz: Reτ = ρ¯w uτ l/μw = l+ , Mτ = uτ / γRTw (1) with l = h, R and mean values of the dynamic viscosity and the speed of sound computed at the constant wall temperature, Tw . The mean density at the wall, ρ¯w , and the wall shear stress, τ¯w , are a result of the computation. The friction velocity reads uτ = τ¯w /ρ¯w . We follow common practice and apply a tilde and an overbar to deﬁne Favre and Reynolds averages, and double and single dashes to specify Favre and Reynolds ﬂuctuations, e.g., u , ρ . Channel and pipe ﬂow are driven by a uniform body force, f¯x , which replaces the mean ∂ p¯ pressure gradient, ∂x , and allows for periodic boundary conditions of all ﬂow variables in streamwise x-direction. Non-dimensionalized with inner variables the mean pressure gradient is the inverse of the Reynolds number − p¯+ x = −

1 μw ∂ p¯ = ρ¯w uτ τ¯w ∂x aReτ

(2)

with a = 1 for channel ﬂow and a = 0.5 for pipe ﬂow. The pressure gradient has an eﬀect on the whole ﬂow, the importance of which decreases as the Reynolds number increases. This follows from the mean streamwise momentum balance, which after integration in wall-normal y-direction and neglecting correlations with viscosity ﬂuctuations, leads to the linear relation for the total stress in both ﬂows: μ ¯ d¯ u+ ρu v y+ − = 1 + ay + p¯+ x = 1− + + μw dy τ¯w l

(3)

y = 0 denotes the wall. In the case of pipe ﬂow, y = R − r and v is positive when pointing away from the wall. In the high Reynolds number limit,

DNS of Compressible Turbulent Flows

555

Table 1. Flow parameters. Flow

Reτ

Mτ

Rem

Mm

Channel Pipe

241 245

0.082 0.075

2850 3250

1.37 1.36

Reτ → ∞, equation (3) takes the same form as for zero-pressure-gradient boundary layers. The values of the ﬂow parameters for the present channel and pipe ﬂows are listed in Table 1, along with the corresponding bulk Reynolds and Mach numbers which are deﬁned as Rem = ρ¯m u ¯m l/μw , Mm = u¯m / γRTw (4) ρ¯m , u ¯m are the cross-sectionally averaged mean or bulk density and velocity. The ﬁfth-order compact low-dissipation upwind scheme of Adams and Shariﬀ [1] is used to discretize the convection terms of the compressible Navier–Stokes equations, the compact sixth-order scheme of Lele [6] is applied to the molecular transport terms, and a third-order low-storage Runge–Kutta scheme advances the solution in time. Equidistant grids are chosen in streamwise and spanwise (circumferential) directions. In the wall-normal direction the grids are clustered. The computational domain to compute channel ﬂow has a size of 4πh × 4πh/3 × 2h and 10R × 2πR × R for pipe ﬂow. The Cartesian grid for channel ﬂow comprises 192 × 128 × 151 points in (x, z, y)-directions and that for pipe ﬂow 256 × 128 × 91 points in (x, φ, r)-directions. The singularity due to the axis of the cylindrical coordinate system is avoided by placing no grid point on the axis, as suggested by Mohseni and Colonius [8]. 2.1 Mean ﬂow variables A key to the understanding of fully developed compressible turbulent channel and pipe ﬂow lies in the large changes in mean ﬂuid properties, ρ¯ and μ ¯, caused by viscous heating. As shown in Fig. 1, the steep near-wall gradients and the core values of mean temperature and density do not coincide in both ﬂows (here and in the following ﬁgures dashed/solid lines represent channel/pipe ﬂow). The reasons for this are obvious when the integrated mean energy balance is written down for the pipe. It diﬀers from the corresponding equation for channel ﬂow primarily by the presence of transverse curvature terms which disappear only in the limit Reτ → ∞ . In Fig. 2 we have plotted the mean velocity for both ﬂows against y/l. The curves seem to collapse in the nearwall region. However, in reality this is not the case, due to the diﬀerent mean density and viscosity distributions. In the outer layer the velocity in the channel is lower than in the pipe which explains diﬀerences in the bulk velocity and the friction coeﬃcient. Figure 3 shows proﬁles of the mean pressure. The pressure variations from the wall to the centerline are small compared to the

556

R. Friedrich and S. Ghosh 1.5

T¯/Tw

1.4 1.3 1.2 1.1

ρ/ ¯ ρ¯w

1 0.9 0.8 0.7 0.6 0

0.2

0.4

0.6

0.8

1

y/l Fig. 1. Mean density and temperature.

20

u ¯

+

15 10 5 0

0

0.2

0.4

0.6

0.8

1

y/l

Fig. 2. Mean velocity u ¯+ . 1 0.999 0.998

p¯/ p¯w

0.997 0.996 0.995 0.994 0.993 0.992

0

0.2

0.4

0.6

0.8

1

y/l

Fig. 3. Mean pressure.

corresponding variations of the mean density and temperature. The diﬀerences in pressure distributions between channel and pipe ﬂow are again a result of transverse curvature. The integrated wall-normal momentum balance for the pipe reads: y/R d(y/R) p¯ = p¯w − ρv v − (5) (ρv v − ρw w ) 1 − y/R 0

DNS of Compressible Turbulent Flows

557

The integral term in equation (5) does not appear in the corresponding equation for the channel. It disappears, however, in the limit Reτ → ∞, which means that the pressure distributions then match in both ﬂows. Note that for channel ﬂow the velocity components are (u,v,w) in (x,y,z)-direction. The same components are used for pipe ﬂow in (x, r, φ)-direction. 2.2 Reynolds stresses and budgets A comparison of the Reynolds shear stress and the viscous stress is presented in Fig. 4, using the normalizations of equation (3). Both stresses behave similarly in both ﬂows. The minor diﬀerences might be due to the imperfect match of the friction Mach numbers, see Table 1. The outer scale, τ¯w , is used to normalize the normal Reynolds stresses as well. Figures 5 and 6 show the streamwise and wall-normal Reynolds stresses plotted against the semi-local coordinate, y ∗ = yu∗τ /¯ ν with, u∗τ = τ¯w /ρ¯, suggested by Huang et al. [5]. The stresses in both ﬂows match fairly well in the near-wall regions of both ﬂows. In the core regions the wall-normal stresses clearly deviate due to reasons discussed in the context of the mean pressure. Non-dimensionalizing the terms in the streamwise Reynolds stress budget with τ¯w2 /¯ μ, leads to the proﬁles shown in Fig. 7. The source/sink terms, namely production, dissipation and pressure–

ρu v /τw,

μ ¯ du+ μw dy +

1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

y/l Fig. 4. Reynolds and viscous shear stresses. 9

ρu u /τw

8 7 6 5 4 3 2 1 0

0

20

40

60

80 100 120 140 160 180

y∗

Fig. 5. Streamwise Reynolds stress.

558

R. Friedrich and S. Ghosh 0.8

ρv v /τw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20 40 60 80 100 120 140 160 180

y∗

Fig. 6. Wall-normal Reynolds stress. 0.04 0.03 0.02 0.01 0 –0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07

0.25 0.2 0.15

PR

0.1 0.05

VD

0 – 0.05

DS

– 0.1 – 0.15 0

20

40

y∗

60

80

100

TT MF

PS

0

20

40

60

80

100

y∗

Fig. 7. Streamwise Reynolds stress budget. Left: Production (PR), dissipation (DS), viscous diﬀusion (VD). Right: Pressure–strain correlation (PS) turbulent transport (TT), mass ﬂux variation (MF).

strain correlation reveal diﬀerences between channel and pipe ﬂow in the core regions. Only the diﬀerences in the pressure–strain correlation seem to be dynamically important and mean that the wall-normal and spanwise stresses in the channel receive less energy by redistribution than those in the pipe, see Fig. 6, e.g. This can be traced back to the lower mean density in the channel via a Green’s-function-based analysis of the pressure ﬂuctuations as in Foysi et al. [4]. Another interesting aspect is the higher level of dissipation and viscous diﬀusion close the channel wall as compared to the pipe. Similar eﬀects are also observed (but not shown) in the wall-normal and transverse components of the dissipation rates, pressure–strain (and pressure–diﬀusion) correlations and the viscous diﬀusion terms. They might be due to the splatting eﬀect in which high-speed ﬂuid is carried to the wall (sweep event) and creates a ﬂow pattern similar to an impinging jet with a net energy transfer to the streamwise and spanwise velocity components. Since transverse curvature inhibits the spreading of ﬂuid in the circumferential direction, the discussed terms have lower levels in the pipe.

DNS of Compressible Turbulent Flows

559

3 Supersonic nozzle and diﬀuser ﬂow In fully developed channel and pipe ﬂow with a mean body force replacing the mean streamwise pressure-gradient, there is negligible mean dilatation, ˜ 0. It is, therefore, a challenge to investigate ﬂows where mean hence div u dilatation eﬀects appear as a result of compressibility, and to study in which way they modify the turbulence structure. We have chosen fully developed supersonic pipe ﬂows as inﬂow conditions to nozzle and diﬀuser ﬂows. The DNS of pipe and nozzle (diﬀuser) ﬂow run simultaneously and are coupled via MPI routines using the concept of characteristics. Non-orthogonal curvilinear coordinate systems are used to compute the ﬂows in the nozzle and the diﬀuser with a treatment of axis-singularities as suggested by Mohseni and Colonius [8]. The space-time discretisation methods described in Section 2 apply here as well. Nozzle and diﬀuser have a length of L = 10R0 . R0 is the constant radius of the pipe upstream. At the end of the nozzle its circular cross-section has increased by a factor of 2.5, while the diﬀuser cross-section has decreased by 8%. The computational domains of pipe and nozzle are resolved in (x, φ, r)directions by 256×128×91 points and those of pipe and diﬀuser by 256×256× 140 and 384 × 256 × 140 points respectively. The Reynolds and Mach numbers of the fully developed turbulent pipe ﬂow entering the nozzle and the diﬀuser are listed in Table 2. The Reynolds numbers are based on R0 . In all cases the Prandtl number is 0.71, and the ratio of speciﬁc heats, γ, and the viscosity exponent, n, have values 1.4 and 0.7. The wall temperatures of nozzle and diﬀuser are kept constant at the same values as the inﬂow-generating pipes. The Clauser parameter, takes mean values of −1.6 in the nozzle and 5 in the ﬁrst half of the diﬀuser. What happens in the second half of the diﬀuser, where the ﬂow develops transonic features is not discussed here. In the nozzle the bulk Mach number reaches a value of 2.3 at its end, while the bulk Reynolds number decreases by about 40%. In the diﬀuser the bulk Mach number drops from a value of 1.48 to slightly below 1 at x/L = 0.5, while the bulk Reynolds number stays nearly constant. Figure 8 shows radial proﬁles of mean density and temperature at three positions in the nozzle and the diﬀuser. The rapid downstream drop of the temperature in the core below its constant wall value is associated with a decrease in density and pressure. Due to the normalization with local wall values, the density ratio increases, however. In Fig. 9 the van Driest transformed mean velocity, normalized with the local friction velocity, shows the typical features of acceleration in the nozzle and deceleration in the diﬀuser. From the viewpoint of turbulence modelling, it is of Table 2. Parameters of nozzle and diﬀuser inﬂow. Flow

Reτ

Mτ

Rem

Mm

Nozzle Diﬀuser

245 306

0.076 0.0858

3524 4268

1.25 1.48

560

R. Friedrich and S. Ghosh

1.3

T¯/Tw

T¯/Tw

1.4

1.2 1.1

0.9

ρ/ ¯ ρ¯w

ρ/ρ ¯ ¯w

1

0.8 0.7 0

0.2

0.4

0.6

0.8

1

1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0

y/R(x) = 1−r/R(x)

0.2

0.4

0.6

0.8

1

y/R(x)

25

30

20

25

15

u+ VD

+

uVD

Fig. 8. Radial proﬁles of mean density and temperature at positions x/L = 0(......), 0.45 (—), 0.8(− − −) in the nozzle (left) and at x/L = 0(......), 0.13 (—), 0.4(− − −) in the diﬀuser (right).

