Closure strategies for turbulent and transitional flows

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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 - isbn-10 0-521-79208-8 hardback

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´c and S. Jakirli´c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 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¨ ohlich and W. Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 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

. . . . . . . . . . . . . . . 522

Combusting Flows

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´c and S. Kenjereˇs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 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´e de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex, France [email protected] J-P. Bonnet, Universit´e de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex, France [email protected] C. Cambon, Laboratoire de M´ecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, 36 avenue Guy de Collongue, BP 163, 69131 Ecully Cedex, France [email protected] G.N. Coleman, Aeronautics and Astronautics, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK [email protected] T.J. Craft, Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK [email protected] P.A. Durbin, Department of Mechanical Engineering, Stanford University, Stanford CA 94305-3030 USA [email protected] J. Fr¨ ohlich, Institut f¨ ur Hydromechanik, Universit¨ at Karlsruhe, Kaiserstr.12, D-76128 Karlsruhe, Germany [email protected] T.B. Gatski, Computational Modeling & Simulation Branch, Mail Stop 128, NASA Langley Research Center, Hampton VA 23681-2199, USA [email protected] B.J. Guerts, Faculty of Mathematical Sciences, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands [email protected] K. Hanjali´c, Department of Applied Physics, Thermal and Fluids Sciences, University of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands [email protected] B. Ilyushin, Institute of Thermophysics SD RAS, Lavrentyev Avenue 1, 630090 Novosibirsk, Russia [email protected] R. Jacobs, TenFold Corporation, Draper, UT 84020, USA [email protected] S. Jakarli´c, Institute for Fluid Mechanics and Aerodynamics, Darmstadt University of Technology, Petersenstr. 30, D-64287, Darmstadt, Germany [email protected]

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 [email protected] S. Kenjereˇs, Department of Applied Physics, Thermal and Fluids Sciences, University of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands [email protected] B.E. Launder, Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK [email protected] D.R. Laurence, Department of Mechanical Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK; and Electricit´e de France [email protected] A. Leonard, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena CA 91125, USA [email protected] M.A. Leschziner, Imperial College of Science Technology and Medicine, Aeronautics, Department, Prince Consort Rd., London SW7 2BY, UK [email protected] F-S. Lien, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada [email protected] Y. Nagano, Department of Environmental Technology, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan [email protected] T.W.J. Peeters, Corus Research, Development & Technology P.O. Box 10000, 1970 CA IJmuiden, The Netherlands [email protected] B.A. Petterson-Reif, Norwegian Defence Research Establishment, N-2025 Kjeller, Norway [email protected] W. Rodi, Institut f¨ ur Hydromechanik, Universit¨ at Karlsruhe, Kaiserstr.12, D-76128 Karlsruhe, Germany [email protected] D. Roekaerts, Department of Applied Physics, Thermal and Fluids Sciences, University of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, Netherlands [email protected] C.L. Rumsey, Computational Modeling & Simulation Branch, Mail Stop 128, NASA Langley Research Center, Hampton VA 23681-2199, USA [email protected] N.D. Sandham, Aeronautics and Astronautics, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK [email protected]

x

Contributors

A.M. Savill, Department of Engineering, University of Cambridge, Trumpington St., Cambridge CB2 1PZ, UK [email protected] H.A. Wouters, Corus Research, Development & Technology, P.O. Box 10000, 1970 CA IJmuiden, The Netherlands [email protected] X. Wu, Department of Mechanical Engineering, Stanford University, Stanford CA 94305-3030, USA xiaohua [email protected]

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´e de France 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

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

xiii

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´erospatiales 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´et´e Nationale d’Etudes et de Conception de Moteurs 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´c and Jakirli´c provide an overview of the important 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¨ ohlich and Rodi (Chapter [8]) introduce 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´c and Kenjereˇs where the large-scale structures in Rayleigh–B´enard convection are resolved by employing a time-dependent solution 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´c and Kenjereˇs [22] report that the problem of Rayleigh–B´enard convection (in which a horizontal layer of ﬂuid, conﬁned 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 εˆ 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, 1 T →∞ T

t0 +T

f (x) = lim

f (x, t) dt,

(2.2)

t0

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 1 Sij = 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

∂Ui ν ∂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

3.1

13

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 2 dU , (3.4) νt = ρl 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

νti = l

Cebeci–Smith:

2

1 2

νti = l2 2Wij Wij ,

Baldwin–Lomax: where Wij is the rotation tensor 1 Wij = 2

∂Ui ∂Ui ∂xj ∂xj

∂Ui ∂Uj − ∂xj ∂xi

(3.5) (3.6)

,

(3.7)

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)

(3.8)

νto = 0.0269Fwk FK (y; ym /0.3).

