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This page intentionally left blank CLOSURE STRATEGIES FOR TURBULENT AND TRANSITIONAL FLOWS The Isaac Newton Institute of Mathematical Sciences of the University of Cambridge exists to stimulate research in all branches of the mathematical sciences, including pure mathematics, statistics, applied mathematics, theoretical physics, theoretical computer science, mathematical biology and economics. The research programmes it runs each year bring together leading mathematical scientists from all over the world to exchange ideas through seminars, teaching and informal interaction. This book, which has grown out of a two-week instructional conference at the Newton Institute in Cambridge, is designed to serve as a graduate-level textbook and, equally, as a reference book for research workers in industry or academia. CLOSURE STRATEGIES FOR TURBULENT AND TRANSITIONAL FLOWS edited by B.E. Launder UMIST and N.D. Sandham University of Southampton Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521792080 © Cambridge University Press 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 - isbn-13 978-0-511-06939-0 eBook (EBL) - isbn-10 0-511-06939-1 eBook (EBL) - isbn-13 978-0-521-79208-0 hardback - 0-521-79208-8 hardback isbn-10 Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS Contributors Preface Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Introduction B.E. Launder and N.D. Sandham ....................................... 1 Part A. Physical and Numerical Techniques 1. Linear and Nonlinear Eddy Viscosity Models T.B. Gatski and C.L. Rumsey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Second-Moment Turbulence Closure Modelling K. Hanjali´ and S. Jakirli´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 c c 3. Closure Modelling Near the Two-Component Limit T.J. Craft and B.E. Launder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4. The Elliptic Relaxation Method P.A. Durbin and B.A. Pettersson-Reif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5. Numerical Aspects of Applying Second-Moment Closure to Complex Flows M.A. Leschziner and F.-S. Lien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6. Modelling Heat Transfer in Near-Wall Flows Y. Nagano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7. Introduction to Direct Numerical Simulation N.D. Sandham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8. Introduction to Large Eddy Simulation of Turbulent Flows J. Fr¨hlich and W. Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 o 9. Introduction to Two-Point Closures C. Cambon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 10. Reacting Flows and Probability Density Function Methods D. Roekaerts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 vi Contents Part B. Flow Types and Processes and Strategies for Modelling them Complex Strains and Geometries 11. Modelling of Separating and Impinging Flows T.J. Craft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 12. Large-Eddy Simulation of the Flow past Bluﬀ Bodies W. Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13. Large Eddy Simulation of Industrial Flows? D. Laurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .392 Free Surface and Buoyant Eﬀects on Turbulence 14. Application of TCL Modelling to Stratiﬁed Flows T.J. Craft and B.E. Launder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15. Higher Moment Diﬀusion in Stable Stratiﬁcation B.B. Ilyushin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 By-Pass Transition 16. DNS of Bypass Transition P.A. Durbin, R.G. Jacobs and X. Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 17. By-Pass Transition using Conventional Closures A.M. Savill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 18. New Strategies in Modelling By-Pass Transition A.M. Savill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Compressible Flows 19. Compressible, High Speed Flows S. Barre, J.-P. Bonnet, T.B. Gatski and N.D. Sandham Combusting Flows . . . . . . . . . . . . . . . 522 20. The Joint Scalar Probability Density Function Method W.P. Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 21. Joint Velocity-Scalar PDF Methods H.A. Wouters, T.W.J. Peeters and D. Roekaerts . . . . . . . . . . . . . . . . . . . . . . 626 Contents vii Part C. Future Directions 22. Simulation of Coherent Eddy Structure in Buoyancy-Driven Flows with Single-Point Turbulence Closure Models K. Hanjali´ and S. Kenjereˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 c s 23. Use of Higher Moments to Construct PDFs in Stratiﬁed Flows B.B. Ilyushin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 24. Direct Numerical Simulations of Separation Bubbles G.N. Coleman and N.D. Sandham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 25. Is LES Ready for Complex Flows? B.J. Geurts and A. Leonard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 26. Recent Developments in Two-Point Closures C. Cambon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 CONTRIBUTORS S. Barre, Universit´ de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex, e France stephane.barre@lea.univ-poitiers.fr J-P. Bonnet, Universit´ de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers e Cedex, France bonnet@univ-poitiers.fr C. Cambon, Laboratoire de M´canique des Fluides et d’Acoustique, Ecole Centrale e de Lyon, 36 avenue Guy de Collongue, BP 163, 69131 Ecully Cedex, France cambon@mecaflu.ec-lyon.fr G.N. Coleman, Aeronautics and Astronautics, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK gnc@soton.ac.uk T.J. Craft, Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK tim.craft@umist.ac.uk P.A. Durbin, Department of Mechanical Engineering, Stanford University, Stanford CA 94305-3030 USA pdurbin@stanford.edu J. Fr¨hlich, Institut f¨r Hydromechanik, Universit¨t Karlsruhe, Kaiserstr.12, D-76128 o u a Karlsruhe, Germany froehlich@ifh.uni-karlsruhe.de T.B. Gatski, Computational Modeling & Simulation Branch, Mail Stop 128, NASA Langley Research Center, Hampton VA 23681-2199, USA t.b.gatski@larc.nasa.gov B.J. Guerts, Faculty of Mathematical Sciences, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands b.j.geurts@math.utwente.nl K. Hanjali´, Department of Applied Physics, Thermal and Fluids Sciences, University c of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands hanjalic@ws.tn.tudelft.nl B. Ilyushin, Institute of Thermophysics SD RAS, Lavrentyev Avenue 1, 630090 Novosibirsk, Russia ilyushin@itp.nsc.ru R. Jacobs, TenFold Corporation, Draper, UT 84020, USA rjacobs@10fold.com S. Jakarli´, Institute for Fluid Mechanics and Aerodynamics, Darmstadt University c of Technology, Petersenstr. 30, D-64287, Darmstadt, Germany j.suad@hrzpub.tu-darmstadt.de Contributors ix W.P. Jones, Department of Mechanical Engineering, Imperial College of Science Technology and Medicine, University of London, Exhibition Road, London SW7 2BX, UK w.jones@ic.ac.uk S. Kenjereˇ, Department of Applied Physics, Thermal and Fluids Sciences, University s of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands kenjeres@ws.tn.tudelft.nl B.E. Launder, Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK brian.launder@umist.ac.uk D.R. Laurence, Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK; and Electricit´ de France e dominique.laurence@umist.ac.uk A. Leonard, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena CA 91125, USA tony@galcit.caltech.edu M.A. Leschziner, Imperial College of Science Technology and Medicine, Aeronautics, Department, Prince Consort Rd., London SW7 2BY, UK Mike.Leschziner@ic.ac.uk F-S. Lien, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada fslien@sunwise.uwaterloo.ca Y. Nagano, Department of Environmental Technology, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan nagano@heat.mech.nitech.ac.jp T.W.J. Peeters, Corus Research, Development & Technology P.O. Box 10000, 1970 CA IJmuiden, The Netherlands tim.peeters@corusgroup.com B.A. Petterson-Reif, Norwegian Defence Research Establishment, N-2025 Kjeller, Norway bre@ffi.no W. Rodi, Institut f¨r Hydromechanik, Universit¨t Karlsruhe, Kaiserstr.12, D-76128 u a Karlsruhe, Germany rodi@ifh.uni-karlsruhe.de D. Roekaerts, Department of Applied Physics, Thermal and Fluids Sciences, University of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands roekaerts@tnw.tudelft.nl C.L. Rumsey, Computational Modeling & Simulation Branch, Mail Stop 128, NASA Langley Research Center, Hampton VA 23681-2199, USA c.l.rumsey@larc.nasa.gov N.D. Sandham, Aeronautics and Astronautics, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK n.sandham@soton.ac.uk x Contributors A.M. Savill, Department of Engineering, University of Cambridge, Trumpington St., Cambridge CB2 1PZ, UK ams3@eng.cam.ac.uk H.A. Wouters, Corus Research, Development & Technology, P.O. Box 10000, 1970 CA IJmuiden, The Netherlands huib.wouters@corusgroup.com X. Wu, Department of Mechanical Engineering, Stanford University, Stanford CA 94305-3030, USA xiaohua wu@hotmail.com PREFACE Material for this volume ﬁrst began to be assembled in 1999 when a six-month Programme on Turbulence was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge. The Programme had its own origins in an Initiative on Turbulence by the UK Royal Academy of Engineering, which had identiﬁed the prediction of turbulent ﬂow as a key technology across a range of industrial sectors. Researchers from diﬀerent disciplines gathered together at the Newton Institute building in Cambridge to work on aspects of the turbulence problem, one of the most important of which was closure of the averaged (or ﬁltered) turbulent ﬂow equations. An Instructional Workshop on Closure Strategies for Modelling Turbulent and Transitional Flows was held from April 6 to 16th. The aim of the workshop was for the experts on closure to present material leading up to the current state-of-the-art in a form suitable for research students and others requiring a broad overview of the ﬁeld, together with an appreciation of current issues. A sequence of 38 lectures by 19 diﬀerent lecturers provided background to techniques, examples of current applications, and reﬂections on possible future developments. Recognising that a gathering of so many experts from a variety of backgrounds was somewhat unusual, it was felt that a more polished version of the lecture material would be of interest to a wider audience, as a reﬂection upon the current state of prediction methods for turbulent ﬂows. The result has taken rather longer to appear than originally intended, but the opportunity has been taken for contributions to be updated wherever possible, to take account of the most recent developments. The Editors would like to thank the contributors for their eﬀorts to improve the papers, with a view to making them more accessible to the intended audience, which remains that of the original workshop. Especial thanks are due to Dr T. Gatski who shouldered the not inconsiderable task of shaping three separate contributions into a single chapter on compressible ﬂow. Thanks are also due to the Newton Institute for hosting the workshop, and the main sponsors of the Turbulence Programme, including the then British Aerospace (now BAE Systems), Rolls–Royce, The Meteorological Ofﬁce, British Gas Technology, British Energy, and the UK Defence Evaluation Research Agency. Finally we record our appreciation to Mrs C. King who provided invaluable secretarial support throughout the Workshop and in the editing of the present volume. ACRONYMS ABL APG ASM BL C/D CDF CERT CE CFD CMC CPU DES DIA DNS DSM EASM ECL EDF EDQNM EMST ER ERCOFTAC ESRA EVM FFT GGD/GGDH GLM IEM ILDM IP/IPM ISAT KH LDA LDV LES LHS LIA LIPM LMSE LRR LSES MCM MDF Atmospheric Boundary Layer Adverse Pressure Gradient Algebraic Stress Model [1] Binomial Lagrangian [21] Coalescence/Dispersion [21] Cumulative Distribution Function [20] ´ Centre d’Etudes et de Recherches, Toulouse Constrained Equilibrium [21] Computational Fluid Dynamics Conditional Moment Closure [10] Central Processing Unit Detached Eddy Simulation [8] Direct Interaction Approximation Direct Numerical Simulation [7] Diﬀerential Second-moment-closure Model (also SMC) [2] Explicit Algebraic Stress Model [1] Ecole Centrale de Lyon Electricit´ de France e Eddy-Damped Quasi-Normal Markovian [9] Euclidean Minimum Spanning Tree [20] Elliptic Relaxation [4] European Research Community On Flow, Turbulence and Combustion Extended SRA [19] Eddy Viscosity Model [1] Fast Fourier Transform Generalized Gradient Diﬀusion (Hypothesis) [2] Generalized Langevin Model [21] Interaction-by-exchange-with-the-mean [21] (also known as LMSE) Intrinsic Low-Dimensional Manifold [21] Isotropization of Production (Model) [2] In Situ Adaptive Tabulation [20,21] Kelvin–Helmholtz Laser Doppler Anemometer Laser Doppler Velocimetry Large Eddy Simulation [8] Left Hand Side Linear Interaction Approximation [19] Lagrangian IP Model [21] Linear Mean Square Estimation [20] (also known as IEM) Launder, Reece and Rodi [2] Large-Scale Eddy Structures Mapping Closure Model [21] Mass Density Fraction [21] Acronyms xiii MPP MUSCL NASA NLEVM NSE ONERA PBL PDF PIV POD PTM QI QUICK RANS RDT RHS RLA RMS RNG RSM/RSTM SDM SGD SGS SIG SIMPLE SLM SLY SM SMC SNECMA SRA SSG SSM SST TCL TFM TKE TPC T-RANS T-S T3A, T3B . . . TVD UMIST UMIST UTS VLES Massively Parallel Processing Monotone Upwind Schemes for Scalar Conservation Laws National Aeronautics and Space Administration Nonlinear Eddy Viscosity Model [1] Navier–Stokes Equations ´ Oﬃce National d’Etudes et de Recherches A´rospatiales e Planetary Bounday Layer Probability Density Function [10] Particle Image Velocimetry Proper Orthogonal Decomposition Production-Transition Modiﬁcation [18] Quasi-Isotropic Quadratic Upstream-Interpolation for Convection Kinematics [5] Reynolds Averaged Navier–Stokes [1] Rapid Distortion Theory [9] Right Hand Side Ristorcelli, Lumley and Abid [4] Root Mean Square Renormalization Group Theory [2] Reynolds Stress (Transport) Model [2] Semi-deterministic Method [22] Simple Gradient Diﬀusion [2] Sub-Grid Scale [8] Special Interest Group Semi-Implicit Method for Pressure-Linked Equation [5] Simpliﬁed Langevin Model [21] Savill, Launder and Younis (a second-moment transition model due to Savill [17]) Smagorinsky Model [8] Second Moment Closure [2] ´ Soci´t´ Nationale d’Etudes et de Conception de Moteurs ee d’Avions Strong Reynolds Analogy [19] Sarkar, Speziale and Gatski [1] Scale Similarity Model [8] Shear Stress Transport [1] Two-Component Limit [3] Test Field Model [9] Turbulence Kinetic Energy Two-Point Closure [26] Time-dependent RANS [22] Tollmien–Schlichting A series of transition test cases [17] Total Variation Diminishing University of Manchester Institute of Science and Technology Upstream Monotonic Interpolation for Scalar Transport [5] University of Technology, Sydney Very Large Eddy Simulation Introduction B.E. Launder and N.D. Sandham Although Computational Fluid Dynamics (CFD) has developed to a point where it is a routine tool in many applications, several diﬃculties remain. Numerical issues, such as grid generation, are often diﬃcult and costly, in the sense that much time and eﬀort has to be devoted to the task, but they are manageable. The other main problem concerns the realistic physical modelling of turbulent and transitional ﬂow, and is much less tractable. The aim of this volume is to provide a reasonably comprehensive, up-todate and readable account of where the numerical computation of industrially important, single-phase turbulent ﬂows has reached. Turbulent ﬂow appears in such a diversity of guises that no single model used for engineering calculations can expect to mimic all the observed phenomena to the level of approximation sought. Thus, diﬀerent levels and types of modelling are adopted according to the nature of the physical situation under study, the type of information to be extracted, and the accuracy required. The book has been organized within three main sections. In Part A the focus is on techniques (with applications serving to illustrate the appropriateness of the technique adopted) while Part B examines particular types of ﬂow, usually adopting a single preferred modelling strategy. Finally, in Part C, some current research approaches are introduced. Throughout, references to other articles in the book are given by their chapter number in square brackets. The individual articles themselves are sequenced broadly in terms of increasing complexity, at least within Parts A and B. The nomenclature undergoes some variation across the chapters, reﬂecting the diﬀerences habitually adopted in the journal literature over the diﬀerent themes covered in the volume. Nomenclature for core variables is deﬁned at the start of Chapter [1] and this is essentially common for Part A. Additional deﬁnitions or variants are provided in the individual chapters as needed. Part A: Physical and Numerical Techniques Chapters [1]–[6] present a sequence of articles on single point closure. These represent the core of what is usually understood by ‘turbulence modelling’. Chapter [1] by Gatski and Rumsey considers linear and non-linear models of eddy-viscosity type. It begins with algebraic variants of the mixing-length hypothesis and considers in turn various elaborations up to conventional twoequation models and the k-ε-v 2 extension. The chapter closes with an extensive discussion of non-linear eddy-viscosity models, a closure level which appreciably enlarges the range of ﬂows that may successfully be modelled, usually for 1 2 Introduction little additional cost. However, when transport or force-ﬁeld eﬀects on the turbulent ﬂuctuations are large, a formal second-moment closure is usually to be preferred. Thus, one solves transport equations for all the ‘second moments’ i.e. the non-zero turbulent stresses and, in non-isothermal ﬂows, the heat ﬂuxes too. In Chapter [2] Hanjali´ and Jakirli´ provide an overview of the important c c modelling issues at this level and some of the modelling strategies adopted over the last 25 years. The chapter concludes by presenting an impressive range of test ﬂows that have been computed with the form of closure adopted by the authors’ group. Chapters [3] and [4] which follow, by Craft and Launder (CL) and Durbin and Petterson-Reif (DP) provide, in greater detail, particular modelling strategies in second-moment closure especially for the crucially important pressurestrain terms. Both are motivated by the aim of replacing the widely adopted, though limited, algebraic ‘wall-reﬂection’ scheme that attempts to account for modiﬁcations to the pressure ﬂuctuations brought about by a wall. The DP chapter reviews the current form of the ‘elliptic-relaxation’ method which replaces the algebraic scheme by a set of relatively simple partial diﬀerential equations. That by CL reviews the ‘two-component-limit’ strategy; their aim is partly to remove the need for wall reﬂection and partly to achieve a wider applicability of the model in free ﬂows by adopting a more elaborate treatment for the case where walls are absent. If one is going to adopt a model at second-moment closure level, one’s output comprises point values of the stresses rather than the value of the eddy viscosity. The recommended strategies for incorporating such models into the computer code in order to achieve rapid convergence of the numerical solver are the subject of Chapter [5] by Leschziner and Lien. Finally, from among this examination of single-point closures, Chapter [6] by Nagano considers the problem of turbulent heat (or mass) diﬀusion. The discussion covers both second-moment and eddy viscosity approaches with particular focus being placed on an equation for the dissipation rate of mean square temperature ﬂuctuations. A major requirement for heat transport modelling is that, besides gaseous ﬂow where the molecular diﬀusivities of heat and momentum are of a similar magnitude, one also needs to cope with Prandtl numbers both much less than (liquid metals) and much greater than (oils) unity. Sandham (Chapter [7]) and Fr¨hlich and Rodi (Chapter [8]) introduce o simulation-based approaches, dealing respectively with Direct Numerical Simulation (DNS), where all scales of turbulence are resolved, and Large-Eddy Simulation (LES), where large scales are resolved and small scales modelled. These approaches are becoming increasingly realistic as computer performance continues to improve. DNS provides reference solutions for simple canonical ﬂows, against which turbulence closure assumptions can be checked, whilst LES is developing towards a practical method of prediction. Limitations on Introduction 3 Reynolds number, due to the range of turbulence scales that need to be resolved, are emphasised in these contributions. An alternative perspective on turbulence closure is provided by Cambon in Chapter [9]. Here the single-point approaches discussed in [1]–[5] are placed into the context of multi-point and higher-order closures. Though mathematically more demanding, such approaches contain more of the physics of turbulent ﬂow and provide useful insight into fundamental phenomena, such as nonlinearity and non-locality. Emphasis is placed on two-point closures, with practical examples of rotation and stratiﬁcation used to illustrate the insight that can be obtained with this approach. Part A concludes with Chapter [10], in which Roekaerts gives an introduction to the modelling of reacting ﬂows. In this application mass-weighted averaging is introduced for the ﬁrst time (this form of averaging is also used in Chapter [19], when compressible, high-speed ﬂows are discussed). To account for chemistry eﬀects, methods based on probability density functions (PDFs) are introduced. Applications of one-point scalar PDF methods and joint velocity-scalar PDFs appear later, in Chapters [20] and [21]. Part B: Flow Types and Processes Part B of the volume begins with a consideration of the capability of singlepoint closures and LES in tackling ﬂows with separated ﬂow regions and strong streamline curvature. Craft in [11] examines the strengths and, all too often, the weaknesses of single-point closure when applied to separated and impinging ﬂows. This article, read in conjunction with the applications reported in [3]–[5], provides an overview of the performance achieved by the diﬀerent modelling levels. In Chapters [12] and [13], Rodi and Laurence discuss the capabilities of LES, and we have our ﬁrst glimpse of a current debate concerning the extent to which LES will replace single-point closure approaches for practical problems. The topic is revisited in [25] of Part C, but we see already in Chapter [12] the potential of LES, compared to single-point methods, for simulation of a laboratory experiment of ﬂow around a bluﬀ-body, dominated by separation and strong vortex shedding. Diﬀerences between techniques require further investigation, but the chapter ends on an optimistic note that LES will ‘soon become aﬀordable and ready for practical applications’. Laurence in [13], however, damps some of the high expectations, by suggesting that in many industrial problems the increase in computer power will simply result in more complete single-point predictions, and we may be waiting many years to see LES widely used. The application of second- and third-moment closure to problems of horizontal shear ﬂows aﬀected by buoyancy is the theme of Chapters [14] and [15] by Craft and Launder, and Ilyushin. Stably stratiﬁed horizontal ﬂows turn out to be far more diﬃcult to capture than vertical mixed convection, 4 Introduction where even a linear eddy viscosity model does fairly well in reproducing the observed phenomena. The reason for this diﬀerence in ease of predictability is that, for vertical ﬂow, buoyant eﬀects in the mean momentum equation introduce additional shear which is usually the dominant feature of any change in turbulence structure. In the horizontal shear ﬂow the only important eﬀects of stratiﬁcation arise through the impact of buoyancy on the turbulent ﬁeld itself. Because it is the vertical velocity ﬂuctuations that are mainly aﬀected by the stratiﬁcation, second-moment closure is usually seen as the best starting point for closure. Yet, as both sets of authors point out, situations arise where second-moment closure is inadequate, though agreement with observation may be restored if, instead, closure is eﬀected at third-moment level. Looking ahead, an alternative route for dealing with this type of problem is developed in [22] by Hanjali´ and Kenjereˇ where the large-scale structures in c s Rayleigh–B´nard convection are resolved by employing a time-dependent solue tion of the Reynolds equations using just a (highly) truncated second-moment closure. The problem of ‘by-pass’ transition has been the subject of single-point turbulence modelling since the early 1970s. The rationale was originally provided by the fact that at least some low-Reynolds-number two-equation eddyviscosity models reproduced the reversion of a turbulent boundary layer back to (or towards) laminar when subjected to a severe acceleration. In view of that, it was conjectured that forward transition (from laminar to turbulent ﬂow) in the presence of a turbulent external stream could also be predicted by the same model . . . and so it proved. Since those early days the appreciation of the detailed processes taking place in by-pass transition has come a long way, progress being greatly assisted by DNS/LES studies of the type provided by Durbin, Jacobs and Wu in Chapter [16]. Savill’s survey of modelling approaches is divided into two parts, Chapter [17] dealing with the use of conventional closures that have been designed for fully turbulent ﬂows while Chapter [18] considers special modelling features. A major aim of current research eﬀorts is to drive down the level of external-stream turbulence at which accurate prediction can be made and it is this goal that has led to the use of intermittency parameters and other devices discussed in [18]. Compressible ﬂows, which form the subject of Chapter [19], in fact contain several diﬀerent phenomena requiring the modellers’ attention. The ﬁrst is the question of how one should perform the averaging process in a ﬂuid where the density is itself varying in time. From there, issues concerning the eﬀects of density ﬂuctuations on the diﬀerent processes provide a major challenge. Finally structural changes to turbulence passing through a shock wave need to be considered. All these topics are addressed by Barre, Bonnet, Gatski and Sandham. It is noted that questions of numerical solution, addressed in Chapter [5], have taken account of the requirements of compressible ﬂow. Indeed that chapter shows an application to a supersonic three-dimensional Introduction 5 ﬂow with a bow shock present. In [20] Jones presents a review of the one-point scalar PDF approach, applied to ﬂows with chemical reaction. It is argued that in the exact equation for a scalar PDF it is the term representing molecular mixing which presents the chief diﬃculty, and various approaches are described. Applications to a jet diﬀusion ﬂame illustrate the current state of the art. Extensions to a joint velocity-scalar PDF, solved by means of a Monte Carlo method, are described by Wouters, Peeters and Roekaerts in Chapter [21]. This approach has a uniﬁed treatment of all the terms in the averaged equations that must be closed, including conventional Reynolds stresses. The method is expensive, but results for a bluﬀ-body stabilized diﬀusion ﬂame are promising. Part C: Future Directions In this ﬁnal part of the volume, space is allocated to some of the strategies that have not yet found an established place in the hierarchy of modelling or, as in [25], to issues of what directions are ready for further exploitation. As has been signalled earlier, Hanjali´ and Kenjereˇ [22] report that the probc s lem of Rayleigh–B´nard convection (in which a horizontal layer of ﬂuid, cone ﬁned within the space between horizontal planes, is heated from below) is captured much better with a truncated second-moment closure if one adopts a time-dependent rather than a steady-state numerical solution. Essentially what results from the time-dependent simulation is a close replica of the large eddy-simulation of the same ﬂow. Put another way, the TRANS simulation is eﬀectively a coarse-grid LES that uses a higher level of sub-grid model and is grid independent. Clearly, for the problem chosen it would seem that this fusion of RANS and LES strategy is wholly satisfactory and superior (from the standpoint of accuracy or cost) to either a steady state RANS or a conventional LES. Ilyushin in Chapter [23] also develops inter-linkages between two approaches to turbulence that are usually viewed as discrete. In this case he shows, among other contributions, how a knowledge of just the second- and third-order moments enable the probability density functions to be approximated. Coleman and Sandham review the latest direct simulations of separation bubbles in Chapter [24]. Turbulent separation bubbles are at the current limits of computer power, with severe Reynolds number restrictions. However, DNS of transitional separation bubbles, an important phenomenon that in some cases controls the performance of aerofoils, are already at the Reynolds numbers encountered in applications. Guerts and Leonard consider, in Chapter [25], recent developments in LES and the issues facing LES that need to be addressed for it to be developed into a reliable predictive tool. The guidelines for developing reliable LES listed in section 5 complement those of Chapter [7] for DNS, and should be borne in mind by anyone interested in using LES 6 Introduction for complex ﬂow problems. Closure methods will continue to require guidance from experiment and theory, and in Chapter [26] we conclude the volume with a review by Cambon of the potential for further insight coming from recent developments in two-point closures. Part A. Physical and Numerical Techniques 1 Linear and Nonlinear Eddy Viscosity Models T.B. Gatski and C.L. Rumsey 1 Introduction Even with the advent of a new generation of vector and now parallel processors, the direct simulation of complex turbulent ﬂows is not possible and will not be for the foreseeable future. The problem is simply the inability to resolve all the component scales within the turbulent ﬂow. In the context of scale modeling, the most direct approach is oﬀered by the partitioning of the ﬂow ﬁeld into a mean and ﬂuctuating part (Reynolds 1895). This process, known as a Reynolds decomposition, leads to a set of Reynoldsaveraged Navier–Stokes (RANS) equations. Although this process eliminates the need to completely resolve the turbulent motion, its drawback is that unknown single-point, higher-order correlations appear in both the mean and turbulent equations. The need to model these correlations is the well-known ‘closure problem.’ Nevertheless, the RANS approach is the engineering tool of choice for solving turbulent ﬂow problems. It is a robust, easy to use, and cost eﬀective means of computing both the mean ﬂow as well as the turbulent stresses and has been overall, a good ﬂow-prediction technology. From a physical standpoint, the task is to characterize the turbulence. One obvious characterization is to adequately describe the evolution of representative turbulent velocity and length scales, an idea that originated almost 60 years ago (Kolmogorov 1942). The physical cornerstone behind the development of turbulent closure models is this ability to correctly model the characteristic scales associated with the turbulent ﬂow. This chapter describes incompressible, turbulent closure models which (can) couple with the RANS equations through a turbulent eddy viscosity (velocity × length scale). In this context both linear and nonlinear eddy viscosity models are discussed. The descriptors ‘linear’ and ‘nonlinear’ refer to the tensor representation used for the model. The linear models assume a Boussinesq relationship between the turbulent stresses or second-moments and the mean strain rate tensor through an isotropic eddy viscosity. The nonlinear models assume a higher-order tensor representation involving either powers of the mean velocity gradient tensor or combinations of the mean strain rate and rotation rate tensors. Within the framework of linear eddy viscosity models (EVMs), a hierarchy of closure schemes exists, ranging from the zero-equation or algebraic models to the two-equation models. At the zero-equation level, the turbulent velocity and 9 10 Gatski and Rumsey Nomenclature bij , b ∗ Cµ , Cµ D/Dt D, Dij k L l P, p P R R2 Sij , S T T(n) Ue Ui uτ Wij , W ∗ Wij , W∗ Reynolds stress anisotropy tensor, (ui uj /2k) − δij /3 eddy viscosity calibration coeﬃcient material derivative (= ∂/∂t + Uj ∂/∂xj ) represents the combined eﬀect of turbulent transport and viscous diﬀusion turbulent kinetic energy (≡ τii /2) characteristic length scale in wall proximity mixing length mean pressure turbulent kinetic energy production term symmetric, traceless tensor in algebraic stress equation ﬂow parameter (≡ −{W2 }/{S2 }) mean strain rate tensor (≡ (∂Ui /∂xj + ∂Uj /∂xi )/2) characteristic time scale in wall proximity tensor basis element edge velocity mean velocity component friction velocity mean rotation rate tensor in noninertial frame (≡ (∂Ui /∂xj − ∂Uj /∂xi )/2) modiﬁed mean rotation rate tensor in inertial frame W ij X xi αn ε ε ˆ εij δ δ∗ η κ ρ σij Πij ν ∗ νt , νt νti , νto τ τij , τ Ωij Ωr ω mean rotation rate tensor in transformed frame orthogonal transformation matrix coordinate direction in inertial (Cartesian) frame (x, y, z) tensorial expansion coeﬃcients isotropic turbulent energy dissipation rate near-wall modiﬁed dissipation rate dissipation rate tensor boundary layer thickness displacement thickness √ scalar invariant (≡ Sik Ski ) von Karman constant density viscous stress tensor pressure strain rate correlation kinematic viscosity turbulent eddy viscosity inner and outer eddy viscosity turbulent time scale (= k/ε) Reynolds stress tensor (≡ ui uj ) arbitrary time-independent rotation rate of noninertial frame rotation rate of noninertial frame dissipation rate per unit kinetic energy [1] Linear and nonlinear eddy viscosity models 11 length scales are speciﬁed algebraically whereas, at the two-equation level, differential transport equations are used for both the velocity and length scales. Within the framework of nonlinear eddy viscosity models (NLEVMs), the characterizing feature is the (polynomial) tensor representation for the secondmoments or Reynolds stresses. However, the method of determining the expansion coeﬃcients diﬀers among models. In some methods the expansion coeﬃcients are determined through calibrations with experimental or numerical data and the imposition of dynamic constraints. In other methods, the expansion coeﬃcients are related directly to the closure coeﬃcients used in the full diﬀerential Reynolds stress equations. The models derived using these latter methods are sometimes referred to as explicit algebraic stress models. Over the years, there has been a multitude of models at the EVM and NLEVM levels proposed for the RANS equations. No attempt is made (since we would surely fail) to be all inclusive with the choice of models for each level of closure discussed. Our goal, however, is to provide the reader with a broad perspective on the development of such models, so that, with this broader view, he or she will be better prepared to assess the viability of using a particular closure scheme. 2 Reynolds-averaged Navier–Stokes formulation As a prelude to the discussion of the linear and nonlinear eddy viscosity models, it is desirable to describe the Reynolds averaging procedure and the resulting form of the mean momentum and continuity equations. In the Reynolds decomposition, the ﬂow variables are decomposed into mean and ﬂuctuating components as f =f +f . (2.1) The average of a ﬂuctuating quantity is zero f = 0, and the mean quantity f can be extracted if a statistically steady or a statistically homogeneous turbulence is assumed. For example, if the turbulence is stationary, f (x) = lim 1 T →∞ T t0 +T f (x, t) dt, t0 (2.2) and the average of the product of two quantities is f g = f g + f g . The velocity (ui ) and pressure (p) ﬁelds can be decomposed into their mean (Ui , P ) and ﬂuctuating parts (ui , p), and the resulting Reynolds-averaged Navier–Stokes (RANS) equations can be written as ∂Ui 1 ∂P ∂σij ∂τij DUi ∂Ui =− + − . = + Uj Dt ∂t ∂xj ρ ∂xi ∂xj ∂xj (2.3) For an incompressible ﬂow, the mass conservation equation reduces to the mean continuity equation, ∂Uj = 0. (2.4) ∂xj 12 Gatski and Rumsey The viscous stress tensor σij for a Newtonian ﬂuid and incompressible ﬂow is given by σij = 2νSij , (2.5) where ν is the kinematic viscosity, and Sij is the strain rate tensor Sij = 1 2 ∂Ui ∂Uj + ∂xj ∂xi . (2.6) As equation (2.3) shows, for closure the RANS formulation requires a model for the second-moment (or Reynolds stress) τij (= ui uj ). 3 Linear eddy viscosity models Using continuity, equation (2.4), and the deﬁnition of the viscous stress, equation (2.5), the RANS equation can be written in the form DUi 1 ∂P ∂τij ∂ =− + + Dt ρ ∂xi ∂xj ∂xj ν ∂Ui ∂xj . (3.1) For linear eddy viscosity models (linear EVMs), the equation is closed by using a Boussinesq-type approximation between the turbulent Reynolds stress and the mean strain rate 2 τij = kδij − 2νt Sij , (3.2) 3 where k (= τii /2) is the turbulent kinetic energy, and νt is the turbulent eddy viscosity. In Section 3.4, it will be shown that such a closure model can be extracted from an analysis of a simple shear ﬂow in local equilibrium. When equation (3.2) is used as the turbulent closure in linear EVMs, equation (3.1) can be rewritten as DUi 1 ∂p ∂ ∂Ui = (ν + νt ) , + Dt ρ ∂xi ∂xj ∂xj (3.3) where the isotropic part of the closure model, 2k/3, is assimilated into the pressure term so that p = P + 2k/3. In the EVM formulation, the turbulence ﬁeld is coupled to the mean ﬁeld only through the turbulent eddy viscosity, which appears as part of an eﬀective viscosity (ν + νt ) in the diﬀusion term of the Reynolds-averaged Navier–Stokes equation. Since in general, νt > ν, this formulation of the problem can be rather robust numerically, especially when compared to the alternative form of retaining the stress gradient ∂τij /∂xj explicitly in equation (3.1). In the remainder of this section, a hierarchy of linear eddy viscosity models will be presented ranging from the least complex (algebraic) to the most complex (diﬀerential transport) means of specifying the turbulent eddy viscosity νt . [1] Linear and nonlinear eddy viscosity models 13 3.1 Zero-equation models The zero-equation model is so named because the eddy viscosity required in the turbulent stress-strain relationship is deﬁned from an algebraic relationship rather than from a diﬀerential one. The earliest example of such a closure is Prandtl’s mixing-length theory (Prandtl 1925). By analogy with the kinetic theory of gases, Prandtl assumed the form for the turbulent eddy viscosity in a plane shear ﬂow with unidirectional mean ﬂow U1 (x2 ) = U (y) and shear stress τ12 = τxy = −νt dU/dy. The eddy viscosity was assumed to have the form dU , (3.4) νt = ρl2 dy where l is the mixing length that requires speciﬁcation for each ﬂow under consideration. In a free shear ﬂow, the mixing length would be a characteristic measure of the width of the shear layer. In a planar wall-bounded ﬂow, the mixing length l in the near-wall region would be proportional to the distance from the wall. These relationships, though simple, give rise to signiﬁcant insights about the structure of turbulent ﬂows. In the case of wall-bounded ﬂows, the law of the wall and the structure of the outer layer of the boundarylayer ﬂow can be deduced. Several texts and reviews in the literature provide an insightful description of the physical and mathematical basis for this type of modeling. These include Tennekes and Lumley (1972), Reynolds (1987), Speziale (1991), and Wilcox (1998). Two of the most popular and versatile algebraic models are the Cebeci– Smith (see Cebeci and Smith 1974) and the Baldwin–Lomax (see Baldwin and Lomax 1978) models. Even though the original development of these models was motivated by application to compressible ﬂows, no explicit account was taken of compressibility eﬀects. Density eﬀects are simply accounted for through a variable-mean-density extension of the incompressible formulation (µt = ρνt ). These are two-layer mixing-length models that have an inner layer eddy viscosity given by Cebeci–Smith: Baldwin–Lomax: where Wij is the rotation tensor Wij = 1 2 ∂Ui ∂Uj − ∂xj ∂xi , (3.7) νti = l 2 ∂Ui ∂Ui ∂xj ∂xj 1 2 (3.5) (3.6) νti = l2 2Wij Wij , and 2Wij Wij represents the magnitude of the vorticity. An outer layer eddy viscosity is given by Cebeci–Smith: Baldwin–Lomax: νto = 0.0168Ue δ ∗ FK (y; δ) νto = 0.0269Fwk FK (y; ym /0.3). (3.8) (3.9) 14 Gatski and Rumsey The mixing length is deﬁned similarly in both models for zero-pressure-gradient ﬂows. That is, + + (3.10) l = κy 1 − e−y /A , where κ = 0.41 is the von Karman constant, A+ = 26 is the Van Driest damping coeﬃcient, and y + is the distance from the wall in wall units (uτ y/ν). In the expressions for the outer layer eddy viscosity, δ is the boundary-layer thickness, δ ∗ is the displacement thickness, and Ue is the edge velocity. In general, the damping coeﬃcient A+ can be a function of the pressure gradient, but for present purposes it will be assumed to be constant. Throughout this subsection, attention will be focused on the form of the models for zeropressure-gradient ﬂows; extensions that include pressure gradient eﬀects can be found in the references cited for the particular algebraic models. The functions FK and Fwk are an intermittency and a wake function, respectively. The Klebanoﬀ intermittency function FK is given by FK (y; ∆) = 1 + 5.5 and the wake function Fwk is given by 2 Fwk = min ym Fm ; ym Udif /Fm y ∆ 6 −1 , (3.11) (3.12) 1 max (l 2Wij Wij ) . (3.13) κ y In the above, ym is the distance from the body surface where Fm occurs, ∆ is the boundary-layer thickness δ in the Cebeci–Smith model, and ∆ is ym /0.3 in the Baldwin–Lomax model. The quantity Udif is the diﬀerence between the maximum and minimum total velocity in the proﬁle. Unlike the Cebeci– Smith model, the Baldwin–Lomax model does not need to know the location of the boundary-layer edge. As equation (3.13) suggests, the Baldwin–Lomax model bases the outer layer length scale on the vorticity in the layer rather than on the displacement thickness, as in the Cebeci–Smith model. Extensions and generalizations to more complex ﬂows can be found in Cebeci and Smith (1974), Degani and Schiﬀ (1986), and Wilcox (1998). A disadvantage of the Cebeci–Smith and Baldwin–Lomax turbulence models is that they possess an inherent dependency on the grid structure: quantities are evaluated and searched for along grid lines ‘normal’ to walls. This dependency can be problematic for unstructured grids or for multiple-zone structured grids. Also, it has been shown that these models, in their original form, generally do not predict separated ﬂows well. For example, when strong shock-induced separation is present, these models tend to predict the shock position too far aft. However, the Baldwin–Lomax model with the DeganiSchiﬀ modiﬁcation is often still used in industry for three-dimensional vortical Fm = with [1] Linear and nonlinear eddy viscosity models 15 ﬂow applications because other models (including some of the one- and twoequation ﬁeld equation models) can diﬀuse vortices excessively. 3.2 ‘Half-equation’ models The motivation for the development of the Johnson–King model (Johnson and King 1985) was primarily the need to solve a particular class of ﬂows – turbulent boundary layer ﬂows in strong adverse pressure gradients – rather than the development of a universal model. The model was developed to account for strong history eﬀects that were observed to be characteristic of turbulent boundary layers subjected to rapid changes in the streamwise pressure gradient. Johnson and King felt that the simple algebraic models (as outlined in Section 3.1) could be modiﬁed suﬃciently, without recourse to the more elaborate diﬀerential transport formulations (such as the two-equation formulation to be discussed in Section 3.4), to better predict ﬂows with massive separation. Thus, advection eﬀects were deemed essential, whereas turbulent transport and diﬀusion eﬀects were assumed to have much less importance. This level of closure derives its name somewhat subjectively because an ordinary diﬀerential equation is solved instead of a partial diﬀerential equation. Nevertheless, this level of closure does generalize the algebraic models by specifying a smooth functional behavior for the eddy viscosity across the boundary layer and by accounting in a limited way for history (relaxation) eﬀects by solving a ‘transport equation’ for the maximum shear stress. Since the inception of the Johnson–King model, it has undergone some modiﬁcation (Johnson 1987, Johnson and Coakley 1990) to improve its predictive capabilities for a wider class of ﬂows, and in particular, for compressible ﬂows. For the present purpose, only the simpler incompressible formulation will be outlined. The Johnson–King model is also a two-layer model; however, in this model, the eddy viscosity changes in a prescribed functional manner from the inner layer form to the outer layer form. This functional form is given by (Johnson and King 1985) νt = νto [1 − exp (νti /νto )] . (3.14) In the later form of the model (Johnson and Coakley 1990), which was also used in the solution of transonic ﬂow problems, this functional dependency was based on a hyperbolic tangent function. The inner layer eddy viscosity is given by τxy |m νti = l2 , (3.15) κy where l is the mixing length deﬁned in equation (3.10) with A+ = 15, and the subscript m denotes maximum value along a grid coordinate line normal to a solid wall surface. In zero-pressure-gradient, two-dimensional ﬂows in which the law of the wall holds, this expression for νti corresponds to the Cebeci– Smith inner layer eddy viscosity given in equation (3.5). 16 The outer layer eddy viscosity is given by νto = 0.0168Ue δ ∗ FK (y; δ)σ(x), Gatski and Rumsey (3.16) which is the Cebeci–Smith form, equation (3.8), with the addition of the factor σ(x) that accounts for streamwise evolution of the ﬂow. At each streamwise station, σ(x) is adjusted so that the relation νt |m = −τxy |m ∂U/∂y|m (3.17) is satisﬁed. The remaining quantity that is needed is τxy |m ≡ τm , and this is determined from a transport equation for the shear stress τxy . Unlike conventional Reynolds-stress closures in which the transport equation for the turbulent shear stress contains modeled pressure-strain correlations and turbulent transport terms, this turbulent shear stress equation is extracted from the turbulent kinetic energy equation (cf. equation (3.30)) by assuming that the shear stress anisotropy b12 = −τxy /2k = 0.125 is constant at the point of maximum shear. The log-layer of an equilibrium turbulent boundary layer ﬂow is a constant stress layer; therefore, the assumption used here is not without merit in an equilibrium ﬂow. It is interesting to see that such an assumption does not adversely impact the model performance for the class of separated ﬂows for which it was developed. If the viscous diﬀusion eﬀects are neglected, the evolution equation for τm is Um √ dτm τm τm √ − Cdif 1 − σ 1/2 (x) , (3.18) = b12 τmeq − τm dx Lm (0.7δ − ym ) 3/2 where τmeq is the equilibrium value (σ(x) = 1) for the shear stress, Cdif = 0.5 for σ(x) ≥ 1 and zero otherwise, and Lm is the dissipation length scale given by Lm = κy, ym /δ ≤ 0.09/κ (3.19) Lm = 0.09δ, ym /δ > 0.09/κ. (3.20) Because σ(x) is not known a priori at each streamwise station, it is necessary to iterate on the equation set at each station to determine its value. While the discussion here has focused on two-dimensional ﬂows, extensions have been proposed for three-dimensional ﬂows (e.g., Savill et al. 1992) which have also yielded good ﬂow ﬁeld predictions. The Johnson–King model suﬀers from the same disadvantage as the Cebeci– Smith and Baldwin–Lomax models: it relies on the grid structure because quantities are evaluated and searched for along lines ‘normal’ to walls. For this reason, the model has received less attention in the last decade with the increased use of unstructured and multiple-zone structured grids, for which ﬁeld-equation turbulence models are more ideally suited. [1] Linear and nonlinear eddy viscosity models 17 3.3 One-equation models Up to this point, both the zero- and half-equation models have focused on the speciﬁcation of an eddy viscosity (which is the underlying basis of the development of single-point closure schemes) rather than on a speciﬁcation of either a turbulent velocity or length scale individually. At the one-equation level of closure, a transport equation is introduced, which in the earliest models that date back to Prandtl was for the turbulent velocity scale (turbulent kinetic energy), with an algebraic prescription for the turbulent length scale. Modern-day approaches have evolved beyond this formulation to the solution of transport equations for the turbulent Reynolds number or the turbulent eddy viscosity (velocity scale × length scale). Some of these formulations will be discussed here, and the interested reader can also refer to the text by Wilcox (1998) for additional information. Spalart and Allmaras (1994) devised a one-equation model based primarily on empiricism and on dimensional analysis arguments. Unlike the zero- and half-equation models discussed previously, this one-equation model is local; that is, the equation at one point does not depend on the solution at other points. Therefore, it is easily usable with any type of grid: structured or unstructured, single block, or multiple blocks. The eddy viscosity relation is given by νt = ν fv1 , ˜ (3.21) ˜ ˜ where fv1 = χ3 /(χ3 + c3 ), and χ = ν /ν. The variable ν is determined by v1 using the transport equation D˜ ν ˜ ˜ ˜ 1 ∂ (ν + ν ) ∂ ν ˜ = cb1 (1 − ft2 )S ν + Dt σ ∂xk ∂xk − (cw1 fw with auxiliary relations ˜ S = fv2 fw ν ˜ 2Wij Wij + 2 2 fv2 κ d χ = 1− 1 + χfv1 1 + c6 = g 6 w3 g + c6 w3 1/6 ˜ cb2 ∂ ν 2 σ ∂xk ν ˜ cb1 − 2 ft2 κ d + 2 (3.22) (3.23) (3.24) −1/6 g −6 + c−6 w3 = 1 + c−6 w3 (3.25) (3.26) (3.27) (3.28) g = r + cw2 (r6 − r) ν ˜ r = ˜ 2 d2 Sκ ft2 = ct3 exp(−ct4 χ2 ), where d is the minimum distance to the nearest wall. The closure coeﬃcients are given by: cb1 = 0.1355, cb2 = 0.622, σ = 2/3, κ = 0.41, cw1 = cb1 /κ2 + (1 + 18 Gatski and Rumsey cb2 )/σ, cw2 = 0.3, cw3 = 2, ct3 = 1.2, ct4 = 0.5, and cv1 = 7.1. Although not discussed here, Spalart and Allmaras (1994) also developed an additional term that is used to trip the solution from laminar to turbulent at a desired location. This feature may be important as the subsequent downstream predictions can critically depend on the appropriate choice for the onset of turbulence. Over the years since its introduction, the Spalart–Allmaras model has become popular among industrial users due to its ease of implementation and relatively low cost. Even though this one-equation level of closure is based on empiricism and dimensional analysis, with characterizing ﬂow features usually accounted for on a term-by-term basis using phenomenological based models, it has tended to perform well for a wide variety of ﬂows. As the study by Shur et al. (1995) has shown, the model can even outperform some two-equation models in separating and reattaching ﬂows. Recently, Spalart and Shur (1997) have developed a ‘rotation function,’ which multiplies the production term and sensitizes the Spalart–Allmaras model to the eﬀects of rotation and curvature. This function is based on the rate of change of the principal axes of the strain rate tensor. Other contemporary one-equation models using the eddy viscosity have been proposed. One is the Gulyaev et al. (1993) model, which is an improved version of the model developed by Sekundov (1971). It has been shown in the Russian literature to solve a variety of incompressible and compressible ﬂow problems (see Gulyaev et al. 1993 for selected references). Another is the model by Baldwin and Barth (1991), that has its origins in the k-ε two-equation formulation and was a precursor to the Spalart–Allmaras model. In their original forms, both the Spalart–Allmaras and Baldwin–Barth models are known to cause excessive diﬀusion in regions of three-dimensional vortical ﬂow. Dacles-Mariani et al. (1995) proposed the use of a modiﬁed form of the production term; rather than basing it on the magnitude of vorticity ( 2Wij Wij ) alone, the following functional form is assumed: 2Wij Wij + 2 min(0, 2Sij Sij − 2Wij Wij ). (3.29) This method was shown to help for a particular application using the Baldwin– Barth model, but it is not a universally accepted ﬁx. The problem of excessive diﬀusion in some vortical ﬂow applications by these models, in general, still persists. 3.4 Two-equation models While the previous closure models discussed have focused on the speciﬁcation of a turbulent eddy viscosity to be used directly in the RANS equation (3.3), the two-equation level of closure attempts to develop transport equations for both the turbulent velocity and length scales of the ﬂow. Many variations on [1] Linear and nonlinear eddy viscosity models 19 this approach exist, but the most common approaches use the transport equation for the turbulent kinetic energy for the turbulent velocity scale equation. On the other hand, the length scale equation has generally been the most controversial element of the two-equation formulation. For the present purposes, attention will be focused in this subsection on the k-ε and k-ω formulations, where ε is the turbulent energy dissipation rate and ω is the dissipation per unit turbulent kinetic energy. The turbulent kinetic energy equation k is easily derived from the ﬂuctuating momentum equation for ui by forming the transport equation for the scalar product ui ui /2. The resulting equation can be written as Dk = P − ε + D, (3.30) Dt where the right-hand side represents the transport of k by the turbulent production P = −τik ∂Ui /∂xk , the isotropic turbulent dissipation rate, ε, and the combined eﬀects of turbulent transport and viscous diﬀusion D. When equation (3.2) is used, the turbulent production term can also be written in terms of the eddy viscosity as P = 2νt (Sik Ski ) = 2νt η 2 , (3.31) where the velocity gradient tensor is decomposed into the sum of the symmetric strain rate tensor Sij and the antisymmetric rotation rate tensor Wij , η 2 = Sik Ski (or η 2 = {S2 } in matrix notation), and the trace Sik Wki = {WS} = 0. In such a formulation, the behavior of the individual stress components is governed by the Boussinesq relation given in equation (3.2), which is an isotropic eddy viscosity relationship. In general, the evolution of the individual stress components is not isotropically partitioned among the components. For this eﬀect to be accounted for, higher-order closures are required such as the nonlinear eddy viscosity models to be discussed later in this chapter or the Reynolds stress formulation to be discussed in Chapter [2]. Nevertheless, the Boussinesq relation is not without physical foundation. For example, in (thin) simple shear ﬂow where an equilibrium layer exists, it is assumed that τxy = Cµ k (3.32) with Cµ the model constant. In the region of local equilibrium, the energy production and dissipation rates are in balance, so that in a thin shear ﬂow, the kinetic energy equation reduces to P = −τxy ∂U =ε= ∂y τxy Cµ k 2 ε (3.33) where equation (3.32) has been used. This yields the familiar closure model for the turbulent shear stress, τxy = −Cµ k 2 ∂U . ε ∂y (3.34) 20 Gatski and Rumsey Dimensional analysis considerations dictate that the eddy viscosity νt be given by the product of a turbulent velocity scale and a turbulent length scale. With the velocity scale given by k 1/2 , the remaining task is the development of the scale variable. In this chapter, two such alternatives are considered. The ﬁrst is the turbulent energy dissipation rate ε which implies that k 3/2 /ε is proportional to the length scale, and the second is the speciﬁc dissipation rate1 , ω, which implies that k 1/2 /ω is proportional to length scale. Thus, the eddy viscosity νt is given by the relation νt = Cµ k2 = Cµ kτ, ε τ= k ε (3.35) for the k-ε two-equation model, and νt = k ω (3.36) for the original k-ω two-equation model. The modeling coeﬃcient Cµ usually assumes a value of 0.09. (Note that this value is slightly larger than the value assumed in the derivation of the half-equation model in Section 3.2.) For the k-ω formulation, the kinetic energy equation (3.30) is suitably modiﬁed by using the substitution ε = Cµ kω (see Wilcox 1998). The coeﬃcient σk in equation (3.37) is σk = 1 for the k-ε model, whereas σk = 2 for the k-ω model. (The reader should be aware that the most recent version of the k-ω model, as proposed by Wilcox (1998), is diﬀerent from the original Wilcox version. The necessary references are provided in Wilcox 1998.) Consistent with the simpliﬁed form of a two-equation formulation, a gradient-transport model for the turbulent transport is usually used in the kinetic energy equation, ∂ νt ∂k D= ν+ , (3.37) ∂xj σk ∂xj where the ﬁrst term on the right is the viscous contribution, and the second is the model for the turbulent transport. The coeﬃcient σk is an eﬀective Prandtl number for diﬀusion, which is taken as a constant in incompressible ﬂows. The value of σk is dependent on the particular scale variable used. The resulting simple form of the modeled turbulent kinetic energy equation is an obvious appeal of the formulation. There are several variations to the modeled form of the transport equation for the isotropic dissipation rate ε. A rather general expression (Jones and Launder 1972) from which many of the forms can be derived and which can be integrated to the wall is given by Dε ∂ 1 = (Cε1 P − Cε2 ε) + Dt τ ∂xk 1 ν+ νt σε ∂ε , ∂xk (3.38) i.e., dissipation rate of kinetic energy (k) per unit k. [1] Linear and nonlinear eddy viscosity models 21 where Cε1 ≈ 1.45 is usually ﬁxed from calibrations with homogeneous shear ﬂows, and Cε2 is usually determined from the decay rate of homogeneous, isotropic turbulence (≈ 1.90). The closure coeﬃcient σε acts like an eﬀective Prandtl number for dissipation diﬀusion and is speciﬁed to ensure the correct log-law slope of κ−1 , κ2 . (3.39) σε = Cµ (Cε2 − Cε1 ) During the late 1990s, the two-equation ‘shear stress transport’ (SST) model of Menter (1994), has gained increasing favor among industrial users, due primarily to its robust formulation and improved performance for separated ﬂows over traditional two-equation models. One of the primary features of Menter’s model is that it is a blend of Wilcox’s original k-ω formulation near walls and a k-ε formulation in the outer region and in free shear ﬂows. Thus, the model does not have to contend with the problems often encountered by k-ε models near walls (see Section 3.5), while it still retains the k-ε predictive capabilities in free shear ﬂows. Since the transport equation for the turbulent kinetic energy k has been given previously in equation (3.30), only the transport equation for the speciﬁc dissipation rate of turbulence kinetic energy ω for the SST model is given here: Dω ∂ γ = P − βω 2 + Dt νt ∂xk ν+ νt σω ∂ω 1 − F1 ∂k ∂ω +2 . ∂xk σω2 ω ∂xk ∂xk (3.40) The function F1 is the blending function that is used to ‘switch’ between the k-ω (F1 = 1) and the k-ε (F1 = 0) formulations, F1 = tanh(Γ4 ), where Γ = min max k 500ν ; Cµ ωd ωd2 √ 4σω2 k , CDkω d2 (3.41) ; (3.42) and CDkω represents the cross-diﬀusion term (the last term in the ω equation (3.40)), limited to be positive and greater than some very small arbitrary number. In the derivation of the modiﬁed ω equation, Menter neglects a set of diﬀusion terms that are demonstrated to be small (Menter 1994) and also neglects the molecular viscosity in the cross-diﬀusion term. The model constants σk , σω , β, and γ model constants are evaluated from (σk , σω , β, γ)T = F1 (σk1 , σω1 , β1 , γ1 )T + (1 − F1 )(σk2 , σω2 , β2 , γ2 )T . (3.43) The other important feature of the SST model (which represents a departure from the component k-ω and k-ε models) is a modiﬁcation to the deﬁnition of the eddy viscosity to account for the eﬀect of the transport of the principal 22 Gatski and Rumsey turbulent shear stress. The deﬁnition of the eddy viscosity νt in the model is altered from the forms given previously in equation (3.36): νt = 2b12 k , max(2b12 ω; 2Wij Wij F2 ) (3.44) where b12 (= 0.155) is the shear stress anisotropy (see Section 3.2). The blending function F2 is given by F2 = tanh(Γ2 ), 2 where √ 2 k 500ν Γ2 = max ; Cµ ωd ωd2 (3.45) . (3.46) Without the modiﬁed form of equation (3.44), most k-ω and k-ε linear eddy viscosity models have been generally found to yield poor results for separated ﬂows. The constants for Menter’s SST model are given by: σk1 = 1.17647, σω1 = 2, β1 = 0.075, σk2 = 1, σω2 = 1.16822, and β2 = 0.0828. The constant γ1 is a function of β1 and σω1 whereas γ2 is a function of β2 and σω2 as follows: γ1,2 = β1,2 /Cµ − κ2 /(σω1,2 Cµ ). (3.47) Notice that the value of σk1 has been recalibrated by Menter from its original (Wilcox k-ω model) value of 2 to recover the correct ﬂat-plate log-law behavior when using the modiﬁed eddy viscosity equation (3.44). The other coeﬃcients σω1 , β1 , and γ1 are the same as those in Wilcox’s original model. The constants σk2 , σω2 , β2 , and γ2 have a direct correspondence with the k-ε coeﬃcients: σk2 = σk β2 = Cµ (Cε2 − 1) σω2 = σε γ2 = Cε1 − 1. (3.48) (3.49) Menter uses the Launder–Sharma (1974) coeﬃcients: Cµ = 0.09, Cε1 = 1.44, Cε2 = 1.92, κ = 0.41, σk = 1, with σε computed by way of equation (3.39). Although Menter’s SST model uses two heuristic blending functions (both of which rely on distance to the nearest wall), they have held up well under a great number of applications and still remain in many production codes as the same functions cited in the 1994 reference. It is worth mentioning that many two-equation models, including Menter’s SST model, are sometimes implemented by using an approximate production term P = νt (2Wij Wij ), (3.50) where 2Wij Wij is the square of the magnitude of vorticity. This form is a convenient approximation because the magnitude of vorticity is often readily [1] Linear and nonlinear eddy viscosity models 23 available in many CFD codes. However, this approximation, while often found to be valid for thin-shear-dominated aerodynamic ﬂows, may lead to serious errors in some ﬂow situations. On the other hand, use of the full production term can sometimes cause problems such as overproduction of turbulence and/or negative normal stresses near stagnation points or shocks. One method commonly used for alleviating this problem is through the use of a limiter such as ˜ P = min(P, 20D) (3.51) on the production term in the k-equation. See Durbin (1996) for a more thorough discussion and alternate strategies. Much more could be discussed about two-equation models in general and the k-ε and k-ω models in particular because of the ease with which they can be applied and the widespread use they have enjoyed. The interested reader is referred to the book by Mohammadi and Pironneau (1994), which is devoted entirely to the k-ε turbulence model, and the book by Wilcox (1998) which is primarily focused on the k-ω model. Additional references are also provided in reviews by Hanjalic (1994), Gatski (1996), and So and Speziale (1998). 3.5 Near-wall integration In the discussion of the lower order zero- and half-equation models, it was clearly seen that the models were constructed for direct integration to the wall through the two-layer structure for the eddy viscosity. In addition, while less explicit about its suitability for direct integration to the wall, the oneequation formulation, and speciﬁcally the Spalart–Allmaras model, was also developed with the capability of being used unaltered in wall-bounded ﬂows. However, in the two-equation formulation, equations (3.35) and (3.38) from the k-ε model, in the high-Reynolds number form, do not provide an accurate representation in the near-wall, viscosity aﬀected region. In addition, the destruction-of-dissipation rate term Cε2 ε2 /k, is singular at the wall since ε is ﬁnite, and the turbulent kinetic energy k = 0. There has been an extensive list of near-wall modiﬁcations over the last two decades for both the eddy viscosity and the transport equation for the eddy viscosity. The near-wall turbulent eddy viscosity has taken the form νt = fµ Cµ k2 , ε (3.52) where fµ is a damping function. The transport equation for the dissipation rate has been generalized so that Dˆ ε ε ˆ ε2 ˆ ∂ = f1 Cε1 P − f2 Cε2 + Dt k k ∂xk ε = ε + D, ˆ ν+ νt σε ∂ε ˆ +E ∂xk (3.53) (3.54) 24 Gatski and Rumsey where f1 is a damping function, f2 is used to ensure that the destruction term is ﬁnite at the wall, and D and E are additional terms added to better represent the near-wall behavior. A partial list of the various forms for these functions can be found in Patel et al. (1985), Rodi and Mansour (1993), and Sarkar and So (1997). While the list is not all inclusive, it does provide the functional forms which are used today for these near-wall functions. As seen in Section 3.4, the SST model solves the near-wall problem by ‘switching on’ the k-ω form near the wall. Another alternative to the introduction of damping functions is the elliptic relaxation approach that was ﬁrst proposed by Durbin (1991). This approach has been extended to a full Reynolds stress closure Durbin (1993a); however, in the context of the current chapter, the description of the approach will be limited to the two-equation k-ε formulation. In the context of a linear eddy viscosity framework, this is a three-equation model for the turbulent kinetic energy k, turbulent dissipation rate ε, and the normal stress component τ22 . (The model has been referred to as the k − ε − v 2 or v 2 − f model and is based on an elliptic relaxation approach. The notation used here will be slightly diﬀerent, in keeping with our attempt to have a uniﬁed notation throughout the chapter.) A major assumption of the model is that the eddy viscosity νt should be given by νt = Cµ τ22 T, (3.55) where T is the applicable characteristic timescale of the ﬂow in the proximity of the wall ν 1/2 T = max τ, 6 (3.56) ε and the coeﬃcient Cµ ≈ 0.2. Since the timescale τ = k/ε → 0 as the wall is approached, equation (3.56) reﬂects the physical constraint that the characteristic time scale T should not be less than the Kolmogorov time scale ν/ε. With this assumption, the dissipation rate equation (3.38) is rewritten as Dε ∂ 1 ∗ = (Cε1 P − Cε2 ε) + Dt T ∂xk ν+ νt σε ∂ε ˆ ∂xk (3.57) to account for the variability of the characteristic timescale. With the exception ∗ of the production-of-dissipation rate coeﬃcient Cε1 , the closure coeﬃcients are ∗ assigned values close to the standard ones (Durbin 1993b, 1995), but Cε1 has assumed diﬀerent non-constant values (Durbin 1993b, 1995) to optimize the predictive capability of the model. The modeled equation for τ22 is approximated from the Reynolds stress transport equation and is given by ε Dτ22 ∂ = kf22 − τ22 + Dt k ∂xk ν+ νt σk ∂τ22 , ∂xk (3.58) [1] Linear and nonlinear eddy viscosity models where the function f22 is, in general, obtained from L2 ∇2 f22 − f22 = Π22 25 (3.59) as discussed in Chapters [2] and [4]. The characteristic length scale L is deﬁned in a manner analogous to the timescale so that k 3/2 , Cη L = CL max ε ν3 ε 1/4 , (3.60) where CL ≈ 0.25 and Cη ≈ 80. Since the length scale CL k 3/2 /ε → 0 as the wall is approached, equation (3.60) reﬂects the physical constraint that the characteristic length scale L should not be less than the Kolmogorov length scale (ν 3 /ε)1/4 . This approach to the near-wall integration problem clearly diﬀers from the standard damping function approach used in two-equation modeling. To date there have been several applications of the methodology to a variety of ﬂow problems. Continued application and reﬁnement may lead to a more extensive adaptation of this technique for near-wall model integration. The interested reader is encouraged to review the cited references for additional details and motivation. 4 Nonlinear eddy viscosity models The linear eddy viscosity models just discussed have proven to be a valuable tool in turbulent ﬂow-ﬁeld predictions. However, inherent in the formulation are several deﬁciencies which do not exist within the broader Reynolds stress transport equation formulation. Two of the most notable deﬁciencies are the isotropy of the eddy viscosity and the material-frame indiﬀerence of the models. The isotropic eddy viscosity is a consequence of the Boussinesq approximation which assumes a direct proportionality between the turbulent Reynolds stress and the mean strain rate ﬁeld. The material frame-indiﬀerence is a consequence of the sole dependence on the (frame-indiﬀerent) strain rate tensor. These deﬁciencies preclude, for example, the prediction of turbulent secondary motions in ducts (isotropic eddy viscosity) and the insensitivity of the turbulence to noninertial eﬀects such as imposed rotations (material frame-indiﬀerence). Remedies for these deﬁciencies can be made on a case-bycase or ad hoc basis; however, within the framework of a linear eddy viscosity formulation such defects cannot be ﬁxed in a rigorous manner. The category of nonlinear eddy viscosity models (NLEVMs) simply extends in a rigorous or general manner the one-term tensor representation in terms of the strain rate (see equation (3.2)) used in the linear EVMs to the generalized form n 2 (n) τij = kδij + αn Tij . (4.1) 3 n=1 26 Gatski and Rumsey Since one of the advantages of the NLEVMs is to be able to capture some eﬀects of the stress anisotropies that occur at the diﬀerential second-moment level of closure, it is helpful to recast some of the equations in terms of the stress anisotropy bij given by bij = τij δij − . 2k 3 (4.2) For example, in terms of bij , equation (4.1) can then be rewritten as n bij = n=1 (n) αn Tij , (n) (4.3) where Tij are the tensor bases and αn are the expansion coeﬃcients which need to be determined. As was shown in Section 3, the linear EVMs, through the Boussinesq approximation, couple to the RANS equations through a simple additive modiﬁcation to the diﬀusion term (see equation (3.3)). In the case of nonlinear EVMs, this coupling can be more complex. The coupling can be either through the direct use of equation (4.1) in equation (3.1) or through a modiﬁed form of equation (3.3) given by DUi 1 ∂p ∂ ∂Ui = (ν + νt ) + S.T. + Dt ρ ∂xi ∂xj ∂xj (4.4) where S.T. are nonlinear (source) terms from the tensor representation (4.1). The degree of complexity associated with the nonlinear source terms is dependent on both the number and form of the terms chosen for the tensor representation. The choice of the proper tensor basis is, of course, dependent on the functional dependencies associated with the Reynolds stress τij or the corresponding anisotropy tensor bij . As seen from the transport equation for the Reynolds stresses (e.g. Speziale 1991), the only dependency on the mean ﬂow is through the mean velocity gradient. Thus, it has been generally assumed in developing turbulent closure models for the Reynolds stresses, that, in addition to the functional dependency on the turbulent velocity and length scales, the dependence on the mean velocity gradient be included as well. The turbulent velocity scale is usually based on the turbulent kinetic energy k and the turbulent length scale on the variable used in the corresponding scale equation (which for the purposes here will be the isotropic turbulent dissipation rate ε). The continuum mechanics community has dealt with such questions on tensor representations for several decades (e.g., Spencer and Rivlin 1959). Within this context, the stress anisotropy tensor is considered here with the functional dependencies bij = bij (k, ε, Skl , Wkl ), (4.5) [1] Linear and nonlinear eddy viscosity models 27 where the dependence on the mean velocity gradient has been replaced by the equivalent dependence on the strain rate tensor (see equation (2.6)) and the rotation rate tensor (see equation (3.7)). (In dealing with tensor representations, it is sometimes better, for notational convenience, to use matrix notation to eliminate the cumbersome task of accounting for several tensor indices. For this reason, both tensor and matrix notation will be used in describing the models in this and subsequent sections.) Equation (4.5) can be rewritten in matrix notation as b = b(k, ε, S, W). (4.6) In the case of fully three-dimensional mean ﬂow, a symmetric, traceless tensor function b of a symmetric tensor (S) and an antisymmetric tensor (W) can be represented as an isotropic tensor function of the following ten (integrity) (n) tensor bases T(n) (= Tij ): T(1) T(2) T(3) T(4) T(5) =S = SW − WS = S2 − 1 {S2 }I 3 = W2 − 1 {W2 }I 3 = WS2 − S2 W T(6) = W2 S + SW2 − 2 {SW2 }I 3 T(7) = WSW2 − W2 SW T(8) = SWS2 − S2 WS T(9) = W2 S2 + S2 W2 − 2 {S2 W2 }I 3 T(10) = WS2 W2 − W2 S2 W. (4.7) The expansion coeﬃcients αn associated with this representation can, in general, be functions of the invariants of the ﬂow αn = αn ({S2 }, {W2 }, k, ε, Ret ), (4.8) where {S2 } = Sij Sji , {W2 } = Wij Wji are the strain rate and rotation rate invariants, respectively, and Ret = k 2 /νε is the turbulent Reynolds number. Of course, a smaller number of terms could be used for the representation, such as in three-dimensions where the minimum number is ﬁve to have an independent basis; however, the expansion coeﬃcients will then be more complex (Jongen and Gatski 1998) and could possibly be singular. An advantage of using the full integrity basis is that the expansion coeﬃcients will not be singular. The linear term T(1) is the strain rate S, as in the linear EVM case, and its coeﬃcient α1 is now the turbulent eddy viscosity νt , which is used in equation (4.4). The nonlinear source terms are the remaining terms T(n) (n ≥ 2) in the polynomial expansion. In the remainder of this section, the development of the models will be categorized based on the methodology used to determine the expansion coefﬁcients αn . First, what is usually termed nonlinear eddy viscosity models are discussed. These are models in which a polynomial expansion is assumed that is a subset of equation (4.7), and the expansion coeﬃcients are determined based on calibrations with experimental or numerical data and physical constraints. Second, what is usually termed algebraic stress models or algebraic 28 Gatski and Rumsey Reynolds stress models are discussed. These are models in which a polynomial expansion is, once again, assumed from equation (4.7), but the expansion coeﬃcients are derived in a mathematically consistent fashion from the full diﬀerential Reynolds stress equation. In both cases, an explicit tensor representation for b is obtained in terms of S and W. 4.1 Quadratic and cubic tensor representations In this subsection, a few examples of nonlinear eddy viscosity models represented by the tensor expansions in equations (4.1) or (4.3) are discussed. The models examined include expansions where both terms quadratic (n = 2) and cubic (n = 3) in the mean strain rate and rotation rate tensors are retained. Each of these examples (while not all inclusive) provide insight into the variety of assumptions required in identifying the expansion coeﬃcients αn needed in the algebraic representation of the Reynolds stresses. As a ﬁrst example of a nonlinear eddy viscosity model using an explicit representation for the Reynolds stress anisotropy, consider the quadratic model proposed by Speziale (1987). Speziale’s approach, while also motivated by the need to include Reynolds stress anisotropy eﬀects into a linear eddy viscosity type of formulation, diﬀered in its development of the tensor representation for the Reynolds stress anisotropy. Speziale assumed that the anisotropy tensor bij was of the form δSkl bij = bij k, ε, Skl , , (4.9) δt where ∂Ul δSkl DSkl ∂Uk Sml − Smk = − δt Dt ∂xm ∂xm (4.10) is the Oldroyd convective derivative (e.g., Aris 1989, p. 185). This dependency on the convective derivative was used to ensure that the nonlinear polynomial approximation would be frame-indiﬀerent, in keeping with the frameindiﬀerent properties of the anisotropy tensor itself. Calibrations were based on fully developed channel ﬂow predictions using both k-l and k-ε two-equation models. The resulting tensor representation (using the strain rate and rotation rate tensor notation) was DS 1 b = α1 S + α2 (SW − WS) + α3 S2 − {S2 }I + αD , 3 Dt (4.11) where αD (k, ε) was a closure coeﬃcient determined from the calibration. Written in this form, it can be seen that the introduction of the frame-indifferent convective derivative simply modiﬁes two of the tensor bases given in equation (4.7). Preliminary validation studies were done for a rectangular duct ﬂow and for a backstep ﬂow to highlight the improved predictive capabilities of the nonlinear model over the corresponding linear eddy viscosity k-l and k-ε forms. [1] Linear and nonlinear eddy viscosity models Next consider the quadratic model proposed by Shih et al. (1995), b = α1 S + α2 (SW − WS) . 29 (4.12) The αi coeﬃcients were determined by applying the rapid distortion theory constraint to rapidly rotating isotropic turbulence, and the realizability constraints τββ ≥ 0, no sum (4.13) and 2 τβγ ≤ τββ τγγ , Schwarz inequality (4.14) to the limiting cases of axisymmetric expansion and contraction. The coeﬃcients were optimized by further comparison with experiment and numerical simulation of homogeneous shear ﬂow and the inertial sublayer. Initial validation studies were run on rotating homogeneous shear ﬂow, backward-facing step ﬂows, and conﬁned jets with overall improved predictions over the linear eddy viscosity models. It was also found that the standard wall function approach yielded better predictions than any of the low-Reynolds number k-ε models. The algebraic representation given in equation (4.12) for the Reynolds stresses was coupled with a standard k-ε two-equation model given in equations (3.30) and (3.38). The values used for the coeﬃcients and other details of the calibration process are given in Shih et al. (1995). While quadratic models have been widely used, some have argued (e.g., Craft et al. 1996) that the range of applicability of such models is limited and that higher-order terms are needed to be able to predict ﬂows with complex strain ﬁelds. Craft et al. (1996) considered a model of the form 1 b = α1 S + α2 S2 − {S2 }I 3 1 +α3 (SW − WS) + α4 W2 − {W2 }I (4.15) 3 2 +α5 W2 S + SW2 − {SW2 }I + α6 WS2 − S2 W . 3 (The form given here diﬀers slightly from the form presented in Craft et al., although the two representations can be shown to be equivalent.) Calibration of the closure coeﬃcients was based on an optimization over a wide range of ﬂows. These included plane channel ﬂow, circular pipe ﬂow, axially rotating pipe ﬂow, fully developed curved channel ﬂow, and impinging jet ﬂows. This algebraic representation for the Reynolds stresses was then coupled with lowReynolds number forms for the kinetic energy and dissipation rate equations (see equations (3.30) and (3.52)–(3.54)). Attempts at extending the model to ﬂows far from equilibrium have been undertaken and are discussed in Craft et al. (1997). Other cubic models have been proposed, for example, by Apsley and Leschziner (1998) and Wallin and Johansson (2000), and the interested reader is referred to these papers for further details on their development and application. 30 Gatski and Rumsey 4.2 Algebraic stress models The identifying feature of algebraic stress models (ASMs) is the technique used to obtain the expansion coeﬃcients αn . As noted previously, these coeﬃcients have a direct relation to the Reynolds stress model used, or more speciﬁcally, to the pressure-strain rate correlation model. The algebraic stress model used here is based on the model originally developed by Pope (1975) for two-dimensional ﬂows, and later extended by Gatski and Speziale (1993), to three-dimensional ﬂows. The implementation has since been reﬁned, and the formulation to be presented is based on recent work by Jongen and Gatski (1998b). 4.2.1 Implicit algebraic stress model The starting point for the development of ASMs is the modeled transport equation for the Reynolds stress anisotropy tensor bij (see Gatski and Speziale 1993) given by Dbij 1 = Dt 2k = −bij Dτij τij Dk − Dt k Dt P 2 2 − ε − Sij − bik Skj + Sik bkj − bmn Smn δij (4.16) k 3 3 Πij τij 1 Dij − + D . 2k 2k k + (bik Wkj − Wik bkj ) + where Πij is the pressure-strain rate correlation, and Dij is the combined eﬀect of turbulent transport and viscous diﬀusion (D = Dii /2). While it is outside the scope of this chapter to discuss the modeling of the pressure-strain correlation Πij , it is necessary for the development of the algebraic stress model to specify a form for the pressure-strain rate model. For the purposes here, the SSG model (Speziale, Sarkar, and Gatski 1991) will be used, and can be written in the form 0 1 Πij = − C1 + C1 P ε εbij + C2 kSij (4.17) 2 + C3 k bik Sjk + bjk Sik − bmn Smn δij − C4 k (bik Wkj − Wik bkj ) , 3 where the closure coeﬃcients can, in general, be functions of the invariants of the stress anisotropy. It should be noted that the functional form given in equation (4.17) is representative of any linear pressure-strain rate model which could be used as well. Substituting equation (4.17) into equation (4.16) [1] Linear and nonlinear eddy viscosity models and rewriting yields Dbij τij 1 Dij − − D Dt 2k k bij 2 = − + a3 bik Skj + Sik bkj − bmn Smn δij a4 3 − a2 (bik Wkj − Wik bkj ) + a1 Sij . 31 (4.18) The coeﬃcients ai are directly related to the pressure-strain correlation model by a1 = 1 4 − C2 , 2 3 1 a2 = (2 − C4 ) , 2 (4.19) 1 a3 = (2 − C3 ) , 2 and g= 1 C1 P C0 +1 + 1 −1 2 ε 2 a4 = gτ, −1 = γ0 P + γ1 ε −1 , (4.20) 0 1 where C1 = 3.4, C1 = 1.8, C2 = 0.36, C3 = 1.25, and C4 = 0.4. An implicit algebraic stress relation is obtained from the modeled transport equation for the Reynolds stress anisotropy equation (4.18) when the following two assumptions ﬁrst proposed by Rodi (1976) are made: Dbij = 0, Dt and or Dτij τij Dk = , Dt k Dt (4.21) τij D. (4.22) k Equation (4.21) is equivalent to requiring that the turbulence has reached an equilibrium state, Db/Dt = 0, and equation (4.22) invokes the assumption that any anisotropy of the turbulent transport and viscous diﬀusion is proportional to the anisotropy of the Reynolds stresses. Both these assumptions impose limitations on the range of applicability of the algebraic stress model. Later in this section, some alternative assumptions will be proposed that will improve the range of applicability of the ASM. With these assumptions, the left side of equation (4.18) vanishes, and the equation becomes algebraic: Dij = bij 2 + a3 bik Skj + Sik bkj − bmn Smn δij a4 3 −a2 (bik Wkj − Wik bkj ) + a1 Sij = 0, (4.23) 32 or rewritten using matrix notation − Gatski and Rumsey 2 1 b − a3 bS + Sb − {bS}I + a2 (bW − Wb) = R. a4 3 (4.24) For linear pressure-strain rate models and an isotropic dissipation rate, it follows that R = a1 S. However, the generalization implied by using R is intended to indicate that the right-hand side of equation (4.24) can contain any known symmetric, traceless tensor (Jongen and Gatski 1998). Equation (4.24) has to be solved for b and is an implicit equation. Such an equation can be solved numerically in an iterative fashion. Unfortunately, such procedures can be numerically stiﬀ, depending on the complexity of the ﬂow to be solved. It is desirable to obtain an explicit solution to this equation which still retains its algebraic character. The ﬁrst attempt at this was by Pope (1975) who obtained an explicit solution of equation (4.24) using a three-term basis (cf. equations (4.3) and (4.7)) for two-dimensional mean ﬂows 1 b = α1 S + α2 (SW − WS) + α3 S2 − {S2 }I , 3 (4.25) where the αi are scalar coeﬃcient functions of the invariants S2 and W2 . Gatski and Speziale (1993) derived a corresponding expression for three-dimensional mean ﬂows which required all ten terms from the integrity basis given in equation (4.7). A general methodology will now be presented that allows for the systematic identiﬁcation of the coeﬃcients αi from an implicit algebraic equation such as that given in equation (4.24). 