20 15

10 10

5

5

0 1

10 y+

100

0

1

10

100

y+

Fig. 9. Radial proﬁles of Van Driest transformed mean velocity in the nozzle (left) and in the diﬀuser (right). Positions as in Fig. 8.

interest to see an O(1) increase of the solenoidal and compressible dissipation rates in the diﬀuser, cf. Fig. 10. Yet, the contribution of the dilatational dissipation rate to the transport of turbulence kinetic energy remains negligibly small. As an example of the changes in turbulence structure, we solely discuss the behaviour of the axial Reynolds stress, its production (Pxx ) and pressure– strain correlation terms (Πxx ) in cylindrical coordinates. The production term is split into production by mean shear, dilatation and extra rate of strain. The pressure–strain correlation contains a deviatoric and a dilatational part, viz: ˜x 1 ∂ u˜l 1 ∂ u˜l ∂ u˜x ∂ u Pxx = −ρux ur − − ρux ux −ρux ux , ∂x 3 ∂xl ∂r 3 ∂xl shear

Πxx = p

∂ux ∂x

mean dilatation

−

d 3

1 + p d 3

extra rate of strain

(6)

Enthalpic production is negligibly small. Figure 11 shows the dramatic attenuation and enhancement of the axial Reynolds stress in the nozzle and

DNS of Compressible Turbulent Flows

561

0.00035

1.2

0.0003

1

0.00025

τw2 /¯ μ d /¯

μ τw2 /¯ s /¯

0.8 0.6

0.0002 0.00015

0.4

0.0001

0.2 0

5e-05 0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

y/R(x)

0.6

0.8

1

y/R(x)

Fig. 10. Radial proﬁles of solenoidal (left) and dilatational dissipation rates (right) in the diﬀuser. Positions as in Fig. 8. 30

9 8

25

ρu u /τw

ρu u /τw

7 6 5 4 3 2

15 10 5

1 0

20

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.6

0.8

1

y/R(x)

y/R(x)

Fig. 11. Radial proﬁles of axial Reynolds stress in the nozzle (left) and in the diﬀuser (right). Positions as in Fig. 8. 0.7

0.25

0.6

0.2

0.5

0.15

sh

0.3

0.1 0.05

sh

0

dil es

– 0.05

0

0.2

sh

0.4 0.2

sh es

0.1

dil

0 –0.1 0.4

0.6

y/R(x)

0.8

1

0

0.2

0.4

0.6

0.8

1

y/R(x)

Fig. 12. Radial proﬁles of axial Reynolds stress production by mean shear (sh), extra rate of strain (es), and mean dilatation (dil) in the nozzle (left) at x/L = 0(......), 0.45 (—) and in the diﬀuser (right) at x/L = 0(......), 0.25 (—).

the diﬀuser, respectively. In the nozzle this is partly due to a corresponding reduction in the production by mean shear and a negative production by mean dilatation and extra rate of strain (Fig. 12). In the diﬀuser production

562

R. Friedrich and S. Ghosh 0.05

0

0

–0.005

–0.05

Πxx /¯ τw2 /¯ μ

Πxx /¯ τw2 /¯ μ

0.005

–0.01 –0.015 –0.02 –0.025

–0.1 –0.15 –0.2 –0.25

–0.03

–0.3 0

0.2

0.4

0.6

y/R(x)

0.8

1

0

0.2

0.4

0.6

0.8

1

y/R(x)

Fig. 13. Radial proﬁles of axial pressure–strain correlations in the nozzle (left) and in the diﬀuser (right). Positions as in Fig. 8.

by mean shear is enhanced, while production by mean dilatation and extra rate of strain provide a positive contribution. Note that proﬁles are shown at two positions only. An order of magnitude reduction of Πxx in the nozzle and a similar increase in the diﬀuser are observed in Fig. 13. In all cases the pressure–dilatation term in equation (6) is negligibly small and need not be modelled.

4 Conclusions Using DNS it is observed that subtle diﬀerences exist between compressible channel and pipe ﬂows in the near-wall region, e.g., in the dissipation rates, pressure–strain and pressure–diﬀusion correlations as well as viscous diﬀusion terms. They might be linked to the eﬀect of transverse curvature on the splatting mechanism. Larger diﬀerences in the core region for mean pressure, radial Reynolds stress and the quantities mentioned before are due to the diﬀerences in mean temperature and density. DNS of supersonic nozzle and diﬀuser ﬂow with fully developed turbulent pipe ﬂow as inﬂow condition has proven that remarkable changes happen in the turbulence structure due to mean dilatation eﬀects, even at moderate changes in the bulk Mach numbers. These eﬀects appear explicitly in Reynolds stress production terms and implicitly in pressure–strain correlations. It remains to be shown to what extent mean dilatation modiﬁes pressure ﬂuctuations and in turn pressure–strain correlations. Eﬀects of increasing intrinsic compressibility (e.g., the compressible dissipation rate) are observed in the diﬀuser, but are not strong enough to justify a need for incorporating these eﬀects in statistical turbulence modelling of supersonic ﬂows.

DNS of Compressible Turbulent Flows

563

References 1. N. A. Adams and K. Shariﬀ. A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. Journal of Computational Physics, 127:27–51, 1996. 2. P. Bradshaw. The eﬀect of mean compression or dilatation on the turbulence structure of supersonic boundary layers. Journal of Fluid Mechanics, 63: 449–464, 1974. 3. G. N. Coleman, J. Kim, and R. D. Moser. A numerical study of turbulent supersonic isothermal-wall channel ﬂow. Journal of Fluid Mechanics, 305: 159–183, 1995. 4. H. Foysi, S. Sarkar, and R. Friedrich. Compressibility eﬀects and turbulence scalings in supersonic channel ﬂow. Journal of Fluid Mechanics, 509:207–216, 2004. 5. P. G. Huang, G. N. Coleman, and P. Bradshaw. Compressible turbulent channel ﬂows: DNS results and modelling. Journal of Fluid Mechanics, 305:185–218, 1995. 6. S.K. Lele. Compact ﬁnite diﬀerence schemes with spectral-like resolution. Journal of Computational Physics, 103:16–42, 1992. 7. S.K. Lele. Compressibiliy eﬀects on turbulence. Annual rev. ﬂuid mech., 26: 211–254, 1994. 8. K. Mohseni and T. Colonius. Numerical treatment of polar coordinate singularities. Journal of Computational Physics, 157:787–795, 2000. 9. J. Sesterhenn. A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes. Computers and Fluids, 30:37–67, 2001. 10. Smits A. J. and Dussauge J. P. Turbulent shear layers in supersonic ﬂow, 2nd edition. Spinger, 2006.

Large-Eddy Simulation of Transonic Buﬀet over a Supercritical Airfoil E. Garnier and S. Deck ONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190 Meudon, France, [email protected]; [email protected]

1 Introduction The transonic buﬀet is an aerodynamic phenomenon that results in a large-scale self-sustained periodic motion of the shock over the surface of the airfoil. The time scale associated to this motion is much slower than the one of the wall bounded turbulence. It is then an appropriate case for URANS approaches and ﬁrst attempts with these methods have been reasonably successful in reproducing the mean features of such ﬂows. Nevertheless, as shown by Thiery and Coustols [8] results are very sensitive to the turbulence model. Moreover, with some models, it is necessary to increase the angle of attack with respect to the experiment to obtain an unsteady ﬂow. Furthermore, they have evidenced a signiﬁcant sensitivity of the results to the conﬁnement due to the wind tunnel walls. The ﬁrst hybrid RANS/LES computation on this conﬁguration was performed by Deck [3] who has demonstrated that zonal DES (ZDES) generally improves the results with respect to URANS computations carried out with the Spalart–Allmaras model. In particular, the spectral content of the pressure ﬂuctuations in the separated zone is much more closer to the experimental data with ZDES than with URANS. In this latter computation, the shock/boundary layer interaction was treated in RANS mode and one of the purpose of the present study is to assess the improvement that may result from a fully turbulent treatment of the boundary layer on the suction side of the airfoil by means of LES. More generally, the main objective of this study is to assess the capabilities of LES to capture the buﬀet phenomenon. The large amount of data provided by these simulations could then support the progress in the physical understanding of such ﬂows. The validation of the computation is performed against the very comprehensive experiment performed at ONERA by Jacquin et al. [5] which was also used by Thiery and Coustols [8] and Deck [3].

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 81, c Springer Science+Business Media B.V. 2010

565

566

E. Garnier and S. Deck

2 Description of the computation The supercritical OAT15A airfoil was computed in the same ﬂow conditions than in the experiment by Jacquin et al. [5]. This airfoil has a chord of 230 mm and a relative thickness of 12.3%. Its angle of attack is equal 3.5◦ . The free-stream Mach number was set to 0.73 and the Reynolds number based on the chord length is equal to 3 · 106 . The ﬂow solver is the structured multiblock code FLU3M developed at ONERA. It is second-order accurate in space and time. The numerical scheme dedicated to the computation of the convective ﬂuxes is based on a Roe scheme which was modiﬁed to adapt locally its dissipation using the Ducros et al. sensor [4]. The Selective Mixed Scales Model has been chosen for this study [6]. The time step has been imposed to 3 · 10−7 s in order to ensure the convergence of the subiterative process of the temporal implicit scheme using ﬁve subiterations. In order to limit the required computational eﬀort, the ﬂow is computed in RANS mode on the pressure side of the airfoil and in LES mode on the suction side and in the wake. Moreover, RANS zones are treated in 2D. The grid reﬁnement criteria commonly used in LES of attached ﬂows are satisﬁed (Δx+ ≈ 50 in the longitudinal direction, Δz + ≈ 20 in the spanwise direction + and Δymin ≈ 1 in the wall-normal direction). Despite the zonal treatment of the ﬂow, 20.8 millions of cells are necessary to compute a domain width of only 3.65% of chord in the grid A (N z = 140). The span and consequently the number of points were doubled to construct the grid B (N z = 280). This may be insuﬃcient but the grid size results from a compromise with the long integration time required to capture few buﬀeting periods.

3 Mean ﬁeld analysis After a transient of two periods, the ﬂow has been averaged over only one period of the buﬀet phenomenon for the case A. The span was then doubled to generate the grid B and, after a transient of one period, the statistics were collected over four additional periods. Figure 1 presents an isovalue of the Q criterion colored by the longitudinal velocity. The separation occurs after the location of the shock identiﬁed by one isovalue of the pressure (in purple). On this snapshot which corresponds to a situation where the shock moves downstream, the ﬂow is separated under the lambda shock and near the trailing edge. Figure 2 (left) shows the averaged pressure distribution on the airfoil. The buﬀet zone is shifted downstream by 6% of chord with respect to the experiment. This shift appears more clearly on the pressure ﬂuctuation distributions presented in Fig. 2 (right). The cause of this shift has not yet been identiﬁed. The use of a doubled span (grid B) signiﬁcantly reduces the ﬂuctuations near the trailing edge. The analysis of the instantaneous ﬁelds obtained on

Large-Eddy Simulation of Transonic Buﬀet over a Supercritical Airfoil

567

Fig. 1. Q criteria near the wing wall and one isovalue of the pressure to mark the shock location. 0.5

−2 LESA LESB Exp.

-Kp

−1 −0.5 0

LESA LESB Exp.

0.4 Prms/Qo

−1.5

0.5

0.3 0.2 0.1

1 1.5

0 0

0.2

0.4 0.6 x/c

0.8

1

0.2

0.4

0.6

0.8

x/c

0.014

0.014

0.013

0.013 y(m)

y(m)

Fig. 2. Averaged pressure coeﬃcient distribution (left) and rms pressure distribution (right).

