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, 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

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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

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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

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

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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

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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

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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

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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

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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

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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

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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

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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 

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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.

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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

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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.

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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)

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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

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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]).

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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

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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

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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

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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 

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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

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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

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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

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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).

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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

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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,

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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 

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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

• •

•

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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

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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

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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

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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.

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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

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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 

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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].

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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◦

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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.

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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.

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1/3

1/3

1/3

1/6

1/6

1/6

0

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1/3

0

(IIIb /2)1/3

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(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

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2

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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

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0.99

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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

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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 

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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

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0

45

d11

dr11

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1 0.4 0.8

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fr12

r12

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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

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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

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r33

dr33

d33

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fr11

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0

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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

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r22

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r33-MHD r22-MHD

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d11

d11-MHD

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y/h

2

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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

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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.

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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.

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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 

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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

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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).

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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

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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 

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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

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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.

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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.

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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.

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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 

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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

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(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.

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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.

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Fig. 4. Dependence of frictional the Reynolds number.

and form drags

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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.

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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 

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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 .

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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 .

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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

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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.

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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.

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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

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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.

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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

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110 100 90 80 SPL

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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.

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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 

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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)

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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],

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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)).

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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.

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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 ) +

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in Q

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in Q

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on Ω

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t

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Q ⊂ R 3 × R+

tn+1 Qn

tn

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t2 t1 0

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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)

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

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(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 .

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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.

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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.

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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

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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

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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

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200 y+

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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.

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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

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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.

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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

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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.

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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

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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

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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

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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)

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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 bec