4.2.2 Explicit solution While it is possible to implement the following methodology by using any number of terms in the tensor representation T(n) , it is diﬃcult to obtain closed form analytic expressions beyond n = 3. Thus, the discussion here is limited to n = 3, and the three-term basis T(1) , T(2) , and T(3) (exact for two-dimensional ﬂows) from equation (4.7) is used for the representation, that is, 3 b= n=1 αn T(n) , (4.26) with the same three-term tensor basis T(n) shown in equation (4.25). Equation (4.24) can be solved ` la Galerkin by projecting this algebraic a relation onto the tensor basis T(m) itself. For this solution, the scalar product of equation (4.24) is formed with each of the tensors T(m) , (m = 1, 2, . . . , n). This procedure leads to the following system of equations: n 1 αn − (T(n) , T(m) ) − 2a3 (T(n) S, T(m) ) + 2a2 (T(n) W, T(m) ) a4 n=1 = (R, T(m) ), (4.27) [1] Linear and nonlinear eddy viscosity models 33 where the scalar product is deﬁned as (T(n) , T(m) ) = {T(n) T(m) }. In a more compact form, n αn Anm = (R, T(m) ), (4.28) n=1 where the n × n matrix A is deﬁned as 1 Anm ≡ − (T(n) , T(m) ) − 2a3 (T(n) S, T(m) ) + 2a2 (T(n) W, T(m) ). a4 For a two-dimensional mean ﬂow ﬁeld, the matrix A is (4.29) Anm = 2a2 η 4 R2 1 2 − a4 η −2a2 η 4 R2 − 2 4 2 η R a4 0 − 1 a3 η 4 3 , 0 1 4 − η − 1 a3 η 4 3 (4.30) 6a4 which, when inverted, leads to the following expressions for the representation coeﬃcients a4 ({RS} + 2a2 a4 {RWS} − 2a3 a4 {RS2 } , (4.31) α1 = − α0 η 2 {RWS} α2 = a4 a2 α1 + , (4.32) η 4 R2 α3 = −a4 2a3 α1 + 6{RS2 } , η4 (4.33) where α0 = 1 − 2 a2 a2 η 2 + 2a2 a2 η 2 R2 , and R2 (= − W2 / S2 ). The ﬂow 2 4 3 3 4 2 is a dimensionless variable that is useful for characterizing the parameter R ﬂow (Astarita 1979, Jongen and Gatski 1998a); for example, for a pure shear ﬂow R2 = 1, whereas for a plane strain ﬂow R2 = 0. This set of equations is the general solution valid for two-dimensional mean ﬂow and for any arbitrary (symmetric traceless) tensor R. As noted previously, when a linear pressure-strain correlation model is assumed, as well as an isotropic dissipation rate, then R = a1 S. This expression leads to a right-hand side for equation (4.28) proportional to {RS} a1 η 2 (m) (R, T ) = −2{RWS} = 0 . {RS2 } 0 (4.34) Using equation (4.34) in equations (4.31)–(4.33) and substituting into equation (4.26) leads to the representation for the Reynolds stress tensor τ 1 2 τ = kI + 2kα1 S + a2 a4 (SW − WS) − 2a3 a4 S2 − {S2 }I 3 3 . (4.35) 34 Gatski and Rumsey In equation (4.31), a4 is a function of P/ε (and therefore α1 ). Gatski and Speziale (1993) simpliﬁed this expression by assuming the coeﬃcient g (see equation (4.20)) and therefore P/ε to be constant in the analysis. However, we follow the approach proposed by Ying and Canuto (1996) and Girimaji (1996), in which the value of g is not ﬁxed; the variation of the productionto-dissipation-rate ratio in the ﬂow is accounted for in the formulation. It is easily shown that the production-to-dissipation rate ratio is given by P = −2 {bS} τ, ε (4.36) and that the invariant {bS} is directly related (Jongen and Gatski 1998a) to the coeﬃcient α1 appearing in the tensor representation through {bS} = α1 η 2 . (4.37) From equations (4.19) and (4.20), the coeﬃcient a4 can then be written as a4 = γ1 − 2γ0 α1 η 2 τ −1 τ. (4.38) The dependency of a4 on the production-to-dissipation rate ratio through α1 makes both sides of equation (4.31) functions of α1 . This dependency results in a cubic equation for α1 given by 2 3 γ0 α1 − γ0 γ1 2 1 γ 2 − 2τ 2 γ0 {RS} − 2η 2 τ 2 α η 2 τ 1 4η 4 τ 2 1 γ1 {RS} + 2τ a2 {RWS} − a3 {RS2 } a2 3 − R2 a2 2 3 = 0. α1 (4.39) + 1 4η 6 τ The expansion coeﬃcients of the nonlinear terms, α2 and α3 , retain the same functional dependency on α1 as before. When expressed in terms of the production-to-dissipation rate ratio with R = a1 S, equation (4.39) can be shown (Jongen and Gatski 1998a) to be equivalent to earlier results (Ying and Canuto 1996, Girimaji 1996). Previously (Ying and Canuto 1996, Girimaji 1996), the selection of the proper root for the solution of equation (4.39) was done on the basis of continuity arguments. Here, the proper choice for the solution root is based on the asymptotic analysis of Jongen and Gatski (1999). It was found that the root with the lowest real part leads to the correct choice for α1 . The explicit tensor representation given in equation (4.35) with α1 , α2 , and α3 determined by using equations (4.39), (4.32), and (4.33), respectively, is coupled with a k-ε two-equation model: Dk ∂ =P −ε+ Dt ∂xk ν+ νt σk ∂k , ∂xk νt σε ∂ε , ∂xk (4.40) (4.41) Dε ε ε2 ∂ = Cε1 P − fε Cε2 + Dt k k ∂xk ν+ [1] Linear and nonlinear eddy viscosity models 35 where ν is the kinematic viscosity and νt = Cµ kτ is an ‘equilibrium’ eddy viscosity. Also, fε = 1 − exp − Rek 10.8 , Rek = k 1/2 d , ν (4.42) σk = 1.0, σε = Cε1 = 1.44, κ2 , κ = 0.41, Cµ (Cε2 − Cε1 ) Cε2 = 1.83, Cµ = 0.096, (4.43) and d is the distance to the nearest wall. Note in equation (4.35) that α1 = ∗ ∗ −Cµ τ , where Cµ is the term that appears in the deﬁnition of the eddy viscosity ∗ ∗ ∗ νt = Cµ kτ . In most standard k-ε models, Cµ is taken to be a constant value ∗ = C (near 0.09). On the other hand, this explicit algebraic stress solution Cµ µ ∗ has the eﬀect of yielding a variable Cµ in the linear component of the stress, in addition to yielding nonlinear components proportional to SW − WS and S2 − 1 {S2 }I. 3 4.2.3 Curvature eﬀects and the equilibrium assumption In a recent study of two-dimensional ﬂow in a U-bend (Rumsey et al. 1999), the formulation of the explicit algebraic stress model just discussed was unable to correctly predict the turbulence second-moments in the strongly curved regions of the ﬂow. However, it was shown that a second-moment closure model could correctly predict the behavior of the turbulence. Since the ASM is closely coupled with the corresponding second-moment model, the poor performance of the ASM could be traced to the two underlying assumptions used in the development, that is, equations (4.21) and (4.22). The poor performance was attributed to the equilibrium assumption shown in equation (4.21). It was found from the second-moment closure computation of the ﬂow that some components of Dbij /Dt were not zero and that their magnitudes were as large as the components of a1 S (see equation (4.24)). Since the assumption aﬀecting the turbulent transport and viscous diﬀusion, equation (4.22), was eﬀectively satisﬁed throughout the ﬂow, any deﬁciency in algebraic stress model prediction can be attributed directly to the imposition of the equilibrium assumption equation (4.21). Similar deﬁciencies have been recognized previously. For example, Fu et al. (1988) showed that in free shear ﬂows with swirl the approximated advection terms were independent of swirl whereas the exact form in the second-moment equation was not. Thus, the equilibrium assumption precluded an accurate representation of a key dynamic feature. If the equilibrium condition Dbij /Dt = 0 is not correct, then can another condition on Dbij /Dt be chosen such that the results of the second-moment closure formulation are replicated? For the answer, it is only necessary to recall that the turbulence second-moment equations are not frame-indiﬀerent 36 Gatski and Rumsey and that Dbij /Dt = 0 in one frame does not imply the same in another frame (D/Dt is only frame-indiﬀerent under an extended Galilean transformation, Speziale 1979, 1998). The answer, then, is to ﬁnd a coordinate frame in which to apply the equilibrium condition Dbij /Dt = 0 (Gatski and Jongen 2000). Under a Euclidean transformation (Speziale 1979), the transformation of the turbulence anisotropy tensor bpq from a Cartesian base system is simply given by ¯pq = Xi bij Xj , b (4.44) p q where X(t) = ∂x/∂x is a proper orthogonal tensor. It then follows that the material derivative of equation (4.44) yields D¯pq b Dbij j D D Xi bij Xj + Xi bij Xj . = Xi Xq + p p q p Dt Dt Dt Dt q (4.45) (Note that in a general Euclidean transformation, a constant shift of the time variable is allowed. This shift is neglected here because it plays no relevant role in the analysis.) Equation (4.45) shows that the material derivative of the Reynolds stress anisotropy is not frame-indiﬀerent under arbitrary timedependent rotations. Since the equilibrium condition Dbij /Dt = 0 is a ﬁxed point of equation (4.18) (neglecting the contribution from the turbulent transport and viscous diﬀusion), this result shows that the ﬁxed point is also not frame-indiﬀerent. It is assumed at this point that equation (4.45) has transformed to a coordinate frame in which the equilibrium condition D¯pq /Dt = 0 b does hold so that equation (4.45) can be rewritten as Dbij = bik Ωkj − Ωik bkj , Dt where Ωij = Xi k (4.46) D Xj (4.47) Dt k and Ωij is a tensor related to the rate of rotation between the barred and unbarred (Cartesian) systems. The equilibrium assumption given in equation (4.21) can now be replaced with equation (4.46), and the resulting implicit algebraic stress equation is (cf. equation (4.23)) bij 2 + a3 bik Skj + Sik bkj − bmn Snm δij a4 3 (4.48) −a2 ∗ bik Wkj − ∗ Wik bkj + a1 Sij = 0, where ∗ Wij = Wij + 1 Ωij . a2 (4.49) The result in equation (4.49) shows that by accounting for curvature eﬀects through a modiﬁcation of the condition on Dbij /Dt, the resulting implicit [1] Linear and nonlinear eddy viscosity models 37 algebraic stress equation (in the Cartesian base frame) is only altered through a change to the mean rotation rate tensor. At this point in the analysis, we have not yet speciﬁcally identiﬁed the transformation X(t) between the two systems. The choice for the transformed (barred) system becomes clearer after the traditional analogy between curvature and rotation is exploited. Two ﬂows which have been studied by using the algebraic stress formulation and which ﬁt into the general formulation just discussed are rotating homogeneous shear and fully developed rotating channel ﬂow. Both of these have rigid-body rotation Ω about the axis (z-axis) perpendicular to the plane of shear (x,y plane). Of relevance here is the formulation used in solving these problems. In both cases, the ﬂow ﬁelds are described in the noninertial (barred) frames where the condition D¯ij /Dt = 0 holds exactly. Since equation (4.48) b is an implicit equation for bij in the inertial (base) frame, it needs to be transformed to the noninertial (barred) frame which is given by ¯ij b 2 b b b + a3 ¯ik S kj + S ik¯kj − ¯mn S mn δij a4 3 (4.50) ∗ b −a2 ¯ik W kj − ∗ b W ik¯kj + a1 S ij = 0, where now W ij = W ij − ∗ ijr Ωr + 1 Ωij , a2 Ωij = DXk j i Xk Dt (4.51) and Ωr = (0, 0, Ω). The frame-invariance properties of the strain-rate tensor and the stress anisotropy equation (4.44) have been used, and the corresponding lack of frame-invariance of the rotation-rate tensor is shown by the appearance of the term ijr Ωr . Since the noninertial eﬀects are imposed through a rigid-body rotation perpendicular to the plane of shear, the tensor Ωij is simply related to the rotation rate Ωr by Ωij = − so that ∗ ijr Ωr (4.52) W ij = W ij − 1 + 1 a2 ijr Ωr , (4.53) where ijr is the permutation tensor. Equation (4.53) is the intrinsic rotation rate tensor and is the form used in previous algebraic stress formulations of rotating homogeneous shear and rotating channel ﬂow. With this example, the role and description of Ωij , as a tensor related to the rate of rotation between the barred and unbarred systems, becomes clear. The issue of relative rotation rate also arises in the study of non-Newtonian constitutive equations (e.g., Schunk and Scriven 1990, Souza Mendes et al. 1995). There, the measure is based on the principal axes of the strain rate 38 Gatski and Rumsey tensor, which are a mutually orthogonal set of axes that rotate at the angular velocity associated with the rotation rate tensor Ωij . In this coordinate frame, it will be assumed that the condition D¯ij /Dt = 0 will hold. Spalart and Shur b (1997) also used the principal axes frame of reference to account for system rotation and curvature in sensitizing a one-equation model. In the principal axes coordinate frame, the transformation matrix X(t) can be deﬁned through the relation between the unit vectors of the ﬁxed and principal axes system i (j) (i) ek = Xj ek , (4.54) where superscript (i) (= 1, 2, 3) is the particular unit vector, and subscript k(= 1, 2, 3) is a component of the unit vector (i). Since in this analysis, the (j) j ﬁxed (unbarred) system is a Cartesian system where ek = δk , the unit vectors in the principal axes frame are simply given by ek = Xk . (i) i (4.55) When equation (4.55) is substituted into equation (4.47), the eigenvectors in the principal axes frame can then be expressed in terms of Ωij , Ωij = ei (k) Dej (k) Dt . (4.56) With the speciﬁcation of Ωij in equation (4.56), an explicit tensor representation for the implicit algebraic equation for bij , equation (4.48), can be obtained. In contrast to the previous representation in terms of S and W, the new representation is now in terms of S and W∗ . Figure 1 shows the computed Reynolds shear stress near the inner (convex) wall at 90◦ in the bend of a U-duct in comparison with experimentally measured values (Monson and Seegmiller 1992). In the ﬁgure, the Reynolds shear stress is nondimensionalized with respect to the square of the reference velocity, and the distance with respect to channel width. The experiment indicates a suppression of the Reynolds shear stress due to convex curvature. The original EASM does not show this turbulence suppression, but both the second-moment closure model and the EASM, with modiﬁcation to the equilibrium condition, do show the eﬀect. 4.2.4 Turbulent transport and viscous diﬀusion assumption Even though algebraic stress models ﬁrst appeared nearly three decades ago, their applicability was limited because of poor numerical robustness and poor predictive performance in some ﬂows. Launder (1982) recognized a deﬁciency in the early algebraic stress formulations due to the poor behavior of the effective eddy viscosity in some regions of wake and jet ﬂows. A remedy was [1] Linear and nonlinear eddy viscosity models 39 EASM (Dbij/Dt = 0) EASM (Dbij/Dt ≠ 0) Second-moment closure Experiment 0.3 0.2 y 0.1 0 -0.004 −τ12 0 0.004 0.008 Figure 1: Comparison of the Reynolds shear stress at 90◦ in the bend of a U-duct. Dbij /Dt = 0 condition obtained from equation (4.46). proposed which modiﬁed the turbulent transport hypothesis used in formulating the algebraic stress model. Later, Fu et al. (1988) concluded that in free shear ﬂows, where transport eﬀects are signiﬁcant, full diﬀerential stress models should be used rather than algebraic stress models. This conclusion was reached based on the poor predictive performance in both plane and round jet ﬂows. In a recent study by Carlson et al. (2001) on the prediction of wake ﬂows in pressure gradients, poor mean ﬂow predictions in the vicinity of the wake centerline, where transport eﬀects dominate, were also found. In this subsection, the assumption applied to the viscous and turbulent transport term, equation (4.22), is re-examined and a modiﬁcation is proposed. Equation (4.22) assumes that the anisotropy in Dij is directly related to the anisotropy in the Reynolds stresses themselves. A diﬀerent constraint on Dij can be found by ﬁrst rewriting equation (4.22) in terms of the anisotropy tensor bij , τij 2 (4.57) Dij − D = Dij − Dδij − 2Dbij . k 3 The right-hand side of equation (4.57) is now the sum of the deviatoric part of Dij and is a term proportional to the anisotropy tensor bij with scalar coeﬃcient D. If equation (4.57) is now substituted into equation (4.18), the diﬀerential anisotropy equation can be written as Dbij 2 1 Dij − Dδij − Dt 2k 3 bij 2 = − ∗ + a3 bik Skj + Sik bkj − bmn Smn δij a4 3 − a2 (bik Wkj − Wik bkj ) + a1 Sij . (4.58) 40 Gatski and Rumsey The coeﬃcients ai have been given previously in equation (4.19), with the exception of a∗ , which is now given by 4 1 1 1 = + D a∗ a4 k 4 or, using equation (3.30) for the turbulent kinetic energy, a∗ = 4 γ1 + 1 Dk + 1 − 2 (γ0 − 1) α1 η 2 τ ε Dt −1 (4.59) τ. (4.60) To obtain an implicit algebraic stress equation from equation (4.58), it is once again necessary to assume the equilibrium condition on the anisotropy tensor Dbij /Dt = 0 and to apply the new constraint, 2 Dij − Dδij = 0, 3 (4.61) on the viscous and turbulent transport terms. Equation (4.61) simply states that the deviatoric part of the tensor Dij is zero. In the case of homogeneous shear at equilibrium, an algebraic stress model should yield the same results as the full second-moment closure from which it is derived. Thus, any modiﬁcation to the algebraic stress model is constrained by this consistency condition. As equation (4.60) shows, the modiﬁcation in the formulation is partially through the term ε−1 Dk/Dt, which for the homogeneous shear case at equilibrium, can be related through the condition on the turbulent time scale, Dτ 1 Dk k Dε = − 2 = 0, (4.62) Dt ε Dt ε Dt to the production-to-dissipation rate ratio P 1 Dk k Dε = 2 = Cε1 ε Dt ε Dt ε − Cε2 . (4.63) ∞ At equilibrium, the production-to-dissipation rate ratio is given by the relation P ε = Cε2 − 1 Cε1 − 1 (4.64) ∞ which, when used in equation (4.63), gives the new relation for a∗ 4 ∗ ∗ a∗ = γ1 − 2γ0 α1 η 2 τ 4 −1 τ, (4.65) where ∗ γ0 = γ0 − 1, ∗ γ1 = γ1 + 1 + (4.66) (4.67) Cε2 − Cε1 . Cε1 − 1 [1] Linear and nonlinear eddy viscosity models 41 The coeﬃcients Cε1 and Cε2 retain the same values used in the EASM formulation with the original assumption on the anisotropy of the turbulent transport and viscous diﬀusion equation (4.22). This modiﬁcation meets the requirement that the homogeneous shear results are unaltered and only slightly aﬀect the results for the log-layer where the value of Cµ now takes the value 0.0885. The only other alterations to the algebraic stress formulation are that now a∗ , 4 ∗ ∗ γ0 , and γ1 are used instead of the coeﬃcients a4 , γ0 , and γ1 in the original formulation. ∗ Figure 2 shows the theoretical values of Cµ (= α1 /τ ) as a function of R for various levels of P/ε for the EASM with the original turbulent transport and viscous diﬀusion assumption. Note that for regions in an equilibrium log-layer, ∗ where R = 1 and P/ε = 1, the value of Cµ is approximately 0.096, which is the ‘equilibrium’ level employed in the model. However, in other regions of ∗ the ﬂow ﬁeld, Cµ can assume unreasonably high levels. For example, near the centerline of a wake, P/ε can be small and R tends toward zero. In this case, ∗ the original scheme yields unrealistically large levels of Cµ near 0.68! As a result, the turbulent eddy viscosity produced near the center of a wake is very high, and the velocity proﬁles tend to be somewhat ‘ﬂattened.’ This behavior is consistent with results obtained by Fu et al. (1988). 0.7 P/ε=0.01 0.6 0.5 0.10 0.4 C* µ 0.3 0.50 0.2 1.0 1.5 2 3 0.1 4 20 0 -2 -1 0 1 2 R ∗ Figure 2: Values of Cµ as a function of R for various levels of P/ε, EASM with the original turbulent transport and viscous diﬀusion assumption. Theoretical results, using the EASM with the modiﬁed viscous diﬀusion and ∗ turbulent transport assumption (see Fig. 3), show Cµ ≈ 0.0885 in the log-layer (again corresponding to the ‘equilibrium’ level employed in the model), and also more reasonable levels when P/ε is small. For example, the maximum level ∗ of Cµ is less than 0.19, as opposed to 0.68 for the original model. Consequently, the resulting wake proﬁles using the modiﬁed model are more realistic, as 42 Gatski and Rumsey shown in Fig. 4 for a wake generated by a splitter plate (Carlson et al. 2001) developing in zero pressure gradient. 0.7 0.6 0.5 0.4 C* µ 0.3 P/ε=0.01 0.10 0.2 2 1.5 1.0 4 3 20 0.50 0.1 0 -2 -1 0 1 2 R ∗ Figure 3: Values of Cµ as a function of R for various levels of P/ε, EASM with the modiﬁed turbulent transport and viscous diﬀusion assumption. 0.01 0.008 modified EASM original EASM 0.006 y, m 0.004 0.002 0 20 21 u, m/s 22 23 24 Figure 4: Eﬀect of modiﬁed turbulent transport and viscous diﬀusion assumption on wake velocity proﬁle. [1] Linear and nonlinear eddy viscosity models 43 5 Summary As this chapter has shown, a wide variety of linear and nonlinear eddy viscosity models have been proposed over the last three decades. The continuous development of closure models has been motivated by the equally continuous identiﬁcation of turbulent ﬂow ﬁelds which cannot be predicted to suﬃcient accuracy by the currently available models. However, a review of the literature shows that many new models are nothing more than straightforward extensions of existing models that attempt to account for particular physical eﬀects in the individual ﬂow ﬁelds studied. Unfortunately, such developments lead to a confusing array of closure models which in reality are not dynamically diﬀerent from one another. The purpose of this chapter has been to examine two broad classes of models, namely the linear and nonlinear eddy viscosity classes, and to brieﬂy analyze representative models within each class. In addition, an attempt was made to provide a cohesive presentation within each class to emphasize the commonality amongst the models both within and across the two classes. Unnecessary proliferation of the number of models, without signiﬁcant increase in the predictive capability of important dynamic features of each ﬂow, only undermines the credibility of turbulence closure modeling within the framework of a Reynolds-averaged Navier–Stokes approach. Nevertheless, linear and nonlinear eddy viscosity models have been, are, and will continue to be a popular choice among computational ﬂuid dynamicists for the solution of practical engineering turbulent ﬂow ﬁelds. References Apsley, D.D., and Leschziner, M.A. (1998) ‘A new low-Reynolds-number nonlinear two-equation turbulence model for complex ﬂows,’ Int. J. Heat and Fluid Flow 19, 209–222. Aris, R. (1989). Vectors, Tensors, and the Basic Equations of Fluid Mechnaics, Dover, New York. Astarita, G. (1979) ‘Objective and generally applicable criteria for ﬂow classiﬁcation,’ J. Non-Newtonian Fluid Mech. 6, 69–76. Baldwin, B.S., and Barth, T.J. 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(1972) A First Course in Turbulence MIT Press. Wallin, S., and Johansson, A.V. (2000) ‘An explicit algebriac Reynolds stress model for incompressible and compressible turbulent ﬂows,’ J. Fluid Mech. 403, 89–132. Wilcox, D.C. (1998) Turbulence Modeling for CFD, Second Edition, DCW Industries, Inc., La Ca˜ada, California. n Ying, R., and Canuto, V.M. (1996) ‘Turbulence modeling over two-dimensional hills using an algebraic Reynolds stress expression,’ Boundary-Layer Meteorol. 77, 69– 99. 2 Second-Moment Turbulence Closure Modelling K. Hanjali´ and S. Jakirli´ c c 1 Introduction Diﬀerential second-moment turbulence closure models (DSM) represent the logical and natural modelling level within the framework of the Reynoldsaveraged Navier–Stokes (RANS) equations. They provide the unknown secondmoments (turbulent stresses ui uj and turbulent ﬂuxes of heat and species, θuj ) by solving model transport equations for these properties. Hence, instead of modelling directly the second-moments, as is done with eddy-viscosity/diﬀusivity schemes, the modelling task is shifted to unknown higher-order correlations which appear in their diﬀerential transport equations. The main advantage of DSM is in the exact treatment of the turbulence production terms, be it by the mean strain or by body forces arising from thermal buoyancy, rotation or other forces. In addition, the solution of a separate transport equation for each component of the turbulent stress enables, in principle, an accurate prediction of the turbulent stress ﬁeld and its anisotropy, which reﬂects the structure and orientation of the stress-bearing turbulent eddies and, thus, plays a crucial role in turbulence dynamics in complex ﬂows. This role can be identiﬁed in the process of turbulence energy production, particularly if pure strain (compression and dilatation) is dominant such as in stagnation regions, or in energy redistribution and consequent enhancement or damping e.g. in rotating and swirling ﬂows. Stress anisotropy is an important source of secondary motion, and plays a role in controlling the dynamics of longitudinal vortices. Accurate prediction of the wall-normal stress component is also important in reproducing the wall phenomena, such as the wall shear stress or heat and mass transfer. Further, capturing stress anisotropy also enables a more realistic modelling of the scale-determining equation (dissipation rate or other variable). DSMs have long been expected to replace the currently popular two-equation k-ε and other eddy viscosity models as the industrial standard for Computational Fluid Dynamics (CFD). However, despite more than three decades of development and signiﬁcant progress, these models are still viewed by some as a development target rather than as a proven and mature technique for solving complex ﬂow phenomena. Admittedly, DSMs do not always show superiority over two-equation EVM models. One reason for this is that more terms need 47 48 Hanjali´ and Jakirli´ c c to be modelled. While this oﬀers an opportunity to capture the physics of various turbulence interactions better, the advantage may be lost if some of the terms are modelled wrongly. The use of a DSM also puts a greater demand on computing resources, and requires greater skills of the code user, because the model transport equations are now strongly coupled. However, the advantages and potential of DSMs for complex ﬂows have been generally recognized and the numerical diﬃculties are now to a large extent resolved. The demand on computer resources (memory, time) is not excessive: it is roughly twice as large as for two-equation EVMs for high Re number ﬂows using wall functions. These advances, together with the growing awareness among industrial CFD users of the limitations of two-equation eddy viscosity models and the need to model complex ﬂows with higher accuracy, will lead in future to a much wider use of DSMs in CFD1 . We begin this section by considering ﬁrst the basic model, and move later to discuss recent trends and advances. Major advantages and inherent potential of the DSMs are then discussed by focussing on some speciﬁc features of complex ﬂows, which are usually intractable with standard linear two-equation models. Attention will be conﬁned to nonreacting, single-phase turbulent ﬂows and to discussion of some basic issues related to turbulence modelling. The superiority of DSMs is demonstrated by a series of computational examples using either the same or very similar computational methods and model(s). Examples include several nonequilibrium ﬂows – attached and with separation and reattachment, ﬂow impingement and stagnation, secondary motion, swirl and system rotation. The modelling of molecular eﬀects, both near and away from a solid wall and associated laminar-to-turbulent and reverse transition are also discussed in view of the need for an advanced closure approach, particularly when wall phenomena are in focus. 2 2.1 The Basic Linear Second-Moment Closure Model for High-Re-number Flows The model equation for ui uj The exact transport equation for ui uj for an incompressible ﬂuid, including eﬀects of rotation and body force, may be written as Dui uj ∂ui uj ∂Uj ∂Ui ∂ui uj + Uk = − ui uk + uj uk = Dt ∂t ∂xk ∂xk ∂xk Lij 1 + (fi uj + fj ui ) Gij Cij Pij There is currently substantial activity in reviving the idea of nonlinear eddy viscosity (NLEVM) models and their ‘relatives’, the algebraic stress models (ASM). These models oﬀer substantial improvement over linear EVMs, see e.g., Speziale (1991), Craft, Launder and Suga (1995, 1996), Wallin and Johansson (2000), as well as sections on NLEVMs and ASMs in [1]. [2] Second-moment turbulence closure modelling 49 −2Ωk (uj um ikm Rij + ui um jkm ) + p ρ ∂ui ∂uj + ∂xj ∂xi φij − 2ν ∂ui ∂uj ∂xk ∂xk εij . ∂ + ∂xk ∂ui uj ν ∂x k ν Dij − ui uj uk t Dij − p (ui δjk + uj δik ) ρ p Dij (2.1) Dij Terms in boxes must be modelled. Note that Ωk represents the system rotation (angular) velocity, which should be distinguished from the rotation-rate vector of a ﬂuid element Wi = 1 ijk Wkj associated with the rotation-rate (‘vorticity’) 2 ∂Uj i tensor Wij = 1 ( ∂Uj − ∂xi ), and the ﬂuid vorticity ωi = ijk Wkj . 2 ∂x Each term has been given a short-hand alias so that in further discussion we may refer just to the symbolic representation of the stress transport equation: p ν t Lij + Cij = Pij + Gij + Rij + φij − εij + (Dij + Dij + Dij ), (2.2) where Lij represents the local change in time; Cij the convective transport; Pij the production by mean-ﬂow deformation; Gij the production by body force; Rij the production/redistribution by rotation force; φij the stress redistribution due to ﬂuctuating pressure; εij the viscous destruction; and Dij the diﬀusive transport. The modelling of the ui uj and ε equations follows the principles for modelling the k-ε equations, using the characteristic turbulence time scale τ = k/ε and length scales L = k 3/2 /ε, except that ui uj itself does not need to be modelled. The principal task is modelling the pressure-strain term φij and the stress dissipation rate εij . The standard modelling practice for the basic model is outlined below. 2.1.1. Stress dissipation. At high Reynolds numbers the large scale motion is unaﬀected by viscosity, while the ﬁne-scale structure is locally isotropic, i.e. unaﬀected by the orientation of large eddies. Consequently, the correlation ∂ui ∂u εij = 2ν ∂xk ∂xj — which is associated with smallest eddies — should reduce k to zero if i = j, while for i = j all three components should be equal. Hence, a common way to model the viscous destruction of stresses for high Re-number ﬂows is: 2 εij = εδij , (2.3) 3 ∂ul l where ε = ν ∂xk ∂uk and δij is the Kronecker unit tensor. It should be men∂x tioned that at high Reynolds numbers ε can be interpreted as the amount of 50 Hanjali´ and Jakirli´ c c energy exported by large (energy containing) eddies and transferred through the spectrum towards smaller eddies until ultimately it is dissipated. Hence, although ε represents essentially a viscous process, its value (‘the dissipation rate’) is governed by large, energy-containing eddies. Moreover, the above assumption about the isotropy of εij is not very appropriate in non-homogeneous ﬂow regions such as in the vicinity of a solid wall. Nonetheless, this assumption is widely used in standard DSM models and its deﬁciency is supposedly compensated by the model of the pressure-strain term φij , which accounts for the turbulence anisotropy. 2.1.2. Turbulent diﬀusion. The form of Dij is such that V Dij dV = 0 over a closed domain bounded by impermeable surfaces (as follows from a Gaussian transformation of the volume integral into a surface integral). Hence ν the Dij term is of a transport (diﬀusive) nature. The molecular diﬀusion Dij can be treated exactly but at high turbulence Re numbers it is negligible. The remaining two parts, the turbulent diﬀusion by ﬂuctuating velocity and p t ﬂuctuating pressure, Dij and Dij , need to be modelled. The most popular t is the generalized gradient diﬀusion (GGD), known also as the model for Dij ∂φ Daly–Harlow model: ϕuk = −Cφ τ uk ul ∂xl . The application of GGD to the turbulent velocity diﬀusion of stress yields: t Dij = ∂ ∂ (−ui uj uk ) = ∂xk ∂xk k ∂ui uj Cs uk ul ε ∂xl . (2.4) A simpler variant is simple gradient diﬀusion (SGD), with an isotropic (scalar) eddy diﬀusivity (Shir 1973). t Dij = ∂ ∂ (−ui uj uk ) = ∂xk ∂xk Cs k 2 ∂ui uj ε ∂xk . (2.5) More advanced treatments are discussed in Section 3. The turbulent transport by pressure ﬂuctuations has a diﬀerent nature (transport by the propagation of disturbances) and none of the gradient transp p port forms is applicable to modelling Dij . Yet, it is common to ‘lump’ Dij with t and to adjust the coeﬃcient C . In many ﬂows the pressure transport is Dij s smaller than the velocity transport so that this approximation often brings no adverse consequences. However, in ﬂows driven by thermal buoyancy (e.g. Rayleigh–B´nard convection) this is not the case and such models are not e appropriate. 2.1.3. Pressure-strain interaction. Pressure ﬂuctuations act to ‘scramble’ the turbulence structure and redistribute the turbulent stress among components to make turbulence more isotropic. Some insight into the physics and hints for modelling can be gained from the exact Poisson equation for the [2] Second-moment turbulence closure modelling 51 pressure ﬂuctuations (obtained after diﬀerentiating the equation for ul with respect to xl ): ∂2p ∂2 ∂Ul ∂um ∂fl =− (ρul um − ρul um ) − 2ρ +ρ , 2 ∂xl ∂xm ∂xm ∂xl ∂xl ∂xl which can be integrated to yield p at x: p 1 = ρ 4π ∂2 ∂Ul ∂um ∂fl dV (x ) (u u − ul um ) + 2 − , ∂xl ∂xm l m ∂xm ∂xl ∂xl |r| (2.7) (2.6) V ∂ui ∂uj + ∂xj ∂xi the following exact expression for φij : where r = x − x. Multiplication by (at x) and averaging yields p φij = ρ ∂ui ∂uj + ∂xj ∂xi 1 = 4π V ∂ 2 ul um ∂xl ∂xm ∂ui ∂uj + ∂xj ∂xi φij,1 +2 ∂Ul ∂xm ∂um ∂xl φij,2 ∂ui ∂uj + ∂xj ∂xi − ∂fm ∂xm ∂ui ∂uj dV (x ) + ∂xj ∂xi |x − x | φij,3 1 + 4π 1 ∂ p A r ∂n ∂ui ∂uj + ∂xj ∂xi −p φw ij ∂ui ∂uj + ∂xj ∂xi ∂ ∂n 1 dA . r (2.8) Diﬀerent terms in (2.8) can be associated with diﬀerent physical processes, which can be modelled separately. The most common approach is to split the term into the following parts: φij = φij,1 + φij,2 + φij,3 + φw + φw + φw , ij,1 ij,2 ij,3 where φij,1 is the return to isotropy of non-isotropic turbulence, (‘slow term’). In the absence of the mean rate of strain Sij and a body force, and away from any boundary constraint, pressure ﬂuctuations will force turbulence to approach an isotropic state. φij,2 is the ‘isotropization’ of the process of stress production due to Sij (‘rapid term’). Pressure ﬂuctuations will slow down a preferential feeding of turbulence by Sij into a particular component imposed by the active strain-rate components. (2.9) 52 Hanjali´ and Jakirli´ c c φij,3 is the ‘isotropization’ of stress production due to a body force; φw , φw , φw is the wall blockage (‘eddy splatting’) and pressure reij,1 ij,2 ij,3 ﬂection eﬀect associated with φij,1 , φij,2 and φij,3 respectively. The ﬁrst process, which is dominant, impedes the isotropizing action of the pressure ﬂuctuations. The pressure reﬂection acts in fact in the opposite way (the pressure wave reﬂected from a solid surface enhances eddy scrambling), but this eﬀect is smaller in comparison with the wall-blockage eﬀect. Jones and Musonge (1988) argued that the mean strain rate should appear also in the exact transport equation for the slow term φij,1 and that separate modelling of each part of φij may not be fully justiﬁed. Their ‘integral’ model for the complete φij diﬀers, however, only slightly from other models in the values of the coeﬃcients (see next section). However, splitting the terms, even if not fully justiﬁed, enables us to distinguish some physical eﬀects from others and gives some basis for their modelling. For that reason we follow the conventional approach here. The model of φij,1 (‘the slow term’). Based on the idea that the pressure ﬂuctuations tend to reduce turbulence anisotropy, Rotta (1951) proposed a simple linear model by which φij,1 is proportional to the stress anisotropy tensor itself (of course with a negative sign). The expression is known as the linear return-to-isotropy model: ui uj 2 φij,1 = −C1 εaij = −C1 ε (2.10) − δij , k 3 where aij is known as the stress anisotropy tensor; it is just twice the value of the tensor bij introduced in [1]. To promote a reduction in anisotropy the coeﬃcient C1 must be greater than 1; the most common value is C1 = 1.8. The models of φij,2 and φij,3 (‘the rapid terms’). Without going deeper into the physics, we recall at this point that φij,2 is associated with the mean rate of strain, which is usually the major source of turbulence production. Hence the pressure scrambling action can be expected to modify the very process of stress production. Following this idea, Naot et al. (1970) proposed a model of φij,2 analogous to Rotta’s model of the slow term, known as the ‘Isotropization of Production’ (IP) model: 2 φij,2 = −C2 Pij − δij P . (2.11) 3 An analogous approach to the pressure eﬀect on stress generation due to body forces leads to: 2 φij,3 = −C3 Gij − δij G . (2.12) 3 Note that P ≡ 1/2Pkk and G ≡ 1/2Gkk [2] Second-moment turbulence closure modelling 53 Interdependence of coeﬃcients. The above listed coeﬃcients C1 , C2 and C3 have been obtained mainly from selected experiments where only one of the processes can be isolated (e.g. C1 from experiments on free return to isotropy of initially strained turbulence, C2 from rapid distortion theory). Of course, the coeﬃcients have also been tuned through subsequent validation in a series of experimentally well documented ﬂows. Values other than those quoted above have also been proposed as more suitable for some classes of ﬂows. However, the validation revealed that a change in one coeﬃcient also requires an adjustment of others in order to reproduce fully the total eﬀect of the pressure-strain term. A useful correlation between C1 and C2 is: C1 ≈ 4.5(1 − C2 ). (2.13) The most frequently used values are: C1 = 1.8, C2 = 0.6 and C3 = 0.55. Modelling the wall eﬀects on φij . Solid walls and free surfaces ‘splat’ neighbouring eddies, which leads to a larger turbulence anisotropy. Wall impermeability (blocking eﬀect) damps the velocity ﬂuctuations in the wall-normal direction. On the other hand, pressure ﬂuctuations reﬂecting from bounding surfaces will enhance the pressure scrambling eﬀect. Both these eﬀects are non-viscous in nature and are essentially dependent on the wall distance and the wall conﬁguration. The blockage eﬀect is stronger resulting in an impediment of the isotropizing action of pressure ﬂuctuations. As a consequence, the stress anisotropy in the near-wall region is higher than in free ﬂows at similar strain rates (Launder et al. 1975). Thus, for a simple shear with x1 the ﬂow direction and x2 the direction of shear: Homogeneus shear ﬂow (Champagne et al.) Near-wall region of a boundary layer a11 a22 a33 a12 0.30 −0.18 −0.2 −0.33 0.55 −0.45 −0.11 −0.24 Wall damping is expected to aﬀect both the ‘slow’ and the ‘rapid’ pressureinduced stress-redistribution processes and we can decompose φw into two ij terms: φw ‘corrects’ the values of the stress components in such a way as ij,1 to slow down the redistribution and diminish the wall-normal component to the beneﬁt of the streamwise and spanwise ones, and impedes the reduction of the shear stress. Likewise, φw modiﬁes the processes of stress production ij,2 in the near-wall region. The wall correction should attenuate with distance from the wall: this is usually arranged through an empirical damping function, fw = L/CL y, where L is the turbulence length scale. Close to a wall L ∝ y and the constant CL is chosen so that fw ≈ 1. At larger wall-distances L ≈ const so that fw → 0 Based on the above reasoning several models of φw and φw have been ij,1 ij,2 proposed, the most frequently employed schemes being: 54 • Shir (1973): w φw = C1 ij,1 Hanjali´ and Jakirli´ c c ε 3 3 uk um nk nm δij − ui uk nk nj − uk uj nk ni · fw ; k 2 2 (2.14) • Gibson and Launder (1978): 3 3 w φw = C2 φkm,2 nk nm δij − φik,2 nk nj − φjk,2 nk ni · fw ; ij,2 2 2 w w where C1 = 0.5, C2 = 0.3, fw = (2.15) 0.4 k 3/2 , xn is the normal distance to the εxn wall and nk is the unit vector of the coordinate normal to the wall. These schemes have been and still are being successfully applied for attached ﬂows on nearly plane, continuous surfaces but they have been superseded for more complex ﬂows by alternatives discussed later in this chapter and in [3] and [4]. 2.2 The model equation for ε The exact equation for ε is not of much help as a basis for modelling except to give some indication of the meanings and importance of various terms. The dissipation rate appears as the exact sink term in the k equation and needs to be provided to close the model (εij in ui uj equation). However, what is needed for closing other terms is the characteristic time and length scale of the energycontaining eddies and the equation for energy transfer from these eddies down the spectrum towards the smaller eddies. This energy transfer rate coincides with the dissipation rate only under the conditions of spectral equilibrium. Nevertheless, it is instructive to look at the exact transport equation for ε: Dε ∂ε ∂ε + Uk = −2ν = Dt ∂t ∂xk Lε Cε ∂ui ∂uk ∂ul ∂ul + ∂xl ∂xl ∂xi ∂xk Pε1 +Pε2 ∂Ui ∂xk −2νuk ∂ui ∂ 2 Ui ∂fi ∂ui + ν ∂xl ∂xk ∂xl ∂xl ∂xl Pε3 Gε 2 −2ν ∂ui ∂ui ∂uk ∂ 2 ui − 2 ν ∂xk ∂xl ∂xl ∂xk ∂xl Pε4 Y + ∂ ∂xk ν ∂ε ∂xk ν Dε − uk ε t Dε − 2ν ∂p ∂uk . (2.16) ρ ∂xi ∂xi p Dε Dε [2] Second-moment turbulence closure modelling 55 The physical meaning of the terms can be inferred from comparison with the transport equation for the mean square of ﬂuctuating vorticity ωi2 (‘enstrophy’), since for homogeneous turbulence ε ≈ 2νωi2 (Tennekes and Lumley 1972). At high Reynolds numbers the source terms in the third row (Pε4 and Y ) are dominant, while the other production terms can be neglected as smaller 1/2 (Pε1 + Pε2 and Gε by Ret and Pε3 by Ret , where Ret = k 2 /νε is the turbulence Reynolds number). Of course, for low-Re-number regions of ﬂow, these terms need to be taken into account. Note that all terms in boxes must be modelled. In the DSM closures the same basic form of model equation for ε is used as in the k-ε model, except that now ui uj is available (and also θui if the secondmoment closure level is also used for the thermal and other scalar ﬁelds). This has the following implications: • The P and G (production of kinetic energy) in the source term of ε are treated in exact form; • The generalized gradient-diﬀusion hypothesis is used to model turbulent diﬀusion k ∂ε ∂ ∂ t Dε = (−uk ε) = Cε uk ul . (2.17) ∂xk ∂xk ε ∂xl Hence, the model equation for ε has the form (Hanjali´ and Launder 1972): c Dε ∂ = Dt ∂xk k ∂ε Cε uk ul ε ∂xl + (Cε1 P + Cε3 G + Cε4 k ∂Uk ε − Cε2 ε) , ∂xk k (2.18) where the coeﬃcients have the same values as in the k-ε model except for the new coeﬃcient Cε = 0.18 (which replaces σε ) The following values of coeﬃcients have been recommended: Cs 0.2 C1 1.8 C2 0.6 w C1 0.5 w C2 0.3 Cε 0.18 Cε1 1.44 Cε2 1.92 Cε3 1.44 Cε4 −0.373 2.3 Second-moment closure for scalar ﬁelds for high-Pecletnumber ﬂows The second-moment closure models for scalar ﬁelds (thermal, species concentration) follow essentially the same principles as the modelling of the velocity ﬁeld (Launder 1976). This means that the transport equations for the scalar ﬂux hui , θui , cui are modelled starting from their exact parent equations (here h is the ﬂuctuating enthalpy, θ is the ﬂuctuating temperature and c is the ﬂuctuating concentration; for a multi-component mixture each species concentration is considered separately, i.e. c is replaced by c(i) ). Because the principles are the same, except for the source terms, which usually require no special consideration (nor modelling), we consider here only the equation for the turbulent heat ﬂux θui . 56 2.3.1 The Model Equation for Scalar Flux θui Hanjali´ and Jakirli´ c c The exact transport equation for the turbulent heat ﬂux vector for high Peclet numbers (P e = Re.P r) can be derived in a manner analogous to the stress equation: Dui θ Dt = −ui uk Θ Piθ ∂Θ p ∂θ ∂Ui −uk θ −βgi θ2 + ∂xk ∂xk ρ ∂xi U Piθ Giθ φiθ −(α + ν) ∂θ ∂ui ∂ − ∂xk ∂xk ∂xk ui uk θ + p t Diθ +Diθ pθ δik . ρ (2.19) −εiθ The physical meaning of the various terms can be inferred by comparison with the Reynolds-stress equation: Θ Piθ is the ‘thermal’ production (nonuniform temperature ﬁeld interacting with turbulent stresses), U Piθ is the ‘mechanical’ production (mean ﬂow deformation interacting with the turbulent heat ﬂux), Giθ is the gravitational production (gravitation interaction with the ﬂuctuating temperature ﬁeld), φiθ is the pressure–temperature-gradient correlation, εiθ is the molecular destruction, Diθ is the diﬀusive transport (where ‘t’ denotes diﬀusion by turbulent velocity and ‘p’ by pressure ﬂuctuations). All three production terms can be treated in exact form, but an additional transport equation needs to be provided for the temperature variance θ2 . Other terms need to be modelled. Following the same modelling principles we can express the pressure scrambling term as: φiθ = φiθ,1 + φiθ,2 + φiθ,3 = −C1θ ui θ U − C2θ Piθ − C3θ Giθ . k (2.20) Θ It is noted that the term corresponding to Piθ is absent because the mean temperature gradient does not appear in the exact Poisson equation for φiθ . The turbulent diﬀusion by velocity and pressure ﬂuctuations is modelled by GGD. It should be noted that the viscous diﬀusion needs also to be modelled, except in the case of Prandtl numbers of O(1). For high Pe numbers εiθ can [2] Second-moment turbulence closure modelling 57 be neglected. Hence, the model equation for the scalar ﬂux (with wall eﬀects omitted2 ) is: Dui θ Dt = −ui uk ∂Ui ∂Θ − (1 − C2θ )uk θ − (1 − C3θ )βgi θ2 ∂xk ∂xk ui θ ∂ k ∂ui θ −C1θ ε uk ul . + Cθ k ∂xk ε ∂xl (2.21) 2.3.2 The model equation for scalar variance θ2 The model equation for θ2 resembles closely the k equation and can be modelled in the same manner. It contains a single production term which can be treated exactly. The turbulent transport is modelled in the usual gradienttransport form. The only problem is the sink term (molecular destruction) ∂θ εθ = 2α ∂xj . A transport equation for εθ can be derived, resembling the ε equation, except that it has twice as many terms, so that its modelling poses a lot of uncertainty. (Proposals for closing the εθ equation are given in Chapter [6].) The usual approach, based on the assumption that the ratio of the thermal to mechanical time scale τθ /τ = R is constant (where τθ = θ2 /2εθ and τ = k/ε) leads to the simple approximation 2 εθ = Hence, the model equation for θ2 is ε θ2 . R 2k (2.22) ∂ ∂Θ 1 θ2 Dθ2 − = −2ui θ ε + Cθ2 Dt ∂xi 2R 2k ∂xj 2.3.3 ui uj k ∂θ2 ε ∂xi . (2.23) Summary of coeﬃcients for scalar ﬂux model The following values of coeﬃcients can be recommended for the scalar ﬂux model for high Peclet number ﬂows: Cθ2 0.2 Cθ 0.15 C1θ 3.5 C2θ 0.55 C3θ 0.55 R 0.5 2.4 The algebraic stress/ﬂux models (ASM/AFM) A considerable simpliﬁcation of the diﬀerential equation for ui uj can be achieved by eliminating the transport terms in individual stress components in terms Wall-reﬂection eﬀects have been considered in Gibson and Launder (1978) and Launder and Samaraweera (1979). More elaborate models than equation (20) appear to avoid the need for explicit wall corrections (see [3], [14], [15].) 2 58 Hanjali´ and Jakirli´ c c of transport terms of the kinetic energy. The common approach is to assume the so-called weak non-equilibrium hypothesis (Rodi 1976) by which the time and space evolution of the stress anisotropy tensor is equal to zero, i.e.: Daij = 0. Dt The expansion of aij = ui uj 2 − δij leads to: k 3 ui uj Dk ui uj Dui uj − Dij = − Dk = (P + G − ε). Dt k Dt k (2.24) (2.25) Each stress component ui uj can now be expressed in terms of an algebraic expression: 2 k 2 2 ui uj = δij k + α1 Pij − Pδij + α2 Gij − Gδij 3 ε 3 3 , (2.26) where α1 and α2 are functions of P/ε and G/ε (containing also the coeﬃcients from the modelled expressions for the pressure-strain terms). ASMs have some advantages such as a reduction of computing time in comparison with the full (diﬀerential) DSM, they usually give better results than the linear k-ε model where stress anisotropy is strong and important, e.g. secondary ﬂows in the straight ducts. However, they are usually derived from a presumed DSM model and they can at best perform as well as the parent DSM provided the ﬂow evolution is slow. A major shortcoming is the inability to reproduce the evolution of the stress anisotropy in nonequilibrium ﬂows, when the ﬂow ‘history’ and development cannot be fully accounted for by transport terms in the k- and ε-equations. The same approach can be applied to derive an algebraic ﬂux model (AFM) for a scalar ﬁeld, in which case the weak equilibrium hypothesis is applied to the scalar-ﬂux correlation coeﬃcient, i.e. D(ui θ/ (kθ2 )/Dt = 0. The AFM/ASM approach has even greater appeal if convection of scalars is considered, because even for a single scalar ﬁeld the number of diﬀerential transport equations in a full second-moment closure may be as large as 17. The above ASM is implicit in ui uj . Besides, the functions α have expressions in the denominator, which may become very small or even zero, leading to singularities and numerical instability. In order to overcome the numerical problems, several explicit nonlinear ASMs and AFMs have recently been proposed in the literature (Speziale and Gatski 1993, Wallin and Johansson 2000). Modelling at this level is more extensively considered in [1]. 3 Advanced Diﬀerential Second-Moment Closures The basic diﬀerential second-moment closure models (DSM) have proved to perform better than linear Eddy Viscosity Models (EVM) in many ﬂows, but [2] Second-moment turbulence closure modelling 59 not always, and not by a convincing margin. Over the past two decades there has been much activity aimed at improving the basic DSM. All improvements lead, inevitably, to more complex models which may pose additional computational diﬃculties (numerical instabilities, slower convergence). The model developers often focus on only one or two crucial terms in the ui uj and ε equation and propose more sophisticated expressions which better satisfy physical and mathematical constraints (realizability, two-component limit, vanishing and inﬁnite Reynolds numbers, etc.). Validation is usually performed in a limited number of test cases that display particular features which are the focus of the new development. The resulting complex model is often out of balance with the usually much simpler models adopted for the rest of the unknown processes. We conﬁne our attention here to only a few advancements, which seem to bring desirable improvement and yet retain the form of the model expressions at a manageable level of sophistication. The focus is on the models of turbulent diﬀusion and pressure scrambling in the ui uj equation and on some proposals to improve the ε equation. More complex models of the processes are dealt with in other chapters in this volume. 3.1 Some improvements to the modelled ui uj equation Turbulent diﬀusion of ui uj . A coordinate-frame invariant model of Dij can be derived by tensorial expansion of the GGD hypothesis. Alternatively, a truncation of the model transport equation for triple velocity correlation ui uj uk , retaining only the ﬁrst order terms, yields (Hanjali´ and Launder c 1972): t Dij = k ∂uj uk ∂uk ui ∂ui uj ∂ ∂ (−ui uj uk ) = Cs ui ul + uj ul + uk ul ∂xk ∂xk ε ∂xl ∂xl ∂xl . (3.1) Application of the moment-generating function leads to still more complex expressions (e.g. Lumley 1978, Cormac et al. 1978, Magnaudet 1992). In addition to the invariant expression (3.1), the above-mentioned expressions contain additional terms, some including the gradients of the turbulent kinetic energy and its dissipation rate, and even the mean rate of strain. While these more general expressions lead to some improvements, a comparison with the DNS results for a plane channel ﬂow showed that none of the above mentioned models satisﬁes all stress components (Hanjali´ 1994, Jakirli´ 1997). Nagano c c and Tagawa (1991) proposed a new way to treat triple velocity and scalar correlations by solving the transport equations for triple moments for each of the velocity ﬂuctuations, u3 , while evaluating the mixed triple moments i from algebraic correlations. The latter were derived from structural characteristics of the shear-generated turbulence. This approach led to improvements in near-wall ﬂows, but it is not coordinate invariant. 60 Hanjali´ and Jakirli´ c c It should be noted, however, that all the more general expressions give rise to a large number of component terms, particularly in non-Cartesian coordinates. Because in many ﬂows with a strong stress production the turbulent transport is relatively unimportant, a simpler model usually suﬃces. A simpler expression which satisﬁes the coordinate-frame invariance and still retains a relatively simple form is the expression proposed by Mellor and Herring (1973): t Dij = k2 ∂ ∂ (−ui uj uk ) = Cs ∂xk ∂xk ε ∂uj uk ∂uk ui ∂ui uj + + ∂xi ∂xj ∂xk . (3.2) Pressure-strain interaction: The ‘slow’ term. The return to isotropy is in fact a nonlinear process: a tensorial expansion making use of the Cayley– Hamilton theorem leads to a quadratic model of the ‘slow’ term (Lumley 1978, Reynolds 1984, Fu, Launder and Tselepidakis 1987, Speziale, Sarkar and Gatski 1991); 1 φij,1 = −ε[C1 aij + C1 (aik ajk − δij A2 )], 3 (3.3) where C1 , C1 are, in general, functions of turbulence Re-number and stress anisotropy invariants A2 = aij aij , A3 = aij ajk aki and the ‘ﬂatness’ parameter A = 1 − 9 (A2 − A3 ). Speziale, Sarkar and Gatski (1991) (SSG) proposed 8 C1 = −1.05 and this value has been generally accepted in the framework of the complete quadratic pressure-strain model (see below). The UMIST group (Craft and Launder 1991) proposed a similar expression (validated in free ﬂows, but subsequently also used in near-wall ﬂows): φij,1 = −C1 ε 1 1+ C10 C1 1 aij + C1 aik akj − A2 δij 3 , (3.4) with C1 = 3.1(A2 A) 2 , C1 = 1.2 and C10 = 1 Earlier, Lumley (1978) and Shih and Lumley (1993) discussed the quadratic expression, but due to the lack of evidence, they discarded the second term and proposed C1 in the form of a function dependent on Reynolds number and stress invariants. The appearance of DNS data for each part of the pressure-strain term makes it possible not only to verify the proposed expression, but also the values of the coeﬃcients. Equation (3.3) contains two unknowns, C1 and C1 . Using any pair of experimental data for φij,1 for two components enables candidate values for C1 and C1 to be obtained. In fact, because four components of φij,1 are available in a channel ﬂow, the problem is overdeﬁned and diﬀerent solutions can emerge for diﬀerent combinations, if expression (3.3) is not unique. Such a test in a plane channel ﬂow, using the DNS results of Kim et al. (1987) showed that both coeﬃcients, C1 and C1 vary strongly across the ﬂow. However, the [2] Second-moment turbulence closure modelling 61 Figure 1: Variation of C1 and C1 in a plane channel ﬂow. Symbols: evaluation 11-22, 11-33, 22-33. Lines: from pairs of DNS components of φij,1 , diﬀerent models, - - - HL, —– HJ, – - – SSG, – – CL, · · ·· LT, · – · LRR (for acronyms see Table 1). results for diﬀerent pairs of φij,1 collapse on one curve in the region close to a wall for y + < 60 for Rem = 5600, though departing substantially away from the wall, Figure 1. The data also show that a simple expression C1 = −A2 C1 (3.5) matches the DNS data well in the near-wall region. It should be noted, however, that C1 changes sign at y + ≈ 12, exhibiting a peak at y + ≈ 6. Most models do not reproduce such a behaviour, but impose a monotonic approach of C1 to zero at the wall. Figure 1 shows the variation of C1 and C1 given by diﬀerent model proposals, discussed above. Pressure-strain interaction: The ‘rapid’ term. The basis for modelling of the ‘rapid’ term is the general expression ∂Ul mi (b + bmj ), (3.6) li ∂xm lj which represents in a symbolic form the corresponding element of equation (2.8) after the mean velocity gradient is taken out of the volume integral by φij,2 = 62 Hanjali´ and Jakirli´ c c assuming local mean ﬂow homogeneity. The modelling task is reduced to expressing the fourth-order tensor bmi in terms of available second-order tensors lj (turbulent stress tensor ui uj and Kronecker unit tensor δij ). A convenient and more general way is to formulate the complete φij,2 in the form of a tensorial expansion series in terms aij , Sij and Wij . The complete expression (closed by the Cayley–Hamilton theorem) contains terms up to fourth order in aij (Johansson and H¨llback 1994), though cubic models have also been designed a to meet most constraints. An (incomplete) cubic model can be written in the general form: 2 φij,2 = −C2 Paij + C3 kSij + C4 k aik Sjk + ajk Sik − δij akl Skl 3 +C5 k(aik Wjk + ajk Wik ) −−−−−−−−−−−−−−−−−−−−−−−−−−−−− +C6 k (aik akl Sjl + ajk akl Sil − 2akj ali Skl − 3aij akl Skl ) +C7 k (aik akl Wjl + ajk akl Wil ) 3 +C8 k[a2 (aik Wjk + ajk Wik ) + ami anj (amk Wnk + ank Wmk )], (3.7) mn 2 1 ∂Ui ∂Uj 1 ∂Ui ∂Uj + and Wij = − 2 ∂xj ∂xi 2 ∂xj ∂xi The line separates the quasi-linear (in aij ) from nonlinear expressions (‘quasilinear’, because C2 P aij is, in fact, quadratic since P contains aij , while all other terms are linear). Although nonlinear models are claimed to satisfy better the mathematical constraints and physical requirements, the large number of terms currently reduce their appeal for industrial applications. In addition to the rudimentary IP model introduced earlier, equation (2.11), two other popular models that still retain a simple form are the linear Quasi-Isotropic model of Launder, Reece and Rodi (1975), denoted as LRR-QI, and the quasilinear model of Speziale, Sarkar and Gatski (1991), denoted as SSG.3 The models diﬀer in the values of coeﬃcients. Unlike the IP model, the LRR-QI is capable of reproducing some – though insuﬃcient – stress anisotropy in the near-wall region even without any wall reﬂection term φw and was shown to ij perform slightly better in some homogeneous and free thin shear ﬂows (Launder et al. 1975). However, subsequent broader testing showed that the IP with the Shir (1973) and Gibson and Launder (1978) wall-echo corrections, (2.14 and 2.15), perform generally more satisfactorily. The SSG model contains the above mentioned quasi-linear term, which is absent from LLR-QI model, and one of the coeﬃcients is formulated as a function of the second stress invariant A2 . A summary of the coeﬃcients for some of the models of φij found in the literature is given in Table 1. where Sij = The early model of Hanjali´ and Launder (1972) also contained the nonlinear term c C2 Paij with the same value of the empirical coeﬃcient, though with opposite sign (Table 1). 3 [2] Second-moment turbulence closure modelling 63 Table 1: Summary of coeﬃcients in pressure-strain models φij,1 Authors Linear Quadratic C1 HL LRR IP LRR QI SSG 2.8 1.8 1.8 1.7 C1 0 0 0 −1.05 C2 −0.9 0 0 0.9 Linear C3 0.8 0.8 0.8 0.8− 1/2 0.625A2 1/2 φij,2 Quadratic Cubic C4 C5 C6 C7 0 0 0 0 C8 0 0 0 0 0.71 0.582 0 0.6 0.6 0 0.873 0.655 0 0.65 0.2 0 LT 6.3AA2 0.7C1 0 0.8 0.6 0.866 0.2 0.2 1.2 Abbreviations: HL: Hanjali´ and Launder (1972), LRR: Launder, Reece and Rodi c (1975), SSG Speziale, Sarkar and Gatski (1991), LT: Launder and Tselepidakis (1994). The model of the rapid pressure-strain term can alternatively be written k in terms of Sij , Pij and Dij = − ui uk ∂Uj + uj uk ∂Uk (Launder et al. 1975). ∂x ∂xi The quasi-linear model, corresponding to the ﬁrst line of equation (3.7) reads ∗ ∗ ∗ φij,2 = −C2∗ Paij − C3 kSij − C4 (Pij − 2/3Pδij ) − C5 (Dij − 2/3Pδij ). (3.8) The conversion of the coeﬃcients follows from the following relationships ∗ ∗ ∗ (Hadˇi´ 1999): C2∗ = C2 , C3 = 4/3C4 − C3 , C4 = 1/2(C4 + C5 ), and C5 = zc 1/2(C4 − C5 ). It should be mentioned that the LRR-IP and LRR-QI models require the use of wall-echo terms φw deﬁned earlier, whereas the SSG model ij does not. Apparently, the extra quasi-linear term and the function C3 account for the stress redistribution modiﬁcation by a solid wall making the wall-echo terms redundant. While this statement is not fully true (LRR-IP+wall echo terms reproduce better the stress anisotropy in a near-wall region, as can be seen in Hadˇi´ 1999), the SSG has some appeal as a compromise between the zc desired accuracy and computational economy: it is more practical than LRRQI with wall-correction term φw , and more accurate than both the IP and ij LRR-QI models if used without the φw . ij 64 Hanjali´ and Jakirli´ c c The quasi-linear term can be interpreted as an extension of the slow term φij,1 with the coeﬃcient C1 replaced by a function C1 (1 + C2 P/ε). The ratio P/ε has been used earlier in several models, even at the two-equation level, to account for departures from local energy equilibrium. The inclusion of nonlinear terms brings more ﬂexibility and potential to model better various extra strain eﬀects, as well as to satisfy mathematical and physical constraints. For example, the more complete two-componentlimit model, Craft and Launder (1996), which retains all cubic terms yet with only two adjustable coeﬃcients, was reported to perform very well both in near-wall and free-ﬂow regions in several complex ﬂows. More details of this approach can be found in [3] and [14]. Elliptic relaxation concept. Instead of wall damping deﬁned in terms of local ﬂow and turbulence variables, wall topology and distance, Durbin (1991, 1993) proposed providing the required damping through an elliptic relaxation (ER) equation, which accounted for the non-viscous wall blocking eﬀect. In the framework of eddy viscosity models, this approach requires one elliptic relaxation equation for the wall damping function f , which is used to model an additional transport equation for another scalar property denoted as v 2 (which essentially reduces to the wall-normal turbulent stress in the nearwall region of wall-parallel ﬂows). The resulting k-ε-v 2 -f model (with νt = D Cµ v 2 /τ and τ = max[k/ε, Cτ (ν/ε)], where the latter scale-switch accounts for viscous eﬀects) allows integration up to the wall. While this model seems to perform signiﬁcantly better than the conventional high- or low-Re-number eddy viscosity models, it still shows deﬁciencies in reproducing strongly nonequilibrium ﬂows where the stress anisotropy changes fast and is aﬀected by the ﬂow history. Durbin (1993) extended his elliptic relaxation approach to full second-moment closure by proposing the following elliptic equation for the tensorial damping function fij corresponding to the stress tensor ui uj : L2 ∇2 fij − fij = − φh ij , k (3.9) where φh is the ‘homogeneous’ (far from a wall) pressure-strain model, for ij which, in principle, any known model (without wall correction) can be used4 . The function fij is then used to obtain φD = k fij . With viscous eﬀects acij counted for by imposing Kolmogorov scales as lower bounds on both the time and length scales of turbulence (just as in the k-ε-v 2 -f model), the model allows the integration up to a solid wall (for further comments on the implementation of viscous eﬀects, see Section 5). A fuller account of the physical basis and the associated analysis for the ER approach, together with examples of applications is provided in [4]. Note that Durbin used φD = (Πij − 1/3Πkk δij ) − (εij − ui uj /kε) instead of the convenij p tional pressure-strain φij , where Πij = Dij −φij is the velocity–pressure-gradient correlation. 4 [2] Second-moment turbulence closure modelling 65 The ER approach to second-moment closure requires solution up to six elliptic relaxation equations, one for each component of the turbulent stress ui uj . Despite the demonstrated success in reproducing several types of ﬂows, the unavoidable additional computational eﬀort, together with some problems experienced in deﬁning and implementing wall boundary conditions for each fij , have limited wider testing of this model. While still utilizing the elliptic relaxation concept within the second-moment closure framework a signiﬁcant simpliﬁcation can be achieved by solving a single elliptic equation. It is recalled that the elliptic relaxation equation essentially accounts for geometrical eﬀects (wall conﬁguration and topology) and provides a continuous modiﬁcation of the homogeneous pressure-strain process as the wall is approached to satisfy the wall conditions. Hence, it should be possible to deﬁne this transition by a single variable. Manceau and Hanjali´ (2000b) have proposed a c ‘blending model’ which entails the solution of the elliptic relaxation equation for a blending function α, 1 L2 ∇2 α − α = − , k (3.10) with boundary conditions kα|w = 0 and kα|∞ → 1. This blending function is then used to provide a transition between the homogeneous (far-from-the-wall) and inhomogeneous (near-wall) pressure-strain model φ∗ = (1 − kα)φw + kαφh ij ij ij (3.11) and between the isotropic and near-wall nonisotropic stress dissipation rate εij = (1 − Akα) ui uj 2 ε + Akα εδij . k 3 (3.12) Here φh can be any known homogeneous model of φij , whereas the inhomoij geneous part (satisfying the wall constraints) is deﬁned as φw = −5 ij ε 1 ui uk nj nk + uj uk ni nk − uk ul nk nl (ni nj − δij ) . k 2 ∇α . ||∇α|| (3.13) The unit normal vectors are obtained from n= (3.14) The testing of this model in a plane channel and in ﬂow over a backwardfacing step showed very good agreement with the DNS data, though further validation in more complex ﬂows still remains to be undertaken. 3.2 Some modiﬁcations to the ε equation The rudimentary form of the ε equation is used in practically all industrial CFD codes to provide the sink term in the kinetic energy equation, and to 66 Hanjali´ and Jakirli´ c c supply the turbulence scale by which the turbulent diﬀusion is modelled. This equation has too simple a form for such a task for any turbulent ﬂow signiﬁcantly out of local energy equilibrium. The inability of EVMs to generate accurately the normal stresses is the root of the failure of such schemes to model the production of ε due to normal straining, if these are signiﬁcant (ﬂow acceleration and deceleration, ﬂow recovery after reattachment, etc.) The use of second-moment closure with the same ε equation at least obviates this problem, because the normal-stress components are then (hopefully) well reproduced. However, any extra strain rates or departure from equilibrium may require additional modiﬁcations to the ε equation. There have been several proposals to modify and upgrade the simple ε equation, though most of them were aimed at curing a speciﬁc deﬁciency related to a particular ﬂow class. Speciﬁc tuning usually resulted in achieving the aim, but in most cases subsequent validation in some other ﬂows produced unwanted eﬀects. A case in point is the modiﬁcation which emerged from the application of the Renormalization Group Theory (RNG). The outcome of this approach is the insertion of an extra term in the ε equation: SRN G = η(1 − η/η0 ) Pε , 1 + βη 3 k (3.15) where η ≡ Sk/ε is the relative strain parameter (in fact the ratio of the turbulence time scale and the scale of mean rate of strain) and S ≡ Sij Sji is the strain-rate modulus. This term aimed to distinguish large from small strain rates by increasing or decreasing the source of dissipation – depending on whether the strain parameter is larger or smaller, respectively, than what is believed to be a typical value for homogeneous equilibrium shear ﬂow, η0 = 4.8. While the term indeed improved the predictions in the recirculation zone, as well as in the stagnation region, it proved to be harmful in many other ﬂow cases, particularly when the normal straining is dominant. The reason is that the term does not distinguish the sign of the strain Sij , and produces the same eﬀect for the same strain intensity irrespective of its sign, i.e., whether the ﬂow is subjected to acceleration or deceleration, compression or expansion (e.g. in reciprocating engines). Figure 2 shows the eﬀect of including the RNG term (3.15) with the LRR stress model to ﬂow in an axisymmetric contraction and expansion at roughly the same strain η = 62.2 and 86.6 respectively, compared with DNS results of Lee (1985) (Hanjali´ 1996). The standard model c gives poor results. The RNG modiﬁcation indeed improves the ﬂow predictions in the expansion, but actually leads to worse agreement in the contraction. Hence, the use of such a remedy, particularly by inexperienced users, can yield adverse, instead of beneﬁcial, eﬀects. The strain-rate modulus S and the analogue mean-vorticity modulus W = Wij Wij have been used by some authors to deﬁne variable coeﬃcients in the ε equation (primarily Cε1 ), aimed at accounting for nonequilibrium eﬀects and dissipation anisotropy (e.g. Speziale and Gatski 1997). Such modiﬁcations suﬀer from the same deﬁciency as RNG, [2] Second-moment turbulence closure modelling 67 Figure 2: Predictions of kinetic energy evolution in an axisymmetric contraction (left) and expansion (right) with the basic LRR model and RNG modiﬁcations in the ε equation (Hanjali´ 1996). c because of the inability to distinguish the sign of mean strain rate and vorticity. A sounder approach is to use turbulent stress invariants or other turbulence parameters to modify the coeﬃcients, or to deﬁne extra source terms. Craft and Launder (1991) proposed to modify Cε2 as Cε2 = 0 Cε2 1/2 , (3.16) (1 + 0.65A A2 ) 0 (which also requires modifying Cε1 ). This modiﬁcation was shown to perform well in several ﬂows, but within the framework of a nonlinear pressure-strain model; its use in connection with conventional models may require additional tuning and validation. Simpler remedies to improve the prediction of complex industrial ﬂows without having to redeﬁne the rest of the model, are also possible: for example, modiﬁcations that involve introducing two extra terms in the standard ε equation, SΩ and Sl , discussed below. Both terms have local eﬀects only in the ﬂow regions where a remedy is needed, while doing no harm in ﬂows where the con- 68 Hanjali´ and Jakirli´ c c ventional dissipation equation serves well. The term SΩ , deﬁned as: SΩ = Cε5 kWij Wij , (3.17) was introduced long ago by Hanjali´ and Launder (1980) to enhance the efc fects of irrotational straining on the production of ε (note that the coeﬃcient Cε5 = 0.1 while Cε1 is now 2.6, instead of the conventional 1.44!). Although not helpful in dealing with streamline curvature, as originally intended, the term proved to be helpful in reproducing ﬂows with very strong adverse and favourable pressure gradients (Hanjali´ et al. 1997). It is noted that Shimoc mura’s (1993) proposal to account for system rotation has the same form, except that one of the vorticity vectors is replaced by the intrinsic vorticity, as discussed later. The second new term Sl is deﬁned as Sl = max 1 ∂l Cl ∂xn 2 −1 1 ∂l Cl ∂xn 2 ;0 ε˜ ε A, k (3.18) where l = k 3/2 /ε is the turbulence length scale and Cl = 2.5. This term resembles the so-called Yap correction, but has a coordinate-invariant form. It has been introduced to compensate for excessive growth of the length scale, and it has proved beneﬁcial in the prediction of separating and reattaching ﬂows. The term has indeed a local character, as demonstrated for example, in a ﬂow behind a backward-facing step, where the standard IP, QI or SSG models produce anomalous behaviour of the streamline pattern around ﬂow reattachment (Hanjali´ 1996). c 4 Potential of Diﬀerential Second-Moment Closures for Modelling Complex Flow Phenomena As mentioned earlier, the major advantage of second-moment closure is that the Reynolds stress ui uj need not be modelled directly, but is provided from the solution of a model diﬀerential transport equation. Other beneﬁts can be deduced from the inspection of the exact transport equation for ui uj . First, it contains several more terms. While this poses a new challenge (a need for modelling) and brings additional uncertainty (unavoidable new empirical coeﬃcients), it enables the exact treatment of several important turbulent interactions and, thus, it also enables more subtle features of turbulent ﬂows to be captured. Other terms can be modelled in a diﬀerent, more appropriate way than with EVMs because of the availability of ui uj , rather than simply k. Some of these features and terms in the equation are considered below focussing on the potential of DSM, as compared with EVM, to reproduce the physics of the most common ‘complexities’ in turbulent ﬂows. [2] Second-moment turbulence closure modelling 69 Stress production. The ﬁrst beneﬁt comes from the possibility of treating the stress generation in its exact form. Turbulent stresses are generated at the expense of mean ﬂow energy by mean ﬂow deformation, Pij , and by body forces (buoyancy, electromagnetic, etc), Gij , and rotation, Rij . Second moments are explicitly present in all generation terms: Pij and Rij contain ui uj and Gij = fi uj is usually replaced by βgi θuj for buoyancy generation, (θ is the ﬂuctuating temperature, or concentration, and β is the corresponding expansion coeﬃcient). The advantage of obtaining second moments from their own transport equations, instead of from an eddy viscosity/diﬀusivity model becomes particularly obvious when comparing, e.g., the production of kinetic energy in both EVM and DSM in a two-dimensional ﬂow: P EV M P DSM ∂U1 2 ∂U2 2 ∂U1 ∂U2 + 2µt + µt + ∂x1 ∂x2 ∂x2 ∂x1 ∂U1 ∂U2 ∂U1 ∂U2 = −u2 − u2 − u1 u2 + . 1 2 ∂x1 ∂x2 ∂x2 ∂x1 = 2µt 2 (4.1) (4.2) In thin shear ﬂows (dominated by simple shear ∂U1 /∂x2 ) both expressions give a similar value of P, because the eﬀect of normal straining is negligible. In 1 2 2 more complex ﬂows, ∂U1 , ∂U2 , ∂U1 , . . . can have signiﬁcant values and diﬀer∂x ∂x ∂x DSM and P EVM may be very diﬀerent. This becomes more ent signs. Hence, P evident (and more important) in ﬂows with a complex strain-rate ﬁeld, i.e. when the ‘extra strain rate’ originates from streamline curvature, ﬂow skewing, lateral divergence, bulk dilation. Unlike EVMs, DSMs account exactly for stress production by each component of the strain rate. Even a small ‘extra strain rate’ can have a signiﬁcant eﬀect on stress production. For example, in 2 a thin shear ﬂow with a mild curvature, such as in a ﬂow over an airfoil, ∂U1 ∂x is much less than ∂U1 ∂x2 , but u2 is much greater than u2 near the airfoil surface, 1 2 1 2 hence both terms in P12 = − u2 ∂U2 + u2 ∂U1 are of importance (Bradshaw 2 ∂x 1 ∂x et al. 1981). The problem becomes even more serious in ﬂows fully dominated by pure (normal) strain, because the expression for P EVM is always positive and cannot diﬀerentiate the sign of the strain rate, i.e. it cannot distinguish dilatation from compression, or ﬂuid acceleration from deceleration. The exact contribution to P DSM caused by normal straining is the interaction between speciﬁc components of turbulent normal stress (positive quantities) with corresponding components of the normal strain-rate that can be either negative or positive. A simple example is a ﬂow in a nozzle and diﬀuser of the same shape (of equal contraction and expansion ratio), where, depending on the inﬂow stress anisotropy (for the same k), very diﬀerent ﬂow development may be expected in the two cases (see, for example, the DNS of Lee and Reynolds 1985, Hanjali´ 1996). The standard k-ε EVM yields the same results for the same initial c level of k and ε for both the compression and expansion. Other, industrially more relevant examples are stagnation regions, ﬂow impingement on a solid 70 Hanjali´ and Jakirli´ c c Figure 3: A schematic of stress component interactions in a thin shear ﬂow. surface, boundary layer recovery after reattachment, recirculating ﬂows, the central region in the cylinder of a reciprocating engine where the ﬂow is subjected to a cyclic compression and expansion, etc. In all these and other cases the computations with a linear EVM can never be accurate (e.g. Hadˇi´ et al. zc 2001). Stress interaction and anisotropy. Another important turbulence feature that can be reproduced only by models based on the stress-transport equation, is the stress interaction. In ﬂows with a preferential orientation of the velocity ﬁeld, the dominant strain rate component feeds energy into selective stress component(s). Pressure ﬂuctuations redistribute a part of the largest stress into other components (and also reduce the shear stress) making turbulence more isotropic. An illustration of stress interaction and the role of pressure ﬂuctuations is given in Figure 3 for a simple thin shear ﬂow, which shows a general ﬂow chart of turbulence energy (stress) components. The exact treatment of the stress generation and the possibility of accounting for stress-component interactions gives a better prospect for modelling the important turbulence parameter, the stress anisotropy, which governs, to a large extent, the wall heat and mass transfer. This is particularly the case in regions with either small or no wall shear, such as around impingement, separation or reattachment, where the transport of mean momentum and heat transfer do not show any correlation or analogy. In these regions, the heat and mass transport are governed by the wall-normal turbulent stress component, and its accurate prediction is a crucial prerequisite for computing accurately the transport phenomena at a solid surface. Streamline curvature. Most complex ﬂows involve strong streamline curvature, which may occur locally even if the ﬂow boundaries are not curved (e.g. the curved shear layer and separation bubble behind a step, or on a plane wall due to adverse pressure gradient), or the whole ﬂow can be curved by the [2] Second-moment turbulence closure modelling 71 Figure 4: Illustration of eﬀects of streamline curvature: ﬂow in a curved pipe. imposed boundaries. In the latter case, the centrifugal force aﬀects the ﬂow and turbulence in much the same way as the Coriolis force in rotating ﬂows (Ishigaki 1996). The streamline curvature generates an extra strain rate, which can exert a signiﬁcant eﬀect on stress production. Streamline curvature attenuates the turbulence when the mean ﬂow angular momentum increases with curvature radius (e.g. in ﬂow over a convex surface – stabilizing curvature), whereas it ampliﬁes the turbulence in the opposite situation (e.g. over a concave surface – destabilizing curvature). The criterion can be related to the direction of the ﬂuid rotation vector or the vorticity ωk = ijk Wji (which deﬁnes the sign of the shear stress) with respect to the mean angular momentum vector: if these two vectors have the same direction, the turbulence will be attenuated, if opposite directions, the turbulence will be enhanced, Figure 4. Eﬀects of streamline curvature are directly related to the stress production and stress interactions. Linear EVMs fail to account for curvature eﬀects in most turbulent ﬂows because of their inability to reproduce normal stresses which appear explicitly in the production term Pij . Of course, one can model the eﬀects of streamline curvature with simple models by ad hoc corrections. This has been done in EVM schemes, e.g., by expressing the coeﬃcients in the sink term of the dissipation equation in terms of a ‘curvature Richardson number’ Rit = (k/εR)2 Uθ ∂(RUθ )/∂R, where R is the local radius of the streamline curvature, and Uθ is the resultant mean velocity. Although modiﬁcation of the scale-determining (ε equation) may be needed to deal more appropriately with streamline curvature, such a remedy cannot compensate for the deﬁciency of the k equation as a replacement for the eﬀects of normal stress. Note that Rit changes its sign and magnitude in accordance with the sign of the curvature, thus producing stabilizing or destabilizing eﬀects. However, DSMs capture these eﬀects via exact production terms in the stress equations, which show a selective sensitivity to streamline curvature. A highly curved shear layer may serve as an example of a ﬂow where streamline curvature has dominant eﬀect. 72 Hanjali´ and Jakirli´ c c Figure 5: Illustration of system rotation: a rotating plane channel. Computations by Gibson and Rodi (1981) by EVM and DSM showed that the basic DSM model reproduces the eﬀect much better than the EVM k-ε model. Illustrative examples of computations of ﬂows with bulk curvature are the ﬂows in U-bends and S-bends in circular or square-sectioned pipes, reported by Iacovides and Launder (1985), Anwer et al. (1989), Iacovides et al. (1996), Luo and Lakshminarayama (1997), and others. All these computations underline the superiority of DSM or ASM schemes over a linear EVM. System rotation. The next ﬂow feature which can be better captured by DSMs is system rotation. The bulk-ﬂow rotation aﬀects both the mean ﬂow and turbulence by the action of the Coriolis force FiC = −2ρΩj Uk ijk , where Ωj is the system angular velocity. System rotation directly inﬂuences the intensity of stress components through stress redistribution. It is also known that rotation aﬀects the turbulence scales. As mentioned earlier, the stress transport equation contains the exact rotational production term Rij . Because the Coriolis force acts perpendicular to the velocity vector Uk , it has no direct inﬂuence on the k-budget (Rii = 0), but redistributes the stress among components and thus modiﬁes the net production of individual stress components. Rotation also causes a decrease in dissipation, even in isotropic turbulence (see the DNS by Bardina et al. 1985). The eﬀects of rotation are illustrated in a plane channel rotating about an axis perpendicular to the main ﬂow direction with an angular velocity [2] Second-moment turbulence closure modelling 73 vector (‘system vorticity’) Ωj (here Ω3 ), Figure 5. Note that Ωj is aligned with the vector of the mean shear vorticity ωk = ijk Wji (here ω3 = W21 − 1 2 W12 = − ∂U2 , since ∂U1 = 0). The ﬂuctuating Coriolis force ampliﬁes the ∂x ∂x turbulence on the pressure side, destabilizing the ﬂow, while near the suction side the tendency is opposite. Comparison of the directions of the angular velocity and shear vorticity vectors gives a convenient indication of the eﬀects on turbulence. If Ω3 and ω3 have the same direction (Ω3 ↑↑ ω3 ), rotation attenuates the turbulence (stabilizing eﬀect, ‘suction side’). In contrast, if Ω3 and ω3 have opposite directions (Ω3 ↑↓ ω3 ), rotation will amplify the turbulence (destabilizing eﬀect, ‘pressure side’). At high rotation rates, the stabilizing eﬀect may completely damp the turbulence on the suction side causing local laminarization. The eﬀect of rotation is often deﬁned in terms of the local Rossby number, deﬁned as the ratio of the mean shear vorticity to 1 /∂x the system vorticity Ro = ωl /Ωl (here Ro = − ∂U2Ω3 2 ), or its reciprocal S = 1/Ro. The sign of Ro or S indicates whether the eﬀect on turbulence will be amplifying (−) or attenuating (+). It is noted that for a general classiﬁcation, the bulk rotation number N = 2Ω h/Ub (the inverse of the bulk Rossby number – not to be confused with the local one deﬁned above), is often used, where h is the channel half width and Ub is the bulk velocity. The bulk rotation number serves only to indicate the intensity of rotation, not a stability criterion, nor as a parameter for modifying the turbulence model. The local Rossby number Ro (or S) has served in the past to modify the simple k-ε model by expressing one of the coeﬃcients in the ε equation in terms of Ro. However, the stress transport equation accounts exactly for the eﬀects of rotation through the exact rotational generation term Rij (stress generation by ﬂuctuating Coriolis force) in equation (1). The table below lists the components of Rij for the plane channel example (note, Ω3 = Ω). ij 11 22 33 12 dU1 1 Pij −2u1 u2 dx2 0 0 −u2 dU2 2 dx Rij 4Ωu1 u2 −4Ωu1 u2 0 −2Ω(u2 − u2 ) 1 2 In addition to the exact Rij term in the DSM, system rotation aﬀects also the stress redistribution induced by ﬂuctuating pressure. Hence, the eﬀect of rotation should be accounted for in the model of the pressure-strain term. A simple way to do this is to replace Pij in φij,2 (see below) by the total stress generation Pij + Rij . However, in order to ensure the material frame indiﬀerence, Pij should be replaced by Pij + 1 Rij (e.g. Launder et al. 1987). So 2 far, no convincing proof has been provided as to which of the two modiﬁcations performs better, but both versions have led to substantial improvement in predicting rotating ﬂows. Because Rii = 0, the basic k-ε model, without modiﬁcations, cannot mimic the rotation eﬀects on turbulence. Modiﬁcations are introduced usually via additional term(s) in the ε-equation, analogous to the modiﬁcations for curvature 74 Hanjali´ and Jakirli´ c c eﬀects, in terms of rotation Richardson number Rir = −2Ω(k/ε)2 (∂U1 /∂x2 ), or by making the coeﬃcient in the eddy-viscosity a function of a rotation parameter. In fact, the ε-equation should, in principle, be modiﬁed to account for the eﬀects of rotation on turbulence scale even in conjunction with DSM models. Bardina et al. (1987) suggested an additional term: −CΩ ε 1 Xij Xij 2 1/2 , (4.3) where CΩ = 0.15 and Xij = Wij + kji Ωk is the ‘intrinsic’ mean vorticity tensor.5 Shimomura (1993) proposed a slightly diﬀerent formulation, −CΩ k ωl Ωl , which improved predictions of ﬂow in a rotating plane channel (Jakirli´ et al. c 1998). It is interesting to note that this term resembles closely that proposed by Hanjali´ and Launder (1980) for nonrotating ﬂows to enhance the eﬀect of c irrotational straining on ε (energy transfer through the spectrum, see below), Cε3 k ωl ωl . This term can be combined with that of Shimomura to yield a joint term which would be eﬀective in both nonrotating and rotating ﬂows, i.e. Cε3 kωl (ωl − CΩ /Cε3 Ωl ). Successful DSM computation of rotating channel ﬂows with no modiﬁcation of ε-equation were reported by Launder et al. (1987) for moderate rotation. These results indicate that the eﬀect of rotation is stronger on the stresses than on the turbulence scale, which illustrates further the advantage of DSM, as compared with EVM. The importance of integration up to the wall (using a low-Re-number model) in ﬂows where the rotation is suﬃciently strong to cause laminarization on the suction side, has been demonstrated by Jakirli´ et c al. (1998) and Pettersson and Andersson (1997), from whose work examples are drawn in [4], (see also Dutzler et al. 2000). Swirl. Swirling ﬂows can be regarded as a special case of ﬂuid rotation with the axis usually aligned with the mean ﬂow direction (longitudinal vortex) so that the Coriolis force is zero. Swirl enhances the turbulent mixing and often induces recirculation. These features are exploited in internal combustion (IC) engines and gas-turbine combustors, and for heat transfer enhancement (vortex generators). A common feature of such ﬂows is that the swirl is strong and conﬁned to short cylindrical enclosures, whose length is of the same order as the duct diameter. Another type of conﬁned swirl, usually of weak intensity, is encountered in long tubes, either imposed at the tube entrance, or selfgenerated by secondary motions as a consequence of upstream multiple tube bends (double- or S-bends in diﬀerent planes). A third kind is the swirling motion inside rotating cylinders or pipes, either developing or fully developed. 5 or, in terms of rotation (‘dual’) vectors, −CΩ ε (Xl Xl )1/2 , where Xl = 1 ωl + Ωl 2 [2] Second-moment turbulence closure modelling 75 a. b. c. Figure 6: A schematic of tangential ﬂow velocity in (a) rotating pipe ﬂow, (b) swirling pipe ﬂow and (c) in a free swirl. A special case is the spin-down ﬂow when a rotating pipe is suddenly brought to a standstill – used sometimes in studying experimentally the eﬀect of swirl in a piston-cylinder assembly related to IC engines. Unconﬁned free swirling ﬂows such as swirling jets, represent yet another category relevant to swirling burners, which possess some speciﬁc features and are known to modify substantially the ﬂow characteristics even at a very low swirl intensity. Although all these cases deal with essentially the same phenomenon – rotating ﬂuid in axisymmetric geometries – their predictions pose diﬀerent challenges. To illustrate the eﬀect of swirl on turbulence one may follow the same arguments as outlined earlier in the context of curved ﬂows and system rotation, Figure 6, i.e. by considering the directions of the bulk ﬂow rotation vector and of the ﬂuid element vorticity. In short, the turbulence will be damped in regions where the rotation vector of the ﬂuid vorticity in the ﬂow cross-section (ωz in Figure 6) has the same direction as the bulk ﬂow rotation vector Ωz , while turbulence will be enhanced in regions where these two vectors have opposite signs. For example, in a rotating pipe, the ﬂuid rotation will damp turbulence over the whole cross-section, with greatest eﬀect near the pipe wall, which may cause the ﬂow to laminarize locally, if this eﬀect is stronger than the turbulence production by shear due to the radial gradient of axial velocity. If the pipe wall is stationary, with a swirl imposed at its entrance (or generated by the sudden stopping of pipe rotation, ‘spin-down’), in the outer region close to the pipe wall the turbulence production will be enhanced because ωz has an opposite sign to Ωz . In the core region the orientation of ωz is the same as that of Ωz and turbulence will be damped. In a strong swirl the core may even be laminarized. The same arguments apply for swirling jets. EVM computations employ swirl-dependent coeﬃcients in the modelled equations, but generally with limited success. In fact, the original DSM computations did not greatly improve the prediction of a swirling jet. Note that the rotational term Rij is absent in an inertial coordinate frame and the eﬀect of swirl should be accounted for by either modifying the models of some terms, adding extra terms in model equations, or by expressing coeﬃcients in terms of swirl parameters. The simple ‘Isotropization of Production’ (IP) model of 76 Hanjali´ and Jakirli´ c c φij,2 (see below) seems to perform better than the LRR QI (quasi-isotropic) model, (Launder and Morse 1979). By recognizing that a major deﬁciency lies in φij,2 , Fu, Leschziner and Launder (1987), proposed the inclusion of convection Cij to ensure material frame indiﬀerence, (negligible eﬀects in nonswirling ﬂows): 2 2 φij,2 = −C2 Pij + Rij − P δij − Cij − C δij . (4.4) 3 3 It was found, however, that this modiﬁcation produces little eﬀect (at least in weak swirls) since the last term in equation (4.4) is smaller than other terms in the expression. Still better predictions are expected with improvements of the scale equation. Several cases of swirling ﬂows reported in the literature show obvious superiority of the DSM model. This is particularly the case for strong swirl in a combustion-chamber type of geometry (e.g. Hogg and Leschziner 1989; Jakirli´ c 1997). Similar improvements are achieved for a swirl in a long pipe (Jakirli´ c 1997). Mean pressure gradient. The transport equations for turbulent stresses and scale properties do not contain mean pressure. However, the mean pressure gradient modiﬁes the mean rate of strain and – depending on the sign – ampliﬁes or attenuates turbulence. Extreme cases are laminarization of an originally turbulent ﬂow, when dp/dx 0 (severe acceleration) and ﬂow separation when dp/dx 0 (strong deceleration). Both extreme cases represent a challenge to turbulence modelling. Turbulent ﬂows subjected to periodic variations of pressure gradient or other external conditions (pulsating and oscillating ﬂows) fall into the same category with an additional feature: a hysteresis of the turbulence ﬁeld lagging in phase behind the mean ﬂow perturbations. Linear EVMs cannot capture these features; DSMs perform generally better, though additional modiﬁcations (mainly in the scale equation, see below) are needed. Wall functions are inapplicable for specifying boundary conditions and integration up to the wall, with appropriate modiﬁcations of the model, is essential to reproduce these phenomena accurately. Predictions with a low-Renumber DSM of the turbulence evolution and decay in an oscillating ﬂow in a pipe at transitional Re numbers, displaying a visible hysteresis of the stress ﬁeld, were reported by Hanjali´ et al. (1995). An overview of the performance c of DSMs in ﬂows with diﬀerent pressure gradients involving separation is given in Hanjali´ et al. (1999). c Secondary currents and longitudinal vortices. This term refers usually to a secondary motion with longitudinal, streamwise vorticity ω1 , superimposed on the mean ﬂow in the x1 -direction. Skew-induced (pressure-driven) ω1 (Prandtl’s 1st kind of secondary ﬂows) is essentially an inviscid process, generated by the bending of existing mean vorticity. Viscous and turbulent [2] Second-moment turbulence closure modelling 77 Figure 7: Schematic of wing-tip vortices and secondary currents in conduits of noncircular cross-section. stresses cause ω1 to diﬀuse. Turbulent-stress-induced ω1 (Prandtl’s 2nd kind of secondary ﬂow), is generated by the turbulent stresses due to anisotropy of the Reynolds-stress ﬁeld. Secondary motions can arise in the form of a ‘cross-ﬂow’ such as in 3D thin shear ﬂows (ω1 ≈ ∂U3 /∂x2 ), or in the form of recirculating ‘cross-currents’ such as occur in noncircular ducts (ω1 = ∂U3 /∂x2 − ∂U2 /∂x3 ), see Figure 7. Of course, a secondary current can be imposed on the ﬂow in order to enhance mixing or heat and mass transfer, such as in the case of vortex generators. Skew-induced secondary velocity can be high, while the stress-induced currents are usually weak, though still very important for turbulent transport. The importance of turbulence in the dynamics of mean-ﬂow vorticity can be illustrated by considering the vorticity transport equation, which, for the ω1 (streamwise) component, reads: Dω1 Dt = ν ∂ 2 ω1 + ∂xk ∂xk ∂2u 2 u3 ∂x2 2 ω1 ∂U1 ∂x1 ∂2 + ω2 ∂U1 ∂U1 + ω3 ∂x2 ∂x3 vortex stretching ‘skew-induced’ ω1 (vortex bending) − + ∂2u 2 u3 ∂x2 3 + ∂x2 ∂x3 (u2 − u2 ) . 2 3 ω1 (4.5) ‘stress-induced’ generation of Skew-induced secondary motion, driven essentially by mean-ﬂow deformation and by the mean pressure ﬁeld does not require a complex turbulence model. However, stress-induced motion cannot be handled with an EVM and requires a model which can compute individual turbulent stress components (DSM or ASM). In fully developed ﬂows in ducts of non-circular cross-section the application of ASM is suﬃcient to capture the stress-induced secondary motion. Illustrations of the prediction of secondary currents in square ducts have been published inter alia by Demuren and Rodi (1984), in pipe bends by Anwer et al. (1989), and in U-ducts by Iacovides et al. (1996). See also Chapters [1, 3, 4]. Three-dimensionality. Finally, a few remarks should be added concerning the ﬂow three-dimensionality eﬀects. Even a mild three-dimensionality of the mean ﬂow produces signiﬁcant changes in turbulence structure. In strong 78 Hanjali´ and Jakirli´ c c Figure 8: Schematic of mean velocity proﬁles in ﬂow with a longitudinal vortex and in a pressure-skewed ﬂow. cross-ﬂows the eﬀects can be dramatic as, for example, in the case of a unidirectional ﬂuid stream with a superimposed longitudinal vortex, such as is produced by a vortex generator to enhance wall heat transfer. The resulting mean velocity proﬁles may look very skewed, as shown in Figure 8, which is diﬃcult to reproduce by simple eddy-viscosity or similar models. Other examples of relatively simple 3D boundary layers include wing-body (or blade-rotor) junctions, ﬂows encountering laterally moving walls imposing a transverse shear, such as in the stator-rotor assembly in turbomachinery. In fully 3D separating ﬂows the problem is even more challenging. Current turbulence models have been developed on the basis of our knowledge of 2D ﬂows. Plausible extensions to the third dimension do not always yield satisfactory results. This is particularly the case for linear eddy-viscosity models. Even in a simple 3D boundary layer, the eddy viscosity is not isotropic, as discussed earlier, i.e.: u1 u2 u3 u2 = ; ∂U1 /∂x2 ∂U3 /∂x2 (4.6) This ﬁnding illustrates that complex ﬂows require turbulence models of a higher order than the EVM. Further illustrations of the inadequacy of the eddy viscosity concept in 3D ﬂows can be found in Hanjali´ (1994) and elsewhere. c 5 5.1 Advanced Models for Near-wall and Low Re-number Flows The wall-function approach and its deﬁciency All industrial CFD codes use ‘wall functions’ to serve as the wall boundary conditions. The viscosity-aﬀected near-wall region is bridged by placing the [2] Second-moment turbulence closure modelling 79 ﬁrst grid point outside the viscous layer in the fully turbulent region: such high-Reynolds-number turbulence closures can thus be used for ﬂows at high bulk Reynolds numbers. This simpliﬁes to a great degree both the modelling and computational tasks. By obviating the need to resolve steep gradients very close to the wall, a relatively coarse numerical grid often suﬃces for reaching grid-independent solutions. However, the wall functions have been derived on the basis of wall scaling in attached boundary layers in local turbulent energy equilibrium. Their use in nonequilibrium ﬂows is therefore considered inappropriate, particularly in separating and recirculating ﬂows, around reattachment, in strong pressure gradients and in rotating ﬂows. Figure 9 shows two examples of the deviation of the mean velocity proﬁle from the conventional logarithmic law which serves as a basis for most wall functions for the mean velocity. The upper three proﬁles correspond to three locations in a ﬂow behind a backwardfacing step in the recovery zone downstream from reattachment. Even greater deviation is exhibited in the recirculation zone. The lower ﬁgure shows axial and tangential velocity in a swirl generated by cylinder rotation some time after the rotation was stopped (spin-down). Good agreement with experiments was achieved only with the application of a low-Re-number DSM integrated up to the wall. Various modiﬁcations have been proposed to improve and extend the validity of wall functions to non-equilibrium and separating ﬂows, but none of the proposals showed general improvement. The incorporation of pressure gradient is most straightforward and can easily be done by extending the near-wall ﬂow analysis, e.g. Ciofalo and Collins (1989), Kiel and Vieth (1995), Kim and Choudhury (1995). Such modiﬁcations generally lead to some improvement of attached thin shear wall ﬂows with pressure gradient (with convection still being neglected), but their validity is conﬁned only to such situations. A more general two-layer approach, based on splitting the wall layer in viscous and nonviscous parts with assumed variation of shear stress and kinetic energy in each layer, was earlier proposed by Chieng and Launder (1980) and Johnson and Launder (1982). The assumed proﬁles for uv and k enable the stress production and dissipation over the ﬁrst control volume next to a wall to be integrated separately, instead of assuming wall-equilibrium values. However, despite some improvement of wall friction and heat transfer behind a back step and sudden pipe expansion, the approach still has serious deﬁciencies. More general wall functions that would be applicable to various complex ﬂows (with separation, stagnation, laminar-to-turbulent transition, buoyancy and other eﬀects) are still awaited. Development of such wall functions has recently been taken up again by the UMIST group, and the initial results are encouraging (Craft et al. 2001, 2002). 