0.012

0.011

0.011

0.01

0.012

0

50 100 150 200 250 300 350 400 U(m/s)

0.01

0

20

40

60

80

Urms(m/s)

Fig. 3. Mean longitudinal velocity (left) and longitudinal velocity ﬂuctuations (right) proﬁles at x/c=0.35 (Grid B).

grid A has evidenced that this overestimation was due to the presence of intense two-dimensional coherent structures developing when the ﬂow separates from the shock up to the trailing edge. The span of the grid B allows the three-dimensionalisation of these structures which limits their intensity and subsequently the wall pressure ﬂuctuations. The proﬁles of averaged and ﬂuctuating longitudinal velocity at x/c = 0.35 are plotted in Fig. 3. These data evidence that, upstream of the interaction,

E. Garnier and S. Deck 0.03

0.03

0.02

0.02 y(m)

y(m)

568

0.01

0

0.01

0

50 100

150 200 250 300 U(m/s)

0

0

50

100

150

Urms(m/s)

Fig. 4. Mean longitudinal velocity (left) and longitudinal velocity ﬂuctuations (right) proﬁles at x/c=0.75 (Grid B).

the velocity ﬁeld is well estimated. This result was far from being trivial since the ﬂow undergoes a numerically forced transition at the same station than in the experiment (x/c = 0.07). Downstream from the interaction (at x/c = 0.75), one can observe in Fig. 4 that the agreement of the LES with both averaged and ﬂuctuating longitudinal velocity proﬁles is more than satisfactory. It is however worthwhile to notice that between x/c = 0.4 and x/c = 0.6, experimental and numerical velocity proﬁles diﬀer signiﬁcantly since the shock is not located at the correct mean position. Nevertheless, it is believed that the results quality is suﬃcient to initiate a physical analysis of the ﬂow.

4 Spectral analysis Due to the short duration of the LES simulations, an auto-regressive (AR) model method has been used to compute the Power Spectral Density of the pressure. Indeed, this method is well adapted to study short data that are known to consist of sinusoids in white noise [9]. The AR parameters are obtained with Burg’s method [1]. The pressure spectrum for x/c = 0.9 is compared to experiment in Fig. 5. The occurrence of strong harmonic peaks highlights the periodic nature of the motion. On the experimental side, the main peak at 69 Hz represents the frequency of the self-sustained motion of the shock over the airfoil. A slightly higher frequency near 71 Hz is found in the computation. The two ﬁrst harmonics of the main peak are also correctly captured numerically.

5 Space and time scales Once, the main statistical and spectral features of the ﬂow have been found, it is worthwhile to study the kinematics of these pressure waves. To this end, let us consider the ﬂuctuating pressure at diﬀerent stations. The two-point

Large-Eddy Simulation of Transonic Buﬀet over a Supercritical Airfoil

569

exp.S3Ch LES

180

SPL

160 140 120 100 1

10

2

3

10

4

10

10

frequency(Hz)

Fig. 5. PSD of pressure ﬂuctuations.

−0.3

∞

71U

0.8

−0.2

41U

∞

1

Δξ /c

0.6 6.1

0.4

11 0 −2 U∞ 10 −2 U∞

7.1 4

0.2 0 0

5 τmaxU ∞ /c

Fig. 6. Propagation velocities obtained by a least square ﬁtting of the linear relation between the separation distance Δξ and time delay τ (ﬁlled symbol: exp, solid line: upper side of the airfoil).

two-time correlation coeﬃcient: R (Δξ, τ )

=

√P

(x1 ,t)P (x1 +Δξ,t−τ )

(P 2 (x1 ))

√

(P 2 (x1 +Δξ))

establishes the correlation between two signals located at abscissa x1 et x1 + Δξ and separated by a time delay τ . The convection velocity can be obtained as the slope of the linear ﬁtting of the Δξ versus τmax (τmax represents the delay where the correlation coeﬃcient reached its maximum), as illustrated in Fig. 6. On the upper-side of the airfoil, a downstream propagation velocity equal 6.11 10−3 U∞ is clearly identiﬁed for the LES and appears to be slightly lower than in the experiment. On the lower side of the airfoil, a forward motion at velocity 0.341 10−3 U∞ is evidenced. The latter velocity is close to the upstream travelling acoustic waves on the lower side of the airfoil.

570

E. Garnier and S. Deck

6 Discussion To assess the frequency of the motion, Lee [7] proposed that the period of the shock oscillation should agree with the time it takes for a disturbance to propagate from the shock to the trailing edge added to the time needed for an upstream moving wave to reach the shock from the trailing edge. A simpliﬁed model has been used in reference [3] to assess the total duration to c−xs c−xs complete such a loop: T = vdownstream + |vupstream | where c is the chord and xs is the mean location of the shock wave. xS can be obtained by noting the ﬁrst abscissa where the skewness of pressure ﬂuctuations is zero. One gets (xs /c)LES = 0.52 while (xs /c)exp = 0.45. The velocity of upstream-travelling acoustic waves is vupstream = a(M − 1) where a is the local speed of sound in the ﬁeld outside the separated area. With M = 0.8 and a = 330 m/s, the Lee’s equation gives f = 1/T ≈ 110 Hz which is higher than the frequency FLES ≈ 70 Hz. More recently, Crouch et al. [2] advocated that transonic buffet results from global instability where the unsteadiness is characterized by phase-locked oscillations of the shock and the separated shear layer. Within this scenario, the region downstream from the shock is not the only region contributing to the feedback loop. Indeed, an upstream travelling acoustic motion has been highlighted on the lower surface of the airfoil (see Fig. 6). A deeper investigation of these phenomena will follow the present work.

Acknowledgments This work has been partly sponsored by the French National Research Agency (project ANR-07-CIS7-009-04).

References 1. Burg, J.P. (1978) In Modern Spectrum Analysis, Ed. D.G. Childers, 34–41, IEEE Press, New-York 2. Crouch, J.D. and Garbaruk, A. and Magidov, D. and Jacquin, L. (2008), Proceedings of IUTAM conference, Corfou, GREECE 3. Deck, S. (2005) AIAA J. 43:1556–1566 4. Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C., and Poinsot, T. (1999) J.Comput. Phys. 152:517–549 5. Jacquin, L., Molton, P., Deck, S., Maury, B., Soulevant, D. (2005) AIAA paper 2005–4902 6. Lenormand, E., Sagaut, P., Ta Phuoc, L., and Comte, P. (2000) AIAA J. 38:1340–1350 7. Lee, B.H.K. (1990) Aeronautical Journal, 143–152 8. Thiery, M. and Coustols, E. (2004) Flow, Turbulence and Combustion 74:331–354 9. Trapier, P. and Duveau, P. and Deck, S. (2006) AIAA J. 44:2354–2365

Detached-Eddy and Delayed Detached-Eddy Simulation of Supersonic Flow over a Three-Dimensional Cavity V. Togiti1 , H. L¨ udeke1 , and M. Breuer2 1

2

Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), 38108 Braunschweig, Germany, vamshi.togiti/[email protected] Department of Fluid Mechanics (PfS), Institute of Mechanics, Helmut–Schmidt–University, 22043 Hamburg, Germany, [email protected]

1 Introduction Cavity ﬂows are geometrically simple but the ﬂuid dynamics in such ﬂows is complex. This kind of ﬂows is common in many applications such as ﬂow around weapon bays, landing gear wells of an aircraft or in scram-jet combustion chambers. Complex ﬂow features that prevail in supersonic cavity ﬂows are boundary layer separation, expansion/compression waves, compressible free shear layers and acoustic/pressure waves. Understanding these complex ﬂows allows to reduce potentially dangerous pressure oscillations. In the past, detached-eddy simulation (DES) and large-eddy simulation (LES) have been used to study supersonic cavity ﬂows, where the free-stream Mach number was less than two [2, 4]. The computational results which include unsteady pressure spectra, the overall sound pressure level (OASPL) and the mean pressure distribution along the cavity ﬂoor were compared with available experiments, but to the authors’ best knowledge not the ﬂow ﬁeld, due to inadequate experimental data concerning the mean ﬂow ﬁeld. In these studies the results were found to be in good agreement with the experiments. The main objective of the present work is to apply the standard DES [6], the modiﬁed DES formulation (DES-MB) by Breuer et al. [1] and the delayed DES (DDES) formulation by Spalart et al. [5] to supersonic cavity ﬂows and to evaluate their performance. In detail, the goal is to study the inﬂuence of the near-wall corrections and the delayed switching of LES on grids which have almost isotropic cells in the free shear layer and to investigate the inﬂuence of grid resolution and momentum thickness of the on-coming boundary layer on the cavity ﬂow dynamics and ﬁnally to compare the computational results with the experimental data of Zhuang [7].

V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 82, c Springer Science+Business Media B.V. 2010

571

572

V. Togiti et al.

2 Numerical methodology In the current investigation the DLR-TAU code is used, which is an unstructured compressible ﬂow solver. Details about the ﬁnite-volume ﬂow solver can be found in [3]. In the present study the convective ﬂuxes are approximated by the advection upstream splitting method AUSM+ -UP, and the diﬀusion ﬂuxes are evaluated by a central scheme. The time advancement is done using an implicit dual time-stepping scheme of second-order accuracy with an implicit lower-upper symmetric Gauß–Seidel algorithm, LUSGS.

3 Details of the test case The conﬁguration investigated is based on the experimental work done by Zhuang [7]. In the experiments, the cavity has an aspect ratio of L/D = 5.2 and L/W = 5.8 and the Mach number is 2. The Reynolds number based on the length of the cavity is 2.8 · 106 . In the present study, the experimental conditions of Zhuang are reproduced. The computational domain has an inﬂow boundary at 5.2 D upstream of the cavity, where supersonic inﬂow conditions are applied. This distance was required to match the experimental boundary layer thickness of about 0.1 D at the leading edge of the cavity. The momentum thickness at this position is θ = 0.2 mm. The upper and outﬂow boundaries are placed at 10 D above the cavity and 6 D downstream of the cavity, respectively. Supersonic outﬂow conditions are applied at the outﬂow plane and far-ﬁeld conditions at the upper boundary. In spanwise direction, periodic boundary conditions are used.

4 Results 4.1 Comparison of DES, DES-MB and DDES Instantaneous numerical schlieren images (magnitude of density gradient) produced by DES, DES-MB and DDES are shown in Fig. 1. All versions of DES are able to predict large-scale structures in the free shear layer and the shear layer breakup. In the case of DES-MB, more vortical structures

DES

DES-MB Fig. 1. Numerical schlieren images.

DDES

DES & DDES of Supersonic Flow over 3D Cavity

573

are apparent, which is due to a low level of eddy viscosity in the shear layer predicted by this model. Time-averaged streamwise velocity contours in the midplane are depicted for diﬀerent versions of DES in Fig. 2 and compared with the experiment. The contours show a recirculation region inside the cavity with very high velocity which is as high as 40% of the free-stream velocity. The level of recirculation velocity and the extent of the recirculation predicted by diﬀerent DES versions are almost the same, but they are slightly overpredicted compared with the experiment. Proﬁles of the resolved streamwise Reynolds stresses predicted by diﬀerent version of DES are depicted in Fig. 3 and compared with the experiment. The

DES

DES-MB

DDES

Experiment [7]

Fig. 2. Contours of mean streamwise velocity u.

b

8

2

( y-y0.5) /θ

( y-y0.5) /θ

4

0 −2

c

8

DDES DES DES-MB Exp.

6

8

6

6

4

4

2

2

( y-y0.5) /θ

a

0

−2

0

−2

−4

−4

−4

−6

−6

−6

−8

−8 0

0

10 20 30 40 σ u /U

X/L = 0.1

10 20 30 40 σ u /U

X/L = 0.5

−8 0

10 20 30 40 σ u /U

X/L = 0.9

Fig. 3. Resolved streamwise turbulent intensity in %.

V. Togiti et al.

a

b

180

DDES exp. mode1 mode2 mode3 mode4

SPL [dB]

170 160 150 140 130 120

200 DES DES-MB DDES Exp.

190 OASPL [dB]

574

180 170 160 150

1000

10000 Frequency [Hz]

SPL spectra

0

0.2

0.4

0.6

0.8

1

X/L

OASPL

Fig. 4. SPL spectra (a) and overall sound pressure level (b).

proﬁles show that the stress level predicted is higher in the initial region of the free shear layer, and the level comes closer to the experimental data further downstream. In the case of the modiﬁcations by Breuer et al. [1] slightly higher stresses are predicted compared to other DES versions. The reason is a lower level of eddy viscosity in the free shear layer as mentioned above. The spectrum of the unsteady pressure monitored along the base of the cavity near the trailing edge is shown in Fig. 4a. For the sake of brevity solely the pressure spectrum predicted by DDES is displayed here. The spectrum shows peaks at the modes experimentally determined and the dominant mode in the cavity, i.e., mode 4, is also predicted correctly. The overall sound pressure level (OASPL) along the cavity ﬂoor is depicted in Fig. 4b. The results show that the sound pressure level is overpredicted by about 5 dB compared with the experiment while almost the same level of overall sound pressure is delivered by the diﬀerent versions of DES. The reason for higher overall sound pressure levels along the cavity ﬂoor and higher resolved stresses in the shear layer in the predictions are more large-scale structures in the free shear layer, possibly due to inaccurate inﬂow conditions (see below). 4.2 Inﬂuence of grid resolution In this section, results obtained by DDES on grids with diﬀerent resolutions are discussed and the inﬂuence of the grid resolution is studied. The timeaveraged momentum thickness θ of the shear layer is shown in Fig. 5a. A larger momentum thickness is predicted in the initial region of the shear layer on the ﬁne grid. Almost the same level of shear layer thickness is predicted on the medium and ﬁne grid downstream of the initial free shear-layer region (X/L > 0.5). Larger momentum thickness of the free shear layer on the medium and ﬁne grid compared with the coarse grid are due to more resolved eddies on the medium and ﬁne grid, which lead to higher ﬂuid entrainment and higher reverse velocity. The overall sound pressure levels predicted on the diﬀerent grids are displayed in Fig. 5b. The predictions yield that the sound pressure level is not much inﬂuenced by the grid resolution.

DES & DDES of Supersonic Flow over 3D Cavity

a

b

5 FG MG CG

3

θ [mm]

200 CG MG FG Exp.

190 OASPL [dB]

4

2 1

575

180 170 160 150

0

0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

X/L

0.6

0.8

1

X/L

Momentum thickness θ

OASPL

Fig. 5. Time-averaged shear layer momentum thickness (a) and overall sound pressure level (b); FG: ﬁne grid, MG: medium grid and CG: coarse grid.

b 200

0.3 θ0.2 θ0.4 Exp.

Y/L

0.2

θ0.2 θ0.4 θ0.4-slip Exp.

190 OASPL [dB]

a

0.1 0

−0.1

180 170 160 150

−0.2

0

0.2

0.4

0.6 X/L

Shear layer

0.8

1

0

0.2

0.4

0.6

0.8

1

X/L

OASPL

Fig. 6. u = 0 in the shear layer (a) and overall sound pressure level (b).

4.3 Inﬂuence of momentum thickness of the boundary layer at the cavity leading edge In this section, results obtained by DDES with the same boundary layer thickness at the leading edge of the cavity, but diﬀerent momentum thicknesses are discussed. For that purpose θ is increased from 0.2 mm in the previous cases to 0.4 mm. Concerning the mean velocity distribution a strong inﬂuence of the momentum thickness is not observed. This is visualized by the time-averaged zero streamwise velocity component proﬁle in the free shear layer along the cavity in Fig. 6a. The proﬁle shows that the predictions in the region X/L = 0.2 to 0.8 match with the experiments, whereas diﬀerence near the leading and trailing edge prevail. The overall sound pressure level along the cavity ﬂoor for diﬀerent boundary layers along with the experiment is depicted in Fig. 6b. A lower sound pressure level and a closer agreement with the experiment in the region of X/L of 0.2 to 0.8 is observed in the case of the boundary layer with a larger momentum thickness. The inﬂuence of the boundary condition applied in the lateral direction on the OASPL is also studied by enforcing slip-wall condition. In Fig. 6b,

576

V. Togiti et al.

predictions for the boundary layer with larger momentum thickness, θ0.4 -slip, unveil that the sound pressure level is not much aﬀected by the boundary condition applied.

5 Summary In the current investigation diﬀerent versions of DES were applied to a supersonic cavity ﬂow and the inﬂuence of DES model modiﬁcations, grid resolution and boundary conditions, especially the boundary layer momentum thickness at the cavity leading edge, were investigated. Concerning the mean velocities the prediction by diﬀerent versions of DES on the same grid followed the trends of the experiments. In all cases, the resolved streamwise and normal stresses were overpredicted compared with the experiment. The overall sound pressure level along the cavity ﬂoor at the base of the leading and trailing edge was 5 dB higher than in the experiments. All three versions of DES gave almost similar results on the same grid The inﬂuence of the grid resolution on the ﬂow ﬁeld was studied using DDES. The predictions showed a larger amount of ﬂuid entrainment inside the cavity and higher stresses in the free shear layer as the grid was reﬁned. The reason for this trend is thought to be more resolved eddies in the free shear layer which lead to a larger shear layer momentum thickness. The inﬂuence of the momentum thickness of the on-coming boundary layer was studied using DDES on a coarse grid. In the case of a larger boundary layer momentum thickness the predictions showed improved agreement with the experiment, but still deviations prevailed though diﬀerent boundary conditions in the lateral direction were applied. The possible reason for these deviations is the on-coming boundary layer. Since in the computations a steady inﬂow condition was applied, the boundary layer at the cavity leading edge might slightly deviate from the experimental conditions.

References 1. Breuer, M., Jovici´c, N., Mazaev, K. (2003). Comparison of DES, RANS and LES for the Separated Flow Around a Flat Plate at High Incidence. Int. J. Num. Meth. Fluids 41:357–388. 2. Hamed, A., Basu, D., Das, K. (2003). Detached-Eddy Simulation of Supersonic Flow Over Cavity. 41st AIAA Conf., Reno, Nevada, 2003–0549. 3. Mack, A., Hannemann, V. (2002). Validation of the Unstructured DLR TAUCode for Hypersonic Flows, AIAA Paper, 2002–3111. 4. Smith, B.R. (2001). Large-Eddy Simulation of a Supersonic Cavity Flow with an Unstructured Grid Flow Solver. DNS/LES Progress and Challenges. Proc. of the Third AFOSR Int. Conf. on DNS/LES, Arlington Texas. 5. Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M.Kh., and Travin, A. (2006). A New Version of Detached-Eddy Simulation, Resistant to Ambiguous Grid Densities. J. Theor. Comput. Fluid Dynamics, 20:181–195.

DES & DDES of Supersonic Flow over 3D Cavity

577

6. Spalart, P.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. Advances in DNS/LES, 1st AFSOR Int. Conf. on DNS/LES, Greyden Press, Col., OH. 7. Zhuang, N. (2007) Experimental Investigation of Supersonic Cavity Flows and Their Control. PhD Thesis, Florida State University.

A WALE-Similarity Mixed Model for Large-Eddy Simulation of Wall Bounded Compressible Turbulent Flows G. Lodato, P. Domingo, and L. Vervisch INSA-Rouen, UMR-CNRS-6614-CORIA, Campus du Madrillet, Avenue de l’Universit´e, BP 8, 76801 Saint Etienne du Rouvray Cedex, France, [email protected]; [email protected]; [email protected] Abstract Wall-jet interaction is studied with Large Eddy Simulation (LES) in which a mixed similarity Sub-Grid Scale (SGS) closure is combined with the WallAdapting Local Eddy-viscosity (WALE) model for the eddy-viscosity term. Reduced macrotemperature and macropressure are introduced to deduce a weakly compressible form of the mixed similarity model and the relevant formulation for the energy equation is deduced accordingly. LES prediction capabilities are assessed by comparing ﬂow statistical properties against experiment of an unconﬁned impinging round-jet at Reynolds number of 23,000 and 70,000.

1 Introduction Impinging jets are a relatively simple conﬁguration which can be characterized by three main regions: (a) free jet, (b) stagnation and (c) wall jet ﬂows. Partly due to this heterogeneity and partly due to some peculiar features, like the eﬀect of strong curvature over the wall jet turbulence development, the impinging jets represent a particularly tough test bench for turbulence modeling. Most of the studies which can be found in literature regarding LargeEddy Simulation (LES) of impinging round-jets are focused on wall thermal exchange, without particular attention on Sub-Grid Scale (SGS) modeling performances within the most critical near-wall region [10, 11]. When wall boundaries are present, most sub-grid scale models are generally inadequate in reproducing the correct wall-scaling for unresolved quantities. The dynamic procedure [9, 19] can be used to recover the correct wall scaling, but some spatial or lagrangian [18] averaging need to be adopted. Similarity models (see [17, 21] for a review) remove the hypothesis of alignment between SGS stress and resolved shear stress tensor, thus achieving strong correlation between expected and measured SGS stresses. These models provide correct wall scaling on each component of the SGS stress tensor [14] and predict energy backscatter. On the other hand, similarity models are generally not enough dissipative and the addition of an eddy-viscosity term may be V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 83, c Springer Science+Business Media B.V. 2010

579

580

G. Lodato et al.

advisable [3]. The resulting closures, the similarity-mixed models, generally resort to the classical Smagorinsky eddy viscosity hypothesis and to dynamic modeling, in order to retain the correct wall-scaling [1, 2, 22, 23]. In this context, the WALE model [20] is of interest since by construction, it retains the correct wall-scaling. Along these lines, a compressible version of the similarity mixed model is discussed below, where the eddy-viscosity term is computed by the WALE approximation. The model is tested on turbulent impinging round-jets with Reynolds numbers of 23,000 and 70,000; results are compared against available experimental data [4, 7]. A comparison with the standard WALE model and a compressible extension of the Lagrangian Dynamic Smagorinsky model [18] is also presented at ReD = 23, 000.

2 Mathematical formulation The problem is described by the ﬁltered compressible set of Navier–Stokes equations for a Newtonian ﬂuid following the ideal single-component gas law. Introducing the ﬁlter operator bar and the Favre-ﬁlter operator tilde, the set of equations is derived with analogous reasoning as in [13]; in the perspective of mixed similarity modeling [3] a slightly diﬀerent deﬁnition of macropressure and macrotemperature is used here as part of the spherical part of the SGS stress, namely the spherical part of the modiﬁed Leonard term Lkk [8], is computed from the resolved ﬂow ﬁeld. We then talk about reduced macropressure and macrotemperature as they include the spherical part of the true Reynolds term only. The ﬁltered compressible Navier–Stokes equations in compact form read: ∂t ρ + ∂j (ρuk ) = 0, 1 ij = ∂j τ d , ∂t (ρui ) + ∂j ρui u j + δij % + 3 Lkk − 2μA ij μcp j − 2μ uk Ajk − Pr ∂j ϑ = ∂j qj , ∂t (ρe) + ∂j ρe + % u

(1) (2) (3)

where the SGS terms on the RHS are expressed using the mixed similarity assumption extended to compressible ﬂows [6, 22] (superscript ‘d’ indicates the deviatoric part) d d − τijd ρ u j + ρui uj , $ j − u i u iu

k + γρeI uk , (4) −qk γρ e$ k − e I u Iu

while % = p + ρuk uk /3, and ϑ = T + ρuk uk /(2ρcv ) are the reduced macropressure and macrotemperature respectively. Under the assumption that the 2 true SGS Mach number, Msgs = ρuk uk /(γp) is suﬃciently small [13], the ﬁltered equation of state results formally identical to its unﬁltered counterpart, the pressure and temperature being replaced by the reduced macropressure and macrotemperature respectively. Consequently the resolved total energy ρe (kinetic + thermal) and the resolved internal energy ρeI may be uk u k /2 + Lkk /2 and ρeI = ρcv ϑ = %/(γ − 1). expressed as: ρe = ρcv ϑ + ρ