80 Hanjali´ and Jakirli´ c c Figure 9: Mean velocity proﬁles at selected locations in the recovery region behind a backward-facing step ﬂow (Hanjali´ and Jakirli´ 1998) (above) and c c proﬁles of axial and tangential velocity in a spin-down ﬂow in an engine cylinder at the beginning of compression (below), (Jakirli´ et al. 2000). c 5.2 Models with near-wall and low-Re-number modiﬁcations Integration up to the wall is a more generally applicable strategy than adopting wall functions. This approach requires ﬁrst the introduction of substantial modiﬁcations to the turbulence models in order to account for complex wall eﬀects, primarily for viscosity (‘low-Re-number models’), but also for nonviscous ﬂow ‘blocking’ and pressure-reﬂection by a solid wall. This in turn requires a much ﬁner grid resolution in and around the viscosity-aﬀected wall sublayer, and, consequently, increases demands for computational resources, with often formidable requirements on the numerical solver to ensure convergent solutions. Low-Re-number models are at present available both at the EVM and DSM level. While primarily developed for treating the near-wall [2] Second-moment turbulence closure modelling 81 viscous region, some of the models are reasonably successful also in predicting turbulence transition. In fact, general low-Re-number models, which can distinguish appropriately between the non-viscous blocking and viscous wall eﬀects are indispensable for predicting the laminar-to-turbulent and reverse transition (at least the forms of transition which can be handled within the framework of the Reynolds averaging approach, such as by-pass transition, or the revival of inactive background turbulence and its laminarization). By-pass transition is the subject of [17] and [18]. A number of proposals for modifying DSMs to account for low-Reynoldsnumber and wall-proximity eﬀects can be found in the literature. These modiﬁcations are based on a reference high-Re-number DSM which serves as the asymptotic model to which the modiﬁcations should reduce for suﬃciently high Reynolds number and at a suﬃcient distance form a solid wall. Hanjali´ and Launder (1976), Launder and Shima (1989), Hanjali´ and Jakirli´ c c c (1993), Shima (1993), Hanjali´ et al. (1995) base their modiﬁcations on the c basic DSM with linear pressure-strain models, in which the coeﬃcients are deﬁned as functions of turbulent Reynolds number and invariant turbulence parameters. Earlier models also use the distance from a solid wall. More recent models are based on DNS data and term-by-term modelling, which ensures model realizability, compliance with the near-wall stress two-component limit, as well as with the limit of vanishing turbulence Reynolds number. Launder and Tselepidakis (1991), Craft and Launder (1996), Craft (1997) use the cubic pressure-strain model in which the coeﬃcients were determined by imposing, in addition to basic constraints discussed earlier, the two-component limit. A larger number of terms with associated coeﬃcients reduces the need for introducing functional dependence and additional turbulence parameters. As mentioned earlier, Durbin (1991, 1993) uses elliptic relaxation to account for non-viscous blockage eﬀect of a solid wall, whereas a switch of time and length scale from the high-Re-number energy-containing scales to Kolmogorov scales (lower bound), when the latter become dominant, accounts for viscosity eﬀects6 . All models require a very ﬁne numerical mesh within the viscosity aﬀected wall sub-layer, so that the computation becomes time-consuming and impractical for more complex ﬂow cases. Durbin’s elliptic relaxation approach seems to be somewhat less demanding in this respect, but at present it cannot be used to predict transitional phenomena. In principle, all modiﬁcations involve the following: • Inclusion of viscous diﬀusion in all equations; Jakirli´ (1997, p. 44) has noted that the current ER proposal appears to be inconsistent: c the time scale and the length-scale switching (and the introduction of viscous eﬀects in each) occur at very diﬀerent distances from a wall: e.g. in a plane channel ﬂow at Rem = 14000 at y + ≈ 6 and at y + ≈ 150, respectively. Moreover, the length scale switch implies that the Kolmogorov scale (and, consequently, the viscous eﬀect) prevails over almost the whole ﬂow width for this Reynolds number. 6 82 • Provision of a non-isotropic model for εij ; Hanjali´ and Jakirli´ c c • Addition of further terms to the ε-equation (supposedly to model Pε3 ) • Replacement of certain constant coeﬃcient by functions of turbulent Re number, Ret = k 2 /(νε), dimensionless wall distance and/or other turbulence parameters. 5.2.1 A low-Re-number DSM An example of a low-Re-number second-moment closure (DSM) is the model proposed by Hanjali´ et al. (1995) (see also Hanjali´ et al. 1997, Jakirli´ 1997, c c c Hadˇi´ 1998). This model has been used with reasonable success in a number zc of 2D and some 3D high- and low-Re-number wall ﬂows including cases of severe acceleration (laminarizing 2D and 3D turbulent boundary layers), bypass and separation-induced laminar-to-turbulent transition, oscillating ﬂows at transitional and high Re numbers, rotating and separating ﬂows. Some examples will be shown in the next subsection. The model is based on the basic DSM (Section 2) in which a low-Re-number version of the ε equation is used. The coeﬃcients in the ui uj equation are expressed as functions of Ret and invariants of the stress and dissipation rate tensors to account for the viscous and inviscid wall eﬀect respectively. The model satisﬁes the two-component and vanishing Reynolds number limits, thus enabling integration up to the wall. Viscosity has a scalar character (it dampens all stress components and is independent of the wall distance and its topology), and it is only indirectly related to the wall presence via no-slip conditions. Hence, its eﬀect can be conveniently accounted for through the turbulent Reynolds number Ret = k 2 /(νε). This should be formulated in a general manner to be applicable both close to a wall and away from it. Inviscid eﬀects are basically dependent on the distance from a solid wall and its orientation, as seen from the Poisson equation for ﬂuctuating pressure (‘Stokes term’). This term accounts for the wall blockage and pressure reﬂection. However, the DNS data for a plane channel show that this term decays fast with the wall distance and becomes insigniﬁcant outside the viscous layer. Yet, a notable diﬀerence in the stress anisotropy between a homogeneous shear ﬂow and an equilibrium wall boundary layer for comparable shear intensity shows that the eﬀect of the wall’s presence extends much further from the wall into the log-layer. This indicates an indirect wall eﬀect through the strong inhomogeneity of the mean shear rate, a fact that is ignored by almost all available pressure-strain models7 . In view of above discussion, the use of wall distance through the function fw and wall orientation represented by the unit normal vector ni in the adopted 7 An exception is the model of Craft and Launder (1996), see [3]. [2] Second-moment turbulence closure modelling 83 models for φw seems reasonable, despite some opposing views in the literature. ij However, these modiﬁcations, introduced for and tuned in high-Re-number wall attached ﬂows, cannot account for inviscid eﬀects closer to a wall (in the ‘buﬀer’ and viscous layer). Wall impermeability imposes a blocking on the ﬂuid velocity and its ﬂuctuations in the normal direction, causing the stress ﬁeld to be strongly non-isotropic. This fact has been exploited by Hanjali´ et al. (1995) c by introducing, in addition to Ret , invariants of both the turbulent-stress and dissipation-rate anisotropy, aij = ui uj /k − 2/3δij and eij = εij /ε − 2/3δij , respectively, A2 , A3 , E2 and E3 , as parameters in the coeﬃcients. This enables one to account separately for the wall eﬀect on the anisotropy of the stressbearing and dissipative scales, shown by the DNS data to be notably diﬀerent (Hanjali´ et al. 1997, 1999), Figure 10a. The sensitivity of stress invariants to c pressure gradient is illustrated in Figure 10b, where Lumley’s two-component (‘ﬂatness’) parameter A is plotted for boundary layers in zero, favourable and adverse pressure gradients. Also, the predictions of A with the here presented low-Re-number second-moment closure model are shown. Based on the above arguments, the following modiﬁcations were introduced: Stress transport equation: φij : Linear models are adopted for the slow, rapid and wall terms, equations (2.10), (2.11), (2.14) and (2.15), in which the coeﬃcients are deﬁned as follows: √ C = 2.5AF 1/4 f ; F = min{0.6; A2 } C1 = C + AE 2 ; f = min C2 = 0.8A1/2 ; where 9 A = 1− (A2 −A3 ); 8 9 E = 1 − (E2 − E3 ); 8 A2 = aij aji ; E2 = eij eji ; A3 = aij ajk aki ; E3 = eij ejk eki ; aij = eij = ui uj 2 − δij k 3 εij 2 − δij . ε 3 Ret 150 3/2 ;1 ; fw = min k 3/2 ; 1.4 2.5εxn w C2 = min(A; 0.3), w C1 = max(1−0.7C; 0.3); εij , the stress dissipation rate model: 2 εij = fs ε∗ + (1 − fs ) δij ε ij 3 ε [ui uj + (ui uk nj nk + uj uk ni nk + uk ul nk nl ni nj )fd ] ε∗ = ij u u k 1 + 3 pk q np nq fd 2 √ fs = 1 − AE 2 ; fd = (1 + 0.1Ret )−1 . 84 Hanjali´ and Jakirli´ c c Figure 10: Lumley’s two-component (’ﬂatness’) parameter for turbulent stress (A) and its dissipation rate (E) in the recirculation region behind a backward facing step (above); Stress ﬂatness parameter in boundary layers in zero, favourable and adverse pressure gradients (below). ε equation: • Equation (2.18) for ε is modiﬁed to the form: Dε Dt = ∂ ∂xk k ∂ε νδkl + Cε uk ul ε ∂xl ε ∂Uk − Cε2 fε ε ˜ ∂xk k ∂ 2 Ui ∂ 2 Ui ν k +Cε ν uj uk + SΩ + Sl , ε ∂xj ∂xl ∂xj ∂xl + Cε1 P + Cε3 G + Cε4 k where fε = 1 − Ret Cε2 − 1.4 exp − Cε2 6 2 (5.1) , ε = ε − 2ν ˜ ∂k 1/2 ∂xn , and SΩ and Sl have been deﬁned earlier, equations (3.17) and (3.18). [2] Second-moment turbulence closure modelling 85 6 Some Illustration of DSM Performance Many complex ﬂows contain several ‘types’ of strain rate and it is not easy to distinguish their individual eﬀects on turbulence. Moreover, diﬀerent types of strain or other eﬀects may dominate diﬀerent regions so that improvements in one region can lead to a deterioration in others. Improvements can often be achieved with diﬀerent remedies and it is not always clear which modiﬁcations have a better physical meaning. Some illustrations of the eﬀects of various model modiﬁcations are provided in Figure 11 for the case of a turbulent jet issuing from a circular tube and impinging normally on a plane wall. In this ﬂow the eﬀect of mean pressure is dominant so that the inﬂuence of the turbulence model on the predicted mean velocity ﬁeld is not strong, at least in the stagnation region, Figure 11a. However, the predicted turbulence depends greatly on the model applied, as shown in Figure 11b, where the performance of model variants is shown. The standard k-ε model yields a far too high kinetic energy, because of poor modelling of the normal stress production, as discussed earlier. The next three curves illustrate the eﬀect of the model of the pressure-strain term in the various DSMs. The basic DSM (BDSM) is that of Launder, Reece and Rodi (1975) (LRR-IP) with Gibson and Launder (1978) wall-echo correction (hereafter denoted LRRG), performs better, though still not satisfactorily owing to the inadequacy of the wallecho term for impinging ﬂows. Indeed, better results are obtained by simply omitting the wall-echo term. Further improvement is achieved when using the pressure-strain model of Speziale, Sarkar and Gatski (1991), denoted as SSG, which contains no extra wall-echo term. The application of the cubic model of Craft and Launder (1991) was reported to perform best (not shown here), but at the expense of greater complexity. While the above discussion clearly demonstrates the importance of the pressure-strain model, it would be wrong to conclude that this is the sole cause of unsatisfactory performance. The scale equation (here ε) in its simplest form is clearly inadequate to model complex ﬂows with extra strain rates. The eﬀects of three possible modiﬁcations, each involving an extra term, are illustrated by the last three curves in Figure 11b, showing further possibilities for improving the predictions. Accounting for the eﬀect of irrotational strain (term SΩ ) together with the control of length scale in the near-wall region, brings the results almost into accord with the experiments. We show now a series of examples of external and internal wall-bounded ﬂows dominated by various types of strain rates. Because the stress interaction plays a dominant role, most of these ﬂows cannot be accurately predicted by linear EVMs. Furthermore, all ﬂows considered are far from energy equilibrium. As discussed in Section 5.1. the use of conventional wall-functions for deﬁning wall-boundary conditions in such ﬂows is inappropriate, and the solutions presented here have been obtained by integration of the model equations up to the wall. 86 Hanjali´ and Jakirli´ c c Figure 11: Predicted mean velocity (above) and turbulence kinetic energy (below) at r/D = 0.5 for an axisymmetric jet impinging on a plane wall, obtained with diﬀerent models (Basara et al. 1997). The ﬁrst example is a boundary layer on a ﬂat plate developing in an increasingly adverse pressure gradient. It is known that in incompressible ﬂow the pressure gradient aﬀects turbulence indirectly through the modulation of the mean strain. The strongest eﬀects occur in the near-wall region permeating even through the viscous sublayer and invalidating the equilibrium inner wall scaling (Uτ , ν/Uτ ) for turbulence properties. Diﬀerent stress components respond at diﬀerent rates to stress production, redistribution and turbulent transport, modifying strongly the stress anisotropy and consequently the mean ﬂow ﬁeld. In contrast to a favourable pressure gradient, the positive dP/dx shifts the stress anisotropy maximum away from the wall. All these eﬀects can be reproduced only with a DSM, and only if the integration is performed up to the wall, using a model with adequate modiﬁcations for near wall and viscosity eﬀects. Figure 12 illustrates basic features of the turbulent stress ﬁeld and consequent eﬀects on mean ﬂow properties for two boundary layers, one in a gradually increasing adverse pressure gradient (both dP/dx and d2 P/dx2 are positive), (Samuel and Joubert 1974) and the second subjected to a sudden, [2] Second-moment turbulence closure modelling a. b. 87 c. d. Figure 12: Boundary layer in increasingly adverse pressure gradient: (a) – friction factors for two ﬂows, (b) – production of turbulence kinetic energy, (c) – rms of streamwise velocity ﬂuctuations, (d) – shear stress (Hanjali´ et c al. 1999). though moderate adverse pressure gradient (Nagano et al. 1993). Computations were obtained with the low-Re DSM described in Section 5.2.1 (for more details, see Hanjali´ et al. 1999). Figure 12a shows the evolution of friction c factor in the two ﬂows to be in good agreement with experiments. Figures 12c and 12d illustrate the evolution of the turbulent stress ﬁeld: the rms of the streamwise ﬂuctuations, the shear stress, and Figure 12b the production of the kinetic energy of turbulence. The last diagram shows a dramatic modiﬁcation in the wall region, strongly inﬂuenced by the interaction of normal stresses and irrotational strain, a feature that cannot be captured by any EVM. In the outer region there is almost no eﬀect of pressure gradient except far downstream at x = 1400mm (for which no experimental data are available), illustrating the above noted role of pressure gradient. A further test of the ability of a turbulence model to respond to an imposed strong variation of the pressure gradient is an oscillating ﬂow produced by a succession of favourable and adverse pressure gradients. Oscillating and pulsating ﬂows are encountered in various engineering applications, as well as in physiological ﬂows. A particular challenge occurs for ﬂow at transitional Reynolds numbers when the imposed favourable pressure gradient during the acceleration phase may cause ﬂow laminarization, followed by a sudden ‘revival’ of very weak decaying turbulence at the onset of the deceleration phase – all within 88 a. Hanjali´ and Jakirli´ c c b. Figure 13: Oscillating boundary layer over a range of Re numbers, based on Stokes thickness δ = 2ν/ω and maximum free stream velocity: (a) – friction factor, (b) – phase lead of maximum wall shear stress vs maximum free stream velocity, (c) – cycle variation of wall shear stress (Hanjali´ et al. 1995). c a single cycle. The phase angle at which this sudden transition occurs is very sensitive to the local Reynolds number, based on the Stokes thickness (which includes the frequency of the oscillations of the imposed pressure gradient, or free stream velocity). Figure 13c shows the computed variation of the wall shear stress over a cycle for diﬀerent Reynolds numbers, based on the Stokes thickness. It is interesting to note that the model reproduces well both the shear stress values and the transition phase angle for diﬀerent Re numbers, in accord with the available DNS and experiments. Figures 13a and 13b show the variation of the wall friction factor and the phase lead of the maximum wall shear stress with respect to the maximum free stream velocity over a range of Reynolds numbers, indicating clearly the change from the laminar to the turbulent regime). The next example shows a three-dimensional boundary layer, illustrating the model response to a sudden transverse shear: a two-dimensional boundary layer developing axially along a stationary cylinder encounters a laterally moving wall (rotating aft part), a situation similar to a stator-rotor assembly [2] Second-moment turbulence closure modelling 89 c. Figure 13: Continued. in axial turbomachinery (Hanjali´ et al. 1994). Figure 14 shows the response c of diﬀerent normal stress components at two diﬀerent wall distances within the boundary layer. The transverse shear generates a strong additional production, primarily of streamwise and spanwise ﬂuctuations, which is reﬂected very close to the wall (y = 2.5mm) in a sudden increase in these stress components and a consequent increase in stress anisotropy. This naturally results in a sudden increase in the total friction factor. Further from the wall the turbulence ﬁeld reacts more gradually. The next illustration shows some examples of swirling ﬂows. A distinction is made between the strong swirls in short cylindrical containers such as combustion chambers, and those in long pipes. The latter ﬂow, even with a weak swirl, seems more diﬃcult to capture with an EVM. Figure 15 compares the axial and tangential velocities computed by several models with experimental data (Jakirli´ 1997). Both the high- and low-Re-number DSM reproduce the c ﬂow features much better than the low-Re-number k-ε model of Chien (1982). The importance of integration up to the wall with the use of the low-Renumber model is illustrated further in Figure 16, which shows a comparison of DSM computations and experiments for a spin-down operation of a rapid 90 Hanjali´ and Jakirli´ c c Figure 14: Response of turbulent stress ﬁeld in a boundary layer encountering transverse wall movement (Hanjali´ et al. 1994). c compression cylinder at two time instants (Jakirli´ et al. 2000). It is worth c noting that the DSM reproduced the vortex breakdown and the complex shape of axial velocity with high peaks and change of sign, all in accord with the DNS data base. A further example is provided in Figure 17, which shows a recent computation of the mean ﬂow development in a single-stroke compression engine with swirl. Here the swirl eﬀect is combined with compression. The computation with the low-Re-number DSM and the integration up to the wall yields a ﬂow pattern in good agreement with DNS (available only for up to 50% compression). Further support is provided by comparison with the experiments: computed tangential velocities at diﬀerent crank angles agree well with the experimental results. The last example in a series of relatively simple ﬂows, which show speciﬁc turbulence features that pose a challenge to modelling, is the separation bubble on a ﬂat wall created by imposed suction and blowing along the boundary of the computational domain opposite to the wall. The adverse pressure gradient created by suction causes the incoming boundary layer to separate, whereas [2] Second-moment turbulence closure modelling 91 Figure 15: Proﬁles of axial and tangential velocity in swirl ﬂow in a long pipe, computed by low-Re-number k-ε and DSM models (Jakirli´ 1997), compared c with experiments of Steenberger (1995). the subsequent blowing forces the bubble to reattach. Predicting the locations of separation and reattachment, and the bubble length and thickness requires an accurate reproduction of the complex turbulence dynamics, integration up to the wall, with accurate numerical resolution of ﬂow and turbulence details in the near-wall region. The DNS of this ﬂow has been provided by Spalart and Coleman (1997). Figure 18 shows the computed friction factor and streamlines compared with the DNS results. It should be mentioned that the bubble shows a tendency to split into two parts, as is visible from the friction factor behaviour. The computed results are in good agreement with the DNS. A DNS of a similar conﬁguration, but with incoming laminar ﬂow, was recently reported by Spalart and Strelets (2000) and Alam and Sandam (2000). Good reproductions of these simulations with the same low-Re-number DSM were also obtained (for details see Hadˇi´ and Hanjali´ 2000). zc c 92 Hanjali´ and Jakirli´ c c DSM: DNS: Figure 16: Streamlines and axial velocity proﬁles computed with the low-Re DSM (Jakirli´ 1997) and DNS (Pascal 1998) at two time instants in a spinc down operation of a rapid compression machine. [2] Second-moment turbulence closure modelling 93 Figure 17: Velocity vectors and proﬁles of swirl velocity in a single-shot compression engine: computations with the low-Re-number DSM (Jakirli´ et al. c 2000). 94 Hanjali´ and Jakirli´ c c Figure 18: Pressure-induced separation on a ﬂat wall: above: friction factor; below: streamlines. Computations with the low-Re-number DSM (Hadˇi´ 1999) zc and DNS results of Sapalart and Coleman (1997). 6.1 Some comments on low-Re-number DSMs The above examples indicate that the prediction of wall phenomena (wall shear, heat and mass transfer) in complex ﬂows can at present be achieved only by applying an advanced low-Re-number model with integration up to the wall. However, if wall phenomena are not especially of interest, it seems that even the classical wall functions, in conjunction with better high-Re-number models (i.e. second-moment closures), can reproduce reasonably well the general ﬂow pattern, particularly if external eﬀects, (e.g. pressure gradient) are dominant. Figure 19 shows the proﬁles of shear stress in a boundary layer in adverse pressure gradient discussed earlier (see Figure 12d), but this time computed by the standard DSM and wall functions. As compared with Figure 12d, here the ﬁrst two computed points are very erroneous, but in the outer region (for y + > 30) the computations agree well with experimental results, much as those obtained with a low-Reynolds number model integrated up to the wall. Recirculation bubbles behind a back-step and sudden expansion ﬂows are other examples (see Hanjali´ and Jakirli´ 1998): the streamline patterns and c c locations of reattachment obtained with the low- and high-Reynolds-number DSMs (the latter using wall functions) look very similar and are both in good [2] Second-moment turbulence closure modelling 95 Figure 19: Shear stress distribution in a boundary layer in adverse pressure gradient, computed with the standard DSM and wall functions (Jakirli´ 1977). c Compare with Figure 12d. Figure 20: Friction factor at step-wall in a sudden pipe expansion, computed with a high- and low-Re-number DSM (Hanjali´ and Jakirli´ 1998). c c agreement with experiments. However, the friction factors are very diﬀerent, as illustrated in Figure 20 for a sudden pipe expansion: the low-Re-number predicts the friction factor much better. The low-Re-number DSM discussed above and others have also been successfully used to compute some more complex ﬂows at high bulk Reynolds numbers (Craft 1998, Hanjali´ 1999). It should be noted, however, that the application c of advanced low-Re-number models to complex high-Re-number industrial 3D ﬂows, where a non-orthogonal body-ﬁtted grid is needed, may still require a computer budget that would be unacceptable for many industries. A middle 96 Hanjali´ and Jakirli´ c c of the road option may be a ‘two-layer’ (zonal) approach in which a simpler model (two-equation or one-equation EVM) is applied within the viscous sublayer, matched with a DSM or other advanced model in the rest of the ﬂow (fully turbulent regime). Yet, such a strategy introduces simpliﬁcations which are inconsistent with the model applied in the rest of ﬂow. Simple two- or one-equation low-Re-number models can be designed to account for viscous (scalar) eﬀects, but they can hardly reproduce the wall-topology-dependent non-viscous blocking and the consequent stress anisotropy. This inevitably restricts the generality of this approach to near-equilibrium situations. A better approach is to employ low-Re-number ASMs which can be obtained by conventional truncation of the parent diﬀerential low-Re-number model. Such a model seems justiﬁed in the near-wall region where the convection and diﬀusion of the turbulent stress are much smaller than the source terms. Such an algebraic stress model can also be used for near-wall ﬂows in combination with large eddy simulation (LES) in the outer regions, as implied by the Detached Eddy Simulation (DES) or other hybrid RANS/LES approaches (e.g. Travin et al. 2000). However, irrespective of the level of modelling used in the viscosity aﬀected near-wall region, the need for a ﬁne grid in the wall-normal direction still remains if the viscous sublayer is to be resolved. Simple low-Re-number models may be computationally more robust, but demands on computation resources are still very high for 3D ﬂows. Practical ﬂows will, for the foreseeable future, probably rely on the use of wall functions. Further improvement and generalization of wall functions is currently viewed by the industrial CFD community as one of the most urgent tasks. 7 Concluding Remarks • Diﬀerential Second-Moment (Reynolds-stress) turbulence models (DSM) are the natural and logical level (within the Reynolds averaging framework) for closing the equations governing turbulent ﬂows. In contrast to Eddy Viscosity Models (EVM), DSMs have a sounder physical basis and treat some important turbulence interactions, primarily the stress generation, exactly. This allows better capturing of the evolution of the turbulent stress ﬁeld and its anisotropy, and such mean-ﬁeld inﬂuences as the eﬀects of streamline curvature, ﬂow and system rotation and ﬂow three-dimensionality. • The potential of the DSM, although long recognized, has so far neither been fully explored nor exploited, mainly due to persisting numerical diﬃculties, and uncertainties in modelling some of the processes, such as pressure-scrambling, which do not appear in two-equation EVMs. • Numerical problems, associated with the implementation of advanced turbulence models, and unavoidably increased demands on computing resources still discourage their wider application for the computation of [2] Second-moment turbulence closure modelling 97 complex ﬂows. However, over the past few years these problems have been considerably reduced. DSMs have already been incorporated in some commercial CFD codes and used to solve some very complex ﬂows. It is likely that they will be more widely used in the near future. • Integration up to the wall and a ﬁne resolution of the viscous sublayer cannot be avoided if low-Re-number ﬂows, transition phenomena and accurate wall friction and heat transfer are to be solved. • The need to resolve thin ﬂow regions near walls and interfaces requires model modiﬁcations (‘low-Re-number’ models), highly non-uniform (and possibly a local self-adapting) grid, and a ﬂexible robust solver. For these reasons, integration up to the wall may still not be a viable option for complex industrial ﬂows at high-Reynolds numbers. However, standard wall functions in conjunction with second-moment closure yield reasonable predictions of the ﬂow pattern and pressure distribution, except in the near vicinity of the separation and reattachment, and can be used if wall/interface phenomena are not of primary importance. • Further developments, which will make DSMs more appealing, are expected in the near future. In addition to model improvements, new numerical solvers are in the oﬃng – speciﬁcally targeted at solving the transport equations for turbulent stresses and their coupling with the Reynolds-averaged Navier–Stokes equations – which will not be burdened by the eddy-viscosity tradition. New wall functions are also needed for complex nonequilibrium ﬂows, which should reproduce more accurately the wall phenomena and yet bridge the viscous sublayer and dispense with the need for a ﬁne grid resolution of the near-wall region. • Even so, the present level of development and acquired know-how already permits and, indeed, calls for a wider use of such advanced models in industrial applications. Acknowledgement. The authors acknowledge the computational contributions used for illustration by Dr. I. Hadˇi´, from the Technical University zc Hamburg-Harburg, Germany. Thanks are also due to Mr. Bart Hoek from TU Delft for providing several drawings. References Alam, M. and Sandham, N. 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Fluid Mech. 403, 89–119. 3 Closure Modelling Near the Two-Component Limit T.J. Craft and B.E. Launder 1 Introduction Most widely-used turbulence models have been developed and tested with reference to ﬂows near local equilibrium, where there are only moderate levels of Reynolds stress anisotropy. The present contribution considers the development of models which are designed to give the correct behaviour in much more extreme situations, where the turbulence approaches a 2-component state. To illustrate the type of ﬂow situation to be considered, Figure 1 illustrates the ﬂow in the vicinity of a wall. While all turbulent velocity components must vanish at the wall, the normal ﬂuctuations, v, must vanish more rapidly since by continuity ∂v/∂y must always be zero there (as ∂u/∂x and ∂w/∂z both vanish), Figure 2. A similar two-component structure arises close to the free surface of a liquid ﬂow where again ﬂuctuating velocities normal to the free surface become negligible compared with ﬂuctuations lying in the plane Figure 1: Near-wall ﬂow. Figure 2: Normal Reynolds stress components in plane channel ﬂow, from the DNS of Kim et al. (1987). 102 [3] Closure modelling near the two-component limit 103 of the free surface. Clearly, the turbulence structure in such a ﬂow will be very diﬀerent from that found in free ﬂows, where the stress anisotropy is much smaller. Consequently, it might be expected that simple models developed and tuned for the latter ﬂows are unlikely to give good predictions in near-wall or free-surface regions, or other ﬂows which are close to the 2-component limit. The importance of explicitly respecting this two-component limit in turbulence modelling originated from two papers from the 1970s. First, a short note by Schumann (1977) advocated that modelling proposals should make it impossible for unrealizable values of the turbulence variables to be generated (such as negative values for the mean square velocity ﬂuctuations in any direction). Shortly thereafter, Lumley (1978) remarked that if such realizability was to be ensured one needed to focus on the behaviour of the model at the moment when one of the velocity components had just fallen to zero. When this two-component state has been reached one must ensure that, for the normal stress that has fallen to zero, its rate of change also vanishes. That is essential to prevent the stress ﬁeld achieving unrealizable values at the next instant of time. Shih and Lumley (1985) were the ﬁrst to apply realizability constraints to the modelling of the pressure correlation terms in both the Reynolds stress and scalar ﬂux transport equations. However, while the work initially adopted a rigorous analytical path, they later (Shih et al. 1985) had to include additional higher order correction terms to gain agreement with simple shear ﬂow experiments. In later work at UMIST, Fu et al. (1987), Fu (1988), Craft et al. (1989) and Craft (1991) showed that by applying a slightly diﬀerent constraint to the scalar ﬂux model, a realizable model was obtained which gave good agreement with experiments for a range of shear ﬂows. The model has since been extended further by Launder and Tselepidakis (1993), Launder and Li (1994), Craft and Launder (1996) to include viscous and inhomogeneity eﬀects found in near-wall or surface regions. Consideration of these eﬀects is, however, deferred until [11] on Impinging and Separated Flows. A further class of ﬂows where turbulence approaches the two-component state is where a strongly stabilizing force ﬁeld is applied, whether due to buoyancy, rotation, or electro-magnetic eﬀects. The extension of the methodology to such cases is developed in [14]. 2 A TCL closure of the Reynolds stress transport equations From [2], equation (1), the stress transport equations, in the absence of any external force ﬁeld, can be written symbolically as Dui uj = Pij + φij + dij − εij . Dt (2.1) 104 Craft and Launder The exact generation term Pij ≡ −ui uk ∂Uj /∂xk − uj uk ∂Ui /∂xk , whilst the remaining terms, which require modelling, are the pressure-strain correlation φij , diﬀusion dij and dissipation rate εij . In the shear ﬂows discussed here, diﬀusion is a relatively unimportant process, so the simple gradient diﬀusion model of Daly and Harlow (1970), dij = ∂ ∂xk k νδlk + cs uk ul ε ∂ui uj , ∂xl (2.2) is often employed. The modelling of the remaining terms is made so as to ensure compliance with the two-component limit (TCL). Broadly, two strategies are adopted, which are exempliﬁed in the modelling of the pressure-strain processes. 2.1 Pressure-Strain Processes In many ﬂows, it is the pressure-strain correlation φij which is the most important term requiring modelling, and consequently much eﬀort has been put into developing improved models for this process. As reported in [2], equation (6) et seq., integration of the Poisson equation for pressure ﬂuctuations for regions where the surface integral is unimportant leads to φij ≡ p ρ ∂uj ∂ui + ∂xj ∂xi 1 4π ∂ 3 uk ul uj ∂ 3 uk ul ui + ∂rl ∂rk ∂rj ∂rl ∂rk ∂ri φij1 (2.3) dV |r| = − V ol − 1 2π V ol ∂ 2 ul uj ∂ 2 ul ui + ∂rk ∂rj ∂rk ∂ri φij2 ∂Uk dV , ∂rl |r| where non-primed quantities are evaluated at the point where φij is being determined, whilst primed quantities are evaluated at positions within the integration volume at a displacement of r from this point. From this, it can be seen that there are two distinct contributions to φij : one involving interactions between ﬂuctuating quantities and one dependent on mean strain rates. In buoyancy-aﬀected ﬂows there is a further contribution, which will be considered separately in [14]. The volume of integration in equation (2.3), although formally being the entire ﬂuid domain, can in practice be regarded as the region where the time averaged two-point correlations are non-zero, corresponding to some relatively small region surrounding the point at which φij is being evaluated with a radius typically of the local integral lengthscale. [3] Closure modelling near the two-component limit 105 It is noted that equation (2.3) contains no surface integral discussed both in [2] and, in more detail, in [4]. This represents a fundamental diﬀerence of strategy between closure schemes which are otherwise of the same type. The philosophy explored at UMIST is that if one ensures compliance of the model of φij with the two-component limit, there should, in many cases, be no requirement for any further wall correction - at least if one remains outside the buﬀer layer where the eﬀects of very rapid spatial variations must be accounted for. 2.1.1 Mean-Strain (or ‘Rapid’) Part of φij If the mean strain is assumed to vary much more slowly in space than the twopoint correlation gradients in equation (2.3), it can be regarded as uniform over the volume of integration, so that φij2 can be modelled as lj li φij2 = Xkj + Xki ∂Uk , ∂xl (2.4) li where the tensor Xkj represents the integral of the two-point velocity-derivative correlations: li Xkj = − 1 2π V ol ∂ 2 ul ui dV . ∂rk ∂rj |r| (2.5) This approach has been employed by Naot et al. (1973) and Launder et al. li (1975) to derive models of φij2 in which Xkj is simply a linear function of the Reynolds stresses: li Xkj = αul ui δkj + β (ui uj δlk + ul uj δik + ul uk δij + ui uk δlj ) +γuk uj δil + ξk (δlj δik + δlk δij ) + ηkδil δkj , (2.6) where the Greek symbols are coeﬃcients to be determined. One can note that the exact integral in equation (2.5) satisﬁes li • Continuity: Xki = 0 li • Normalization: Xkk = 2ul ui . By applying these two constraints, all the coeﬃcients except one in equation (2.6) can be determined and the resultant model, known as the ‘Quasi Isotropic (QI) Model’, may be written as φij2 = − γ+8 30γ − 2 (Pij − 1/3δij Pkk ) − k 11 55 8γ − 2 − (Dij − 1/3δij Dkk ), 11 ∂Ui ∂Uj + ∂xj ∂xi (2.7) where Dij ≡ −ui uk ∂Uk /∂xj − uj uk ∂Uk /∂xi . 106 Craft and Launder However, it is not possible to choose a value of γ that will enable this linear form to satisfy the 2-component limit, which requires φαα2 = 0 if uα = 0. An obvious extension of the approach is to recognise that the two-point correlations appearing in the integral of equation (2.5) will not depend linearly li on the second-moments, and thus to allow Xkj to be a nonlinear function of the Reynolds stresses. If one includes both quadratic and cubic terms, then the most general expression satisfying the required symmetry properties can be written as li Xkj = λ1 δli δkj + λ2 (δlj δki + δlk δij ) k +λ3 ali δkj + λ4 akj δli + λ5 (alj δki + alk δij + aij δlk + aki δlj ) +λ6 ali akj + λ7 (alj aki + alk aij ) + λ8 alm ami δkj + λ9 akm amj δli +λ10 (alm amj δki + alm amk δij + aim amj δlk + akm ami δlj ) +λ11 amn amn δli δkj + λ12 amn amn (δlj δki + δlk δij ) +λ13 ali akm amj + λ14 akj alm ami +λ15 (alj akm ami + alk aim amj + aij alm amk + aki aml amj ) +λ16 amn anp apm δli δkj + λ17 amn anp apm (δlj δki + δlk δij ) +λ18 amn amn ali δkj + λ19 amn amn akj δli +λ20 amn amn (alj δki + alk δij + aij δlk + aki δlj ). (2.8) The approach outlined below, following the analysis of Fu (1988), is to assume the coeﬃcients λ1 , . . . , λ20 to be constants, and to apply the continuity, normalization and 2-component-limit constraints in order to determine as many of the coeﬃcients as possible. li Applying the continuity constraint (Xki = 0), and making use of the Cayley– Hamilton theorem, leads to six equations: λ1 + 4λ2 = 0 λ3 + λ4 + 5λ5 = 0 λ6 + λ7 + λ8 + λ9 + 5λ10 = 0 λ10 + λ11 + 4λ12 = 0 λ13 + λ14 + 4λ15 + 2λ18 + 2λ19 + 10λ20 = 0 λ16 + 4λ17 + 1/3(λ13 + λ14 + 2λ15 ) = 0 (2.9a) (2.9b) (2.9c) (2.9d) (2.9e) (2.9f) Similarly, the normalization constraint leads to a further six equations: 3λ1 + 2λ2 = 4/3 3λ3 + 4λ5 = 2 2λ7 + 3λ8 + 4λ10 = 0 λ9 + 3λ11 + 2λ12 = 0 λ13 + 2λ15 + 3λ18 + 4λ20 = 0 4λ15 + 9λ16 + 6λ17 = 0 (2.10a) (2.10b) (2.10c) (2.10d) (2.10e) (2.10f) [3] Closure modelling near the two-component limit 107 The 2-component-limit constraint is most conveniently handled in principal axes of the Reynolds stresses, where ui uj = 0 if i = j. If u2 is taken as the 2 vanishing component, then the other two normal stresses can be written as u2 = (1 + δ)k and u2 = (1 − δ)k. The 2-component limit requires that 1 3 l2 Xk2 ∂Uk = 0, ∂xl (2.11) and substituting the above values for the stresses into this equation leads to a further four relations between the model coeﬃcients: 2 7 4 10 4 11 λ1 + λ2 − (λ3 + λ4 ) − λ5 + λ6 + λ7 + (λ8 + λ9 ) + λ10 3 3 9 9 9 9 2 8 34 2 + (λ11 + λ12 ) − (λ13 + λ14 ) − λ15 − (λ16 + λ17 ) 3 27 27 9 4 14 − (λ18 + λ19 ) − λ20 = 0 9 9 3λ5 − 2λ7 + 2λ10 + 2λ20 = 0 2λ20 = 0. (2.12a) (2.12b) (2.12d) 3λ10 − 6(λ11 + λ12 ) − 2λ15 − 6(λ16 + λ17 ) + 4(λ18 + λ19 ) + 6λ20 = 0 (2.12c) These equations can be solved, leaving four undetermined parameters: λ1 = λ3 = λ4 = λ5 = λ6 = λ7 = 8 15 14 4 15 + 3 t 1 11 15 + 3 t 1 5 −t 1 10 + 8t − 2p 3 − 10 − 3 t + p 2 λ8 = 1 5 + t − 2p λ15 = 3s − 9 t 4 λ16 = −2s + t λ17 = s λ18 = 15 4 t 2 λ2 = − 15 λ9 = − 15 t − 2p 2 λ10 = p λ11 = 3t + p λ12 = −3t 4 − λ13 = −6s − λ14 = −6s + 1 2p 27 4 t 33 4 t −r λ19 = r + 3r λ20 = 0. − 3r However, it can be shown that the contributions to φij2 arising from the terms with coeﬃcients s and p are identically zero. The resulting model can thus be written as φij2 = −0.6 (Pij − 1/3δij Pkk ) + 0.3aij Pkk uk uj ul ui ∂Uk ∂Uj ul uk ∂Ui ∂Ul − −0.2 + ui uk + uj uk k ∂xl ∂xk k ∂xl ∂xl −c2 [A2 (Pij − Dij ) + 3ami anj (Pmn − Dmn )] 7 A2 +c2 − (Pij − 1/3δij Pkk ) 15 4 +0.1 [aij − 1/2 (aik akj − 1/3δij A2 )] Pkk − 0.05aij alk Pkl 108 Craft and Launder uj um ui um ul um Pmj + Pmi − 2/3δij Pml k k k ul ui uk uj 1 ∂Ul ul um uk um ∂Uk +0.1 − /3δij + 6Dlk + 13k k2 k2 ∂xk ∂xl ul ui uk uj + 0.2 (Dlk − Plk ) , k2 +0.1 where A2 = aij aij . There are two free coeﬃcients, c2 and c2 , which can be set by tuning the model to simple shear ﬂows. Fu et al. (1987) recommended values of c2 = 0.6, c2 = 0, which considerably simpliﬁes the task of implementing the model in a computer code. Later, however, Fu (1988) concluded that slightly better agreement for free shear ﬂows could be obtained with c2 = 0.55 and c2 = 0.6, values which greatly improved the performance in near-wall ﬂows since in many cases if one remains outside the viscosity-aﬀected sublayer, no wall corrections of the type described in [2] are then needed, Launder and Li (1994). 2.1.2 Turbulence (or ‘Slow’) Part of Pressure-Strain No-one has so far devised a successful analytical route for modelling φij1 analogous to that for φij2 . Thus the two-component limit is imposed empirically through stress invariants of which, for a second rank tensor, there are two independent parameters. One of these, A2 has already appeared in the expression for φij2 . The natural second parameter might be thought to be A3 ≡ aij ajk aki . (2.13) However, Lumley (1978) showed that, for modelling purposes, a combined invariant A, deﬁned as A ≡ 1 − 9/8(A2 − A3 ), (2.14) was a particularly powerful choice because, in the limit of two-component turbulence, the parameter always goes to zero.1 By including the parameter A in a model for φij1 , one may thus arrange that the model of φij1 is consistent with the two-component limit. Thus, for φij1 , a nonlinear extension of the return to isotropy model of Rotta (1951) could be written as: φij1 = −c1 εaij − c1 ε(aik akj − 1/3A2 δij ) − c1 εA2 aij 1 (2.15) We can conveniently map the range of attainable states of the turbulent stress ﬁeld as an A2 -A3 plot (Lumley 1978), Figure 3a. All realizable states fall within or on the boundary of this triangle, the upper line corresponding to two-component turbulence while the two curved lines represent axisymmetric turbulence (that is, where two of the normal stresses are equal). The origin corresponds to isotropic turbulence. Alternatively, on an A2 -A plot, two-component turbulence corresponds with states lying on the A2 axis, Figure 3b. [3] Closure modelling near the two-component limit 109 (Although it might appear that the cubic term (aik akl alj − 1/3A3 δij ) should also be included in equation (2.15), the Cayley–Hamilton theorem means that the term is proportional to A2 aij , and its inclusion would not thus add any more generality to the form shown). Figure 3: The anisotropy invariant map in A2 -A3 and A2 -A space. The approach followed is simply to make the coeﬃcients functions of the invariants A and A2 , to ensure that they vanish in the 2-component limit. The UMIST group, for example, have employed the form φij1 = −c1 ε aij + c1 (aij ajk − 1/3A2 δij ) − fA εaij , where c1 = 3.1(A2 A)1/2 c1 = 1.1 fA = A1/2 . (2.16) 2.2 Dissipation Since the dissipative processes arise predominantly from the smallest scales of turbulence, εij is normally considered to be essentially isotropic, even if the stress ﬁeld is signiﬁcantly anisotropic. From this assumption, εij is often modelled as 2/3εδij . However, local isotropy is not consistent with the 2-component limit which requires ε22 to vanish at a wall. A simple way of ensuring compliance with this limit is to devise a model where εij ∝ (ui uj /k)ε close to a wall (or, indeed, in other circumstances where the stress ﬁeld is near the two-component limit). A transition function, based on the ‘ﬂatness’ parameter A, can be employed to switch between the two forms: εij = 2/3εδij fε + ui uj ε(1 − fε ). k (2.17) 110 Craft and Launder If fε takes a value of unity in isotropic turbulence, far from walls (where A ≈ 1), but becomes zero when A vanishes, then εij will switch between the two desired limits. Whilst such a simple form does satisfy some of the conditions required of εij , it does not show the correct limiting behaviour for all components, nor does it behave correctly near a free surface. These aspects will be brieﬂy considered in [11]. Of course, in practical calculations the dissipation rate ε also has to be modelled, and this is generally done by solving a separate transport equation for it. The most widely employed model can be written Dε εPkk ε2 ∂ = cε1 − cε2 + Dt 2k k ∂xk k νδlk + cε uk ul ε ∂ε , ∂xl (2.18) with coeﬃcients cε1 = 1.44, cε2 = 1.92. At UMIST, workers have retained this general form, but have included some account of the eﬀects of diﬀerent stress anisotropy on ε by allowing cε2 to be a function of A and A2 , and reducing the value of cε1 . The recommended form for the coeﬃcients is: cε1 = 1.0 cε2 = 1.92/(1 + 0.7A2 A). 1/2 (2.19) 3 Applications to the computation of dynamic ﬁeld The performance of the model described in Section 2 is now considered, ﬁrst for free ﬂows, then for ﬂows near a single plane wall or free surface and then, ﬁnally, for composite walls. To provide as accurate an impression as possible of the capabilities of the TCL approach, we limit attention to computations made with a single form of the model. Consequently, earlier TCL forms adopted by Tselepidakis (1991) (see Launder and Tselepidakis 1993) and Launder and Shima (1989) (see also Shima 1993, 1998) have not been included here even though the latter, in particular, has been successfully applied to a wide range of two- and three-dimensional boundary layers near walls. Nor do we include the very recent publications by Leschziner and his group of the TCL model applied to transonic and supersonic ﬂows (Batten et al. 1999b,a). 3.1 Free Shear Flows The free coeﬃcients in the model were originally assigned to secure satisfactory agreement in various homogeneous shear ﬂows and plane strains. Thus one can hardly claim to be predicting these ﬂows since they formed part of the overall optimization process. A typical example is the homogeneous shear ﬂow considered in Figure 4. This particular example is a severe test as it starts from isotropic turbulence. The growth of the anisotropy is predicted broadly as the DNS of the four non-zero stress components indicates, although the [3] Closure modelling near the two-component limit 111 Figure 4: Development of stress anisotropies in homogeneous shear ﬂow. Lines: TCL model predictions, Symbols: DNS of Matsumoto et al. (1991). Flow Plane jet Round jet Plane wake Experimental value 0.110 0.093 0.086 Basic model 0.100 0.105 0.070 TCL model 0.110 0.101 0.069 Table 1: Predicted and measured spreading rates of some self-preserving free shear ﬂows. initial growth of the anisotropy of u2 and u2 is rather too weak. It is, however, 1 2 considerably better than that achieved by the Basic Model, presented in [2]. Turning to inhomogeneous shear ﬂows, Table 1 reports the computed and measured asymptotic growth rates of three free shear ﬂows: the plane and round jets and the plane wake. These three ﬂows collectively provide a severe test for any model. The table indicates that the TCL scheme again comes close to mimicking the growth rates of all three, whereas the Basic Model does badly for both the plane wake and the round jet. It is worth noting that these results were obtained with a full elliptic solution of the transport equations rather than the usual thin-shear-ﬂow approximation. This practice led to a reduction in growth rates (compared with a thin-shear-ﬂow treatment) of about 12% for the round jet and about 4% for the plane jet (El Baz et al. 1993). This diﬀerence reﬂected the rapid axial decay of the round jet. There was negligible diﬀerence between the two treatments for the wake. There are similar improvements in the prediction of growth rates for buoyantly-driven plumes, considered in [14]. 112 Craft and Launder Finally, Figure 5 compares the development of plane wakes created by two diﬀerent bodies, thus providing diﬀerent initial states of turbulence energy and dissipation, but with the same momentum deﬁcit. The experiments show that the eﬀects of the diﬀerent initial conditions carry over very far downstream – well beyond the region of measurement. The Basic Model, however, quickly forgets about the diﬀerent initial conditions, showing identical growth for the two cases. In contrast, the TCL scheme clearly displays a very similar development to that recorded. 0.4 0.4 Airfoil Expt. Pred. Solid Strip Expt. Pred. u2 /(∆Uc2) c u2 /(∆Uc2) c 0.3 0.3 Airfoil Expt. Pred. Solid Strip Expt. Pred. 0.2 0.2 0.1 0.1 0.0 0 500 1000 1500 0.0 x/θ 2000 (y1/2/θ )2 600 0 500 1000 1500 x/θ 2000 (y1/2/θ )2 600 Airfoil Expt. Pred. Solid Strip Expt. Pred. 450 450 Airfoil Expt. Pred. Solid Strip Expt. Pred. 300 300 150 150 0 0 500 1000 1500 0 x/θ 2000 0 500 1000 1500 x/θ 2000 Figure 5: Development of the centreline streamwise Reynolds stress and the half-width of the plane wake behind two diﬀerent bodies. Left hand graphs: Basic Model, right hand graphs: TCL Model. From El Baz (1992). 3.2 Flows Near Plane Surfaces If one applies a log-law boundary condition for velocity and analogous localequilibrium conditions for the near-wall stresses it is possible to apply the model discussed so far to wall ﬂows as well as free ﬂows without introducing any form of ‘wall-reﬂection’ correction to φij . This is a very great beneﬁt! The case of fully-developed ﬂow in a plane channel is shown in Figure 6, where agreement with data is seen to be satisfactory. To integrate all the way to the [3] Closure modelling near the two-component limit 113 wall across the viscous sublayer does require a correction to account for the very rapid change of the mean velocity gradient within the ‘buﬀer region’ as well as the inclusion of viscous eﬀects. These elaborations are discussed in [11] concerned with impinging and separated ﬂows. Figure 6: Reynolds stress proﬁles in fully developed channel ﬂow at Re = 20000. From Li (1992). Solid line TCL model; broken line Basic Model with wall-reﬂection terms added; symbols experiments. The ﬂow in the vicinity of a free liquid surface (that is, a gas-liquid interface) traditionally requires the application of a ‘wall’ correction if the Basic Model is adopted (McGuirk and Papadimitriou 1985). Again, such corrections are dispensed with when the TCL closure is adopted. Figure 7 compares the development of a 3D surface jet adopting these two second-moment closures: the Basic Model, including ‘wall-reﬂection’ at the free surface and the TCL model (Craft et al. 2000). Quite clearly the development of the shear ﬂow is much better captured by the latter scheme. Finally, the corresponding case of a 3-dimensional wall jet (Craft and Launder 1999, 2001) is summarized in Table 2. Firstly, it is noted from the experiments that the lateral spreading is markedly greater than that normal to the wall. This eﬀect is due to an induced secondary ﬂow that draws ﬂuid down to the wall and ejects it parallel to the wall. The driving source for the secondary ﬂow is the anisotropy of the turbulent stress ﬁeld in the jet’s cross-section. Now, a linear eddy-viscosity model predicts isotropic normal stresses when there is negligible normal straining; consequently, this unequal growth rate is entirely missed at this level of modelling. Both second-moment closures, on the other hand, exhibit strongly anisotropic growth patterns – indeed appreciably too large (especially the Basic Model, whose growth rate is more than three times that reported experimentally). The reason appears to be that this ﬂow takes much longer to reach full development than the experimenters had believed. If instead of the fully-developed value, one examines the spreading rate at around 70 jet diameters downstream (the downstream limit of the 114 Craft and Launder experiments) the TCL model accords closely with the experimental growth (the corresponding developing-ﬂow value for the Basic Model is not available, though there is no doubt that it would still be considerably too high). Figure 7: Development of the three-dimensional free-surface jet half-widths normal to the free-surface (y1/2 ) and in the lateral direction (z1/2 ). Computations of Craft et al. (2000), ——: TCL model; – - –: Basic model; Symbols: experiments of Rajaratnam and Humphries (1984). From Craft et al. (2000). dy1/2 /dx 0.065 0.079 0.053 0.060 0.055 dz1/2 /dx 0.32 0.069 0.814 0.51 0.308 z1/2 /y1/2 ˙ ˙ 4.94 0.88 15.3 8.54 5.6 Expt. (Abrahamsson et al, 1997) Linear EVM Basic model TCL model TCL model at 70 diameters Table 2: Spreading rates normal to the wall (dy1/2 /dx) and in the lateral direction (dz1/2 /dx) in the 3-dimensional wall jet. 3.3 Flow Over Complex Surfaces Figure 8 shows axial velocity contours over the cross section of a straight rectangular sectioned duct where, over the lower wall, two regions, symmetrically located relative to the centre-plane of the duct, have been roughened. The [3] Closure modelling near the two-component limit 115 roughness creates a complex Reynolds stress pattern which, in turn, induces an appreciable secondary ﬂow, with an upwelling of ﬂuid in the vicinity of the mid-plane, which distorts the axial velocity contours as shown in the ﬁgure. Again, the source of this streamwise vorticity is the anisotropy of the in-plane Reynolds stresses. No linear eddy-viscosity turbulence model can create such streamwise vorticity. We see, however, that the TCL computations (Launder and Li 1994) mimic the measured distribution very closely. The Basic Model (with wall-reﬂection corrections) gets the correct sense of the secondary motion, but the detailed prediction is evidently not as successful as with the TCL model. A similar story unfolds in the case of ﬂow through a smooth square-sectioned U-bend (Iacovides et al. 1996). In this case, a strong secondary ﬂow is induced by the bend curvature. Figure 9 shows the variation of shear stress between the inner and outer curved walls at 45◦ into the bend. On the symmetry plane both TCL and Basic models achieve reasonable agreement with the measured stress proﬁle. As one moves progressively towards the top wall of the duct, however, the TCL scheme takes account of the inﬂuence of this upper wall much better than the Basic Model even though the former model has no explicit means (such as distance to the upper wall) of sensing the presence of that boundary. Figure 8: Flow through a rectangular-sectioned duct with a partially roughened lower wall. Computations of Launder and Li (1994), experiments of Hinze (1973). (a) Contours of mean streamwise velocity. (b) Predicted secondary ﬂow patterns. From Launder and Li (1994). 116 Craft and Launder Figure 9: Turbulent shear stress proﬁles across the duct at 45◦ around a squaresectioned U-bend. Computations of Iacovides et al. (1996), ——: Basic model; – - –: TCL model; Symbols: experiments of Chang et al. (1983). From Iacovides et al. (1996). 4 Scalar ﬂux modelling Similar considerations relating to the two-component-limit behaviour can be applied to the modelling of the scalar-ﬂux transport equations. The exact transport equations (see [2], equation (2.19)) can be written symbolically as Dui θ = Piθ + φiθ + diθ − εiθ , Dt (4.1) [3] Closure modelling near the two-component limit 117 where the production Piθ ≡ −ui uk ∂Θ/∂xk − uk θ ∂Ui /∂xk which does not require modelling, whilst φiθ represents the pressure-scalar gradient correlation, diθ the diﬀusion and εiθ the dissipation rate of the scalar ﬂux. In this case, the assumption of isotropic dissipation leads to εiθ = 0, and consequently the main modelling task is to approximate φiθ . An analytical expression similar to equation (2.3) can be obtained for φiθ : φiθ = − 1 4π ∂ 3 uk ul θ dV 1 − ∂rl ∂rk ∂ri |r| 2π φiθ = φiθ1 + φiθ2 , ∂ 2 ul θ ∂Uk dV , ∂rk ∂ri ∂rl |r| (4.2) V ol V ol from which φiθ is traditionally modelled as (4.3) where φiθ1 represents the turbulence interactions and φiθ2 depends on the mean strains. In buoyancy-aﬀected ﬂows there is a further contribution which will be discussed in [14]. 4.1 Mean-Strain (or ‘Rapid’) Part of Pressure-Scalar Gradient Correlation: φiθ2 Again, if the mean strain is assumed to be essentially constant over the volume of integration in equation (4.2), the φiθ2 process can be modelled as φiθ2 = 2bl ki ∂Uk , ∂xl (4.4) where the tensor bl represents the integral: ki bl = − ki 1 4π ∂ 2 ul θ dV . ∂rk ∂ri |r| (4.5) V ol Adopting the same approach to modelling this tensor as was done for φij2 , bl can be modelled in terms of the Reynolds stresses and scalar ﬂuxes. Howki ever, the linearity principle (noting that the integral in equation (4.5) is linear in the scalar θ) requires that an expansion for bl , whilst possibly being nonlinki ear in the Reynolds stresses, should only depend linearly on the scalar ﬂuxes. Including all possible terms which satisfy the required symmetry in i and k, such an expansion up to cubic order can be written as bl = α1 ul θδik + α2 uk θδli + ui θδlk ki +α3 ul θaik + α4 uk θali + ui θalk +α5 um θaml δik + α6 um θ (amk δli + ami δlk ) +α7 um θaml aik + α8 um θ (amk ail + ami akl ) +α9 ul θami amk + α10 aml uk θaim + ui θakm +amn amn α11 ul θδik + α12 ui θδlk + uk θδli +un θamn α13 aml δik + α14 (amk δli + ami δlk ) . (4.6) 118 Craft and Launder Constraints similar to those applied in the modelling of φij2 can now be applied to determine as many of the model coeﬃcients as possible. The equivalent continuity and normalization conditions give: • Continuity: bk = 0 ki • Normalization: bl = ul θ kk and applying them leads to the eight relations: α1 + 4α2 = 0 α3 + α4 + α5 + 4α6 = 0 α7 + α8 + α9 + α10 + α13 + 4α14 = 0 α10 + α11 + 4α12 = 0 3α1 + 2α2 = 1 2α4 + 3α5 + 2α6 = 0 2α8 + 2α10 + 3α13 + 2α14 = 0 α9 + 3α11 + 2α12 = 0. (4.7a) (4.7b) (4.7c) (4.7d) (4.7e) (4.7f) (4.7g) (4.7h) Before considering what 2-component limit constraint should be applied, note that if only the linear terms are retained (so only α1 and α2 are nonzero) the above equations yield α1 = 0.4, α2 = −0.1, leading to the linear QI model of Launder (1973) (see also, Launder 1975; Lumley 1975): φiθ2 = 0.8uk θ ∂Ui ∂Uk − 0.2uk θ . ∂xk ∂xi (4.8) Returning to the question of satisfying the 2-component limit, Shih and Lumley (1985) applied a constraint that ensured the Schwarz inequality, uα θ 2 ≤ u2 θ 2 , α (4.9) could not be violated. They did this by imposing the condition that the rate 2 of change of the diﬀerence uα θ − u2 θ2 should be zero when equality held α or, mathematically, 2uα θ 2 Dθ2 Du2 Duα θ α = u2 + θ2 , α Dt Dt Dt (4.10) when uα θ = u2 θ2 . α However, this relation links the models for φij2 and φiθ2 , and the outcome was that not only did Shih and Lumley (1985) determine all the coeﬃcients in bl , but the above constraint also led to both free coeﬃcients in the TCL ki model of φij2 being determined as zero. [3] Closure modelling near the two-component limit 119 Unfortunately, it was the c2 and c2 terms that enabled good agreement with simple shear ﬂow experiments. Without them, Shih and Lumley were forced to add some additional, arbitrary, higher-order correction terms to their model in order to get the correct stress levels in shear ﬂow. There is, moreover, a further objection to the Shih and Lumley formulation, in that, if a genuine passive scalar is being considered, the thermal ﬁeld modelling should not inﬂuence the modelling of the underlying dynamic ﬁeld. For these reasons, workers at UMIST have adopted an alternative approach, by ensuring that the nett contribution to the ui θ transport equation, φ2θ +P2θ , vanishes when u2 is zero (Craft 1991). For φiθ2 , this condition translates to 2 ∂Uk l 1 ∂U2 b = ul θ ∂xl k2 2 ∂xl (4.11) when u2 = 0. By again considering the situation in principal axes of the stresses, this condition leads to the six equations 2 1 2 4 2 1 α1 − α3 + α5 − α7 + α9 + α11 + α13 3 3 9 9 3 9 2 2 α5 − α7 + α13 3 3 2α11 + α13 2 1 2 4 2 1 α2 − α4 + α6 − α8 + α10 + α12 + α14 3 3 9 9 3 9 2 2 α6 − α8 + α14 3 3 2α12 + α14 = 1 2 (4.12a) (4.12b) (4.12c) (4.12d) (4.12e) (4.12f) = 0 = 0 = 0 = 0 = 0. Solving these, together with the earlier continuity and normalization equations, leads to the result α1 α2 α3 α4 α5 α6 = = = = = = 0.4 −0.1 −1/6 −1/6 1/15 1/15 α8 α9 α10 α11 α12 α13 α14 = = = = = = = 1/8 − 1/2α7 −1/8 + α7 −1/2α7 1/20 − 1/2α7 −1/80 + 1/4α7 −1/10 + α7 1/40 − 1/2α7 . with, apparently, one free coeﬃcient. However, the term multiplied by α7 can be shown to be identically zero, and hence the resulting model for φiθ2 can be 120 written: φiθ2 = 0.8uk θ ∂Ui ∂Uk 1 ε P − 0.2uk θ + /3 ui θ − 0.4uk θail ∂xk ∂xi k ε ∂Um ∂Ul + ∂xl ∂xm Craft and Launder ∂Uk ∂Ul + ∂xl ∂xk +0.1uk θaik aml +0.15aml − 0.1uk θ (aim Pmk + 2amk Pim ) /k ∂Uk ∂Ul + ∂xl ∂xk amk ui θ − ami uk θ −0.05aml 7amk ui θ ∂Uk ∂Ui + uk θ ∂xl ∂xl ∂Ui ∂Ui + amk ∂xk ∂xl , (4.13) −uk θ aml where there are no free coeﬃcients. 4.2 Turbulence (or ‘Slow’) Part of Pressure-Scalar Gradient Correlation: φiθ1 To model φiθ1 in a similar manner to φij1 , the obvious extension to the linear model of Monin (1965), would be to employ an expression of the form ε ε ε ε φiθ1 = −cθ1 ui θ − cθ1 aij uj θ − cθ1 aik akj uj θ − cθ1 A2 ui θ. k k k k (4.14) Considering this expression in principal axes of the stresses, it is clear that such a model satisﬁes the condition that φ2θ1 should vanish when u2 = 0 regardless 2 of the values of the model coeﬃcients. The UMIST group has therefore employed a form similar to this, allowing the coeﬃcients to be functions of the stress invariants, and tuning them to a range of free shear ﬂows. The exact expression for φiθ1 does not depend explicitly on mean scalar gradients, and these have thus not traditionally appeared in the modelled process. However, Jones and Musonge (1983) argued that the ﬂuctuating quantities do, nevertheless, depend on mean gradients, and thus included a term in their model for φiθ which did explicitly contain the mean scalar gradient. Craft (1991) also found it beneﬁcial to include some explicit mean scalar gradient dependence in order to capture simple homogeneous shear ﬂows at diﬀerent strain rates. The form employed for φiθ1 in this latter work was φiθ1 = −cθ1 r1/2 ε ∂Θ ui θ(1 + cθ1 A2 ) + cθ1 aik uk θ + cθ1 aik akj uj θ − c∗ rkaij , θ1 k ∂xj (4.15) [3] Closure modelling near the two-component limit where cθ1 = −0.8, cθ1 = 1.7 1 + 1.2(A2 A)1/2 ∗ = −0.2A1/2 , cθ1 cθ1 = 0.6, cθ1 = 1.1, 121 and the timescale ratio r is deﬁned2 as r = (2εθ /θ2 )(k/ε). In this case the factor A1/2 in the coeﬃcient c∗ ensures that this part of the model also satisﬁes the θ1 2-component limit. The parameter r represents the ratio of mechanical to thermal timescales, where 2εθ is the dissipation rate of the scalar variance θ2 . A common approach is to assume a constant value for r, although available data shows that it takes signiﬁcantly diﬀerent values in diﬀerent ﬂows, and that such an approach does not, therefore, have a wide range of applicability. Craft et al. (1996) proposed modelling r as a function of the scalar ﬂux invariant A2θ ≡ ui θ ui θ/(kθ2 ), taking r = 1.5(1 + A2θ ) (4.16) The above form was shown to give good predictions in a range of shear ﬂows, including buoyancy-aﬀected ﬂows. Such a correlation does, nevertheless, have its limitations and the most reliable route for obtaining r would be to solve a suitable transport equation for the dissipation rate εθ . A number of such equations have been proposed (see, for example Newman et al. 1981; Jones and Musonge 1983; Shih et al. 1985; Craft and Launder 1989; Nagano et al. 1991) although it must be conceded that few of these have been applied over a very wide range of ﬂows, and there is thus relatively little agreement on the exact form that such an equation should take. In the examples below, in order to focus attention on the modelling of the scalar ﬂuxes, the timescale r has either been prescribed (from available data) of obtained from the correlation of equation (4.16). A comprehensive account of recent approaches to modelling the εθ equation in near-wall heat transport is provided in [6]. 5 Applications to the computation of the scalar ﬁeld in free shear ﬂows Many important applications where scalar transport is of interest involve the prediction of heat or mass transfer rates to or from a solid surface. In such situations, however, the overall scalar transport is dominated by the ﬂow behaviour in the near-wall sublayer, where viscous eﬀects must be considered. Since, in this chapter, only high-Reynolds-number modelling has been considered, the examples presented relate only to free ﬂows: in particular, the scalar ﬁeld development in simple shear ﬂows and in the plane and round jets. 2 r is the reciprocal of the timescale ratio R introduced in Chapter [2]. 122 Craft and Launder Figure 10: Thermal ﬁeld development in weakly strained homogeneous shear ﬂow. Solid line: TCL model, Broken line: Basic RSM. From Craft (1991). Figure 11: Thermal ﬁeld development in moderately strained homogeneous shear ﬂow. Solid line: TCL model, Broken line: Basic RSM, Symbols: measurements of Tavoularis and Corrsin (1981). From Craft (1991). Figures 10 and 11 show scalar ﬁeld results in homogeneous shear ﬂows, with a mean scalar gradient applied in the same direction to the shear. The ﬁgures plot the development of the ratio of streamwise to cross-stream scalar ﬂuxes, and the turbulent Prandtl number, deﬁned as σt = (uv dΘ/dy)/(vθ dU/dy), against non-dimensional distance along the wind tunnel, τ = (x/U )dU/dy. Figure 10 corresponds to a case with a relatively low mean strain, resulting in turbulence not too far from local equilibrium, whilst Figure 11 relates to the case measured by Tavoularis and Corrsin (1981) at a higher mean strain rate. Although both the TCL and the widely used linear Basic Model give reasonable predictions when the ﬂow is close to local equilibrium, the additional terms built into the TCL model clearly give much better predictions of the scalar ﬂuxes at the higher strain rate. [3] Closure modelling near the two-component limit 123 As was seen in Section 3.1, the TCL model resulted in a better prediction of the hydrodynamic spreading rates of free jets than did the Basic Model. Table 3 shows the predicted scalar spreading rates in the plane and axisymmetric jets, obtained with both the TCL and the Basic models, together with experimental values. The predicted values are certainly not unreasonable, and the TCL model arguably returns slightly better predictions, although there is clearly room for further improvement. As discussed by Craft (1991), however, the TCL results can be improved by a more elaborate modelling of the timescale ratio r. Experiment Basic Model TCL Model Plane Jet 0.140 0.145 0.132 Round Jet 0.110 0.131 0.127 Table 3: Scalar ﬁeld spreading rates of free jets in stagnant surroundings. Further applications of the models, to buoyancy-aﬀected ﬂows and to separated and impinging ﬂows, will be presented in [14] and [11]. References Batten, P., Craft, T.J., Leschziner, M.A. (1999a), ‘Reynolds-stress modeling of afterbody ﬂows’, in Turbulence and Shear Flow Phenomena 1 (S. Banerjee, J. Eaton, eds.), Begell House, New York. Batten, P., Craft, T.J., Leschziner, M.A., Loyau, H. (1999b), ‘Reynolds-stresstransport modeling for compressible aerodynamics applications’, AIAA J., 37, 785– 796. Chang, S.M., Humphrey, J.A.C., Modavi, A. (1983), ‘Turbulent ﬂow in a strongly curved U-bend and downstream tangent of square cross section’, Phys. Chem. Hydrodyn., 4, 243–269. Craft, T.J. (1991), ‘Second-moment modelling of turbulent scalar transport’, Ph.D. thesis, Faculty of Technology, University of Manchester. Craft, T.J., Fu, S., Launder, B.E., Tselepidakis, D.P. (1989), ‘Developments in modelling the turbulent second-moment pressure correlations’, Tech. Rep. Report TFD/89/1, Dept. of Mech. Eng., UMIST. Craft, T.J., Ince, N.Z., Launder, B.E. (1996), ‘Recent developments in second-moment closure for buoyancy-aﬀected ﬂows’, Dynamics of Atmospheres and Oceans, 23, 99– 114. Craft, T.J., Kidger, J.W., Launder, B.E. (2000), ‘Second-moment modelling of developing and self-similar three-dimensional turbulent free-surface jets’, Int. J. Heat Fluid Flow, 21, 338–344. 124 Craft and Launder Craft, T.J., Launder, B.E. (1989), ‘A new model for the pressure/scalar-gradient correlation and its application to homogeneous and inhomogeneous free shear ﬂows’, in Proc. 7th Turbulent Shear Flows Symposium, Stanford University. Craft, T.J., Launder, B.E. (1996), ‘A Reynolds stress closure designed for complex geometries’, Int. J. Heat Fluid Flow, 17, 245–254. Craft, T.J., Launder, B.E. (1999), ‘The self-similar, turbulent, three-dimensional wall jet’, in Turbulence and Shear Flow Phenomena 1 (S. Banerjee, J. Eaton, eds.), Begell House, New York. Craft, T.J., Launder, B.E. (2001), ‘On the spreading mechanism of the threedimensional wall jet’, J. Fluid Mech. 435, 305–326. Daly, B.J., Harlow, F.H. (1970), ‘Transport equations in turbulence’, Phys. Fluids, 13, 2634–2649. El Baz, A., Craft, T.J., Ince, N.Z., Launder, B.E. (1993), ‘On the adequacy of the thin-shear-ﬂow equations for computing turbulent jets in stagnant surroundings’, Int. J. Heat Fluid Flow, 14, 164–169. El Baz, A.M.R. (1992), ‘The computational modelling of free turbulent shear ﬂows’, Ph.D. thesis, Faculty of Technology, University of Manchester. Fu, S. (1988), ‘Computational modelling of turbulent swirling ﬂows with secondmoment closures’, Ph.D. thesis, Faculty of Technology, University of Manchester. Fu, S., Launder, B.E., Tselepidakis, D.P. (1987), ‘Accommodating the eﬀects of high strain rates in modelling the pressure-strain correlation’, Tech. Report TFD/87/5, Dept. of Mech. Eng., UMIST. Hinze, J.O. (1973), ‘Experimental investigation on secondary currents in the turbulent ﬂow through a straight conduit’, Appl. Sci. Res., 28, 453. Iacovides, H., Launder, B.E., Li, H.-Y. (1996), ‘Application of a reﬂection-free DSM to turbulent ﬂow and heat transfer in a square-sectioned U-bend’, Exp. Thermal and Fluid Science, 13, 419–429. Jones, W.P., Musonge, P. (1983), ‘Modelling of scalar transport in homogeneous turbulent ﬂows’, in Proc. 4th Turbulent Shear Flow Symposium, 17.18–17.24 Karlsruhe. Kim, J., Moin, P., Moser, R. (1987), ‘Turbulence statistics in fully developed channel ﬂow at low Reynolds number’, J. Fluid Mech., 177, 133–166. Launder, B.E. (1973), ‘Scalar property transport by turbulence’, Tech. Report HTS/73/26, Mech. Eng. Dept., Imperial College, London. Launder, B.E. (1975), Course notes, Lecture Series No 76, von Karman Inst. RhodeSt.-Gen`se, Belgium. e Launder, B.E., Li, S.-P. (1994), ‘On the elimination of wall-topography parameters from second-moment closure’, Phys. Fluids, 6, 999–1006. Launder, B.E., Reece, G.J., Rodi, W. (1975), ‘Progress in the development of a Reynolds stress turbulence closure’, J. Fluid Mech., 68, 537. Launder, B.E., Shima, N. (1989), ‘Second-moment closure for the near-wall sublayer: development and application’, AIAA J., 27, 1319–1325. [3] Closure modelling near the two-component limit 125 Launder, B.E., Tselepidakis, D.P. (1993), ‘Contribution to the modelling of near-wall turbulence’, in Turbulent Shear Flows 8 (F. Durst, R. Friedrich, B.E. Launder, F.W. Schmidt, U. Schumann, J.H. Whitelaw, eds.), Springer-Verlag, New York. Li, S.-P. (1992), ‘Predicting riblet performance with engineering turbulence models’, Ph.D. thesis, Faculty of Technology, University of Manchester. Lumley, J.L. (1975), ‘Prediction methods for turbulent ﬂows – Introduction’, VKI Course notes, no. 76, von Karman Inst. Rhode-St-Gen`se, Belgium. e Lumley, J.L. (1978), ‘Computational modelling of turbulent ﬂows’, Adv. Appl. Mech., 18, 123. Matsumoto, A., Nagano, Y., Tsuji, T. (1991), ‘Direct numerical simulation of homogeneous turbulent shear ﬂow’, in Proc. 5th Symposium on Computational Fluid Dynamics, Tokyo. McGuirk, J.J., Papadimitriou, C. (1985), ‘Buoyant surface layers under fully entraining and internal hydraulic jump conditions’, in Proc. 5th Symposium on Turbulent Shear Flows, Cornell University. Monin, A.S. (1965), ‘On the symmetry of turbulence in the surface layer of air’, Izv. Atm. Oceanic Phys., 1, 45. Nagano, Y., Tagawa, M., Tsuji, T. (1991), ‘An improved two-equation heat transfer model for wall turbulent shear ﬂows’, in Proc. ASME/JSME Thermal Engng. Joint Conference, 3, 233–240, Reno, USA. Naot, D., Shavit, A., Wolfshtein, M. (1973), ‘Two-point correlation model and the redistribution of Reynolds stresses’, Phys. Fluids, 16, 738. Newman, G.R., Launder, B.E., Lumley, J.L. (1981), ‘Modelling the behaviour of homogeneous scalar turbulence’, J. Fluid Mech., 111, 217–232. Rajaratnam, N., Humphries, J.A. (1984), ‘Turbulent non-buoyant surface jets’, J. of Hydraulic Research, 22, 103–115. Rotta, J. (1951), ‘Statistische Theorie nichthomogener Turbulenz’, Zeitschrift f¨r u Physik, 129, 547. Schumann, U. (1977), ‘Realizability of Reynolds stress turbulence models’, Phys. Fluids, 20, 721–725. Shih, T.-S., Lumley, J.L. (1985), ‘Modeling of pressure correlation terms in Reynolds stress and scalar ﬂux equations’, Tech. Rep. Report FD-85-03, Sibley School of Mechanical and Aerospace Eng., Cornell University. Shih, T.-S., Lumley, J.L., Chen, J.-Y. (1985), ‘Second order modelling of a passive scalar in a turbulent shear ﬂow’, Tech. Rep. Report FD-85-15, Sibley School of Mechanical and Aerospace Eng., Cornell University. Shima, N. (1993), ‘Prediction of turbulent boundary layers with a second-moment closure: Part 1. eﬀects of periodic pressure gradient, wall transpiration and freestream turbulence’, J. Fluids Eng., 115, 56–63. Shima, N. (1998), ‘Low-Reynolds-number second-moment closure without wallreﬂection redistribution terms’, Int. J. Heat Fluid Flow, 19, 549–555. 126 Craft and Launder Tavoularis, S., Corrsin, S. (1981), ‘Experiments in nearly homogeneous turbulent shear ﬂow with a uniform mean temperature gradient. part 1.’, J. Fluid Mech., 104, 311– 347. Tselepidakis, D.P. (1991), ‘Development and application of a new second-moment closure for turbulent ﬂows near walls’, Ph.D. thesis, Faculty of Technology, University of Manchester. 4 The Elliptic Relaxation Method P.A. Durbin and B.A. Pettersson-Reif 1 Non-local wall eﬀects The elliptic nature of wall eﬀects was recognized early in the literature on turbulence modeling (Chou 1945) and has continued to inﬂuence thoughts about how to incorporate non-local inﬂuences of boundaries (Launder et al. 1975). In the literature on closure modeling the non-local eﬀect is often referred to as ‘pressure reﬂection’ or ‘pressure echo’ because it originates with the surface boundary condition imposed on the Poisson equation for the perturbation pressure, p. The Poisson equation is ∇2 p = −2 ∂uj ∂ui ∂Uj ∂ui ∂uj ∂ui − + ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj (1.1) (we are considering constant density ﬂow ρ ≡ 1); the boundary condition is usually taken to be ∂p/∂xn = 0, ignoring a small viscous contribution. The boundary condition inﬂuences the pressure of the interior ﬂuid through the solution to (1.1). Mathematically this is quite simple: the solution to the linear equation (1.1) consists of a particular part, forced by the right-hand side, and a homogeneous part, forced by the boundary condition. The fact that the boundary condition adds to the solution interior to the ﬂuid can be described as a non-local, kinematic eﬀect. Figure 1 schematizes non-locality in the Poisson equation as a reﬂected pressure wave, but for incompressible turbulent ﬂuctuations the wall eﬀect is instantaneous, though non-local. Pressure reﬂection enhances pressure ﬂuctuations; indeed, Manceau et al. (2001) show that pressure reﬂection can increase redistribution of Reynolds stress anisotropy. Redistribution is due to the pressure-strain correlation: the notion that it is increased by the wall eﬀect is contrary to most second moment closure (SMC) models, which represent pressure echo as a reduction of the redistribution term. The idea of associating inviscid wall eﬀects with pressure reﬂection is natural, because the pressure enters the Reynolds stress transport equation through the velocity-pressure gradient correlation. Suppression of the normal component of pressure gradient by the wall should have an eﬀect on the rate of redistribution of variance between those components of the Reynolds stress tensor that contain the normal velocity component – i.e., un ui , where n denotes the wall-normal direction. This eﬀect enters the evolution equation for the Reynolds stress (equation (1.3) below). 127 128 Durbin and Pettersson-Reif PRESSURE REFLECTION IMAGE VORTICITY Figure 1: Schematic representations of non-local wall inﬂuences. However, there is another notion about how anisotropy of the Reynolds stress tensor is altered non-locally by the presence of a wall. The inviscid boundary condition on the normal component of velocity is the no-ﬂux condition u · n = 0. This constraint on the normal velocity produces another non-local, elliptic inﬂuence of the boundary. If the vorticity, ωk , is known the velocity solves the kinematic equation ∇2 ui = − ijk ∂ωk . ∂xj (1.2) The boundary condition u · n = 0 alters the ﬂow interior to the ﬂuid. At a plane boundary this is termed the ‘image vorticity’ eﬀect (ﬁgure 1); for instance, (1.2) can be solved by extending the tangential vorticity components anti-symmetrically inside the boundary, and extending the normal component symmetrically into the boundary. Again, the wall alters the ﬂow interior to the ﬂuid through non-local kinematics. This perspective on ellipticity is often referred to as ‘kinematic blocking’ (Phillips 1955, Hunt and Graham 1978). Kinematic blocking is not an alteration of the redistribution tensor. It can be perceived as a continuity eﬀect: instead of (1.2), suppose that a homogeneous ﬁeld of turbulence u∞ exists, and instantaneously a wall is inserted. Then instantaneously the velocity will be altered to u∞ − ∇φ, where φ is a velocity potential. Incompressibility, ∇ · u = 0 implies that ∇2 φ = 0. The boundary condition is n · ∇φ = u∞ · n. Indeed, without invoking a continuity equation, it is diﬃcult to locate kinematic blocking. The Reynolds stress transport equations are moments of the momentum equations alone; continuity does not add extra single-point moment equations. Hence, the blocking eﬀect has no direct representation in single-point models. The concept of non-local, elliptic wall eﬀects originates in exact kinematics, but the practical question of how to incorporate non-locality into single-point [4] The elliptic relaxation method 129 moment closures is somewhat elusive. Exact expressions can be derived, but they are unclosed because the single-point statistics are found to be functions of two-point correlations. Hence, it is not possible to include the exact formulations in a Reynolds stress transport model. Research in this area has sought circuitous methods to represent wall inﬂuences. The aim of research into modeling is to invoke closure assumptions that lead to tractable analytical formulations. These must be suited to the task of predicting the mean ﬂow and Reynolds stresses. Three approaches may be said to exist on this subject: the ‘wall echo’ device, ﬁrst introduced by Shir (1973) and adopted in alternative forms in most models for the subsequent two decades; the use of a nonlinear expansion with coeﬃcients tuned to comply with the two-component limit (see [3] and [14]) and the elliptic-relaxation strategy that forms the focus of the present contribution. The wall echo formulation of Launder et al. (1975) invokes an additive correction to the homogeneous redistribution model. The additive term is a function of turbulence length scale, L, divided by wall distance, d. This functional dependence is used to reduce the wall correction to zero at distances far enough from the surface that ellipticity no longer alters the turbulence statistics. In the Reynolds stress transport equation, Dui uj + φij − εij + Tij = Pij Dt production redistribution dissipation transport the redistribution tensor is written φij = φh + φw ij ij (1.4) + ν∇2 ui uj viscous (1.3) where φh is a homogeneous pressure-strain model, such as the simple IP model, ij 1 φh = −c1 εaij − c2 Pij − Pkk δij . ij 3 (1.5) Here aij ≡ ui uj /k − 2 δij is the anisotropy tensor. The term φw in (1.4) is ij 3 the additive wall correction. In addition to wall distance, d, it is a function of the wall normal n. The latter must be used interior to the ﬂuid, where it is ill deﬁned. (For example, if it is deﬁned as the normal vector at the nearest wall location then it is discontinuous on a surface emanating from corners. While one might deﬁne a ‘wall normal’ function to be the gradient of a smooth ‘wall distance’ function, that has not been used in the literature.) For concreteness, formulas that have been used in conjunction with the IP model (Gibson and Launder 1978), can be cited. Their basic type is φw = cw ij 1 ε L 3 3 um ul nm nl δij − ui um nm nj − uj um nm ni + ··· k 2 2 d (1.6) 130 Durbin and Pettersson-Reif Here · · · indicates that further terms are contained in the formulas that have been used in the literature. Expression (1.6) illustrates that wall corrections are tensorial operators that act on the Reynolds stress tensor. The ni dependence of these operators has to be adjusted to properly damp each component of ui uj (Craft et al. 1993). The formula for the correction function, φw , has to ij be readjusted in this manner for each homogeneous redistribution model to which it is applied (Lai and So 1990). Elliptic relaxation (Durbin 1993) is a rather diﬀerent approach to wall effects. Instead of adding a wall echo term, the homogeneous redistribution model is used as the source in a non-homogeneous, modiﬁed Helmholtz equation1 equation: φh ij 2 2 L ∇ fij − fij = − . (1.7) k Here fij is an intermediate variable, related to the redistribution tensor by φij = kfij . The turbulent kinetic energy, k, is used as a factor in order to enforce the correct behavior, φij → 0, at a no-slip boundary. The anisotropic inﬂuence of the wall on Reynolds stresses interior to the ﬂuid arises by imposing suitable boundary conditions on the components of the ui uj − fij system. For instance, at a no-slip surface the normal intensity un un = O(d4 ) and tangential intensity ut ut = O(d2 ) can be given their correct asymptotic limits. The wall normal now enters only into the wall boundary condition. The precise form of elliptic relaxation models can be found in Durbin (1993), Wizman et al. (1996) and Pettersson and Andersson (1997). In these references the general formulation is applied to several particular φh models: IP, LRR ij (Launder et al. 1975), SSG (Speziale et al. 1991), FLT (Fu et al. 1987) and RLA (Ristorcelli et al. 1995). While the wall echo approach (1.6) requires a completely new formulation for each model, the elliptic relaxation equation (1.7) is unchanged, except for the source on the right side. Implementation of a new model into a computer code can be done entirely in a subroutine that deﬁnes φh . ij 2 A justiﬁcation The elliptic relaxation formulation can be justiﬁed by a modiﬁcation to the usual rationale for pressure-strain modeling. The analysis in this section is in the vein of formulating a template for elliptic relaxation, rather than being a derivation per se. For simplicity write the Poisson equation for the pressure ﬂuctuation (1.1) as ∇2 p = S(x). (2.1) The ‘modiﬁed’ Helmholtz equation is ∇2 φ − k2 φ = 0; the unmodiﬁed equation has a + sign. 1 [4] The elliptic relaxation method 131 The zero normal derivative condition on p can be imposed on a plane boundary by extending the source, S(x), symmetrically into the surface. Let that extension be understood. Then a formal solution to the Poisson equation is obtained by inverting (2.1) with its free-space Green function: p(x) = − 1 4π S(x ) 3 d x. |x − x | (2.2) The redistribution term in the exact, unclosed second moment closure equations includes the velocity-pressure gradient correlation. Diﬀerentiating (2.2) with respect to xj , then integrating by parts, give ui ∂p 1 (x) = j ∂x 4π ui (x) ∂S(x ) 3 ∂xj d x. |x − x | (2.3) after multiplying by ui and averaging. The integrand of (2.3) contains a twopoint correlation. This embodies one origin of the lack of closure of the singlepoint moment equations: single-point moment equations depend on two-point statistics. If the turbulence is homogeneous, the non-local closure problem is masked by translational invariance of two-point statistics. Analyses for homogeneous turbulence (Chou 1945) note at this point in the derivation that ui (x) ∂S(x ) ∂xj is a function of x − x alone, so that the integral in (2.3) is a constant second order tensor, φh : ij ui ∂p 1 (x) = j ∂x 4π F cnij (x − x ) 3 d x = φh ij |x − x | (2.4) where φh is not a function of x. ij The original form of the source in (1.1) motivates the standard practice of splitting φh into slow and rapid parts, ij φh = Nij + Mijkl ij ∂Ul . ∂xk (2.5) The second (rapid) term is motivated by the ﬁrst term in (1.1); the ﬁrst (slow) term is motivated by the second, nonlinear term in (1.1). The split (2.5) is invoked in the homogeneous redistribution models that are used on the right side of (1.7); for instance the ﬁrst term of (1.5) is the slow part, with Nij = −c1 εaij , and the second is the rapid part, with Mijkl = −c2 (ui uk δjl + uj uk δil − 2 uk ul δij ). 3 If the turbulence is not homogeneous (2.4) is not applicable and the role of non-homogeneity in (2.3) must be examined (Manceau et al. 2001). This requires a representation of the spatial correlation function in the integrand. To 132 Durbin and Pettersson-Reif this end, an exponential function will be used as a device to introduce the correlation length of the turbulence into the formulation. Letting ui (x) ∂S(x ) = ∂xj Rij (x )e−|x−x |/L in (2.3) gives ui ∂p (x) = ∂xj e−|x−x |/L 3 d x. 4π|x − x | Rij (x ) (2.6) The representation of non-locality in this formula can be described as both geometrical and statistical. (•) e−|x−x |/L statistical decorrelation 4π|x − x | d3 x . geometrical spreading The kernel in the above equation is the Green’s function for the modiﬁed Helmholtz equation (1.7); the large dot stands for the source, Rij in the present case. In other words, if L is constant, then (2.6) is the solution to ∇2 u i ∂p ∂p 1 − 2 ui = Rij . ∂xj L ∂xj (2.7) Of course, the speciﬁc equation (1.7) could be obtained by dividing (2.3) by k(x) and letting fij = ui ∂p/∂xj /k and Rij = φh /kL2 . However, the present ij rationalization of (2.7) is meant only to suggest a template for (1.7). Ultimately the motivation for the elliptic relaxation method is to enable boundary conditions and anisotropic wall eﬀects to be introduced into the second-moment closure model in a ﬂexible and geometry-independent manner. The elliptic relaxation procedure accepts a homogeneous redistribution model on its right side and operates on it with a Helmholtz type of Green’s function, imposing suitable wall conditions. The net result can be a substantial alteration of the near-wall behavior from that of the original redistribution model. Figure 2 illustrates how elliptic relaxation modiﬁes the SSG (Speziale et al. 1991) homogeneous redistribution model. The dashed lines in this ﬁgure are the homogeneous SSG model for φh , evaluated in plane channel ﬂow with ij friction velocity Reynolds number of Rτ = 395. When this φh is used as the ij source term in (1.7) the dark solid line is obtained as solution. The circles are DNS data for the redistribution term (Mansour et al. 1988). The redistribution model is shown for the uv, v 2 and u2 components. The elliptic relaxation solution alters the redistribution term rather dramatically when y + < 60. The magnitude and sign of φh are quite wrong; however 12 ∼ φ12 predicted by elliptic relaxation agrees quite well with the data. Equation (1.7) is linear, so its general solution can be written as a particular part, forced by the source, plus a homogeneous part that satisﬁes the boundary condition. The particular part would tend to have the same sign as the source and could [4] The elliptic relaxation method 0.1 0.20 133 uv component 0 v2 component 200 0.15 -0.1 0.10 -0.2 ℘12 via elliptic relaxation h φ 12 SSG formula DNS data 0.05 ℘12 -0.3 0 100 0 0 100 200 y+ 0 y+ u2 component -0.1 -0.2 -0.3 0 100 200 y+ Figure 2: The eﬀect of elliptic relaxation on the SSG formula for φh . Compuij tation of plane channel ﬂow with Rτ = 395. not cause the sign reversal shown by the solution for uv. So it is the homogeneous part of the general solution that causes the sign of φ12 to be opposite to φh near the wall, and brings φ12 into agreement with the DNS data. 12 A similar reduction in magnitude, and drastic improvement in agreement with the data, is seen also for φ11 and φ22 . Near the wall, the homogeneous model shows too large transfer out of u2 and into v 2 (and w2 ). The elliptic relaxation procedure greatly improves agreement with the data. This includes creation of a slightly negative lobe near the wall in the φ22 proﬁle, corresponding with the data. The negative lobe is required if v 2 is to be non-negative (see (3.5)). It is rather intriguing that the elliptic relaxation equation is able to automatically produce such improvement to the redistribution model. How this occurs is not well understood mathematically. 