A WALE-Similarity Mixed Model for Compressible Flows

581

In the above relations, ρ is the ﬂuid mass density, uk (k = 1, 2, 3) are the velocity components, γ = cp /cv is the ratio between speciﬁc heat at constant pressure and temperature and is set equal to 1.4; Pr = 0.72 is the Prandtl number, Aij is the deviator of the deformation tensor and the dynamic viscosity μ is computed using the Sutherland’s law for air. 2.1 The WALE-similarity model (WSM) Within the framework of similarity mixed modeling [3], the SGS contributions are modeled by means of an eddy-viscosity term, which is computed from resolved quantities, and the modiﬁed Leonard term, which involves explicit ﬁltering of resolved quantities. With reference to the Leonard term, scale invariance is assumed to postulate that the structure of the velocity ﬁeld at scales below a certain length-scale Δ is similar to that at scales above Δ [21]. Consequently, the SGS closures are, more generally, rewritten as: d ρνt ij − ρ u τijd = 2ρνt A u j k , (5) = j − u i and qk = γ Pr ∂k eI − γρ e= k − e I u iu Iu t the hat operator representing ﬁltering at cutoﬀ length αΔ, with α ≥ 1. The turbulent Prandtl number, Prt , is set constant and equal to 0.9. In the present paper we adopt the same assumption as in [1], where the SGS model is designed to distinguish between two contributions: (a) the local interactions near the cutoﬀ length scale which are responsible for intense and coherent regions of forward and reverse energy transfer and (b) the non-local interactions which are responsible for a low-intensity forward energy transfer. The former is represented by the modiﬁed Leonard term, measuring the interactions between the resolved scales and a narrow band of sub-grid wave numbers between Δ and 4/3Δ (α = 4/3), while the latter is closed with the WALE eddy-viscosity model [20], which automatically recovers proper scaling in the near-wall region: d d 3/2 νt = Cw2 Δ2 ( sij sij ) / (Sij Sij )5/2 + ( (6) sdij sdij )5/4 , where Cw = 0.5 is a true model constant, Sij is the strain rate tensor of the resolved ﬁeld and sdij is the traceless symmetric part of the square of the 2 = gik gkj . resolved velocity gradient tensor gij 2.2 Flow conﬁguration and numerics The ﬂow conﬁguration under study consists of an unconﬁned impinging roundjet; the jet-nozzle to wall distance is equal to twice the jet diameter D. Two

582

G. Lodato et al.

values of jet Reynolds number have been studied: ReD = 23, 000 and ReD = 70, 000. For both the test-cases, the Mach number of the jet is 0.4. The computational domain is a cartesian grid stretched toward the wall to ensure accuracy in the wall-layer. Sub-layer scaled grid spacings are Δ+ w ∼ 5.5, + + Δ+ ∼ 75.8 and Δ ∼ 7.0, Δ ∼ 139.5 for Re = 23, 000 and 70, 000 reD w r,φ r,φ spectively, subscripts w, r and φ indicating wall-normal, radial and azimuthal directions. These grids produced an average SGS to laminar viscosity ratio below 5 for the lower Reynolds number and below 12 for the higher one. Additional details about LES resolution, as well as a grid sensitivity analysis are presented in [16], where the inﬂuence of mesh resolution is further investigated at ReD = 23, 000. The bottom boundary is an adiabatic no-slip wall, while the lateral sides and the top surface are open boundaries. The jet axis is aligned along x1 and the coordinate system origin is located at the impingement wall. A parallel solver based on an explicit Finite Volumes (FV) scheme was used. Spatial discretization is fourth-order with convective terms computed with a centered skew-symmetric-like scheme [5]. Time integration is performed using a thirdorder Runge-Kutta scheme. 3D-NSCBC [15] treatment is imposed at boundaries. The open boundaries are non-reﬂecting outﬂows, while in the inﬂow section, a non-reﬂecting inlet is applied with prescribed Power Law proﬁle for turbulent pipe ﬂow and a correlated random noise [12].

3 Results and discussion The WALE-Similarity Model (WSM) behavior has been assessed by comparison against the experimental results [4, 7] in terms of resolved average and ﬂuctuating velocities. Additional results are discussed in [16]. Results for the ReD = 23, 000 test-case are shown in Fig. 1. The WSM is compared to the standard WALE model and the Lagrangian Dynamic Smagorinsky model (LDSM). With regards to the average velocity proﬁles (cf. [16]), all the three models perform equally well. With regards to the streamwise ﬂuctuations, the LDSM model predicts fairly well measurements far from the wall, but produces a signiﬁcant overestimation in the near-wall region (Fig. 1a). This overestimation is even stronger with the standard WALE model which is also too dissipative far from the wall. The introduction of the modiﬁed Leonard term in the WSM has a strong impact on the resolved streamwise velocity ﬂuctuations, these last being slightly underestimated far from the wall in the wall-jet region. Vertical ﬂuctuations proﬁles reproduce fairly well the Laser-Doppler (LDA) measures (Fig. 1c); note that hot-wire (HWA) and LDA measurements of vertical ﬂuctuations are diﬀerent, the ﬁrst being signiﬁcantly lower than the last. Good agreement is also observed on radial ﬂuctuations (Fig. 1d). Turbulent shear stresses are fairly well reproduced

A WALE-Similarity Mixed Model for Compressible Flows

a

b

y/D=0.5

0.04

583

0.02

0

y/D=1.0

0.04 0.02 0

0.005

y/D=1.5

0.04

y/D=0.5

0

0.02

–0.005

0

–0.01 0.005 0

y/D=2.0

0.04 0.02 0

–0.005 –0.01 0.005

y/D=2.5

0.04

y/D=1.0

y/D=2.5

0 0.02

–0.005

0

–0.01 0.005 0

y/D=3.0

0.04 0.02

y/D=3.0

–0.005 –0.01 0.00

0 0.00

0.05

0.10

0.15

0.20 x/D

0.25

0.30

0.35

0.40

0.05

u u /Ub2

0.10

0.15

0.20 x/D

0.25

0.30

0.35

0.40

u v /Ub2

d

c

0.18

0.18 y/D=0.5

0.12

y/D=0.5

0.12 0.06

0.06 0 0.12

0 y/D=1.0

0.08

y/D=1.0

0.12

0.04 0 0.00

0.18

0.06 0.20

0.40

0.60

0.80 x/D

v v /Ub

1.00

1.20

1.40

0 0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

x/D

ur ur /Ub

Fig. 1. ReD = 23, 000 test-case. Streamwise ﬂuctuations (a), turbulent shear stress (b), wall-normal RMS velocity (c) and radial RMS velocity (d): solid line, WSM; dashed line, WALE; dot-dashed line, LDSM; circles, HWA measurements [4]; squares, LDA measurements [7].

(Fig. 1b), especially far from the jet’s axis, the proposed WSM giving the best agreement. The results obtained with the WSM on the ReD = 70, 000 test-case are shown in Fig. 2. Solid curves refer to statistical quantities extracted directly from the resolved ﬂow ﬁeld (as in the previous test-case), while dashed lines have been obtained including the deviator of the SGS stress only [21]. As far as average velocity proﬁles are concerned (not shown), the matching between experimental data and computed solution is nearly perfect. The turbulent kinetic energy (Fig. 2a) is well represented by the WSM. The inclusion of the SGS part produces better agreement but, still, since the spherical part of the SGS stress tensor plays the key role in this context, the improvement is modest. Turbulent shear stresses (Fig. 2b), again, are well predicted and

584

G. Lodato et al.

a

b

y/D=1.0

0.03 0.015 0

y/D=1.5

0.03

y/D=0.5

0

0.015 – 0.01

0

y/D=2.0

0.03

y/D=1.0

0

0.015

– 0.01

0

y/D=2.5

0.03

y/D=2.5

0

0.015

– 0.01

0

y/D=3.0

0.03

0 0.00

y/D=3.0

0

0.015

– 0.01

0.05

0.10

0.15

1 (u u 2

0.20 x/D

0.25

0.30

+ v v )/Ub2

0.35

0.40

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

x/D

u v /Ub2

Fig. 2. ReD = 70, 000 test-case. Turbulent kinetic energy (a) and turbulent shear stress (b): solid line, resolved ﬂuctuations; dot-dashed line, resolved ﬂuctuations plus SGS contributions; circles, HWA measurements [4].

the inclusion of the SGS part, which in this case accounts for all the terms, produces a relatively signiﬁcant improvement, the SGS part accounting for about 10% of the turbulent shear stress.

References 1. Akhavan R., Ansari A., Kang S., Mangiavacchi N. (2000). J. Fluid Mech., 408: 83–120. 2. Anderson R., Meneveau C. (1999). Flow Turbul. Combust., 62(3): 201–225. 3. Bardina J., Ferziger J., Reynolds W. (1984). Improved turbulence models based on LES of homogeneous incompressible turbulent ﬂows. Report TF-19, Thermosciences Division, Dept. Mechanical Engineering, Stanford University. 4. Cooper D., Jackson D., Launder B., Liao G. (1993). Int. J. Heat Mass Tran., 36(10): 2675–2684. 5. Ducros F., Laporte F., Soul`eres T., Guinot V., Moinat P., Caruelle B. (2000). J. Comput. Phys., 161: 114–139. 6. Erlebacher G., Hussaini M., Speziale C., Zang T. (1992). J. Fluid Mech., 238: 155–185. 7. Geers L., Tummers M., Hanjali´c K. (2004). Exp. Fluids, 36(6): 946–958. 8. Germano M. (1986). Phys. Fluids, 29(7): 2323–2324. 9. Germano M., Piomelli U., Moin P., Cabot W. (1991). Phys. Fluids A-Fluid, 3(7): 1760–1765. 10. Hadˇziabdi´c M., Hanjali´c K. (2008). J. Fluid Mech., 596: 221–260. 11. H¨ allqvist T. (2006). Large Eddy Simulation of Impinging Jets with Heat Transfer. Ph.D. thesis, Royal Institute of Technology, Department of Mechanics, S-100 44 Stockholm, Sweden.

A WALE-Similarity Mixed Model for Compressible Flows

585

12. Klein M., Sadiki A., Janicka J. (2003). J. Comput. Phys., 186: 652–665. 13. Lesieur M., M´etais O., Comte P., Large-Eddy Simulations of Turbulence (Cambridge University Press, 2005). 14. Liu S., Meneveau C., Katz J. (1994). J. Fluid Mech., 275: 83–119. 15. Lodato G., Domingo P., Vervisch L. (2008). J. Comput. Phys., 227(10): 5105–5143. 16. Lodato G., Vervisch L., Domingo P. (2009). Phys. Fluids, 21(3): 035102. 17. Meneveau C., Katz J. (2000). Annu. Rev. Fluid Mech., 32(1): 1–32. 18. Meneveau C., Lund T., Cabot W. (1996). J. Fluid Mech., 319: 353–385. 19. Moin P., Squires K., Cabot W., Lee S. (1991). Phys. Fluids A-Fluid, 3(11): 2746–2757. 20. Nicoud F., Ducros F. (1999). Flow Turbul. Combust., 62(3): 183–200. 21. Sagaut P., Large Eddy Simulation for Incompressible Flows: An Introduction (Springer-Verlag Berlin Heidelberg, 2001), 2nd edn. 22. Salvetti M., Banerjee S. (1995). Phys. Fluids, 7(11): 2831–2847. 23. Zang Y., Street R.L., Koseﬀ J.R. (1993). Phys. Fluids A-Fluid, 5(12): 3186–3196.

Parametric Study of Compressible Turbulent Spots J.A. Redford, N.D. Sandham, and G.T. Roberts Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK, [email protected]; [email protected]; [email protected] Abstract The compressible Navier–Stokes equations are solved for turbulent spots at Mach 3 and 6, with both adiabatic and cold (Tw = T∞ ) wall boundary conditions on a ﬂat plate. From the results turbulent spot growth and propagation rates are derived which will help to quantify the length of laminar to turbulent transition. The simulations also yield skin-friction and heat-transfer data.