3 Its use with Reynolds Stress Transport equations In practice one solves (1.3) and (1.7) with a model for the transport term Tij to complete the closure. The Daly and Harlow (1970) formula Tij ≡ − ∂ ∂uk ui uj ∂ = νTkl ui uj ∂xk ∂xk ∂xl (3.1) 134 Durbin and Pettersson-Reif is commonly used. Here νTij is a turbulent eddy viscosity tensor that is given by (3.2) νTij = cµ ui uj T and T is the turbulence time-scale T = max k/ε, 6 ν/ε . (3.3) This deﬁnition of the time-scale simply uses the Kolmogoroﬀ scale, ν/ε, as a lower bound, applicable near the wall, and the integral scale k/ε outside the viscous wall layer. It is not meant to be valid in truly low Reynolds number turbulence (such as the ﬁnal period of decay), to which Kolmogoroﬀ scaling does not apply. The complete set of closed Reynolds stress transport and redistribution equations that have been used in most elliptic relaxation models to date are Dui uj + Ωl ( Dt ikl uk uj + jkl uk ui ) +ε ui uj k (3.4) ∂ ∂ ν ui uj + ν∇2 ui uj ∂xk Tkl ∂xl φh aij ij L2 ∇2 fij − fij = − − k T = Pij + φij + where Pij = −ui uk ∂Uj ∂Ui − uj uk − Ωl ( ikl uk uj + jkl uk ui ) ∂xk ∂xk is the production tensor and φij = kfij . The coordinate system rotation vector Ωl has been included; the equations are written in a rotating frame of reference for use in §4. For non-rotating frames Ωl = 0. It can be seen that one modiﬁcation has been made to the formulation described in §1: εij has been eliminated from (1.3) by adding εui uj /k to the left side of the transport equation and −aij /T to the right side of the elliptic relaxation equation. If the turbulence is homogeneous, then the solution to the second of equations (3.4) is φij = φh + ε ij ui uj 2 − δij . k 3 When this is inserted into the ﬁrst of (3.4), εui uj /k cancels from both sides. So, the aggregate eﬀect of this last modiﬁcation is to replace εij by the isotropic formula 2 δij ε in the quasi-homogeneous limit. This is the behavior far from 3 walls. Near to surfaces, anisotropic dissipation is lumped with the redistribution model: φij → φij + εij − εui uj /k. Equation (3.4) can be understood to model this term in aggregate. The mathematical motive for inserting εui uj /k on the left side of (3.4) is to ensure that all components of ui uj go to zero at least as fast as k as the wall [4] The elliptic relaxation method 135 is approached (Durbin 1993). In particular, the tangential components of ui uj are O(d2 ) as d → 0. The normal component becomes O(d4 ) if the boundary condition εun un (3.5) fnn = −5 lim d→0 k2 is imposed on d = 0. This condition is derived by noting that as d → 0 the dominant balance for the normal stress in the ﬁrst of (3.4) becomes ε un un ∂ 2 un un . = φnn + ν k ∂x2 k Assuming un un ∼ (d4 ) gives ε un un un u − 12ν 2 n = φnn = kfnn . k d But k → εd2 /2ν, so the left side is −5εun un /k, giving (3.5). [The asymptote k → εd2 /2ν follows from the limiting behavior of the k-equation ε = ν∂ 2 k/∂x2 d upon integrating twice with the no-slip condition k = ∂k/∂xd = 0 on d = 0.] Boundary conditions on slip surfaces can be derived in a similar manner, although there has been little work on that subject. A similar analysis shows that the boundary condition on the oﬀ-diagonal fnt must also be (3.5). It is not possible to impose the condition un ut = O(d3 ). However, this asymptotic behavior is only valid as d+ → 0, where molecular transport dominates over turbulent mixing. So, the precise power is less important than the condition un ut ν(∂Ut /∂xn ) as d+ → 0, that is met by the model. The tangential components ft1 t1 and ft2 t2 are only required to be O(1) as d → 0. This is because the dominant balance ε ut ut ∂ 2 ut ut + O(d2 ) = φtt + ν k ∂x2 d causes the viscous and dissipation terms to cancel: with ut ut ∼ (d2 ) and k/ε = d2 /2ν this equation gives φtt = O(d2 ). Guaranteeing the d2 behavior was the motivation for writing φij = kfij . As long as ftt = O(1) as d → 0 the correct tangential balance will be achieved. Demuren and Wilson (1995) use the condition ft1 t1 = ft2 t2 = −1/2fnn to ensure that φij is trace-free. In twodimensional ﬂows Durbin (1993) used ftt = 0 to obtain the Reynolds stresses in the x–y plane, and computed the third normal stress from w2 = 2k−u2 −v 2 . The Demuren and Wilson (1995) condition is probably more satisfactory in general (see also Durbin (1991), Appendix B). 136 Durbin and Pettersson-Reif The length scale in (3.4) is prescribed by analogy to (3.3) as L = max cL k 3/2 ε , cη ν3 ε 1/4 . (3.6) Although most implementations of elliptic relaxation to date have used these simple formulas for L and T , they are not crucial to the approach. The only important feature is that L and T do not vanish at no-slip surfaces. If they vanished then the equations would become singular. But it would also be unphysical for the turbulence scales to vanish at a wall: they represent the correlation length and time of turbulence, not the intensity, which are not zero. In fully turbulent ﬂow it has been found from direct numerical simulations that the Kolmogoroﬀ scaling collapses near-wall data quite eﬀectively. 3.1 Variants The elliptic relaxation approach can be invoked in a variety of manners, as is already implied by (3.4). In principle, there is freedom to choose the source term φh in the elliptic equation; the only constraint is that the model relax ij to φij − εij = φh − 2 εδij in the quasi-homogeneous limit. However, to date ij 3 the primary variations of this ilk have been to substitute diﬀerent existing closures—such as IP or SSG—for the source term. Variants of the elliptic operator have been explored by Wizman et al. (1996) and by Dreeben and Pope (1997). Wizman et al. (1996) note that in the constant stress, or logarithmic, layer the source term in (1.7) is proportional to 1/y and hence fij is as well. This means the the Laplacian of fij does not vanish. If fij is redeﬁned via φij = kfij /L then the source term becomes Lφh /k, making it and fij constant in the log-layer. This makes the Laplacian ij vanish. Wizman et al. (1996) entitle this the ‘neutral formulation’. They also consider replacing the Laplacian by ∇·(L2 ∇f ) = L2 ∇2 f +2L(∇L)·∇f . These modiﬁcations were explored in the interest of improving predictions in the central region of channel ﬂow. Research into such variants of the formulation is currently in progress (Manceau et al. 2001). Dreeben and Pope (1997) invoked the representation 2 φij = ui uk fkj + uj uk fki − δij ul uk fkl 3 in order to make elliptic relaxation compatible with Langevin stochastic models. This formulation automatically satisﬁes the redistribution property φii = 0. However, fij is then no longer symmetric. Although attractive in concept, this variant adds greatly to the computational complexity of the model. Elliptic relaxation has been simpliﬁed into a scalar eddy viscosity formulation by the v 2 -f model (Durbin 1995, Parneix et al. 1998a). To this end the [4] The elliptic relaxation method 137 set of equations for the Reynolds stress tensor is replaced by a pair of equations for a velocity scalar v 2 and a function f that is somewhat analogous to redistribution. The governing equations are v2 Dv 2 +ε Dt k L ∇ f −f 2 2 ∂U = kf + ∂v 2 ∂ νT + ν∇2 v 2 ∂xk ∂xk v2 2 /T − k 3 P = −c2 + c1 k ∂U (3.7) j j i where P = νT ∂xi + ∂Uj ∂xi . These are solved in conjunction with the k-ε ∂x equations, which are needed to obtain the length and time scales. The boundary condition on f is that of equation (3.5), for the normal component of Reynolds stress: f = −5 lim d→0 εv 2 ν 2v2 = −20 lim d→0 εd4 k2 (3.8) for a wall located at d = 0. The eddy viscosity is predicted by the formula νT = cµ v 2 T . The motivation behind (3.7) and (3.8) is to represent the tendency of the wall to suppress transport in the normal direction. The variable v 2 is a scalar, not the normal component of a tensor, but the boundary condition on f makes it behave like un un near to solid walls. The v 2 -f variant is motivated by the need for a practical prediction method. In this model the mean ﬂow is computed with an eddy viscosity. The transport equations for k, ε, and v 2 are solved to obtain the spatial distribution of eddy viscosity. Elliptic relaxation is not a panacea, but it has intriguing properties. Other avenues to geometry-independent near-wall modeling are treated elsewhere in this book. In particular, tensorally nonlinear representations for the twocomponent limit (Launder and Li 1994) are discussed in [3] and [14]. 4 4.1 Applications Insensitivity to the homogeneous model Figure 2 illustrated that near to walls the elliptic relaxation closure overwhelms the homogeneous redistribution model. There is an extent to which this makes the prediction of surface properties insensitive to the detailed homogeneous redistribution model. Figure 3 shows the friction coeﬃcient in ﬂow over a backward facing step using both the IP and SSG models. The two models give very similar solutions for Cf . The same is true of the entire mean ﬂow ﬁeld. Although this is not always the case, it is clear that in this particular calculation elliptic relaxation largely determines the solution. The speciﬁcs 138 Durbin and Pettersson-Reif 6 4 10 3 Cf 2 0 IP SSG DNS data -2 -5 0 5 10 15 20 25 x/H Figure 3: Friction coeﬃcient in ﬂow over a backward facing step. The step is at x = 0. Model solutions compared to DNS data from Le et al. (1997). of the homogeneous redistribution model are less important, although they are not irrelevant. In other ﬂows they play a role and the complexity of the homogeneous model also has an impact on numerical tractability An under-prediction of the minimum Cf is seen in ﬁgure 3 near x/H = 4. This is a common feature of predictions of most second-moment transport models in ﬂow over a backward step, irrespective of the near wall treatment (Parneix et al. 1998b). As a second example, consider the ﬂow over a convex bump. This geometry is characterized by substantial favorable and adverse pressure gradients. While that might seem at ﬁrst to make it a stringent test, in fact the experimental data of Webster et al. (1996) are predicted well by many models. Velocity proﬁles computed with the IP model via elliptic relaxation are shown on the left side of ﬁgure 4. Predictions by the SSG model are virtually indistinguishable from those portrayed. The closure model predictions are in excellent accord with the data. Although the data do not extend to the upper wall, the predicted boundary layer thickness must be correct; otherwise the central velocity would be wrong, due to the need to maintain a constant mass ﬂux. The friction coeﬃcient, Cf (x), and pressure coeﬃcient, Cp (x), predictions by either SSG or IP are also in good agreement with data; again, only predictions by the IP are shown in ﬁgure 4. In this case, as well as for the backstep, elliptic relaxation makes the results relatively insensitive to the homogeneous SMC model. Transport of mean momentum towards a solid boundary is controlled largely by a thin layer immediately next to the wall. Success in computing wallbounded ﬂows therefore depends strongly on the model used to account for the non-local wall eﬀects that occur in the proximity of solid boundaries. As [4] The elliptic relaxation method 139 0.5 0.4 0.3 0.75 0.50 0.25 100Cf Cp Y 0 0.2 -0.25 0.1 0 -0.5 -0.50 0 0.5 1 1.5 2 2.5 U profiles -0.75 -0.5 0 0.5 x 1.0 1.5 2.0 Figure 4: Velocity proﬁles, skin friction and surface pressure coeﬃcients in ﬂow over a bump. Solutions to the IP model with elliptic relaxation are compared to experimental data from Webster et al. (1996). elucidated in §3, the boundary conditions for the elliptic relaxation equation (3.4) assures a proper suppression of the normal component of turbulent transport in the vicinity of the surface. This has a large inﬂuence on skin friction predictions in the two examples considered so far. In strongly curved or rotating ﬂows the near-wall modeling inﬂuences relaminarization of the viscous wall layer. The majority of turbulent ﬂows of engineering interest are characterized by nonequilibrium near-wall turbulence as well as by eﬀects of inertial forces arising from streamline curvature or system rotation. The most natural level of closure modeling to adopt in these cases is full Reynolds stress transport models, which in a natural and systematic way account for rotational eﬀects. The next sections contain examples to illustrate ﬂows in which rotation and curvature are important. 4.2 Rotating cylinder Pettersson et al. (1996) considered the turbulent boundary layer around an inﬁnitely long, axially rotating cylinder in a quiescent ﬂuid. The cylinder rotates with a constant angular velocity ω as deﬁned in ﬁgure 5. The mean ﬂow equation in this case is ur uθ = R 2 τw ∂Uθ + νr 2 r ∂r with Uθ = ωR on the surface and Uθ → 0 as r → ∞. In the inviscid region ur uθ > 0 if τw > 0. The second moment closure model is required to predict ur uθ . Figure 5 compares predictions by the IP model in conjunction with both elliptic relaxation and with the Launder–Shima low-Reynolds number, second moment closure (Launder and Shima 1989). The Launder–Shima model 140 Durbin and Pettersson-Reif 1.0 IP Launder & Shima Exp ω R Uθ (r) r U/Uw 0.0 0.0 0.2 (r-R)/R 0.4 Schematic of ﬂow conﬁguration 0.4 Mean velocity distribution uv/2k 0.3 IP Launder & Shima Exp 0.2 0.1 0.0 10-3 10-2 Structural parameter Figure 5: Rotating cylinder in a quiescent ﬂuid Re ≡ ωR2 /ν = 20,000. Experimental data of Andersson et al. (1991). utilizes IP with the additive wall-echo methodology described prior to equation (1.6). Diﬀerences between the model predictions therefore can be attributed mainly to the method of near-wall modeling. In the limit as the solid boundary is approached, the ‘structural parameter’ uv/k should tend to 0 due to kinematic blocking. However, the additive wall-correction in Launder–Shima does not provide suﬃcient suppression of turbulent normal and shear stresses; the structural parameter predictions in ﬁgure 5 behave incorrectly near the surface. The mean ﬂow prediction in the upper part of ﬁgure 5 also is incorrect. When the same IP closure is used in conjunction with elliptic relaxation the predictions improve dramatically. Both the mean ﬂow and the structure parameter are brought into good agreement (r-R)/R 10-1 [4] The elliptic relaxation method 141 (a) (b) z Ω Uwall x y Figure 6: Schematic of plane channel ﬂow in orthogonal mode rotation. (a) Poiseuille ﬂow (b) Plane Couette ﬂow. with the experimental data. The importance of modeling suppression of the wall-normal intensity is made clear by this example. 4.3 Rotating Channel Flows Non-inertial frames of reference are encountered in a wide variety of engineering ﬂows. When the momentum equations are transformed to a rotating frame, a Coriolis acceleration, 2Ω × u, is added. One half of the Coriolis acceleration comes from transforming the time-derivative, the other half comes from rotation of the velocity components relative to an absolute frame. When the Reynolds stress transport equations (3.4) are similarly transformed, the frame rotation adds the same term, −Ωl ( ikl uk uj + jkl uk ui ), to both the time derivative and to the production tensors, by this same reasoning. It is important to distinguish these two contributions because the production tensor, Pij , often appears in closure models; the IP formula (1.5) is a case in point. Only the contribution of rotation to the production tensor should be added to the closure formula. If this is not done correctly the equations will not be coordinate frame independent. The Coriolis acceleration profoundly aﬀects turbulent ﬂow. Depending on magnitude and orientation of the rotation vector, Ω, relative the mean ﬂow vorticity, ω = ∇×U, turbulence can be augmented or reduced: the turbulence is suppressed if the imposed rotation is cyclonic (that is, if the mean ﬂow vorticity is parallel to the rotation vector); the turbulence is enhanced if the imposed rotation is anticyclonic. An imposed system rotation may also contribute to the formation of organized large scale structures. These roll-cells are found in turbulent ﬂows subjected to anticyclonic rotation (Andersson 1997). The presence of a rotationinduced secondary mean ﬂow inevitably alters the turbulence ﬁeld as well. 142 Durbin and Pettersson-Reif U/Ub 1.0 Ro = 0.05 Ro = 0.10 Ro = 0.20 -uv/u*2 0 1.0 DNS RLA 0.0 0.0 Ro = 0.05 0.0 0.0 0.0 DNS RLA Ro = 0.10 Ro = 0.50 0.0 Ro = 0.20 0.0 1.0 Ro = 0.50 0.0 -1.0 0.0 y/h -1.0 0.0 y/h 1.0 Mean velocity distribution 0.5 Turbulent shear stress distribution 0.0 Ro = 0.05 -uv/k 0.0 Ro = 0.10 0.0 Ro = 0.20 0.0 DNS RLA Ro = 0.50 0.0 -1.0 y/h 1.0 Structural parameter Figure 7: Spanwise rotating Poiseuille ﬂow at Re∗ ≡ u∗ h/ν = 194; Ro ≡ Ω2h/Ub . DNS: Kristoﬀersen and Andersson (1993). The mean ﬂow ﬁeld can be directly altered by the imposed rotation, or it can respond indirectly in consequence of alterations to the turbulence. Test cases where the latter dominates are attractive because the response of the closure model to system rotation is then all important. Fully developed turbulent ﬂow between two inﬁnite parallel planes in orthogonal mode rotation constitutes one such example. Experimental (Johnston et al. 1972) and DNS data bases (Kristoﬀersen and Andersson 1993; Lamballais et al. 1996) are available to assist the model development. These data have contributed to making this particular ﬂow a standard benchmark test case. The most frequently adopted conﬁguration is pressure-driven (Poiseuille) ﬂow subjected to spanwise rotation (case a of ﬁgure 6). In the absence of [4] The elliptic relaxation method 143 rotation—and in laminar ﬂow, even with rotation—the mean velocity proﬁle is symmetric about the middle of the channel. Imposed rotation breaks the mean ﬂow symmetry. The mechanism is indirect; rotation alters the turbulence ﬁeld, producing asymmetry in the Reynolds shear stress. Because the mean ﬂow vorticity ω changes sign across the channel, the ﬂow ﬁeld is simultaneously subjected to both cyclonic and anticyclonic rotation. The side on which the turbulence is suppressed (enhanced) is usually referred to as the stable (unstable) side of the channel. In ﬁgure 6 the vorticity in the upper part of the channel is counter-rotating with the frame; this is destabilizing. The increase in turbulent mixing on that side steepens the velocity proﬁle. Asymmetry develops as shown in the ﬁgure; it can be seen more clearly in ﬁgure 7. Note that y is increasing downward in ﬁgure 6. The proﬁles in the upper left of ﬁgure 7 are consistent with ﬁgure 6 if the former are rotated clockwise by 90◦ . At high cyclonic rotation rates, the ﬂow ﬁeld tends to relaminarize. Nearwall modeling then plays a crucial role. The commonly used wall-function approach assumes fully developed turbulence and must be abandoned in this case. Figure 7 displays model predictions by Pettersson and Andersson (1997) of unidirectional, fully developed rotating Poiseuille ﬂow, U = [U (y), 0, 0] and Ω = [0, 0, Ω(y)]. The transport of mean momentum is governed by 0 = − ∂P ∗ ∂y ∂P ∗ ∂U ∂ ν + − uv ∂x ∂y ∂y ∂v 2 = −2ΩU − ∂y (4.1) where P ∗ = P − 1 Ω2 (x2 + y 2 ) is the mean reduced pressure. The Coriolis ac2 celeration does not directly aﬀect the mean ﬂow: it is balanced by the pressure and turbulent normal stress in the y-direction, as stated in the y-momentum equation. In the x-momentum equation ∂P ∗ /∂x is a constant. If the ﬂow were laminar, U would be independent of Ω. In turbulent ﬂow Coriolis accelerations appear in the Reynolds stress transport equations; rotation thereby alters the mean ﬂow through uv. The computations of ﬁgure 7 were performed with the highly complex Ristorcelli et al. (1995) model (RLA). They are compared with DNS of Kristoffersen and Andersson (1993). The model predictions exhibit many of the eﬀects of the Coriolis acceleration upon the turbulence and mean ﬂow ﬁelds. These include the almost irrotational core region where Ω ≈ 2dU/dy, the diminution of the Reynolds stresses with increased rotation on the y/h = 1 side of the channel. The stable side of the channel has essentially laminarized when Ro ≈ 0.5. Frame rotation sometimes causes secondary ﬂows. Longitudinal roll-cells on the unstable side of rotating Poiseuille channel ﬂow have been seen experimentally (Johnston et al. 1972) and numerically (Kristoﬀersen and Andersson 144 Durbin and Pettersson-Reif 1.0 〈U z /Uw 〈 DNS RLA Ro = 0.0 0.5 Ro = 0.1 0.0 0.0 0.0 0.0 1.0 Ro = 0.2 y/h 2.0 Spanwise averaged mean streamwise velocity U(2h)= Uw y z U(0)= 0 Ω Figure 8: Spanwise rotating plane Couette ﬂow at Re ≡ Uw h/ν = 2600 and Ro ≡ Ω2h/Uw = 0.1. Vectors: secondary ﬂow ﬁeld in a plane perpendicular to the streamwise direction; Contours: streamwise mean velocity. From Andersson et al. (1998). 1993). However, the observed roll-cell patterns were not steady. In this case the turbulence model is assumed to represent the entire spectrum of unsteady motion. But, in rotating Couette ﬂow, which is case b of ﬁgure 6, steady roll cells have been observed. A computation of such ﬂow should include all three velocity components to allow such cells to form. They were obtained in the following computations of plane turbulent Couette ﬂow subjected to spanwise rotation. In contrast to the pressure-driven channel ﬂow, when the ﬂow ﬁeld is driven solely by a moving wall the mean vorticity is single signed and the mean velocity proﬁle is antisymmetric (ﬁgure 6b). Hence the ﬂow is exposed entirely [4] The elliptic relaxation method 145 either to cyclonic or to anticyclonic rotation. The antisymmetry of the mean ﬂow ﬁeld is therefore preserved in a noninertial frame of reference. Pettersson and Andersson (1997) computed plane Couette ﬂow subjected to anticyclonic rotation, assuming a unidirectional mean ﬂow ﬁeld and ignoring the roll cells. That assumption met with limited success. It was conjectured that the signiﬁcant discrepancies that existed between model predictions and DNS data could be attributed to organized large scale structures. The latter were observed in the DNS of Bech and Andersson (1997). In that study it was found that the rotation induced streamwise vortices within an intermediate range of rotation numbers. The vortices spanned the entire channel, from top to bottom. In contrast to rotating Poiseuille ﬂow, these roll-cells were observed to be steady. They should truly be considered part of the mean ﬂow. Correct usage of Reynolds averaged closure then requires that the vortices be computed explicitly; the turbulence model is only responsible for the incoherent portion of the ﬂuid motion. Andersson et al. (1998) therefore adopted a more correct and physically appealing approach: they treated the secondary ﬂow as an integral part of a two-dimensional, three-component mean velocity ﬁeld. The ﬂow ﬁeld was of the form U = [U (y, z), V (y, z), W (y, z)]. The turbulence model was then left to represent only the real turbulence. The mean ﬂow roll-cells appeared automatically in the computation. In this case the governing mean momentum equations for the two-dimensional, three-component ﬂow ﬁeld are 0 = 2ΩV + ∂P ∗ ∂y ∂P ∗ ∂z ∂ ∂U ∂ ∂U ν ν − uv + − uw ∂y ∂y ∂z ∂z ∂V ∂ ∂V ∂ ν ν − v2 + − vw ∂y ∂y ∂z ∂z = −2ΩU + = ∂W ∂ ∂W ∂ ν ν − vw + − w2 ∂y ∂y ∂z ∂z ∂V ∂W + ∂y ∂z (4.2) 0 = All six components of the Reynolds stress tensor are required in this ﬂow. Even though u2 does not appear in the mean ﬂow equations (4.2), it enters the computation because the model depends on k. The full set of Reynolds stress transport and elliptic relaxation equations (3.4) were solved. Figure 8 shows the predicted secondary ﬂow in a cross-plane of the channel. The maximum value of the secondary ﬂow at Ro = 0.1 is approximately 13% of the mean streamwise bulk velocity. The width of each of the roll-cells is about 146 Durbin and Pettersson-Reif Figure 9: Axially rotating pipe ﬂow at Re ≡ 2RUb /ν = 20000. N ≡ Uθwall /Ub . Experiment by Imao et al. (1996). equal to the height of the channel. The dimensions of the cells are relatively insensitive to the width of the computational domain, as long as it is large enough to capture the pair of vortices; the particular vortices illustrated here are from a computation with a domain wide enough to capture two complete pairs. The predicted spanwise averaged, mean streamwise velocity is shown in the upper part of ﬁgure 8; it is in excellent agreement with the DNS data. A word of warning: it should be emphasized that the validity of this approach requires a truly steady roll-cell pattern. When such steady secondary ﬂow is present it should be computed as part of the mean ﬂow. Similarly if coherent, periodic vortex shedding is present, it too will be part of the mean ﬂow computation. Turbulence models are not designed to represent determin- [4] The elliptic relaxation method 147 istic structures. But if large-scale, incoherent unsteadiness is present it must be regarded as part of the turbulence and to lie within the province of the model.2 4.4 Axially Rotating Pipe Fully developed turbulent ﬂow inside an axially rotating pipe constitutes another interesting test case. It has relevance to internal cooling in turbomachinery. The ﬂow ﬁeld can be assumed to be one-dimensional with two velocity components: U = [0, Uθ (r), Uz (r)] — see ﬁgure 9. The imposed circumferential wall velocity, Uθ,wall , is stabilizing. It suppresses the turbulence across the entire pipe. The direction of the mean velocity changes rapidly through the wall-layer, from being circumferential at the wall, to becoming nearly axial away from the vicinity of the wall. This rapid turning of the mean ﬂow direction close to a solid boundary is analogous to what is seen in three-dimensional turbulent boundary layers. It produces skewing between axes of the Reynolds stress and of the mean rate of strain. Therefore it is desirable to integrate the governing set of equations all the way to the wall. Pettersson and Andersson (1997) employed several pressure-strain models in conjunction with elliptic relaxation to compute this ﬂow. The mean momentum equations in this case are ∂P ∂ 2 Uz 1 ∂(rur uz ) 1 ∂Uz 0 = − − +ν + ∂z ∂r r ∂r r ∂r (4.3) 2U ∂ θ 1 ∂Uθ 1 ∂(rur uθ ) 0 = ν + − . ∂r2 r ∂r r ∂r As in the previous case, all six components of the Reynolds stress tensor are non-zero and the full set of transport equations has to be solved. Figure 9 displays the mean axial and mean circumferential velocity components across the pipe using the SSG and IP models. The SSG model captures the departure of the circumferential velocity from solid body rotation (Uθ ∝ r) much better than the IP model. The departure from solid body rotation is a feature that is peculiar to turbulent rotating pipe ﬂow. It originates in Reynolds stress production terms created by the rotation: in particular, ur uθ is the solution to (4.4) 0 = Prθ − Crθ + φrθ + Trθ . The crucial terms in this equation are production and convection: Prθ = −u2 r ∂Uθ Uθ + u2 θ ∂r r and Crθ = (u2 − u2 ) r θ Uθ . r Under solid body rotation ∂rUθ /∂r = Uθ /r. Hence the shear stress responsible for the departure from solid body rotation has its origin in normal stress anisotropy, u2 = u2 . r θ 2 Editors’ note. Readers will note that a diﬀerent view is taken in [22]. 148 Durbin and Pettersson-Reif 2.0 x 2h y z/h 1.0 2h z 0.0 0.0 Schematic of square duct 1.0 y/h 2.0 Figure 10: Fully developed ﬂow in a straight square duct at Re∗ ≡ 2hu∗ /ν = 600. Vectors: Secondary mean ﬂow ﬁeld; Contours: Mean streamwise velocity. SSG model. (Petterssson-Reif and Andersson 1999.) The second of (4.3) implies that in the inviscid limit ur uθ = 0 since it must be non-singular at the center of the pipe. The Uθ proﬁle must then adapt itself such that ur uθ = 0 is consistent with (4.4). This requirement determines the speciﬁc departure from solid body rotation, which is strongly a function of the particular homogeneous redistribution model φrθ . That model dependence is apparent in ﬁgure 9. Although departure from solid body rotation is a characteristic feature of rotating pipe ﬂow, it is actually quite weak. The engineering importance of predicting this and analogous weak secondary ﬂows is sometimes questioned— with good cause. In complex geometries stronger secondary ﬂows are usually generated by cross-stream pressure gradients. In swirling ﬂows, such as occur in swirl combustors, rotation is usually imparted by guide vanes and is a primary, not a secondary, ﬂow. The structural parameter shown in the lower left of ﬁgure 9 indicates a signiﬁcant reduction of turbulent shear stress by the pipe rotation. The accompanying reduction of wall shear stress constitutes the most important eﬀect of the imposed pipe rotation. An important feature of wall-bounded turbulent ﬂows in noninertial frames of reference is the departure from the equilibrium value of uv/k ≈ 0.3 in the log-layer. The value 0.3 is inherent in the usual derivation of the ‘law-of-the-wall’ boundary condition. Therefore the classical wall-function fails in strongly rotating ﬂows. In the examples presented in this subsection, the elliptic relaxation approach has proven to be viable in strongly rotating wall-bounded ﬂows, including cases in which relaminarization occurs. [4] The elliptic relaxation method 149 Figure 11: Fully developed ﬂow in a straight square duct at Re∗ ≡ 2hu∗ /ν = 600. DNS data of Huser and Biringen (1993). (From Petterssson-Reif and Andersson 1999.) 4.5 Square Duct The majority of geophysical and engineering ﬂows exhibit a non-zero mean ﬂuid motion in the plane perpendicular to the primary ﬂow direction. The generation of this secondary motion can be attributed to two fundamentally diﬀerent mechanisms: (i) quasi-inviscid deﬂection of the mean ﬂow ﬁeld due to body forces (such as the Coriolis force discussed above); (ii) an imbalance between gradients of the Reynolds-stress components. The latter mechanism is termed Prandtl’s secondary ﬂow of the second kind. The standard illustration of secondary ﬂow of the second kind is turbulent ﬂow inside noncircular ducts. That is the subject of this ﬁnal example. From a modeling perspective, the square duct constitutes a demanding test case for any turbulence model. Regions approximating one, two and threecomponent mean ﬂow exist and the model must perform well in all of them simultaneously. Figure 10 shows a computation in this geometry. The pattern of velocity vectors has 8-fold symmetry with respect to rotations and reﬂections. The secondary ﬂow depicted in ﬁgure 10 can be attributed to streamwise vorticity. The transport equation for streamwise vorticity is V ∂Ωx ∂Ωx +W =ν ∂y ∂z ∂ 2 Ωx ∂ 2 Ωx + ∂y 2 ∂z 2 + ∂ 2 (v 2 − w2 ) ∂ 2 vw ∂ 2 vw + (4.5) − ∂y∂z ∂y 2 ∂z 2 150 Durbin and Pettersson-Reif where Ωx = ∂W − ∂V . The two last terms are referred to as turbulent source ∂y ∂z terms: the ﬁrst involves normal stress anisotropy; the second involves the secondary Reynolds shear stress. The ﬁrst has traditionally been recognized as driving the streamwise vorticity. However, recent DNS of Huser et al. (1994) indicate, to the contrary, that the net source term is dominated by the shear stress contribution. Pettersson-Reif and Andersson (1999) computed the fully developed turbulent ﬂow in a straight square duct. A full SMC was for the ﬁrst time integrated all the way to the wall in this ﬂow. The secondary ﬂow had the correct eightfold symmetry about the two wall mid-planes and the corner bisectors of the ﬁeld shown in ﬁgure 10. The discrepancy most noted by Pettersson-Reif and Andersson (1999) is that the predicted secondary ﬂow was even weaker than that of the data. However, the secondary ﬂow is quite weak, being only a few percent of the centerline velocity, and its prediction is of questionable practical importance. Figure 11 compares model predictions with DNS data from Huser and Biringen (1993) at Re∗ ≡ 2hu∗ /ν = 600. As in previous examples, the elliptic relaxation method is able to capture much of the near-wall ﬂow, including the corner region, which is the focus in this case. Pettersson-Reif and Andersson (1999) also include computations at a higher Reynolds number, compared to laboratory experiments. The agreement of the primary velocity and the turbulent Reynolds stresses with data is generally good. References Andersson, H.I., Johansson, B., L¨fdahl, L. and Nilsen, P.J. (1991). ‘Turbulence in o the vicinity of a rotating cylinder in a quiescent ﬂuid: experiments and modelling’. In Proc. 8th Symp. Turbulent Shear Flows, Munich, Germany, 30.1.1.–30.1.6. Andersson, H.I. (1999). ‘Organized structures in rotating channel ﬂow’. In IUTAM Symp. on Simulation and Identiﬁcation of Organized Structures in Flows, Lyngby, Denmark, J.N. Frensen, E.J. Hopﬁnger and N. Aubry (eds.), Kluwer. Andersson, H.I, Pettersson, B.A. and Bech, K.H. 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(1998a). ‘Computation of 3D turbulent boundary layers using the v 2 -f model’, Flow, Turbulence and Combustion 10, 19– 46. Parneix, S., Laurence, D. and Durbin, P.A. (1998b). ‘A procedure for using DNS databases’, ASME J. Fluids Eng. 120, 40–47 Phillips, O.M. (1955). ‘The irrotational motion outside a free turbulent boundary’, Proc. Camb. Phil. Soc. 51, 220–229. Pettersson, B.A., Andersson, H.I. and Hjelm-Larsen, O. (1996). ‘Analysis of nearwall second-moment closures applied to ﬂows aﬀected by streamline curvature’. In Engineering Turbulence Modelling and Measurements 3, W. Rodi and G. Bergeles (eds.), Elsevier, 49–58. Pettersson, B.A. and Andersson, H.I. (1997). ‘Near-wall Reynolds-stress modelling in noninertial frames of reference’, Fluid Dyn. Res. 19, 251–276. Pettersson, B.A., Andersson, H.I. and Brunvoll, A.S. (1998). ‘Modeling near-wall effects in axially rotating pipe ﬂow by elliptic relaxation’, AIAA J. 36, 1164–1170. Pettersson-Reif, B.A. and Andersson, H.I. (1999). ‘Second-moment closure predictions of turbulence-induced secondary ﬂow in a straight square duct’. In Engineering Turbulence Modelling and Measurements 4, W. Rodi and D. Laurence, (eds.), Elsevier, 349–358. Ristorcelli, J.R., Lumley, J.L. and Abid, R. (1995). ‘A rapid-pressure covariance representation consistent with the Taylor-Proudman theorem materially frame indifferent in the two-dimensional limit’, J. Fluid Mech. 292, 111–152. Shir, C.C. (1973). ‘A preliminary numerical study of atmospheric turbulent ﬂows in the idealized planetary boundary layer’, J. Atmos. Sci. 30, 1327–1339. Speziale, C.G., Sarkar, S. and Gatski, T.B. (1991). ‘Modeling the pressure-strain correlation of turbulence: an invariant dynamical systems approach’, J. Fluid Mech. 227, 245–272. Webster, D.R., DeGraaﬀ, D.B. and Eaton, J.K. (1996). ‘Turbulence characteristics of a boundary layer over a two-dimensional bump’, J. Fluid Mech. 320, 53–69. Wizman, V., Laurence, D., Durbin, P.A., Demuren, A. and Kanniche, M. (1996). ‘Modeling near wall eﬀects in second moment closures by elliptic relaxation’, Int. J. Heat and Fluid Flow 17, 255–266. 5 Numerical Aspects of Applying Second-Moment Closure to Complex Flows M.A. Leschziner and F.-S. Lien Abstract The incorporation of Reynolds-stress closure into general ﬁnite-volume schemes, in which the discretization of convection is minimally diﬀusive, presents a number of algorithmic problems not encountered in schemes containing eddyviscosity models. The main problem is low numerical stability, arising from the general stiﬀness of the turbulence-model equations, the absence of the numerically stabilizing second-order derivatives associated with the eddy viscosity, and – in the case of a fully collocated storage of all variables – a decoupling between stresses and strains. The chapter presents a number of algorithmic measures designed to enhance the stability and rate of convergence of incompressible as well as compressible-ﬂow solvers, the latter based on modern Riemann schemes. It also discusses aspects of the incorporation of wall boundary conditions for the Cartesian stress components in conjunction with wall laws which are formulated in wall-oriented coordinates. Two examples are included for complex 3D ﬂows, one incompressible and the other compressible (transonic). 1 Introduction Despite decades of research into the formulation, improvement and validation of second-moment closure models, the large majority of RANS codes applied in practice continue to use linear eddy-viscosity models to represent the effects of turbulence on the mean ﬂow. The appeal of such models is rooted in their simplicity, favourable numerical characteristics and surprisingly good predictive capabilities over a fair range of conditions, especially if the basic model forms are augmented by ad hoc corrections to counteract a number of fundamental weaknesses. While there is no argument about the fundamental superiority of secondmoment closure and the mechanisms responsible for it, there is no consensus on the degree to which this fundamental strength translates itself into practical predictive advantages and broad generality. One important source of predictive variability is the approximation of the terms responsible for redistributing turbulence energy among the normal stresses and for reducing the shear stresses 153 154 Leschziner and Lien in opposition to strain-induced generation. This problem is by-passed in eddyviscosity models owing to the absence of these terms in the turbulent-energy equation which is, in the large majority of eddy-viscosity models, the basis of the turbulence-velocity scale on which the eddy-viscosity depends. Other sources for inconsistent performance include diﬃculties with boundary conditions, especially at walls, the greater sensitivity of the solutions obtained with second-moment closure to numerical grid disposition and discretization errors, and the far greater scope for coding errors. The greater sensitivity of second-moment models to numerical resolution, while arguably a disadvantage on resource grounds, is associated with the tendency of the models to return lower levels of diﬀusion in complex strain, which is, however, precisely the reason why these models often gives superior predictions. Other factors contributing to the above-noted sensitivity include the far larger number of source-like terms that need to be approximated by numerical integration (usually single-point quadrature) and the fact that diffusion is also usually included in the form of explicit source-like fragments – so that convection becomes a more dominant process, in numerical terms. Some of the issues noted above in relation to the sensitivity to numerical resolution also contribute to several other numerical diﬃculties which are regarded as disincentives for the adoption of second-moment closure in practice. Among them, numerical ‘brittleness’, due to the absence of stabilizing second-order diﬀusion terms associated with the unconditionally positive eddy viscosity, is a major issue. Another is the often substantially greater computerresource requirements brought about by the larger number of model equations and the slower rate of convergence arising from the more complex coupling between the equations, the algorithmically explicit treatment of the many source-like terms and the strong under-relaxation that is often required to procure stability. Notwithstanding the modelling and numerical challenges posed by secondmoment closure, vigorous research continues in this area, and new model forms are emerging (e.g. Craft and Launder 1996, Craft 1998, Jakirli´ and Hanjali´ c c 1995, and Batten et al. 1999). The rationale underlying these eﬀorts is that second-moment closure is, at present, the only general foundation oﬀering a fundamentally ﬁrm route to achieving improved predictive performance over a broad range of practical ﬂows. It is, speciﬁcally, the only route which oﬀers, through the retention of the formally exact generation terms, the prospect of an accurate representation of the complex interactions between diﬀerent types of strain and stress components, and among heat and mass ﬂuxes, mean-scalar gradients, stresses and strains. While nonlinear eddy-viscosity models have recently received much attention (see Chapter [1] and, for example, Apsley et al. 1998, and Loyau et al. 1999), this is mainly a reﬂection of pressure from CFD practitioners to devise models which are better than linear eddyviscosity forms, but which have similarly favourable numerical characteristics. [5] Applying Second-Moment Closure to Complex Flows 155 The fact that several of the more elaborate nonlinear eddy-viscosity models have been derived from drastically simpliﬁed forms of second-moment closure is consistent with the statement made above on the fundamental ﬁrmness of the latter framework. The main purpose of this chapter to deal with some numerical aspects of second-moment closure – speciﬁcally, its incorporation into ﬁnite-volume procedures. While there are some general principles pertaining to all types of algorithm, there are also some substantial diﬀerences in detail arising from the diﬀerent variable-storage arrangement adopted – e.g. cell-centred, staggered, cell-vertex storage – and also from the diﬀerent approaches taken to approximating convection and determining the pressure in incompressible and compressible ﬂows. Following a summary statement on the closure equations used as a basis for conveying numerical issues, consideration is given to incompressible ﬂows computed with non-orthogonal structured grids. Aspects speciﬁc to compressible ﬂows are covered separately later. The chapter ends with two application examples, illustrating that complex 3D ﬂows can now be computed with second-moment closure without major diﬃculties. 2 Second-moment Closure A Reynolds-stress-transport model (RSTM) consists of a set of diﬀerential transport equations governing the distribution of related turbulent stresses. Each equation represents, in essence, a balance between stress transport, generation, destruction and redistribution (return to isotropy). It is primarily the exact representation of stress generation, which varies greatly from stress to stress, that gives this type of closure the ability to return a realistic statement on stress anisotropy. Although there exist about a dozen major variants of second-moment closure, all have a broadly similar mathematical structure, in so far as the stresstransport equations contain convection and diﬀusion ﬂuxes and a large number of source-like terms arising from stress production, dissipation and redistribution. Hence, in terms of implementation, it is suﬃcient to consider a representative variant in detail and follow this, as appropriate, by comments which pertain to diﬀerences associated with model variants. The ‘generic’ closure adopted below is the well-known and most widely-used variant of Gibson and Launder (1978), in which stress diﬀusion is represented by the generalized-gradient-diﬀusion hypothesis (GGDH), while pressure-strain interaction is approximated by additive linear ‘return-to-isotropy’ and ‘isotropization-of-production’ models and associated wall-related corrections. In terms of Cartesian tensor notation, the closure may be written as follows: ∂(ρUk ui uj ) 1 = dij + Pij + (Φij,1 + Φij,2 + Φw ) − εδij , ij ∂xk 3 (1) 156 where Leschziner and Lien Cs ρk ∂ui uj uk ul , ε ∂xl ∂Uj ∂Ui + ρuj uk , Pij = − ρui uk ∂xk ∂xk ε 2 Φij,1 = −C1 ρ ui uj − δij k , k 3 (2) 1 Φij,2 = −C2 Pij − Pkk δij , 3 3 3 Φw = Φkl nk nl δij − Φik nj nk − Φjk ni nk f, ij 2 2 ε Φij = C1w ρ ui uj + C2w Φij,2 , k and f is a ‘wall-distance function’ to be deﬁned later. For further considerations, it is advantageous to treat the stress components and the related equations explicitly, i.e. as separate entities, and to adopt a general non-orthogonal coordinate framework; this is, in fact, the environment into which the closure has been implemented. Without loss of generality, in terms of numerical-implementation issues, and to enhance clarity, attention is conﬁned to plane 2D ﬂow, for which the set of equations (1) may be written, after some manipulation and insertion of (2) into (1), as follows: dij = ∂ ∂xk 1 J ∂ 1 ∂ 1 ρUc ϕ − (q11 ϕξ + q12 ϕη ) + ρVc ϕ − (q12 ϕξ + q22 ϕη ) ∂ξ J ∂η J ρε = α1 P11 + α2 P22 + α3 P12 + α4 Pk + α5 u2 + α6 v 2 + α7 uv + α8 ρε, k ϕ = u2 , v 2 , uv, (3) where J is the Jacobian linking the (x, y) and (ξ, η) systems, the subscripts ξ and η denote diﬀerentiation with respect to these coordinates, q11 = q22 q12 Cs ρk 2 2 u yη − 2uvxη yη + v 2 x2 , η ε Cs ρk 2 2 = u yξ − 2uvxξ yξ + v 2 x2 , ξ ε Cs ρk 2 = − u yξ yη − uv(xξ yη + xη yξ ) + v 2 xξ xη , ε (4) (5) (6) the contravariant velocities Uc and Vc are given by: Uc = U yη − V xη , Vc = V xξ − U yξ , (7) and the turbulence production Pij arise as: P11 = − 2ρ 2 u (Uξ yη − Uη yξ ) + uv(Uη xξ − Uξ xη ) J (8) [5] Applying Second-Moment Closure to Complex Flows P22 = − P12 2ρ 2 v (Vη xξ − Vξ xη ) + uv(Vξ yη − Vη yξ ) J ρ = − u2 (Vξ yη − Vη yξ ) + v 2 (Uη xξ − Uξ xη ) J +uv(Uξ yη + Vη xξ − Uη yξ − Vξ xη ) , 157 (9) (10) and Pk = 0.5(P11 + P22 ). The coeﬃcients αi in the above expressions for the individual stress components are summarized in Table 1. In this table, the ‘wall-damping functions’ fx , fy and fxy for any wall are: fx = n2 f, 1 fy = n2 f, 2 fxy = n1 n2 f, (11) where, as shown in Figure 1, (n1 , n2 ) are directional cosines relating the global system unit-vectors (e1 , e2 ) to the local wall-normal vector e2 , and f = (k 3/2 /ε)/(Cl ln ), with ln being the wall-normal distance. Figure 1: Global and local wall-aligned systems of unit vectors for any curved wall. Other closures, e.g. the low-Re versions of Jakirli´ and Hanjali´ (1995) and c c Craft and Launder (1996), contain additional terms associated with the ﬂuid viscosity and/or with higher-order approximations for the pressure-strain interaction terms. The implementation issues discussed below apply to all variants, except for aspects pertaining to the imposition of wall boundary conditions – an issue requiring more careful attention in high-Re models coupled to wall laws. 3 3.1 Numerical Issues General Considerations Experience shows that the incorporation of second-moment closure into generalﬂow solvers is a non-trivial task, mainly because the equations exhibit a numerically stiﬀ behaviour. Speciﬁcally, the equations contain large source-like terms, are highly nonlinear and are strongly coupled. In addition, momentum 158 Leschziner and Lien Table 1: Coeﬃcients associated with the sources of the Reynolds-stress equations. ϕ = u2 α1 α2 α3 α4 ϕ = v2 α1 α2 α3 α4 ϕ = uv α1 α2 α3 α4 1.5C2 C2w fxy 1.5C2 C2w fxy 1 − C2 + 1.5C2 C2w (fx + fy ) −2C2 C2w fxy α5 α6 α7 α8 −1.5C1w fxy −1.5C1w fxy −[C1 + 1.5C1w (fx + fy )] 0 −C2 C2w fy 1 − C2 + 2C2 C2w fx C2 C2w fxy 2 (C2 − 2C2 C2w fx + C2 C2w fy ) 3 α5 α6 α7 α8 C1w fx −(C1 + 2C1w fy ) −C1w fxy 2 (C1 − 1) 3 1 − C2 + 2C2 C2w fx −C2 C2w fy C2 C2w fxy 2 (C2 − 2C2 C2w fx + C2 C2w fy ) 3 α5 α6 α7 α8 −(C1 + 2C1w fx ) C1w fy −C1w fxy 2 (C1 − 1) 3 diﬀusion does not arise in the form of second-order derivatives of the subject variable (momentum), and this can have a dramatically adverse eﬀect on the stability of the solution. The impact on stability depends critically on the manner in which the transported variables are stored spatially, both relative to one another and relative to the cells over which conservation is satisﬁed (cell-vertex/cell-centred, staggered/collocated storage), the type and disposition of the numerical grid (structured/unstructured, skewness, aspect ratio), the degree of implicitness and inter-variate coupling maintained by the solution algorithm, and, to some extent, also on the complexity of the ﬂow and its geometry (e.g. boundary-layer vs. massively separated ﬂow). There are some major diﬀerences between semi-implicit, pressure-based schemes (e.g. SIMPLE), which are used extensively for incompressible ﬂows, and density-based schemes used for compressible ﬂows. In the latter category, two main groups of schemes are conventional explicit, decoupled, timemarching formulations (e.g. Runge–Kutta), which solve the conservation laws [5] Applying Second-Moment Closure to Complex Flows 159 in their basic form in a segregated manner, and the more modern Riemannsolver-based schemes which account for the propagation of waves and which are usually combined with implicit time-integration. The degree of implicitness and coupling in the latter framework can vary greatly, however: it can encompass the entire set of transport equations or only subsets; it can include only coupling via the ﬂuxes or via the source terms; it can involve coupling between variables at each point or include spatial coupling. All the above factors inﬂuence stability, in general, and the degree of diﬃculty encountered in the incorporation of second-moment closure, in particular. The diversity of approaches and the resulting diﬀerences in their numerical properties means that there are no universally applicable rules or algorithmic measures for a stable and eﬃcient implementation of second-moment closure. Rather, stability-promoting practices have evolved in response to the need to achieve stability or increase the convergence rate of diﬀerent algorithms. In what follows, some advantageous stability-promoting practices, which are used for incompressible and compressible ﬂows, are introduced. None of the practices is essential in all circumstances, and no practice can be claimed to secure stability unconditionally. 3.2 Collocated-storage, Pressure-based Algorithms Apart from the problems associated with non-orthogonality, especially at boundaries, the main diﬃculty in combining a stress-transport model with a collocated ﬁnite-volume scheme for complex geometries arises from the fact that storage of all variables at the same spatial location tends to lead to chequerboard oscillations, caused by an inappropriate decoupling of velocity and Reynolds stresses when linear interpolation is used to approximate cell-face stresses in terms of nodal values. The practices outlined below re-establish coupling by use of interpolation methods analogous to those proposed by Rhie and Chow (1983) for momentum interpolation in the solution of the meanﬂow equations within a pressure-based strategy. While the considerations to follow assume the ﬂow to be incompressible, the practices introduced also apply, subject to some minor variations, to compressible ﬂows computed with pressure-based schemes. Computational studies on transonic ﬂows combining pressure-based schemes with Reynolds-stress-transport models include those of Lien and Leschziner (1993b) and Leschziner and Ince (1995). To highlight the underlying rationale, it is instructive to focus ﬁrst on the Boussinesq relationship, −ρui uj = µT ∂Ui ∂Uj + ∂xj ∂xi 2 − δij ρk. 3 (12) As is evident, ρu2 is ‘driven’ by ∂U/∂x, while ρuv is ‘driven’ by ∂U/∂y and ∂V /∂x. To retain this physical coupling in the numerical representation, it is necessary to store u2 between U -velocity locations approximating ∂U/∂x. 160 Leschziner and Lien Figure 2: Staggered Reynolds-stress storage locations. Similarly, uv needs to be stored between U - and V -velocity nodes approximating ∂U/∂y and ∂V /∂x, respectively. This rationale is reﬂected by the arrangement shown in Figure 2. In a 3D environment, a total of seven separate control volumes are thus required if the Reynolds-stress model is used. Examples of 3D ﬂows computed with this arrangement may be found in Lin and Leschziner (1993) and Iacovides and Launder (1985), the latter using an algebraic Reynolds-stress model. The above methodology is, of course, untenable in a collocated arrangement. To retain coupling, a nonlinear interpolation practice has been devised which prevents stress-related odd-even oscillations. In order to facilitate transparency, the method is explained ﬁrst by reference to a 2D Cartesian arrangement having a uniform mesh ∆x = ∆y, with generalization pursued later. With the Reynolds-stress model of Gibson and Launder (1978) chosen to represent turbulence transport, it may be shown that the discretized form of the transport equation for the normal stress u2 at the location P may be written: u2 = P m=E,W,N,S HP Am u2 + SC /AP + µP m 11 (Uw − Ue )P , ∆x (13) where 4 8 2 2 − C2 + C2 C2w fx + C2 C2w fy ρu2 3 3 3 AP ∆x∆y P µP 11 = , (14) and SC includes a cross-diﬀusion term arising from Daly and Harlow’s stressdiﬀusion model (1970) and some fragments of the production, pressure-strain and dissipation processes. Analogous expressions for u2 and u2 are: e E u2 = E (Uw − Ue )E HE + µE , 11 AE ∆x u2 = e (UP − UE ) He + µe , 11 Ae ∆x (15) [5] Applying Second-Moment Closure to Complex Flows 161 with µe = (µP + µE )/2. If the H/A terms are also linearly interpolated, the 11 11 11 resulting form of u2 is: e u2 = e 1 2 u + u2 E 2 P linear interpolation (16) 1 + 2 µP + µE (UP − UE ) − µP (Uw − Ue )P − µE (Uw − Ue )E /∆x, 11 11 11 11 U −velocity smoothing which is identical to the form proposed by Obi et al. (1989). The above practice of extracting apparent viscosities (e.g. µ11 associated with the mean strain ∂U/∂x) had already been suggested by Huang and Leschziner (1985) who employed the staggered Reynolds-stress arrangement. Their objective was to enhance the iterative stability by increasing the magnitude of the diagonal coeﬃcient AP . In contrast, it may be seen from (16) that the same apparent-viscosity approach applied to the collocated arrangement for all Reynolds stresses introduces fourth-order smoothing, here depending on the U -velocity component rather than, as is the case with Rhie and Chow’s interpolation, on pressure. Equation (16) has been derived for steady-state conditions without the use of under-relaxation in the solution sequence. The generalization to unsteady conditions and with under-relaxation included requires care to ensure that the solution will not depend on the under-relaxation factor, and that the cell-face interpolants do not depend on the time step. Details on this extension may be found in Lien and Leschziner (1994a). In deriving the above expression for the apparent viscosity, an element of uncertainty is that the level of this viscosity is a strong function of the manner in which the source in the u2 -equation is transformed into the equivalent quasilinear form via: JSu2 ← SC + SP u2 . (17) P To appreciate the nature of the problem, it must ﬁrst be pointed out that µ11 depends strongly on AP via (14). But AP contains SP , and this value depends on how the terms encountered in Su2 are treated. It is a general objective here to maximize the magnitude of the (negative!) fragment SP . If any term in Su2 is found to be negative, but does not contain u2 as a factor, it may nevertheless be allocated to SP by dividing that term by u2 (the previous iterate) and then adding the result to SP . Because SP varies greatly from node to node, the averaging process (16) may yield a value for Ae which gives an inappropriate level of µe (which equals 0.5[µP + µE ]). This leads to 11 11 11 the conclusion that it would be desirable to construct a scheme which allows µ11 (and other apparent viscosities) to be determined without reference to the discretization process and its details. To this end, attention is directed towards 162 Leschziner and Lien the diﬀerential equation governing u2 . This may be written in the following compressed form: 2 C11 − D11 = −A − B + P11 + Φ11 − ρε + A + B , 3 with A = B = ρε (C1 + 2C1w fx ) u2 k ∂U 4 8 2 2 − C2 + C2 C2w fx + C2 C2w fy ρu2 , 3 3 3 ∂x (19) (20) (18) and C11 , D11 , P11 and Φ11 being, respectively, convection, diﬀusion, production and redistribution. As regards terms A and B, suﬃce it to say here that both arise naturally from fragments of P11 and Φ11 . Combination of (18) to (20) yields: ∂U −ρu2 = µ11 + ∂x with µ11 k ε 2 C11 − D11 − (P11 + Φ11 − ρε + A + B) 3 , C1 + 2C1w fx (21) 2 4 2 − C2 + C2 C2w (4fx + fy ) ρku2 3 3 = . C1 + 2C1w fx ε (22) Note that µ11 in (21) and (22) is associated with the diﬀerential gradient; in contrast, µP in (13) is applicable to the gradient approximation at grid-node 11 P . The derivation of µ22 follows a path analogous to (18) to (22). Hence, only the end result is given, namely: 2 4 2 − C2 + C2 C2w (4fy + fx ) ρkv 2 3 3 = . C1 + 2C1w fy ε µ22 (23) Attention is turned next to the interpolation formula for the shear stress uv. A treatment consistent with the one applied in relation to u2 would involve ∂U ∂V extracting a viscosity µ12 by reference to and . This is not possible, ∂y ∂x ∂U ∂V however, because the fragments multiplying and in the stress model ∂y ∂x are not identical. Hence, here, uv is only sensitized to one of the two strains; which one is chosen is dictated by the direction of the derivative of the shear ∂ρuv stress. Since in the U -momentum equation the shear-stress gradient is , it ∂y ∂U is natural to relate ρuv to , leading to the stability-promoting second-order ∂y ∂(µ12 ∂U/∂y) . The same rule can be applied to the V -momentum derivative ∂y [5] Applying Second-Moment Closure to Complex Flows 163 ∂(µ21 ∂V /∂x) equation, resulting in the diﬀusion term . In order to derive the ∂x two apparent viscosities µ12 and µ21 from the uv-equation, this equation is ﬁrst written as follows: C12 − D12 = − ρε 3 C1 + C1w (fx + fy ) uv k 2 3 − 1 − C2 + C2 C2w (fx + fy ) 2 v2 ∂U ∂V + u2 ∂y ∂x . (24) Then (24) can be reformulated in either of the two forms: ∂U −ρuv = µ12 + ∂y or ∂V + ∂x k ε ∂U 3 C12 − D12 + 1 − C2 + (fx + fy ) ρv 2 2 ∂y , 3 C1 + C1w (fx + fy ) 2 k ε ∂V 3 C12 − D12 + 1 − C2 + (fx + fy ) ρu2 2 ∂x , 3 C1 + C1w (fx + fy ) 2 (25) −ρuv = µ21 (26) where 3 1 − C2 + C2 C2w (fx + fy ) 2 3 C1 + C1w (fx + fy ) 2 3 1 − C2 + C2 C2w (fx + fy ) 2 3 C1 + C1w (fx + fy ) 2 ρkv 2 ε ρku2 ε µ12 = (27) µ21 = . (28) To extend the above concepts to the general curvilinear environment, attention is next focused on the U - and V -momentum equations, written in terms of the general coordinates (ξ, η): ∂(Uc ρU ) ∂(Vc ρU ) + ∂ξ ∂η (29) ∂(P + ρu2 )yξ ∂(ρuv)xξ ∂(P + ρu2 )yη ∂(ρuv)xη =− + + − ∂ξ ∂η ∂ξ ∂η ∂(Uc ρV ) ∂(Vc ρV ) + ∂ξ ∂η 2 )x ∂(P + ρv 2 )xξ ∂(ρuv)yξ ∂(ρuv)yη ∂(P + ρv η − − + . = ∂ξ ∂η ∂ξ ∂η (30) 164 Leschziner and Lien It is clear from (29) and (30) that no physical (second-order) diﬀusion terms arise naturally. In order to extract apparent viscosities from the Reynoldsstress equations in terms of the (ξ, η) coordinate system, a tedious, but otherwise rather straightforward manipulation of the transformed equations, analogous to that in the Cartesian framework, may be carried out. Interestingly, the ﬁnal expressions are identical to equations (22), (23), (27) and (28), except that the wall-related damping function in the pressure-strain model assumes diﬀerent forms. Thus, for a single x-directed wall in the Cartesian framework, fy is given by fy = (k 3/2 /ε)/(Cl y), while, in contrast, fy along a curved surface becomes fy = n2 (k 3/2 /ε)/(Cl ln ) (see equation (11)). Introduction of the 2 apparent viscosity into (29) and (30) leads to: ∂ Uc ρϕ − ∂ξ where for ϕ = U : u 2 q1 = µ11 yη + µ12 x2 , η u 2 q2 = µ11 yξ + µ12 x2 , ξ ϕ q1 J ∂ϕ ∂ + Vc ρϕ − ∂ξ ∂η ϕ q2 J ∂ϕ = JSϕ ∂η (31) (32) yξ (33) JSu = − ∂P ∂ξ yη + ∂P ∂η yξ − ∂ρ(u2 )uξ ∂ξ xξ , yη + ∂ρ(u2 )uη ∂η ∂ρ(uv)uξ + ∂ξ with ρ(u2 )uξ = ρu2 + ρ(uv)uξ xη − ∂ρ(uv)uη ∂η µ11 yξ ∂U µ11 yη ∂U , ρ(u2 )uη = ρu2 − , J ∂ξ J ∂η µ12 xξ ∂U µ12 xη ∂U = ρuv − , ρ(uv)uη = ρuv + , J ∂ξ J ∂η (34) while for ϕ = V : v 2 q1 = µ21 yη + µ22 x2 , η v 2 q2 = µ21 yξ + µ22 x2 , ξ (35) xξ (36) JSv = ∂P ∂ξ xη − ∂P ∂η xξ + yη + ∂ρ(v 2 )vξ ∂ξ ∂ρ(uv)vη ∂η xη − yξ , ∂ρ(v 2 )vη ∂η ∂ρ(uv)vξ − ∂ξ with ρ(v 2 )vξ = ρv 2 − ρ(uv)vξ µ22 xξ ∂V µ22 xη ∂V , ρ(v 2 )vη = ρv 2 + , J ∂ξ J ∂η µ21 yξ ∂V µ21 xη ∂V = ρuv + , ρ(uv)vη = ρuv − . J ∂ξ J ∂η (37) [5] Applying Second-Moment Closure to Complex Flows 165 3.3 Analysis of Fourth-Order Smoothing While equation (31), incorporating (32)–(37), is mathematically identical to the original RANS equation written in a curvilinear coordinate system, important diﬀerences between the two forms arise from particular choices of discretization practices. One speciﬁc practice is to approximate both the apparent diﬀusion term on the LHS of equation (31) and the source terms (JSϕ ) on the RHS by appropriate second-order central diﬀerences. The result is the introduction of an artiﬁcial fourth-order smoothing term which prevents the stress-velocity decoupling that would arise if equation (31) were written in its original form, without the apparent diﬀusion terms. The mechanism by which this is achieved is best conveyed by considering equation (31) in its Cartesian-coordinate form: ∂(ρU ϕ) ∂(ρV ϕ) ∂ ϕ ∂ϕ q1 + − ∂x ∂y ∂x ∂x + ∂ ϕ ∂ϕ q2 ∂x ∂y = Sϕ (38) apparent diﬀusion where, for ϕ = U , u q1 = µ11 , u q2 = µ12 Su = − with ∂P ∂ρ(u2 )ux ∂ρ(uv)uy − − , ∂x ∂x ∂y ∂U , ∂x u ρ(uv)uy = ρuv + q2 (39) u ρ(u2 )ux = ρu2 + q1 ∂U . ∂y (40) By reference to the collocated arrangement shown in Figure 3, the ﬁrst ‘apparent diﬀusion’ term on the LHS of equation (38) may be approximated by: ∂ ϕ ∂ϕ q1 ∂x ∂x ←− u u u u (q1 )e UE − [(q1 )e + (q1 )w ] UP + (q1 )w UW . (∆x)2 (41) The normal-stress gradient on the RHS of equation (39) is also approximated by a central diﬀerence, yielding: ∂ ρ(u2 )ux ∂x ←− ρ(u2 )ux ∂U ∂x E − ρ(u2 )ux u − q1 E W 2∆x u q1 = With u q1 ∂U ∂x (42) W ∆ + ρu2 ∆x . 2∆x ∂U ∂x = E u (q1 )E (UEE − UP ) , 2∆x u q1 ∂U ∂x = W u (q1 )W (UP − UW W ) , 2∆x (43) 166 Leschziner and Lien Figure 3: Collocated cell storage arrangement. equation (42) becomes: u u u u ∆ ρu2 ∂ (q1 )E UEE − [(q1 )E + (q1 )W ] UP + (q1 )W UW W 2 )ux ←− (u + . ∂x 4(∆x)2 ∆x (44) Note that approximation (41) and the penultimate term of equation (42) sup∂ϕ ∂ qϕ . If posedly represent one and the same second-order derivative, ∂x 1 ∂x u is a constant, it is evident that the diﬀerence between the one assumes that q1 two approximations, DU , is proportional to: DU ∝ UEE − 4UE + 6UP − 4UW + UW W , (45) which is a fourth-order velocity-based smoothing. This is analogous to the smoothing term in equation (16), except that the apparent viscosity µ11 is here derived from the diﬀerential form, instead of the discretized form of the second-moment closure. 3.4 Source-Term Linearization Integration of equation (3) over the ﬁnite volume shown in Figure 3, application of the Gauss Divergence Theorem and approximation of the ﬂuxes with appropriate second-order schemes yields the following algebraic expression for the cell-centroidal value of any scalar ﬂow property ϕ: AP ϕp = m=E,W,N,S Am ϕm + JSϕ . (46) The form of the coeﬃcients AP and Am (m = E, W, N, S) depends upon the precise nature of the ﬂux-approximation schemes (see Lien and Leschziner [5] Applying Second-Moment Closure to Complex Flows 167 1994a, 1994b). To ensure (or promote) positivity of ϕ, in the case of the Reynolds normal stresses, turbulence kinetic energy and turbulence dissipation rate, the source term JSϕ at the node P is linearized as follows: JSϕ = SC + SP ϕP , (47) where SC ≥ 0 and SP ≤ 0, so that SP can be absorbed into AP , i.e. AP ← (AP − SP ), in order to increase diagonal dominance of the coeﬃcient matrix. As a result, numerical stability is greatly enhanced. Note that JSϕ contains the RHS of equation (3) and contributions associated with q12 ϕξ and q12 ϕη – the cross diﬀusion terms on the LHS. To secure the above constraints on SC and SP for ϕ = u2 , say, the source term is decomposed into the following two parts: JSu2 = JSu2 + J ρε ρε (C1 + 2C1w fx )u2 − J (C1 + 2C1w fx ) u2 , P P k k A B (48) with SC = max(A, 0), SP = min(A, 0) u2 + 10−15 P + B. (49) The addition of 10−15 in equation (49) is merely to avoid a computational singularity in the limiting case of zero normal stress. Similar expressions relate to v 2 , w2 , k and ε. 3.5 Wall Conditions Walls pose particular challenges in the context of turbulent-ﬂow computations, because spatial variations in the near-wall turbulence structure are intense due to the combined inﬂuence of viscosity and wall-induced anisotropy. When lowReynolds-number models are applied, numerical integration encompasses the entire near-wall region including the viscous sublayer. Hence, in this case, the implementation of wall conditions is (numerically) straightforward, consisting simply of imposing no-slip and impermeability relations. The treatment is more diﬃcult when log-law-based ‘wall laws’ are adopted in conjunction with high-Re models to bridge the viscous sublayer. Particular diﬃculties arise when Reynolds-stress modelling is applied in conjunction with curved walls; it is this aspect which is of particular interest here. Attention is directed to a general near-wall volume abutting a curved wall, as shown in Figure 4. The point P is assumed to be in the log-law region, with the logarithmic variation assumed to prevail normal to the wall at any tangential velocity position. The resultant shear force Fs shear acting on the s cell’s southern face Area is: s Fs shear = τw Area (50) 168 Leschziner and Lien Figure 4: Boundary cell at a curved wall. where τw = ρP kP Cµ κ ln E 1/2 l k ∗ n P 1/2 1/4 Ut (51) ν (52) U =U−U , t n ln = (rP − rs ) · n. Here, ln is the normal distance away from the wall, and Ut and Un denote, respectively, the tangential and normal velocity components (here treated as vectorial quantities). The unit normal vector pertaining to the cell in Figure 4 is n = ∇η/|∇η|, or in expanded form: (n1 , n2 , n3 ) = where 2 2 2 ηx + ηy + ηz = Areas . (ηx , ηy , ηz ) 2 2 2 ηx + ηy + ηz , (53) (54) Hence, the components of Ut in (52) can be written as: t t t (Ux , Uy , Uz ) = (U, V, W ) − (U n1 + V n2 + W n3 )n, (55) with t Ux = (1 − n2 )U − n1 n2 V − n1 n3 W, 1 t Uy t Uz (56) (57) (58) = (1 − = (1 − n2 )V − n1 n2 U − n2 n3 W, 2 n2 )W − n1 n3 U − n2 n3 V. 3 Once the tangential velocity components are resolved, the coeﬃcient AS in the discretized equation pertaining to the near-wall cell is ﬁrst nulliﬁed, and then the source SC is modiﬁed in such a manner as to explicitly include the shear force imposed on the southern cell face as follows: [5] Applying Second-Moment Closure to Complex Flows for the x-momentum equation, SC ← S C − ρP kP Cµ κ ln E for the y-momentum equation, SC ← SC − ρP kP Cµ κ ln E for the z-momentum equation, S C ← SC − ρP kP Cµ κ ln E 1/2 l k ∗ n P 1/2 1/4 t Uz Areas . 1/2 l k ∗ n P 1/2 1/4 t Uy Areas ; 1/2 l k ∗ n P 1/2 1/4 t Ux Areas ; 169 (59) ν (60) ν (61) ν To implement (59)–(61), the correct value of kP is needed. In the near-wall cell, this is essentially governed by the balance between volume-averaged production and dissipation. Both must be evaluated in a manner consistent with the log-law variation in the cell. If the entire cell is assumed to reside within the log-law region, the average k-production arises as: Pk = ln(ln /lv ) ρP κ∗ ln kP 1/2 (τw · τw ), (62) 2 where lv is the viscous sublayer thickness. With ε = 2ν(k/ln ) in the viscous 3/2 /κl in the log-law region, the average value of ε becomes: sublayer and ε = k n k ε= P ln 3/2 2ν 1/2 lv kP + ln(ln /lv ) . Cl (63) Since ε is unconditionally positive (but preceded by a minus sign in the k-equation), iterative stability can be enhanced by the replacement: SP ← S P − ρP ε J. kP (64) The aforementioned modiﬁcations in the near-wall region suﬃce for the implementation of the k-ε model. The extension of the above treatment to the Reynolds-stress model is less straightforward than might seem at ﬁrst sight. In a Cartesian framework, the average near-wall stress productions are well approximated by P11 = 2Pk , P22 = 0, P33 = 0, (65) 170 while the shear stress itself is given by −ρuv τw . Leschziner and Lien (66) A problem here is, however, that additional average pressure-strain terms (Φij ) need to be evaluated, because these contribute substantially to the balance equations, unlike in the case of turbulence energy. The pressure-strain terms contain products of stresses and strains, and the variation of the former across the sublayer is both uncertain and highly inﬂuential to the averaging process. Considerable further complications arise in a non-Cartesian environment, because of the tensorial nature of the stresses, productions and Φij , and the consequent transformation involved in determining productions and contributions of Φij in terms of wall-oriented coordinates. These diﬃculties provide motivation for an alternative route. In this, the k-equation, incorporating the production and diﬀusion terms appropriate to the second-moment closure, is solved together with (62) to (64), rather than the equations for u2 , v 2 , and w2 . Then, the near-wall values of the Reynolds-stress components, in terms of wall-oriented coordinates, are determined from local-equilibrium forms of the Reynolds-stress equations from which transport terms are omitted and in which the log-law is used to approximate the shear strain, assumed to be the only strain. The end result is a closed set of algebraic equations for the wall-oriented stresses in terms of the turbulence energy: (u2 )w = 1.098kP , P (v 2 )w = 0.247kP , P (uv)w = −0.255kP . P (67) Finally, the Cartesian stress components are determined from the wall-oriented Reynolds stresses through a local coordinate transformation as follows (see Lien and Leschziner 1993a): u2 P 2 vP = (u2 )w t2 + (v 2 )w n2 + 2(uv)w t1 n1 P 1 P 1 P = (u2 )w t2 + (v 2 )w n2 + 2(uv)w t2 n2 P 2 P 2 P = (u2 )w t1 t2 + (v 2 )w n1 n2 + 2(uv)w (t1 n2 + t2 n1 ), P P P (68) (69) (70) uvP where ti and ni are the components of the tangential and wall-normal unit vectors, respectively. 3.6 Approximation of Turbulence Convection It is generally assumed that the numerical approximation of turbulence convection is of subordinate importance, because the associated equations are dominated, in most shear ﬂows, by source and sink terms arising from generation, dissipation and redistribution. This is often put forward as a justiﬁcation for the use of the ﬁrst-order upwind scheme to approximate turbulence convection, so as to increase the diagonal dominance of the discretized sets of [5] Applying Second-Moment Closure to Complex Flows 171 equations and thus enhance the stability of the solution. Although the upwind scheme introduces a substantial amount of artiﬁcial turbulence diﬀusion, the argument is that this diﬀusion only makes a small contribution to the balance expressed by the equations. The principal requirement for numerical stability in the solution of the turbulence-model equations is that the numerical scheme should not introduce artiﬁcial minima which could cause negative values for (physically) unconditionally positive quantities (e.g. normal stresses and dissipation) and hence instability. While this requirement is met by the ﬁrst-order upwind scheme, a much less diﬀusive approach which is entirely satisfactory is to use a modern Total Variation Diminishing (TVD) scheme. This in a nonlinear method, in that the approximation is sensitive to the local solution, introducing just enough artiﬁcial diﬀusion to eliminate extremes which the basic nonTVD scheme would normally provoke. An example is the UMIST (Upstream Monotonic Interpolation for Scalar Transport) scheme of Lien and Leschziner (1994b), implemented in combination with second-moment closure and the foregoing stability-promoting measures into a general 3D non-orthogonal-grid algorithm for complex ﬂows. The UMIST scheme is a monotonic form of the QUICK scheme of Leonard (1979) and has been constructed along the lines of van Leer’s (1979) MUSCL scheme. Applications reported in Lien and Leschziner (1994b) conﬁrm that many ﬂows are largely insensitive to the accuracy of approximating turbulence convection. However, there are ﬂows in which this accuracy is important. One group include boundary layers undergoing bypass transition, in which the precise representation of the evolution of the turbulence quantities can have a substantial impact on the position of transition. Figure 5, taken from Chen et al. (1998), illustrates this sensitivity for the case of a transitional ﬂat-plate boundary layer computed with a low-Re linear eddy-viscosity model. The general message is that it is always advisable to apply the most accurate discretization scheme, subject to stability constraints, to the turbulence equations. This is especially pertinent to second-moment closure, because stress convection can become inﬂuential in some ﬂows in which turbulence-damping processes (e.g. swirl and density stratiﬁcation) diminish generation and redistribution relative to transport. 3.7 Multigrid Acceleration Multigrid relaxation has become a well-established method for accelerating the convergence of elliptic and hyperbolic ﬂows, in the wake of Brandt’s pioneering work in the 1970s (Brandt 1977). In its simplest form, a two-level multigrid scheme transfers (‘restricts’) residuals from the (ﬁne) working grid to a coarser grid, and then returns (‘prolongates’) incremental changes (corrections) resulting from a reduction in the coarse-grid residuals to the ﬁner working grid. This process is followed by a few solution (relaxation) steps on the ﬁner grid to yield 172 Leschziner and Lien Figure 5: Sensitivity of predicted transition of a ﬂat-plate boundary layer to the approximation of turbulence convection (from Chen et al. 1998). the ﬁnal solution. As residuals decay more rapidly on coarse grids than on ﬁne grids, this double transfer, with residual relaxation (‘smoothing’) eﬀected on the coarser grid, tends to give much faster convergence than a relaxation on the working grid only. In practice, elaborate multigrid ‘cycles’ are used, in which residuals, corrections and actual solutions are transferred through a sequence of coarsening and reﬁning grids. Figure 6 shows, schematically, three alternative cycles, including a ‘full multigrid V-cycle’. Many applications have been reported in which variants of the multigrid method have been exploited to accelerate the convergence of computations for high-speed inviscid and laminar ﬂows. In these relatively simple conditions, the CPU costs tend to increase in proportion to N log N , where N is the number of grid nodes. This (almost) linear increase compares with quadratic or even cubic rates of increase for conventional single-grid relaxation schemes. Applications to complex turbulent ﬂows are relatively rare. Experience shows that, with some special steps in the prolongation and restriction operations applied to the turbulence-transport equations (see Lien and Leschziner 1994c), the eﬀectiveness of the multigrid scheme is generally maintained, if turbulence eﬀects are represented by eddy-viscosity models, provided the grid is not too distorted and the cell-aspect ratio does not exceed O(10) to O(50). Hardly any experience exists on the performance of multigrid schemes in computations using second-moment closure. Lien and Leschziner (1994c) have undertaken an extensive study of the performance of multigrid schemes in a wide range of ﬂows, including turbulent ﬂows computed with two-equation eddy-viscosity models (both high-Re [5] Applying Second-Moment Closure to Complex Flows 173 Figure 6: Alternative multigrid cycles. and low-Re variants) and second-moment closure. Their experience was that the convergence acceleration achieved with second-moment closure, although worthwhile was well below that attained for simpler models. Figure 7 shows results for the separated ﬂow in a 2D plane diﬀuser computed with the k-ε model and the Gibson–Launder (1978) Reynolds-stress model. Also included are convergence histories for the U -momentum residual, in terms of ‘work units’ (WU), for computations with a single grid of 160 × 40 nodes and sequences of 2, 3 and 4 grids. As seen, convergence is much faster with the multigrid scheme. However, it is found that the N log N behaviour is no longer maintained. 3.8 Density-based Scheme Although pressure-based schemes have been used in combination with secondmoment closure to compute compressible ﬂows, including shocks (e.g. Lien and Leschziner 1993b, Leschziner and Ince 1995), the usual approach is to solve the mass-conservation equation directly to yield the density. There are numerous density-based schemes for compressible ﬂow. Probably the simplest and most widely used is that due to Jameson et al. (1981), which solves the conservation equations for mass, momentum and energy in a segregated, explicit manner (e.g. by a Runge–Kutta method) using centred approximations for the ﬂuxes in conjunction with stabilizing artiﬁcial second/fourth-order dissipation. Most modern upwind schemes are now based on Riemann solvers which exploit the 174 Leschziner and Lien Figure 7: Predicted solutions and multigrid convergence histories for turbulent ﬂow in a 2D plane diﬀuser (from Lien and Leschziner 1994c). characteristics of the conservation equations and resolve the characteristic lines along which acoustic and contact waves propagate. These characteristics arise upon the diagonalization of the Jacobian ﬂux matrices {A}, {B} and {C} of the conservation set, normally written in the form: ∂Q ∂(F − Fv ) ∂(G − Gv ) ∂(H − Hv ) + + + ∂t ∂ξ ∂η ∂ζ ∂Q ∂Q ∂Q ∂Q = + {A} + {B} + {C} = S, ∂t ∂ξ ∂η ∂ζ (71) [5] Applying Second-Moment Closure to Complex Flows 175 where Q is the vector of dependent variables, F, G and H are convective ﬂuxes, Fv , Gv and Hv are diﬀusive ﬂuxes and S is the source vector. The diagonalization gives the eigenvalues that deﬁne the characteristic lines (waves) and hence identiﬁes the ‘upwind’ directions. The characteristics, representing shock, rarefaction and contact waves, are determined as part of an exact or approximate solution to the ‘Riemann problem’. A Riemann solver determines the incremental change in state of the ﬂow at a particular node, typically the centre of a cell, as one traverses waves in a direction normal to the interface joining the cell to its neighbour. The state of the Riemann problem at cell faces (the interface state, determined as a function of left and right states), allows the interface ﬂux to be evaluated according to the directions of the waves arising within the Riemann solution. The resulting equations, including the ﬂux approximations in terms of dependent variables, can then be solved (marched) in time by an explicit or implicit solution method (e.g. Euler-implicit), the latter entailing a coupled solution of the equations – for example, using a Newton linearization and a relaxation or factorization (ADI) solution. In the case of implicit schemes, a natural and potentially very stable route is to incorporate the turbulence-model equations into the set of conservation equations which are then solved implicitly. Within a Riemann-solver-based scheme, the extended Jacobian ﬂux matrices (7 × 7 for 2-equation models in 3D, and 12 × 12 for Reynolds-stress models in 3D, per direction) have to be linearized. If source terms are integrated based on cell-centred data, this does not introduce any more characteristics than arise in the inviscid equation set (eigenvalues associated with the propagation of the turbulence-model quantities are the same as that for the contact or shear wave), although the characteristics are modiﬁed slightly by the turbulence equations – for example, through the appearance of the turbulence energy in the total-energy ﬂuxes. Then, a full linearization of ﬂux balances and source terms (i.e. non-ﬂux terms), can be handled by some implicit (e.g. Newton) method. A fully-coupled solution is very challenging, as coupling arises at three levels: point-wise coupling of the ﬂuxes, point-wise coupling via the source terms and spatial coupling (in 3D!). In the case of two-equation models, Barakos and Drikakis (1998) have adopted a solution of all equations within an unfactored scheme based on a linearized Rieman solver using a point-implicit relaxation method. In the case of Reynolds-stress closures, the task of a fully-implicit solution is formidable. Morrison (1992) has presented an approximate factorized solution process which requires the inversion of block penta-diagonal systems, with blocks consisting of 12 × 12 matrices for the Reynolds-stress model. However, the scheme is not fully implicit, in that source terms of the turbulence equations associated with production, redistribution and dissipation are only point-coupled. Further simpliﬁcations introduced by Morrison led to a scheme in which the turbulence-model equations are, in eﬀect, decoupled from the mean-ﬂow equations, except for coupling eﬀected through the presence of the 176 Leschziner and Lien turbulence energy (normal stresses) in the ﬂuxes, via the Jacobian ﬂux matrices. Probably the most elaborate solution scheme has been employed by Vallet (1995), who adopted a ﬂux-vector-splitting technique (an early upwind scheme which is somewhat more diﬀusive than most Riemann solvers when applied to contact discontinuities, shear waves and boundary layers). Vallet adopted a point-implicit method, in which coupling among ﬂuxes as well as source terms was accounted for across the entire set of 12 conservation equations. Close examination of the presentation of the method – especially the structure of the ﬂux and source Jacobians – suggests, however, that coupling between the turbulence-equation and mean-ﬂow subsets is rather weak. Thus, the meanﬂow characteristics are aﬀected by the stresses and the stresses are aﬀected by the contact wave, but the turbulence-model sources are not strongly coupled to the mean-ﬂow equations. However, coupling among the turbulence-model equations is established through a full linearization of all sources with respect to all turbulence quantities. The subset is then solved by sub-iterations rather than full inversion (which would require very large amounts of storage). There is no unambiguous evidence that this strong level of coupling is beneﬁcial to stability and convergence. For example, Vallet reports convergence histories for a transonic channel ﬂow which show that convergence stalls after a reduction in residuals by less than two orders of magnitude. A somewhat simpler approach, adopted by Batten et al. (1997) in conjunction with Reynolds-stress modelling and a nonlinear (HLLC) Riemann solver, is to solve the mean-ﬂow equations by means of a block-coupled implicit scheme and solve the set of turbulence-transport equations as a segregated set. Thus, the ﬂux F of any turbulence variable (ρϕ) is assembled by reference to the contact-wave velocity and the (Riemann problem) interface state of the density, and then an implicit, decoupled equation is derived by including only diagonal components of the convective, diﬀusive and source Jacobians in the equation, the remaining terms being treated explicitly. In eﬀect, the ﬂux F in the turbulence equation is decomposed into a sum of implicit and explicit contributions, the latter treated as a deferred correction: F = FImplicit + FCorrection . (72) While this approach is not necessarily TVD in transients, the above treatment is suﬃcient to ensure positivity. The sum of the ﬂux corrections is treated as a source term and linearized via the following approach, due to Patankar (1980). The uncoupled, implicit equation for any conserved scalar quantity, ρϕ, may be written as: J (ρϕ)n+1 − (ρϕ)n − ∆t f n+1 ds = JSt + Sc , (73) f n+1 denotes the sum of all implicitly discretized ﬂuxes (including where diﬀusion and convection terms), St are the source terms arising from the turbulence model, and Sc represents the sources from any deferred-corrections. [5] Applying Second-Moment Closure to Complex Flows 177 Linearizing convective and diﬀusive ﬂuxes and splitting all sources into positive and negative contributions gives: J − ∆t ∂f ∆(ρϕ) = ∂(ρϕ) + − + − f n + J St + St + Sc + S c . (74) Positive source terms are treated explicitly, whilst negative contributions are scaled by (ρϕ)n+1 /(ρϕ)n and moved to the left-hand side of the equations. Introducing a small positive constant, δ = 10−30 , to prevent division by zero gives: J − ∆t − − ∂f JSt + Sc ∆(ρϕ) − (ρϕ)n+1 = ∂(ρϕ) (ρϕ)n + δ + + f n + JSt + Sc . (75) To retain the ‘∆’ form, the term − − JSt + Sc (ρϕ)n (ρϕ)n + δ is added to both sides of equation (75) to give: J − ∆t = − − ∂f JSt + Sc ∆(ρϕ) − ∂(ρϕ) (ρϕ)n + δ − − JSt + Sc + + f n + JSt + Sc + (ρϕ)n . (ρϕ)n + δ (76) In the above form, all ﬂux-balance and source terms appear on the right-hand side. However, the small parameter, δ, also appears in the denominator of the S − terms on both left-hand and right-hand sides of the equation. One might expect that the (ρϕ)n terms on the right-hand side could be cancelled by ignoring this small parameter. However, since it is possible for ϕ → 0, this term cannot be ignored, since (ρϕ)n < δ will again lead to small negative values of ϕ in subsequent iterations. The above procedure has no eﬀect on a converged solution, but it ensures that positivity is preserved on relevant data, such as the normal stresses, turbulence energy, and dissipation rate, even if the turbulence model itself is not strictly realizable. No such procedure was found necessary for any mean-ﬂow equation. In this case, the deferred-correction terms FCorrection were simply treated explicitly, with the exception of those terms relating to the Reynolds-stress traction vectors, which were treated using apparent viscosities in essentially the same manner as that described in Section 3.2 for incompressible ﬂow. 4 4.1 Application Examples Overview Of the many incompressible and compressible ﬂows which have been computed with second-moment closure over the past decade, two 3D ﬂows have been 178 Leschziner and Lien chosen here to illustrate the performance of second-moment closure in complex strain and to comment on numerical aspects. One is incompressible and has been computed with a pressure-correction scheme. The other is transonic and has been computed with a Riemann-solver-based implicit upwind scheme. The reader interested in broader expositions of the physical aspects and the predictive performance of second-moment closure over a wide range of ﬂows is referred to Chapters [1] and [2] as well as review articles by Launder (1989), Leschziner (1990, 1994, 1995), Hanjali´ (1994) and Leschziner et al. (1999). c 4.2 Prolate Spheroid This is a ﬂow around an elliptical body of axes ratio 6:1 and inclined at 10◦ or 30◦ to an oncoming, uniform stream. It was one of several test cases in the European-Commission-funded international validation exercise ECARP (Haase et al. 1996). The geometry represents the group of external ﬂows around streamlined bodies which feature vortical separation that arises from an oblique ‘collision’ and subsequent detachment of boundary layers on the body’s leeward side. Of the two ﬂows, that at 30◦ and Re = 6.5×106 (based on chord) is much more challenging, but poses signiﬁcant uncertainties due to a complex pattern of natural transition on the windward surface. Experimental data have been obtained by Meier et al. (1984) and Kreplin et al. (1985) for pressure, skin friction and mean-velocity, the last with a 5-hole probe. Corresponding computations have been performed by Lien and Leschziner (1995) with the second-moment closure of Gibson and Launder (1978), coupled to a low-Re linear eddy-viscosity model (EVM) in the viscous sublayer. A secondorder TVD scheme was used on a high-quality conformal mesh of 98 × 82 × 66 nodes, with the y + -value closest to the wall being kept to 0.5 to 1 across the entire surface. Test calculations with a nonlinear EVM on a 1283 grid have shown only the skin friction to change slightly at this level of grid reﬁnement. Figure 8 contains comparisons of azimuthal velocity proﬁles at one streamwise location at 10◦ incidence, while Figure 9 gives, for 30◦ incidence, one azimuthal pressure distribution, one skin-friction distribution and one velocity ﬁeld, the last showing the leeward separation and the associated transverse vortex. In Figure 8, the ‘truncated RSTM’ is the Gibson–Launder model without the wall correction Φw to the rapid part of the pressure-strain term. This ij,2 truncation was motivated by the observation, made in computations for separated aerofoil ﬂows, that the correction tended to increase rather than decrease the level of wall-normal turbulence intensity and shear stress as the boundary layer approaches separation. In general, the nonlinear EVM and the secondmoment closure give similar results which are closer to the experimental data that those obtained with the linear EVM. However, the improvement is not uniformly pronounced across all ﬂow properties, and the uncertainties associated with transition in the 30◦ case do not warrant a categorical statement on [5] Applying Second-Moment Closure to Complex Flows 179 Figure 8: Prolate spheroid at 10◦ incidence: azimuthal velocity proﬁles above leeward side at one axial position (from Lien and Leschziner 1995). model performance for this very complex ﬂow. In terms of numerical performance, the use of a two-stage continuation strategy, in which the Reynolds-stress model was applied following an initial stage of partial convergence with the linear EVM, resulted in CPU times of the order of only 50% in excess of that needed with the linear EVM. 4.3 Fin-Plate Junction This is one of the most complex compressible ﬂows predicted so far with second-moment closure. The geometry, shown in Figure 10, was the subject of a recent Europe–US workshop on high-speed ﬂows. A Mach 2 ﬂat-plate boundary layer collides with the rounded normal ﬁn, producing a complex shock/boundary-layer interaction and multiple horseshoe vortices. Experimental data are available for surface pressure, LDA velocity, skin-friction patterns 180 Leschziner and Lien Figure 9: Prolate spheroid at 30◦ incidence: (a) skin-friction lines; (b) circumferential pressure variations; (c) circumferential variations of skin-friction direction; (d) structure of vortices above rear leeward side (from Lien and Leschziner 1995). [5] Applying Second-Moment Closure to Complex Flows SST 181 MCL Figure 10: Supersonic ﬁn-plate-junction ﬂow: (a) ﬂow structure in shockaﬀected region ahead of ﬁn; (b) plate-pressure distributions at various spanwise (y) stations; (c) skin-friction lines on lower wall (from Batten et al. 1999). and Reynolds stresses (Barbaris and Molton 1992, 1995). While the geometry and ﬂow are well controlled, some minor uncertainties arise because of lack of detail in the measured boundary layer well upstream of the ﬁn (only its thickness was given) and the presence of leakage between the ﬁn tip and one wall of the windtunnel. The latter poses some uncertainty about the boundary conditions on the computational boundary plane above the lower ﬂat plate along which the interaction takes place. Computations were performed by Batten et al. (1999) with eddy-viscosity models, the Jakirli´–Hanjali´ (1995) linear Reynolds-stress model and the c c 182 Leschziner and Lien Figure 10: Continued. Craft–Launder cubic model, modiﬁed by Batten et al. (1999). A ﬁn-adapted 80 × 80 × 70 C-type grid was used, with the y + closest to the wall being of order 0.5. The results shown in Figure 10 illustrate that only the second-moment closure is able to reproduce the multiple separation/reattachments ahead of the ﬁn which is observed in the experiment, although the patterns are not identi- [5] Applying Second-Moment Closure to Complex Flows 183 Figure 10: Continued. cal. However, the size of the separated zone and the associated pressure ﬁeld, which are closely linked to the predicted strength of the shock/boundary-layer interaction, are strongly dependent on which model variant is used. As seen, the Jakirli´–Hanjali´ model signiﬁcantly underestimates the interaction, thus c c predicting a delayed boundary-layer separation. In contrast, the cubic model does signiﬁcantly better, returning pressure distributions close to the experimental variations. Finally, Figure 11 shows the convergence histories of computations with the two Reynolds-stress models mentioned above, in contrast to that of the SST model of Menter (1994), which is a linear EVM variant popular in CFD for 184 Leschziner and Lien Figure 11: Supersonic ﬁn-plate-junction ﬂow: convergence histories with different models (from Batten et al. 1999). Aeronautical Engineering and combines the k-ε model near the wall with the k-ω model in internal regions. As is the case in the previous incompressible ﬂow, the computational penalty associated with Reynolds-stress modelling is of the order 50%. 