1 Introduction There is a continuing drive to ﬁnd technology and materials that will make sustained hypersonic ﬂight feasible. The transition from laminar to turbulent ﬂow in a boundary layer is important when trying to quantify surface heat transfer and drag in a supersonic or hypersonic ﬂow; these are important factors when deciding upon suitable materials. Turbulent spots play an important role in transition as they are one route by which a strongly perturbed boundary layer ﬁnally becomes turbulent. The spot has a smaller spread rate at higher Mach numbers [1], lengthening the transition process and making the prediction of transition more important. Data quantifying turbulent spot growth under a variety of conditions will help to improve transition predictions. Experimental studies of incompressible turbulent spots in ﬂat plate boundary layers are quite common but high Mach number experiments are more difﬁcult to perform. The increasing availability of high power computer resources means that it is now possible to simulate high Mach number turbulent spots [2, 3]. However, the Krishnan and Sandham [2] Mach 6 simulation was underdeveloped, only containing structures associated with Mack mode instability rather than turbulence. Here we rectify this with a higher Reynolds number simulation using a ﬁner grid. Compressibility tends to initially stabilise a laminar boundary layer. An incompressible boundary layer is unstable through the two-dimensional Tollmien–Schlichting instability. Compressibility causes the primary instability V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 84, c Springer Science+Business Media B.V. 2010

587

588

J.A. Redford et al.

to become oblique until approximately Mach 3, while at still higher Mach numbers a Mack (second) mode of instability becomes the most unstable. Krishnan and Sandham [2] found that their Mach 6 spot simulation consisted of structures reminiscent of the Mack mode instability. At high Mach number there is considerable heating within the boundary layer and in real life applications it may be necessary to cool the surface. In the present investigation we consider a cold wall condition, where Tw = T∞ , as well as an adiabatic wall temperature. The diﬀerence between the cold and adiabatic temperature is a factor of 7.0 at Mach 6, so we can expect a signiﬁcant change in the local Mach and Reynolds number resulting in changes to the turbulent spot ﬂow structure and growth characteristics.

2 Method The code used here is the same low-dissipation method that Krishnan and Sandham [2] used. It is desirable to avoid simulating the ﬂow around the plate leading edge, because of the ﬁne grid spacing that would be required near the surface. Also, at high Mach numbers the leading edge ﬂow becomes more complicated with the introduction of a shock wave due to viscous-inviscid interaction. To avoid these problems and save computational eﬀort we use a velocity and temperature proﬁle from a compressible similarity solution [4] as the inlet condition for the computational domain. A two-dimensional simulation is then run to develop a steady state laminar boundary layer which is to be used as the base ﬂow. The lateral boundary condition is periodic, thus it is important that the turbulent spot has suﬃcient room to spread without interfering with its own image. The top boundary and outlet use a characteristic boundary condition to reduce the reﬂections back into the domain. The turbulent spot is initiated at a stable location (Re < Recrit ) in a laminar boundary layer using a vortical disturbance similar to that used by Breuer and Landhal [5]. Symmetry is deliberately broken by rotating the initial disturbance by 1◦ , which eﬀectively doubles the sample size when measuring the spot spreading angle. The Reynolds number at the domain inlet is varied because of the stabilizing eﬀect of compressibility and wall temperature. Simulations at greater Mach number or wall temperature both require an increased Reynolds number to become turbulent. In the current simulations at Mach 3 the inlet displacement thickness Reynolds number is Reδ∗ = 1,500 and 1,000 for the adiabatic and cold wall respectively, while the Mach 6 spots simulations have Reδ∗ = 5,500 and 3,000. The grid (Nx ×Ny ×Nz ) and domain (Lx × Ly × Lz ) size of each turbulent spot simulation are shown in Table 1, The Prandtl number is set to 0.72 and we use a Sutherland’s law to calculate the viscosity with a Sutherland constant of 110 K at a reference temperature of 288 K.

Parametric Study of Compressible Turbulent Spots

589

Table 1. Details of each simulation including grid size (Ni ), domain size (Li ), and the averaging width (±Δz) for the skin-friction and heat transfer measurements. M

Re∗δ

Tw /T∞

Nx

Ny

Nz

Lx

Ly

Lz

±Δz

3 3 6 6

1,500 1,000 5,500 3,000

2.5 1.0 7.0 1.0

1,101 1,321 801 1,001

111 111 141 141

191 321 201 201

700 700 600 750

40 40 30 30

120 120 60 60

12.5 7.5 6.0 6.0

Fig. 1. Visualisation of compressible turbulent spots at t = 400 with isosurfaces of Π = −8 × 10−4 in black, and u = u − ulam = ±0.02 in medium and light gray. (a) M = 3 adiabatic; (b) M = 3 cold; (c) M = 6 adiabatic; (d) M = 6 cold.

3 Results The turbulent spots are illustrated in Fig. 1 by isosurfaces of Π = (∂ui /∂xj ) (∂uj /∂xi ), and the streamwise velocity perturbation u from laminar ﬂow. Each spot consists of a turbulent core highlighted by Π, accelerated ﬂuid (u > 0) in the calmed region at the rear, and deﬁcit regions (u < 0) at the lateral extremities of the spot. The presence of turbulence is conﬁrmed by the familiar hairpin structures that are to be found in the centre of each spot. All the simulations result in spots with a turbulent core region. The wall temperature has a marked eﬀect on the structure of the turbulent spot, particularly in combination with an increase in Mach number. At Mach 3 the cold wall causes the formation of spanwise-oriented structures at the sides and under the front overhang of the spot. For the Mach 6 cold wall spot

590

J.A. Redford et al.

Spot spreading half angle (degrees)

these structures are more prevalent and have spread to the calmed region. A similar pattern is found in [3] for a Mach 5 cold wall spot. The lateral structures are attributed to a Mack mode instability. The cold wall reduces the threshold Mach number for such instabilities since the local sound speed reduces with temperature. Movies of the simulations in a frame of reference traveling at the free stream velocity show that new structures are added in the wing tip region. The rear of the spot contains low speed near wall structures that convect out of the spot core. The turbulence at the front of the spot is relatively inactive as it is washed downstream, and it is here that some remnants of the initial condition are found. Triggering the turbulent spots using the method of [5] tends to create horned structures protruding from the front of the spot. Previous studies of compressible turbulent spots [1–3] have found the growth rate to be reduced with an increased free stream Mach number. The same eﬀect is observed in our simulations (Fig. 2). Spot growth rates are computed by plotting a plan view of the wall normal vorticity ωy = 0.06 for the same spot at two separate times and manually ﬁtting lines either side of the spot. The process is repeated at diﬀerent times (keeping the same spacing) creating a sample of 24 measurements, and error bars of one standard deviation are added to Fig. 2. The variation in individual measurements is wide due to the subjective procedure for taking the measurements, but the sample is large so the mean should be reliable. The current results are consistent with the experimental measurements collected by Fischer [1]. At Mach 6 the adiabatic wall turbulent spot spreading rate is 50% greater than in [2], because the spot now contains well developed turbulence (Fig. 1c). Figure 2 shows that the cold wall condition reduces the spread angle of the turbulent spot by 20–30%, a fact that is consistent with the ﬂow visualisations 12 10 8 6 4 2 0 0

1

2

3

4

5

6

7

M

Fig. 2. Collected measurements of spot spreading half-angle. Current data are ﬁlled symbols, • adiabatic wall results, cold wall results. Krishnan and Sandham [2] adiabatic wall . Jocksch and Kleiser [3] adiabatic wall +, cold wall ×. The shaded area was derived by Fischer [1] from experimental data.

Parametric Study of Compressible Turbulent Spots

591

(Fig. 1) which show more slender spots for the cold wall case. Lower wall temperature reduces the sound speed within the boundary layer (increasing the local Mach number), so it is perhaps unsurprising that the reduced wall temperature has a similar eﬀect to an increased free-stream Mach number on spot growth rate. It is normally presumed that once the Reynolds number has been increased beyond a threshold the fully developed spot growth will not depend on Reynolds number. We ﬁnd that the spreading angle has no dependence on the location at which the measurement is taken. However, Jocksch and Kleiser [3] found that the spreading angle of their turbulent spot was increased with Reynolds number at the spot initiation point. It is therefore possible that the Reynolds number at the initiation point, along with the spot initiation method, aﬀects the later development in these simulations. This is to be investigated further in future work. Figure 3 shows the skin-friction coeﬃcient for each spot at one time instant. To improve the sample size we have averaged over a spanwise line of length 2Δz given in Table 1. Comparing simulation results with the Eckert [6]

a 3

b

x10−3

x10−3 4

2.5 3 Cf

Cf

2 1.5

2

1 1

0.5 0

c

2

4

6 Rex

8

0

10 x

2

3

4

105

d

x10−3

5 Rex

6

7

8 x 105

−3

2 x10

1.5 1

Cf

Cf

1.5

0.5 0

1

0.5

0.5

1

1.5

2 Rex

2.5

3

3.5

x 105

0

0.5

1

1.5 Rex

2

2.5 x 105

Fig. 3. Skin-friction coeﬃcient (—) of adiabatic wall spots compared with the Eckert laminar (– –) and turbulent (– · –) correlations. (a) Mach 3 adiabatic (t = 680); (b) Mach 3 cold (t = 640); (c) Mach 6 adiabatic (t = 400); and (d) Mach 6 cold (t = 700).

592

J.A. Redford et al.

compressible boundary layer correlations shows how well developed the ﬂow is within the turbulent spot. In doing so we are assuming that the turbulence within a spot is similar to the turbulence found in a fully turbulent boundary layer that develops from the plate leading edge. In three of the four cases there is a reasonable match between the spot skin-friction and the fully turbulent boundary layer. The Mach 6 cold wall case is about two thirds of the way to the turbulent correlation, and is probably still developing.

4 Conclusion Compressible turbulent spot growth rates and structure are found to be affected by the Mach number and wall temperature. In changing from an adiabatic to a cold-wall boundary condition the spreading angle is reduced by 20 to 30 %. At Mach 3 the cold wall also causes the introduction of Mack mode structures that protrude from the sides of the spot. These structures are also present in both simulations at Mach 6.

Acknowledgment This work is funded by the EU through the ATLLAS project. Time on the UK HPCx supercomputer was provided by the UK Applied Aerodynamics Consortium (EPSRC Grant EP/F005954/1).

References 1. Fischer MC (1972) Spreading of a turbulent disturbance AIAA J. 10(7):957–959. 2. Krishnan L, Sandham ND (2006) Eﬀect of Mach number on the structure of turbulent spots J. Fluid Mech. 566:225–234. 3. Jocksch A, Kleiser L (2007) Growth of turbulent spots in high-speed boundary layers, TSFP 5, Garching, Germany, 837-842. To appear in the Int. J. Heat Fluid Flow. 4. White FM, (1991) Viscous Fluid Flow, 2nd edition, McGraw-Hill. 5. Breuer KS, Landahl MT (1990) The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances J. Fluid Mech. 87(4): 641–672. 6. Eckert ERG (1955) Engineering relations for skin friction and heat transfer to surfaces in high velocity ﬂows J. Aero. Sci. 22:585–587.

Azimuthal Resolution Eﬀects in LES of Subsonic Jet Flow and Inﬂuence on Its Noise F. Keiderling and L. Kleiser Institute of Fluid Dynamics, ETH Zurich, 8092 Zurich, Switzerland, [email protected]; [email protected]

Abstract In the direct-computation approach of aerodynamically generated noise an extended computational domain is employed, which heavily increases the computational cost. Reynolds numbers of practical applications come into reach only by using large-eddy simulations (LES) combined with high-order accurate schemes, optimized time-integration methods and eﬀective subgrid-scale (SGS) models. These simulations are still very costly and therefore parameter studies (physical or numerical) are usually not performed. In particular, grid-independence studies are seldom found in the literature. Rare exceptions, for comparatively low Reynolds numbers, are certain simulations for which results from direct numerical simulations (DNS) are available. In the present work, the azimuthal resolution eﬀect on the directly computed noise of subsonic jet ﬂow at a Reynolds number (based on the jet diameter) of Re = 4.5 · 105 and a Mach number of Ma = 0.9 is investigated. The results for this study are more complicated to compare because of the unavoidable lack of reference DNS data at this Reynolds number. Therefore, conclusions about SGS model eﬀects on the simulation results cannot be drawn. Nevertheless, as it is wellknown that jet ﬂows are very sensitive to changes of the initial conditions we want to assess the predictive quality of our simulation results and thus an investigation of resolution eﬀects seems most appropriate.