5 Concluding Remarks The application of second-moment closure to complex ﬂows is, unavoidably, more challenging than that of eddy-viscosity models. The challenges are not merely rooted in numerical diﬃculties, but also in the sheer complexity of the governing equations, signiﬁcant uncertainties in inﬂuential closure approximations – reﬂected by a signiﬁcant level of predictive variability – and diﬃculties in relation to boundary conditions. Low numerical stability, slow convergence and high storage and CPU demands tend to discourage the widespread adoption of second-moment closure for industrial applications. However, much can be done to counteract these numerical diﬃculties by use of the algorithmic practices outlined in this chapter. Not all measures are essential or, indeed, always advantageous. Nor do they guarantee favourable numerical behaviour. In the most favourable circumstances, a combination of the reported practices with a continuation strategy, in which the second-moment calculation is preceded by an eddy-viscosity computation, the CPU and memory resources can be depressed to around 1.3 to 1.5 times the resource for an eddy-viscosity prediction. The overhead is therefore relatively small, and the potential gain in predictive realism can be very signiﬁcant. [5] Applying Second-Moment Closure to Complex Flows 185 References Apsley, D., Chen, W-L, Leschziner, M. A. and Lien, F-S. (1998). ‘Non-linear eddyviscosity modelling of separated ﬂows’, IAHR J. of Hydraulic Research 35 723–748. Barberis, D. and Molton, P. (1992). ‘Shock wave-turbulent boundary layer interaction in a three dimensional ﬂow – laser velocimeter results’. Technical Report, ONERA TR.31/7252AY. Barberis, D. and Molton, P. (1995). ‘Shock wave-turbulent boundary layer interaction in a three dimensional ﬂow’. AIAA-95–0227, Reno, Nevada. 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DFVLR Goettingen Report IB 222–84 A11. Menter, F.R. (1994). ‘Two-equation eddy-viscosity turbulence models for engineering applications’, J. AIAA 32 1598–1605. Morrison, J.H. (1992). ‘A compressible Navier–Stokes Solver with two-equation and Reynolds stress turbulence closure models’. NASA Contractor Report 4440. Obi, S., Peric, M. and Scheuerer, G. (1989). ‘A ﬁnite-volume calculation procedure for turbulent ﬂows with second-order closure and collocated variable arrangement’. Proc. 7th Symp. Turbulent Shear Flows, Stanford Univ., 17.4.1–17.4.6 Patankar, S.V. (1980). Numerical Heat Transfer and Fluid Flow. McGraw-Hill. Rhie, C.M. and Chow, W.L. (1983). ‘Numerical study of the turbulent ﬂow past an airfoil with trailing edge separation’, J. AIAA 21 1525–1532. Vallet, I. (1995). A´rodynamique Numerique 3D Instationnaire avec Fermature Base Reynolds au Second Ordre. Doctoral Thesis, Universit´ de Paris 6. e van Leer, B. (1979). ‘Towards the ultimate conservative diﬀerence scheme, V. A second-order sequel to Godunov’s methods’, J. Comp. Phys. 32 101–136. 6 Modelling Heat Transfer in Near-Wall Flows Y. Nagano Abstract Recent developments in turbulence models for heat transfer are presented, focusing on near-wall behavior of thermal turbulence in ﬂows with diﬀerent Prandtl numbers. First, we outline a phenomenological two-equation heattransfer model for gaseous ﬂows along with an accurate prediction of wall turbulent thermal ﬁelds. The model reproduces the correct wall-limiting behavior of velocity and temperature under arbitrary wall thermal conditions. The model appraisal is given with four diﬀerent typical thermal ﬁelds, which often occur in engineering applications, in wall turbulent shear ﬂows. Secondly, we describe the methodology of how to construct a rigorous two-equation heattransfer model with the aid of the most up-to-date direct numerical simulation (DNS) data for wall turbulence with heat transfer. The DNS data indicate that the near-wall proﬁle of the dissipation rate, εθ , for the temperature variance, kθ (= θ2 /2), is completely diﬀerent from the previous model predictions. We demonstrate the results of a critical assessment of existing εθ equations for both two-equation and second-order closure models. Based on these assessments, we construct a new dissipation rate equation for temperature variance, taking into account all the budget terms in the exact εθ equation. Also, we present a similarly reﬁned kθ equation, which is linked with this new εθ equation, to constitute a new two-equation heat-transfer model. Comparisons of the reﬁned model predictions with the DNS data for a channel ﬂow with heat transfer are given, which shows excellent agreement for the proﬁles of kθ and εθ themselves and the budget in the kθ and εθ equations. The only limitation is that this reﬁned model is applicable only to gaseous ﬂows such as air streams. Thus, ﬁnally, we present the development of a heat-transfer model for a variety of Prandtl-number ﬂuids. This model incorporates new velocity and time scales to represent various sizes of eddies in velocity and thermal ﬁelds with diﬀerent Prandtl numbers. Fundamental properties of the reconstructed kθ -εθ model are ﬁrst veriﬁed in basic ﬂows under arbitrary wall thermal boundary conditions and next in backward-facing-step ﬂows at various Prandtl numbers through a comparison of the predictions with the DNS and measurements. 1 Introduction The turbulence model for heat transfer is a set of diﬀerential equations which, when solved with the mean-ﬂow and turbulence Reynolds-stress equations, 188 [6] Modelling heat transfer in near-wall ﬂows 189 allows calculations of relevant correlations and parameters that simulate the behavior of thermal turbulent ﬂows. Like the classiﬁcation of turbulence models for the Reynolds stresses, phenomenological turbulent heat-transfer models are classiﬁed into zero-equation, two-equation, and heat-ﬂux equation models. The zero-equation heat-transfer model is a typical and most conventional method for analyzing the turbulent heat transfer, in which the eddy diﬀusivity for heat αt is prescribed via the known eddy viscosity νt together with the most probable turbulent Prandtl number P rt , so that αt = νt /P rt . Thus, in this formulation the analogy is tacitly assumed between turbulent heat and momentum transfer (Launder 1988). Many previous experimental studies have, however, revealed that there are no universal values of P rt even in simple ﬂows (Kays 1994), e.g. at the same streamwise location, a value of P rt close to the wall is diﬀerent from that away from the wall (Cebeci 1973; Antonia 1980). The recent sophisticated renormalization group (RNG) theory for turbulence based on an iterative averaging method (Nagano and Itazu 1997) indicates that the turbulent Prandtl number changes according to the molecular Prandtl number (Itazu and Nagano 1998). This lack of universality restricts the applicability of a zero-equation model. On the other hand, a heat-ﬂux equation model ought to be more universal, at least in principle. This model, however, is still rather primitive and extensive further research is in progress. For the velocity ﬁeld, the linear eddy viscosity k-ε model of turbulence is still regarded as a powerful tool for predicting many engineering ﬂow problems including jets, wakes, wall ﬂows, reacting ﬂows, and ﬂows with centrifugal and Coriolis forces (Rodi 1984). For scalar turbulence, Nagano and Kim (1988) developed a corresponding two-equation model for heat transport (hereinafter referred to as the NK model). They modelled the eddy diﬀusivity for heat αt using the temperature variance kθ = (θ2 /2) and the dissipation rate of temperature ﬂuctuations εθ , together with k and ε. The NK model is applicable to thermal ﬁelds where the real value of P rt is unknown, and thus is more widely applicable than the conventional zero-equation model. A weakness is that the NK model has been developed mainly for heat transfer under uniform-walltemperature conditions. Consequently, in order to analyse heat transfer problems under various wall thermal conditions, we need further improvements to the NK model or the development of a more sophisticated kθ -εθ heat-transfer model. Thus, a modiﬁed kθ -εθ model (Youssef, Nagano and Tagawa 1992), maintaining the original concept of the NK model has been developed. Using a Taylor-series expansion for the energy equation in the near-wall region, they have made it clear how the wall-limiting behavior of turbulence quantities in a thermal ﬁeld varies with the thermal-wall condition, and then constructed the basic modelled equations to satisfy these requirements. This heat-transfer turbulence model was tested by application to turbulent boundary layers with four diﬀerent wall thermal ﬁelds; namely, a uniform wall temperature, a uniform wall heat ﬂux, a stepwise change in wall temperature, and a constant heat ﬂux followed by an adiabatic wall. 190 Nagano Fortunately, recent direct numerical simulations (DNS) for wall shear ﬂows provided the details of turbulent quantities near the wall (e.g., Kim et al. (1987), and Kasagi et al. (1992)). It was shown from these DNS data that the near-wall proﬁles of the dissipation rates of turbulent kinetic energy and temperature variance were completely diﬀerent from the previous model predictions. Recently, Rodi and Mansour (1993), and Nagano and Shimada (1995a) have improved the k-ε model using DNS databases in which all the budget terms in the exact ε equation were incorporated in the modelled ε equation. The performance of the existing ε-equation models was assessed by Nagano and Shimada (1994; 1995b), and a rational ε equation was reconstructed by Nagano et al. (1994). Similarly, two-equation heat-transfer models (kθ -εθ ) have been improved (Nagano et al. 1991; Youssef, Nagano and Tagawa 1992; Hattori, Nagano and Tagawa 1993; Sommer et al. 1992; Abe, Kondoh and Nagano 1995), since Nagano and Kim (1988) proposed the ﬁrst model for wall turbulent shear ﬂows. The two-equation heat-transfer model is a powerful tool for predicting the heat transfer in ﬂows with almost complete dissimilarity between velocity and thermal ﬁelds. Also, the characteristic time scale for a thermal ﬁeld needed in a second-order closure model may now be calculated with the two-equation heat-transfer model (Shikazono and Kasagi 1996). In the present chapter, as the ﬁrst example of modelling heat transfer, we develop a rigorous kθ -εθ model. In particular, modelling of the εθ equation is performed by taking into account all the budget terms in the exact εθ equation. First, we make a critical assessment of previous εθ equations for both two-equation and second-order closure models. Second, we reconstruct a more sophisticated εθ equation reﬂecting the assessment results. Then, we propose a set of kθ -εθ models to match with the present rigorous εθ equation. Finally, we verify a set of model equations using DNS data and experimental data. In order to show the performance of the proposed two-equation heat-transfer model, we analyze the case of a sudden change in thermal wall condition, which is hard to measure using conventional tools; and we then investigate the physical phenomena of the system using the results of analysis. As mentioned above, after the ﬁrst proposal by Nagano and Kim (1988), several kθ -εθ models have been proposed (Youssef et al. 1992; So and Sommer 1993; Hattori, Nagano and Tagawa 1993). Most of the previous models have adopted a dimensionless parameter y + = uτ y/ν, which is normalized by the viscous length ν/uτ consisting of the friction velocity uτ and the kinematic viscosity ν, to represent the distance from the wall, y. However, as recently pointed out on a number of occasions, the viscous length ν/uτ becomes inﬁnity (in other words uτ becomes zero) at the separation and reattachment points, so that the introduction of a parameter without the viscous length is indispensable in order to extend the applicability of the turbulence model for complex engineering problems. Abe et al. (1994) introduced a new parameter [6] Modelling heat transfer in near-wall ﬂows 191 Table 1: Constants and fuctions for the k-ε model. ∗ σk ∗ σε ft fw1 fw2 Cε1 Cε2 Cε3 Cε4 1.4/ft 1.3/ft 1 + 6fw1 fw (4) fw (26) 1.45 1.9 0.005 0.5 (1 + f2 )(1 − fw1 ){1 − 0.6 exp[−(Rt /45)1/2 ]} 13 0.5 exp(−2 × 10−4 Rv )[1 − exp(−2.2Rv )] (k/ε)[1/(1 + νt /ν)](1/Rt )fw1 (1 − fw2 ){1 + (60/Rt ) exp[−(Rt /55)1/2 ]} 3/4 1/2 f2 f2 Rv fµ using the Kolmogorov length η = (ν 3 /ε)1/4 in place of the viscous length ν/uτ as the characteristic length, and succeeded in predicting the correct reattachment points in backward-facing-step ﬂows at various Reynolds numbers. Thus, in this chapter, we use the same normalization as in Abe et al. (1994). As the second example of modelling heat transfer, we construct a reﬁned two-equation heat-transfer model based on information obtained from the DNS databases for turbulent heat and ﬂuid ﬂows with diﬀerent Reynolds and Prandtl numbers. In addition, in modelling the turbulence transport processes, we consider length and time scales characterizing a range from small to larger eddies in both the velocity and thermal ﬁelds, and we try to combine the eﬀects of those various scales in each ﬁeld with the present two-equation heat-transfer model without employing the viscous length ν/uτ . 2 Two-equation model of turbulence for velocity ﬁeld A velocity ﬁeld is described with the following continuity and momentum equations ∂Ui = 0, (2.1) ∂xi DUi ∂ 1 ∂P + =− Dt ρ ∂xi ∂xj where D/Dt ≡ ∂/∂t + Uj ∂/∂xj . ν ∂Ui − ui uj , ∂xj (2.2) 192 Nagano In the k-ε model, the Reynolds stress ui uj in (2.2) (see Nagano and Tagawa 1990a) can be obtained from the following set of equations −ui uj = νt ∂Ui ∂Uj + ∂xj ∂xi 2 − δij k, 3 (2.3) νt = Cµ fµ Dk ∂ = Dt ∂xj Dε ∂ = Dt ∂xj ν+ ν+ νt σε νt σk k2 , ε ∂k ∂Ui − ui uj − ε, ∂xj ∂xj (2.4) (2.5) (2.6) ∂ε ε ∂Ui ε2 − Cε1 ui uj − Cε2 fε . ∂xj k ∂xj k As indicated by Myong and Kasagi (1990), and by Nagano and Tagawa (1990a), imposing the rigid boundary condition (i.e. no-slip) at the wall does not necessarily lead to the correct asymptotic solutions of k ∝ y 2 , −uv ∝ y 3 , νt ∝ y 3 , and ε ∝ y 0 for y → 0, unless the wall-limiting behavior of turbulence is properly incorporated in the turbulence model adopted. In the present study, for the basic formulation in the k-ε model, we adopt the Nagano–Shimada model (Nagano and Shimada 1995a) (hereinafter referred to as the NS model), which reproduces strictly the limiting behavior of wall. The model also reproduces the turbulent energy and its dissipation rate (including their budgets) very closely for the cases of wall bounded ﬂows: Dk Dt ∂k ∂Ui −ε − ui uj ∂xj ∂xj k ∂ε ∂ fw1 , 0 , + max −0.5ν ∂xj ε ∂xj ∂ε ε ∂Ui ε2 ∂ νt − Cε2 f2 = ν+ ∗ − Cε1 ui uj ∂xj σε ∂xj k ∂xj k = ∂ ∂xj ν+ νt ∗ σk +fw2 ννt +Cε4 ν ∂ 2 Ui ∂xj ∂xk 2 (2.7) Dε Dt + Cε3 ν k ∂k ∂Ui ∂ 2 Ui ε ∂xk ∂xj ∂xj ∂xk (2.8) ∂ ε ∂k fw1 . (1 − fw1 ) ∂xj k ∂xj The NS model employed the wall friction velocity uτ in the wall reﬂection function fw . However, here we introduce the Kolmogorov velocity uε in this function described as followed: fw (ξ) = exp − y∗ ξ 2 , (2.9) where y ∗ = uε y/ν (= y/η) is the dimensionless distance from the wall based on the Kolmogorov velocity scale uε = (νε)1/4 (or the Kolmogorov length [6] Modelling heat transfer in near-wall ﬂows 193 scale η = (ν 3 /ε)1/4 ). This model function is more useful for analysis of various complex ﬂows, as conﬁrmed by Abe et al. (1995) and Nagano et al. (1997). The model constants and functions were optimized for the proposed function. These are listed in Table 1. Note that, from (2.9), fw (4) = exp − (y ∗ /4)2 and fw (26) is similarly deﬁned. 3 3.1 Two-equation model for heat transfer (DNS-based modelling) Governing Equations The energy equation may be written DΘ ∂ = Dt ∂xj α ∂Θ − uj θ . ∂xj (3.1) In (3.1), the term on the right-hand side, the turbulent heat ﬂux uj θ, is described using the concept of eddy diﬀusivity for heat αt (see, Nagano and Kim 1988; Nagano et al. 1991), ∂Θ −ui θ = αt , (3.2) ∂xi where αt = Cλ fλ kτm . (3.3) As a time-scale equivalent to the relative ‘lifetime’ of the energy-containing eddies or temperature ﬂuctuations, we adopt the mixed or hybrid time-scale τm which is a function of the time-scale ratio R = τθ /τu , where τu = k/ε and τθ = kθ /εθ (kθ = θ2 /2) are the dynamic and scalar time-scales, respectively. Obviously, τm blends both thermal and mechanical contributions. The characteristic length scale (i.e. the spatial extent of a ﬂuctuating temperature) can hence be written as Lm = k 1/2 τm , and the eddy diﬀusivity for heat can be modelled as αt ∝ k 1/2 Lm = kτm . The present expression for αt can be regarded as a generalized form for the eddy diﬀusivity introduced by Nagano and Kim (1988). Accordingly, we incorporate near-wall eﬀects on the thermal ﬁeld in the model function fλ . The optimal value of eddy diﬀusivity for heat αt can be expressed as a function of the state of both velocity and thermal ﬁelds by solving the transport equations for k, ε, kθ , and εθ . The exact transport equations for kθ and εθ are symbolically expressed as follows (Nagano and Kim 1988): Dkθ = Dkθ + Tkθ + Pkθ − εθ , Dt Dεθ 1 2 3 4 = Dεθ + Tεθ + Pεθ + Pεθ + Pεθ + Pεθ − Υεθ . Dt (3.4) (3.5) 194 The terms on the right-hand sides in (3.4) and (3.5) are identiﬁed as Nagano Molecular diﬀusion 2k 2ε ∂ θ ∂ θ , Dεθ = α , Dkθ = α ∂xj ∂xj ∂xj ∂xj Turbulent diﬀusion ∂uj kθ ∂uj εθ , Tεθ = − , T kθ = − ∂xj ∂xj Mean gradient production ∂uj ∂θ ∂Θ ∂θ ∂θ ∂Uj ∂Θ 1 2 Pkθ = −uj θ , Pεθ = −2α , Pεθ = −2α , ∂xj ∂xk ∂xk ∂xj ∂xk ∂xj ∂xk (3.6) Gradient production ∂θ ∂2Θ 3 Pεθ = −2αuj , ∂xk ∂xj ∂xk Turbulent production ∂uj ∂θ ∂θ 4 Pεθ = −2α , ∂xk ∂xk ∂xj Destruction 2 2θ ∂ 2 , Υεθ = 2α ∂xk ∂xj where kθ = θ2 /2 and εθ = α(∂θ/∂xk )2 , respectively. 3.2 Wall-Limiting Behavior of Velocity and Temperature The behavior of the turbulent quantities of velocity and thermal ﬁelds near the wall can be inferred from a Taylor series expansion in terms of y, together with the continuity, momentum and energy equations, namely, 2 3 U + = y + + a1 y + + a2 y + + · · · 2 + b y3 + · · · u = b1 y + b 2 y 3 2 + c y3 + · · · v = c1 y 2 2 + d y3 + · · · w = d1 y + d2 y 3 2 + d2 )/2]y 2 + (b b + d d )y 3 + · · · k = ui ui /2 = [(b1 1 2 1 2 1 3 + (b c + c b )y 4 + · · · uv = b1 c1 y 1 2 1 2 (3.7) εw = ν(∂ 2 k/∂y 2 )w = ν(b2 + d2 ) 1 1 2 3 Θ+ = P ry + + g1 y + + g2 y + + · · · 2 + ··· θ = θ w + h1 y + h2 y 2 + 2h θ )/2]y 2 + · · · 2 kθ = θ2 /2 = θw /2 + [(h1 2 w 2 + (c θ + c h )y 3 + · · · vθ = c1 θw y 2 w 1 1 2 k /∂y 2 ] = α(h2 + 2h θ ) εθw = α[∂ θ w 2 w 1 [6] Modelling heat transfer in near-wall ﬂows 195 In (3.7), in view of the correspondence between k and kθ proﬁles near the wall, a smooth change in temperature variance kθ in the immediate vicinity of the wall is assumed, i.e. (∂kθ /∂y)w = 0, which is exact in the case of both uniform wall temperature and uniform wall heat ﬂux. From (3.7), in the vicinity of the wall, we obtain the following relations: U + = y + , u ∝ y, v ∝ y 2 , w ∝ y, Θ+ = P ry + , and θ ∝ y p/2 (where, p = 2 : without ﬂuctuations in wall temperature, θw ; p = 0 : with θw ﬂuctuations). These asymptotic relations provide the representation for the wall-limiting behavior of turbulence given as: k ∝ y 2 , −uv ∝ y 3 , ε ∝ y 0 , θ2 ∝ y p , vθ ∝ y 2+p/2 and εθ ∝ y 0 . Note that, as may be seen from (3.2), vθ and αt vary as the same power of y near the wall. Consequently, from the wall-limiting behavior of turbulence, we have the following two regimes according to the wall thermal conditions αt ∝ y 3 for p = 2 (without θw ﬂuctuations) αt ∝ y 2 for p = 0 (with θw ﬂuctuations). (3.8) As discussed later, the eddy diﬀusivity αt should be modelled to satisfy the above requirements consistently. 3.3 3.3.1 Assessment of Modelled εθ Equations Modelled εθ equations The modelled dissipation rate equations for temperature variance used in the current kθ -εθ models for wall shear ﬂows are written in one of two ways: εθ εθ Dεθ ∂ 2 εθ + Tεθ + CP 1 fP 1 Pkθ + CP 2 fP 2 Pk =α Dt ∂xj ∂xj kθ k −CD1 fD1 εθ 2 εθ ε − CD2 fD2 + additional term, kθ k (3.9) εθ εθ D εθ ∂ 2 εθ + Tεθ + CP 1 fP 1 Pkθ + CP 2 fP 2 Pk =α Dt ∂xj ∂xj kθ k −CD1 fD1 εθ 2 εθ ε − CD2 fD2 + additional term, kθ k (3.10) where Pk = −ui uj (∂Ui /∂xj ) is the production rate of k. The turbulent diﬀusion term Tεθ in (3.5) is generally modelled as follows: ∂ αt ∂εθ : at a two-equation level, ∂xj σφ ∂xj Tεθ = (3.11) ∂ k ∂εθ Cs fR ui uj : at a second-order closure level. ∂xj ε ∂xi The Tεθ term is also modelled in the same manner as in (3.11). 196 Nagano Table 2: Existing εθ and εθ equation models. Nagano-Kim(1988) Cλ Cs CP 1 CP 2 CD1 CD2 σφ τm fλ Aµ Aλ fR fP 1 fP 2 fD1 fD2 Additional term 0.11 — 0.9 0.72 1.1 0.8 1.0 (k/ε)(2R)1/2 [1 − exp (−y + /Aλ )] √ — (30.5/ P r)(Cf /2St) — 1.0 1.0 1.0 1.0 ααt (1 − fλ )(∂ 2 Θ/∂xj ∂xk )2 R = (kθ /εθ )/(k/ε) 2 Nagano-Tagawa-Tsuji(1991) 0.1 — 0.85 0.64 1.0 0.9 1.0 (k/ε) (2R)2 + 3.4(2R)1/2 /Rt 2 3/4 [1 − exp (−y + /Aλ )] — √ 26/ P r — 1.0 1.0 2 [1 − exp(−y + /5.8)] 2 (1/CD2 )(Cε2 fε − 1) [1 − exp(−y + /6)] — Cε2 = 1.9 fε = 1 − exp[−(Rt /6.5)2 ] In (3.10), quantities ε and εθ , called the isotropic dissipation rates of k and kθ , are deﬁned by the following equations, respectively: ε = ε − ε, √ (3.12) (3.13) εθ = ε θ − ε θ , √ where ε = 2ν(∂ k/∂y)2 , εθ = 2α(∂ ∆kθ /∂y)2 , and ∆kθ = kθ − kθw . 3.3.2 Assessment procedure In (3.9) and (3.10), εθ and εθ are the only unknown variables, and all turbulence quantities except εθ and εθ are supplied directly from the DNS data; i.e., Ui , ui uj , k, ε, ε, Θ, and kθ are not calculated from any modelled equation, but are given as the ‘true’ values from the DNS data. We perform the model assessment in a fully developed channel ﬂow with heat transfer (Hattori and Nagano 1998) for which a trustworthy DNS database (Kasagi et al. 1992) is available. The Reynolds number based on the friction velocity and a channel half-width, Reτ , is 150 and the Prandtl number is 0.71. [6] Modelling heat transfer in near-wall ﬂows 197 Table 2: (continued) Shikazono-Kasagi(1996) Cλ Cs CP 1 CP 2 CD1 CD2 σφ — 0.3 0.8 0.3 1.0 0.3 — Abe-Kondoh-Nagano(1995) 0.1 — 1.9 0.6 2.0 0.9 1.6 3/4 τm — (k/ε) fR + [3(2R)1/2 /(Rt P r)]fd fλ — [1 − exp (−y ∗ /Aµ )] [1 − exp (−y ∗ /Aλ )] — 14 Aµ √ — Aµ / P r Aλ 2R/(0.7 + R) 2R/(0.5 + R) fR 1.0 [1 − exp(−y ∗ )]2 fP 1 fP 2 1.0 1.0 fD1 1.0 [1 − exp(−y ∗ )]2 fD2 1.0 (1/CD2 )(Cε2 fε − 1) [1 − exp(−y ∗ /5.7)]2 Additional — 2αCw2 (kθ /εθ )v 2 (∂ 2 Θ/∂y 2 )2 − εθ εθ /kθ term fd = exp[−(Rt /200)2 ] Cw2 = max[0.1, 0.35 − 0.21P r] Cε2 = 1.9 fε = 1 − exp[−(Rt /6.5)2 ] 3.3.3 Models for assessment We assess the six temperature dissipation rate equations proposed by Nagano and Kim (NK) (1988), Hattori, Nagano and Tagawa (HNT) (1993) and Shikazono and Kasagi (SK) (1996), which are the εθ equations, and those by Nagano, Tagawa and Tsuji (NTT) (1991), Abe, Kondoh and Nagano (AKN) (1995) and Sommer, So and Lai (SSL) (1992), which are the εθ equations. The abbreviations in parentheses are introduced for ease of reference. The details of the above six modelled equations are listed in Table 2. It should be mentioned that the SSL model has partly introduced ε and εθ to prevent divergence in the calculation caused by ﬁnite values of ε and εθ at the wall, while the NTT and AKN models avoid it by introducing the proper fD1 and fD2 functions. In the SSL model, the turbulent heat-ﬂux ui θ in the Pkθ term is modelled using αt , but the Reynolds shear stress ui uj and turbulent diﬀusion term Tεθ are modelled at a second-order closure level [see (3.11)]. The AKN model puts fP 1 = fD1 to avoid divergence in the calculation of ﬂows with complete dissimilarity between velocity and thermal ﬁelds. In the SK model, where the kθ -εθ model is employed to calculate the time scale of the thermal ﬁeld, the tur- 198 Nagano Table 2: (continued) Sommer-So-Lai(1992) Cλ Cs CP 1 CP 2 CD1 CD2 σφ τm 0.11 0.11 0.9 0.72 1.1 0.8 — 1/4 (k/ε) (2R)1/2 + 0.1(2R)1/2 /Rt ×(fεt /fλ ) fλ 1 − exp −y + /Aλ 1 − exp −y + /Aµ 1 − exp −y + /Aλ Aµ — 30/(1 + 11.8P + ) 30 Aµ /P r1/3 Aλ fR 1.0 — 1.0 1.0 fP 1 fP 2 1.0 1.0 εθ /εθ 1.0 fD1 ε/ε (1/CD2 )(Cε2 fε − 1) fD2 Additional fεt [(CD1 − 2)(εθ /kθ )εθ + CD2 (ε/k)εθ ∗ ααt (1 − fw )(∂ 2 Θ/∂xj ∂xk )2 −(ε∗ 2 /(2kθ ) + (1 − CP 1 )(εθ /kθ )Pθ θ term fεt = exp[−(Rt /80)2 ] fd = exp[−(Rt /120)2 ] fw = [1 − exp{−y + /(30/P r1/3 )}]2 ε∗ = εθ − 2α(kθ /y 2 ) θ ∗ Pθ = uθ(∂Θ/∂x) P + = ν(dP/dx)/ρu3 τ Cε2 = 1.9 2 fε = 1 − 0.3 exp[−Rt ] 2 Hattori-Nagano-Tagawa(1993) 0.1 — 0.85 0.64 1.0 0.9 1.0 3/4 (k/ε) (2R)1/2 + [7.9(2R)1/2 /Rt ]fd bulent diﬀusion and production terms are modelled at a second-order closure level. A characteristic time scale τm , whose importance was demonstrated by Nagano and Kim (1988), has been used in all the two-equation heat-transfer models for wall shear ﬂows. It can be shown that the eddy diﬀusivity for heat αt is governed near the wall by the Kolmogorov microscale in the NTT, HNT and AKN models, and by the Taylor microscale in the SSL model. 3.3.4 Assessment results The results of assessment for εθ - and εθ -equation models at a two-equation level and those at a second-order closure level are shown in Figure 1, where in εθ -equation modelling εθ is obtained from εθ = εθ + εθ . The resultant characteristic time scale τθ is assessed in Figure 2. To assess the NK model, the time scale τu in αt is given by k/ε from the DNS, because the NK model is usually combined with the Nagano and Hishida model (1987) (ε-equation model). [6] Modelling heat transfer in near-wall ﬂows 199 Figure 1: Assessment of εθ and εθ equations: (a) εθ equations (εθ = εθ + εθ ) at a two-equation level; (b) εθ equations at a two-equation level; (c) εθ - and εθ equations at a second-order closure level. As can be seen from Figures 1 and 2, the results of assessment for εθ and εθ -equation models indicate that none of the four models can reproduce accurately the DNS behavior. Especially, predicted εθ tends to increase in the buﬀer layer (5 < y + < 40). In Figures 1(c) and 2(c), only the SK model qualitatively and quantitatively reproduces a trend similar to DNS. However, the constants CP 1 and CP 2 in the SK model do not satisfy the relation for a 200 Nagano Figure 2: Proﬁles of time scale τθ : (a) in εθ equations at a two-equation level; (b) in εθ equations at a two-equation level; (c) in εθ and εθ equations at a second-order closure level. ‘constant stress and constant heat ﬂux layer’, namely κ2 /P rt CP 1 − CD1 + + CP 2 − CD2 = 0, R Cµ (3.14) where κ = 0.39 − 0.41, P rt = 0.9, Cµ = 0.09 and R = 0.5 are typical values in wall-bounded ﬂows. [6] Modelling heat transfer in near-wall ﬂows 201 Next, we discuss the gradient of εθ at the wall. The near-wall behavior of εθ without θw ﬂuctuations can be inferred from a Taylor series expansion in terms of y as follows (Youssef et al. 1992): εθ = h1 + 4h2 y + O(y 2 ), (3.15) where the coeﬃcients h1 and h2 are independent of the y coordinate. On the other hand, from (3.4), the molecular diﬀusion term balances the dissipation term at y = 0: εθ = α ∂ 2 kθ ∂y 2 with α ∂ 2 kθ = h1 + 6h2 y + O(y 2 ). ∂y 2 (3.16) From (3.15) and (3.16), the coeﬃcient h2 [= (1/4)(∂εθ /∂y)|w ] should be zero. This can be seen, of course, in the DNS data. In theory, the wall-limiting behavior of εθ must be εθ ∝ y 2 . In the εθ equation of the NK and HNT models, however, the molecular diﬀusion term balances with CD1 εθ 2 /kθ term at y = 0. As a result, the wall-limiting behavior of εθ becomes εθ ∝ y 1 . Therefore, in the εθ equation, adding the extra term to reproduce the correct wall-limiting behavior of εθ is of the ﬁrst importance to obtain the correct proﬁle of εθ near the wall. The SK model has an additional term to balance the molecular diﬀusion term in the εθ equation at the wall, as suggested by Kawamura and Kawashima (1994). The budget data for the εθ and εθ equations are shown in Figure 3. The budget in√ the εθ equation is represented by α(∂ 2 εθ /∂y 2 ) = α(∂ 2 εθ /∂y 2 ) + 2α2 (∂ 2 [(∂ ∆kθ /∂y)2 ]/∂y 2 ). Obviously, the two-equation model predictions are not in agreement with the DNS data. As seen from Figure 3(c), the sum total of the budget in the SK model is the closest to the DNS. This is a consequence of the smaller model constants CP 2 and CD2 used, which render the production and destruction terms smaller in magnitude. In the εθ -equation models, the NTT model is rather close to the DNS. This is because the NTT model has no additional production term. From these assessments it becomes clear that solutions for εθ are signiﬁcantly inﬂuenced by any additional production term, the values ascribed to the model constants, and the formulation of the characteristic time scale. 3.4 3.4.1 Construction of a Rigorous kθ -εθ Model Modelling the eddy diﬀusivity for heat αt Thermal eddy diﬀusivity αt given by (3.3) must be adequately modelled with the dominant characteristic velocity and time scales responsible for scalar transfer. Thus, it is important to reﬂect the inﬂuence of the time scales for both velocity and thermal ﬁelds. Previous αt models have been based on the concept of a single time scale, e.g., the assumption of the turbulent Prandtl 202 Nagano 1 2 3 4 Figure 3: Budgets of modelled εθ equations (Pεθ + Pεθ + Pεθ + Pεθ + Tεθ − Υεθ ) 1 2 3 4 and εθ equations (Pεθ + Pεθ + Pεθ + Pεθ + Tεθ − Υεθ ): (a) two-equation level εθ equations; (b) two-equation level εθ equations; (c) second-order closure level εθ - and εθ equations. √ 2 number P rt or of a mixed time scale, e.g., τm = τu τθ or τm = τθ /τu . However, as is frequently pointed out, the former fails to predict heat transfer in ﬂows with a dissimilarity between velocity and thermal ﬁelds, while the latter compromises the accuracy of the predicted near-wall turbulence quantities because it relates τu and τθ , which characterize large-scale motions, to the region adjacent to the wall where the dissipative motion is dominant. Therefore, a [6] Modelling heat transfer in near-wall ﬂows 203 further development of αt , reﬂecting the eﬀect of various time scales in velocity and thermal ﬁelds, is needed. Recently, Abe et al. (1995) have proposed the following multiple-time-scale τm using the hybrid time scale τh = τu R/(Cm + R) (i.e., 1/τh = 1/τu + Cm /τθ with Cm as a model constant): τm k = ε √ 2R 2R 3 exp − + 0.5 + R P r R3/4 t Rt 200 2 , (3.17) where Rt = k 2 /(νε) is the turbulent Reynolds number and P r is the molecular Prandtl number. Using the multiple-time-scale similar to (3.17), we adopt the following representation for αt to satisfy the wall-limiting behavior of thermal turbulence indicated by (3.8): αt = Cλ fλ kτm = Cλ fλ k k ε √ 2R 2R 26 Rh exp − + 4/3 3/4 0.5 + R P r R 220 t , (3.18) fλ = [1 − fw (Aµ )]1/2 [1 − fw (Aλ )]1/2 (3.19) where Rh = kτh /ν = Rt [2R/(0.5+R)] is the turbulent Reynolds number based on the harmonic-averaged time scale τh = (k/ε)[2R/(0.5 + R)]. Note that τh becomes identical to τu = k/ε in local equilibrium ﬂows with R = 0.5. 3.4.2 Modelling the εθ equation As shown in the previous section, none of the existing εθ and εθ models at a two-equation level give qualitative and quantitative agreement with the DNS. Hence, we will construct an εθ -equation model based on the NTT model by taking into account all the budget terms in the exact εθ equation. 1 2 4 (a) Modelling of Pεθ , Pεθ , Pεθ and Υεθ 1 2 4 The Pεθ , Pεθ , Pεθ and Υεθ terms can be modelled in a way similar to the NK (Nagano and Kim 1988) and NTT models (Nagano et al. 1991): 1 2 4 Pεθ + Pεθ + Pεθ − Υεθ = −CP 1 fP 1 εθ εθ 2 εθ ∂Ui εθ ε ∂Θ uj θ − CD1 fD1 − CP 2 fP 2 ui uj − CD2 fD2 . kθ ∂xj kθ k ∂xj k (3.20) 1 2 The DNS data indicates that the Pεθ and Pεθ terms exert a great inﬂuence on the production of εθ near the wall. The modelling given by (3.20), however, is 204 Nagano 4 based on Pεθ and Υεθ , so that the inﬂuence of other terms is not suﬃciently 1 2 reﬂected. Therefore, we model the contributions from the Pεθ and Pεθ terms using an order-of-magnitude analysis, as done in modelling ε by Rodi√and Mansour (1993) and Nagano and Shimada (1995a). With kθ and θ = kτθ (thermal turbulence length scale), we can estimate an order of magnitude of 1 2 4 the Pεθ , Pεθ and Pεθ terms as √ k kθ λθ 1 G , Pεθ = O λ θ √ kkθ 2 (3.21) S , Pεθ = O θ kkθ θ 4 , Pεθ = O 2 λ θ where G = [(∂Θ/∂xj )(∂Θ/∂xj )]1/2 represents the mean temperature gradient, S = [(∂Ui /∂xj )(∂Ui /∂xj )]1/2 is the mean strain rate, and λ = kν/ε and λθ = kθ α/εθ are the Taylor microscales for the velocity and temper√ 1 4 ature ﬁelds, respectively. The above relations give Pεθ /Pεθ ≈ (λθ / kθ )G = √ 2 4 G/(εθ /α)1/2 , Pεθ /Pεθ ≈ (λ/ k)S = S/(ε/ν)1/2 . Consequently, we deﬁne the parameters RT and RU as (∂Θ/∂xj )2 RT = (∂θ/∂xj ) 2 1/2 1/2 = G , (εθ /α)1/2 (3.22) (∂Ui /∂xj )2 RU = (∂ui /∂xj )2 1/2 1/2 = S . (ε/ν)1/2 (3.23) These parameters represent the ratio of the gradient of mean ﬂow to that of 1 4 2 4 ﬂuctuating components. Apparently, the relations Pεθ /Pεθ ≈ RT and Pεθ /Pεθ ≈ RU hold. Since the structure of turbulent shear ﬂows near the wall is governed mainly by the gradient of the mean ﬂow (see, e.g., Hinze (1975)), 1 2 contributions of Pεθ and Pεθ terms must appear when RT > 1 and RU > 1. We replace the mean temperature gradient G and the strain rate parameter S with the well-known relations for the constant heat ﬂux layer [G = (qw /ρcp )/(α + αt ) = uτ θτ /(α + αt )] and the constant stress layer [S = (τw /ρ)/(ν + νt ) = uτ 2 /(ν + νt )]. Then, (3.22) and (3.23) lead to RT = uτ θτ (ν/P r + αt ) (P rεθ /ν)1/2 fw (6), (3.24) RU = u2 τ fw (6). (ν + νt )(ε/ν)1/2 (3.25) [6] Modelling heat transfer in near-wall ﬂows 205 1 2 The contributions of Pεθ and Pεθ can now be included in the model functions fP 1 and fP 2 as follows: fP 1 = (1 − fP 1 )fp , fP 1 = exp(−7 × 10−5 RT 10 )[1 − exp(−2.2RT 1/2 )], fP 2 = (1 − fP 2 )fp , fP 2 = exp(−7 × 10−5 RU 10 )[1 − exp(−2.2RU 1/2 )], (3.26) (3.27) where fp is introduced for correcting overproduction near the wall, and the wall reﬂection function fw (6) in (3.24) and (3.25) is given by (2.9) with ξ = 6. In the following equations (3.28) and (3.29), fw (12) and fw (3) are similarly deﬁned. 3 (b) Modelling of Pεθ 3 1 2 4 The Pεθ term is negligibly small in comparison with Pεθ , Pεθ , Pεθ and Υεθ . 1 +P 2 +P 4 −Υ , However, when compared with the sum of these terms, i.e., Pεθ εθ εθ εθ 3 3 the Pεθ term becomes of the same order, so modelling Pεθ is also important. In the present model, we adopt the following form similar to (2.8) in the k-ε model: 3 Pεθ = ααt fw (12) ∂2Θ ∂xj ∂xk 2 ∂k ∂Θ ∂ 2 Θ k + CP 3 α fR , ε ∂xk ∂xj ∂xj ∂xk (3.28) where fR = 2R/(0.5 + R). It should be noted that the hybrid turbulent Reynolds number Rh and the corresponding time scale τh in (3.18) can be written as Rh = fR Rt and τh = fR τu . (c) Modelling of Tεθ A gradient-type diﬀusion plus convection by large-scale motions may eﬀectively represent turbulent diﬀusion for a scalar (see, e.g., Hinze (1975)). Thus, considering the relation εθ 2(ε/k)kθ at R 0.5 and the near-wall-limiting behavior of Tεθ , we write Tεθ as Tεθ = ∂ ∂xj αt ∂εθ σφ ∂xj + Cεθ α ∂ ∂xj [1 − fw (3)]3/2 ε ∂kθ fw (3) . k ∂xj (3.29) (d) Modelled εθ -equation To sum up, the proposed εθ -equation can be written as Dεθ Dt = ∂ ∂xj α+ αt σφ ∂εθ εθ εθ ∂Ui ∂Θ − CP 1 fP 1 uj θ − CP 2 fP 2 ui uj ∂xj kθ ∂xj k ∂xj ∂2Θ ∂xj ∂xk 2 ε2 εθ ε −CD1 fD1 θ − CD2 fD2 + ααt fw (12) kθ k 206 ∂k ∂Θ ∂ 2 Θ k +CP 3 α fR ε ∂xk ∂xj ∂xj ∂xk ε ∂kθ ∂ +Cεθ α fw (3) . [1 − fw (3)]3/2 ∂xj k ∂xj The wall reﬂection function fw (ξ) is given by (2.9). Nagano (3.30) (e) Model functions and constants From (3.30), the molecular diﬀusion term balances the dissipation terms at y = 0: ε2 εθ ε ∂ 2 εθ α 2 = CD1 fD1 θ + CD2 fD2 . (3.31) ∂y kθ k Considering the limiting behavior of wall turbulence, fD2 ∝ y 2 and fD1 ∝ y 2 (without θw ﬂuctuations) or fD1 ∝ y n where n > 0 (with θw ﬂuctuations) are required to satisfy (3.31). In free turbulence, as described next (see (3.41)), the limiting behavior requires CD2 fD2 = Cε2 fε − 1. (3.32) In the present model, the following equations are thus proposed to meet the requirements for both wall and free turbulence fD1 = 1 − exp − (y ∗ /12)2 = 1 − fw (12) fD2 = (1/CD2 )(Cε2 fε − 1) [1 − fw (12)] /6.5)2 ]. (3.33) (3.34) with fε = 1 − 0.3 exp[−(Rt The constants appearing in the present two-equation heat-transfer model are determined as follows. Firstly, we specify a value of Cλ in (3.18) deﬁning the eddy diﬀusivity for heat, αt . In the log-law region where the molecular diﬀusion is negligible, i.e. fµ = fλ = 1, Cλ may be given from (2.4) and (3.18), together with the turbulent Prandtl number P rt = νt /αt , by Cλ = Cµ / (P rt fR ) with fR = 2R/(0.5 + R) (3.35) thus, substituting the typical values of Cµ = 0.09, R = 0.5, and P rt = 0.9 (Nagano and Kim 1988; Launder 1988), we obtain Cλ = 0.10. We determine the constants CD1 and CD2 in the equation for εθ , (3.30) from the decay law of homogeneous turbulence. In a homogeneous decaying turbulent ﬂow, (2.5), (2.6), (3.4) and (3.30) become simply dk dx dε U dx dkθ U dx dεθ U dx U = −ε, = −Cε2 fε = −εθ , = −CD1 fD1 ε2 εθ ε θ − CD2 fD2 , kθ k ε2 , k (3.36) (3.37) (3.38) (3.39) [6] Modelling heat transfer in near-wall ﬂows 207 where the x axis is taken in the ﬂow direction. On the other hand, it is known that the time-scale ratio R = (kθ /εθ )/(k/ε) does not change in the ﬂow direction in homogeneous grid-generated turbulence (Newman et al. 1981; Warhaft and Lumley 1978), thus, rewriting (3.39) in terms of R and substituting (3.37)– (3.39) into this equation, we obtain U dεθ 1 = dx R ε2 kθ ε2 kθ εθ ε − Cε2 fε 2 − 2 k k k =− ε2 εθ ε θ − (Cε2 fε − 1) . kθ k (3.40) Equations (3.39) and (3.40) give the following relations CD1 fD1 = 1, CD2 fD2 = Cε2 fε − 1. (3.41) Equation (3.41) is also valid for the initial period (fε = fD1 = fD2 = 1) in decaying turbulent ﬂows, and hence we have CD1 = 1 and CD2 = Cε2 −1 = 0.9. The model constants CP 1 and CP 2 for the production terms in the εθ equation (3.30) are determined by considering the characteristics of the loglaw region (constant stress-heat-ﬂux layer) in wall turbulence. In this region, the convection terms in the transport equations k, ε, kθ , and εθ can all be ignored, and the production terms for k and kθ balance with the respective dissipation terms, thus, with (3.18), rewriting (3.30) gives ε2 εθ ∂Θ εθ ∂U εθ ε vθ − CP 2 uv − CD1 θ − CD2 = 0. kθ ∂y k ∂y kθ k (3.42) With the above-mentioned characteristics of constant stress-heat-ﬂux layer, the following relation is obtained from (3.42) Cλ ∂ σφ ∂y k 2 ∂εθ fR ε ∂y − CP 1 1/2 CP 2 = (CD1 − CP 1 )/R + CD2 − (κ2 /P rt )/(σφ Cµ ), (3.43) where κ is the von K´rm´m constant. Equation (3.43) is similar to the wella a known relation in the k-ε model given by 1/2 Cε1 = Cε2 − κ2 /(σε Cµ ). (3.44) The value CP 2 = 0.77 is then obtained if we substitute the foregoing values of CD1 , CD2 , R, P rt , and Cµ for (3.43), together with κ = 0.39 − 0.41 and CP 1 = 0.9 which is determined on the basis of computer optimization. Note that the present value of CP 1 = 0.9 is exactly the same as the NK model constant. (It is noted that Jones and Musonge (1988) developed a transport equation for εθ similar to the NK model and assigned the value of CP 1 = 0.85 and CP 2 = 0.7.) The model constants and functions in the present εθ -equation at a twoequation level are listed in Table 3. 208 Nagano Table 3: Model constants and functions in the present εθ models. Cλ Cs CP 1 CP 2 CP 3 CD1 CD2 Cεθ σφ τm fλ fw (ξ) Aµ Aλ fR fP 1 fP 2 fD1 fD2 Additional term fd Cε2 fε fP 1 fP 2 fp RT RU 0.1 — 0.9 0.77 0.05 1.0 0.9 1.6 1.8 (k/ε) fR + [26(2R)1/2 /(P r4/3 Rt )]fd [1 − fw (Aµ )]1/2 [1 − fw (Aλ )]1/2 exp − (y ∗ /ξ)2 28 Aµ /P r1/3 2R/(0.5 + R) (1 − fP 1 )fp [1 − fw (12)] (1 − fP 2 )fp 1 − fw (12) (1/CD2 )(Cε2 fε − 1)[1 − fw (12)] 3 (= Pεθ ) exp(−Rh /220) 1.9 1 − 0.3 exp[−(Rt /6.5)2 ] 10 exp(−7 × 10−5 RT )[1 − exp(−2.2RT )] 1/2 1/2 10 exp(−7 × 10−5 RU )[1 − exp(−2.2RU )] 1 + 0.75 exp[−(Rh /40)1/2 ] fw (6)uτ θτ /[(α + αt )(εθ /α)1/2 ] fw (6)uτ 2 /[(ν + νt )(ε/ν)1/2 ] 3/4 (f ) Second-order closure modelling In second-order closure modelling, the turbulent diﬀusion term Tεθ and the 3 production term Pεθ should be slightly modiﬁed, since the second-order closure model needs neither νt nor αt . Hence, the gradient parameters RT and RU are changed as follows: uτ θτ + vθ RT = fw (6), (3.45) (εθ ν/P r)1/2 [6] Modelling heat transfer in near-wall ﬂows 209 Figure 4: Assessment of the proposed εθ -equation models: (a) proﬁles of εθ near the wall; (b) proﬁles of time scale τθ near the wall; (c) budget of the 1 2 3 4 proposed εθ -equation models (Pεθ + Pεθ + Pεθ + Pεθ + Tεθ − Υεθ ). RU = u2 + uv τ fw (6). (εν)1/2 (3.46) The model functions fP 1 and fP 2 in (3.26) and (3.27) are deﬁned by fP 1 = exp(−7 × 10−5 RT 10 )[1 − exp(−1.1RT 1/2 )], fP 2 = exp(−7 × 10−5 RU 10 )[1 − exp(−1.1RU 1/2 )]. (3.47) 210 Nagano Figure 5: Budget of temperature variance kθ . The turbulent diﬀusion term, Tεθ , can be written as Tεθ = ∂ ∂xk k ∂εθ Cs fR uj uk ε ∂xj + Cεθ α ε ∂kθ fw (3) , k ∂xj (3.48) are exactly the same as in the two-equation ∂ ∂xj [1 − fw (3)]3/2 where Cs = 0.11, and fR and Cεθ heat-transfer model. 3 The Pεθ term may be written [see (3.28)] as k ∂2Θ ∂2Θ k ∂uj u ∂Θ ∂ 2 Θ 3 + CP 3 α fR , (3.49) Pεθ = CP 4 αuj uk fR ε ∂xk ∂x ∂x ∂xj ε ∂xk ∂x ∂xj ∂xk with CP 3 = 0.1 and CP 4 = 0.25. (g) Assessment of proposed εθ equation models Figure 4 shows the solutions obtained from new εθ equations. As shown previously, the existing two-equation-level models have never reproduced the correct near-wall behavior of εθ , whereas the present predictions give excellent agreement with the DNS data. Owing to the inclusion of the model for αt , the proposed model at a two-equation level gives predictions slightly diﬀerent from those at a second-order closure level. The overall predictions, however, are much better than with the existing models. It should also be noted that, for the budget balance in the εθ equation (Figure 4(c)), excellent agreement is now achieved. 3.4.3 Modelling the kθ equation Figure 5 shows the budget of temperature variance predicted by the NTT model (Nagano et al. 1991), which is the basis for the proposed model and the [6] Modelling heat transfer in near-wall ﬂows 211 Table 4: Model constants and functions in the present kθ model. Cθ ∗ σh 0.1 1.8/[1 + 0.5fw (28)] AKN model (Abe et al. 1995). Obviously, the model predictions are diﬀerent from DNS data near the wall because of the solution given by the εθ equation and the modelling of the turbulent diﬀusion term in the kθ equation. Therefore, in the kθ equation given by (3.4), it is the turbulent diﬀusion term Tkθ that should be modelled. We adopt the foregoing turbulent diﬀusion modelling, and write Tkθ as T kθ = ∂ ∂xj αt ∂kθ ∂ +Cθ ∗ ∂x σh j ∂xj √ σuk u dk n ej [1 − fw (28)]1/2 k kθ [fw (28)]1/2 , (3.50) where dk , n and ej are unit vectors in the streamwise, wall-normal and xj directions, respectively, and σuk u is a sign function, ﬁrst introduced by Nagano and Tagawa (1990b). The sign function σuk u is necessary to make a model independent of the coordinate system, and is deﬁned as σx = 1 (x ≥ 0), −1 (x < 0). (3.51) The ﬁnal formulation of the kθ -equation model is written as follows: Dkθ Dt = ∂ ∂xj α+ αt ∗ σh ∂kθ ∂xj +Cθ √ ∂ σuk u dk n ej [1 − fw (28)]1/2 k kθ [fw (28)]1/2 ∂xj (3.52) +Pkθ − εθ . The model functions and constants in the present kθ model are listed in Table 4. 3.5 Discussion of Predictions with Proposed Models In general, a turbulence model must give predictions of good accuracy in both fundamental internal and external ﬂows. If the model does not indicate good agreement for both cases, one can hardly rely on it to predict complex ﬂows of technological interest. In this study, the modelling takes into account all the key turbulence quantities and their budgets obtained by DNS results, so that we can conﬁrm the precision of the model prediction in both ﬁelds. Then we assess the proposed model performance in ﬂow ﬁelds for diﬀerent thermal boundary conditions at the wall. 212 3.5.1 Numerical scheme Nagano The numerics sometimes aﬀect the results of the turbulence models, both in the algorithm chosen and in the number and distribution of grid points (Kline 1980). Therefore, special attention was paid to the numerics to enable a more meaningful model appraisal. The numerical technique used is a ﬁnite-volume method, as used by Hattori and Nagano (1995). The coordinate for regions of very large gradients should be expanded near the wall. Thus, for internal ﬂows, a transformation is introduced so that η = (y/h)1/2 . For external ﬂows, the following nonuniform grid (Nagano and Tagawa 1990a) across the layer is employed: yj = ∆y1 (K j − 1)/(K − 1), (3.53) where ∆y1 , the length of the ﬁrst step, and K, the ratio of two successive steps, are chosen as 10−5 and 1.03, respectively. For both internal and external ﬂows, 201 cross-stream grid points were used to obtain grid-independent solutions. To conﬁrm numerical accuracy, the cross-stream grid interval was cut in half for internal ﬂow cases. No signiﬁcant diﬀerences were seen in the results. √ The boundary conditions are Uw = kw = kθw = 0, εw = 2ν(∂ k/∂y)2 , √ εθw = 2α(∂ ∆kθ /∂y)2 and Θw or qw are determined by experimental or DNS data at a wall, ∂U/∂y = ∂k/∂y = ∂ε/∂y = ∂Θ/∂y = ∂kθ /∂y = ∂εθ /∂y = 0 at the axis for internal ﬂows (symmetry); U = Ue , k = ε = kθ = εθ = 0 and Θ = Θe at the outer edge of the boundary layer, where Ue and Θe are prescribed from experiments. The criterion for convergence is max |X (i+1) − X (i) /X (i) | < 10−5 , (3.54) where X = U , k, ε, Θ, kθ , and εθ , and i denotes the number of iterations. The computations were performed on a personal computer and a DEC Alpha workstation. 3.5.2 Channel ﬂow with heat transfer (constant-heat-ﬂux wall and constant-temperature wall) It is important to predict the velocity ﬁeld precisely for relevant temperature ﬁeld prediction. The framework of the proposed k-ε model is based on the NS model, which has been conﬁrmed to show highly accurate prediction of wallbounded turbulent ﬂows (Nagano and Shimada 1995a). In this study, however, the wall reﬂection function is, as noted above, now based on (2.9), so that the model was tested in the channel ﬂow calculated for the DNS conditions of both Moser et al. (1999) (Reτ = 395) and of Kasagi et al. (1992) (Reτ = 150) shown in Figure 6. From Figure 6, it can be seen that the mean velocity and turbulent energy are predicted quite successfully for both cases. Next, we assess the constructed two-equation heat-transfer model with the k-ε model in a fully developed channel ﬂow under both constant-temperature [6] Modelling heat transfer in near-wall ﬂows 213 Figure 6: Channel ﬂow predictions: (a) mean velocity; (b) turbulent energy. (Kim and Moin 1989) (Reτ = 180 and P r = 0.71) and constant-heat-ﬂux wall conditions (Kasagi et al. 1992) (Reτ = 150 and P r = 0.71). Comparisons of the predicted mean temperature, turbulent heat ﬂux, temperature variance, near-wall behavior of temperature variance and turbulent heat ﬂux with DNS are shown in Figures 7(a), 7(b), 7(c), 7(d) and 7(e) respectively. The model predictions are in almost perfect agreement with the DNS data and reproduce exactly the wall-limiting behavior near the wall for both thermal wall conditions, i.e., θw = 0 and θw = 0. Figures 8 and 9 show the predicted budget of temperature variance and its dissipation rate, compared with the DNS data. Obviously, agreement of each term in both budgets with DNS is also very good. An important point of the present study is the modelling for the turbulent diﬀusion term, Tkθ , in the kθ equation. From a comparison of Figure 5 with Figure 8, the calculated budget of the proposed model is seen to improve on previous models near the wall (y + < 15). These facts indicate that the modelling of a gradient-type diﬀusion plus convection by large-scale motions is eﬀective for the turbulent diﬀusion term, and that the proposed modelling is appropriate for construction of a set of heat-transfer models. 214 Nagano (a) (b) (c) (d) (e) Figure 7: Thermal ﬁeld predictions in channel ﬂow: (a) mean temperature; (b) turbulent heat ﬂux; (c) temperature variance; (d) near-wall behavior of temperature variance; (e) near-wall behavior of turbulent heat ﬂux. 3.5.3 Boundary-layer ﬂows with uniform-temperature or uniformheat-ﬂux wall In the following, we assess the present two-equation heat-transfer model in boundary-layer ﬂows under diﬀerent thermal conditions. The most basic situations encountered are the heat transfer from a uniform-temperature or uniform-heat-ﬂux wall. The results of thermal ﬁeld calculations under a constantwall-temperature or constant-wall-heat-ﬂux condition along a ﬂat plate, compared with experimental data of Gibson et al. (uniform-temperature wall) (1982) and of Antonia et al. (uniform-heat-ﬂux wall) (1977), are shown in Figure 10. It is known that the NTT model for reference gives good prediction of turbulent thermal ﬁelds under these wall thermal conditions (Youssef et [6] Modelling heat transfer in near-wall ﬂows 215 Figure 8: Budget of temperature variance in channel ﬂow. Figure 9: Calculated budget of εθ in channel ﬂow. al. 1992), and the present predictions also indicate good agreement with the experimental data. 3.5.4 Constant wall temperature followed by adiabatic wall The next test case for which calculations have been performed is concerned with a more complex thermal ﬁeld in a boundary layer along a uniformly heated wall followed by an adiabatic wall. Figure 11 shows a comparison of 216 Nagano Figure 10: (a) mean temperature; (b) turbulent heat ﬂux; (c) rms temperature. the predicted results with the experimental data (Reynolds et al. 1958) of temperature diﬀerences between the wall and the free-stream ∆Θ(= Θe −Θw ). It can be seen that the proposed model gives generally good predictions for the rapidly changing thermal ﬁeld. Also by comparison, the present model gives no prediction inferior to the AKN model (Abe et al. 1995). [6] Modelling heat transfer in near-wall ﬂows 217 Figure 11: Comparison of the predicted variations of wall temperature with the measurement. Figure 12: Comparison of the predicted rms temperature proﬁles and measurements (sudden decrease in wall heat ﬂux). 3.5.5 Constant heat ﬂux followed by adiabatic wall To further verify the eﬀectiveness of the present model for calculating various kinds of turbulent thermal ﬁelds, we have carried out the calculation of a boundary layer ﬂow along a uniform heat-ﬂux wall followed by an adiabatic wall, which has been reported in detail by Subramanian and Antonia (1981). The calculated distributions of rms temperature ﬂuctuations normalized by temperature diﬀerence between the free stream and the wall, ∆Θc , at a step change in surface thermal condition, are shown in Figure 12, compared with the experimental data (Subramanian and Antonia 1981) and the prediction of the AKN model. Both models indicate a slight underprediction of the peak value of rms temperature. The proposed model, however, shows the variation of physical phenomena of rms temperature in the thermal layer along a uniform heat-ﬂux followed by an adiabatic wall. In particular, the rapid decrease 218 Nagano Figure 13: Comparison of the predicted variations of wall temperature and Stanton number with the measurements (double-pulse heat input) in temperature ﬂuctuations from the inner region has been captured by the proposed model. 3.5.6 Double-pulse heat input As a ﬁnal test case, we have calculated the more complex thermal case, where the heat input is spatially intermittent in a double-pulse manner. Then we have investigated the mechanism of turbulent heat transfer in such a rapidly changing thermal layer. The temperature diﬀerence between the free stream and the wall, ∆Θ = Θw − Θe , and the Stanton number reported by Reynolds et al. (1958), are shown in Figure 13 compared with the prediction of the present model. It is indicated that both the velocity and the thermal ﬁelds are well predicted, and the turbulent heat transfer characteristics in the thermal entrance region are reproduced very well. Figure 14(a) shows how the turbulent near-wall thermal layer changes when the heat input is intermittent, where ∆Θ = Θ − Θe is normalized by the temperature diﬀerence between the wall and the free stream, ∆Θc = Θwc − Θe , just before the ﬁrst heat input/cutoﬀ point. It can be seen that a very abrupt decrease and increase in mean ﬂuid temperature occurs in the wall region, which is a consequence of the no-heat-input condition followed by heat input, i.e., ∂Θ/∂y|w = 0 → ∂Θ/∂y|w = constant. Within a short distance from the abrupt change-over, the mean temperature proﬁle becomes uniform over most of the thermal layer. The following discussion deals with how these phenomena aﬀect other turbulent quantities. Figure 14(b) shows the distribution of turbulent heat ﬂux normalized by uτ and ∆Θc . Just after the ﬁrst heat input/cutoﬀ point, with vanishing mean temperature gradient near the wall, the turbulent heat ﬂux, vθ, decreases rapidly. Just before the second heat input point, vθ has greatly decayed with its maximum occurring in the outer layer. Over the reheated wall, vθ again shows a rapid increase near the [6] Modelling heat transfer in near-wall ﬂows 219 Figure 14: Variations of turbulent quantities for double-pulse heat input: (a) mean temperature; (b) turbulent heat ﬂux; (c) rms temperature. wall. This is qualitatively consistent with the experimental result (Antonia et al. 1977) for the thermal entrance region of the boundary layer on a ﬂat plate. Next, variations of the rms temperature are shown in Figure 14(c). Just after the heat cutoﬀ point, distributions of the rms temperature tend to be similar to the experimental evidence obtained by Subramanian and Antonia 220 Nagano Figure 15: Budget of temperature variance for double-pulse heat input: (a) x = 0.517 [m]; (b) x = 0.579 [m]; (c) x = 0.876 [m]; (d) x = 0.936 [m]. (1981) and discussed in the previous section. It can be seen that just before the second heat input point, the rms temperature remains in the outer region only, and that it increases very rapidly near the wall beyond that point. Figures 15(a)–(d) show budgets of temperature variance at locations just before the ﬁrst heat cutoﬀ (x = 0.517 m), just after the heat cutoﬀ (x = 0.579 m), just before the second heat input (x = 0.876 m), and just after the second heat input (x = 0.936 m), respectively. In these ﬁgures, each term is normalized by the peak value of the production Pkθ at the respective locations. Since the mean temperature gradient vanishes near the wall as shown in Figure 14(a) at x=0.579 m, the peak value of the production term tends to increase in the outer region and the rapidly decreasing temperature ﬂuctuation is restrained by an increase of the convective term there. Consequently, the ﬂuctuating temperature is transported actively by the turbulent diﬀusion from the outer region to the wall, though the dissipation also increases away from the wall. Since the molecular diﬀusion and the dissipation preserve the near-wall structure and √ temperature ﬂuctuations are created by the mean temperature no gradient, kθ is virtually nonexistent just before the second heat input point, as shown in Figure 14(c). From the above, after the ﬁrst cutoﬀ point, it is understandable that the near-wall structure of thermal turbulence is preserved mainly by diﬀusion from the outer to the inner region, and the temperature [6] Modelling heat transfer in near-wall ﬂows 221 ﬂuctuation decreases remarkably. Then, just after the second heat input point, the near-wall proﬁle of temperature ﬂuctuation returns rapidly to the unperturbed initial proﬁle. The remaining ﬂuctuation in the outer region does not participate in the reproduction. Since the proposed model is rigorously constructed by considering the budget proﬁles of turbulence quantities obtained by DNS, we may expect that the model could be used to investigate the detailed mechanism of heat transfer in complex applications, as illustrated in this section. 4 4.1 Two-equation Model for heat transfer (eﬀects of Prandtl number) Construction of kθ -εθ Model Eddy diﬀusivity αt for kθ -εθ model 4.1.1 As mentioned in the foregoing, the eddy diﬀusivity for heat, αt , is generally given by (3.3). Features of αt in (3.3) with (3.17) are summarized as follows: √ in the near-wall region, αt yields the relation αt ∝ kη R/P r ∝ νt R/P r, so it is possible to adequately capture the behavior of dissipative motions and, in the region far from the wall, the αt model consists only of time scales of the energy-containing eddies through the hybrid time scale τh (see section 3.4.1). In the present kθ -εθ model (Nagano and Shimada 1996), we adopt a multipletime-scale similar to (3.17): τm = k ε 2R + Cm + R 2R Bλ fη P r Rt θ , (4.1) where Cm is the weighting constant of the composite time scale, Bλ is the model constant that represents the eﬀectiveness of dissipation eddies, and fηθ is the model function limiting the Bλ -aﬀected region. As for the model function fλ in (3.3), we introduce the following formulation: ∗ fλ = 1 − exp −Aλ yθ n y ∗2−n = 1 − exp −A∗ y ∗2 , λ where √ A∗ = Aλ (1 + Cη P r)n . λ (4.2) ∗ Here, Aλ and Cη denote model constants, and the dimensionless parameter yθ is deﬁned using the mixed length scale ηm as √ ∗ yθ = y/ηm = (1 + Cη P r)y ∗ , ηm = 1 Cη + η ηθ −1 (4.3) η √ . = 1 + Cη P r 222 Nagano The length scale η = (ν 3 /ε)1/4 represents the Kolmogorov microscale deﬁned previously, and ηθ = (α2 ν/ε)1/4 is the Batchelor microscale (Batchelor 1959). y ∗ is deﬁned as y ∗ = y/η. From (4.3), the characteristic length scale ηm gives weight to the Kolmogorov microscale η for lower Prandtl number ﬂuids (P r < 1), while for higher Prandtl number ﬂuids the Batchelor microscale ηθ becomes dominant. [For low Prandtl number ﬂuids, the so-called Obukhov microscale (Obukhov 1949) ηo = (α3 /ε)1/4 = ηP r−3/4 could be used as the characteristic length scale of dissipation eddies. However, it is easily understood that η is always smaller than ηo because of the relation ηo /η = P r−3/4 .] It should also be noted that fλ given by the above equation diﬀers slightly from (3.19). This is because we here incorporate Prandtl-number eﬀects on the thermal ﬁeld in the model function fλ . Also note that the relation between ε and ε is represented by (3.12). ∗ We note that the fact that the sum of the power of yθ and y ∗ in (4.2) is 2 results from a restriction in the wall-limiting behavior of αt (Youssef et al. 1992; Abe et al. 1995). We determine the power n, the constant Aλ , and the weighting constant Cη from the following procedure: we calculate algebraically √ A∗ = Aλ (1 + Cη P r)n so that the maximum values of kθ at P r = 0.1, 0.71 λ and 2.0 agree with those of the DNS data, while the remaining constants and functions are ﬁxed. As a result, we obtain n = 1/4, Aλ = 7×10−4 , and Cη = 2. The weighting constant Cm plays an important role in giving weight to a shorter time scale between τu and τθ . Thus, we consider the relationship between τu and τθ before determining Cm . It is clear that, in decaying homogeneous ﬂows with high Prandtl number ﬂuids, the time-scale ratio R(= τθ /τu ) is greater than unity in the high Rt region (Iida and Kasagi 1993). Also, it is easily veriﬁed that for turbulent wall ﬂows, R strictly equals P r at the wall. (Note that R = P r is ensured for the case of θw = 0). Hence, the characteristic time scale suitable for high Prandtl number ﬂuids is τu . In the case of low Prandtl number ﬂuids, on the other hand, since the DNS data (Iida and Kasagi 1993; Kasagi et al. 1992; Kasagi and Ohtsubo 1992) indicate that R is always smaller than unity irrespective of the types of ﬂow ﬁelds, τθ is appropriate for the characteristic time scale. When P r 1 (e.g., air ﬂow), the wavenumbers related to peak intensities of both velocity and temperature ﬂuctuations are close to each other, so that the inﬂuence of both τu and τθ becomes signiﬁcant. For the reasons mentioned above, the weighting constant Cm would be expected to change with P r, and hence we decide that Cm = 0.2/P r1/4 . As for the model function fηθ , Sato et al. (1994) pointed out that fηθ exerts a signiﬁcant inﬂuence on the prediction of high Prandtl number ﬂuids in which the dissipation scale becomes much smaller. Thus, we write the model function ∗ ∗ fηθ using the foregoing parameter yθ as fηθ = exp[−(yθ /25)3/4 ]. Finally, we consider the model constant Bλ representing the eﬀectiveness of dissipation eddies. In order to obtain the correct wall-limiting behavior of αt independent of wall thermal conditions, and to reproduce the fact that the [6] Modelling heat transfer in near-wall ﬂows 223 ratio of the respective time scales for dissipation eddies in the velocity and R/P r (Youssef et al. 1992; Abe et al. 1995), thermal ﬁelds becomes equal to √ Bλ is set to the value 120/(1 + 2 P r)1/4 . Here, the value of 120 is chosen so that the wall-limiting behavior of the calculated vθ agrees with that of the DNS. As a result, the near-wall behavior of αt is proportional to νt R/P r, similar to that given by Abe et al. (1995). To sum up, the proposed αt model is written as follows: αt = Cλ fλ k2 ε 2R + 0.2/P r1/4 + R y∗ 120 2R 1 √ exp − θ P r (1 + 2 P r)1/4 Rt 25 3/4 , (4.4) ∗ fλ = 1 − exp(−7 × 10−4 yθ 1/4 y ∗7/4 ), √ ∗ yθ = (1 + 2 P r)y ∗ . The model constant Cλ is assigned the standard value 0.10 (see (3.35)). 4.1.2 Modelling the kθ -equation The kθ -equation necessary to determine the time scale τθ (or the time scale ratio R) is given by (3.4). As mentioned before, the only term to be modelled in (3.4) is the turbulent diﬀusion Tkθ , which plays a signiﬁcant role in the accurate prediction of εθ . The turbulent diﬀusion term Tkθ is generally modelled using the generalized gradient diﬀusion hypothesis (GGDH). However, as pointed out in section 3, GGDH modelling for Tkθ causes an imbalance in the budget for the kθ -equation and produces an incorrect behavior of εθ . This discrepancy is mainly due to the fact that GGDH modelling represents the turbulent diﬀusion caused by relatively small-scale (higher wavenumber) eddies. (It can be readily understood that the formulation of the modelled turbulent diﬀusion through GGDH is similar to that of the molecular diﬀusion Dkθ which is a small-scale phenomenon in a turbulent ﬂow.) As a result, an underestimation of turbulent diﬀusion occurs in the buﬀer layer where relatively lower wavenumber eddies are dominant, and the behavior of εθ is incorrectly reproduced. Thus, in order to obtain the correct behavior of εθ , the eﬀect of large-scale structures must be reﬂected in the turbulent diﬀusion modelling. In the present study, the turbulent diﬀusion term, including the contribution from lower wavenumber eddies, is modelled by using the following proposal similar to that of Hattori and Nagano (1998) (see (3.50)): T kθ = ∂ ∂x αt σh ∂kθ ∂ + Cθ ∂x ∂x √ k kθ fwθ σui uj eSi eN j e , (4.5) σh = σh0 /fh , where σh0 , Cθ , fh , and fwθ are the model constants and functions, respectively. (Note that, as already mentioned, the sign function σui uj and the unit vectors 224 Nagano Table 5: Model constants and functions of the proposed kθ -εθ model. Cλ 0.10 ∗ CD3 Cm 0.2 P r1/4 Cτ 3.0 σh 1.6 fh fλ (4.2) Cθ 0.1 fh (4.6) σφ 1.8 fwθ (4.7) CP 1 0.825 fP 1 1.0 CP 2 0.9 fP 2 1.0 CD1 1.0 fD1 1.0 CD2 fD2 (4.15) ∗ fD3 0.025 fw eSi , eN j , e are needed to make the model independent of the coordinate system.) The model constants σh0 and Cθ are assigned the values 1.6 and 0.1, respectively, and the model functions fh and fwθ are given as follows: fh = Cm + R 2R 1+ 5 Rt exp − Pr 100 , (4.6) (4.7) 1/4 fwθ = (1 − fw )2 fw with fw = exp (−y ∗ /8) , where fw is the wall reﬂection function similar to (2.9), Rt = k 2 /(ν ε) is the turbulent Reynolds number based on ε, and (Cm + R)/2R = τu /(2τh ) in (4.6) is necessary for fh to ensure the balance in the order of magnitude between the modelled turbulent diﬀusion term through GGDH [∂(αt /σh · ∂kθ /∂xj )/∂xj ∼ O(αt /σh · θ 2 / 2 ), where ( ) denotes the rms value and implies the integral length scale] and the strict turbulent diﬀusion term [Tkθ ∼ O(u θ 2 / )] in the kθ -equation in the local equilibrium state. The ﬁnal formulation of the kθ equation model is written as follows: √ Dkθ ∂ αt ∂kθ + Cθ k kθ fwθ σui uj eSi eN j e = α+ + Pkθ − εθ . Dt ∂x σh ∂x (4.8) One may note that (4.8) is almost identical to (3.52). 4.1.3 Modelling the εθ -equation In the two-equation modelling of a thermal ﬁeld, the dissipation rate εθ of temperature variance must be determined from its transport equation. The use of the εθ -equation involves the same problems as those already mentioned in the use of the ε-equation, so instead of the εθ -equation, we adopt the following εθ -equation similar to that proposed by Nagano and Kim (1988): D εθ Dt = ∂ ∂xj α+ αt σφ ∂ εθ εθ + (CP 1 fP 1 Pkθ − CD1 fD1 εθ ) ∂xj kθ ∂2Θ ∂xj ∂xk 2 εθ + (CP 2 fP 2 Pk − CD2 fD2 ε) + ααt (1 − fλ ) k , (4.9) [6] Modelling heat transfer in near-wall ﬂows 225 where σφ , CP 1 , CP 2 , CD1 , CD2 are model constants, and fP 1 , fP 2 , fD1 , fD2 are model functions. In the present study, these model constants and functions are basically identical to those adopted by previous kθ -εθ models (Nagano and Kim 1988; Youssef et al. 1992; So and Sommer 1993; Hattori et al. 1993) except for some model functions, and are listed in Table 5. The relation between εθ and εθ is deﬁned by (3.13). In the proposed model, we use the above εθ -equation (4.9) as is, but its relative time scales are reconsidered below. The physical role of a transport equation is to correctly express how transported turbulence quantities change through the eddy motions with various scales. Indeed, the previous modelling of the source and sink terms included in (4.9) was made by the formulation εθ × (time scale)−1 . Thus, we classify terms in the transport equation into the following three groups: (i) Comparatively slow motion due to the existence of mean velocity and temperature gradients (contributing to the generation of turbulence quantities); (ii) Relatively rapid motion due to the eﬀect of large (energy-containing) eddies; (iii) remarkably rapid motion caused by dissipation eddies. First, we consider the time scale dominating the motion (i). After investigating the coherent structure of wall turbulence using the DNS database (Robinson 1991), it was clariﬁed that the dissipation reaches its maximum in the ‘internal shear layer’ (or near-wall shear layer (Robinson 1991)) occurring near the wall. This means that a close relationship exists between the production process of ε (or εθ ) and the formation of the internal shear layer. Also, it is well known that an internal shear layer occurs when surrounding higherspeed ﬂuid impacts on the upstream edge of a kinked low-speed streak and/or on the low-speed ﬂuid ejected away from a wall by streamwise vortices in the viscous sublayer, and that the lifetime of the internal shear layer is strongly inﬂuenced by the characteristic time scale of the mean ﬁeld. This also means that the characteristic time scale in producing ε or εθ can be closely related to the lifetime of an internal shear layer (or the time scale of the mean ﬁeld). Thus, the characteristic time scales for the production of ε and εθ are expected to be represented through those related to mean velocity and thermal ﬁelds, e.g., mean velocity gradient τU = 1/(∂U/∂y). Put another way, the shear rate parameter S ≡ 2Sij Sij , where Sij ≡ (∂Ui /∂xj + ∂Uj /∂xi )/2 denotes the mean velocity gradient (strain) tensor, seems to be suitable as the characteristic time scale to be considered. However, since turbulence must be generated by the production terms Pk and Pkθ in the k- and kθ -equations through interactions between mean and ﬂuctuating 226 Nagano ﬁelds, the time scales contributing to the generation of turbulence should be directly expressed by using Pk and Pkθ as τU = τΘ = 1 = Pk /k 1 = Pkθ /kθ −ui uj ∂Ui k ∂xj −uj θ ∂Θ kθ ∂xj −1 , −1 (4.10) . It is interesting to note that (4.10) is clearly identiﬁed with the time scales of the production processes previously introduced by Nagano and Kim (1988) [see (4.9)]. Next, let us examine ﬂuid motion (ii). It is already well known that the motion of energy-containing eddies makes little contribution to generating turbulence but rather acts to supply energy to smaller eddies through the energy-cascade process. The characteristic time scales of the energy-containing eddies for both velocity and thermal ﬁelds are closely connected with those of eddies with wavenumbers related to the maximum values of the spectral distributions for k and kθ , and these are generally represented as follows: τu = k/ε, τθ = kθ /εθ . (4.11) Note that the eﬀects of (4.11) are already explicitly included in (4.9). Finally, we discuss the motion (iii) with rapid change. It is known that almost all the energy fed by the mean ﬁeld is accumulated in the energycontaining eddies and, subsequently, that it is successively supplied to smaller eddies (or dissipation eddies) through the deformation work of larger eddies. The dissipation eddies have vorticity much higher than that of the energycontaining eddies, so that the characteristic time scale of the former eddies becomes shorter compared with that of the latter. Almost all the modelled εθ or εθ -equations proposed so far have applied only τu and τθ in (4.11) to all eddy motions. In the vicinity of the wall, however, it is clear that there are always dissipation eddies. Also, we can easily imagine that, in ﬂows with P r 1 or P r 1, the size of the dissipative (or destruction) eddies in one ﬁeld develops at a similar rate to that of energy-containing eddies in another ﬁeld. Therefore, the dissipation-eddy time scales become very signiﬁcant factors in model construction. The representative time scale of the dissipation eddies in a velocity ﬁeld is generally expressed by the following well-known Kolmogorov time scale: τηu = ν/ε. (4.12) In a similar way, the representative time scale of the dissipation eddies in a thermal ﬁeld can be deﬁned as follows: τηθ = α/εθ kθ /k = R/P rτηu . (4.13) [6] Modelling heat transfer in near-wall ﬂows 227 It should be noted that the Taylor microscales λ = νk/ε and λθ = αkθ /εθ for velocity and thermal ﬁelds, which are often used as the scales of the smallest eddies, √ strongly linked to the characteristic time scales of dissipation eddies are √ as λ = kτηu and λθ = kτηθ . In the present study, as mentioned previously, the characteristic time scales of dissipation eddies [(4.12) and (4.13)] are written through the following composite time scale τηm : 1 τηm = 1 Cτ 1 + = τηu τηθ τηu 1 + Cτ Pr R . (4.14) where Cτ is a weighting constant. Now, (4.14) is not directly incorporated into the present εθ -equation [(4.9)], but is indirectly reﬂected in the model constant CD2 and the model function fD2 as ∗ ∗ ∗ ∗ CD2 fD2 = CD2 fD2 1 + CD3 fD3 Rt 1 + Cτ Pr R , (4.15) with ∗ ∗ CD2 fD2 = (Cε2 f2 − 1) 1 − exp −y ∗2 . (4.16) ∗ ∗ ∗ ∗ Here, CD3 , Cτ , and fD3 are CD3 = 0.025, Cτ = 3.0, and fD3 = fw , respectively. [We would like to emphasize that the proposed εθ -equation model with (4.15) can improve the accuracy of prediction without modifying the existing numerical scheme.] Equation (4.16) is established to reproduce the limiting behavior of free turbulence (see Youssef et al. (1992)). The term in square brackets on the right-hand side in (4.16) is needed to ensure that the molecular diﬀusion term in (4.9) will strictly balance with the destruction term directly related to the thermal ﬁeld, i.e., CD1 fD1 εθ εθ /kθ , with the correct wall-limiting behavior of εθ ∝ y 2 (Kawamura and Kawashima 1994; Shikazono and Kasagi 1996). 4.2 Model Performance in Thermal Fields In this section, we examine the validity of the proposed kθ -εθ model for various ﬂow conditions as follows. Case A. Channel ﬂows with internal heat source. Case B. Channel ﬂows with uniform wall heat ﬂux. Case C. Channel ﬂows with injection and suction. Case D. Reynolds and Prandtl number dependence for internal ﬂows. Case E. Boundary layer ﬂows under arbitrary wall thermal boundary conditions. 228 Table 6: Boundary conditions of the proposed kθ -εθ model for various ﬂow ﬁelds. Case const w A const α ∂Θ ∂y =− qw ρcp B C D E (Θw =const) E (qw =const) α ∂Θ ∂y F =− w Θw const α ∂Θ ∂y =− w qw ρcp Θw|suc on suction side qw ρcp kθw εθ w 0 0 0 0 0 0 0 0 Θw|inj on injection side 0 0 0 and ∂kθ /∂y = 0 0 0 0 Nagano [6] Modelling heat transfer in near-wall ﬂows Case F. Backward-facing step ﬂow for various Prandtl number ﬂuids. 229 The numerical scheme and the grid system are the same as for the k-ε model, as already mentioned. In the analysis of case D, however, 201 grid points were used for the calculation of relatively low Reynolds and/or Prandtl number ﬂuids, but 1001 ﬁner grid points were used to calculate thermal ﬁelds in high Reynolds and/or Prandtl number ﬂuids with reference to the recent work by Sato et al. (1994). The wall-boundary conditions are collectively listed in Table 6. (Symmetric boundary conditions are imposed on various thermal turbulence quantities at the centerline of internal ﬂows, while free-stream conditions are imposed for external ﬂows, Hattori et al. (1993).) 4.2.1 Channel ﬂow with internal heat source First of all, we conﬁrm the validity of the proposed kθ -εθ model in channel ﬂows with internal heat source. Figures 16(a)–16(c) show the thermal-ﬁeld turbulence quantities, compared with the DNS data of Kim and Moin (1989) at Reτ = 180. These ﬁgures reveal the excellent agreement between the predictions and the DNS data, and the Prandtl number dependence is adequately captured. Predicted budget proﬁles in various Prandtl number ﬂuids are given in Figure 17. From these ﬁgures, we immediately notice the diﬀerent contributions of each term in the kθ -equation under diﬀerent P r conditions. For example, the maximum value of the production term at a relatively high Prandtl number (P r = 2.0) is located around y + = 10, which is identiﬁed with the interface between the viscous sublayer and the buﬀer layer in a velocity ﬁeld, whereas for relatively low Prandtl number ﬂuid (P r = 0.1), the production term becomes a maximum at y + 30, which is almost equal to the interface between the buﬀer layer and the log-law region; this means that these diﬀerences are closely related to the development of the thermal boundary layer. 4.2.2 Channel ﬂow with a uniform wall heat ﬂux Next, the proposed model is applied to ﬂow with thermal boundary conditions at the wall diﬀerent from the above case. The calculated proﬁles of thermal turbulence in a channel ﬂow with a uniform wall heat ﬂux are shown in Figure 18, in comparison with the DNS data of Kasagi et al. (1992) for air (P r = 0.71) and those of Kasagi and Ohtsubo (1992) for mercury (P r = 0.025). The mean temperature proﬁles in Figure 18(a) indicate that the present model eﬃciently predicts low Prandtl number ﬂuids. We notice from Figure 18(b) that, despite + the slight discrepancy in the predicted heat ﬂux vθ for mercury, the overall agreement between the predictions and the DNS is quite good. (Note that the + slight discrepancy in the predicted vθ at P r = 0.025 exerts little inﬂuence on the behavior of the mean temperature proﬁle because of the dominance 230 Nagano Figure 16: Thermal turbulence quantities in various Prandtl number ﬂuids (channel ﬂow with internal heat source): (a) mean temperature; (b) temperature ﬂuctuation intensities; (c) turbulent heat ﬂux. of molecular rather than turbulent diﬀusion.) Also, it can be seen from Figure 19, which shows the budget of the kθ -equation for air ﬂow (P r = 0.71), that the obtained proﬁles agree fairly well with the DNS data, and, in partic√ ular, that the predicted εθ , which is given by εθ = εθ + 2α(∂ ∆kθ /∂y)2 , can [6] Modelling heat transfer in near-wall ﬂows 231 Figure 17: Budget proﬁles of the kθ -equation in channel ﬂow with an internal heat source: (a) P r = 2.0; (b) P r = 0.71; (c) P r = 0.1. satisfactorily reproduce the tendency of the DNS data. 4.2.3 Heat transfer in channel ﬂow with injection and suction We consider a channel ﬂow with injection and suction, in which the thermal boundary condition on one wall is quite diﬀerent from that on the other. Figures 20(a) and 20(b) show the mean temperature normalized by Θw |suc − 232 Nagano Figure 18: Thermal turbulence quantities in air and mercury channel ﬂows with uniform wall heat ﬂux: (a) mean temperature; (b) temperature ﬂuctuation intensities; (c) turbulent heat ﬂux. Θw |inj and turbulent heat ﬂux vθ normalized by uτ and θτ of each wall, together with the DNS data (Sumitani and Kasagi 1995). It is readily seen from the ﬁgures that, though the present kθ -εθ model shows small overpredictions of + Θ on the injection side and of vθ on the suction side, compared with the DNS, + [6] Modelling heat transfer in near-wall ﬂows 233 Figure 19: Budget proﬁle of the kθ -equation in channel ﬂow with uniform wall heat ﬂux at P r = 0.71. Figure 20: Mean temperature and turbulent heat ﬂux proﬁles in channel ﬂow under a diﬀerent wall temperature condition with wall transpiration: (a) mean temperature; (b) turbulent heat ﬂux. the model captures the essential characteristics of this complex thermal ﬁeld. Thus, heat transfer control by injection and suction can be analyzed by the + present model. One should note that the constant heat-ﬂux layer (−vθ 1) does not exist in this type of ﬂow. 4.2.4 Reynolds and Prandtl number dependence for internal ﬂows (a) Reynolds number dependence Here we examine the performance of the proposed model over a wide range of Reynolds and Prandtl numbers. First, we compute mercury pipe ﬂows (P r = 0.025) under the constant-wall-temperature condition to investigate the Reynolds number dependence of the proposed model and compare the predictions with the available experimental data (Borishansky et al. 1964; Hochreiter and Sesonske 1974). As seen from Figure 21(a), the predicted Θ+ proﬁle at 234 Nagano Figure 21: Reynolds-number dependence of various thermal turbulence quantities in mercury pipe ﬂows under a constant-wall-temperature condition: (a) mean temperature; (b) turbulent heat ﬂux. Rem = 105 agrees with the experimental data (Borishansky et al. 1964) quite well. Also, the ﬁgure indicates that there are systematic variations of Θ+ with varying Rem . A thermal ﬁeld at low Reynolds number (Rem = 5 × 103 ) is mainly dominated by heat conduction, while as Rem increases, heat conduction is limited within the near-wall sublayer (y + < 30), and the eﬀect of turbulent convection governs the remainder. We note that these variations are closely related to the activity of turbulence motion inducing strong turbulent heat ﬂux (vθ) at high Rem seen from Figure 21(b). Figure 22 shows temperature ﬂuctuations in a high-Reynolds-number (Rem = 50000) mercury pipe ﬂow (where r0 in the ﬁgure denotes the radius of the pipe). It is clear from the ﬁgure that, though a little overprediction is seen in the central region of the pipe, the peak value of the predicted temperature ﬂuctuation is in reasonable agreement with the measurements (Hochreiter and Sesonske 1974). The Reynolds number dependence of turbulent heat transfer coeﬃcient or the Nusselt number N u in various Prandtl number ﬂuids (P r = 0.004 − 100) is thoroughly investigated in Figure 23. Comparisons are made with several semi-empirical formulas for a pipe ﬂow under a constant-wall-temperature condition (Bhatti and Shah 1987) [Note that Gnielinski’s formula in the ﬁgure implies the modiﬁed Petukhov’s correlation (see Bhatti and Shah (1987)).] Obviously, the predicted N u proﬁles at higher Prandtl numbers (P r ≥ 7) are in fairly good agreement with the Sandall et al. formula over a wide range of Reynolds numbers, and complete correspondence in air ﬂow (P r = 0.71) is obtained between the prediction and the Kays-Crawford formula. For the lower Prandtl number cases (P r = 0.004 and P r = 0.025), the Chen-Chiou formula gives the best ﬁt with the present model. Note that as P r approaches zero, N u 5.5 is obtained from the present model in the lower limit of Reynolds number, whereas some recent empirical formulas (Bhatti and Shah 1987) give N u 5.0 for ﬂows under a constant-wall-temperature condition. [6] Modelling heat transfer in near-wall ﬂows 235 Figure 22: Temperature ﬂuctuation in mercury pipe ﬂow at high Reynolds number (Rem 50000). Figure 23: Reynolds number dependence of turbulent heat transfer coeﬃcient in pipe ﬂows under a constant-wall-temperature condition at diﬀerent Prandtl numbers (0.004 ≤ P r ≤ 100). (b) Prandtl number dependence Next, we investigate the validity of the present model in various Prandtl number ﬂuids. First, the model performance for low Prandtl number ﬂuids (P r ≤ 1) at a constant Reynolds number (Rem = 10000) and constant wall temperature is discussed. Thermal turbulence quantities of mean temperature √ + Θ+ , turbulent heat ﬂux vθ , temperature ﬂuctuation kθ /(Θw −Θ0 ), and turbulent Prandtl number P rt in pipe ﬂow are presented in Figures 24(a)–24(d). Owing to the lack of experimental data for various turbulence quantities at the corresponding Reynolds number, simply the results obtained by the proposed model under varying P r are discussed. It is readily found from those ﬁgures that there are systematic variations with varying P r; for example, Θ+ shown in Figure 24(a) reveals that the conduction sublayer becomes increasingly thick 236 Nagano (a) (b) (c) (d) Figure 24: Prandtl number dependence of various thermal turbulence quantities in pipe ﬂow at Rem = 10000: (a) mean temperature; (b) turbulent heat ﬂux; (c) temperature ﬂuctuation; (d) turbulent Prandtl number. as P r decreases (e.g., the outer edge of the conduction sublayer is y + 8 at P r = 0.71 while y + 150 at P r = 0.025), and for P r > 0.1, the logarithmic temperature distributions can clearly be recognized. It is also clear from Fig+ + ure 24(b) of vθ that the peak location of vθ shifts toward the center of the pipe with decreasing P r and that the peak at P r < 0.1 is around y + 150 in close accordance with the peak of temperature ﬂuctuation mentioned below. A look at the temperature ﬂuctuation proﬁles [Figure 24(c)] immediately reveals some distinct features. For example, its ﬂattened distributions over the whole ﬂow region can be seen for P r ≤ 0.1. When P r increases from 0.1, on the other hand, the temperature ﬂuctuation, reaches its maximum in the vicinity of the wall. As for the turbulent Prandtl number P rt , some interesting trends can be seen from Figure 24(d). It is apparent that the predicted P rt at P r = 0.71 can correctly capture the tendency of the DNS obtained for air channel ﬂow at a similar Reynolds number. It is also clear that the predicted P rt tends to vary systematically as P r decreases: P rt gradually decreases in the nearwall region (1 < y + < 10) and increases in the logarithmic region (y + > 40) from the standard value at P r = 0.71. This can be recognized by investigating the molecular and turbulent transport terms of the energy equation. For fully developed turbulent ﬂows, the energy equation reduces to [6] Modelling heat transfer in near-wall ﬂows 237 Figure 25: The mean temperature proﬁle at a high Prandtl number ﬂuid (P r = 95, channel ﬂow at Rem = 10000). Figure 26: Turbulent heat and mass transfer at various Prandtl and Schmidt numbers (pipe ﬂow at Rem = 10000). q = qw 1 1 νt + P r P rt ν dΘ+ . dy + (4.17) The ﬁrst term in brackets on the right-hand side denotes the molecular conduction and the second denotes the turbulent diﬀusion. The mean temperature distribution near the wall can be expressed as Θ+ = P ry + . (4.18) This equation signiﬁes that the heat transport near the wall is dominated by the molecular conduction, as conﬁrmed by many experiments. In low Prandtl number ﬂuids, it is conﬁrmed from Figures 21 and 24 that the region where the molecular conduction predominates is extended to the logarithmic region of the velocity ﬁeld. In this region, νt /ν in (4.17) is large, so the turbulent Prandtl number should become larger to render the eﬀect of the turbulent diﬀusion negligible. In the viscous sublayer, on the other hand, νt /ν is much smaller than unity so the value of P rt may be on the order of unity. It should be noted that the fact that the obtained P rt at any P r in the immediate vicinity of the 238 Nagano wall approaches a constant value (about 1.21) is due to the near-wall behavior of the proposed νt and αt models irrespective of the variation of P r, i.e., P rt as Cµ Aµ Bµ Cλ Aλ Bλ 2R/P r 1.21, (4.19) y + → 0 (with R → P r). Next, before evaluating the performance of the proposed model in higher Prandtl number ﬂuids (P r ≥ 1.0), we discuss the predictive accuracy at extremely high Prandtl numbers. Owing to the diﬃculty of carrying out turbulent heat transfer experiments in higher Prandtl number ﬂuids, the measurements reported up to now have been very few, and generally limited to mean temperature proﬁles; hence, the predictive accuracy of the present model is veriﬁed through a comparison of the prediction with the following measurements of mean temperature proﬁles. The predicted mean temperature proﬁle in a fully developed channel ﬂow (Rem = 10000) of engine oil (P r = 95) is shown in Figure 25 with the corresponding experimental data. It can be seen from the ﬁgure that the mean temperature proﬁles for both the prediction and the experiment show abrupt changes in the region adjacent to the wall (y + < 10). It is also clear that, though the beginning of the predicted almost constant mean temperature region is displaced slightly away from the wall in comparison with the measurement, the value obtained at the center of the channel agrees well with the experimental one. This means that either the wall friction temperature θτ or the mean temperature gradient at the wall can be accurately obtained from the proposed model, and this is especially signiﬁcant in accurately calculating the heat transfer rate mentioned below. After an evaluation of the proposed model at high P r through a comparison of the prediction with the measurement, we now thoroughly examine the performance of the present model in various high P r ﬂuids. In the high P r case, just as for a low P r, systematic variations for various thermal turbulence quantities are conﬁrmed from Figures 24(a)-24(d). We note that it is suﬃcient to show results up to P r = 10, because fundamental characteristics of thermal turbulence at much higher Prandtl numbers can faithfully be captured at P r = 10. It is found from Figure 24(a) that Θ+ becomes almost constant within the fully turbulent region as P r increases; this implies the thinning of + the √ thermal boundary layer. The heat ﬂux vθ and temperature ﬂuctuation kθ /(Θw − Θ0 ) proﬁles shown in Figures 24(b) and 24(c) also exhibit sys+ 1) tematic variations with increasing P r: the constant heat ﬂux layer (vθ emerges in the wall region; and the maximum kθ location shifts toward the wall. It should be emphasized that the present model, as already mentioned, can mimic the trend of both experiment and various empirical formulas for various Reynolds number ﬂows with high accuracy, so that the predicted trend in various Prandtl number ﬂuids also seems to capture the actual ﬂow ﬁelds well. As for P rt at high P r, ﬁrstly, we immediately notice the opposite trend [6] Modelling heat transfer in near-wall ﬂows 239 (a) (b) Figure 27: Thermal turbulence quantities at various streamwise locations in backward-facing step ﬂow (ReH = 28000, δ0 /H = 1.1) with uniform wall heat ﬂux for P r = 0.71 : (a) mean temperature; (b) turbulent heat ﬂux. with increasing P r [see Figure 24(d)] as found for low P r, in particular, an abrupt increase of P rt in the near-wall region (1 < y + < 10). Kays (1994) has pointed out that it is necessary for P rt in high Prandtl number ﬂuids to abruptly increase in the near-wall region as the wall is approached. Therefore, we conclude that such a behavior of the predicted P rt is in the right direction. Now, we discuss the dependence of the Prandtl number (or the Schmidt number Sc) on turbulent heat and mass transfer coeﬃcients. Predictions of the Nusselt number N u and the Sherwood number Sh in a pipe at Rem = 10000 using the present model are presented in Figure 26. Since the predicted result corresponds fairly well with the various existing experimental data (see Sideman and Pinczewski (1975)) and empirical formulas (Harriott and Hamilton 1965; Azer and Chao 1961), the applicability of the present model to ﬂows for a wide range of Prandtl numbers is duly veriﬁed. 