1 Numerical methods For details of the computational code, which is based on a conservative formulation of the compressible Navier–Stokes equations, we refer to Ref. [2]. The inﬂow condition is imposed by a sponge reference solution which consists of a hyperbolic-tangent velocity proﬁle and a superposition of a set of linearly unstable eigenmodes. As SGS model we use the ADM-RT (relaxation term) model (relaxation coeﬃcient χ = 50; in the evaluation of the relaxation term the deconvolution operator appears with deconvolution order N = 5, see [3; 4] for details). The computational grid of the baseline conﬁguration contains Nr × Nθ × Nz = 237×50×349 points in a domain of dimension Lr = 20r0 and Lz = 40r0 , V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 85, c Springer Science+Business Media B.V. 2010

593

594

F. Keiderling and L. Kleiser

Table 1. Nomenclature and parameters of presented cases: excited azimuthal wavenumbers |n| (magnitude denotes left- and right-turning), total disturbance amplitude A based on jet exit velocity, azimuthal resolution Nθ and total number of grid points N = Nr × Nθ × Nz . Case LR Med-m HR Low-m High-m Med

|n| = nl , . . . , nu

A/%

Nθ

4, . . . , 8 4, . . . , 8 4, . . . , 8 4, . . . , 8 4, . . . , 8 1, . . . , 8

3.0 3.0 3.0 1.5 4.5 3.0

32 50 78 50 50 50

N/106 2.6 4.1 6.4 4.1 4.1 4.1

where r0 denotes the nozzle radius. Comparisons are presented for simulations with coarse (Nθ = 32, case LR) and ﬁne (Nθ = 78, case HR) azimuthal resolution. All remaining parameters are kept constant. The nomenclature and the relevant parameters of the various cases are given in Table 1. At the inﬂow, baseﬂow Qb and disturbances Qn are imposed via sponge technique using the reference solution Qinfl (r, θ, z, t) = Qb (r) +

nu |n|=nl

= Qb (r) + A

Qn (r, θ, z, t) ˆ n (r)ei(αn z+nθ−ωn t+φn ) An Q

. An

Here, Q = [ρ, ρu, ρv, ρw, E]T denotes the conservative state vector. The disturbances Qn are multiplied by the total disturbance amplitude A to scale the linear superposition of the eigenmodes (normalized by |w ˆn | = 1). Phases φn (t) and amplitudes An (t) are varied in time in a random-walk fashion to prevent phase-locking. The chosen numerical approach has two implications for this study. First, grid reﬁnement does not only aﬀect the discretization error but simultaneously shifts the ﬁlter cutoﬀ to higher wavenumbers. In the limit of a ﬁne grid spacing results converge toward a DNS, rather than to a solution of the ﬁltered equations. Second, the jet ﬂow is very sensitive to changes of the inﬂow disturbances which in our numerical investigations of jet mixing noise which are coupled to the artiﬁcially initiated ﬂow transition. As the transition is triggered using eigenmodes that are randomized in time and space a small amplitude component of random noise is generated. This random component, due to its discrete representation, links the inﬂow condition to the numerical resolution. Thus, as the spatial resolution in the azimuthal direction is changed, we have to accept that this has additional – unfortunately inseparable – eﬀects: With increasing resolution the inﬂow disturbances are more

Resolution Eﬀect in LES of Jet Noise

595

eﬀectively imposed on the baseﬂow (comparable to an increase of excitation amplitude across all modes represented on the grid) and the shift of the ﬁlter cutoﬀ modiﬁes (although only minutely) the high-frequency content of the inﬂow disturbance. Based on jet noise studies in the literature we considered our baseline resolution Nθ = 50 as suﬃcient. However, a reduction of Nθ during initial tests showed strong eﬀects on the ﬂow statistics which required further clariﬁcation.

2 Resolution eﬀect In order to allow for a later comparison, results for diﬀerent modal excitations are ﬁrst shown (for a physical interpretation of the results in this and the subsequent sections see Ref. [2]). The mean axial velocity along the centerline and its root-mean-square (RMS) ﬂuctuations are shown in Fig. 1 for the 3%amplitude cases Med (all unstable helical wavenumbers n excited) and Med-m (|n| = 1, 2, 3 excluded). The data are compared to the experiments [5] and [6] and to the LES in Ref. [7]. After the potential core collapse at z/r0 ≈ 11 the mean velocity of case Med decays fastest, the RMS ﬂuctuations show a delayed and more abrupt increase with highest values among the considered cases. 2.1 Eﬀect on the ﬂow ﬁeld To quantify the resolution eﬀect on the inﬂow excitation amplitude, the onedimensional turbulent kinetic energy (TKE) spectra at the inﬂow plane are shown in Fig. 2. With increasing resolution the spectra of the cases LR, Med-m and HR are shifted to higher energy levels as if the disturbance amplitude had been increased. For a quantitative comparison, Fig. 2b shows the same data together with the previously investigated disturbance amplitudes

b

a

0.2

1/2/U

0.8 0.6 0.4

0.1

w c2

wc / Uj

j

1

0 0

10

20 z /r0

30

0

10

20

30

z /r0

Fig. 1. (a) Mean axial velocity wc and (b) RMS of axial velocity ﬂuctuations Med, Med-m. [7], ◦ [5], w2 1/2 along the jet centerline, cases [6].

596

F. Keiderling and L. Kleiser

a

b

10

10–2

–4

E(n)/U2j

E(n)/U 2j

10–2

–6

10

–8

10

10–4 10–6 10–8

1

2

3

4

5

n

6

7

8

9

1

2

3

4

5

n

6

7

8

9

Fig. 2. One-dimensional turbulence energy spectra E in θ-direction along the nozzle LR, Med-m and HR, lip r = r0 at the inﬂow plane z = 0 (a) cases Low-m and High-m in addition. (b) cases

a

b

0.6 0.4

0.2

/Uj 1/ 2

0.8

wc 2

wc /Uj

1

0

10

20 z/r 0

30

0.1

0

0

10

20

30

z/r 0

Fig. 3. (a) Mean axial velocity wc and (b) RMS of axial velocity ﬂuctuations LR, Med-m, HR. [7], ◦ w2 1/2 along the jet centerline, cases [5], [6].

(1.5% and 4.5% for case Low-m and High-m, respectively). The diﬀerences between medium and high disturbance amplitude in the downstream development were found to be only minor (see [2]). Thus, the distinction between the resolution eﬀect and changes in excitation level becomes possible. Figure 3 shows the resolution eﬀect on the mean axial velocity and the RMS ﬂuctuations along the centerline. Two things can be noticed in Fig. 3a: First, case LR exhibits the strongest decay of the axial velocity beyond the termination of the potential core, whereas the baseline conﬁguration Med-m and case HR are found in close agreement. Second, with downstream distance the mean centerline velocities of Med-m and HR deviate slightly (downstream of z/r0 ≈ 18). As pointed out before, we cannot expect the results to perfectly match and diﬀerences are partially related to the increased disturbance amplitude. In Fig. 3b, case LR has highest axial RMS ﬂuctuations, which are similar to case Med with wavenumbers n = 1 through n = 8 excited (Fig. 1b). The results of case HR fall again close to the baseline conﬁguration Med-m. The axial velocity spectra vs. Strouhal number St = ωDj /2πUj at several downstream positions are shown in Fig. 4a and b along the jet centerline and the nozzle lip-line, respectively. Close to the inﬂow, the spectra remain almost

Resolution Eﬀect in LES of Jet Noise

b

10−3 10−4 10−5 10−6 10−7

wˆ w∗ 1/2/Uj2r0

wˆ w∗ 1/2/Uj2r0

a

597

10−3 10−4

0

0.4

0.8 St

1.2

1.6

10−5

0

0.4

0.8 St

1.2

1.6

Fig. 4. Axial velocity spectra at streamwise positions z/r0 = i · 3 (i = 0, . . . , 4, from LR, Med-m bottom to top) along (a) jet centerline and (b) jet lip line, cases and HR. The groups of spectra are shifted by a factor of 10 each for clarity.

unaﬀected by the changes of azimuthal resolution except for the combined amplitude eﬀect, which is noticed for St ≥ 1. At the third location, z/r0 = 6, the most noticeable resolution eﬀect is found in Fig. 4b: The spectrum of case LR develops an additional peak at the fundamental frequency of the instability mode n = 1, St ≈ 0.68 (in addition to the dominant resolutionunaﬀected peak at St ≈ 0.45). In contrast, cases Med-m and HR give virtually identical results for both radial positions with only minor deviations next to the dominant peaks. Further downstream, the diﬀerent behavior of case LR transfers to the centerline where increased levels for all Strouhal numbers are observed and thus the results group as follows: The coarse resolution of case LR modiﬁes the transition to be dominated by low frequencies. The baseline conﬁguration Med-m and the highly resolved case HR are found in close agreement.

2.2 Eﬀect on the acoustic near-ﬁeld Figure 5 shows the frequency-dependent sound pressure levels (SPL) at locations corresponding to polar angles φ = 30◦ and 90◦ measured from the downstream jet axis. For comparison near-ﬁeld data from experiments [8] and the LES study [7] are shown. The diﬀerences in polar distances R are accounted for by assuming a 1/R-decay and taking a common polar distance of R = 18r0 . Under φ = 30◦ case LR develops a broad dominant frequency band below St ≈ 0.5 in which sound pressure levels are increased by approximately 4.5dB compared to the baseline conﬁguration. This increase is related to the enhanced turbulence levels at low frequencies around the end of the potential core (see Fig. 4a). Consistent with the ﬂow results, case HR has a similar pressure spectrum as the baseline conﬁguration, and both cases are found in fair agreement with the reference data. The resolution eﬀects under φ = 30◦

598

a

F. Keiderling and L. Kleiser

b

130

120

120

SPL

SPL

125

115

115 110

110 105 0.1

125

0.2

0.4

St

0.8

1.6

3.2

105 0.1

0.2

0.4

St

0.8

1.6

3.2

Fig. 5. Near-ﬁeld pressure spectra in dB at angles (a) φ = 30◦ (r = 9r0 ; z = 27r0 ) LR, Med-m and HR. and (b) φ = 90◦ (r = 18r0 ; z = 12r0 ), cases [8] at (a) r = 15r0 , z = 30r0 , (b) r = 15r0 , z = 10r0 . [7] at (a) r = 12r0 , z = 29r0 , (b) r = 15r0 , z = 11r0 .

between LR and Med-m/HR are comparable to eﬀects caused by the removal of lower order modes from the forcing [2]. Perpendicular to the jet axis (φ = 90◦ , Fig. 5b) we see a signiﬁcant over prediction of all cases and the reference LES (for an explanation see, e.g., Ref [7]). Nevertheless, the results for case LR are again comparable to a diﬀerent modal excitation (signiﬁcantly enhanced SPL at low frequencies St ≤ 0.8). The presented results for the pressure spectra suggest that a too coarse azimuthal resolution modiﬁes transition to be dominated by low azimuthal wavenumbers and hence, low frequencies dominate the near-acoustic emission.

3 Conclusions The investigated azimuthal resolution in this study is inevitably tied to the imposition of the inﬂow condition. This side-eﬀect has partially been compensated for by means of a separate investigation focusing on disturbanceamplitude eﬀects on the ﬂow and its noise. The following conclusions can be drawn: A coarse azimuthal resolution Nθ = 32 results in a numerically modiﬁed transition, in which low azimuthal wavenumbers are found to dominate although higher modes (azimuthal wavenumbers n = 4, . . . , 8) are excited. For the coarsely resolved case LR this results in a helically (n = 1) dominated transition which alters the noise-generating mechanism. In addition, with increasing radial distance the azimuthal grid spacing becomes too coarse, acting as a grid ﬁlter for the smallest scales. The increase of resolution to Nθ = 78 exhibits no fundamental diﬀerences compared to the results of the standard resolution Nθ = 50. In general, the results of these two cases, Med-m and HR, fall close together. We conclude that consistent results with high predictive quality are obtained but reemphasize that grid-independent results are not to

Resolution Eﬀect in LES of Jet Noise

599

be expected due to the change of ﬁltered wavenumber band connected with an increase of Nθ . The baseline resolution using Nθ = 50 grid points seems to suﬃciently capture the relevant physical processes.

References 1. 2. 3. 4.

5. 6. 7. 8.

Gutmark, E. & Ho, C.-M., (1983) Phys. Fluids 26, 2932. Keiderling, F., (2008) PhD thesis, ETH Z¨ urich, Diss. ETH No. 17955. Stolz, S., Adams, N. A. & Kleiser, L., (2001) Phys. Fluids 13, 2985. Schlatter, P., Stolz, S. & Kleiser, L., (2004) Relaxation-Term Models for LES of Transitional/Turbulent Flows and the Eﬀect of Aliasing Errors, in Direct and Large-Eddy Simulation V, pp. 65–72, Kluwer. Lau, J. C., Morris, P. J. & Fisher, M. J., (1979) J. Fluid Mech. 93, 1. Arakeri, V. H., Krothapalli, A., Siddavaram, V., Alkislar, M. B. & Lourencoa, L. M., (2003) J. Fluid Mech. 490, 75. Bogey, C. & Bailly, C., (2005) AIAA J. 43, 1000. Bogey, C., Barr´ e, S., Fleury, V., Bailly, C. & Juv´ e, D., (2007) Int. J. of Aeroacoustics 6, 73.

Large Eddy Simulations of Compressible MHD Turbulence in Heat-Conducting Fluid A.A. Chernyshov, K.V. Karelsky, and A.S. Petrosyan Theoretical Section, Space Research Institute of the Russian Academy of Sciences, Profsoyuznaya 84/32, 117997, Moscow, Russia, [email protected]; [email protected]

1 Introduction We develop the large eddy simulation (LES) method for study of compressible magnetohydrodynamic (MHD) turbulence in heat-conducting ﬂuid. Turbulent ﬂows in a magnetic ﬁeld are common both in applied areas and in physics of astrophysical and space plasma. Among the engineering applications, possibility of boundary layer control and drag reduction, MHD ﬂow in a channel (for steel-casting processes) and in a pipe (for cooling of nuclear fusion reactors) can be mentioned. In the previous works authors used LES approach for study of incompressible MHD turbulent ﬂow [1] and already applied LES technique for compressible MHD ﬂow of polytropic gas [2–5]. However, in all mentioned papers MHD equations without the equation of energy balance are considered. Applications of LES technique to heat-conducting compressible MHD ﬂows are signiﬁcantly more diﬃcult due to the increased complexity introduced by the need to solve the energy equation. This introduces novel subgrid-scale (SGS) terms due to the presence of magnetic ﬁeld in total energy equations. We develop parameterizations for these SGS stresses in present work. Computations at various Mach numbers are performed for three-dimensional decaying compressible MHD turbulence. Validity of developed LES method is demonstrated by comparison with the direct numerical simulation (DNS) results.

2 LES formulation In order to simplify the equations describing turbulent MHD ﬂow with variable density, we use the Favre ﬁltering to avoid additional SGS terms associated with nonconstant density. The overbar denotes the ordinary ﬁltering and the tilde speciﬁes the Favre ﬁltering. The following notations are used: ρ is the density; p is the pressure; uj is the velocity in the direction xj ; T is the temperature; σij = 2μSij − 23 μSkk δij is the viscous stress tensor; Sij = 1/2 (∂ui /∂xj + ∂uj /∂xi ) is the strain rate tensor; μ is the coeﬃcient V. Armenio et al. (eds.), Direct and Large-Eddy Simulation VII, ERCOFTAC Series 13, DOI 10.1007/978-90-481-3652-0 86, c Springer Science+Business Media B.V. 2010

601

602

A.A. Chernyshov et al.

of molecular viscosity; η is the coeﬃcient of magnetic diﬀusivity; Bj is the magnetic ﬁeld in the direction xj ; j is the current density; δij is the Kronecker delta. After the use of Favre-ﬁltering procedure for system of MHD equations, the inﬂuence of the small-scale turbulence on the ﬁltered part of MHD equations is deﬁned through thefollowing subgrid terms: the SGS stress ten1 ¯ ¯ sor τiju = ρ¯ (u$ u − u ˜ u ˜ ) − B B − B B i j i j i j i j ; magnetic SGS stress tensor Ma2 ˜ τijb = ui Bj − u˜i B¯j − Bi uj − B¯i u˜j ; SGS heat ﬂux Qj = ρ¯ u$ j T − u˜j T ; SGS turbulent diﬀusion Jj = ρ¯ u j uk uk − u˜j u k uk ; SGS magnetic energy ﬂux Vj = Bk Bk uj − Bk Bk u˜j ; SGS energy of the interaction between the mag netic tension and velocity Gj = uk Bk Bj − u˜k B¯k B¯j . We use Smagorinsky model for MHD-case [4, 5] to parameterize the SGS stresses in the ﬁltered momentum and magnetic induction equations: 2 1 τiju = −2Cs ρ¯¯2 |S˜u | S˜ij − S˜kk δij + Ys ρ¯¯2 |S˜u |2 δij ; (1) 3 3 τijb = −2Ds ¯2 |¯j|J¯ij . (2) 1/2 Here, |S˜u | = 2S˜ij S˜ij is the ﬁltered strain-rate magnitude; J¯ij = 1 ¯ ¯ 2 ∂ Bi /∂xj − ∂ Bj /∂xi is the large-scale magnetic rotation tensor. The constants Cs , Ys and Ds are model coeﬃcients, their values being self-consistently computed during run time with the help of the dynamic procedure. The eddy diﬀusivity model is used for the closure of the subgrid-scale heat $ ˜ ﬂux Qj = ρ¯ uj T − u˜j T :

Qj = −Cs

¯2 ρ¯|S˜u | ∂ T˜ , P rT ∂xj

(3)

where Cs is the model coeﬃcient used above in the Smagorinsky model (1)–(2). Let us consider the subgrid-scale turbulent diﬀusion Jj . It is based on an analogy with Reynolds-averaged Navier–Stokes equations and on the assumption that u˜i u˜ ˜i . Then, the model for Jj is written as: u , Jj u˜k τjk

(4)

u where SGS tensor τjk was found above. For the ﬁnal closure of the complete system of compressible MHD equations, it is necessary to parameterize the novel SGS terms in the energy equation that arise because of the presence of the magnetic ﬁeld. We use an approach based on generalized central moments in order to derive these SGS terms. In our work this approach is extended and is applied for MHD case. We present the sum of subgrid-scale tensors V and G in the following form: 1 b Vj − Gj B¯k τjk . (5) 2

LES of Compressible MHD Turbulence in Heat-Conducting Fluid

603

3 Results Since compressibility eﬀects and temporal dynamics of temperature deﬁned from the total energy equation depend nontrivially on the Mach number, we consider three cases: the Mach number Ms = 0.38, that is, the ﬂow is moderately compressible; Ms = 0.70, when compressibility plays an important role in turbulent ﬂow; and the third case where Ms = 1.11, corresponding to appearance of strong discontinuity in essentially compressible ﬂow. The following notations are introduced for DNS and LES runs. Diamond line represents the results obtained by DNS. Solid line is LES data without any SGS closure. Dotted line presents LES results. In order to measure the eﬀect of modeling SGS terms in total energy equation, additional simulation omitting the energy SGS terms is performed. The market “+” represents LES results where only SGS stresses τiju and τijb in momentum and magnetic induction equations are calculated. One can see from Fig. 1(left) and (middle) that taking into account SGS models for kinetic and magnetic energy brings LES curve closer to DNS data and the calculation accuracy is consequently improved. The SGS closures in the total energy equation do not essentially aﬀect the time dynamics of magnetic and kinetic energy. As it can be seen in the ﬁgures, coincidence takes place in two cases, namely, in presence and absence of energy SGS terms. The similar behavior of plots is observed for cross-helicity. The use of SGS models (1) and (2) improves signiﬁcantly the calculation accuracy. SGS closures in the energy equation do not inﬂuence evolution of the crosshelicity. In contrast to dynamics of kinetic and magnetic energy, inclusion of 0.27

0.13 Magnetic energy

Kinetic energy

0.26 0.25 0.24 0.23 0.22

0.126 0.122 0.118

0.21 0.2

0.114 0

0.05 0.1 0.15 0.2 0.25 0.3

0

0.05 0.1 0.15 0.2 0.25 0.3

t

t 0.49

Temperature

0.48 0.47 0.46 0.45 0.44

0

0.05 0.1 0.15 0.2 0.25 0.3 t

Fig. 1. Time evolution of kinetic energy (left), magnetic energy (middle) and temperature (right) obtained from the ﬁltered DNS and from LES results for Ms = 0.38.

604

A.A. Chernyshov et al.

SGS parameterizations in the total energy equation has a distinct eﬀect on the results of numerical experiments (Fig. 1(right)). The most exact results for the temperature are shown if all SGS closures in the system of the ﬁltered MHD equations are accounted for. Ignoring all SGS tensors results in the maximum deviation from DNS results. The next series of the numerical experiments correspond to the case Ms = 0.70 (not illustrated). As before, SGS tensors in the energy equation scarcely aﬀect the evolution of both large-scale kinetic and large-scale magnetic energy. For the kinetic energy, one can see increasing discrepancy between LES results without SGS parameterizations and with SGS closures involved. The rate of dissipation of the magnetic energy decreases as the Mach number grows. The time evolution of temperature shows that there are not no substantial distinctions between all results on the initial time interval. In the next time period DNS results are closer to LES with all SGS parameterizations in ﬁltered MHD equations. Shocks in a turbulent ﬂuid ﬂow arise at Mach number Ms > 1.0. They intensify large gradients, instabilities, the ﬂuctuations of various parameters, describing the motion of compressible MHD ﬂuid. For the case Ms = 1.11, in Fig. 2(left) time dynamics of the kinetic energy is illustrated. For DNS results, the strong oscillations are inherent (for the sake of clarity, it is shown with grey color in Fig. 2). In general, it is possible to make a conclusion that application of SGS models improves the calculation accuracy. LES results with SGS closures are more accurate in comparison with LES without SGS 0.37 0.012 Magnetic energy

Kinetic energy

0.36 0.35 0.34 0.33 0.32

0

0.05

0.1

0.15

0.2

0.0115 0.011 0.0105 0.01

0.25

0

0.05

0.1

0.15

0.2

0.25

t

t 0.5

Temperature

0.4 0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

t

Fig. 2. Time evolution of kinetic energy (left), magnetic energy (middle) and temperature (right) obtained from the ﬁltered DNS and from LES results for Ms = 1.1.

LES of Compressible MHD Turbulence in Heat-Conducting Fluid

605

parameterizations for magnetic energy (Fig. 2(middle)). The DNS results for magnetic energy decreases nonsteadily as plotted in Fig. 2 (middle) by reason of the presence of large gradients in a supersonics ﬂow. In Fig. 2(right), the temperature oscillates together with the kinetic and magnetic energy, due to the presence of shock waves in a supersonic turbulent ﬂow. The temperature decreases in time nonmonotonically with the magnetic energy since the MHD ﬂow is supersonic.

References 1. 2. 3. 4. 5.

M¨ uller W.-C, Carati D. (2002) Phys. Plasmas, 9(3):824–834 Chernyshov A, Karelsky K, Petrosyan A (2006) Phys. Plasmas, 13(3):032304 Chernyshov A, Karelsky K, Petrosyan A (2006) Phys. Plasmas, 13(10):104501 Chernyshov A, Karelsky K, Petrosyan A (2007) Phys. Fluids, 19(5):055106 Chernyshov A, Karelsky K, Petrosyan A (2006) Flow, Turbulence and Combustion, 80(1):21–35