4.2.5 Backward-facing step ﬂow for various Prandtl number ﬂuids (a) Comparison of turbulence quantities in air ﬂow Finally, we assess the proposed model in complicated ﬂow ﬁelds with heat transfer. The backward-facing step ﬂow with a uniformly heated bottom wall downstream of the step, measured by Vogel and Eaton (1984), is arguably the best data for assessing the performance of the proposed model. The mean + temperature (Θ − Θ∞ )/(Θw − Θ∞ ) , turbulent heat ﬂux vθ , and Stanton number St = qw /[ρcp U0 (Θw − Θ0 )] (where cp denotes speciﬁc heat at constant pressure) are illustrated in Figures 27 and 28, in comparison with the experimental data (Vogel and Eaton 1984) of δ0 /H = 1.1, where H is the step 240 Nagano height and δ0 is the upstream boundary-layer thickness at the step location. Note that the predicted results for the corresponding velocity ﬁeld are given in Figure 29. Again, in each ﬁgure, X ∗ denotes the streamwise distance normalized by the reattachment length XR , i.e., X ∗ = (X − XR )/XR . As seen from the mean temperature proﬁles in Figure 27(a), there is good agreement between the present prediction and the measurement at various streamwise locations, though the P rt -constant model gives consistently high values of St. This indicates that the conventional approach using a P rt -constant model has serious problems which cannot be overlooked in calculating a thermal ﬁeld in complex ﬂows of industrial interest. From the proﬁles of vθ in Figure 27(b), we immediately notice quite diﬀerent characteristics between the recirculation (X ∗ < 0) and redeveloping (X ∗ > 0) regions. Within the recirculation region, the prominent peak of vθ appears in the shear layer near separation as well as in the vicinity of the heated wall, as found experimentally by Vogel and Eaton (1984). The ﬁrst peak near the heated wall is mainly caused by the steepness of mean temperature shown in Figure 27(a), while the second peak is induced by the strong turbulent motion in the separated shear layer caused by the steep mean-velocity gradient, as seen from Figure 29; this means that the two peaks of vθ originate from quite diﬀerent physical phenomena. Farther downstream, in the recirculation region, the second peak decreases, but the level within the recirculation region, in particular at y/H 0.5, increases. Vogel and Eaton (1984) have reported that in the middle of the recirculation region, the majority of the thermal transport from a heated wall is eﬀected by large organized motions of the ﬂuid. In the present study, the thermal transport by large organized motions is adequately reﬂected in the modelled kθ -equation through the turbulent diﬀusion model. Thus, the present model can properly reproduce the same tendency of vθ as in the experiment, as already known from the agreement between the predicted and measured Θ proﬁles. Within the redeveloping region (X ∗ > 0) downstream of the reattachment, proﬁles of vθ appear very similar to those of a ﬂat plate boundary layer with the ﬁrst peak remaining constant. The Stanton number St distributions (Figure 28) demonstrate that the prediction is in qualitative and quantitative agreement with the experiment (Vogel and Eaton 1984) within the degree of experimental uncertainty. On the other hand, the behavior of the P rt -constant model is considerably diﬀerent from the experimental one. In particular, the P rt -based model predicts a peak value of St about twice the experiment. The Launder and Sharma model reportedly gives a similar trend (Chieng and Launder 1980). (b) Turbulent heat transfer for various Prandtl number ﬂuids We investigate the Prandtl number eﬀect of turbulent heat transfer in backwardfacing step ﬂows. Figures 30 and 31 show the respective local maximum value [6] Modelling heat transfer in near-wall ﬂows 241 Figure 28: Stanton number variance in back-step ﬂow (P r = 0.71). Figure 29: Streamwise velocity proﬁle in backward-facing step ﬂow at ReH = 28000 and δ0 /H = 1.1. of St and the mean Nusselt number N u in the recirculation region (Kondoh et al. 1993) deﬁned by XR N u(x)dx Nu = 0 , XR over a wide range of Prandtl numbers (P r = 1 × 10−3 − 102 ). (Here, local maximum value means the value that emerges around the reattaching point.) It is readily conﬁrmed from Figure 30 that there are three sub-regions in the ﬁgure: (i) P r < 0.1; (ii) 0.1 ≤ P r ≤ 1 and (iii) 1 < P r. Note that in the numerical analysis of laminar heat transfer in backward-facing step ﬂows (Kondoh et al. 1993), similar characteristics with three diﬀerent modes of behavior can be seen. Categories (i) and (iii) indicate a similar Prandtl number dependence varying approximately with P r−0.5 . On the other hand, within the middle range of Prandtl number [category (ii)], a quite diﬀerent behavior is seen; Stmax becomes constant independent of the Prandtl number. In contrast to the Stmax proﬁle, the conﬁguration of N u indicates that N u varies in proportion to P r for P r < 1.0, while the approximate relation N u ∝ P r0.5 is obtained for P r > 1.0. Furthermore, we ﬁnd that, as P r decreases, N u asymptotes to a constant value of 220; this means that heat conduction becomes overwhelming in the limit of low Prandtl number, as shown earlier in Figure 23 for pipe ﬂow. Now, to further examine this uniqueness of Stmax , 242 Nagano Figure 30: Local maximum value of St in back-step ﬂows of various Prandtl number ﬂuids. Figure 31: Mean Nusselt number N u in the recirculation region at various Prandtl numbers. we present contour maps of the mean temperature (Θ − Θ∞ ) in Figure 32 in which the contour value is normalized by the respective maximum temperature (Θmax − Θ∞ ). The reattachment point is indicated by an arrow. It should be mentioned that the diﬀerence between the maximum and ambient temperature for P r > 10 is too locally concentrated near the step corner to be drawn. In category (i) (P r < 0.1), as seen from Figure 32(a), the thermal boundary layer remains thick even near the reattachment point, e.g., the thickness at P r = 0.01 is about 0.5H. This indicates that the thermal ﬁeld downstream of the step is substantially governed by heat conduction. As P r increases from 0.1, i.e., in regime (ii), within the recirculation region [see Figure 32(b)], the obtained temperature distribution is markedly aﬀected by the ﬂow pattern in the central recirculation bubble: thermal convection becomes more and more dominant. For example, the temperature distribution in the recirculation region just behind the step wall is formed by an upward ﬂow along the step wall, and hence the deterioration of turbulent heat transfer occurs because of the gradual variation of mean temperature, as seen from Figure 28. It is also found from the ﬁgures that Θmax appears at the intersection between the central and the secondary recirculation bubbles, and that the thermal-boundarylayer thickness downstream of the reattachment point becomes rapidly thinner [6] Modelling heat transfer in near-wall ﬂows 243 (a) (b) (c) Figure 32: Temperature distributions in back-step ﬂows of various Prandtl number ﬂuids (↑: reattachment point): (a) P r = 0.01; (b) P r = 0.3; (c) P r = 2.0. with increasing P r. The thinning of the thermal boundary layer immediately induces a decrease in wall temperature or an increase of heat-transfer rate. This improvement of turbulent heat transfer strongly depends on the Prandtl number. The thickness appears to change in proportion to P r, and hence a constant Stmax (or N umax ∝ P r) results. When the Prandtl number further increases [regime (iii)], as shown in Figure 32(c), though Θmax appears at the intersection mentioned above, the temperature distribution associated with the upward ﬂow along the step wall in the recirculation region is diminished. Furthermore, it is clear that the minimum-wall-temperature region spreads around the reattachment point. Therefore, the proﬁles of St at high P r represent a plateau around this region (not shown). 5 Conclusions Using the DNS data for turbulent wall shear ﬂows with heat transfer, we have shown the methodology of how to construct a rigorous near-wall model for the temperature variance and its dissipation rate equations. In the kθ - and εθ -equations, the turbulent diﬀusion terms are represented by gradient-type diﬀusion plus convection by large-scale motions. In the εθ -equation, all of the production and destruction terms are modelled to reproduce the correct behavior of εθ near the wall. Note that, in order to obtain the correct walllimiting behavior of εθ , it is of prime importance to have the correct kθ proﬁle near the wall. It is also shown that the present model works very well for calculating the heat transfer under diﬀerent thermal conditions. Furthermore, the present model reproduces the budget proﬁles of turbulence quantities as accurately as DNS. Thus, we anticipate a practical application of the present model in revealing the underlying physics of turbulent heat transfer in complex ﬂows of technological interest. In kθ -εθ modelling, the contributions from various eddies are also taken into consideration. Results obtained from the proposed kθ -εθ model in channel 244 Nagano ﬂows under arbitrary wall thermal boundary conditions show that the Prandtl number dependence identiﬁed by DNS is satisfactorily captured. 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Sandham Abstract Direct numerical simulation (DNS) of the three-dimensional Navier–Stokes equations provides data to study turbulence, including quantities that cannot be accurately measured experimentally. With the advent of massively parallel computing the range of ﬂows that can be treated by DNS is increasing. This chapter reviews the development of DNS methods and the fundamental limitation on simulation Reynolds number which arises from the basic nature of turbulence. Particular attention is then given to methods of validation of simulations to attain suﬃcient conﬁdence in the data for application to the turbulence closure problem. Diﬀerent uses for the data are outlined using examples taken from simulations of channel and wake ﬂows. 1 Introduction Direct solution of the Navier–Stokes equations has been made possible by the development of fast digital computers, with the ﬁrst simulations of isotropic turbulence appearing in the 1970s. Since that time there have been notable advances in algorithms relating to spectral methods and high-order ﬁnite difference schemes. However, for the most part simulations of more complicated ﬂows have depended for their feasibility on further developments in computer hardware. An illustration of the recent rate of increase in computer performance is given by Figure 1, which shows data from the Top500 supercomputer list (http://www.netlib.org/benchmark/top500.html). On this graph we show the maximum achieved performance for the ‘linpack’ benchmark (Rmax ) of the computers rated number 1 and number 500, together with an average over the top 500 supercomputer sites from the June census each year. The slope of the curve for the average machine shows performance increasing as exp(0.58t) where t is the time in years. This corresponds to roughly an order of magnitude increase every four years, or a doubling every 1.2 years. One needs to interpret some aspects of the curves with care. The peak performance for the linpack benchmark is not very closely approached with ﬂuid mechanics codes, and how close one can get depends on the manufacturer (and particularly upon how much they have optimised the machine for this particular benchmark). The recent rate of performance improvement may also be atypical because of the recent move to massively parallel processing (MPP). Whether a code can eﬀectively use the enhanced capacity depends on the application. However, even taking these issues into account it is clear that there is an ongoing exponential growth in computing capacity which shows no signs of slowing down in 248 [7] Introduction to direct numerical simulation 10 4 249 Performance (Rmax) Gflops 103 number 1 average number 500 10 2 10 1 100 10-1 1993 1994 1995 1996 1997 1998 1999 2000 Year (June census) Figure 1: Increasing performance of supercomputers over the last 8 years from the Top500 supercomputer list. the near future. In particular the Accelerated Strategic Computing Initiative (ASCI) project in the USA is driving the performance of the largest computers further forward. The cost of a numerical simulation that can be carried out will increase with size of the problem in a manner that depends on the particular algorithm employed. Suppose for the sake of illustration we have an algorithm whose cost scales as N log N where N is the number of points in a given spatial direction. With a three-dimensional simulation, such as that required for turbulence we must cube this to get the cost per time step. Also for typical algorithms we must take smaller time steps as N increases to ensure stability (a constant Courant number condition). This introduces another factor of N so that we may say that the computational cost of a simulation on an N 3 grid varies roughly as N 4 (log N )3 . For N of the order of 100 a doubling of N then costs a factor of 25 (reducing slightly as N increases further), which according to the above estimates of computer performance will occur every 5 to 6 years. This is in reasonable agreement with the actual progression in size of the largest simulations from 323 in the mid 1970s to 1283 in the mid to late 1980s to the present day 5123 . There have been a number of milestones along this path to larger simulations as a combination of algorithm and computing power has made new ﬂows feasible for simulation. The beginning of the ﬁeld of turbulence simulation can be associated with the spectral simulation of decaying isotropic turbulence (with three periodic directions) by Orszag and Patterson (1972), which 250 Sandham used a 323 grid. Rogallo (1981) extended the range of turbulent ﬂows that could be investigated by the addition a special transformation to extend the fully spectral approach to simulations of strained and sheared homogeneous turbulence, which he carried out on 1283 grids. Extensions to inhomogeneous ﬂow problems followed research into the development of algorithms for mixed Chebyshev and Fourier spectral methods. Kim, Moin and Moser (1987) simulated a fully developed turbulent channel ﬂow, while Gilbert (1988) simulated the complete transition process in channel ﬂow, beginning with instability waves and ending with fully-developed turbulence. The turbulent boundary layer problem was ﬁrst simulated by Spalart (1988). Most of these ﬂows have been simulated again on ﬁner and ﬁner grids as computers have improved, leading to higher Reynolds number simulations (e.g. Moser et al. 1999). Another development has been the extension of the transformation method of Rogallo to inhomogeneous ﬂow (Coleman et al. 2000) allowing the simulation of the eﬀects of strained near-wall turbulence within a computational domain with one inhomogeneous direction. Other extensions have been to ﬂows that have two inhomogeneous directions. Spalart and Watmuﬀ (1993) used a ‘fringe’ technique to allow simulation of an adverse pressure gradient turbulent boundary layer and made detailed comparisons with experiment. Examples of recent large simulations are wake-induced bypass transition (Wang et al. 1999) and turbulent trailing edge ﬂows (Yao et al. 2000, 2001), which have used 52 and 67 million grid points respectively. For further reading and an extensive bibliography the reader is referred to the review paper by Moin and Mahesh (1998). In the following sections we review the criteria that limit the DNS technique, discuss the validation of data from the simulations, and consider how the data can be used in relation to the closure problem of turbulence. 2 Turbulence scales and resolution requirements To appreciate the limitations of DNS we need to know something about the ﬂows we are trying to compute. Turbulence is a nonlinear phenomenon with a wide range of spatial and temporal scales. The large scales are usually ﬁxed by the geometry of the ﬂow, while the smallest scales are determined by the ﬂow itself. Estimates for the size of the smallest scales are available from simple dimensional reasoning. The Kolmogorov microscale η is deﬁned if we assume that it only depends on the ﬂuid viscosity ν and the rate of dissipation of energy , 1/4 ν3 η= . (1) A connection with ﬂow Reynolds number can be made if we make some further assumptions. For a ﬂow in equilibrium we may take production equal to dissipation. The production can be assumed to scale as U 3 /L where U is a reference bulk velocity and L a length scale of the problem, usually ﬁxed [7] Introduction to direct numerical simulation 251 by the geometry. Both U and L are characteristic of the largest scales of the turbulence. Thus we can write η (2) ∼ Re−3/4 L where Re = U L/ν is the bulk Reynolds number of the ﬂow. The number of grid points N that we will require for a given simulation will be proportional to L/η and hence to Re3/4 . For diﬀerent deﬁnitions of Reynolds number appropriate for diﬀerent ﬂows the exponent changes in the range 0.75 to 0.9 (Reynolds 1989). We can now extend our prediction of increases in N over time from the previous section to increases in Re over time. A doubling of Reynolds number will require a factor of 2.2 to 2.5 increase in N at a total cost increase of a factor of 37 to 67. At the current rate of increase of computer performance, and assuming that algorithms can be made to continue scaling eﬃciently, this leads to a potential doubling of Reynolds number every 6 or 7 years. It is important to appreciate the practical implications of this Reynolds number scaling. Suppose that with present computers we can carry out a simulation of a turbulent boundary layer with displacement thickness Reynolds number of 4000 (double that of Spalart who did his simulations over a decade ago). At this rate it will be another 20 years before we can simulate the Reynolds numbers typical of the ﬂow over the wings of large aircraft, and then only in small computational domains corresponding to perhaps a few centimetres in each direction. The external aerodynamics application is one of the most extreme, but timescales upwards of 20 years appear in a majority of technological application areas. Thus for design purposes one will rely on closure at the RANS and LES levels for many years to come. That said, there are already niche areas where DNS can contribute at Reynolds numbers that are already relevant; examples of these relate to transition in attached or separated ﬂows as discussed elsewhere in this volume. What will be important for further development of RANS and LES is that it will be feasible to provide DNS data over a range of low to medium Reynolds numbers (covering perhaps a factor of four variation), such that Reynolds number trends can be assessed. As we have seen, the number of grid points required depends on the computational box size and the microscale of turbulence η. However this microscale was derived purely on dimensional grounds. From practical calculations of ﬂows away from solid boundaries it appears that the actual resolution required is approximately h = 5η where h is the grid spacing. For ﬂow near walls the turbulence is highly anisotropic. Guidelines for the required resolution are based on wall units using a reference length of ν/uτ where uτ = τw /ρ is the friction velocity, τw being the wall shear stress. These units are given a superscript + and based on this we have guidelines ∆x+ < 15 (streamwise grid spacing) ∆z + < 8 (spanwise) and 10 points in the region y + < 10 (normal). The normal direction is given diﬀerently as the grid is usually stretched in this direction. To develop a more general criterion Manhart (2000) has suggested the use of a directional dissipation scale. 252 Sandham A corresponding Kolmogorov micro-timescale for the small eddies, τ , is given by ν 1/2 . (3) τ= The time step must be selected so that the smallest timescales of turbulence are accurately computed. For the majority of algorithms (fully explicit and mixed explicit/implicit) used for DNS the time step required for stability is already signiﬁcantly smaller than this timescale, so this should be already resolved. Of course generally this should be checked as each new ﬂow is attempted. 3 Numerical methods The governing equations for incompressible ﬂow are the continuity equation, which reduces to a solenoidal constraint on the velocity ∂ui = 0, ∂xi and the three components of the momentum equation ∂ui ∂ui 1 ∂p ∂ 2 ui + =ν . + uj ∂t ∂xj ρ ∂xi ∂xj ∂xj (5) (4) It can be seen that the pressure only appears as a derivative and thus an arbitrary constant can be added to it. A reference pressure must be decided upon for each calculation. The nonlinear terms, shown here in a non-conservative form, may be rearranged before discretisation as discussed later. It is from these terms that turbulence develops its characteristic wide range of spatial scales. To solve the equations on a computer we need to consider methods for time advancement and spatial discretisation. For the time advancement, methods suitable for ordinary diﬀerential equations (Runge-Kutta, Crank–Nicolson, Adams–Bashforth) may be readily obtained from numerical methods texts and will not be discussed further. For many applications an implicit treatment of the viscous terms, in addition to pressure, is desirable. It is rare to ﬁnd fully implicit methods used, since the cost per time step is high with these methods and to resolve the small time scales of turbulence many time steps are required. For the remainder of this section we focus on issues relating to the spatial discretisation and the calculation of derivatives. With the vector computers that dominated scientiﬁc computing in the 1980s and the ﬁrst half of the 1990s computer memory was a major limiting factor and much eﬀort was spent developing eﬃcient methods for simulating a range of scales accurately. A useful idea of the resolving power of a numerical discretisation can be obtained by a simple modiﬁed wavenumber analysis which asks how well a given method calculates single Fourier modes eiκx . [7] Introduction to direct numerical simulation 253 3 2.5 2 exact (spectral) 2nd 4th 6th (compact) hk’ 1.5 1 0.5 0 0 1 2 3 hk Figure 2: Graph of modiﬁed against actual wavenumber for diﬀerent schemes, including the exact spectral solution. For example on substitution into a standard second-order accurate ﬁnite diﬀerence method f (x + h) − f (x − h) f (x) = (6) 2h we have the result sin(κh) iκx (7) f (x) = i e . h This can be compared to the exact result f = iκeiκx . Hence we can think of a modiﬁed wavenumber sin(κh) κ = . (8) h This is shown on Figure 2 together with a fourth-order central scheme f (x) = 2 (f (x + h) − f (x − h)) f (x + 2h) − f (x − 2h) − , 3h 12h (9) and a compact 5-point scheme (Lele 1988) requiring a tridiagonal matrix inversion f (x+h)+3f (x)+f (x−h) = 7 (f (x + h) − f (x − h)) f (x + 2h) − f (x − 2h) + . 3h 12h (10) We can see that these schemes have an increasing range of wavenumbers over which they compare well to the exact result shown by the straight line. In this case a Fourier spectral method gives the exact result directly. 254 Sandham In fact a great deal of eﬀort has been spent on exploiting the properties of spectral methods for turbulence simulation. Subtleties of the methods concern mainly the nonlinear terms. One can make use of fast transforms (FFTs) to reduce the order N 2 convolution to something of order N log N , where N is the number of modes. In these ‘pseudo’ spectral methods the time advance takes place in wave (Fourier transformed) space, while the nonlinear terms are computed in real space, using FFTs for the transformations back and forth. Aliasing errors are a source of concern with spectral methods since the high wavenumber components interact during the calculation of the nonlinear terms producing wavenumbers that are not resolved. These can then reﬂect back and corrupt wavenumbers that are carried by the computation. Solutions to the problem involve various degrees of additional expense, for example mode truncation, mode extension (the 3/2 rule is perhaps the most commonly used approach) and random phase shifting (Rogallo 1981). A subject that has been less well explored is the inﬂuence of the mathematical formulation of the convective terms. In particular skew-symmetric formulations (Zang 1989) have attractive properties. The latest study on the combined eﬀect of convective term formulation and de-aliasing method is by Kravchenko and Moin (1997). For incompressible isotropic turbulence and related ﬂows where all three direction are assumed periodic, spectral methods are well developed. The reader is referred to the book by Canuto et al. (1987) for details. For inhomogeneous incompressible ﬂows the numerical issues relate to the simultaneous satisfaction of the incompressibility (divergence free velocity) condition and the boundary conditions, for example no-slip walls. The inﬂuence matrix technique of Kleiser and Schumann is described in Canuto et al. Another elegant solution is to use divergence free variables (Spalart et al. 1991). Parallel implementation of the Kleiser-Schumann method is discussed in Sandham and Howard (1998). The limitation of spectral methods is the geometry. Fourier methods necessitate periodic boundary conditions. Eﬃcient schemes using Chebyshev polynomials are available for problems with one inhomogeneous direction, allowing channel and boundary layer problems to be computed eﬃciently. However with more than one inhomogeneous direction spectral methods become rather cumbersome. Various tricks with buﬀer regions (e.g. Spalart and Coleman 1998) extend the range of ﬂows that can be computed with periodic boundary conditions, including separation bubble ﬂows, but these zones need tuning for each case. Spectral element methods (see for example the book by Karniadakis and Sherwin 1999) set out to combine the excellent resolution properties of spectral methods with the geometric ﬂexibility of ﬁnite element methods. Cost has been a limiting factor thus far in the use of such methods and they have not yet found wide application to DNS of turbulence. For compressible ﬂow, compact and non-compact high-order ﬁnite diﬀerence schemes are popular and explicit time advance schemes are usually applied. [7] Introduction to direct numerical simulation 255 Severe numerical problems are present when shock waves coexist with turbulence. For particular examples, the reader is referred to the compressible ﬂow chapter of this book. With recent MPP architectures the limitation of computer memory for DNS has eased, leaving run time the critical factor. Eﬃcient spectral turbulence simulations that use all the memory of a 512 processor Cray T3E, for example, would take many months to run on that computer. There is nowadays a perceptible trend to return to lower- (second-) order schemes to allow more complex geometries to be calculated. The extra memory available is used to add extra grid points to compensate, at least in part, for the reduced accuracy of the scheme. Examples are the Le and Moin (1987) backward-facing step calculation, the Wu et al. (1999) bypass transition calculation and the Yao et al. (2000, 2001) trailing-edge ﬂow. For such methods an early rule of thumb was that a factor of 1.5 to 2 more resolution in all directions was required. This requirement gets worse for higher-order statistics but with 50% more resolution in all directions, compared to spectral methods, it is possible to get statistical data from ﬁnite diﬀerence simulations that is suﬃciently accurate for modelling at the second moment level. 4 Validation Issues Producing quality simulation data is by no means a trivial task. Computer programs containing several thousands of lines of code need to be debugged initially, but even a working code needs to be applied properly. Users of the data need to be familiar with the methods by which data has been validated so that they can use it sensibly. Here we list some of the major issues. A. Basic numerical method validation. This might be calculation of exact solutions of the Navier–Stokes equations or of known stability characteristics. For example in a boundary layer ﬂow one might ﬁx a Blasius base ﬂow and compare the growth rates and mode shapes of ﬂuctuations with the eigenvalues and eigenvectors from a separate solution of the Orr-Sommerfeld equation. B. Internal consistency checks on the results. It may be veriﬁed for example that the output statistics from a turbulence simulation conserve mass and momentum and balance the mean and turbulence kinetic energy equations to an acceptable accuracy. If instantaneous ﬂowﬁelds are available it can be veriﬁed that the ﬂow locally satisﬁes the governing equations. C. Resolution, domain size and sample size. This checks the convergence of the solution. Put simply a good simulation should give the same answers to within an acceptable error bound when recomputed with double the number of points in each direction separately, or with half the time 256 Sandham step, or with double the domain size in each direction, or with twice the statistical sample time. With spectral methods it is often convenient to monitor an energy spectrum during a simulation and thus ensure that there is suﬃcient resolution for the energy in the highest wavenumbers to be small (for example six to eight orders of magnitude smaller than the peak). D. Comparisons between diﬀerent codes. This is rarely done before publication, but a published DNS will often become a test case that is recomputed by other researchers using diﬀerent codes. This provides an accumulated conﬁdence in the data. E. Comparison with experimental data. One might expect this to be the most popular validation approach but it is rare to ﬁnd examples of DNS where the main source of conﬁdence in the results is a demonstrated agreement with experiments. This is due primarily to the simple geometries of most DNS to date and the diﬃculty of reproducing (or indeed measuring at all) the exact inﬂow or initial conditions in laboratory experiments. In two cases of the listed milestone simulations (Kim, Moin and Moser 1987, and Gilbert 1988) there was actually a severe disagreement with existing data (Gilbert shows the experiments overestimate the mean ﬂow mean ﬂow, plotted in wall units, by 10% and the spanwise turbulence ﬂuctuations by up to 50%). This was in fact due to measurement diﬃculties in the near-wall region. Later experiments have been in much better agreement with the DNS (Nishino and Kasagi 1989). Another problem is the diﬃculty of obtaining genuinely two-dimensional ﬂows in experiments for comparison with simulations employing periodic spanwise boundary conditions. A majority of supposedly ‘2D’ experiments may not in fact be suitable for any sort of CFD validation (Spalart 2000). None of the above methods are suﬃcient in themselves, and a combination is usually employed. Of most interest to users of DNS databases are the internal checks that can be done on the data after it has been accumulated and stored. 5 Applications Despite restrictions of Reynolds number and the variety of ﬂows available, the use of DNS data is changing the way turbulence models are built and tested. In particular the completeness of information available from simulation, including terms such as pressure-velocity correlation that have not been available from experiments, is enabling the use of DNS data to test closure models at several diﬀerent levels. Poor models can now be rejected very quickly. The sources of errors in existing models can be more readily identiﬁed and detailed budgets can be used to identify missing physics and suggest new modelling strategies. In the following subsections we will highlight a range of uses for DNS, using data [7] Introduction to direct numerical simulation 257 Figure 3: Sketch of plane channel ﬂow. The ﬂow between two inﬁnite plates is driven by a mean pressure gradient in the x direction. from two ﬂow simulations as examples. Data from a wider range of simulations is contained in Jimenez et al. (1998). The ﬁrst case is a standard incompressible channel ﬂow, the basic ﬂow geometry of which is shown on Figure 3. The streamwise and spanwise directions are periodic and resolved using Fourier modes. In the other direction there are two no-slip walls and a Chebyshev discretisation is used. Results for Reynolds number equal to 180, based on friction velocity and channel half width, will be used, corresponding to the original Kim, Moin and Moser (1987) simulation. The computational domain for that calculation was 4π in the streamwise direction and 2π in the spanwise direction, with lengths normalised by the channel half width. This test case, although relatively simple and of limited use in developing more widely applicable closure models, provides a useful demonstration of the potential of simulations. The second case is a blunt trailing-edge conﬁguration, shown on Figure 4. Here a precursor simulation of an incompressible zero pressure gradient turbulent boundary layer was run to generate the inﬂow data for the main incompressible ﬂow simulation. The inﬂow boundary, with Reynolds number 1000 based on the inﬂow displacement thickness, is located upstream of the trailing edge. The main simulation Reynolds number is 1000 based on trailing edge thickness. The downstream boundary uses a convective outﬂow treatment, while a slip-wall condition is used for the upper and lower boundaries. The spatial discretisation is second-order on a staggered grid. Details of the simulation are given in Yao et al. (2000, 2001). Validation included a series of tests on grids up to 1024 × 256 × 128, together with calculation of turbulence kinetic energy and Reynolds stress budgets. This test case provides a more stringent test of turbulence models, and is used to illustrate some of the issues that arise as more complicated ﬂows are simulated. 258 Y Sandham –3h +8.5h +3h O X –7.5h –5h Z 0 +15h Figure 4: Schematic of the arrangement for the trailing edge simulation. A precursor simulation provides inﬂow data. Dimensions are given in terms of the trailing edge thickness. 5.1 Flow visualisation A ﬁrst use of DNS is direct interrogation of the instantaneous ﬂowﬁeld to obtain a qualitative understanding of the characteristics of the ﬂow. Computer visualisation allows the full three-dimensional ﬂowﬁeld to be studied, and the time dependence to be animated. Phenomena such as vortex shedding, separation and ﬂow impingement on solid surfaces, which may have a decisive inﬂuence on the best approach to closure for a whole class of problems, can be identiﬁed from an early stage. There is no agreed deﬁnition of a vortex in turbulent ﬂow, and diﬀerent measures have been used to reveal diﬀerent aspects of instantaneous ﬂowﬁelds. Iso-surfaces of constant pressure, enclosing low pressure regions, identify important coherent structures (Robinson 1994). The second invariant of the velocity gradient tensor ∂ui ∂uj Π= (11) ∂xj ∂xi is also often used to identify swirling motions as regions where rotation rate dominates over strain rate. This is the same as using Q from the P QR scheme described in Chong et al. (1990). This measure tends to highlight smaller scale structures. Another measure recently used by Jeong and Hussain (1995) is to set a threshold on the second eigenvalue of a particular tensor which corresponds to the pressure hessian ∂ 2 p/∂xi ∂xj when viscous eﬀects are ignored. In practice for channel ﬂow this measure turns out to be very similar to the Π-criterion. The setting of a threshold for all the above measures implies some degree of arbitrariness as to what constitutes a vortex. Kida and Miura (1998) [7] Introduction to direct numerical simulation 259 (a) (b) Figure 5: Visualisation of the trailing edge ﬂow. The ﬂow is from left to right. Red surfaces enclose regions of low pressure while blue regions enclose vortices detected by the Π criterion. have instead proposed the construction of a vortex ‘skeleton’, based on the existence of closed streamlines in a plane normal to the axis of the cores of vortices detected as low pressure regions. On Figure 5 we illustrate two of these measures for the trailing edge ﬂow. Enclosed low-pressure regions are shown by the red surfaces. These clearly pick out the large-scale shedding of predominantly spanwise vortices from the trailing edge. Further downstream these can be seen to break up and lose their spanwise coherence. Also shown are Π-vortices detected from the second invariant of the velocity gradient ﬁeld. This measure picks out smaller scale structures, such as the vortices that are subject to a large strain rate in the region between successive spanwise rollers. The threshold that was set for both these measures is larger than the strongest structures in the incoming 260 Sandham Figure 6: Mean velocity of plane channel ﬂow predicted by ﬁve turbulence models (Launder-Sharma, Chien, Kawamura-Kawashima, K-g and Durbin), compared with DNS. For more details of the models, see Howard and Sandham (2000). boundary layer ﬂow, indicating that the structures in the near wake region are more intense. The upstream turbulent boundary layers, near wake and far wake are thus distinct structural regions, which has implications for selecting a closure approach. Initial results from comparisons with closure methods (Yao et al. 2000, 2001) suggest serious limitations of RANS models for this ﬂow. 5.2 A posteriori model testing for LES and RANS The ﬁrst method of model testing consists of using the DNS data as an extension of the experimental database. Boundary conditions and initial or inﬂow conditions are matched to the DNS and then the model is run and predictions for mean ﬂow and turbulence kinetic energy are compared with DNS. This standard method has become known in the LES community as ‘a posteriori’ testing (Piomelli et al. 1988) to distinguish it from a second ‘a priori’ method of using simulation data. As an example of this approach, on Figures 6 and 7 we reproduce from Howard and Sandham (2000) the mean velocity and turbulent kinetic energy computed from the set of two equation models identiﬁed in the ﬁgure caption, compared with the DNS of the turbulent channel ﬂow. It can be seen that variations in the mean ﬂow prediction are of the order of 10%, while the turbulence kinetic energy varies by as much as a factor of two between models. In this usage, the DNS results are used in the same way as experimental data: models have to be changed and constants tuned to improve the mean ﬂow predictions. This has to be done over a range of ﬂows for the resulting model to be practically useful. An advantage of the DNS-based approach is that it is [7] Introduction to direct numerical simulation 261 Figure 7: Turbulence kinetic energy in plane channel ﬂow predicted by ﬁve turbulence models, compared with DNS. much easier to ensure that simulations and model calculations have exactly the same boundary, initial and inﬂow conditions, where appropriate. This is much more diﬃcult with experiments where, for example, inﬂow dissipation proﬁles are not measured. Also automatic optimisation of constants to minimise the error between DNS and model prediction can be readily carried out. This saves time, and means that the modeller can concentrate on ﬁnding the right model formulation to predict a certain range of turbulent ﬂows. This requires some judgement as to what a model is expected to predict, whether mean ﬂow alone, or mean ﬂow plus turbulence kinetic energy, or other turbulence quantities. 5.3 A priori model testing for LES and RANS A priori testing involves testing the basic assumptions and detailed term by term construction of a turbulence model. For application to LES, the DNS ﬁeld can also be ﬁltered and used to test the local accuracy of the sub-grid model. For RANS applications we can take for example the exact k equation, which can be derived from the Navier–Stokes equations as ∂k ∂k ∂Ui ∂u = −ui uj −ν i + Uj ∂t ∂xj ∂xj ∂xj ∂ui ∂ui + ∂xj ∂xj − ∂ p ν J u + Jj + Jj , (12) ∂xj j where the ﬁrst and second terms on the right are the production and dissipation rate of turbulence and the last three terms are turbulence transport u terms. The triple moment term Jj = ui ui uj /2 is usually modelled as gradient p diﬀusion with an eddy viscosity. The pressure transport term Jj = p uj is ν usually ignored, while the viscous term Jj = −ν∂k/∂xj can be included as is 262 Sandham Figure 8: Kinetic energy budgets for the trailing edge ﬂow problem. (P is production, ε dissipation, J transport and R convection.) and need not be modelled. What can done using DNS data is to compare the modelling of each term with the actual term measured from DNS. As an example of a kinetic energy budget we take the trailing edge ﬂow problem. Figure 8 shows budgets at two locations (a) upstream of the trailing edge and (b) downstream of the trailing edge near the end of the recirculation [7] Introduction to direct numerical simulation 263 region. The various terms are labelled according to the above equation. We can see that in the outer region of the boundary layer an approximate balance of production and dissipation is valid, while in the near wall region and downstream of the wake other terms are signiﬁcant. Particular features of this ﬂow that require attention from a modelling point of view are the negative production region immediately downstream of the trailing edge (not shown on ﬁgure) and the pressure transport term, normally neglected, but which in this ﬂow appears as one of the most important terms in the near-wake region. Another example of a priori testing is in the development of damping functions for the eddy viscosity. Assuming an eddy viscosity relation of the form νt = cµ fµ k2 , (13) Rodi and Mansour (1993) extracted the exact form of fµ from DNS and compared this to several models. Existing models did a poor job of representing the actual eddy viscosity damping and new functions were proposed which match the DNS more closely. A problem with a priori testing for RANS is that in fact most terms in the model equations are poor representations of terms in exact transport equations, even though the overall model may do quite a reasonable job of predicting mean velocity proﬁles. It is then not clear how to improve matters. When one term is changed to agree better with DNS the most likely eﬀect is that the overall prediction will get worse. All other terms must then be considered. Even if all terms are modelled correctly a priori it may well be that the actual model calculation converges to a diﬀerent solution, due to coupling between terms, if indeed it is stable at all. Simple models are in fact already well optimised for a range of ﬂows. It is perhaps with more complex models where a priori testing can provide useful guidance as to how individual terms should be modelled. 5.4 Diﬀerential a priori for RANS A middle way between a priori and a posteriori testing has been suggested by Parneix et al. (1998). In this method whole model equations are tested by solving the equation for a particular variable, substituting DNS values for all other terms in the equation. This method is able to pin the blame for a poor prediction on a particular model equation, on which particular attention can then be focused. As an example, for a second moment closure prediction of a backward facing step problem, Parneix et al. show that the u v equation is a more signiﬁcant source of error than the usually blamed equation. 264 Sandham 6 Conclusions Databases of statistical quantities from DNS are already an essential part of the turbulence modeller’s toolkit, whether used as a supplement to experimental data, or as a guide to rational modelling methods. Further examples of the use of DNS data will appear throughout this volume. In the future it is expected that the relevance of DNS to the closure problem will increase as simulations increase in Reynolds number and geometric complexity. While not providing very high Reynolds number data, there will certainly be data available over a suﬃcient range of Reynolds numbers so that the correct trends can be built into future models. References Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1987). Spectral Methods in Fluid Dynamics, Springer-Verlag. Chong, M.S., Perry, A.E., Cantwell, B.J. (1990).‘A general classiﬁcation of 3-dimensional ﬂow-ﬁelds’, Phys. Fluids A-Fluid Dynamics 2, 765–777. Coleman G.N., Kim J. and Spalart, P.R. (2000). ‘A numerical study of strained threedimensional wall-bounded turbulence’, J. Fluid Mech. 416, 75–116. Gilbert, N. (1988). ‘Numerische Simulation der Transition von der laminaren in die turbulente Kanalstr¨mung’. DFVLR-FB 88-55. DFVLR, G¨ttingen, Germany. o o Howard, R.J.A. and Sandham, N.D. (2000). ‘Simulation and modelling of a skewed turbulent channel ﬂow’. Flow, Turbulence and Combustion. 65(1), 83–109. Jeong, J. and Hussain, F. (1995). ‘On the identiﬁcation of a vortex’, J. Fluid Mech. 285, 69–94. Jimenez, J. et al. (1998). ‘A selection of test cases for the validation of large-eddy simulations of turbulent ﬂows’. AGARD-AR-345, North Atlantic Treaty Organisation, April 1998. Karniadakis, G.E.M. and Sherwin, S.J. (1999). Spectral/hp Element Methods for CFD, Oxford University Press. Kida, S. and Miura, H. (1998). ‘Identiﬁcation and analysis of vortical structures’, Eur. J. of Mech. B-Fluids 17, 471–488. Kim, J., Moin, P. and Moser, R.W. (1987). ‘Turbulence statistics in fully-developed channel ﬂow at low Reynolds number’, J. Fluid Mech. 177, 133–166. Kravchenko, A.G. and Moin, P. (1997). ‘On the eﬀect of numerical errors in large eddy simulations of turbulent ﬂows’, J. Comput. Phys. 131, 310–322. Lele, S.K. (1991). ‘Compact ﬁnite-diﬀerence schemes with spectral-like resolution’, J. Comput. Phys. 103, 16–42. Manhart, M. (2000). ‘The directional dissipation scale: a criterion for grid resolution in direct numerical simulations’. In Advances in Turbulence VIII: Proceedings [7] Introduction to direct numerical simulation 265 of the Eighth European Turbulence Conference, C. Dopazo et al. (eds.), CIMNE, Barcelona. Moin, P. and Mahesh, K. (1998). ‘Direct numerical simulation: a tool for turbulence research’, Ann. Rev. Fluid Mech. 30, 539–578. Moser R.D., Kim J. and Mansour N.N. (1999). ‘Direct numerical simulation of turbulent channel ﬂow up to Re-tau=590’, Phys. Fluids 11, 943–945. Nishino, K. and Kasagi, N. (1989). ‘Turbulence statistics measurement in a twodimensional channel ﬂow using a three-dimensional particle tracking velocimeter’. In Proceedings of Seventh Symposium on Turbulent Shear Flows, Stanford University, Stanford, CA, August 1989. Orszag, S.A. and Patterson, G.S. (1972). ‘Numerical simulation of three-dimensional homogeneous isotropic turbulence’, Phys. Rev. Lett. 28, 76–79. Parneix, S., Laurence, D. and Durbin, P.A. (1998). ‘A procedure for using DNS databases’, J. Fluids Eng. 120, 40–47. Piomelli, U., Moin, P. and Ferziger, J.H. (1988). ‘Model consistency in large eddy simulation of turbulent channel ﬂows’, Phys. Fluids 31, 1884–1891. Reynolds, W.C. (1989). ‘The potential and limitations of direct and large eddy simulation’. In Whither Turbulence?, J.L. Lumley (ed.), Lecture Notes in Physics 357, Springer, 313–343. Robinson, S.K. (1992). ‘Coherent motions in the turbulent boundary-layer’, Ann. Rev. Fluid Mech. 23, 601–639. Rodi, W. and Mansour, N.N. (1993). ‘Low-Reynolds-number k- modeling with the aid of direct simulation data’, J. Fluid Mech. 250, 509–529. Rogallo, R. (1981). ‘Numerical experiments in homogeneous turbulence’, NASA TM81315. Sandham, N.D. and Howard, R.J.A. (199). ‘Direct simulation of turbulence using massively parallel computers’. In Parallel Computational Fluid Dynamics, D.R. Emerson et al. (eds.), Elsever. Spalart, P. (1988). ‘Direct simulation of a turbulent boundary-layer up to Rθ = 1410’, J. Fluid Mech. 187, 61–98. Spalart, P. (2000). ‘Trends in turbulence treatments’, AIAA paper 00-2306. Spalart, P. and Coleman, G.N. (1998). ‘Numerical study of a separation bubble with heat transfer’, Eur. J. Mech. B-Fluids 16, 169–189. Spalart, P.R., Moser, R.D. and Rogers, M.M. (1991). ‘Spectral methods for the Navier–Stokes equations with one inﬁnite and 2 periodic directions’, J. Comput. Phys. 96(2), 297–324. Spalart, P. and Watmuﬀ, J.H. (1993). ‘Experimental and numerical study of a turbulent boundary-layer with pressure-gradients’, J. Fluid Mech. 249, 337–371. Wu, X., Jacobs, R.G., Hunt, J.C.R. and Durbin, P.A. (1999). ‘Simulation of boundary layer transition induced by periodically passing wakes’, J. Fluid Mech. 398, 109– 153. Yao, Y, Sandham, N.D, Savill, A.M and Dawes, W.N. (2000). ‘Simulation of a turbulent trailing-edge ﬂow using unsteady RANS and DNS’. In Proc. 3rd Intl. Symp. on Turbulence, Heat and Mass Transfer, Nagoya, January 2000. 266 Sandham Yao, Y., Sandham, N.D., Thomas, T.G. and Williams, J.J.R. (2001). ‘Direct numerical simulation of turbulent ﬂow over a rectangular trailing edge’. Theor. and Comput. Fluid Dyn. 14(5), 337–358. Zang, T.A. (1989). ‘On the rotation and skew-symmetrical forms for incompressibleﬂow simulations’, Appl. Num. Math. 7, 27–40. 8 Introduction to Large Eddy Simulation of Turbulent Flows J. Fr¨hlich and W. Rodi o 1 Introduction This chapter is meant as an introduction to Large-Eddy Simulation (LES) for readers not familiar with it. It therefore presents some classical material in a concise way and supplements it with pointers to recent trends and literature. For the same reason we shall focus on issues of methodology rather than applications. The latter are covered elsewhere in this volume. Furthermore, LES is closely related to direct numerical simulation (DNS) which is also widely discussed in this volume. Hence, we concentrate as much as possible on those features which are particular to LES and which distinguish it from other computational methods. For the present text we have assembled material from research papers, earlier introductions and reviews (Ferziger 1996, H¨rtel 1996, Piomelli 1998), and our a own results. The selection and presentation is of course biased by the authors’ own point of view. Supplementary material is available in the cited references. 1.1 Resolution requirements of DNS The principal diﬃculty of computing and modelling turbulent ﬂows resides in the dominance of nonlinear eﬀects and the continuous and wide spectrum of observed scales. Without going into details (the reader might consult classical text books such as Tennekes and Lumley (1972)) we just recall here that the ratio of the size of the largest turbulent eddies in a ﬂow, L, to that of the 3/4 smallest ones determined by viscosity, η, behaves like L/η ∼ Reu . Here, Reu = u L/ν with u being a characteristic velocity ﬂuctuation and ν the kinematic viscosity. Let us consider as an example a plane channel, a prototype 1/2 of an internal ﬂow. Reynolds (1989) estimated Reu ∼ Re0.9 from u ∼ u cf , cf ∼ Re−0.2 , where Re is based on the center line velocity and the channel height. In a DNS no turbulence model is applied so that motions of all size have to be resolved numerically by a grid which is suﬃciently ﬁne. Hence, the computational requirements increase rapidly with Re. According to this estimate a DNS of channel ﬂow at Re = 106 for example would take around hundred years on a computer running at several GFLOPS. This is obviously not feasible. Moreover, in an expensive DNS a huge amount of information 267 268 Fr¨hlich and Rodi o would be generated which is mostly not required by the practical user. He or she would mostly be content with knowing the average ﬂow and some lower moments to a precision of a few percent. Hence, for many applications a DNS which is of great value for theoretical investigations and model testing is not only unaﬀordable but would also result in computational overkill. 1.2 The basic idea of LES Suppose somebody wants to perform a DNS but the grid that would be required exceeds the capacity of the available computer; so a coarser grid is used. This coarser grid is able to resolve the larger eddies in the ﬂow but not the ones which are smaller than one or two cells. From a physical point of view, however, there is an interaction between the motions on all scales so that the result for the large scales would generally be wrong without taking into account the inﬂuence of the ﬁne scales on the large ones. This requires a so-called subgrid-scale model as discussed below. Hence, LES can be viewed as a ‘poor man’s DNS’. The poor man, however, has to compensate by cleverness in that a model for the unresolved motion has to be devised and an intricate coupling between physical and numerical modelling is generated. On the other hand, the resolution of the large scales of the ﬂow while modelling only the small ones – not the entire spectrum – is an advantage of the LES approach compared to methods based on the Reynolds-averaged Navier–Stokes equations (RANS). The latter methods often have diﬃculties when applied to complex ﬂows with pronounced vortex shedding or special inﬂuences of buoyancy, curvature, rotation or compression. Finally, LES gives access to the dominant unsteady motion so that it can, for example, be used to study aero-acoustics, ﬂuid-structure coupling or the control of turbulence by an appropriate unsteady forcing. 2 Governing Equations and Filtering The Navier–Stokes equations (NSE) constitute the starting point for any turbulence simulation. Here, we consider incompressible, constant-density ﬂuids for which these equations read ∂ui =0 ∂xi ∂ui ∂ (ui uj ) ∂Π ∂(ν 2Sij ) + = , + ∂t ∂xj ∂xi ∂xj (1) (2) where Sij = (∂uj /∂xi + ∂ui /∂xj )/2 is the strain-rate tensor and Π = p/ρ. For later reference we introduce Reynolds averaging which is used in statistical T 1 turbulence modelling (RANS) as time averaging: u = limT →∞ T 0 u dt. Reynolds averaging has the properties u = u, uv = u v . (3) [8] Introduction to large eddy simulation of turbulent ﬂows 269 Figure 1: Illustration of Schumann’s approach to LES as discussed in the text. According to the idea of LES a means is required to distinguish between small, unresolved, and larger, resolved structures. This is accomplished by the operation u → u, deﬁned below. Unlike the above Reynolds time averaging, this is an operation in space. The fact that RANS and LES methods employ averaging in diﬀerent dimensions inhibits an easy link between them. Several attempts have been made to put both in a common framework (Speziale 1998, Germano 1999) but they will not be discussed here. We now turn to the ways of deﬁning u and illustrate them in the one-dimensional case. 2.1 Schumann’s approach The ‘volume-balance approach’ of Schumann (1975) starts from a given ﬁnitevolume mesh. The integral of a continuous unknown u(x) in (1), (2) over one 1 cell is denoted V u = ∆x V u(x)dx as illustrated in Figure 1 (indices referring to cells are dropped). Integrating the NSE over a cell and using Gauss’ theorem relates these values to surface-averaged quantities denoted j · · ·, such as j uv. These need to be expressed in terms of the cell-averages, which is done in two steps. If the discretization is suﬃciently ﬁne, it is possible to replace j uv by j uj v, with only a minor approximation error, as is usual in ﬁnite-volume methods. This is done in DNS. If the grid is not ﬁne enough, however, the diﬀerence can be signiﬁcant and the unresolved momentum ﬂux j uv − j uj v 270 Fr¨hlich and Rodi o has to be accounted for by a model, the so-called subgrid-scale (SGS) model. Subsequently, the j u are related to the V u either by setting them equal to cell averaged quantities if a staggered arrangement is used or by interpolating from neighbouring values. The ﬁnal SGS contribution to be modelled therefore also depends on the expressions used for j u, i.e. on the discretization scheme. To sum up, the equations are discretized and thereby the split into large and small scales is performed, since the latter cannot be resolved by the discretized system. Note that the operations u → V u and u → j u map an integrable function onto discrete values; a continuous function u(x) is not constructed. Thus, with Schumann’s approach, scale separation, discretization, and the SGS model are not separated conceptually but are intimately tied together. This has advantages in that anisotropies and inhomogeneities of the grid can easily be incorporated. However, it renders the analysis of the various contributions to the solution relatively diﬃcult and hence is considered too restrictive by many workers in the ﬁeld. 2.2 Filtering Leonard (1974) proposed deﬁning u by +∞ u(x) = −∞ G x−x u x dx . (4) An integral of this kind is called a convolution. Here, G is a compactly supported, or at least rapidly decaying, ﬁlter function with G(x) dx = 1 and width ∆. The latter can be deﬁned by the second moment of G as ∆ = 12 x2 G(x) dx. Figure 2 displays the Gaussian Filter, GG = 6/π (1/∆) exp(−6x2 /∆ ), and the box ﬁlter deﬁned by GB = 1/∆ if |x| ≤ ∆/2 and GB = 0 elsewhere. In fact, Deardorﬀ (1966) had already used (4) in the special case G = GB . Figures 3a and 3b illustrate the ﬁltering with smaller or larger ﬁlter width: the larger ∆, the smoother is u. According to (4), u is a continuous smooth function as displayed in Figure 3 which can subsequently be discretized by any numerical method. This has the advantage that one can separate conceptually the ﬁltering from the discretization issue. It is helpful to transfer equation (4) to Fourier space by means of the deﬁnition u(ω) = u(x) e−iωx dx, since in Fourier space, where the spatial frequency ˆ ω is the independent variable, a convolution integral turns into a simple product. Equation (4) then reads u(ω) = G(ω) u(ω) . (5) 2 Figure 4 illustrates the ﬁltering in Fourier space. Equation (5) allows the definition of another ﬁlter, the Fourier cutoﬀ ﬁlter with GF (ω) = 1 if |ω| ≤ π/∆ and 0 elsewhere. From (5) it is obvious that only this ﬁlter yields u = u since (GF )2 = GF . In all other cases the identity is not fulﬁlled. This can [8] Introduction to large eddy simulation of turbulent ﬂows 271 G(x) T ¨ GG ¨ ¨G ¨ B E ∆ ' x Figure 2: Gaussian ﬁlter GG and box ﬁlter GB as deﬁned in the text, both plotted for the same ﬁlter width ∆. be appreciated by comparing u and u for the box ﬁlter in Figures 3 and 4. The second relation in (3) is never fulﬁlled except in trivial cases, so that for general ﬁltering we have u = u, uv = u v (6) which distinguishes clearly the ﬁltering in LES from Reynolds averaging (see Germano (1992) for a detailed discussion). The vertical line in Figure 4 represents the nominal cutoﬀ at π/∆ related to the grid. The Fourier cutoﬀ ﬁlter GF would yield a spectrum of u which is equal to that of u left of this line, and zero, right of it. Equation (5) and Figure 4 therefore demonstrate that when a general ﬁlter, such as the box ﬁlter, is applied, this does not yield a neat cut through the energy spectrum but rather some smoother decay to zero. This is important since SGS modelling often assumes that the spectrum of the resolved scales near the cutoﬀ follows an inertial spectrum with a particular slope and a particular amount of energy transported from the coarse to the ﬁne scales on the average. We see that even if u fulﬁlls this property this can be altered by ﬁltering (for further remarks see Section 4.3). Nevertheless, it is convenient and common to use the notion of a simple cutoﬀ as a model in qualitative discussions. Equation (5) is also helpful for illustrating the fact that derivative and ﬁlter operations commute, i.e. ∂u/∂x = (∂u/∂x). Any convolution ﬁlter (4) can be written as in (5) regardless of the choice of G. Diﬀerentiation appears as multiplication by iω in Fourier space (see equation (28) below), which is commutative. Applying the three-dimensional equivalent of the ﬁlter (4) to the NSE (1) and (2), the following equations for the ﬁltered velocity components ui result: ∂ui =0 ∂xi (7) 272 Fr¨hlich and Rodi o (a) u q u E T u E ∆ ' x (b) E u u r rr j T u E ∆ ' x Figure 3: Filtered functions u and u obtained from u(x) by applying a box ﬁlter: (a) narrow ﬁlter, (b) wide ﬁlter. ∂Π ∂(ν 2S ij ) ∂τij ∂ui ∂ (ui uj ) + = − , + ∂t ∂xj ∂xi ∂xj ∂xj where S ij and Π are deﬁned analogously to the unﬁltered case. The term τij = ui uj − ui uj (8) (9) represents the impact of the unresolved velocity components on the resolved ones and has to be modelled. In mathematical terms it arises from the nonlinearity of the convection term, which does not commute with the linear ﬁltering operation. An important property of ui is that it depends on time. Hence, an LES necessarily is an unsteady computation. Furthermore, ui always depends on all three space-dimensions (except for very special cases). Symmetries of the boundary conditions generally produce the same symmetries for the RANS [8] Introduction to large eddy simulation of turbulent ﬂows 273 log E(ω) u u u log ω Figure 4: Eﬀect of ﬁltering on the spectrum. Here the box ﬁlter employed in Figure 3 is used as well, but the curves are similar for other ﬁlters such as the Gauss Filter. u and u are illustrated by the area between the curves for u and u, and u and u, respectively. The vertical line is related to the Fourier cutoﬀ ﬁlter on the same grid. variable ui , e.g. vanishing dependence on a homogeneous direction. However, due to the very nature of turbulence, this does not hold for ui since the instantaneous turbulent motion is always three-dimensional. The fact that a three-dimensional unsteady ﬂow is to be computed makes LES a computationally demanding approach. We ﬁnally note that for any ﬁlter, the term in (9) vanishes in the limit ∆ → 0, since then u → u according to (4), and all scales are resolved so that the LES turns into a DNS. 2.3 Variable ﬁlter size It should be mentioned here that ﬁltering as deﬁned by (4) is not easily compatible with boundary conditions. For instance, applying a box ﬁlter of constant size ∆ yields u = 0 within a distance ∆/2 from the computational domain and raises the question of how to impose boundary conditions for u. This problem is removed by supposing G to be x-dependent and locally asymmetric. However, if G(x − x ) is generalized to some G(x, x ), or if the prolongation of u 274 Fr¨hlich and Rodi o from a ﬁnite domain to the real axis induces discontinuities, the commutation property is lost and additional commutator terms arise in (7), (8) (Ghosal and Moin 1995). In contrast to the usual SGS term τij , which is generated by the nonlinearity of the convection term, the commutator also appears for linear expressions (see the discussion by Geurts (1999) and Section 4.3). This issue is relevant for pronounced grid stretching in the interior of the domain and close to walls but has been disregarded until recently. Studies for a channel ﬂow are reported in Fr¨hlich et al. (1998, 2000). o 2.4 Implicit versus explicit ﬁltering The ﬁltering approach relaxes the link between the size of the computed scales and the size of the grid since the ﬁlter can be coarser than the employed grid. Consequently, the modelled motion should be called subﬁlter- rather than subgrid-scale motion. The latter labelling results from the Schumann-type approach and is frequently used for historical reasons to designate the former. In practice, however, the ﬁlter G does not appear explicitly at all in many LES codes1 so that in fact the Schumann approach is followed. Due to the conceptual advantages of the ﬁltering approach, reconciliation of both is generally attempted in two ways. The ﬁrst observation is that a ﬁnite-diﬀerence method for (7), (8) with a box ﬁlter employs the same discrete unknowns as Schumann’s approach; for example u(xk ) = Vk u with k referring to a grid point. Choosing appropriate ﬁnite-diﬀerence formulae, the same or very similar discretization matrices are obtained in both cases. Another argument is that the deﬁnition of discrete unknowns amounts to an ‘implicit ﬁltering’ – i.e. ﬁltering with some unknown ﬁlter (but one that in principle exists) – since any scale smaller than the grid is automatically discarded. In this way the ﬁlter is used more or less symbolically only to make the eﬀect of a later discretization appear in the continuous equations. This is easier in terms of notation and stimulates physical reasoning for the subsequent SGS modelling. In contrast to implicit ﬁltering one can use a computational grid ﬁner than the width of G and only retain the largest scales by some (explicit) ﬁltering operation. This explicit ﬁltering has recently been advocated by several authors such as Moin (1997) since it considerably reduces numerical discretization errors as the retained motion is always well resolved. On the other hand it increases the modelling demands since for the same number of grid points more scales of turbulent motion have to be modelled and it is not yet completely clear which approach is more advantageous (Lund and Kaltenbach 1995). The ﬁltering approach of Leonard is now almost exclusively used in papers on LES and has triggered substantial development, e.g. in subgrid-scale modelling. In practice, however, it is most often used for conceptual reasons rather than as a precise algorithmic construction. apart from some ﬁltering operations for the dynamic model, discussed below, which is of a somewhat diﬀerent nature 1 [8] Introduction to large eddy simulation of turbulent ﬂows 275 3 3.1 Subgrid-scale modelling Introduction Subgrid-scale modelling is a particular feature of LES and distinguishes it from all other approaches. It is well-known that in three-dimensional turbulent ﬂows energy cascades, in the mean, from large to small scales. The primary task of the SGS model therefore is to ensure that the energy drain in the LES is the same as that obtained with the cascade fully resolved, as in a DNS. The cascading, however, is an average process. Locally and instantaneously the transfer of energy can be much larger or much smaller than the average and can also occur in the opposite direction (‘backscatter’ – see Piomelli et al. 1996). Hence, ideally, the SGS model should also account for this local, instantaneous transfer. If the grid scale is much ﬁner than the dominant scales of the ﬂow, even a crude model will suﬃce to yield the right behaviour of the dominant scales. This is for two reasons. First, the larger the distance, in wavenumber space, between diﬀerent contributions, the looser is their coupling. Second, as a consequence of this, as well as from the energy cascading, the ﬁner scales exhibit a more universal character which is more amenable to modelling. On the other hand, if the grid scale is coarse and close to the most energetic, anisotropic, and inhomogeneous scales, the SGS model should be of better quality. Obviously, there exist two possible approaches; one is to improve the SGS model and the other is to reﬁne the grid. In the limit, the SGS contribution vanishes and the LES turns into a DNS. Reﬁning the grid, however is restricted due to rapidly increasing computational cost. The alternative strategy, for example solving an additional transport equation in a more elaborate SGS model, can be comparatively inexpensive. Another aspect results from the numerical discretization scheme which introduces a diﬀerence between the continuous diﬀerential operators and their discrete equivalents. This diﬀerence is particularly large close to the cutoﬀ scale. For DNS this is not so disturbing, but with LES we will later see that it is precisely these scales which have a substantial inﬂuence on the modelled SGS contribution. Hence, in LES, the discretization scheme and the SGS model have to be viewed together. Indeed, some schemes such as low-order upwind discretizations generate a considerable amount of numerical dissipation as discussed in Section 5.2. Therefore certain authors perform LES without any explicit SGS model (Tamura, Ohta and Kuwahara 1990, Meinke et al. 1998). The grid is reﬁned as much as possible to decrease the importance of the SGS terms, and the energy drain is in one way or another accomplished by the numerical scheme. Although yielding valuable results in some cases, this kind of modelling can barely be evaluated or controlled. Hence, in most LES, central or spectral schemes are used and the SGS term is represented by an explicit model. We shall now turn to the description of some basic SGS models before giving a summary at the end of this section. 276 Fr¨hlich and Rodi o 3.2 Smagorinsky model The Smagorinsky model, SM, (Smagorinsky 1963) was the ﬁrst SGS model and is still widely used. Like most of the current SGS models, it employs the concept of an eddy viscosity, relating the traceless part of the SGS stresses, a τij , to the strain rate Sij of the resolved velocity ﬁeld: 1 a τij = τij − δij τrk = −2νt S ij . 3 (10) The advantage of (10) is that the resulting equation for ui to be solved looks like (2) with ui instead of ui , Π + 1 δij τkk instead of Π, and ν + νt instead 3 of ν. Hence, it is very easy to incorporate this into an existing solver for the unsteady NSE. The second part of this model involves the determination of the eddy viscosity νt . Dimensional analysis yields νt ∝ l qSGS (11) where l is the length scale of the unresolved motion and qSGS its velocity scale. From the above discussion it is natural to use the ﬁlter size ∆ as the length scale, hence we set l = Cs ∆. Similarly to Prandtl’s mixing length model, the velocity scale is related to the gradients of ui expressed by qSGS = l |S| which yields νt = (Cs ∆)2 |S|. (13) This amounts to assuming local equilibrium between the production of the SGS a 3 kinetic energy, P = −τij S ij and dissipation ε expressed by qSGS /l. Introducing (10) and (12) in P = gives (13). The constant Cs can be determined assuming an inertial-range Kolmogorov spectrum for isotropic turbulence which yields Cs = 0.18. This value has turned out to be too large for most ﬂows so that often Cs = 0.1 or even lower values are employed. Close to walls νt has to be reduced to account for the anisotropy of the turbulence. This is generally accomplished by replacing Cs in (13) with Cs D(y + ). Most often the van Driest damping is used D(y + ) = 1 − e−y + /A+ |S| = 2S ij S ij (12) , A+ = 25, (14) which is known from statistical models. However, this yields νt ∝ (y + )2 for small y + while νt should behave like y +3 . The correct behaviour is achieved by the alternative damping function (Piomelli, Moin and Ferziger 1993) D(y + ) = 1 − e−(y + /A+ )3 1/2 . (15) [8] Introduction to large eddy simulation of turbulent ﬂows 277 The main reason for the frequent use of the SM is its simplicity. Its drawbacks are that the parameter Cs has to be calibrated and its optimal value may vary with the type of ﬂow, the Reynolds number, or the discretization scheme. The kind of damping to be applied near a wall is a further point of uncertainty. Also, the SM, like any other model based on (10) with νt ≥ 0, is strictly dissipative and does not allow for backscatter. It is furthermore not appropriate for simulating transition since it yields νt ≥ 0 even in laminar ﬂows. 3.3 Dynamic procedure From the previous section it is apparent that for physical reasons one would prefer to replace the constant value Cs by a value changing in space and time. The dynamic procedure has been developed by Germano et al. (1991) in order to determine such a value from the information provided by the resolved scales, in particular the ones close to the cutoﬀ scale. In fact this procedure can be mod a applied with any model τij (C, ∆, u) for τij or τij containing a parameter C 2 . The basic idea is to employ this model not only on the grid scale, or ﬁlter scale, ∆ but also on a coarser scale ∆ as illustrated in Figure 5. This is the so-called test scale with, e.g., ∆ = 2∆: sub-grid scale stresses (∆-level) : sub-test scale stresses (∆-level) : mod τij = ui uj − ui uj ≈ τij (C, ∆, u) (16) mod Tij = ui uj − ui uj ≈ τij (C, ∆, u). (17) From the known resolved velocities ui the velocities ui can be computed by applying the ﬁlter . . . to ui using an appropriate function G. Similarly, the term Lij = ui uj − ui uj can be evaluated. It is this part of the sub-test stresses Tij which is resolved on the grid ∆ as sketched in Figure 5: The total stresses ui uj in the expression for Tij can be decomposed into the contribution ui uj resolved on the grid ∆ and the remainder τij . Inserting this in (17) gives Tij = Lij + τij (18) known as Germano’s identity. Hence, on one hand Lij can be computed, on the other hand the SGS model yields a model expression when inserting (16),(17) in (18): mod mod Lmod = τij (C, ∆, u) − τij (C, ∆, u). (19) ij Ideally, C would be chosen to yield Lij − Lmod = 0, ij (20) but this is a tensor equation and can only be fulﬁlled in some average sense, minimizing, e.g., the root mean square of the left-hand side as proposed by Lilly (1991). Principally, the consecutive application of G and G to obtain u 2 In this subsection we distinguish between exact and modelled SGS stresses for clarity. 278 resolved scales Fr¨hlich and Rodi o unresolved scales E' Tij log E(ω) ¨ ¨ ¨ ¨¨ ¨ ¨ resolved ¨¨ ¨ ¨ turbulent ¨¨ ¨ ¨ stresses rr ¨¨ ¨ rr r ¨¨ ¨ Lij ¨ ¨¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ E E τij π/∆ π/∆ log ω Figure 5: Illustration of the dynamic modelling idea as discussed in the text. yields an eﬀective ﬁlter of width ∆ = ∆, which generally is even of diﬀerent type to G and G (e.g. when the box ﬁlter is used). This issue is generally neglected in the literature. For that reason and since ∆ = ∆, with the Fourier cutoﬀ ﬁlter presently used for illustration, we write ∆ instead of ∆ in this section. Eﬀectively, it is the ratio ∆/∆ which is required by the dynamic models. We now apply the dynamic procedure to the SM (10),(13) to get Lmod = −2 C ∆2 |S|S ij + 2 C ∆ |S|S ij ij 2 (21) 2 with C = Cs for convenience. Classically, the model is developed by extracting C from the ﬁltered expression in the second term although in fact C will vary in space. The right-hand side of (21) can then be written as −2CMij so that inserting into (20) with the least-squares minimization mentioned above yields C=− 1 Lij Mij . 2 Mij Mij (22) The advantage of (22) or a similar equation is that now the parameter of the SM is no longer required from the user but is determined by the model [8] Introduction to large eddy simulation of turbulent ﬂows 279 itself. In fact, it is automatically reduced close to walls and vanishes for wellresolved laminar ﬂows. Negative values of C are possible and can be viewed as a way of modelling backscatter. The resulting ‘backward diﬀusion’ can however generate numerical instability so that ν +νt ≥ 0 is often imposed. Furthermore, C, determined by (22) as it is, exhibits very large oscillations which generally need to be regularized in some way. Most often Lij and Mij are averaged in spatially homogeneous directions in space before being used in (22). However, this requires the ﬂow to have at least one homogeneous direction. Another way is to relax the value in time according to C n+1 = C + (1 − )C n using C n from the previous time step (Breuer and Rodi 1994). Yet another way is to use the known value C n in the rightmost term of (21) so that it need not be extracted from the test ﬁlter (Piomelli and Liu 1995). This yields smoothing in space without any homogeneous direction required. 3.4 Scale similarity models Scale similarity models (SSM) were created to overcome the drawbacks of eddy viscosity-type models. Filtering the decomposition ui = ui + ui yields the (exact) relation ui = ui − ui . (23) This can be interpreted as equality between the largest contributions of ui and the smallest contributions of ui (see Figure 4). Furthermore, it is computable from ui . Introducing the decomposition of ui into (9) and modelling ui uj ≈ a ui uj and ui uj ≈ ui uj , respectively, yields the model τij = Lm,a with ij Lm = ui uj − ui uj , ij (24) where . . .a indicates the traceless part of a tensor. A model constant is not introduced as this would destroy the Galilean invariance of the expression. For √ a spectral cutoﬀ ﬁlter u is replaced by u with ∆ = 2 ∆ since for this ﬁlter u = u as discussed above. The SSM allows backscatter, i.e. transfer of energy from ﬁne to coarse scales, and does not impose alignment between the SGS stress tensor τij and the strain rate S ij . On the other hand, (24) turns out to be not dissipative enough so that it is generally combined with a Smagorinsky model. Horiuti (1997) subsumes some current SSMs in the model a τij = CL Lm,a + CB LR,a − 2Cν ∆ | S | S ij ij ij 2 (25) with LR = ui uj − ui uj , evaluated using (23). A further step is to combine ij (25) with the dynamic procedure for the determination of the constants: (a) Cν with CL = 1, CB = 0 (Zang, Street and Koseﬀ 1993); (b) CL and Cν with CB = 0 (Salvetti and Banerjee 1995); 280 (c) CB and Cν with CL = 1 (Horiuti 1997). Fr¨hlich and Rodi o Diﬀerent tests in the cited references as well as by Piomelli, Yu and Adrian (1996) show that SSMs, in conjunction with the dynamic procedure, perform quite well for low-order ﬁnite-diﬀerence or ﬁnite-volume methods. Apart from the ability to represent backscatter this may also be due to the fact that no spatial derivatives are involved in the SSM which reduces the impact of numerical discretization errors. 3.5 Further models and comparative discussion Let us sum up a few strategies or concepts which are currently followed in SGS modelling. One, already mentioned in the beginning of this section, is to employ a crude model and to compensate by grid reﬁnement, which decreases the impact of the model. Another strategy is to employ the same approaches as in RANS modelling. The Smagorinsky model, based on an eddy-viscosity and an algebraic mixing-length expression, is the most prominent example. But as with RANS, more elaborate methods can be used to compute the turbulent viscosity, such as a model employing a transport equation for the SGS kinetic energy, kSGS = 1/2τkk , which furnishes a velocity scale, qSGS = kSGS 1/2 , (Schumann 1975, Davidson 1997). Obviously, the ﬁlter width ∆ constitutes an adequate 1/2 reference length so that according to (11) νt = C ∆kSGS is a reasonable model and no second length-scale-determining transport equation is required. Spalart et al. (1997) have developed an approach called Detached Eddy Simulation (DES). They start from a one-equation RANS turbulence model (Spalart and Allmaras 1994) based on a transport equation for νt . In this equation the distance from the wall is introduced as a length scale in the destruction term. Replacing this physical length scale by a resolution-based scale, CD ∆ (where CD is a parameter), turns the model into a SGS model. This method furthermore oﬀers a particular way of wall modelling which is discussed below. Still more complex approaches have been carried over from RANS. Fureby et al. (1997) employ the SGS equivalent of a Reynolds-stress model and obtain satisfactory results in some tests. The cost increase is claimed to be moderate, as solving the pressure equation requires most of the work. A third strategy, applied with SSM and the dynamic procedure, is based on the multiscale nature of turbulence. It could only be developed with scale separation deﬁned independently from the discretization according to (4) since ﬁltering is used as an individual operation. By analyzing experimental ﬂow ﬁelds along these lines Liu, Meneveau and Katz (1994) propose τij = CL Lij (26) with Lij deﬁned in (18). This diﬀers from (24) since two ﬁlters of diﬀerent size are used. [8] Introduction to large eddy simulation of turbulent ﬂows 281 A fourth strategy is to relate SGS models to classical theories of turbulence. An elementary example is the determination of the Smagorinsky constant assuming a Kolmogorov spectrum. This strategy is also pursued when deﬁning a wave-number-dependent eddy viscosity to be employed with a spectral Fourier discretization and using EDQNM theory to determine νt (k) (Chollet and Lesieur 1981). The spectral eddy viscosity model has also been reformulated in physical space for application in complex ﬂows yielding the structure function model (M´tais and Lesieur 1992): e νt = 0.063∆ F2 (∆), F2 (r) = (ui (x + r) − ui (x))2 , (27) with F2 spatially averaged in an appropriate way. Diﬀerent variants have been developed (Lesieur and M´tais 1996). It can be shown that when implemented e in a ﬁnite-diﬀerence context, this yields a Smagorinsky-type model with |S| in (13) replaced by |∂ui /∂xj |. The last strategy we mention concerns the testing of SGS models. Of course, as with other turbulence models, prototype ﬂows can be computed and the results then compared with experimental data or DNS results. This is still the ultimate test to pass. However, another kind of testing particular to LES has been developed, namely the so-called a priori test: a fully resolved velocity ﬁeld from a DNS is used to explicitly compute the terms which have to be modelled in an LES on a coarser grid. The large-scale velocity on that grid is extracted to determine the SGS stresses by means of a SGS model. The diﬀerence between exact and modelled stresses reﬂects the quality of the model. This information should however be taken with some caution as the test involves discretization eﬀects in a substantially diﬀerent way than in the actual LES. Finally, one has to bear in mind that a perfect SGS model is impossible. Assume the exact grid-scale velocity u is known at all points. The perfect SGS model would then amount to inferring from u on the exact instantaneous SGS velocity u to deduce the exact instantaneous SGS stresses at all points. Since, however, inﬁnitely many velocity ﬁelds u are compatible with the same u, even the best SGS model cannot decide which of them is realized in the actual DNS. In fact, the error introduced by missing SGS information propagates in an inverse cascade to larger scales (Lesieur 1997). 4 4.1 Numerical Methods Discretization schemes in space and time With the ﬁltering approach discussed in Section 2, physical modelling and numerical discretization are conceptually independent. Hence any available numerical method can in principle be used to discretize the ﬁltered equations. A minimal requirement for precision and cost-eﬀectiveness is that the discretization scheme is at least of second-order in space and time. Classically, spectral methods were frequently used for LES and are still employed for problems with 282 Fr¨hlich and Rodi o simple geometry. Derivatives are discretized most accurately and ﬁltering and deﬁltering, as discussed below, is naturally applied in this framework. For more complex boundary conditions, ﬁnite-diﬀerence or ﬁnite-volume methods are prefered. Here, one current trend goes to unstructured meshes, another to cartesian grids with special local treatment at the boundary if this has an irregular shape. Some numerical methods favour certain modelling ideas. For example spectral methods allow the use of a spectral eddy viscosity (Chollet and Lesieur 1991) and explicit ﬁltering by means of (5). Others need particular care in certain points. For example if implicit ﬁltering is used together with a type of ﬁnite element that has a diﬀerent number of degrees of freedom for velocity compared to pressure (which is classically the case for stability reasons), this results in a diﬀerent amount of ﬁltering for these quantities and can deteriorate the result (Rollet-Miet, Laurence and Ferziger 1999). Discretizations in space can be selected according to relevant properties such as the ability to treat complex geometries, cost per grid point, etc. If possible, however, equispaced grids are used since the inﬂuence of grid-inhomogeneity and grid-anisotropy on SGS modelling is not yet fully mastered. A comparative study of a structured and an unstructured method for the same problem was undertaken by Fr¨hlich et al. (1998) where, for the particular case considered, adaptivity of o the former method was roughly compensated by higher cost per node. For more complex geometries, unstructured methods are certainly favourable. Concerning the time scheme we already noted that temporal resolution has to be compatible with resolution in space so that C = u ∆t/∆x ≤ O(1). Since this type of limit is equivalent to the stability limit of explicit methods, in many cases the latter are typically used for LES. Adams–Bashforth, RungeKutta or leap-frog schemes are the most popular ones. If the diﬀusion limit is stricter in a computation, semi-implicit time stepping can be more eﬃcient. 4.2 Analysis of numerical schemes for LES The essential feature that distinguishes LES from DNS is that the smallest resolved grid-scale components, which are just a little larger than the cutoﬀ scale, typically carry more energy. Hence, without explicit ﬁltering which employs a ﬁlter coarser than the mesh size, the smallest resolved scales are by deﬁnition substantially aﬀected by the employed numerical scheme. These scales however inﬂuence most strongly the contribution determined by the SGS model. In fact a complex discrete model for the SGS eﬀect on the resolved ﬂow is created which results from physical as well as numerical modelling. Consequently, the order of a method is not necessarily an appropriate notion in the context of LES. It rather has to be supplemented with a reﬁned analysis like the modiﬁed wavenumber concept as, e.g., discussed by Ferziger (1996). Let us illustrate these statements by means of Figure 6. Refering to equation (5), [8] Introduction to large eddy simulation of turbulent ﬂows the exact spatial derivative of u formulated in Fourier space is ∂u (ω) = iω G(ω) u(ω). ∂x 283 (28) The numerical evaluation of ∂u/∂x by a ﬁnite-diﬀerence formula corresponds to replacing the factor ω in (28) by a modiﬁed wavenumber ωeﬀ (ω) which depends on the particular scheme employed. Derivation and formulas are given, e.g., by Ferziger and Peric (1996). For symmetric schemes this is a real quantity, otherwise it is complex. Starting from ωeﬀ (0) = 0, |ωeﬀ | increases and then drops down to zero again. The point ω∆ where this takes place is determined by the numerical grid employed; it is the highest frequency resolved by the grid. In a DNS this point would be pushed as far as possible to the right (Figure 6a). The order of a discretization scheme can be reformulated in terms of the exponent p in limω→0 |ω − ωeﬀ (ω)| ∝ ω p . Obviously, the information about the order of a scheme is suﬃcient only if ω∆ → ∞ so that the solution to be computed is located entirely at ω/ω∆ ≈ 0. If however ω∆ approaches the relevant scales of the solution to be discretized, the behaviour of the whole curve ωeﬀ is decisive, not only the limit ω → 0. It is rather obvious that computing a derivative with ωeﬀ = ω amounts to replacing G(ω) in (28) by Geﬀ = ωeﬀ /ω G(ω). Hence the ﬁnite-diﬀerence formula results in additional ﬁltering applied to the derivative of u (Salvetti and Beux 1998). Figure 6b furthermore shows that the decay of the error obtained with grid reﬁnement in an LES depends on the behaviour of the solution itself, e.g. on the decay rate of its spectrum. This information is indispensable when aiming to assess the numerical error in an LES and to compare it with the size of the SGS term (Ghosal 1996). So far we have discussed the discrete derivative operator which is a building block when discretizing the whole system of equations. Qualitatively, the real part of ωeﬀ /ω in the convection term introduces spurious or numerical dispersion while its imaginary part results in additional numerical dissipation. Analyzing the fully discrete system is much more complicated but can be achieved when disregarding boundary conditions etc. by the modiﬁed equation approach (Hirt 1968). This has been applied by Werner (1991) to a staggered ﬁnite-volume discretization with Adams–Bashforth time scheme, central diﬀerences for the viscous term, and the QUICK convection scheme. Recall that the QUICK scheme (Leonard 1979) is a third-order upwind interpolation scheme for the ﬂux over the surface of a control volume. Werner observed that this combination results in a spurious fourth-order dissipative term proportional to the cell Reynolds number Recell = u∆x/ν. The same analysis for a leap-frog time scheme with second-order central diﬀerencing yields a fourth-order error term which is independent of Recell . Such an analysis nicely shows that the upwind scheme produces excessive damping for large Recell . This, however, is precisely the working range of LES with typically Recell = O(10000), even if ν is replaced by ν + νt . To demonstrate the eﬀect in a real LES, Werner 284 Fr¨hlich and Rodi o (a) (b) ω∆ ω∆ log ω Figure 6: Discetization of derivatives (Sketch) in case of (a) DNS and (b) LES; —– spectrum of u or u, - - - ωeﬀ , · · · ωeﬀ /ω, the additional ﬁlter when numerically computing a derivative as discussed in the text, here for a secondorder central formula. The vertical axis has an arbitrary scale. In the LES case we consider u to be obtained by an ideal low pass ﬁlter for illustration. Observe that due to the logarithmic frequency scale 88% of the discretization points correspond to the range between the maximum of ωeﬀ and ω∆ for a second-order scheme in a three-dimensional computation. also computed a plane channel with Reτ = 1954 employing the Smagorinsky model. With a modiﬁed leap-frog scheme νt /ν = 1.2, C = 0.14, Recell = 1350 on the centerline. Analysis yields |νnum /ν| ≈ 0.2. With the QUICK upwind scheme the corresponding numbers are νt /ν = 0.9 and |νnum /ν| ≈ 180. Hence the numerical dissipation introduced by the upwinding exceeds the one by the SGS model by two orders of magnitude. Similar, though mostly less detailed, experiences have been reported in several papers by comparing the solution obtained with diﬀerent schemes. The QUICK scheme and lower-order upwind schemes gave worse results than a second-order central scheme in LES of a circular cylinder (Breuer 1998). This was, though with decreasing impact, also observed for ﬁfth- and seventh-order upwinding (Beaudan and Moin 1994). Further studies of the numerical error in LES were performed by Vreman, Geurts and Kuerten (1994a), Kravchenko and Moin (1997) and others. Hence, on the one hand upwind schemes can spoil the result by excessive damping. On the other hand, some researchers omit explicit SGS modelling and let the numerical dissipation of the employed scheme remove the energy. The MILES [8] Introduction to large eddy simulation of turbulent ﬂows 285 approach (Boris et al. 1992), e.g., falls into this class. Comte and Lesieur (1998) however found such schemes to be inferior to explicit SGS modelling. In contrast to the numerical dissipation, the dispersion of a scheme is of lower importance as it has no eﬀect on the energy drain which has been claimed to be the principal task of the SGS modelling. Dispersion, however, is related to the generation of spurious wiggles which in some LES of bluﬀ bodies pose problems (Rodi et al. 1997). 4.3 Further developments In order to improve the current status, attempts are being made to separate more clearly the diﬀerent ingredients in an LES. The aim is to study and improve each of them in a separate and controlled way. One of the directions pursued is explicit ﬁltering as mentioned above. A similar approach has been used by Vreman, Geurts and Kuerten (1994b, 1997) who use a value of ∆ larger than the mesh size of the grid, e.g. by a factor of two in the SGS model, which leads to increased SGS dissipation eﬀectively damping the solution in a similar way as explicit ﬁltering. A second direction is the use and improvement of higher-order energyconserving discretization schemes (Morinishi et al. 1998). They ensure that the total dissipation is entirely controlled by the SGS model and not by the discretization. Bearing in mind the uncertainty in SGS modelling, when for example determining the parameter C in the DSM, the practical importance of an energy-conserving scheme is presently not clear. A third direction is to use higher-order methods as they narrow the range of scales which are inﬂuenced by the discretization of the ﬁltered equations. Furthermore, ﬁlters have recently been devised that commute with discrete derivatives (Vasilyev, Lund and Moin 1998). This ideally ensures that apart from the SGS term τij in equation (8) no commutator term arises which would require modelling. Finally, ‘deﬁltering’ has been applied to invert the attenuation of resolved scales by the implicit ﬁltering related to the discretization (Stolz, Adams and Kleiser 1999). It uses an operation like multiplication of u(ω) with G(ω)−1 to devise an estimate u∗ for the true velocity u based on the resolved velocity u, as may be illustrated with equation (5) and Figure 4. This procedure requires higher-order methods and principal adjustments such as restriction to certain scales since inverse ﬁltering is ill-conditioned (imagine obtaining u from u in Figure 3a or 3b by backward diﬀusion). First results look promising. 5 Boundary conditions We have mentioned already the mathematical problems that arise when boundary conditions for ﬁltered quantities have to be deﬁned. From a physical point 286 Fr¨hlich and Rodi o of view, the ﬂow near a solid wall exhibits substantially diﬀerent structures than away from it. In this region the ‘large scales’ – in the sense that they signiﬁcantly determine the overall properties – are of the order of the boundary layer thickness and hence typically much smaller than in the core of the ﬂow, in particular if the Reynolds number is large. In addition, the small scales in this area exhibit substantial anisotropy, and energy transfer mechanisms are diﬀerent compared with the core ﬂow (H¨rtel 1996, Piomelli et al. 1996). This a makes subgrid-scale modelling in the vicinity of walls a diﬃcult task. 5.1 Resolution of the near-wall region The most natural boundary condition at a wall is the no-slip condition. It requires however that the energy-carrying motion is resolved down to the wall. In an attached boundary layer this motion is mainly constituted by the wellknown streaks of spanwise distance λz ≈ 100, resolution of which requires y + < 2, ∆x+ = 50–150, ∆z + = 15–40. (Piomelli and Chasnov 1996). The resulting simulation is in fact a hybrid between an almost-DNS near the wall and an LES in the main part of the ﬂow. If locally a ﬁne grid is required an eﬃcient discretization calls for a block-structured or an unstructured method. Care has to be taken however since, for example, a low-order FV method that locally splits each cell into a number of smaller ones introduces a sudden decrease in the size of the implicit ﬁlter by a factor of two in at least one direction. This may lead to problems with the SGS modelling. Kravchenko, Moin and Moser (1997) have developed a discretization that employs overlapping Bsplines and hence results in a smoother transition of the eﬀective resolution. It was successfully applied to channel ﬂow up to Re = 109, 410. Another possibility is an unstructured ﬁnite element method as used by Rollet-Miet et al. (1999). Due to particular discretization issues discussed in this reference, so far only a very few codes of this kind are capable of LES. With an unstructured code, one is still left with the task of generating a grid that fulﬁlls the needs, in particular with respect to its inﬂuence on SGS modelling. Regardless which method is used to discretize and compute the nearwall region, the wall-resolving approach can result in substantial complexity and computational eﬀort. Spalart et al. (1997) stressed that reﬁnement needs to be performed not only in the wall-normal but also in the streamwise and spanwise directions and estimated that O(1011 ) grid points would be necessary for a wing at Re = 6.5×106 while ‘108 is impressive today’. Also, resolving the ﬂow in space is worthless if it is not also resolved properly in time, hence the CFL number has to be of order unity, a fact that even further increases the computational burden. To conclude: although a wall-resolving LES is appropriate for lower Re and transitional ﬂows, a diﬀerent approach is needed for higher Re, particularly when the interest of a simulation focuses on features away from the wall. [8] Introduction to large eddy simulation of turbulent ﬂows 287 5.2 Wall functions When for higher Re a wall-resolving LES is not possible, the way out is to use a near-wall model approximating the overall dynamic eﬀects of the streaks on the larger outer scales which are resolvable by the LES. The most commonly used models are wall functions for bridging a region very close to the wall, often the viscous sublayer. Such wall functions are classically used in RANS methods, where they take the form: τω = W ( u1 , y1 ). (29) Here, y1 is the distance of the ﬁrst grid point from the wall, τω the average wall shear stress, u1 the average tangential velocity at y1 , and W a functional dependence. Note that any relation u + = f (y + ) can be converted to the form (29). This can be the log-law, the 1/7-power law, a linear viscous law or even a numerical ﬁt to DNS data. An appropriate blending is generally used so that + W is deﬁned from y1 ≈ 0 to y + of several hundreds. Even if many physical properties such as the low-order moments, and hence W , are well-known for a certain ﬂow – this is the case for the developed ﬂow in a plane channel, for example – it is a delicate task to introduce this knowledge in the context of LES. The available information is of a statistical nature whereas the ﬁltered velocity is an instantaneous, ﬂuctuating quantity. On the other hand it has been demonstrated that the inner and outer regions of a turbulent boundary layer are only loosely coupled (Brooke and Hanratty (1993)) so that an artiﬁcal boundary condition bridging the inner layer has a chance of being successful (Piomelli 1998). One of these wall-function methods (Schumann 1975) employs the mean velocity u(y1 ) , which is successively computed during the LES, to determine the average wall shear stress τω from (29) with W being the logarithmic law of the wall. The same proportionality as between u(y1 ) and τω is then assumed to hold also between the instantaneous quantities u(y1 ) and τω ; in particular they are supposed to be in phase. This yields the instantaneous wall stress as τω = τω u(y1 ), u(y1 ) (30) which is used as a boundary condition in the LES. Werner and Wengle (1993) employed the 1/7-power law instead of the log-law to avoid an iterative evaluation of W . Furthermore, they replaced (29) by τω = W (u1 , y1 ) so that the average velocity is no longer needed. Other combinations and variants are possible as well. A generalization of the approach for the subcritical ﬂow around a cylinder is described by Fr¨hlich (2001). In the technically relevant case of a o rough surface, using the wall-function approach is unavoidable since resolving the ﬂow around each roughness element is impossible. The roughness eﬀect is brought in by the roughness parameter in the log-law (Gr¨tzbach 1977). o 288 Fr¨hlich and Rodi o Wall function boundary conditions work reasonably well in simple ﬂows and save substantial CPU time due to the reduced resolution requirements. They have also been applied to some complex ﬂows around obstacles (Rodi et al. 1997) which, however, were found not to be very sensitive to variations in the boundary conditions. 5.3 Other approaches Wall functions establish a relation between the local wall shear stress and the velocity at the wall-adjacent grid point. This can be generalized to the case where information on a line or a whole plane at some distance parallel to the wall is used to generate the wall-shear stress at a certain point. Such information can be introduced as a boundary condition in unsteady turbulent boundary layer equations which are solved along the wall within the wall cell using an embedded grid (Balaras, Benocci and Piomelli 1996) (cf. Figure 7d). In these equations turbulence is modelled with an eddy viscosity depending on the wall distance. Similar work has been done by Cabot (1995,1996) where diﬀerent models of this type were devised and applied to ﬂow in a plane channel and over a backward facing step. Although yielding better results in some computations than wall functions, these methods have not found wide application yet due to their implementational complexity. Another approach that can be introduced more easily in an LES is based on using the no-slip condition which in turn requires reﬁnement in the wallnormal direction. Parallel to the wall, however, the step size of the outer region is maintained leading to substantial savings. The idea then is to replace the unresolved near-wall structures by elements from RANS simulations. Schumann (1975) decomposed τij into an isotropic part for which the Smagorinsky model is used and an anisotropic part resulting from the mean ﬂow gradient. The latter part is modelled with an eddy viscosity νt,an = min(c∆x,z , κd)d u /dy This is a RANS-like model in which close to the wall the size of the grid ∆x,z is replaced by the distance from the wall d as a length scale. A similar switch is used in the DES approach of Spalart et al. (1997) mentioned above so that close to a wall the original RANS model is employed (see Figure 7b). DES, although conceived for diﬀerent applications, has been tested for channel ﬂow by Nikitin et al. (2000). The authors observe a spurious buﬀer layer reﬂecting diﬃculties in connecting the quasi-steady RANS layer close to the wall to the outer unsteady computation. Further adjustments need to be introduced to apply DES in such cases. Bagett (1998) discussed the issue of blending RANS with LES turbulence models and points out the requirements for adequate spanwise resolution in the near-wall region in order to capture the energetically dominant features. If these are not captured, unphysical structures are generated which degrade the result. [8] Introduction to large eddy simulation of turbulent ﬂows 289 Figure 7: Schematical pictures for the diﬀerent approaches close to solid walls: (a) resolving the near–wall structure, (b) blending with a RANS model, (c) application of a wall function, (d) determination of wall stress by boundary layer equation solved along the wall on an imbedded grid. 5.4 Inﬂow and outﬂow conditions After discussing the boundary conditions at solid walls we brieﬂy mention the conditions at artiﬁcial boundaries, an issue shared with DNS. Turbulent outﬂow boundaries are relatively uncritical. Here, damping zones or convective conditions are generally applied which allow vortices to leave the computational domain with only small perturbations of the ﬂow in its interior. A convective condition for a quantity φ reads ∂φ ∂φ + Uconv =0 ∂t ∂n (31) applied on the outlet boundary with n the outward normal coordinate and Uconv an appropriate convection velocity such as the bulk velocity. The diﬃculty posed by turbulent inﬂow conditions stems from the fact that LES computes a substantial part of the spectrum and hence requires speciﬁcation of the inﬂow conditions in all this spectral range, not just the mean ﬂow. The need for this information can be avoided by imposing streamwise periodicity with a suﬃcient periodic length, but this is inapplicable in many practical ﬂows. Imposing the mean ﬂow plus random perturbations is generally not successful since these perturbations are unphysical so that a large upstream distance must be computed to produce the correct turbulence statistics. With 290 Fr¨hlich and Rodi o more sophisticated perturbations the distance can be shortened. This is a subject of current research. If feasible the best solution is to impose some fully developed ﬂow at the inlet. A separate companion LES, e.g. with streamwise periodicity, can then be performed to generate velocity signals at the grid points in the inﬂow plane of the main LES. An example is the ﬂow around a single cube investigated in Rodi et al. (1997). 5.5 Sample computations In order to illustrate the above discussion, we present results from a standard test case for LES calculations, namely fully developed plane channel ﬂow. For this ﬂow between two inﬁnitely extended plates, periodic conditions can be imposed in the streamwise direction x and the spanwise direction z, with typical domain sizes of Lx = 2π and Lz = π, respectively. Reference quantities are the channel half-width δ and the bulk velocity Ub . DNS of this ﬂow has been performed for low and medium Reynolds number of which currently the highest is Reb = Ub δ/ν = 10935 (Moser, Kim and Mansour 1999) employing a high-precision spectral method with 384 × 257 × 384 points. This Reynolds number has been used in the computations below. The results have been obtained with the structured collocated ﬁnite-volume code LESOCC developed by Breuer and Rodi (1994). The dynamic Smagorinsky model was used with test-ﬁltering and averaging in planes parallel to the walls. The bulk Reynolds number was ﬁxed and an external pressure gradient adjusted so as to yield the desired ﬂow rate. Figure 8 shows a computation where resolving the near-wall ﬂow has been attempted, i.e. no wall-function was used. The number of points in the y direction is 65 and a stretching of 11% has been applied to cluster them close to the walls. The ﬁgure shows the average streamwise velocity and the rmsﬂuctuations. The computed shear stress τw yields Reτ = 504 which is much below the Reτ = 590 in the DNS. The value of uτ = τw /ρ determines the scaling of the axes, and in particular the u + = f (y + )-curve is quite sensitive to it. If the v- and w-ﬂuctuations are plotted in outer scaling, i.e. not by uτ , they are even further below the DNS curves. The observed failure occurs because resolving the ﬂow near the wall requires adequate discretization in all three directions, not just normal to the wall. Here in particular the spanwise resolution is too coarse. In Figure 9 the wall-normal resolution has been improved using 159 points in y with clustering in the buﬀer layer accompanied by a substantially better resolution in spanwise direction. The computed shear stress yields Reτ = 598.5 and the Reynolds stresses compare quite satisfactorily with the DNS data. It is obvious that with a structured discretization the grid in the interior of the channel is ﬁner than it really needs to be, due to the requirements near the wall. To avoid this, a method with local reﬁnement is beneﬁcial as discussed above. [8] Introduction to large eddy simulation of turbulent ﬂows 291 Figure 8: Computation without wall function using ∆x+ = 62, ∆z + = + + + 30, ∆y1 = 1.8. Left: u+ , right: u+ , vrms , wrms , − uv + . Continuous lines are rms DNS data, symbols LES. Figure 10 presents a computation using the wall function of Schumann (1975). In the wall-normal direction 39 equidistant volumes are used. In this case the ﬁrst cell centre is still located in the buﬀer layer, but the viscous sublayer is well bridged. The computed wall shear stress gives Reτ = 589.5 which compares very well with Reτ = 590 in the DNS. The Reynolds stresses are well-predicted, the v-ﬂuctuations being somewhat too small. With this approach it is of course not possible to reproduce the peak in the u-ﬂuctuations 292 Fr¨hlich and Rodi o Figure 9: Computation without wall function using ∆x+ = 50, ∆z + = + 16, ∆y1 = 1. Labels as in Figure 8. close to the wall. For an LES using a wall function the present Reynolds number is relatively low. With higher Re the ﬁrst point usually lies beyond the buﬀer layer at y + ≈ 100. In Figure 11 we ﬁnally show cuts of the instantaneous u- and w-velocity of the third case. Straight lines have been inserted connecting the data points. The angles they form show that, as discussed in Section 4, on the grid level the discrete solution is not smooth, i.e. the velocity scales close to the cutoﬀ are [8] Introduction to large eddy simulation of turbulent ﬂows 293 + Figure 10: Wall function computation with ∆x+ = 62, ∆z + = 30, ∆y1 = 31. Labels as in Figure 8. hardly resolved. Hence, any gradient computed from these values by, e.g., a second-order scheme can only be a crude approximation to the ‘true’ gradient. Recall that gradients enter in the contribution of the SGS model. This ﬁgure illustrates the close interplay between the numerical discretization and the subgrid-scale modelling. The amount of SGS dissipation in eddyviscosity models can be monitored by the ratio νt /ν. It varies locally, and in the above computations attains values up to 7 in the last case, which shows the dominance of the SGS dissipation with respect to the resolved dissipation. 294 Fr¨hlich and Rodi o Figure 11: Velocities u (upper curves) and w (lower curves) at three arbitrary cuts x = const., z = const. in the computation of Figure 10. Thin lines connect the instantaneous values, thick lines show the corresponding averages. Further applications of LES, in particular to bluﬀ body ﬂows, are discussed in other chapters of this volume. 6 Concluding Remarks We have described the concept of the Large-Eddy Simulation technique which in fact is extremely simple and makes it appealing. It turns out, however, that several issues are not simple for numerical or physical reasons. We have aimed at making the reader aware of these points and at clarifying related concepts. In practice, LES is characterized by a large number of decisions concerning the numerical and physical modelling which have to be taken and which all inﬂuence the ﬁnal result. Thorough testing is still a major occupation of the community, and this will presumably not change in the near future. On the other hand, LES has potential on several levels. The ﬁrst is the determination of statistical quantities, such as the average ﬂow ﬁeld, with a higher accuracy than obtained by statistical models. This is based on interchanging the order ‘ﬁrst averaging – then computing’ (RANS) to ‘ﬁrst computing – then averaging’ (LES). To pay oﬀ, the drastic increase in cost has to be justiﬁed by an improved quality of the results. The next level is the determination of statistical quantities which are inaccessible to RANS such as two-point correlations. The third level is to use the instantaneous information on the structure of the ﬂow in order to improve the understanding of vortex dynamics, transition phenomena, etc. or to determine dynamic loading. Finally, this [8] Introduction to large eddy simulation of turbulent ﬂows 295 information can be coupled to other physical processes either within the ﬂow ﬁeld, such as the generation of sound, the transport of scalars (temperature, sediment, . . . ), chemical reactions, etc., or to external processes such as the dynamical response of a solid structure. 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Then, some limitations of single-point closures are illustrated by simple examples, and we discriminate ﬂows dominated by production eﬀects and ﬂows dominated by wave eﬀects. A classical spectral description is introduced for the ﬂuctuating ﬂow and its multi-point correlations. Applications to stably-stratiﬁed and rotating turbulence are discussed. Linear eﬀects captured by RDT include dispersivity of gravity waves, whereas the irreversible collapse of vertical motion and subsequent layering of the velocity ﬁeld is only captured by nonlinear theories or high resolution DNS/LES computations. Applications to weak turbulence in compressible ﬂows are touched upon at the very end. 1 Introduction Two-point statistical closures (Direct Interaction Approximation, DIA, Eddy Damped Quasi Normal Markovian, EDQNM, and the Test Field Model, TFM) were initially mainly developed for the special case of homogeneous, isotropic turbulence during the ground-breaking studies of the 60s and 70s (see e.g. Kraichnan 1959, Orszag 1970, Monim and Yaglom 1975, among many others), but have since then been extended to some anisotropic and even inhomogeneous ﬂows, areas in which work continues today. Although such models are aimed at strongly nonlinear turbulence, their mathematical structure is closely related to that of linear or weakly nonlinear theories (see Cambon and Scott 1999, and references therein). For example, the theory of weak turbulence (see Benney and Saﬀman 1966, and [26]), which has recently seen considerable interest in the geophysical context, presents strong similarities with two-point closures, even if very few studies illustrating the connections between the two approaches have appeared to date. Thus, the 299 300 Cambon case of anisotropic, incompressible, homogeneous turbulence subject to diﬀerent anisotropizing inﬂuences, such as rotation or stratiﬁcation, and of weakly compressible turbulence, even in the isotropic case, present challenges which are currently being addressed using both two-point techniques and asymptotic theories of weak turbulence. It is important to extend the domain of applicability of two-point closures by incorporating results from linear theory (RDT, using methods from stability theory) and weakly nonlinear analyses, results which include at least some aspects of the real dynamics of the ﬂow. Other methods, such as renormalisation or homogenisation, may also help in developing two-point closures. Two-point models are intrinsically more realistic than one-point models, describing more of the physics of turbulence, such as the continuum of diﬀerent scales, and providing a correct treatment of pressure ﬂuctuations (via the formalism of projection onto solenoidal modes in the incompressible case). The fact that two-point closures can be used to describe diﬀerent turbulence scales has proved, and will no doubt continue to prove, useful in the construction of subgrid models in LES, but two-point modelling is by no means limited to this single application, important though it may be. The chapter is organised as follows. The general problem of closure is introduced in section 2, with emphasis on both nonlocal and nonlinear aspects. Section 3 gives a brief background on typical single-point and two-point closure approaches, the latter being developed in anisotropic turbulence, with rapid distortion theory used as a building block (section 4). Throughout, our aim is to illustrate the importance of linear mechanisms, mostly in the form of mean velocity gradients, which render the turbulence anisotropic. For this reason, we restrict attention to models capable of handling anisotropy. Typical eﬀects, such as stable stratiﬁcation and rotation are presented in section 5. Finally, some eﬀects of compressibility are considered in section 6 together with concluding comments. 2 Nonlinearity and non-locality Two major problems of closure in the statistical approach to turbulence are nonlinearity and nonlocality. In this section we introduce the issues. The velocity and pressure ﬁelds are ﬁrst split into mean and ﬂuctuating components and equations for their time evolution are derived from the basic equations of motion of the ﬂuid. Assuming incompressibility, as we shall do in this chapter unless explicitly stated otherwise, this gives the mean ﬂow equations ∂Ui ∂Ui ∂p ∂ 2 Ui =− +ν − + Uj ∂t ∂xj ∂xi ∂xj ∂xj ∂Ui =0 ∂xi ∂ui uj ∂xj Reynolds stress term (2.1) (2.2) [9] Introduction to two-point closures and the equations for the ﬂuctuating component ∂u ∂Ui ∂ui ∂ + (u u − ui uj ) = + Uj i + uj ∂t ∂xj ∂xj ∂xj i j Nonlinear term 301 − ∂p ∂xi + ν ∂ 2 ui ∂xj ∂xj (2.3) Pressure term Viscous term and ∂ui =0 ∂xi (2.4) Here, Ui and p are the mean velocity and ‘pressure’ (pressure divided by density), while ui and p are the corresponding ﬂuctuating quantities, usually interpreted as representing turbulence. At various points, we will describe related work in the area of hydrodynamic stability. In so doing, it is recognised that equations (2.3) and (2.4) for the ﬂuctuating ﬂow are essentially the same as those for a perturbation ui , about a basic ﬂow, Ui , with an additional forcing term, ∂ui uj /∂xj , in the inhomogeneous case. Although the aims of stability theory (to characterise growth of the perturbation) and of the theory of turbulence (to determine the statistics of ui ) are diﬀerent, we believe it is nonetheless valuable to draw parallels between the two ﬁelds of study. It is our hope that in so doing we will encourage workers in both areas to become more conversant with each others work. Equation (2.3) is now used to derive equations for the time evolution of velocity moments, i.e. averages of products of ui with itself at one or more points in space. Setting up the equations for the nth-order velocity moments at n points, one discovers that there are two main diﬃculties. Firstly, the term in (2.3) which is nonlinear in the ﬂuctuations leads to the appearance of (n + 1)th-order moments in the evolution equation at nth-order. Secondly, the pressure term introduces pressure-velocity moments. The pressure ﬁeld is intimately connected with the incompressibility condition. Indeed, taking the divergence of (2.3) leads to a Poisson equation ∇2 p = − ∂2 (u Uj + Ui uj + ui uj − ui uj ) ∂xj ∂xj i (2.5) for the ‘pressure’ ﬂuctuations. Solution of this equation by Green’s functions expresses p at any point in space in terms of an integral of the velocity ﬁeld over the entire volume of the ﬂow, together with integrals over the boundaries, the details of whose expression in terms of velocity do not concern us here. Thus, the pressure at a given point is nonlocally determined by the velocity ﬁeld at all points of the ﬂow, resulting in the equations for the velocity moments being integro-diﬀerential when the pressure-velocity moments are expressed in terms of velocity alone. It should be observed that nonlocality is not speciﬁc to the use of statistical methods, but is intrinsic to the physics of incompressible ﬂuids, for which the pressure ﬁeld responds instantaneously 302 Cambon and nonlocally to changes in the ﬂow to maintain incompressibility. The source term in the Poisson equation (2.5) consists of parts which are linear and nonlinear in the velocity ﬂuctuation, feeding through into corresponding components of p and hence of the pressure-velocity terms in the evolution equation for the n-point velocity moments. Both the nonlinear pressure component and the nonlinear term appearing directly in (2.3) contribute to the closure problem, namely that the equation for the nth-order velocity moments involves (n + 1)th-order moments. In consequence, no ﬁnite subset of the inﬁnite hierarchy of integro-diﬀerential equations describing the velocity moments at all orders is complete, reﬂecting the fundamental diﬃculty of the turbulence problem, viewed through the classical statistical description in terms of moments. The origin of the closure problem is nonlinearity of the Navier–Stokes equations, which feeds through into the moment equations, both directly and via the nonlinear part of the pressure ﬂuctuations. Nonlocality, of itself, does not lead to problems, although the technical diﬃculties associated with integro-diﬀerential, rather than diﬀerential equations, are nontrivial. The non-local problem of closure is removed from consideration only in models for multi-point statistical correlations, e.g. double correlations at two points or triple correlations at three points, so that in such models the problem of closure is determined by nonlinearity alone. On the other hand, the knowledge of a probability density function (PDF) for the velocity is equivalent to the knowledge of all the statistical moments up to any order. Hence the problem of the open hierarchy of the momentequations, mentioned above, is avoided in a PDF approach. Accordingly, the problem of closure induced by the nonlinearity is removed from consideration using a PDF approach, but the non-local problem of closure remains, so that the equations for a local velocity PDF involve a two-point velocity PDF, and equations for a n-point velocity PDF involve a (n + 1)-point velocity PDF (Lundgren 1967). Thus, an open hierarchy of equations is recovered using a PDF approach but with respect to a multi-point spatial description! In order to present all the consequences of the above discussion, a synoptic scheme using a triangle is shown in ﬁgure 1 and discussed as follows. The vertical axis bears the ordering of the statistical moments, from 1 (the mean velocity), 2 (second-order moments), until an arbitrary high order. Each vertical order n corresponds to a number of diﬀerent points for a possible multi-point description of the n-order moment under consideration along the horizontal axis, from 1 (single-point), 2 (two-point), until n. In other words, the vertical axis can display the open hierarchy due to nonlinearity, whereas the horizontal one deals with the non-locality. Each point of the triangle can characterize a level of description, for instance the point [3, 2] represents triple correlations at two points (those that drive the spectral energy transfer and the energy cascade). In addition, the problem of closure can be stated by looking [9] Introduction to two-point closures 303 Figure 1: Synoptic scheme for the general closure problem using the statistical description. The notation [m, n] refers to an n-point representation of an mth statistical moment. at the adjacent points (if any) just above and just to the left. For instance, the equation that governs the Reynolds-stress tensor [2, 1] needs extra information (not given by [2, 1] itself, hence the closure problem) on second-order two-point terms [2, 2] (involved in the ‘rapid’ pressure-strain rate term and the dissipation term), triple-order one- [3, 1] and two-point [3, 2] terms (involved in the ‘slow’ pressure-strain rate and diﬀusion terms). Of course, the Reynolds stress tensor [2, 1] is directly derived from second-order correlations at two points [2, 2], illustrating a simple rule of ‘concentration of the information’ from right to left. Recall that the non-local problem of closure is removed from consideration, leaving only the hierarchy due to nonlinearity, when looking only at [n, n] correlations (located on the hypotenuse of the triangle in ﬁgure 1): the equation that governs [2, 2] needs only extra information on [3, 2]; the equation that governs [3, 3] needs only extra-information on [4, 3]; the latter two examples, which are directly involved in classical two-point closures, will be discussed in the next section. The arrow from [n + 1, n] to [n, n] gives an obvious generalization, and illustrates the open hierarchy of equations due to the nonlinearity only. Regarding the PDF approach, we are concerned with the upper horizontal side of the triangle. It seems to be consistent to relate to the point [∞, 1] a description in terms of a local velocity PDF — knowledge of which is equivalent 304 Cambon to the knowledge of all the one-point moments (the complete vertical line below). Accordingly, the arrow from [∞, 2] to [∞, 1] shows the need for extrainformation on the two-point PDF in the equations that govern the local PDF. In the same way, the arrow from [∞, n + 1] to [∞, n] shows the link of n- to (n + 1)-point PDF (Lundgren 1967), and illustrates the open hierarchy of equations due to non-locality only. The last limit concerns the ultimate point [∞, ∞]. It is consistent to consider that the limit of a joint-PDF of velocity values at an inﬁnite number of points is equivalent to the functional PDF description of Hopf (1952). In this case we reach the top left point of the triangle and there is no need for any extra information; accordingly the Hopf equation is closed, and it is possible to derive from it any multi-point PDF or statistical moment. It is interesting to point out that the bottom right point [1, 1] gives the most crude information about the velocity ﬁeld – its mean value – whereas the opposite point [∞, ∞] gives the most sophisticated. The main problem which concerns engineering, when solving Reynolds-averaged Navier–Stokes equations, is expressing the ﬂux of the Reynolds stress tensor, and this is done in the simplest way by a direct relationship (‘zero-equation’) between the latter term and the mean velocity ﬁeld through a ‘turbulent’ viscosity (obtained from a mixing-length approximation). This is expressed by an arrow from [2, 1] to [1, 1] in our synoptic scheme. As a last general comment, our synoptic scheme clearly shows that the problem of closure, which reﬂects a loss of information at a given level of statistical description, can be removed from consideration if additional degrees of freedom are introduced in order to enlarge the conﬁguration-space. For instance, to introduce as a new dependent variable the vector which joins the two points in a two-point second-order description allows removal of the problem of closure due to nonlocality, which is present using a single-point second-order description. The introduction, as a new dependent variable, of the test-value Υi of the random velocity ﬁeld ui in a PDF approach P (Υi , x, t) = δ(ui (x, t) − Υi ) allows removal of the problem of closure due to nonlinearity, which is present in any description in terms of statistical moments. Finally, any problem of closure is removed using the Hopf equation but the price to pay is an incredibly complicated conﬁguration-space! The probabilistic description, which is of practical interest regarding a concentration scalar ﬁeld rather than a velocity ﬁeld, is extensively addressed elsewhere in this volume in the context of combustion modelling, and we will no longer consider it here. [9] Introduction to two-point closures 305 3 3.1 Review of [2,1] and [3,2] models Second-order, one-point [2,1] models In addition to simple closure models for the Reynolds-averaged Navier–Stokes equations, such as models of ‘turbulent’ viscosity using a mixing length assumption, [2,1] models oﬀer both a dynamical and a statistical description of the turbulent ﬁeld, since the governing equations for the Reynolds stress tensor, turbulent kinetic energy, and for its dissipation rate can reﬂect the eﬀects of convection, diﬀusion, distortion, pressure and viscous stresses, which are present in the equations that govern the ﬂuctuating ﬁeld ui . The exact evolution equation for the Reynolds stress tensor Rij = ui uj , derived from (2.3), has the form ∂Rij ∂Rij = Pij + Πij − + Uk ∂t ∂xk ˙ Rij ∂Ui j where Pij = − ∂xk Rkj − ∂xk Rki is usually referred to as the production tensor and is the only term on the right of (3.1) which does not require modelling, since it is given in terms of the basic one-point variables Ui and Rij of the model. The remaining terms are not exactly expressible in terms of the basic one-point variables and heuristic approximations, forming the core of the model, are introduced to close the equations. The second term on the right of (3.1) is associated with the ﬂuctuating i pressure and is given by Πij = p ( ∂xj + ∂xj ), consisting of one-point correlai tions between the ﬂuctuating pressure and rate of strain tensor. As discussed in the introduction, p is nonlocally determined from the velocity ﬁeld by the Poisson equation (2.5) which, in principle, requires multi-point methods for its treatment. It is usual to decompose Πij into three parts ij − ∂ (Dijk ) ∂xk (3.1) ∂U ∂u ∂u Πij = Πij + Πij + Πij (r) (s) (w) (3.2) corresponding to the three components of the Green’s function solution of (2.5). The ﬁrst is known as the ‘rapid’ pressure component and arises from the source terms in (2.5) which are linear in the velocity ﬂuctuation, stemming from ui Uj + uj Ui . Being linear, this component is present in RDT, hence the term ‘rapid’ component. The second term in (3.2) is the ‘slow’ component (w) and comes from the nonlinear source term in (2.5). Finally, Πij is the wall component and corresponds to a surface integral over the boundaries of the ﬂow in the Green’s function solution for p which is additional to the volume integrals expressing the rapid and slow components. The three components of Πij have zero trace, and are conceived of as representing physically distinct mechanisms of turbulent evolution. Hence, they are modelled diﬀerently. 306 Cambon The terms present in the rate equations for Reynolds stress models in homogeneous turbulence can be exactly expressed as integrals over Fourier space of spectral contributions derived from the second-order spectral tensor Φij , which is the Fourier transform of double correlations at two points, and from the third-order ‘transfer’ spectral tensor Tij (details in section 4). All one-point quantities in (3.1) can be expressed as integrals over wavenumber space. Rij is given by integrating Φij , while Πij = 2 and Πij = (s) (r) ∂Um ∂xl κi κm κj κm Φlj + Φil d3 κ 2 κ κ2 Tij d3 κ (3.3) (3.4) express the rapid and slow parts of Πij , and ij = 2ν κ2 Φij d3 κ (3.5) gives the viscous term (see the next subsection for details). In order to avoid confusion the classical turbulent kinetic energy is denoted k with the wavevector denoted by κ, having κ for its modulus. The equation for the dissipation rate = ν ωi ωi (in quasi-homogeneous and quasi-incompressible turbulence), can be derived from the exact equation that governs the ﬂuctuating vorticity ﬁeld ωi . However, the practical procedure for deriving the -equation hardly uses the latter exact equation and consists of ˙ basing the equation for ˙/ on the equation for k/k with adjustable constants. Advantages and drawbacks of single versus multi-point closure techniques can be brieﬂy discussed as follows. Single-point closure models are much more economical and ﬂexible, and can currently address anisotropic and inhomogeneous ﬂows. Nevertheless, they can easily be questioned in the presence of complex anisotropising mechanisms, and in the presence of a modiﬁed cascade with spectral imbalance, even if one restricts the comparison to the pure homogeneous incompressible case (see [26] for more general ﬂows). These weaknesses appear in predicting the dynamics of Rij , when looking at the ‘rapid’ pressurestrain correlation for complex anisotropisation processes, and when looking at the -equation and ‘slow’ pressure-strain tensor for the cascade, sophisticated though the single-point modelling of (3.3), (3.4), (3.5) may be. For the sake of brevity, only the anisotropy problem will be illustrated, by comparing RDT and single-point closures in the same ‘rapid’ limit of homogeneous turbulence in the presence of uniform mean velocity gradients. In this limit, Rij - models seem to work satisfactorily in the presence of an irrotational mean ﬂow, even for time-dependent pure straining processes, such as the successive plane strains addressed by Gence and Mathieu (1979), or, more recently, for cyclic irrotational compression (Le Penven, and Hadzic, Hanjalic and Laurence, private communications). In the same situation, k- models give wrong results [9] Introduction to two-point closures 307 because of the instantaneous relationship between the deviatoric part of Rij and the mean strain-rate tensor, a relationship that is usually known as the Boussinesq approximation. The same contrast between k- and full Rij - , with only the latter working satisfactorily, is found when looking at stabilising-destabilising eﬀects of rotation in a plane channel (only the trends induced by terms present in homogeneous turbulence are analysed). A clue to understand why Rij - can roughly mimic RDT in the cases mentioned above, is their ability to take into account the production term, in a way much more realistic than in k- . For instance, the RDT solution for pure irrotational mean ﬂow exhibits the dominant role of the time-accumulated strain, which is a particular case of the Cauchy matrix, so that correct trends can also be captured by Rij - models even if the rapid pressure-strain is only roughly modelled, e.g. as proportional to the deviatoric part of the production tensor. Regarding the relevance of rotational Bradshaw (or Richarson) numbers for stabilising–destabilising eﬀects of rotation in a shear ﬂow, Leblanc and Cambon (1997) have explained why an apparently 2D and pressure-less analysis (Bradshaw 1969) gave the same criterion as an ‘exact’ linear stability analysis (Pedley 1969). The reason is the dominant role of pure spanwise modes, which are naturally unaﬀected by pressure ﬂuctuations, yielding again a ‘production dominated’ mechanism. Things are completely diﬀerent when rotation interplays with the straining process in a more subtle way, for instance by inducing inertial waves for which anisotropic dispersion relationships aﬀect (3.3), not to mention (3.4) and (3.5) beyond the RDT limit. For instance, Townsend’s equations for homogeneous RDT were shown (Cambon 1982; Cambon et al. 1985) to develop angular peaks of instability in Fourier space if the rotation rate (half the vorticity) of the mean ﬂow is strictly larger than the strain rate, a fact which was recovered by Bayly (1986) in the diﬀerent context of incisively revisiting the gloriﬁed ‘elliptical ﬂow instability’. The reader is referred to Pierrehumbert (1986) for the literature on elliptical ﬂow instability, to Cambon and Scott (1999) for the linkage of stability analysis to RDT, and ﬁnally to Salhi et al. (1997) for more details on the discussion which is touched upon here. Even without additional mean strain, pure rotation induces complex ‘rapid’ and ‘slow’ eﬀects, for which even the basic principles of single-point closures are questionable (see subsection 5.2). Single-point closures look particularly poor since there is no production by the Coriolis force, whereas the dynamics is dominated by waves whose anisotropic dispersivity is induced by ﬂuctuating pressure. This suggests discriminating ‘turbulence dominated by production eﬀects’ from ‘turbulence dominated by wavy eﬀects’. In short, single-point closures are well adapted to simple turbulent ﬂow patterns of the ﬁrst class in rather complex geometry, whereas two-point closures are more convenient for complex turbulent ﬂows in simpliﬁed geometry, as illustrated by the second class. 308 Cambon 3.2 Third-order, two-point [3,2] models Although several diﬀerent approaches exist in the literature, the simplest way to introduce two-point closure models is to look at the governing equations for double velocity correlations at two points and for triple correlations at three points. This level of information removes from consideration non-local eﬀects (ﬁgure 1), so that the exact relationship between pressure and velocity is accounted for. However, it is possible to derive a tractable formalism only for quasi-homogeneous ﬂows where Fourier space is relevant. Accordingly, wavespace is an invaluable tool for these approaches, but it is only a mathematical convenience for treating non-local operators and multi-point correlations. Hence the double correlations ui (x, t)uj (x + r, t) at two points (or [2, 2] in ﬁgure 1) are treated through their Fourier transform with respect to r, denoted Φij (κ, t); this second-order spectral tensor is also proportional to the covariance matrix u∗ uj . The equation that governs the second-order speci tral tensor, possibly in the presence of mean gradients uniform in space (Craya 1958), involves the transfer terms linked to triple correlations at two points (or [3, 2]), which need to be closed. Rather than using an equation for the [3, 2] term, it is more consistent (avoiding again the closure due to nonlocality) to derive the latter from correlations at three points (or [3, 3]), which appear as ui (κ, t)uj (p, t)ul (q, t) with κ+p+q=0 (3.6) in agreement with ‘triadic interactions’ caused by the quadratic nonlinearity seen in Fourier space. Looking at the equations that govern (3.6), or ∂ uuu + exact linear uuu terms = uuuu , ∂t (3.7) a closure can ﬁnally be made by assuming a linear relationship between fourthand third-order cumulants, or uuuu − uu uu = − 1 uuu . θ (3.8) Regarding the structure of the equation given above, it is important to give some preliminary remarks as follows: • The latter two equations are abridged and ‘symbolic’, in the sense that the Fourier velocity components ui at diﬀerent wave vectors are actually involved in place of u, so that uuu in (3.7) and (3.8) would represent (3.6) and uu would represent Φij . Accordingly, the ‘true’ equation abridged by (3.8) allows one to close the equation (3.7) that governs (3.6), and then to close the equation for the second-order spectral tensor: the inﬁnite hierarchy of open equations ([n, n] ← [n + 1, n] in ﬁgure 1) is broken at the fourth order. [9] Introduction to two-point closures 309 • Equation (3.8) is formally consistent with ‘nearly linear’ and ‘nearly Gaussian’ assumptions: since linear operators conserve the gaussianity, pure linear dynamics (reﬂected for instance by the so-called RDT) conserve the Gaussian properties if present in the initial data. Accordingly, all cumulants remain zero in this situation and both right-hand side and left-hand side of (3.8) identically vanish. Hence a linear relationship such as (3.8) is consistent with considering both right-hand side and left-hand side as formal ‘weak’ departures from gaussianity, caused by formal ‘weak’ nonlinearity. The impact of a closure relationship such as (3.8) on the dynamics of triple correlations, that carry the energy cascade, is conventionally seen as follows: since (3.7) can be rewritten as ∂ uuu 1 + exact linear uuu terms + uuu = ∂t θ uu uu , uu uu in the left-hand side acts as a source term for increasing the term triple correlations, so that the contribution to the right-hand side term, which comes from the closure relationship (3.8), appears as a damping term which exhibits the characteristic time denoted θ. An ‘ad hoc’ eddy-damping term is chosen using the EDQNM-type model, but the structure of more sophisticated two-point closure theories (DIA, TFM) is not fundamentally diﬀerent. The originality of the two-point closure models reported in the following sections mainly lies in the straightforward treatment of anisotropising linear operators, for both second- and third-order moments. 4 From RDT to anisotropic two-point closures The simplest multi-point closure consists of the drastic measure of dropping all nonlinear terms in (2.3) before averaging. If one also drops the viscous term, in keeping with the high Reynolds number associated with the large scales of turbulence, the result is known as rapid distortion theory (RDT), introduced by Batchelor and Proudman (1954) (see Townsend 1976; Hunt and Carruthers 1990; and especially Cambon and Scott 1999, sections 2 and 5, for recent reviews). In neglecting nonlinearity entirely, the eﬀects of the interaction of turbulence with itself are supposed to be small compared with those resulting from mean-ﬂow distortion of turbulence. One often has in mind ﬂows such as weak turbulence encountering a sudden contraction in a channel or ﬂows around an aerofoil. Implicit is the idea that the time required for signiﬁcant distortion by the mean ﬂow is short compared with that for turbulent evolution in the absence of distortion. Linear theory can also be envisaged, at least over short enough intervals of time, whenever physical inﬂuences leading to linear terms in the ﬂuctuation equations dominate turbulent ﬂows, such as strongly stratiﬁed or rotating ﬂuid or a conducting ﬂuid in a strong magnetic ﬁeld. For such cases, the term ‘rapid distortion theory’ is probably a little misleading. 310 Cambon Thanks to linearity, time evolution of ui may be formally written as ui (x, t) = Gij (x, x , t, t )uj (x , t )d3 x (4.1) where Gij (x, x , t, t ) is a Green’s function matrix expressing evolution from time t to time t. Whereas ui is a random quantity, varying from realisation to realisation of the ﬂow, Gij is deterministic and can, in principle, be calculated for a given Ui (x, t). From Gij and the initial turbulence, (4.1) may be used to determine later time behaviour. Another simplifying assumption which is often made is that the size of turbulent eddies, , is small compared with the overall length scales of the ﬂow, L, which might be the size of a body encountering ﬁne-scale free-stream turbulence (see e.g. Hunt 1973). In that case, one uses a local frame of reference convected with the mean velocity and approximates the mean velocity gradients as uniform, but time-varying. Thus, the mean velocity is approximated by Ui = λij (t)xj (4.2) in the moving frame of reference. In the example of ﬁne-scale turbulence encountering a body, one may imagine following a particle convected by the mean velocity, which sees a varying mean velocity gradient, λij (t), even when the mean ﬂow is steady. This velocity gradient distorts the upstream turbulence in a manner one would like to determine. Since the time history is diﬀerent depending on the particle considered, separate calculations are needed for the diﬀerent streamlines of a steady mean ﬂow. In this case, linear solutions of equation (2.3) are found as Lagrangian Fourier modes u (x, t) ∝ exp(ıκ(t) · x) Combining elementary solutions of the form (4.3) via Fourier synthesis ui (x, t) = the RDT solution is ui [κ(t), t] = Gij (κ, t, t )uj [κ(t0 ), t0 ] (4.5) ui exp(ıκ · x) d3 κ (4.4) (4.3) and κ(t0 ) is given in terms of κ(t) by a Cauchy matrix. Changes in the wavenumber due to mean velocity gradients are reﬂected as dependence of ui (κ, t) on a diﬀerent wavevector K = κ(t0 ) at time t0 , a process of spectral transfer in wavenumber space. Recalling that the objective is to calculate statistical properties for a random ui representing turbulence, one can use the above solution in terms of Fourier transforms to that eﬀect. This is straightforward if the turbulence is assumed statistically homogeneous. In that case, the second-order, two-point moments [9] Introduction to two-point closures 311 of velocity can be derived from a spectral tensor Φij (κ, t) (Batchelor 1953), related to the Fourier transform ui (k, t) in individual realisations by u∗ (p, t)uj (κ, t) = ui (−p, t)uj (κ, t) = Φij (κ, t)δ(κ − p) i (4.6) from which the two-point moments may be obtained via the Fourier transform ui (x, t)uj (x + r, t) = Φij (κ, t) exp(ıκ.r) d3 κ (4.7) This relation shows that, for homogeneous turbulence, the second-order moments in physical space and the spectral tensor in wavenumber space contain essentially the same information. The one-point moment may be obtained by setting r = 0 as ui uj = Φij (κ, t)d3 κ. (4.8) From (4.5), (4.6), and the fact that Gij is real, it follows that spectral evolution takes the form Φij [κ(t), t] = Gik (κ, t, t0 )Gjl (κ, t, t0 )Φkl [κ(t0 ), t0 ]. (4.9) Given an initial Φij at t = t0 , for instance isotropic, one can calculate it at later times using (4.9), provided the Green’s function Gij (κ, t, t ) is known. The determination of Gij is thus the main problem in applying homogeneous RDT in practice. The Green’s function also will appear in models allowing for nonlinearity through formal solutions of the moment equations, in which the nonlinear terms are treated as forcing of the linear part. Although purely linear theory closes the equations without further ado and simpliﬁes mathematical analysis, it is rather limited in its domain of applicability, ignoring as it does all interactions of turbulence with itself, including the physically important cascade process. Multi-point turbulence models which account for nonlinearity via closure lead to moment equations with a well-deﬁned linear operator and nonlinear source terms. The view taken in this chapter is that, even when nonlinearity is signiﬁcant, the behaviour of the linear part of the model often still has a signiﬁcant inﬂuence. Thus, it is important to ﬁrst understand the properties of the linearised model, an undertaking which is, moreover, mathematically more tractable than attacking the full model directly. As a bonus, linearised analysis often allows a simpliﬁed formulation of the nonlinear model using more appropriate variables. For the sake of brevity, we do not report here various calculations and experiments in the area of homogeneous turbulence subjected to a mean ﬂow of the kind given by (4.2) (Cambon 1982; Cambon et al. 1985; Gence 1983; Cambon and Scott 1999; Leuchter et al. 1992; Leuchter and Dupeuble 1993), including elliptic and hyperbolic cases, nor stability analyses which use essentially the same relationships as (4.2) and (4.5) (Cambon 2001a, and references 312 Cambon therein). It is important, however, to recall that the feedback of the Reynolds stress tensor in (2.1) vanishes due to statistical homogeneity (zero gradient of any averaged quantity), so that the mean ﬂow (4.2) has to be a particular solution of the Euler equations and can be considered as a base ﬂow for stability analysis. In turn, the form (4.2) is consistent with maintaining homogeneity of the ﬂuctuating ﬂow governed by (2.3) and (2.4), provided homogeneity holds for the initial data. This explains why homogeneous RDT can have the same starting point as a rigorous and complete linear stability analysis in this case, before the random initialisation of the ﬂuctuating velocity ﬁeld is considered in (4.5). In general, RDT operators break statistical isotropy at any scale, even if the initial data are strictly isotropic. It should be borne in mind that isotropy imposes a very special form Φij (κ, t) = E(κ, t) κi κj δij − 2 4πκ2 κ (4.10) on the spectral tensor, where E(κ, t), with κ = |κ|, is the usual energy spectrum, representing the distribution of turbulent energy over diﬀerent scales and the quantity in brackets will be recognised as the projection matrix, Pij (κ). Thus, Φij is determined by a single real scalar quantity, E, which is a function of the magnitude of κ alone. Both the form of Φij at a single point and its distribution over κ-space are strongly constrained by isotropy. Given the, in our view, importance of allowing for anisotropy and the associated eﬀects of mean ﬂow gradients, we will not discuss the many isotropic models of spectral evolution which have been proposed (see, for instance, Monin and Yaglom (1975), section 17). Instead we concentrate on a small number of models capable of handling the anisotropic case and which illustrate the way in which linear theory combines with nonlinear closures. Independently of closure, the spectral tensor Φij is not a general complex matrix, but has a number of special properties, including the fact that it is Hermitian, positive-deﬁnite, as follows from (4.6), and satisﬁes Φij κj = 0, obtained from (4.6) and the incompressibility condition κj uj = 0. Taken together, these properties mean that, instead of the 18 real degrees of freedom of a general complex tensor, Φij has only four. Indeed, using a spherical polar coordinate system in κ-space, the tensor takes the form (see Cambon et al. 1997 for details). 0 0 0 Φ= e + Zr 0 Zi + ıH/κ 0 Zi − ıH/κ e − Zr (4.11) where the scalars e(κ, t) and H(κ, t) are real, and Z(κ, t) = Zr + ıZi is complex. The quantity e(κ, t) = 1 Φii is the energy density in κ-space, whereas 2 H(κ, t) = ıκl lij Φij is the helicity spectrum and, along with Z, is zero in the [9] Introduction to two-point closures 313 isotropic case. Anisotropy is expressed through variation of these scalars with the direction of κ, as well as departures of H and Z from zero at a given wavenumber. Whatever spectral closure is used, the number of real unknowns may be reduced to the above four when carrying out numerical calculations, and presentation of the results can be simpliﬁed using these variables, particularly when the turbulence is axisymmetric. Our starting point for closure is the equation for the Fourier transform of the velocity ﬂuctuation, which takes the form ˙ ui + Mij uj + νκ2 ui = si (4.12) ˙ where ui = ∂ ui /∂t − λlm κl ∂ ui /∂κm corresponds to linear advection by the mean ﬂow (4.2), and Mij = λmj (δim − 2κi κm /κ2 ) gathers linear distortion and pressure terms. Once nonlinear and viscous terms are added, (4.12) generalises the linear inviscid equation for which the RDT solution is (4.5). The nonlinear term si is given by si (κ, t) = −ıPijk (κ) p+q=κ uj (p, t)uk (q, t)d3 p (4.13) in terms of a convolution integral, the usual expression of a quadratic nonlinearity, and Pijk = 1 (Pij κk + Pik κj ) which arises from the elimination of 2 pressure using the incompressibility condition κi ui (κ, t) = 0. The evolution equation for Φij (Craya 1958), derived from (4.6) and (4.12), is ˙ Φij + Mik Φkj + Mjk Φik + 2νκ2 Φij = Tij (4.14) where the left-hand side arises from the linear part of (4.12), consisting of ˙ the term Φij , which, as in (4.12), is a convective time derivative in κ-space, together with RDT and viscous components. A system of equations for the set (e, Z, H), using (4.11), can readily be derived from (4.14); it is particularly useful in the presence of solid body rotation (Cambon et al. 1992, 1997; Reynolds and Kassinos 1995). Of course, this equation with a zero right-hand side has the RDT solution (4.9). The detailed expression for the right-hand side of (4.14), which represents nonlinear triadic interactions, consists of an integral over the third-order spectral moments and requires closure. The quasi-normal assumption gives the typical relationship between Tij and Φij , as recalled below. When the result is employed in the forcing term of the evolution equation for the third-order moments and the latter solved by Green’s function techniques, one obtains ∗ Tij (κ, t) = τij (κ, t) + τji (κ, t) (4.15) where τij (κ, t) t = Pjkl (κ) −∞ κ+p+q=0 Gim (κ, t, t )Gkp (p, t, t )Glq (q, t, t ) 314 Cambon 1 Pmnr (κ )Φpr (p , t ) + Ppnr (p )Φmr (κ , t ) d3 p dt , (4.16) 2 Φqn (q , t ) in which the triple product of Green’s functions arises from the Green’s function solution for the third-order moments and the notation κ+p + q = 0 on the integral sign means that q should be replaced by −κ − p throughout the integrand, representing interacting triads of wavenumbers κ, p, q which form triangles. Equation (4.16) can be seen as a solution of the last symbolic equation of section 3. It gives the generic anisotropic structure of most generalised classical theories dealing with two-point closure, provided the basic Green’s function is replaced by a slightly modiﬁed version, for instance including viscous terms and eddy damping as in EDQNM. We should perhaps say a few words about DIA (Kraichnan 1959), which is more complicated than EDQNM, since it is based on spectral tensors involving two times, rather than Φij (κ, t). Furthermore, it introduces an additional response tensor for which, like the spectral tensor, an evolution equation is formulated and closed using heuristic approximations. These approximations are similar in nature to the quasi-normal one introduced above, supposing as they do that the ﬂuctuating velocities have properties similar to those of Gaussian variables, although such assumptions are less explicit in DIA. The ﬁnal evolution equation for the two-time spectral tensor contains an integral whose structure is much the same as the quasi-normal expression (4.16), with terms such as Glq (q, t, t )Φqn (q , t ) replaced by the two-time spectral tensor Φln (q , t, t ), leaving one remaining Green’s function from the three-fold product, which is replaced by the response tensor. One-point moments contain rather limited information compared with Φij (κ, t), but they are nonetheless usually among the ﬁrst quantities to be calculated following an anisotropic spectral calculation, along with correlation lengths in diﬀerent directions. Given Φij , for instance obtained using RDT, evaluation of the integral over 3D Fourier space can be a nontrivial task. For example, the RDT Green’s function can be determined analytically in the case of simple shear, but the integrals in (4.8) are not straightforward and must be evaluated numerically or asymptotically (Rogers 1991; D.J. Bodony and G.A. Blaisdell, unpublished results). 5 5.1 Application to stably stratiﬁed and rotating ﬂuid General features In this section we consider application of the methods to turbulent ﬂows in stably stratiﬁed conditions and to turbulence subjected to rotation. These cases illustrate the ‘ﬂows dominated by wavy eﬀects’ with ‘zero production’ introduced at the end of subsection 3.1. For pure rotating turbulence there is zero production of kinetic energy and for stratiﬁed rotating turbulence there is [9] Introduction to two-point closures 315 zero production of total (kinetic plus potential) energy. The reader is referred to Cambon et al. (1997) and Cambon (2001c) for more details. Linearised solutions of the Navier–Stokes equations, with buoyancy force b within the Bousinesq assumption, are easily obtained in the presence of a uniform mean density gradient and in a rotating frame. For the sake of simplicity, the mean ﬂow is restricted to a uniform vertical gradient of density and to a solid body rotation in the horizontal direction, with typical parameters N (the Brumt– Waisala frequency) and Ω (the angular velocity). No additional mean velocity gradients are considered in the rotating frame. Pressure ﬂuctuations are removed from consideration in the Fourier-transformed equations by using a local frame in the plane normal to the wave vector (Craya 1958), taking advantage of (2.4), so that the problem in ﬁve components (u1 , u2 , u3 , p, b) in physical space is reduced to a problem in three components, two solenoidal velocity components and a component for b, in Fourier space. For mathematical convenience, as in Cambon (1989) and Godeferd and Cambon (1994), the velocity-temperature ﬁeld in three components is ﬁnally gathered into a single vector v, whose 3D Fourier transform, denoted by a ‘hat’ ( ), can be written as 1 κ v =u+ı b . (5.1) N κ (A similar three-component term, denoted WK , is extensively used in Riley and Lelong (2000), p. 626, but it is not a true vector, in contrast to v). The scaling of the contribution from the buoyancy force allows one to deﬁne twice the total energy spectral density as ∗ vi vi = u∗ ui + N −2 b∗ b. i (5.2) Without stratiﬁcation, N −1 has to be replaced by another time scale τ0 in (5.2), but coupling between the velocity and buoyancy (or temperature) ﬁelds vanishes in this case. The linear equation for v is similar to (4.5), but with a constant wavevector κ(t) = κ(t0 ), since the advection by a mean ﬂow with antisymmetric gradient (solid body rotation) amounts to the addition of a Coriolis force when the motion is seen in the rotating frame, only modifying Mij in (4.12). A similar Green’s function can be expressed as Gij (κ, t, t0 ) = =0,±1 − Ni (κ)Nj (κ) exp[ı σκ (t − t0 )], (5.3) in which N0 and N±1 are the eigenmodes, related to quasi-geostrophic motion and inertio-gravity waves respectively, whereas σκ = N 2 (κ⊥ /κ)2 + 4Ω2 (κ /κ)2 (5.4) holds for the absolute value of the frequency given by the dispersion law of inertio-gravity internal waves. Because of the form of the eigenvectors and of 316 Cambon the dispersion law, the structure of G in (5.3) is consistent with axisymmetry around the axis of reference (chosen vertical here), without mirror symmetry, and κ and κ⊥ hold for axial (along the axis) and transverse (normal to the axis) components of κ. Looking only at inviscid RDT (e.g. Cambon 1989; Hanazaki and Hunt 1996; van Haren 1993; van Haren et al. 1996 for pure stratiﬁed turbulence) the following results can be predicted without detailed calculations. • Inviscid RDT solutions are derived from (5.3) and (5.4) for second-order ∗ spectral tensors, which are related to u∗ uj or vi vj . They consist of i sums of steady and oscillating terms, the frequency of oscillations being directly connected to the dispersion law of internal waves. Such solutions are completely reversible. • After integration over κ-space, such as (4.8), including averaging over all directions, oscillating terms for spectral tensors yield damped oscillations. This damping eﬀect, called ‘phase-mixing’ in Kaneda and Ishida (1999), physically reﬂects the anisotropic dispersivity of inertio-gravity waves. It cannot appear for N = 2Ω, or for particular initial data at N = 2Ω, such as the equipartition case considered for nonlinear applications in Godeferd and Cambon 1994). The fact that some terms in u∗ uj are i conserved after integration, whereas other ones are damped, explains the change of anisotropy over time for all single-point correlations. This change is completely determined by the initial distribution in terms of steady and wavy modes. Except for the anisotropisation of two-time single-point correlations (see at the end of [26]), the linear limit exhibits no interesting creation of structural anisotropy. However in practice there is two-dimensionalisation in rotating turbulence and a horizontal layering tendency in the stably stratiﬁed case. In other words, RDT only alters phase dynamics, and conserves exactly the spectral density of typical modes (full kinetic energy for the rotating case, total energy and ‘vortex’, or potential vorticity, energy for the stably stratiﬁed case), so that two-dimensionalisation or ‘two-componentalization’ (horizontal layering), which aﬀect the distribution of this energy, are typically nonlinear eﬀects. In Godeferd and Cambon (1999), RDT results have begun to be compared with the results of a full DNS, in order to have the deﬁnite answer as to what is linear (given by RDT) and what is nonlinear (only given by DNS) in the rotating and stratiﬁed cases. In addition to such numerical comparisons, the eigenmodes of the linear regime, derived from RDT, form a useful basis for expanding the ﬂuctuating velocity-temperature ﬁeld, even when nonlinearity is present, and nonlinear interactions can be evaluated and discussed in terms of triadic interactions between these eigenmodes. Accordingly, the complete anisotropic description of two-point second-order correlations, e.g. (4.11), can be related to spectra and cospectra of these eigenmodes. [9] Introduction to two-point closures 317 5.2 Pure rotation Rotation of the reference frame is an important factor in certain mechanisms of ﬂow instability, and the study of rotating ﬂows is interesting from the point of view of turbulence modelling in ﬁelds as diverse as engineering (e.g. turbomachinery and reciprocating engines with swirl and tumble), geophysics and astrophysics. Eﬀects of mean curvature or of advection by a large eddy can be tackled using similar approaches. From several experimental, theoretical and numerical studies, in which rotation is suddenly applied to homogeneous turbulence, some agreed statements are summarised as follows (Bardina et al. 1985; Jacquin et al. 1990; Cambon et al. 1992,1997; Cambon 2001c). • Rotation inhibits the energy cascade, so that the dissipation rate is reduced. • The initial 3D isotropy is broken through nonlinear interactions modiﬁed by rotation, so that a moderate anisotropy, consistent with a transition from a 3D to a 2D state, can develop. • Both previous eﬀects involve nonlinear or ‘slow’ dynamics, and the second is relevant only in an intermediate range of Rossby numbers as found by Jacquin et al. (1990). This intermediate range is delineated by RoL = urms /(2ΩL) < 1 and Roλ = urms /(2Ωλ) > 1, in which urms is an axial rms velocity ﬂuctuation, whereas L and λ denotes a typical integral lengthscale (macroscale) and a typical Taylor microscale respectively. • If the turbulence is initially anisotropic, the ‘rapid’ eﬀects of rotation (linear dynamics tackled in a RDT fashion) conserve a part of the anisotropy (called directional) and damp the other part (called polarization anisotropy), resulting in a spectacular change of the anisotropy of Rij . These eﬀects, which are not at all taken into account by current one-point second-order closure models (from k- to Rij - models), have motivated new modelling approaches by Cambon et al. (1992, 1997), and to a lesser extent by Reynolds and Kassinos (1995) for linear (or ‘rapid’) eﬀects only. It is worth noticing that the modiﬁcation of the dynamics by the rotation ultimately comes from the presence of inertial waves (Greenspan 1968), having an anisotropic dispersion law, which are capable of changing the initial anisotropy of the turbulent ﬂow and also can aﬀect the nonlinear dynamics. Contrary to a well-known interpretation, the Proudman theorem shows only that the ‘slow manifold’ (the stationary modes unaﬀected by the inertial waves) is the 2D manifold at small Rossby number, but cannot predict the transition from 3D to 2D turbulence, which is a nonlinear mechanism of transfer from all possible modes towards the 2D ones. In Fourier space, the slow — and 2D — manifold corresponds to the wave plane normal to the rotation axis, or κ = 0. In 318 Cambon Figure 2: Correlation coeﬃcient of vertical velocity and temperature in decaying stably-stratiﬁed turbulence. Comparison between some single-point closure models and generalised EDQNM (ﬁgure courtesy van Haren 1993). (5.3), only the wavy modes N±1 , which reduce to the Waleﬀe (1993) helical modes, are present, and therefore form a complete basis for the velocity ﬁeld. Accordingly, the resonant triads σk ± σp ± σq ∼ 0, with κ + p + q = 0, with σκ given by (5.4) for N = 0, are found to dominate nonlinear ‘slow’ motions. 5.3 Pure stratiﬁed homogeneous turbulence The case of stably stratiﬁed turbulence is diﬀerent, even if the gravity waves present strong analogies with inertial waves. An additional element is the presence of the ‘vortex’, or potential vorticity (PV) mode, which is a particular case of the quasi-geostrophic mode N0 , which is related to = 0 steady motion in (5.3). According to its deﬁnition, extended to vertical wave-vectors (Cambon 2001c, Appendix), it is present for any wavevector orientation, from horizontal to vertical, and contains half the total kinetic energy in the isotropic case. In contrast with the poor relevance to rotating turbulence of classical singlepoint closure models, models consistent with the two-component limit (TCL) by Craft and Launder (Chapter [14] in this volume) have been shown to work unexpectedly well when compared to a full nonlinear spectral, EDQNM-type, model, which retains all the complex spectral behaviour of RDT, as shown in ﬁgure 2. In particular, the frequency and the damping of the oscillations is well reproduced by the TCL model, even though it mainly results from the dispersion law of gravity waves and integration over many wavevectors in the spectral calculation. This illustrates how a single-point closure model can succeed, only due to mathematical constraints (realisability, TCL consistency, etc.) even if the details of the physics (here the anisotropic dispersion law for [9] Introduction to two-point closures 319 wave components and diﬀerent dynamics for wave and vortex motions) cannot be accounted for. Focusing on nonlinear eﬀects, pure vortex interactions have been found to be dominant in triggering the loss of isotropy, as a prerequisite to orient the evolution of the initially isotropic velocity ﬁeld towards a two-component state. EDQNM2 (Godeferd and Cambon 1994) and DNS results (Godeferd et al. 1997) have shown that the spectral energy concentrates towards vertical wavenumbers κ⊥ ∼ 0. Because κ and u are perpendicular, these wavenumbers correspond to predominantly horizontal, low-frequency motions. As for the partial transition towards 2D structure shown in pure rotation, a new dynamical insight is given to the collapse of vertical motion expected in stably stratiﬁed turbulence, but the long-time behaviour essentially diﬀers from a two-dimensionalisation. A sketch of the diﬀerent nonlinear eﬀects of pure rotation and pure stratiﬁcation is shown in ﬁgure 3. Previous EDQNM studies (Carnevale and Martin 1982) focused on triple correlation characteristic times modiﬁed by wave frequencies, whereas wave-turbulence theories proposed scaling laws for wavepart spectra. None of them, however, was capable of connecting wave-vortex dynamics to the vertical collapse and layering. Only recently, by re-introducing a small but signiﬁcant vortex part in their wave turbulence analysis, Caillol and Zeitlin (2000) found that ‘The vortex part obeys a limiting slow dynamics equation exhibiting vertical collapse and layering which may contaminate the wave-part spectra’. This is in complete agreement with the main ﬁnding of Godeferd and Cambon (1994), where this result reﬂects a scrambling of any triadic interactions, including at least one wave mode, so that the pure vortex interaction becomes dominant. The corresponding ‘vortex energy transfer’ is strongly anisotropic. It does not yield a classic cascade (which would contribute to dissipate the energy) but instead yields the angular drain of energy which condenses the energy towards vertical wave-vectors, in agreement with vertical collapse and layering. The latter eﬀect is reﬂected in physical space by the development of two diﬀerent integral length scales, as shown in ﬁgure 4. The integral length scale related to horizontal veloc(1) ity components and horizontal separation L11 is shown to develop similarly to isotropic unstratiﬁed turbulence, whereas the one related to vertical sep(3) aration L11 is blocked. In the same conditions, with initial equipartition of potential and wave energy, linear calculation (RDT) exhibits no anisotropy, (1) (3) or L11 = 2L11 . At much larger times, the transfer terms including wave contribution could become signiﬁcant through resonant wave triads, such as the Riley and Lelong (2000) triads, but this would occur in a velocity ﬁeld strongly altered by vertical collapse and layering. Very recently, concentration of total energy towards vertical wave vectors was obtained by Smith (2000) using high resolution DNS, forced randomly at small scale. 320 Ω Cambon Energy containing cone g Energy containing cone Ω g Figure 3: Representation of the spectral angular dependence occuring in stratiﬁed (top left) and rotating (top right) turbulence, with the corresponding schematic physical structures: layered ﬂow (bottom left) for stratiﬁcation; columnar vertically correlated shapes (bottom right) for rotation. The mode related to vertical wave-vectors appears to be very important, since the concentration of spectral energy on it is the best identiﬁcation of the development of vertical collapse and layering. It corresponds to the limit of the wavy mode, when the dispersion frequency tends to zero. Strictly speaking, this mode is a slow mode, which cannot be incorporated in the wave-vortex decomposition of Riley et al. (1981), and more generally is not present in a classical poloidal–toroidal decomposition. It is absorbed in any decomposition based on the Craya–Herring frame (see Cambon 2001c, Appendix), provided that some care is taken to extend by continuity the deﬁnition of the unit vectors (e(1) , e(2) ) towards κ aligned with the polar (vertical here) axis of the frame of reference (Cambon 1982, 2001c). In so doing, the mode related to e(1) coincides with a toroidal, or ‘horizontal vortex’, mode, but for vertical wave vectors, where it includes half the energy of the vertical slow mode. In the same way, the mode related to e(2) coincides with a poloidal mode, aﬀected by the wavy motion, but for vertical wave vectors, where it includes the other half of the energy of the vertical mode. In the diﬀerent context of weakly nonlinear and weakly inhomogeneous RDT approach, and related DNS of Galmiche et al. (2001), the vertical mode is considered as part of the mean ﬂow and is called the mean vertical shear mode. In fact, the DNS use Fourier modes and periodic boundary conditions in all directions, with initial injection of energy onto the largest vertical mode (κ1 = κ2 = 0, κ3 = κmin ), the so-called [9] Introduction to two-point closures 321 Figure 4: a) Isovalues of ∂u2 /∂x3 from a snapshot of 2563 DNS (an illustration (1) of the layering phenomenon). (b) Integral length scales L11 , with horizontal (3) separation (top) and L11 , with vertical separation (bottom), from 2563 DNS. (c) Same quantities from EDQNM2 model. (Courtesy Godeferd and Staquet 2000). ‘mean shear mode’, so that their interpretation in terms of inhomogeneous turbulence and mean-ﬂuctuating interaction is only one possible interpretation. More consistently, these results could be reinterpreted as purely homogeneous and strongly anisotropic, illustrating concentration of energy towards vertical wave-vectors as in the theoretical and numerical works by Godeferd and Cambon (1994), Godeferd and Staquet (2000) and Smith (2000). 322 Cambon 6 Concluding remarks We hope that this chapter has made clear the importance of linear, anisotropising processes even in turbulence which is too strong for strict validity of the rapid distortion approximation. In principle, multi-point closures allow exact treatment of linear terms. On the other hand, single-point closures, being less computationally demanding and having no diﬃculty with inhomogeneity, currently dominate industrial ﬂow calculations, but involve many more heuristic assumptions. The inﬁnite number of degrees of freedom of the spectral tensor are reduced to just k and , or Rij and , in standard single-point models. Together with a large enough number of adjustable constants, this may be suﬃcient to describe the limited class of ﬂows for which the model has been experimentally parameterised, but one is always likely to encounter surprises in new types of ﬂows, as we have seen for the case of rotating turbulence. The latter ﬂow is an interesting example, not only because rotation is important in many practical ﬂows, but because it illustrates the subtle interplay between linear and nonlinear processes and the signiﬁcance of spectral anisotropy. Anisotropy appears in standard one-point models only via departure of ui uj from (2k/3)δij . The quantity ui uj contains limited overall information on the spectral distribution, while anisotropic structuring of the turbulence, leading to axially elongated structures in the rotating case, is not captured at all, despite its physical importance. The good behaviour of the TCL model, however, ought to be underlined for stratiﬁed turbulence. This illustrates that the two-component and 2D limits have to be carefully discriminated. The techniques and models presented in this chapter can be extended to compressible ﬂows, for which linear descriptions retain their importance. In the case of turbulence subject to compression at small Mach number in ﬂow volumes of limited size, for example in the cylinders of piston engines, the mean velocity has nonzero divergence, reﬂecting the eﬀects of compression, whereas the ﬂuctuating velocity may be taken to be solenoidal, as in the incompressible case (Mansour and Lundgren 1990). This description neglects thermal boundary and acoustic eﬀects, but allows the straightforward extension of incompressible models. Indeed, simple spherical compression can be taken into account, including nonlinearity, by transformations of the time, turbulent velocity and position variables without any need for additional modelling (Cambon et al. 1992). More generally, if the turbulent Mach number is not small, the eﬀects of compressibility are much more complicated, since both acoustic and entropy modes are called into play, as well as the vortical mode inherited from the incompressible case (Lele 1994). Irrotational ﬂows have been studied by Goldstein (1978) using an inhomogeneous RDT formulation (which will be discussed again in [26]), while homogeneous RDT for rotational mean ﬂows has shown the im- [9] Introduction to two-point closures 323 portance of the gradient Mach number, S /c, where c is the sound speed and S a measure of the mean velocity gradient (Simone et al. 1997). The latter study helps explain the systematic changes in energy production rate with gradient Mach number found in numerical simulations (Sarkar 1995). They suggest that compressibility mainly alters the one-point properties of turbulence through the pressure-velocity correlation tensor, rather than via bulk viscous dissipation or other explicit compressible terms (e.g. the pressuredilatation term) usually considered in compressible one-point modelling. In the absence of mean shear, interactions between solenoidal, dilatational and pressure modes are purely nonlinear and can be analysed and modelled in pure isotropic homogeneous turbulence. In this context, the spectral model by Fauchet et al. (1997) gave very promising spectral information, as shown in ﬁgure 5. Nonlinear transfer terms have a structure close to (4.16), with a Green’s function, or response tensor, which gathers both a classic linear ‘acoustic wave propagator’ and a nonlinear damping factor which ensures a decorrelation time for triple correlations shorter than in classical EDQNM. The two-point anisotropic description is more powerful, even if homogeneity is assumed, than is generally recognized. In rotating and stratiﬁed turbulence the anisotropic spectral description, with angular dependence of spectra and cospectra in Fourier space, allows quantiﬁcation of columnar or pancake structuring in physical space. Among various indicators of the thickness and width of pancakes, which can be readily derived from anisotropic spectra, integral (l) length scales Lij related to diﬀerent components and orientations are the most useful. It is also worth noting that a possible confusion can be made between inhomogeneity and anisotropy, especially when considering vertically stratiﬁed ﬂows. For instance, DNS and RDT by Galmiche et al. (2001), which are presented as being inhomogeneous, can be reinterpreted in the area of strictly homogeneous strongly anisotropic turbulence. As another illustration (Lee et al. 1990; Salhi and Cambon 1997), the streak-like tendency in shear (1) ﬂows can be easily found in calculating both the L11 component, which gives (3) the streamwise length of the streaks, and L11 , which gives the spanwise separation length of the streaks (as usual, 1 and 3 refer to streamwise and spanwise coordinates, respectively). In pure homogeneous RDT at constant shear rate, both length scales can be calculated analytically and their ratio (elongation parameter) is found to increase as (St)2 , S = ∂U1 /∂x2 being the shear rate. Finally, we would like to underline that a fully anisotropic spectral (or twopoint) description carries a very large amount of information, even if it only concerns second-order statistics. In the inhomogeneous case, the POD (proper orthogonal decomposition, Lumley 1967) has renewed interest in second-order two-point statistics, but this technique is never completely applied to the homogeneous anisotropic case. It is only said that POD spatial modes are Fourier modes in the homogeneous case, but the true spectral eigenvectors corresponding to POD modes are not considered. These can in fact be easily obtained 324 Cambon Figure 5: Sketch of the spectra obtained by the model of Fauchet et al. (1997). From top to bottom, at larger wavenumber, the ﬁgure shows the solenoidal (given) energy spectrum E ss , the incompressible part of pressure variance pp spectrum Einc , the dilatational energy spectrum E dd and the pressure variance spectrum E pp . It can be seen that E pp collapses with E dd (acoustic equilibpp rium) only at smaller wavenumber, whereas it collapses with Einc at larger pp dd . wavenumber, with E E by diagonalising the tensor Φij , using the above e − Z decomposition (4.11), and noting that the angular position of the principal axes, associated with the nonzero principal components e + |Z| and e − |Z|, is ﬁxed by the phase of Z, at each κ. 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(1997). ‘The eﬀect of compressibility on turbulent shear ﬂow: a RDT and DNS study’, J. Fluid Mech. 330, 307–338. Smith, L. (2000). ‘Energy transfer to large scales in rotating and stratiﬁed turbulence forced randomly at small scales’, Conf. on Dispersive Waves and Turbulence, South Hadley, MA, USA, June 11–15. Townsend, A.A. (1956). The Structure of Turbulent Shear Flow. Revised edition 1976. Cambridge University Press Waleﬀe, F. (1993). ‘Inertial transfers in the helical decomposition’, Phys. Fluids A 5, 677–685. 10 Reacting Flows and Probability Density Function Methods D. Roekaerts 1 Introduction In this chapter the statistical description of turbulent reacting ﬂow is considered. In particular second moment closure for variable density ﬂows and the one-point probability density function (PDF) approach are introduced. In turbulence modelling a subset of statistical properties is obtained by calculating a selected set of moments (e.g. mean and variances of quantities at one point in space) from modelled transport equations. For reacting ﬂow it is advantageous to enlarge the subset to involve the complete probability density function of variables deﬁned at one point in space and time. This approach leads to an incomplete description of turbulence properties and closure assumptions have to be made. The structure of the presentation in this chapter is as follows: instantaneous conservation equations are considered ﬁrst because they provide the basis for the averaged equations and also for the transport equation satisﬁed by the probability density function. Next some basic concepts from statistics are considered. Finally the mean conservation equations are introduced and a discussion of the main closure problems and their possible solution in terms of second moment equations and/or PDFs are given. A more detailed description of models and results is given in [20] and [21]. Further information on the PDF approach can be found in Pope (1985), Dopazo (1994), Fox (1996), Pope (2000). 1.1 Instantaneous conservation equations Conservation of mass is expressed by the continuity equation, ∂ρ ∂ρUi = 0, + ∂t ∂xi (1) and conservation of momentum (here in the absence of external forces) by the Navier–Stokes equation, ∂ ∂ ∂p ∂ ρUj Ui = − − Tij , ρUi + ∂t ∂xj ∂xi ∂xj 328 (2) [10] Reacting ﬂows and probability density function methods 329 in which p is pressure and Tij the viscous stress tensor. For a Newtonian ﬂuid the viscous stress tensor contains only simple shear eﬀects and can be written as ∂Ui ∂Uj 2 ∂Uk Tij = −µ + + µ δij , (3) ∂xj ∂xi 3 ∂xk in which µ is the dynamic viscosity. The conservation equations for species mass fractions, denoted here by the scalar vector φ, read ∂ ∂ α ∂ ρUj φα = − J + ρSα (φ), ρφα + ∂t ∂xj ∂xj j (4) α where Jj is the diﬀusion ﬂux and Sα is a chemical reaction source term. Neglecting eﬀects of thermal diﬀusion and external body forces and assuming Fickian diﬀusion, the ﬂux reads α Jj = −ρID ∂φα ∂xj (5) in which ID is the mass diﬀusion coeﬃcient which here for simplicity is assumed to be equal for all species. The enthalpy conservation equation which in general is also needed, is also of the form (4) with the enthalpy source term in particular containing eﬀects of radiative heat transfer. The description is completed with the thermodynamic equation of state and the caloric equation of state (relating enthalpy and composition to temperature). 2 2.1 Basic statistical concepts One-point statistics Consider the variable φ which is a function of space x and time t (denoted by φ(x, t)). The distribution function F of φ is deﬁned as Fφ (ψ; x, t) ≡ P {φ(x, t) < ψ}, (6) in which ψ is the sample space variable of φ which takes all possible values of φ and P {. . .} stands for the probability that the ﬁeld value φ at (x, t) is smaller than ψ. The probability density function (PDF) is deﬁned as: fφ (ψ; x, t) = ∂ Fφ (ψ; x, t) ∂ψ (7) The probability that a realization of φ(x, t) lies between ψ1 and ψ2 is given by P {ψ1 < φ(x, t) < ψ2 } = ψ2 ψ1 fφ (ψ; x, t)dt = Fφ (ψ2 ; x, t) − Fφ (ψ1 ; x, t). (8) The end values of F are appropriately deﬁned as F (−∞) = 0 and F (+∞) = 1. Field values and probability functions in general are considered to be functions 330 Roekaerts of space and time but to simplify the notation the variables x and t are usually omitted without further notiﬁcation. The average or expectation of φ is denoted by E{φ}, φ , or by φ and can be expressed in terms of the PDF by +∞ φ = −∞ ψfφ (ψ)dψ. (9) In the same way, the expectation of any function Q of φ can be deﬁned by +∞ Q = −∞ Q(ψ)fφ (ψ)dψ. (10) With the deﬁnition of the ensemble average or Reynolds average the variable φ can be decomposed into its mean φ and ﬂuctuation φ according to φ= φ +φ. (11) To clarify the notion of one-point statistics further, an example of twopoint statistics is given. Consider the two-point one-time statistics of the ﬁeld variable φ(x, t) deﬁned by Fφ (ψ1 , ψ2 ; x, x+r, t) which describes the probability that φ(x, t) < ψ1 , and φ(x + r, t) < ψ2 . This function cannot be expressed in terms of the one-point statistics of φ. In the limit of |r| → 0 this description reduces to the combined statistics of φ and its gradients. This means that gradient statistics cannot be described in terms of the one-point statistics of a variable and that such terms are unclosed in any one-point closure. 2.2 Joint probabilities We now consider combined statistics of several variables (e.g. velocities and species concentrations). This provides information about correlations between variables. Consider the n stochastic variables φ = φ1 , φ2 , . . . , φn which have a joint probability function deﬁned by: F φ(ψ) ≡ P {φ1 < ψ1 , φ2 < ψ2 , . . . , φn < ψn }. The joint PDF is deﬁned by f φ(ψ) = ∂n F φ(ψ) ∂ψ1 ∂ψ2 · · · ∂ψn (13) (12) From the joint PDF of φ the statistics of one variable φα can be obtained by integration over the other n − 1 directions of the ψ space: +∞ fφα (ψα ) = −∞ ··· +∞ −∞ f φ(ψ) dψ1 · · · dψα−1 dψα+1 · · · dψn (14) This one-variable PDF is also called the marginal PDF of φα . As in the univariate case (see equation (10)) the expectation of any function of the variables φi can be expressed as an integral over the phase space of this function times the joint PDF. [10] Reacting ﬂows and probability density function methods 331 2.3 Conditional probability The conditional probability P {A|B} is deﬁned as the probability that A occurs given that B occurs, and is given by P {A|B} = P {A, B} . P {B} (15) In the same manner the conditional PDF of φ1 |φ2 can be deﬁned as fφ1 |φ2 (ψ1 |ψ2 ) = fφ1 ,φ2 (ψ1 , ψ2 ) . fφ2 (ψ2 ) (16) The expression fφ1 |φ2 (ψ1 |ψ2 )dψ2 now deﬁnes the PDF of φ1 given the fact that ψ2 < φ2 < ψ2 + dψ2 . Conditional statistics are an important concept in PDF modeling. It turns out that all unclosed terms in the PDF equations can be written in terms of conditional averages (e.g. of the ﬂuctuating velocity given a value of the scalar). In other words, these terms can in general not be expressed as a pure function of the describing variables. These conditional averages Q(φ1 , φ2 )|φ2 = ψ2 can be written in terms of the conditional PDF fφ1 |φ2 by +∞ Q(φ1 , φ2 )|φ2 = ψ2 = −∞ Q(ψ1 , ψ2 )fφ1 |φ2 (ψ1 |ψ2 )dψ1 . (17) 2.4 Favre averaging In turbulent ﬂames, density can vary by a factor of ﬁve or more and density ﬂuctuations can have large eﬀects on the turbulent ﬂow ﬁeld. To simplify the equations describing variable density ﬂow it is common to use density-weighted (Favre) averaging. The main advantage of using Favre-averaging instead of ensemble- or Reynolds-averaging is the fact that explicit density correlations are avoided in the equations (Jones 1980). Let the density be denoted as ρ, and let the vector φ denote the thermochemical scalar variables (e.g. species concentrations, temperature). In the low Mach-number limit, where it is assumed that pressure variation does not aﬀect the density, the density is a pure function of the scalar vector φ. Favre averages are deﬁned by Q= ρQ , ρ (18) and Favre-decomposition into mean and ﬂuctuation is deﬁned as Q=Q+Q , (19) 332 Roekaerts in which Q denotes the Favre-ﬂuctuation. Note that with the deﬁnition of Favre- and Reynolds averages and ﬂuctuations Q =0 and in general, Q =0 Q = 0. It is also useful to deﬁne the Favre-probability density function by f φ(ψ) = f φ(ψ) ρ|φ = ψ ρ(ψ) = f φ(ψ) ρ ρ (20) Q = 0, Because the density is a pure function of the scalar space variables, the conditional average reduces to a function in ψ and the second equality holds. Favre-averages can be expressed in terms of the Favre-PDF according to +∞ Q(φ) = −∞ Q(ψ)f φ(ψ)dψ. (21) The Favre-PDF has the same properties as a standard probability density function and all properties discussed in the previous sections (multivariate statistics, conditional statistics) can be applied in a straightforward manner. 3 Averaged equations By averaging the ﬂow equations a set of equations describing the mean ﬂow properties is obtained. Deﬁning ∂ D ∂ , = + Ui Dt ∂t ∂xi (22) and Favre-averaging the momentum and species equations (2) and (4) gives D ∂ p ∂ ∂ Ui = − − Tij − ρ uj ui Dt ∂xi ∂xj ∂xj ∂ ∂ D α φα = − Jj + ρ Sα − ρ uj φα , ρ Dt ∂xj ∂xj ρ (23) (24) in which the last terms of the RHSs of the equations represent the Reynolds stress and the Reynolds ﬂux which occur in unclosed form. These terms contain second moments of the velocity distribution and joint velocity-scalar distribution respectively, and cannot be expressed in terms of the ﬁrst moments or means. Another unclosed term in the equations is the averaged reaction term. Reaction rates are in general highly non-linear functions of composition and temperature and the averaged reaction rate cannot be expressed as a function of mean concentrations. As a framework for turbulence modelling we now consider second moment closure (SMC; Launder et al. 1975, Lumley 1980). The simpler eddy-viscosity [10] Reacting ﬂows and probability density function methods 333 models are still widely used for reacting ﬂow computations and after some modiﬁcations they perform reasonably well for simple jet ﬂows. However, in recent years the increased computer performance has made the use of SMC also for reacting ﬂow computations (Jones 1994) more tractable. Also, using a hybrid Monte Carlo method to compute the joint velocity-scalar PDF (see [21]), it is preferable to use SMC from a theoretical point of view. The conservation equations for the Reynolds stresses and Reynolds ﬂuxes can be derived by standard methods from equations (2) and (23). A common approach to modeling of variable-density ﬂows is to apply the constant-density second-moment closure models to variable-density ﬂows simply by recasting the Reynolds-averaged terms into Favre averages. However, the variabledensity second-moment equations contain additional terms, containing ∂ Uk /∂xk , ui or φα , which are zero in constant-density ﬂows. The full secondmoment equations for variable density ﬂows and modeling of the unclosed terms are reported by Jones (1980), Jones (1994) and references therein. 3.1 Reynolds-stress equations Assuming high Reynolds number, viscous terms are neglected except for the viscous dissipation term ij . The Reynolds stress equations for variable density ﬂows then read ρ D u u Dt i j = −ρui uk − ui −ρ ij ∂ Uj ∂ Ui − ρuj uk ∂xk ∂xk (25a) (25b) (25c) (25d) (25e) (25f) ∂p ∂p ∂p 2 + uj − δij uk ∂xj ∂xi 3 ∂xk ∂ 2 − ρui uj uk + δij uk p ∂xk 3 ∂p ∂p −ui − uj ∂xj ∂xi ∂uk 2 . + δij p 3 ∂xk Here the Reynolds average is denoted by an overbar and the combined use of an overbar and tilde, as in x, is used to denote the Favre average of a long expression. The terms on the RHS are: (25a) the production by mean shear Pij , (25b) the pressure-strain correlation Πij , (25c) the viscous dissipation ij , (25d) the turbulent ﬂux Tij , and two terms which are zero in constant density ﬂows containing (25e) a mean pressure gradient, and (25f) the trace of the ﬂuctuating strain tensor. The production terms are in closed form whereas the ﬂuctuating pressure, dissipation, turbulent ﬂux, and ﬂuctuating density terms have to be modeled. 334 Roekaerts The viscous dissipation ij is modeled by assuming local isotropy at the smallest scales where viscous dissipation takes place. The dissipation model then reads: 2 (26) δij . ij = 3 For the dissipation of turbulent kinetic energy the standard dissipation equation is solved. The triple correlation terms ui uj uk present in the ﬂux terms can be modelled by a generalized gradient diﬀusion model that reads ∂ ui uj k ui uj uk = −Cs uk ul , ∂xl (27) where the constant Cs has a value around 0.25. Modeling of the ﬂuctuating density terms can be found in Jones (1980), Jones (1994). The ﬁnal unclosed term is the pressure-strain redistribution term which has been, and probably will remain, the focal point of Reynolds-stress modeling. This term does not produce or destroy turbulent kinetic energy but only redistributes energy over the components of the stress tensor. Its modelling will be discussed in [21]. 3.2 Reynolds-ﬂux and scalar-variance equations Assuming only one scalar variable, the equations for the turbulent scalar ﬂux or Reynolds ﬂux ui φ and for scalar variance φ φ respectively read ρ D u φ Dt i = −ρuj φ −φ − ∂p ∂xi ∂ Ui ∂φ − ρu i u j ∂xj ∂xj (28a) (28b) (28c) (28d) (28e) ∂ ρu u φ ∂xj j i ∂p −φ ∂xi +ρui S(φ), and, ρ D φ φ Dt = −2ρuj φ −ρ φ ∂φ ∂xj (29a) (29b) (29c) (29d) ∂ ρu φ φ − ∂xj j +2ρφ S(φ). [10] Reacting ﬂows and probability density function methods 335 The terms on the right-hand sides of these equations are, in analogy with the Reynolds-stress equations, the production terms (28a) and (29a), the pressure scrambling term Πφ (28b), the viscous dissipation of scalar variance φ i (29b), the turbulent ﬂuxes (28c) and (29c) and an additional mean pressuregradient term which is zero in constant density ﬂows (28d). Furthermore, for a reacting scalar, unclosed reaction source terms (28e) and (29d) appear in the equations. The dissipation rate of scalar variance g = φ φ is linked to the dissipation rate of mechanical energy by ωφ ≡ φ g = Cφ ω ≡ Cφ . k (30) The empirical constant Cφ has the standard value 2. Although the constant may vary throughout the ﬂow it is reasonably constant in diﬀusion ﬂames where the ﬂuctuations in velocity and scalars are induced by the same process; namely the diﬀerent velocities and concentrations of fuel and oxidizer streams. 4 One-point scalar PDF closure Solving the moment conservation equations for turbulent reacting ﬂows (24), (28a–e) and (29a–d) the averaged reaction rate term poses a great problem. Because of the highly non-linear behavior of this term, the average value cannot be expressed accurately as a function of scalar mean and variances but the full scalar PDF shape is important. Note that the PDF determines the higher moments of the distribution but, in general, a ﬁnite number of higher moments does not specify the PDF. In turbulent diﬀusion ﬂames a ﬁrst simpliﬁcation can be made using the socalled conserved-scalar approach. A theoretical analysis of the conserved-scalar description can be found, for example, in Williams (1985). The Damkohler number Da is deﬁned as the ratio between a characteristic turbulence timescale and a chemical time-scale. For high Damkohler number reaction is fast compared to turbulence and the reaction rate is limited by the turbulent mixing of fuel and oxidizer. Chemistry can then be described by a variable that indicates the degree of mixedness. This mixture fraction is deﬁned as a normalized element mass fraction. Using this deﬁnition, mixture fraction is one in the fuel stream and zero in the oxidizer stream. Assuming equal diﬀusivities for all species, the conservation equation for mixture fraction reads ρ ∂ξ ∂ξ ∂ = + ρUi ∂t ∂xi ∂xj ρI D ∂ξ ∂xj , (31) with zero chemical source term. In other words, mixture fraction is a conserved scalar. The relation between the physical scalar variables and mixture fraction is given by the speciﬁc conserved-scalar chemistry model used (mixed-is-burnt model, equilibrium model). In a more reﬁned description the inﬂuence of local 336 Roekaerts ﬂow conditions (hence ﬁnite Da eﬀects) are taken into account by assuming that the chemical composition can be retrieved from that of a strained laminar ﬂamelet. Then scalar dissipation rate enters as a second independent variable. A detailed explanation can be found in Peters (1984) and Peters (2000). Even when chemistry can be described fully by mixture fraction, the thermochemical variables are still non-linear functions of mixture fraction. To accurately describe the mean thermo-chemistry, the turbulent mixture fraction ﬂuctuations have to be known. A common way to model these ﬂuctuations is by the assumed-shape PDF method. In this method, the scalar PDF of ξ is modeled as a known function of several of its lower moments. For a detailed study of assumed shape PDF methods including an overview of possible assumed PDF models the reader is referred to Peeters (1995). Nowadays in most studies the β-function PDF model is used. The β-function is selected because it can, as a function of its parameters, take various forms that resemble physically realistic scalar PDFs (e.g. single-delta function PDFs in fuel or oxidizer streams, or Gaussian-like PDFs in well mixed situations). Assumed-shape PDF methods are useful for modeling turbulent reacting ﬂows with single-conserved-scalar chemistry and ﬂamelet models. In a situation with multiple reacting scalars, chemistry can inﬂuence their joint PDF shapes dramatically and scalar correlations can have large inﬂuence on the reaction rates. Attempts to model the joint scalar PDF in a functional assumedshape form have not been very successful. Use of multi-scalar chemistry models requires an accurate description of the joint scalar distribution. This detailed statistical information can be obtained by solving the joint scalar PDF transport equation by means of a Monte Carlo method. This approach will be addressed in [20]. An alternative method for closure of the mean reaction rate equations related to both PDF and laminar ﬂamelet methods is conditional moment closure (CMC). It predicts the conditional averages and higher moments of scalar variables, with the condition being the value of the mixture fraction. A complete description can be found in Klimenko and Bilger (1999). 5 One point joint velocity-scalar PDF closure Knowledge of the joint scalar PDF is suﬃcient to close the mean reaction rate but to model the joint scalar PDF equation an assumption has to be introduced on the correlation between velocity ﬂuctuations and ﬂuctuations in the scalar PDF. This can be seen as a generalisation of the closure problem of the Reynolds ﬂux appearing in equation (24). This closure problem would be absent if the joint velocity-scalar PDF would be known. Indeed knowledge of the joint velocity-scalar PDF would at once imply a closure of all unknown terms in the mean transport equations: the Reynolds stress, the Reynolds ﬂux and the mean chemical source term. This observation and the fact that a new class of elegant Lagrangian solution algorithms can be used make calculation of [10] Reacting ﬂows and probability density function methods 337 the velocity-scalar PDF an attractive alternative that will be further explored in [21]. References Dopazo, C. (1994) ‘Recent developments in PDF methods’. In Turbulent Reacting Fows, P. Libby and F. Williams (eds.), Academic Press, 375–474. Fox, R.O. (1996) ‘Computational methods for turbulent reacting ﬂows in the chemical process industry’, Revue de l’Institut Fran¸ais du P´trole, 51(2), 215–243. c e Jones, W.P. (1980) ‘Models for turbulent ﬂows with variable density and combustion’. In Prediction Methods for Turbulent Flows, W. Kollmann (ed.), Hemisphere, 379– 421. Jones, W.P. (1994) ‘Turbulence modelling and numerical solution methods for variable density and combusting ﬂows;. In Turbulent Reacting Flows, P. Libby and F. Williams (eds.), Academic Press, 309–374. Klimenko, A.Y. and Bilger, R.W. (1999) ‘Conditional moment closure for turbulent combustion’, Prog. Energy Comb. Sci., 25, 595–687. Launder, B.E., Reece, G.J. and Rodi, W. (1975) ‘Progress in the development of a Reynolds-stress turbulence closure’, J. Fluid Mech., 68, 537–566. Lumley, J.L. (1980) Second-order modeling of turbulent ﬂows. In Prediction Methods for Turbulent Flows, W. Kollmann (ed.), Hemisphere, 1–31. Peeters, T.W.J. (1995) Numerical Modeling of Turbulent Natural-Gas Diﬀusion Flames. PhD thesis, Delft Universitity of Technology. Peters, N. (1984) ‘Laminar diﬀusion ﬂamelet models in non-premixed turbulent combustion’, Prog. Energy Comb. Sci., 10, 319–339. Peters, Norbert (2000) Turbulent Combustion. Cambridge University Press. Pope, S.B. (1985) ‘PDF methods for turbulent reactive ﬂows’, Prog. Energy Comb. Sci., 11, 119–192. Pope, S.B. (2000) Turbulent Flows. Cambridge University Press. Williams, F.A. (1985) Combustion Theory, second edition. Benjamin/Cummings. Part B. Flow Types and Processes and Strategies for Modelling them 11 Modelling of Separating and Impinging Flows T.J. Craft 1 Introduction Flows involving separation, reattachment and impingement occur widely in many diverse engineering applications. A number of cooling and drying processes rely on the high heat transfer rates that can be obtained by impinging ﬂuid onto a solid surface. Separation and reattachment are, of course, found in numerous situations, including external aerodynamics, ﬂow over obstacles and internal ﬂow through ducts and pipes with rapidly varying cross-section or ﬂow direction. Many of these internal ﬂows are also associated with heat transfer, such as internal cooling passages for gas-turbine blades. Since heat transfer rates are predominantly determined by the ﬂow behaviour in the immediate vicinity of the wall, it is often necessary to employ low-Reynolds-number turbulence models, which can adequately resolve the near-wall region, when computing applications involving wall heating or cooling. However, the turbulence mechanisms near reattachment or impingement zones are signiﬁcantly diﬀerent from those found in simple shear ﬂows where most turbulence models have been developed. In particular, as will be seen, the popular ε based models are often found to predict extremely large lengthscales in such ﬂows, leading to the prediction of excessive heat transfer rates and the necessity of including additional modelling terms to correct for the defect. Furthermore, the irrotational straining found in impinging ﬂows exposes a number of weaknesses in both eddy-viscosity based models and in widely-used stress transport closures. As an example of the problems encountered in computing impinging ﬂows, Figure 1 shows the predicted and measured Nusselt number, plotted against radial distance from the stagnation point, in an axisymmetric impinging jet ﬂow studied experimentally by Baughn and Shimizu (1989) and Cooper et al. (1992). The jet issues from a long length of pipe, at a Reynolds number of 23000, and impinges perpendicularly onto a ﬂat plate at a distance of 2 jet diameters from the pipe exit. The ﬂow has been computed using a zonal modelling approach, with a high-Reynolds-number stress transport scheme in the fully turbulent region, and the Launder–Sharma k-ε model in the near-wall viscosity-aﬀected region. Without additional modiﬁcations, it can be seen that, whilst the predictions are in agreement with the data at large radial distances 341 342 Craft Figure 1: Nusselt number predictions —— Without Yap term; - - - With Yap term; Baughn and Shimizu (1989). in an impinging jet. Symbols: experiments of (where the ﬂow becomes a radial wall jet), the model fails, fairly spectacularly, in the stagnation zone, overpredicting the Nusselt number by a factor of four or more. The following sections consider the problems encountered in the modelling of impinging and reattaching ﬂows, including a discussion of some of the diﬀerent solutions proposed, and example applications. 2 Turbulence Lengthscales in Impingement and Reattachment Regions In an equilibrium boundary layer, the turbulence lengthscale grows linearly with distance from the wall, and the widely-used k-ε model has been tuned to reproduce this behaviour in such a ﬂow. However, in non-equilibrium situations (for example, where there is separation and reattachment, or impingement) the model is known to return signiﬁcantly larger lengthscales (Yap 1987). Figure 2, taken from Craft (1991), shows the predicted normalized lengthscale l/(cl x) (where l = k 3/2 /ε), plotted against distance from the wall x, at two radial positions in the impinging jet for which heat transfer results were presented in Figure 1. As can be seen, at a radial distance of 3 jet diameters, where the ﬂow resembles a simple shear ﬂow, the predicted near-wall lengthscale is only slightly larger than its equilibrium value. However, along the stagnation line r/D = 0, the k-ε model in the near-wall region returns a lengthscale more than six times greater than that found in an equilibrium boundary layer. [11] Large modelling of separating and impinging ﬂows 343 2.1 Lengthscale Correction Terms Yap (1987) studied the case of ﬂow and heat transfer through an abrupt pipe expansion. He noted that the Launder–Sharma k-ε model overpredicted heat transfer rates around the reattachment point, and traced this to the fact that it also returned very large turbulence lengthscales in this region. Figure 3 (taken from Yap 1987) shows the lengthscale predicted by the Launder– Sharma model, plotted against distance from the wall, in the reattachment region, together with the linear equilibrium lengthscale that would be prescribed in a 1-equation model. As can be seen, the standard ε equation returns lengthscales signiﬁcantly larger than those found in an equilibrium situation and, as a result, the heat-transfer (Figure 4) is predicted considerably too high around the reattachment point. To remedy this excessive lengthscale prediction Yap proposed adding an extra source term in the ε equation: 2 2 3/2 /ε 3/2 /ε k k ε Yε = max 0.83 , 0 (2.1) −1 k 2.5y 2.5y where y is the distance to the wall. If the predicted lengthscale l = k 3/2 /ε is larger than the equilibrium value of 2.5y, Yε acts to increase ε, thus decreasing the lengthscale and driving it towards its equilibrium value, as can be seen from the line labelled “damped ε equation” in Figure 3. The eﬀect of this reduction in lengthscale was found to improve the heat transfer predictions signiﬁcantly, as seen in Figure 4. Figure 2: Predicted lengthscales in an impinging jet. —— Without Yap term, - - - With Yap term. Predictions were obtained using a near-wall lowReynolds-number k-ε model and a high-Reynolds-number stress transport model in the outer region: the vertical dashed line represents the interface between the two models. (In this ﬁgure x denotes the normal distance from the wall.) 344 Craft Whilst the Yap correction has been found to be helpful in a wide range of non-equilibrium ﬂows, it does require one to prescribe the wall-normal distance, which can be diﬃcult, or impossible, in complex ﬂow geometries. Hanjali´ (1996) proposed eliminating the explicit dependence on wall-distance by c making use of the gradient of the lengthscale normal to the wall. Iacovides and Raisee (1997) developed this idea further to include the eﬀect of wall damping across the sublayer, and to make the term completely independent of wall geometry. By diﬀerentiating the usual Wolfshtein (1969) equilibrium lengthscale prescription, and then replacing y ∗ by the turbulent Reynolds number Rt , one can obtain an expression for the gradient of the equilibrium lengthscale: De = dle = cl [1 − exp (−Bε Rt )] + Bε cl Rt exp (−Bε Rt ) dy (2.2) with cl = 2.55 and Bε = 0.1069. Iacovides and Raisee then introduced the quantity F = (Dl − De )/cl (2.3) Figure 3: Predicted lengthscales in the reattachment region downstrean of an abrupt pipe expansion. [11] Large modelling of separating and impinging ﬂows 345 Figure 4: Predicted heat transfer downstream of an abrupt pipe expansion. — - — k-ε model without Yap term; - - - k-ε model with Yap term. Symbols, experimental data (Yap 1987). where Dl is the predicted lengthscale gradient deﬁned as Dl = ∂l ∂l ∂xj ∂xj 1/2 (2.4) with l = k 3/2 /ε, and De is the equilibrium lengthscale gradient from equation (2.2). They then proposed replacing the Yap correction with the term SN Y = max 0.83 ε2 F (F + 1)2 , 0 k (2.5) They successfully applied this term to the prediction of heat transfer in a number of rib-roughened ducts and channels, and Figure 5 shows an example of the predicted Nusselt number in a ribbed pipe, using the Launder–Sharma scheme with both the original Yap correction and their own proposal, equation (2.5). 2.2 Alternative Lengthscale Equations To avoid the lengthscale prediction problems associated with the ε equation, one alternative is to consider the use of a variable other than ε as the subject for the lengthscale-determining transport equation. Wilcox (1988, 1991) proposed a k-ω model (where ω ≡ ε/k) which appeared attractive in that it had fewer additional near-wall terms than those required in the ε equation, and was claimed not to need such ﬁne near-wall grids (although ω does have the rather unappealing property that it goes to inﬁnity at a wall). Soﬁalides (1993) tested 346 Craft Figure 5: Predicted heat transfer in a ribbed pipe, using the Launder–Sharma k-ε model. – – with Yap term; —— with equation (2.5). this model in an impinging jet ﬂow and, although the results suﬀered from the failures associated with the linear EVM stress-strain relation (see later section on impingement problems), it did return qualitatively the correct shape for the heat transfer proﬁle in the stagnation region without any additional lengthscale correction terms. He then retuned the modelled equation, employing it in conjunction with a non-linear stress-strain relation, and Figures 6 and 7 (taken from his dissertation) show fully-developed pipe ﬂow results and heat transfer predictions in the impinging jet. For comparison, it should be noted that if the ε-based non-linear eddy-viscosity model of Suga (1995) (for which results will be presented later) is used without the Yap lengthscale correction in the ε equation, the predicted heat transfer shows a signiﬁcantly high peak value at the stagnation point, in contrast to the very ﬂat proﬁle returned by the model of Soﬁalides. Although the model was not widely tested, and was subsequently found to perform rather poorly in transitional ﬂows, the results reproduced here do show some promise, suggesting that this may be an area deserving of further attention. 3 Turbulence Energy Production in Impingement Regions Although the addition of a lengthscale correction to the ε equation can be seen in Figure 1 to bring a signiﬁcant improvement to the prediction of stagnation heat transfer, the predicted Nusselt number is still too high by almost a factor of 2, indicating that further modelling reﬁnements are needed for this type of ﬂow situation. The modelling strategy employed in the calculations of Figure 1 was a zonal approach, with a high-Reynolds-number RSM coupled to a near-wall low- [11] Large modelling of separating and impinging ﬂows 347 Reynolds-number k-ε model. Although one might expect the high predicted levels of heat transfer to be largely due to weaknesses in the k-ε model, since the heat transfer will be strongly aﬀected by the predicted near-wall turbulence, it will be seen later that there are, in fact, signiﬁcant weaknesses in both the k-ε model and the stress transport model employed in these calculations. 3.1 Weaknesses of the Eddy-Viscosity Formulation Figure 8 shows corresponding heat transfer results in the impinging jet ﬂow using four diﬀerent turbulence models: the Launder–Sharma k-ε model throughout the ﬂow domain; the Basic linear RSM, with the Launder–Sharma model in the near-wall region, and two further RSM’s, again employing the Launder– Sharma model in the near-wall layer. The Yap correction is included in all these calculations and, from the k-ε results, it is clear that there is still a weakness in this model, resulting in overprediction of heat transfer in the impingement region. A reason for this overprediction can be seen from examining the wallnormal and radial rms velocities v and u in the stagnation region. Proﬁles of these, plotted against distance from the wall, are shown in Figure 9 for a selection of radial locations. Limiting attention to the k-ε model for the moment, the predicted levels of turbulence energy in the stagnation region are clearly too high, and this will, of course, lead to an overestimation of the heat transfer. If one considers the case of an axisymmetric irrotational mean strain (Figure 10), such as is found along the stagnation line of an impinging jet, the production rate of k can be written Pk = − u 2 ∂U ∂V ∂W + v2 + w2 ∂x ∂y ∂z ∂V = − v 2 − u2 ∂y (3.1) Figure 6: Mean velocity proﬁles at various Reynolds number in fully-developed pipe ﬂow using a non-linear k-ω model. From Soﬁalides (1993). 348 Craft Figure 7: Heat transfer proﬁles in an impinging jet ﬂow using a non-linear k-ω model. From Soﬁalides (1993). Figure 8: Nusselt number predictions in an impinging jet (from Craft et al. 1993). — - — Launder–Sharma k-ε model; – – Basic RSM with Gibson– Launder wall-reﬂection; —— Basic RSM with Craft–Launder wall-reﬂection; - - - high Re number TCL model; Symbols: measurements of Baughn and Shimizu (1989). since ∂U/∂x = ∂W/∂z = −1/2∂V /∂y from continuity and u2 = w2 by symmetry. If a linear EVM is employed, the normal stresses are approximated as v 2 = 2/3k − 2νt ∂V ∂y u2 = 2/3k − 2νt ∂U ∂V = 2/3k + νt ∂x ∂y (3.2) While equation (3.1) shows that the contributions of the two normal stresses to the production of k should be of opposite sign to one another, the EVM formulation of equation (3.2) indicates that in this case, because of the op- [11] Large modelling of separating and impinging ﬂows 349 Figure 9: Proﬁles of rms velocity ﬂuctuations at various radial positions in an impinging jet (from Craft et al. 1993). Lines as in Figure 8; Symbols: measurements of Cooper et al. (1992). Figure 10: Schematic diagram of the axisymmetric expansion ﬂow in an impinging jet. posite signs associated with the velocity gradient factor, both terms give a contribution of the same sign, resulting in a generation term Pk = 3νt ∂V ∂y 2 (3.3) 350 Craft It is this misrepresentation of the normal stress anisotropy, inherent in the linear EVM formulation, that leads to the high predicted levels of turbulence energy, and consequently heat transfer, in a stagnation ﬂow. Although this failure is a direct result of the linear stress-strain relation misrepresenting the normal stresses, there have been several attempts to resolve this weakness within a linear EVM framework, and the following sections consider some alternative methods that have been proposed to alleviate this problem, before attention is turned to more advanced modelling strategies. 3.2 Kato–Launder Modiﬁcation Kato and Launder (1993) computed the ﬂow past a square cylinder, with the k-ε model, and noted that the above stagnation anomaly occurred on the leading face of the cylinder. They also noted that, when using a linear EVM, i the exact production rate of turbulence energy Pk = −ui uj ∂Uj can be written ∂x 2 where S is the non-dimensional strain rate as Pk = cµ εS S= k [2Sij Sij ]1/2 ε and Sij = 1 2 ∂Ui ∂Uj + ∂xj ∂xi (3.4) To improve predictions they investigated replacing this form of generation by Pk = cµ εSΩ where Ω= k [2Ωij Ωij ]1/2 ε and Ωij = 1 2 (3.5) ∂Ui ∂Uj − ∂xj ∂xi (3.6) In a simple shear the strain and vorticity parameters, S and Ω, are identical. However, in an irrotational strain, such as that found in an impinging ﬂow, Ω vanishes which will lead to very low production of k, thus reducing the predicted levels of k in this region. Figure 11 (taken from Kato and Launder 1993) shows predictions of k along the centerline of the cylinder using the k-ε model both with wall-functions, and with the k-ω model as a near-wall layer model. When the standard production term is employed, a substantial increase of k is seen upstream of the cylinder, which is not present with the modiﬁed production of equation (3.5). This replacement, coupled with a low-Reynolds-number treatment of the near-wall region, also returns improved predictions downstream of the cylinder. Although the modiﬁcation does appear to dramatically improve impinging ﬂows it is, unfortunately, not universally helpful: in further testing, Suga (1995) discovered that the replacement of S by Ω in the generation rate worsened the prediction of ﬂow through a curved channel. [11] Large modelling of separating and impinging ﬂows 351 3.3 Durbin’s Modiﬁcation Durbin (1996) suggested an alternative method of remedying the production rate of k in a stagnation region. As noted above, if the eddy-viscosity formulation is employed, then the generation rate can be written as Pk = cµ εS 2 . Durbin argued that towards the stagnation point the timescale k/ε becomes large, leading to high values of S, and the quadratic dependence of Pk on S leads to the observed high values of k. By considering what constraint it would be necessary to impose on the turbulent viscosity in order to prevent one of the predicted normal stresses from becoming negative in an axisymmetric strain ﬁeld, he suggested that the turbulent viscosity should be modelled as νt = cµ kT where the timescale T is taken as T = min 2 k k ,β ε 3cµ ε 3 4|(S)2 | (3.8) (3.7) for some tunable constant β. In regions where the strain rate S is large, such as in the stagnation region, the second term in equation (3.8) prevents excessive growth of T , thus limiting the generation rate of k, which can now be written as Pk = cµ ε ε T S2 k (3.9) Figure 11: Turbulent kinetic energy along the centerline of a square cylinder. —— k-ε /k-ω with modiﬁed production; — - — k-ε /k-ω with standard production; – – k-ε with wall-functions; - - - 2nd moment closure with wallfunctions (Franke and Rodi 1991); Symbols: experiments. 352 Craft When the second term of equation (3.8) is active in T , the generation rate now depends only linearly on strain rate S, and Durbin demonstrated that such a treatment can avoid the excessive generation rates otherwise found in stagnation ﬂows. 4 4.1 Application of Higher-Order Models to Impingement Flows Non-Linear Eddy-Viscosity Models The underlying weakness with the linear EVM in a stagnation ﬂow was seen above to be associated with the predicted normal stress anisotropy. A nonlinear EVM, which includes higher order terms in the stress-strain relation, does, of course, have the potential to return improved predictions of the normal stresses and thus to improve the prediction of k in the stagnation region, via a better representation of its generation rate. In his development of cubic non-linear eddy-viscosity models, Suga (1995) considered the case of ﬂow impingement, including the axisymmetric impinging jet amongst the ﬂows he employed for model tuning. The examples here show results both for the 2-equation NLEVM developed by Suga (Craft et al. 1996), and also his 3equation version, where a transport equation was solved for the anisotropy invariant A2 in addition to those for k and ε (Craft et al. 1997). Figure 12 shows the predicted normal stresses along the stagnation line, showing that the NLEVM’s are capable of returning the correct levels of turbulence energy in this region, leading to improved heat transfer predictions shown in Figure 13. It is worth noting that even with the improved normal stress predictions, both of these models also include a term similar to the Yap lengthscale correction in the ε equation to improve the heat-transfer predictions. In further work on the use of the 2-equation NLEVM referred to above, Craft et al. (1999) demonstrated that a form of lengthscale correction similar to that outlined in equation (2.5) could be employed with the model, yielding a completely geometry-independent model that still returned predictions as good as those shown in Figure 13. 4.2 Reynolds Stress Transport Models Since stress transport models do not employ the eddy-viscosity formulation for ui uj , they do not share quite the same weaknesses in impinging ﬂows as those encountered above with EVM’s. However, equation (3.1) does indicate that if the turbulence energy is to be correctly predicted, it is necessary for the model to return the correct normal stress anisotropy. In a stress transport scheme this suggests that the accurate modelling of the redistribution process φij can be expected to be crucial in capturing such ﬂows. As will be seen below, some [11] Large modelling of separating and impinging ﬂows 353 widely-used stress models fail to represent this process reliably in stagnation ﬂows. In addition to the k-ε model results, Figure 9 also shows the predicted normal stresses employing the Basic linear RSM, with the Gibson and Launder Figure 12: Wall-normal rms velocity component predictions along the stagnation line of an impinging jet using NLEVM’s (From Suga 1995). —– 3-equation NLEVM; — - — 2-equation NLEVM; – – Launder–Sharma k-ε ; Symbols: measurements of Cooper et al. (1992). Figure 13: Impinging jet heat transfer predictions using NLEVM’s. (From Suga 1995) —– 3-equation NLEVM; — - — 2-equation NLEVM; – – Launder– Sharma k-ε ; Symbols: measurements of Baughn and Shimizu (1989). 354 (1978) wall-reﬂection terms: φij1 = −c1 ε (ui uj /k − 2/3δij ) φij2 = −c2 (Pij − 1/3Pkk δij ) ε φw = c1w (ul uk nl nk δij − 3/2ui uk nj nk − 3/2uj uk ni nk ) fy ij1 k φw = c2w (φlk2 nl nk δij − 3/2φik2 nj nk − 3/2φjk2 ni nk ) fy ij2 Craft (4.1) (4.2) (4.3) (4.4) where nk is the unit vector normal to the wall, fy = l/(2.5y), l is the turbulence lengthscale k 3/2 /ε and y the distance to the wall. Since the above stress model is only valid at high-Reynolds-numbers, it is used with the Launder–Sharma k-ε model in the near-wall region, and consequently the very near-wall normal stresses are not shown in the ﬁgure. However, along the stagnation line it is clear that v 2 , the stress normal to the wall, is signiﬁcantly overpredicted by the stress model — in fact returning results only slightly better than those predicted by the linear EVM — and leads to the overprediction of heat transfer seen in Figure 8. In this case the failure can be traced to the wall-reﬂection model employed in φw . In a simple shear ﬂow (with mean strain ∂U/∂y), energy is generated ij2 in u2 by P11 . A proportion is transferred into v 2 by φ222 = 1/3c2 P11 , whilst the wall-reﬂection term φw opposes this transfer, thus leading to the de222 sired damping of v 2 . In an impinging ﬂow, however, v 2 is itself generated by P22 = −2v 2 ∂V /∂y. The pressure-strain element φij2 then removes some of this generation and redistributes it into the other stress components. The wall-reﬂection contribution φw = −2cw φ222 fy , in opposing this redistribu222 tion, thus acts to increase v 2 as the wall is approached, and it is to a large extent this which leads to the overprediction of v 2 seen in Figure 9. To overcome the above problem, Craft and Launder (1992) proposed an alternative wall-reﬂection model to be used in place of the above φw . They ij2 considered all possible combinations of terms linear in the Reynolds stresses and mean strains, and included wall-normal vectors to give directional sensitivity to the model. By ensuring that the resultant model behaved correctly in both wall-parallel shear ﬂows and impinging ﬂows, they arrived at the form: φw = −0.08 ij2 ∂Ul ul uk (nt nt δij − 3ni nj ) (l/2.5y) ∂xk ∂Uj ∂Uk ∂Ui −0.1kalm nl nk δij − 3/2 nl nj − 3/2 nl ni (l/2.5y) ∂xm ∂xm ∂xm ∂Ul nl nm (ni nj − 1/3nk nk δij ) (l/2.5y) (4.5) +0.4k ∂xm Predictions of stresses and heat transfer using this model can be seen in Figures 8 and 9. Clearly the modiﬁed φw has a signiﬁcant eﬀect in the stagnaij2 tion region, reducing turbulence energy levels, and thus predicted heat transfer rates, to values close to those measured experimentally. [11] Large modelling of separating and impinging ﬂows 355 The ﬁnal model combination shown in Figures 8 and 9 is the TCL model, described in [3]. In these calculations it has been implemented in its highReynolds-number form, with the Launder–Sharma k-ε model in the nearwall region, and includes an additional wall-correction term similar to equation (4.5), but with slightly modiﬁed coeﬃcients. It also is seen to return results in reasonable agreement with the experimental data. As noted earlier, heat-transfer predictions in impinging ﬂows are strongly inﬂuenced by the details of the near-wall turbulence modelling. If a full secondmoment closure is employed in the outer, fully turbulent, region, there may be strong arguments for adopting a similar level of modelling in the near-wall region (or, at any rate, using something more complex than a linear EVM). Although the discussion of a TCL (two-component limit) model in [3] addressed only the issues related to devising such a scheme for high-Reynolds-number ﬂows, the closure described has, more recently, been extended to account for low-Reynolds-number and inhomogeneity eﬀects found in near-wall regions. The details are given in Craft and Launder (1996) and Craft (1998a), but essentially involve the inclusion of a number of damping functions and inhomogeneity corrections, which employ the turbulent Reynolds number and the stress anisotropy invariants A and A2 to account for viscous eﬀects and those due to high levels of turbulence anisotropy. Instead of employing the wall-normal vector and distance, lengthscale gradients are used to sensitize the model to regions and directions where there is strong inhomogeneity (such as near a wall), without introducing any geometry-related quantities. The dissipation rate equation employed is, again, an extension of that reported in [3], and includes a lengthscale correction term of the form described in equation (2.5), but with a smaller coeﬃcient than that employed by Iacovides and Raisee (1997) in their eddy-viscosity model. The above low-Reynolds-number TCL closure has been applied by Craft (1998b) to the impinging jet problem, and Figure 14 shows the predicted Nusselt number, which is seen to be in reasonable agreement with the experimental data. The ﬁgure also shows the eﬀect of neglecting the lengthscale correction term in the ε equation, again highlighting the importance of such a term in near-wall stagnation regions. Two more complex applications, employing essentially the same low-Reynolds-number TCL model are shown in Figures 15 and 16. The ﬁrst of these concerns transonic ﬂow (with a freestream Mach number of 0.875) over an axisymmetric bump, studied experimentally by Bachalo and Johnson (1986). Besides predicting the correct location of the shock at x/c ≈ 0.65, the computation needs to resolve the separation bubble at the end of the bump which causes the pressure plateau at around x/c = 1. The wall-pressure distribution in Figure 15 shows the TCL results to be in better agreement with the data than those of either the linear k-ε EVM or the Launder and Shima (1989) stress transport model. The second case presented here is one of the afterbody 356 Craft ﬂows studied by Carson and Lee (1981). The geometry, shown in Figure 16, resulted in signiﬁcant shock-induced ﬂow separation from the boattail region Figure 14: Nusselt number predictions in an impinging jet using the lowReynolds-number TCL stress model (Craft 1998b). —– TCL model with lengthscale correction; - - - TCL model without lengthscale correction; Symbols: measurements of Baughn and Shimizu (1989). Figure 15: Wall-pressure distributions for the axisymmetric bump ﬂow of Bachalo and Johnson (1986) (From Batten et al. 1999b). SST: model of Menter (1994); MLS: model of Launder and Shima (1989); MCL: Low-Re TCL closure. [11] Large modelling of separating and impinging ﬂows 357 Figure 16: IsoMach contours, boattail Cp and internal nozzle p/pT for the jet Carson and Lee (1981) conﬁguration 1 afterbody (From Batten et al. 1999a). SST: model of Menter (1994); JH: model of Jakirli´ and Hanjali´ (1995); MCL: c c Low-Re TCL closure. of the afterbody. The ﬁgure shows that although the models tested by Batten et al. (1999a) returned almost identical results in the interior throat section (where the ﬂow is governed by inviscid processes), the TCL model again returned generally better Cp values over the boattail than did the k-ε EVM and 358 Craft Jakirli´ and Hanjali´ (1995) models. In both of the above cases it might be c c noted that the SST model of Menter (1994) appears to give results very similar to those returned by the TCL closure. The former model, however, employs a linear eddy-viscosity relation for the stresses and so cannot be expected to perform well in other complex ﬂows, such as those where the eﬀects of normal straining are important. This can be seen in Section 3.4 of [5], which presents the application of several models to the highly complex ﬂow at Mach 2 around a ﬁn/plate junction. It appears that only second-moment closures are able to capture the separation and reattachment pattern ahead of the ﬁn, and the above TCL model is again seen to return predictions in reasonable agreement with the data. References Bachalo, W.D., Johnson, D.A. (1986), ‘Transonic, turbulent boundary-layer separation generated on an axisymmetric ﬂow model’, AIAA J., 24, 437–443. Batten, P., Craft, T.J., Leschziner, M.A. (1999a), ‘Reynolds-stress modeling of afterbody ﬂows’, in Turbulence and Shear Flow Phenomena – 1 (S. Banerjee, J. Eaton, eds.), Begell House, New York. Batten, P., Craft, T.J., Leschziner, M.A., Loyau, H. (1999b), ‘Reynolds-stresstransport modeling for compressible aerodynamics applications’, AIAA J., 37, 785– 796. Baughn, J.W., Shimizu, S. (1989), ‘Heat transfer measurements from a surface with uniform heat ﬂux and an impinging jet’, ASME J. Heat Transfer, 111, 1096–1098. Carson, G.T., Lee, E.E. (1981), Tech. Rep. Technical report, NASA TP 1953. Cooper, D., Jackson, D.C., Launder, B.E., Liao, G.X. (1992), ‘Impinging jet studies for turbulence model assessment: Part 1: Flow ﬁeld experiments’, Int. J. Heat Mass Transfer, 36, 2675–2684. Craft, T.J. (1991), ‘Second-moment modelling of turbulent scalar transport’, Ph.D. thesis, Faculty of Technology, University of Manchester. Craft, T.J. (1998a), ‘Developments in a low-Reynolds-number second-moment closure and its application to separating and reattaching ﬂows’, Int. J. Heat Fluid Flow, 19, 541–548. Craft, T.J. (1998b), ‘Prediction of heat transfer in turbulent stagnation ﬂow with a new second moment closure’, in Proc. 2nd Engineering Foundation Conference in Turbulent Heat Transfer, Manchester, UK. Craft, T.J., Graham, L.J.W., Launder, B.E. (1993), ‘Impinging jet studies for turbulence model assessment. part ii: An examination of the performance of four turbulence models’, Int. J. Heat Mass Transfer, 36, 2685. Craft, T.J., Iacovides, H., Yoon, J. (1999), ‘Progress in the use of non-linear twoequation models in the computation of convective heat-transfer in impinging and separated ﬂows’, Flow, Turbulence and Combustion, 63, 59–80. [11] Large modelling of separating and impinging ﬂows 359 Craft, T.J., Launder, B.E. (1992), ‘New wall-reﬂection model applied to the turbulent impinging jet’, AIAA J., 30, 2970–2972. Craft, T.J., Launder, B.E. (1996), ‘A Reynolds stress closure designed for complex geometries’, Int. J. Heat Fluid Flow, 17, 245–254. Craft, T.J., Launder, B.E., Suga, K. (1996), ‘Development and application of a cubic eddy-viscosity model of turbulence’, Int. J. Heat and Fluid Flow, 17, 108–115. Craft, T.J., Launder, B.E., Suga, K. (1997), ‘Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model’, Int. J. Heat Fluid Flow, 18, 15. Durbin, P.A. (1996), ‘On the k-3 stagnation point anomaly’, Int. J. Heat Fluid Flow, 17, 89–90. Franke, R., Rodi, W. (1991), ‘Calculation of vortex shedding past a square cylinder with various turbulence models’, Paper 20.1, Proc. 8th Turbulent Shear Flows Symposium, Munich. Gibson, M.M., Launder, B.E. (1978), ‘Ground eﬀects on pressure ﬂuctuations in the atmospheric boundary layer’, J. Fluid Mech., 86, 491. Hanjali´, K. (1996), ‘Some resolved and unresolved issues in modelling nonc equilibrium and unsteady turbulent ﬂows’, in Engineering Turbulence Modelling and Experiments, 3, W. Rodi, G. Bergeles (eds.), 3–18. Iacovides, H., Raisee, M. (1997), ‘Computation of ﬂow and heat transfer in 2D rib roughened passages’, in Proceedings of the Second International Symposium on Turbulence, Heat and Mass Transfer (K. Hanjali´, T. Peeters, eds.), Delft. c Jakirli´, S., Hanjali´, K. (1995), ‘A second-moment closure for non-equilibrium and c c separating high- and low-Re-number ﬂows’, 23.23–23.30, Proc. 10th Turbulent Shear Flows Symposium, Pennsylvania State University. Kato, M., Launder, B.E. (1993), ‘The modelling of turbulent ﬂow around stationary and vibrating cylinders’, Paper 10–4, Proc. 9th Turbulent Shear Flows. Launder, B.E., Shima, N. (1989), ‘Second-moment closure for the near-wall sublayer: development and application’, AIAA J., 27, 1319–1325. Menter, F.R. (1994), ‘Two-equation eddy-viscosity turbulence models for engineering applications’, AIAA J., 32, 1598–1605. Soﬁalides, D. (1993), ‘The strain-dependent non-linear k-ω model of turbulence and its application’, M.Sc. dissertation, Faculty of Technology, University of Manchester. Suga, K. (1995), ‘Development and application of a non-linear eddy viscosity model sensitized to stress and strain invariants’, Ph.D. thesis, Faculty of Technology, University of Manchester. Wilcox, D.C. (1988), ‘Reassessment of the scale determining equation for advanced turbulence models’, AIAA J., 26, 1299–1310. Wilcox, D.C. (1991), ‘Progress in hypersonic turbulence modelling’, in Proc. AIAA 22nd Fluid Dynamics, Plasmadynamics and Laser Conference, Honolulu. Wolfshtein, M. (1969), ‘The velocity and temperature distribution in one-dimensional 360 Craft ﬂow with turbulence augmentation and pressure gradient’, Int. J. Heat Mass Transfer, 12, 301–318. Yap, C.R. (1987), ‘Turbulent heat and momentum transfer in recirculating and impinging ﬂows’, Ph.D. thesis, Faculty of Technology, University of Manchester. 12 Large-Eddy Simulation of the Flow past Bluﬀ Bodies W. Rodi Abstract The ﬂow past bluﬀ bodies, which occurs in many engineering situations, is very complex, involving often unsteady behaviour and dominant large-scale structures; it is therefore not very amenable to simulation by the RANS method using statistical turbulence models. The large-eddy simulation technique is more suitable for these ﬂows. In this section work in the area of large-eddy simulations of bluﬀ body ﬂows is summarised, with emphasis on work by the author’s research group as well as on experiences gained from two LES workshops. Results are presented and compared for the vortex-shedding ﬂow past square and circular cylinders and for the ﬂow around surface-mounted cubes. The performance, the cost and the potential of the LES method for simulating bluﬀ body ﬂows, also vis-`-vis RANS methods, is assessed. a 1 Introduction In many engineering situations, bluﬀ bodies are exposed to ﬂow, generating complex phenomena such as ﬂow separation and often even multiple separation with partial reattachment, vortex shedding, bi-modal ﬂow behaviour, high turbulence level and large-scale turbulent structures which contribute considerably to the momentum, heat and mass transport. For solving practical problems, there is a great demand for methods for predicting such ﬂows and associated heat and mass-transfer processes, in particular the loading, including dynamic loading on the bodies and the scalar transport in the vicinity of structures. Usually the Reynolds number is high in practical problems so that turbulent transport processes are important and must be accounted for in a prediction method in one way or another. Until recently, mainly RANS based methods were used in which the entire spectrum of the turbulent motion is simulated by a statistical turbulence model. In vortex-shedding situations, unsteady RANS equations are solved to determine the periodic shedding motion and only the superimposed stochastic turbulent ﬂuctuations are simulated with the turbulence model. So far, mainly variants of the k-ε eddy-viscosity turbulence model have been used for calculating the ﬂow around bluﬀ bodies, but some results have been reported that were obtained with Reynolds-stress models. 361 362 Rodi The RANS calculations have shown that statistical turbulence models have diﬃculties with the complex phenomena mentioned above, especially when large-scale eddy structures dominate the turbulent transport and when unsteady processes like vortex shedding and bistable behaviour prevail and dynamic loading is of importance. The large-eddy simulation (LES) approach is conceptually more suitable in such situations as it resolves the large-scale unsteady motions and requires modelling only of the small-scale turbulent motion which is less inﬂuenced by the boundary conditions. An introduction to the LES approach is provided in the companion chapter by Fr¨hlich and Rodi o (2001). This approach is computationally more expensive than the RANS approach, but recent advances in computer performance and numerical methods have made LES calculations feasible for ﬂows around bluﬀ bodies (see discussion on relative computer requirements in the concluding section). It should be added here that proper direct numerical simulations (DNS), in which all scales of the turbulent motion are resolved and no model is introduced, are feasible only for relatively low Reynolds numbers as the number of grid points required for resolution of all scales increases approximately as Re3 . Calculations at higher Reynolds numbers have been reported which were called direct simulations (e.g. Tamura et al. 1990, Verstappen and Veldman 1997) but some of these were really quasi-LES calculations in which the task of the subgridscale model to withdraw energy was taken over by a dissipative numerical scheme (e.g. in the case of Tamura et al.). The chapter presents and discusses LES calculations for the ﬂow around various bluﬀ bodies, in particular for ﬂows around square and circular cylinders and past surface- mounted cubes. The chapter cannot give an exhaustive review of such calculations and is based mainly on the results presented at several workshops and produced in the author’s group. For the bluﬀ bodies mentioned, the results obtained with various LES methods will be compared with each other as well as with experiments and in some cases also with RANS calculations. The capabilities, problem areas and the potential of LES calculations also vis-`-vis RANS calculations will be discussed. a 2 LES Methods Used The LES approach is introduced in the companion chapter of Fr¨hlich and o Rodi (2001) which summarises brieﬂy the most commonly used methods for ﬁltering, subgrid-scale modelling, near-wall treatment and numerical solutions. Here a brief summary is given of the methods used for obtaining the bluﬀ-body results presented in this chapter. In all cases, the resolved scales were deﬁned by the mesh size, i.e. no explicit ﬁltering was used. Basically, the resolved quantity is the average over a cell of the numerical mesh and this corresponds to applying a top-hat ﬁlter with ﬁlter width equal to the mesh size. The subgrid-scale stresses representing interactions between resolved larger scales and unresolved smaller scales were mostly modelled explicitly by a [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 363 subgrid-scale model; in some cases no model was used and the dissipative eﬀect of the stresses was achieved by numerical damping introduced by a third-order upwind scheme for the convection terms. Many calculations were carried out with the Smagorinsky (1963) eddy-viscosity model, which can be considered the LES version of the Prandtl mixing-length model. Values of the Smagorinsky constant in the range 0.1 to 0.2 have been used for the bluﬀbody calculations presented below. Near the wall, the length scale is modiﬁed by a van Driest damping function. The popular dynamic approach of Germano et al. (1991) has also been used in a number of cases, mostly with the Smagorinsky model as a base so that the approach then determines the spatial and temporal variation of the Smagorinsky ‘constant’ Cs by making use of the information available from the smallest resolved scales. Some calculations were also carried out with a one-equation model as base in which the velocity scale of the subgrid-scale stresses is calculated from a transport equation for the turbulent subgrid-scale energy (Davidson 1997, Menon and Kim 1996). Finally, mixed models combining a scale-similarity model based on a double ﬁltering approach with the Smagorinsky model have also been applied. Concerning near-wall treatment, often no-slip conditions have been used at the wall, but at higher Reynolds numbers the resolution in this region was then not good enough for a proper LES. In a number of higher-Reynoldsnumber calculations also wall functions were applied, mainly the Werner– Wengle (1989) approach which assumes a distribution of the instantaneous velocity inside the ﬁrst cell, namely a linear distribution for y + < 11.81 and a 1/7 power law for y + > 11.8. The three-dimensional, time-dependent equations governing the resolved quantities were mostly solved numerically by ﬁnite-volume methods; so far, ﬁnite-element methods are used rarely. The ﬁnite-volume methods employ either a staggered or non-staggered variable arrangement; some can accomodate general curvilinear grids, but for the ﬂows around rectangular bodies mostly Cartesian grids were used. Usually the grids were stretched in order to achieve a better resolution in the near-wall region with high gradients – in one case embedded grids were used to better resolve this region. The schemes are generally explicit with small time steps to resolve the turbulent ﬂuctuations and various time discretization methods were employed – mostly the secondorder Adams–Bashforth scheme but also the Runge-Kutta, Euler and leap-frog methods. For the discretization of the convection terms, mostly second-order central diﬀerencing was employed, but also the QUICK scheme and third-order upwind diﬀerencing (especially when no subgrid-scale model was used) and in some cases also a ﬁfth-order upwind diﬀerencing method. At the outlet the calculations were all done with convective conditions. 364 Rodi 3 Flow Past Long Cylinders The ﬂow past long cylinders exposed to uniform approach ﬂow is an interesting test case because the geometry is simple, but the ﬂow is complex with unsteady separation. Alternating vortices are shed from the cylinder and transported downstream, where they retain their identity in a K´rm´n vortex street for a a a considerable distance. These vortices are predominantly two-dimensional and so is the time-mean ﬂow, but large-scale three-dimensional structures exist which lead to a modulation of the shedding frequency. The approach stagnation ﬂow is basically inviscid and thin laminar boundary layers form on the forward surfaces of the cylinder. Square and circular cylinders are considered here. In the case of the square cylinder, the ﬂow separates at the front edges and a ﬂapping shear layer develops on the sides of the cylinder, which is initially laminar but becomes turbulent fairly quickly. In the case of the circular cylinder, the separation point is not ﬁxed but depends on the boundary layer development before separation. For the subcritical cases considered here, the boundary layer remains laminar up to separation, and again the separated shear layer becomes turbulent fairly quickly. 3.1 Square Cylinder For the square cylinder, the only detailed experiments with phase-resolved measurements are those reported for Re = 22000 by Lyn et al. (1995) and Lyn and Rodi (1994). The situation examined by them has become the standard test case for this ﬂow. RANS calculations obtained with various turbulence models ranging from the algebraic Baldwin–Lomax model to a Reynolds-stress model (RSM) and also a single LES have been reviewed already in Rodi (1993); this LES suﬀered from a small calculation domain of only 2 cylinder widths D in the spanwise direction. The ﬂow was then posed as a test case for an LES workshop held in 1995 in Rottach-Egern, Germany, and 9 groups submitted 16 diﬀerent results that are reported in Rodi et al. (1995) and partly also in a summary paper of Rodi et al. (1997). These calculations were all performed on a computation domain of 4D in the spanwise direction, extending 4.5D upstream of the cylinder, 6.5D on either side of the cylinder (where the tunnel walls were located) and at least 14.5D downstream of the cylinder. As will be shown shortly, there was a great variance in the results, and hence the same test case was posed again for a workshop held at Grenoble, France, in 1996. The same calculation domain was prescribed as in the earlier workshop. 7 groups presented 20 results to the Grenoble workshop (there is an overlap of 2 sets of results from the author’s group – University of Karlsruhe) and these are summarised in Voke (1997). In the following, some sample results from both workshops are presented and compared with each other and with the experiments. The samples are the same as those in the summary reports of Rodi et al. (1997) and Voke (1997) and were chosen to represent the variety [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 365 of methods and to concentrate on those results in which the statistical values were determined by averaging over a suﬃcient number of shedding cycles. Later results of Sohankar (1998) (see also Sohankar et al. 1999) obtained with 3 diﬀerent subgrid-scale models are added. Some details on the various methods like subgrid-scale model, near-wall treatment and grids used are given in Table 1. The labelling was taken over from Rodi et al. (1995) for the Rottach-Egern workshop and from Voke (1997) for the Grenoble workshop (see key at end of text). UKA2, UK1 and UK3 are from the author’s group using diﬀerent grids and near-wall treatment and hence show the inﬂuence of these. Figure 1 shows samples of streamlines at phase 1 of the shedding cycle in comparison with the experimental streamlines. The shedding motion is qualitatively well reproduced, with a vortex that has just shed from the lower side of the cylinder at the phase considered. However, there are considerable diﬀerences in the details such as the size and strength of the shed vortex and also in the recirculation motion near the side surfaces. Unfortunately, in this region near the walls the streamlines could not be obtained from the measurements so that the complex behaviour there with a number of vortices appearing cannot be assessed. The shedding in the experiments and also generally in the LES calculations is not very regular, as can be seen from excerpts of pressure and lift signals shown in Figure 2. Clearly, lower frequency amplitude variations are present, but a clear shedding frequency could still be determined. The low frequency variations are believed to be due to three-dimensional ﬂow structures. In the two-dimensional RANS calculations these eﬀects cannot be simulated and hence the shedding behaviour is generally regular. Table 1 summarises various global parameters such as the dimensionless shedding frequency (Strouhal number St = f D/U0 ), the time-mean drag coeﬃcient cD , the RMS values of the ﬂuctuations of drag and lift coeﬃcients cD and cL , respectively, and the reattachment length lR indicating the length of the time-mean separation region behind the cylinder. Most LES calculations yielded the correct value of St ≈ 0.13, and it appears that St is not very sensitive to the parameters of the simulation; there are, however, a few deviations from this value, notably the calculations without subgrid-scale model yielded a higher value. Concerning the mean drag coeﬃcient, it seems that the LES calculations using wall functions are generally close to the experimental range while those using no-slip conditions tend to produce too high values of cD . There is also considerable variation in the recirculation length which will be discussed in connection with the centre-line velocity distribution in Figure 3. The ﬂuctuations of the force coeﬃcients also show fairly large variation – here no experimental results are available for a comparison. Table 1 also includes results obtained with RANS models, namley by Bosch (1995) with the standard k-ε model and with a modiﬁcation due to Kato and Launder (1993), and by Franke and Rodi (1993) with the Reynolds-stress 366 Rodi Table 1: Global parameters for ﬂow past square cylinder (UW = upwind diﬀ., CD = central diﬀerencing, WF = wall function, NS = no slip; (1) = adjusted for diﬀerent blockage; (2) = outermost mesh with embedded meshes). Calculation method St CD CD CL LR /D grid Nx × Ny × Nz 125 × 78 × 20 140 × 81 × 13 146 × 146 × 20 165 × 113 × 17 LES RottachEgern Rodi et al. (1997) KAWAMU, 3rd UW No SGS, NS UMIST2, CD Dyn, WF UKAHY2, CD Smag, WF TAMU2, Dyn, 3rd UW NS/WF UK1, Smag, WF UK3, Smag, NS NT7, LDM, WF UOI, Dyn., NS IS3, Dyn. mix, NS TIT, Dyn, NS ST5, Smag, NS Smag, NS Dyn, NS OEDSM, NS Std. k-ε, WF Kato–Launder k-ε, WF Two-layer k-ε Two-layer Kato–Launder k-ε CD CD CD 0.15 0.09 0.13 0.14 2.58 2.02 2.30 2.77 0.27 – 0.14 0.19 1.33 – 1.15 1.79 1.68 1.21 1.46 0.94 0.13 0.13 2.20 2.23 0.14 0.13 0.12 1.01 1.02 1.39 1.29 1.68 1.39 1.38 1.32 1.44 1.39 1.20 1.36 1.23 1.02 109 × 105 × 20 146 × 146 × 20 140 × 103 × 32 192 × 160 × 48 112 × 104 × 32 121 × 113 × 127(2) 107 × 103 × 20 LES Grenoble Voke (1997) 0.131 2.05 5th UW 0.13 2.03(1) 0.18 0.36 0.23 0.28 5th UW 0.133 2.79 3rd UW 0.131 2.62 variable 0.161 2.78 3rd UW 0.126 2.21 0.125 2.04 0.129 2.25 0.134 1.64 0.142 1.79 0.137 1.72 0.143 2.0 LES Sohankar (1998) RANS Bosch (1995) CD 0.16 0.20 0.20 1.47 1.22 1.50 ≈ 1.0 185 × 105 × 25 ≈ 0 0.305 2.8 0.012 0.614 2.04 ≈0 0.07 0.426 2.4 1.17 1.25 100 × 76 100 × 76 170 × 170 170 × 170 RANS RMS, WF Franke and Two-layer RSM Rodi (1993) Experiments 0.136 2.15 0.159 2.43 0.27 0.06 1.49 1.3 0.98 1.0 70 × 64 186 × 156 0.132 1.9 to 2.2 1.38 [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 367 Figure 1: Phase-averaged streamlines at phase 1 from some LES simulations submitted to the Rottach-Egern Workshop (Rodi et al. 1995). model of Launder et al. (1975). In each case, either wall functions (WF) or a two-layer approach applying a one-equation model in the near-wall region was used. The Strouhal number is predicted well also by most of the RANS models tested, but the Kato–Launder modiﬁcation (KL) tends to produce somewhat too large values and the two-layer RSM an excessive value. The mean drag coeﬃcient is signiﬁcantly underpredicted by the standard k-ε model, but roughly the correct value was obtained when the KL modiﬁcation was used with the 368 Rodi Figure 2: Time variation of pressure and lift in ﬂow past square cylinder. two-layer approach. The RSM gives the correct value of cD with wall functions but overpredicts it in the two-layer approach. This result is consistent with the signiﬁcant overprediction of the length of the recirculation region by the standard k-ε model and its underprediction by the two-layer RSM. Figure 3 shows the distribution of the time-mean velocity along the cylinder centre-line. The experimental velocity recovers very slowly in the downstream region or even seems to level oﬀ at a value of about 0.6 of the upstream freestream level. The LES calculations exhibit a great variance in this region. Most of them predict the recovery to the free-stream level considerably faster than the experiments, but two (KAWAMU, TAMU2) level oﬀ at roughly the correct value while UOI and ST5 show a decline of the centre-line velocity beyond x/D ≈ 0.5 which seems physically not plausible. Looking now at the near-cylinder region with recirculation zone, it can be seen that the calcu- [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 369 lations with the lowest velocity values in the far-ﬁeld generally exhibit too short a recirculation length. On the other hand, those calculations with too fast recovery in the far-ﬁeld are in fairly good accord with the experimental distribution in the near-ﬁeld up to x/D ≈ 2. The more recent calculations of Sohankar (1998) exhibit a recirculation that is too short (lR ≈ 1.0) and too weak and the centre-line velocity levels oﬀ at too high a value of U (≈ 0.8). There is no great diﬀerence between the 3 subgrid-scale models used. The calculations of Sohankar et al. (1999) with a dynamic one-equation model obtained on a ﬁner grid (265 × 161 × 25) are also not very diﬀerent, but a little closer to the experiment. The RANS calculations all show too fast a recovery in the far-ﬁeld. The standard k-ε model predicts a signiﬁcantly too long recirculation; in the nearﬁeld a fairly good prediction is achieved with the KL modiﬁcation combined with the two-layer approach while the recirculation length is underpredicted with the RSM. Figure 4 presents the distribution of the total resolved (periodic plus turbulent) ﬂuctuating kinetic energy along the centre-line for the Rottach-Egern workshop calculations plus the UOI results from the Grenoble workshop. Here the various LES results show an even wider variation with an almost fourfold diﬀerence in the peak level of ktot , but the picture is not entirely consistent. TAMU2 yields excessive ﬂuctuations which cause underprediction of the separation length while KAWAMU produces too small ﬂuctuations which explain the excessive separation length. It is diﬃcult to understand why the UMIST2 calculations with an even lower ﬂuctuation level lead to an underprediction in the separation length. It is generally to be observed that the total ﬂuctuations are underpredicted while the drag coeﬃcient and separation length are reasonable. The variance of the total ﬂuctuation results is somewhat smaller for the Grenoble calculations but there is still a factor of about 2, as can be seen from Figure 5 which shows the distributions of the (resolved) total ﬂuctuation components and along the centre-line in comparison with Lyn et al.’s (1995) measurements. None of the calculations is entirely satisfactory. For these quantities, the results of Sohankar (1998) obtained with the Smagorinsky model and with a one-equation dynamic model agree quite well with the experiments. The RANS calculations are not very satisfactory concerning the total ﬂuctuations. The KL k-ε model version with the two-layer approach, which predicted best the velocity distribution, yields the correct shape of the ﬂuctuating energy but overpredicts the level by about 40%. Considerably worse are the RANS predictions of the turbulent kinetic energy component of the ﬂuctuations (periodic ﬂuctuations subtracted) as shown in Figure 6. All RANS calculations strongly underpredict the k-level, while the only LES result available for the turbulent component (UKAHY2) is roughly in accord with the measurements. This means that in RANS calculations where the total ﬂuctuations are realistic or too high, the periodic ﬂuctuations are 370 Rodi Figure 3: Time-mean velocity u along centre-line of square cylinder. (a) Results from Rottach-Egern Workshop (Rodi et al 1995). (b) Results from Grenoble Workshop - complete wake; — experiments, simulations: UK1 ◦, UK3 O, UOI *, IS3 +, NT7 ×, TIT ·, ST5 • (from Voke 1997). (c) Results from Grenoble Workshop - near wake; key as for (b) (from Voke 1997). [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 371 Figure 4: Total kinetic energy of ﬂuctuations (periodic and turbulent) along centre-line of square cylinder. Results from Rottach-Egern Workshop and UOI from Grenoble Workshop. overpredicted. The very diﬀerent behaviour of RANS and LES calculations of k is most likely due to the fact that the fairly high turbulent kinetic energy stems from contributions of low frequency ﬂuctuations as indicated in Figure 2. In the experiments these originate from three-dimensional large-scale structures. LES calculations can capture these structures and count any lowfrequency ﬂuctuations originating from them as turbulence, while of course the two-dimensional RANS calculations cannot and determine k from solving the turbulence model equations. The overall behaviour of the vortex-shedding ﬂow is determined largely by the prediction of the evolution of the separated shear layers on the sides of the cylinder. It was found from the workshop calculations (e.g. Rodi et al. 1997) that the very thin (≈ 0.1D) reverse-ﬂow region near the wall is clearly not well resolved. The calculations with the best resolution near the wall had a mesh dimension of the ﬁrst cell adjacent to the wall of 0.01D, but many used a coarser mesh so that sometimes the ﬁrst mesh point comes to lie at the maximum of the negative velocity. One of the main conclusions at both workshops was that higher resolution is required at the walls. It is surprising that near the side walls of the cylinder, the resolution was not improved from the ﬁrst workshop to the second (in both cases the best resolution was (∆/D = 0.01). None of the methods produced good agreement with experiments in the near wake with recirculation zone and at the same time for the recovery of the ﬂow in the far wake as well as 372 Rodi Figure 5: Total u and v ﬂuctuations (periodic and turbulent) along centreline of square cylinder. Results from Rottach-Egern Workshop and UOI from Grenoble Workshop; key as Figure 3(b) (from Voke 1997). the drag coeﬃcient. There is some indication that the use of stretched grids in the far wake caused numerical inaccuracies: Pourqui´ (1996) reported that the e UK calculations improved signiﬁcantly in this region when a ﬁner grid with low stretching was used. Kogaki et al. (1997) found that numerical dissipation is signiﬁcant when upwind schemes are used, even for 5th order schemes. They have also shown that the spanwise resolution may have an important inﬂuence, and it appears that this was inadequate in many of the calculations presented. Voke (1997) therefore recommends that future studies of this test case should use a minimum number of 32 grid points over a spanwise extent of 4D, and perhaps even this extent should be enlarged. 3.2 Circular Cylinder The only detailed phase-resolved measurements of the ﬂow around a circular cylinder were carried out by Cantwell and Coles (1983) at Re = 140000. This ﬂow is still subcritical with laminar boundary layers developing until separation, but the separated free-shear layer becomes turbulent quickly. As the laminar boundary layer is very thin at this high Reynolds number, it is diﬃcult to resolve in a numerical simulation and hence the ﬁrst and most extensive large-eddy simulations were carried out for a much lower Reynolds number of 3900. Starting with Beaudan and Moin (1994, hereafter referred [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 373 Figure 6: Turbulent component of kinetic energy along centre-line of square cylinder. (1) Pourqui´ (1996), (2) Franke and Rodi (1993), (3) Bosch (1995), e (4) Lyn et al. (1995). to as BM) a series of LES calculations were performed at Stanford for this case (further calculations by Mittal and Moin 1997, Kravchenko and Moin 2000, hereafter referred to as KM) and the case was also calculated by Breuer (1998) and Fr¨hlich et al. (1998) using the same computer code. Further, DNS o calculations for this case are reported in Xia et al. (1998). The near-ﬁeld of this ﬂow with recirculation region was studied experimentally with PIV by Lourenco and Shih (1993, hereafter referred to as LS), and hot-wire measurements downstream of the recirculation region were carried out by Ong and Wallace (1996). Global parameters are available from various other measurements at either Re = 3900 or a reasonably close Re (see KM). Most of the LES calculations were carried out with an O-grid; in this context KM used an embedded grid near the cylinder and in the wake region. On the other hand, Mittal and Moin (1997) employed a C-grid. In each case the spanwise extent of the calculation domain was πD. The number of grid points in radial, circumferential or streamwise, and spanwise direction is given in Table 2 together with the discretization scheme and subgrid-scale model employed. Fr¨hlich o et al. (1998) used the same grid as Breuer (the coarser version) but also a ﬁner one with 48 points in the spanwise direction and performed calculations with and without the Smagorinsky model. Breuer tested various discretization schemes for convection and so did the Stanford group. They found that up- 374 Rodi Table 2: Global parameters for ﬂow past circular cylinder at Re = 3900 (note: the two values from Breuer’s calculations are for the two grids used). Author Discret. grid SGS θ/x, r, z model St cD Cpb LR /D θ Beaudan and Moin (1994) 5th order – .216 upwind 144 × 136 Smag. .209 ×48 Dyn. .203 CDS 401 × 120 Dyn. ×48 .207 .96 .92 1.00 1.00 .89 .81 .95 .93 1.56 1.74 1.36 1.40 85.3 84.8 85.8 86.9 Mittal and Moin (1997) Kravchenko and B-spline 185 × 205 Dyn. Moin (2000) ×48 – 165 × 165 Smag. ×32/64 Dyn. .210 1.04 .94 1.35 88.0 Breuer (1998) CDS .22 .22 .21 .215± 0.005 1.14/1.16 1.11/1.16 1.00/.87 88.6/89.3 1.10/1.10 1.05/1.07 1.11/1.04 87.9/88.5 1.07/1.02 1.01/.94 1.20/1.37 87.7/87.4 0.98± 0.05 0.9± 0.05 1.33± 0.2 (1.19) 85± 2 Experiments (origin Fr¨hlich et al. 1998, o Kravchenko and Moin 2000) wind schemes, even of higher order, produced too much dissipation and that central diﬀerencing schemes are to be preferred. The most accurate scheme is the B-spline method used by KM. These authors also investigated the inﬂuence of the numerical resolution, in particular the spanwise resolution and of the extent of the spanwise calculation domain. Further, BM and Breuer carried out also two-dimensional calculations (without subgrid-scale model) and found that these yielded unrealistic results in that an attached recirculation region behind the cylinder is absent and the U -velocity is always positive along the centre-line of the cylinder. Typical results of the following global parameters are compiled in Table 2 and compared with experiments: Strouhal number St, mean drag coeﬃcient cD , separation angle θ and length of time-mean separation zone LR /D. The results of Fr¨hlich et al. (1998) with and without Smagorinsky model are not o listed as they are very similar to those of Breuer’s coarser-grid results. The table shows that for most global parameters all the results are rather similar; the greatest diﬀerences can be found in the separation length LR . The distribution of the time-mean velocity along the centre-line of the cylinder is displayed in Figure 7 while Figure 8 compares Breuer’s results for u 2 and v 2 [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 375 at x/D = 1.54 obtained with his ﬁnest grid and various subgrid-scale models with the LS measurements. Figure 9 provides proﬁles of u 2 and u v further downstream as obtained by the Stanford group in their various calculations using the dynamic SGS model and compares them with the measurements of Ong and Wallace (1996). Even the simulations without subgrid-scale model gave reasonable agreement for the mean quantities. Introducing a subgridscale model has relatively little eﬀect on the mean velocity and the global parameters, but improves the agreement with experiments, especially for the stresses and spectra, with the best improvement when the dynamic model is employed. Breuer found that the best results are obtained with a ﬁne grid using the dynamic model, but these calculations produced a separation length LR larger than measured by LS. This is conﬁrmed by the Stanford group calculations, but there are diﬀerences in the velocity results which point to the inﬂuence of details of the numerical treatment. KM studied quite extensively the grid eﬀects and found that underresolved LES produced shorter separation lengths which are in agreement with the LS experiment, and this is conﬁrmed by Breuer’s coarser-grid calculations. KM argue that any agreement of such calculations with the LS experiment is fortuitous. They found that inadequate grid resolution can cause early transition of the separated shear layers and that in the LS experiment such early transition was probably caused by external disturbances. They also found that the more accurate numerical treatment achieved with the B-spline method improved the calculations further downstream and that the power spectra obtained with this method are in good agreement with the measurements while this is not the case with other discretization methods. The main message of these calculations is that the inﬂuence of the subgridscale model is not very strong at this relatively low Reynolds number (although the spectra decay too slowly without a model) and that the best results are obtained with a ﬁne-resolution simulation using a dynamic SGS model, which altogether are in good agreement with experiments. The DNS calculations of Xia et al. (1998) indicate similarly good agreement, but for a fuller comparison and assessment, more results of these calculations need to be reported. As was indicated already, the cylinder ﬂow in the upper subcritical regime (Re = 140, 000) studied in detail experimentally by Cantwell and Coles (1983, hereafter referred to as CC) is much more demanding. The boundary-layer √ thickness behaves like 1/ Re so that the grid spacing should be smaller by a factor of 6 in order to resolve the laminar boundary layers with the same quality as for the Re = 3,900 case. However, this is so far not feasible as grid stretching should not be excessive. Fr¨hlich et al. (1998) also report prelimio nary calculations of the Re = 140,000 case, and more extensive calculations with the same code were again carried out by Breuer (1999), studying the inﬂuence of grid ﬁneness, spanwise extent of the calculation domain (1D − πD) and subgrid-scale model. Various results obtained on the coarser grid with 376 Rodi Figure 7: Time-mean velocity u along centre-line of square cylinder at Re = 3900. (a) Calculations of Breuer (1998) on 165 × 165 × 64 grid: D1, without model; D2, Smagorinsky model; D3, dynamics model’ (b) From Kravchenko and Moin (2000): —, their B-spline calculations; - - -, CDS simulations of Mittal and Moin (1997); · · · upwind simulations of Beaudan and Moin (1994); ◦, experiment of Ong and Wallace (1996); experiment of Lourenco and Shih (1993). various subgrid-scale models are given for global ﬂow parameters in Table 2 and for distributions of mean velocity and ﬂuctuations along the centre-line in Figure 10 where they are compared with experiments. Table 2 also includes the results of Fr¨hlich et al. (1998) obtained on a similar grid with the Smagorino sky model. Most global parameters and the mean velocity agree reasonably [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 377 Figure 8: Total resolved Reynolds stresses u 2 (left) and v 2 (right) in wake of circular cylinder (Re = 3900) at x/D = 1.54; calculations of Breuer (1998) on 165 × 165 × 64 grid. For key, see Figure 7a. Figure 9: Total Reynolds stresses u 2 and u v in wake of circular cylinder (Re = 3900); from Kravchenko and Moin (2000). For key, see Figure 7b. 378 Rodi Table 3: Global parameters for ﬂow past circular cylinder at Re = 140, 000. Author Zmax grid SGS model Calculations Fr¨hlich et al. 1D o (1998) Breuer (1999) 2D 166×206×64 Smag.CS = .1 .217 1.157 1.33 1.398 1.411 0.677 .42 .577 .416 .712 93.8 96.37 95.16 94.58 St cD −Cpback LR /D θsep 165×165×64 Dynamic .204 1.239 Smag.CS = .1 .217 1.218 Smag.CS = .065 .247 .707 Experiments Cantwell and Coles (1983) Others (see Fr¨hlich et al. 1998, Breuer 1999) o .179 1.237 .2 ≈ 1.2 1.21 1.34 ≈ 0.5 79 Figure 10: Flow past circular cylinder at Re = 140,000: (a) time-mean velocity u along centre-line; (b) total ﬂuctuations u 2 along centre-line; (c) total ﬂuctuations v 2 along centre-line; (d) total shear stress u v at x/D = 1, calculations of Breuer (1999) on 165 × 165 × 64 grid: A1 dynamic model; A2 Smagorinsky model with CS = 0.1; A3 Smagorinsky model with CS = 0.065. [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 379 Figure 11: Flow around surface-mounted cube according to Martinuzzi (1992). well with the experiments. The predicted Strouhal number is higher than that measured by CC, but is in reasonable agreement with other measurements at a similar Reynolds number. In the far wake the velocity development is not predicted so well which is most likely due to the O-grid being too coarse in this region. Further, the v -ﬂuctuations are predicted too high and the separation angle θ is too large compared with observations, but the latter quantity seems to be very sensitive to small changes in the ﬂow conditions in this upper critical region (Achenbach 1968). In general it appears that the calculated ﬂow state seems to be closer to the critical state than in the CC experiments, i.e. that in a sense more ‘turbulence’ is present and this may be due to numerical oscillations as observed by Fr¨hlich et al. (1998). One clear conclusion of Breuer o (1999) was that the subgrid-scale model has considerably more inﬂuence in this high-Reynolds number case than at Re = 3900. He also concluded that the dynamic model gave the best results but was not decisively better than the Smagorinsky model for this ﬂow. He observed also that the quality of the calculations on the whole rather deteriorates when a ﬁner grid (325 × 325 × 64) is used. The reasons for this behaviour are not understood and altogether there are still considerable problems with this case. It should be added that the LES calculations are, however, much better than the RANS calculations of Franke (1991) using two-layer k-ε and Reynolds-stress models. 4 4.1 Flow over Surface-Mounted Cubes Single Cube Results are reviewed next for the ﬂow over a cube mounted on the lower wall of a plane channel; the cube height H is half that of the channel and the approach ﬂow is developed channel ﬂow. This was also a test case for the Rottach-Egern LES workshop, for which calculations for two Reynolds numbers Re = UB H/ν = 3, 000 and 40,000 were invited. Experiments are available only for the Re = 40, 000 case, namely ﬂow visualisation studies and detailed LDA measurements (Martinuzzi 1992, Martinuzzi and Tropea 1993). 380 Rodi The LES calculations for the workshop have revealed that the ﬂow behaviour at the two Reynolds numbers is very similar. From his visualisation studies and the detailed measurements, Martinuzzi (1992) devised the ﬂow picture given in Figure 11 which shows clearly the very complex nature of the ﬂow in spite of the simple geometry. In contrast to the square-cylinder case, the time-mean ﬂow is now also three-dimensional. The ﬂow separates in front of the cube; in the mean there is a primary separation vortex and also a secondary one. The main vortex is bent as a horse-shoe vortex around the cube into the wake where it has a typical converging-diverging behaviour. The ﬂow separates at the front corners of the cube on the roof and side walls. In the mean, it does not reattach on the roof. A large separation region develops behind the cube which interacts with the horse-shoe vortex. Originating from the ground plate, an arch vortex develops behind the cube. Predominant ﬂuctuation frequencies were detected sideways behind the cube, which were traced to some vortex shedding of the ﬂow past the side walls. Further, bimodal behaviour of the ﬂow separation, and in particular of the vortices in front and on the roof were observed. For the Re = 3, 000 case, three groups submitted four results to the RottachEgern workshop. All were in fairly close agreement and show the main features discussed above. For the Re = 40, 000 case also, three groups submitted four results. Of these, one set showed clearly insuﬃcient averaging and hence is not included here. Information on the remaining submissions is provided in Table 4 (all used wall functions), with the same labelling used as in Rodi et al. (1997). RANS calculations for this cube ﬂow were performed by Lakehal and Rodi (1997) with various versions of the k-ε model as also listed in Table 4. The calculation domain extended 3.0H and 3.5H upstream of the cylinder, 6H and 10H downstream and 7H and 9H laterally for all the LES and RANS calculations, respectively. With each method, developed channel ﬂow was calculated ﬁrst, and the results were then used as inﬂow conditions. Periodic or no-slip conditions were used on the lateral boundaries. The grids employed are listed in Table 4; they are generally non-uniform with ﬁner resolution near the walls. The height of the near-wall cells was 0.0125H in the UKAHY–LES calculations from the author’s group, 0.01H in the RANS calculations with wall functions and 0.001H in the two-layer RANS calculations. Figure 12 compares the streamlines in the plane of the symmetry (left) and near the channel ﬂoor (right) for three LES calculations and the RANS calculation with the standard k-ε model and wall functions. Table 4 compares various lengths of separation regions deﬁned in Figure 11. There is now much closer agreement among the various LES calculations than in the case of the square cylinder. The streamline picture in Figure 12 shows that on the whole LES is able to simulate this complex ﬂow very well. The size of the recirculation zone in front of the cube is predicted correctly by the LES methods, and the length of the large separation region behind the cube is simulated in good agreement [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 381 Figure 12: Streamlines in the symmetry plane (left) and near the channel ﬂoor (right) for ﬂow around a single cube at Re = 4000. Experiment of Martinuzzi (1992), LES calculations from Rottach-Egern Workshop (Rodi et al. 1995), k-ε model calculations from Lakehal and Rodi (1997). with the experiments by the Smagorinsky model, while the dynamic models predict this length somewhat too short. As was shown by Breuer and Rodi (1996), using no-slip conditions instead of wall functions in the Smagorinsky model predictions did not change noticeably the symmetry-plane streamlines. On the other hand, the k-ε model overpredicts the separation length considerably, and the diﬀerence to the experiments becomes even larger when the 382 Rodi Table 4: Global parameters for single cube calculation. Calculation method xF 1 xT xR1 grid Nx × Ny × Nz 165 × 65 × 97 165 × 65 × 97 144 × 58 × 88 110 × 32 × 32 110 × 32 × 32 142 × 84 × 64 142 × 84 × 64 LES Rodi et al. (1997) UKAHY3, Smag. UKAHY4, Dyn. UBWM2, Smag. Std. k-ε WF KL-k-ε-WF Two-layer k-ε Two-layer KL-k-ε 1.29 1.00 0.81 0.65 0.64 0.95 0.95 1.04 – – 0.837 0.43 – – – – 1.70 1.43 1.72 2.18 2.73 2.68 3.40 1.61 RANS Lakehal and Rodi (1997) Exp. Martinuzzi (1992) Kato–Launder modiﬁcation or the two-layer approach are introduced (see Table 4). On the roof, the UKAHY–LES calculations do not predict reattachment in the mean, as was also found in the experiment, and the extent of the separation region is well reproduced. On the other hand, the other LES calculation yielded reattachment and so did the standard k-ε model calculation. Switching to the Kato–Launder modiﬁcation and to the two-layer approach improves the predictions of the ﬂow on the roof and in front of the cube, but increases the separation length behind the cube even more, as was mentioned already. The LES clearly do a better job in the lee of the cube. This may be explained by the fact that, in the experiments, some shedding from the side walls was observed which enhances the momentum exchange in the wake and can reduce signiﬁcantly the length of the separation region behind obstacles. Even though there was no clear shedding detected in the LES results, the resolution of the large-scale unsteady motions in these calculations seems to produce the correct eﬀect, while RANS calculations can of course not account for such eﬀects, explaining possibly the overprediction of the separation region. The complex behaviour of the surface streamlines near the channel ﬂoor as observed in the experimental oil ﬂow pictures is well reproduced by both UKAHY LES calculations, including such details as the convergent-divergent behaviour of the horse-shoe vortex, the primary and secondary separation in front of the cube, the arch vortices behind the cube and the reattachment line bordering the reverse-ﬂow region. The convergent-divergent behaviour of the horse-shoe vortex is best predicted with the dynamic model. It is weaker with the Smagorinsky model and disappears when no-slip conditions are used with [12] Large-eddy simulation of the ﬂow past bluﬀ bodies 383 this instead of wall functions (run UKAHY5 of Breuer and Rodi 1996, not shown here). The UBWM2 LES calculations also show a somewhat simpler ﬂow pattern without the converging-diverging nature of the horse-shoe vortex and with the arch vortices ﬁlling basically the entire separation region behind the cube. A similar picture resulted from the standard k-ε model calculations, but with the separation region predicted unrealistically long. In RANS calculations, the ﬁner details of the complex near-wall ﬂow could only be attained with the two-layer approach. More recently, Krajnovic et al. (1999) have also published calculations for the test case with Re = 40, 000. They reported results obtained with two dynamic one-equation subgrid-scale models and without using any SGS model on a fairly coarse grid. They did not calculate developed channel ﬂow ﬁrst but used the experimental proﬁle (constant in time) as inﬂow condition. Probably because of this, the results show some unrealistic features like considerably too slow a bending of the horse-shoe vortex around the cube and a strange streamline pattern in the separation region behind the cube. When a subgridscale model is used, the separation lengths and the velocity proﬁles in the symmetry plane are not too badly predicted but the calculations without a model show poor results. This indicates that in this case with fairly high Reynolds number the inﬂuence of the subgrid-scale model is quite important. 4.2 Matrix of Cubes Another test case chosen for a series of ERCOFTAC/IAHR workshops, the last one held in Helsinki in 1999, concerns the ﬂow around a matrix of surfacemounted cubes. This case was studied experimentally by Meinders et al. (1997) who placed a matrix of 25 × 10 cubes on one of the walls of a two-dimensional channel. The cubes were H = 15mm high and the channel had a depth of h = 51mm. The channel Reynolds number was Reh = 1.3 × 104 and the Reynolds number based on the cube height H was 3,823, which is much lower than in the case of the single cube of Section 4.1 (namely 40,000). Measurements were performed around a cube in the 18th row from the inlet where the ﬂow was developed and periodic. Hence, in the computations only the ﬂow around a single cube needed to be calculated with periodicity conditions employed in the mid-planes in the downstream and spanwise directions. In the experiment, one cube was heated and the temperature and heat-transfer distributions along the cube walls were measured. In the proceedings of the Helsinki workshop (Hellsten and Rautaheimo 1999), detailed calculation results obtained with two LES methods and one DNS method as well as with various RANS methods are presented and compared with the experiments. These results include velocity and Reynolds-stress proﬁles at various streamwise locations in the symmetry plane and in one horizontal plane as well as heat-transfer and temperature distributions around the cube walls. The LES calculations of Mathey et al. (see also Mathey et al. 1998, 1999) are nearly 384 Rodi identical to the DNS results of van der Velde, Verstappen and Veldman, and the LES results of Niceno and Hanjalic’ are also very close. In general, there is very good agreement with experiments. Figure 13 compares Mathey et al.’s (1998, 1999) calculated streamlines in the vertical symmetry plane and very near the bottom channel wall and in a horizontal plane at mid-height with the experimental observations. The calculations were carried out on a 1003 grid using the Smagorinsky model and no-slip conditions. However, virtually the same results were obtained with the dynamic model and also in calculations not using any subgrid-scale model. This shows again that in this case of fairly low Reynolds number the subgridscale model has little inﬂuence and explains the very good agreement with the DNS calculations of van der Velde et al. In the symmetry plane, some diﬀerences to the single-cube ﬂow pattern can be observed due to the interaction of the cubes; the separated ﬂow vortex in front of the cube is stronger and larger while the vortex on the roof is weaker and the ﬂow reattaches in the mean. Between the two cubes there is no clear reattachment at the bottom, a rather concentrated vortex forms in the upper part of the separation region behind the cube and a kind of a saddle line in the lower part. The separation region has, however, about the same length as in the single-cube case. In the horizontal mid-height plane, there is a recirculating separation region behind the cube. The streamlines at the bottom, which are compared with the experimental oil-ﬂow picture, exhibit similar features as in the single-cube case, but are even more complex, showing nicely the interaction of the two cubes displayed in the picture, mainly the displacement of the horse-shoe vortex formed on the ﬁrst cube by the horse-shoe vortex of the second cube. All these complex features are very well reproduced by the LES. In the experiment some vortex shedding from the sides of the cube was observed and this was also obtained in the LES calculations with the correct frequency. Figure 14 compares the temperature distributions around the cube as c