(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

y FK (y; ∆) = 1 + 5.5 ∆

6 −1

,

(3.11)

and the wake function Fwk is given by

2 /Fm Fwk = min ym Fm ; ym Udif

with

(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 =

[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

Gatski and Rumsey The outer layer eddy viscosity is given by νto = 0.0168Ue δ ∗ FK (y; δ)σ(x),

(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

√ √ τm dτm τm − Cdif 1 − σ 1/2 (x) , (3.18) = b12 τmeq − τm dx Lm (0.7δ − ym ) 3/2

Um

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

3.3

17

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 + c3v1 ), and χ = ν˜/ν. The variable ν˜ is determined by using the transport equation

D˜ ν 1 ∂ ∂ ν˜ (ν + ν˜) = cb1 (1 − ft2 )S˜ν˜ + Dt σ ∂xk ∂xk

− (cw1 fw

cb2 ∂ ν˜ 2 σ ∂xk 2 ν˜ cb1 − 2 ft2 κ d +

(3.22)

with auxiliary relations

fv2

ν˜ 2Wij Wij + 2 2 fv2 κ d χ = 1− 1 + χfv1

S˜ =

fw

1 + c6 = g 6 w3 g + c6w3

1/6

g = r + cw2 (r6 − r) ν˜ r = ˜ Sκ2 d2 ft2 = ct3 exp(−ct4 χ2 ),

g −6 + c−6 w3 = 1 + c−6 w3

(3.23) (3.24) −1/6

(3.25) (3.26) (3.27) (3.28)

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 ofthe 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 ∂U P = −τxy =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+

i.e., dissipation rate of kinetic energy (k) per unit k.

νt σε

∂ε , ∂xk

(3.38)

[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

νt ν+ 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

√

k 500ν Γ = min max ; Cµ ωd ωd2

(3.41)

4σω2 k ; , CDkω d2

(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(Γ22 ), where

√

(3.45)

2 k 500ν Γ2 = max ; Cµ ωd ωd2

.

(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 εˆ εˆ2 ∂ = f1 Cε1 P − f2 Cε2 + Dt k k ∂xk

ε = εˆ + D,

νt ν+ σε

∂ εˆ +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 1/2 ν 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 charac teristic 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 ∗ P − Cε2 ε) + = (Cε1 Dt T ∂xk

νt ν+ σε

∂ εˆ ∂xk

(3.57)

to account for the variability of the characteristic timescale. With the exception ∗ , the closure coeﬃcients are of the production-of-dissipation rate coeﬃcient Cε1 ∗ has assigned values close to the standard ones (Durbin 1993b, 1995), but Cε1 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

νt ν+ σk

∂τ22 , ∂xk

(3.58)

[1] Linear and nonlinear eddy viscosity models

25

where the function f22 is, in general, obtained from L2 ∇2 f22 − f22 = Π22

(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 bij =

n

(n)

αn Tij ,

(4.3)

n=1 (n)

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 − 13 {S2 }I = W2 − 13 {W2 }I = WS2 − S2 W

T(6) = W2 S + SW2 − 23 {SW2 }I T(7) = WSW2 − W2 SW T(8) = SWS2 − S2 WS T(9) = W2 S2 + S2 W2 − 23 {S2 W2 }I 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

29

Next consider the quadratic model proposed by Shih et al. (1995), b = α1 S + α2 (SW − WS) .

(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

4.2

Gatski and Rumsey

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

Dτij τij Dk − Dt k Dt

= −bij

P 2 2 − ε − Sij − bik Skj + Sik bkj − bmn Smn δij (4.16) k 3 3

+ (bik Wkj − Wik bkj ) +

Πij τij 1 Dij − + D . 2k 2k k

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

Πij = − C10 + C11

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

31

and rewriting yields

Dbij τij 1 Dij − − D Dt 2k k bij 2 = − + a3 bik Skj + Sik bkj − bmn Smn δij a4 3

(4.18)

− a2 (bik Wkj − Wik bkj ) + a1 Sij . 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=

a4 = gτ, −1

C11 P C0 +1 + 1 −1 2 ε 2

P = γ0 + γ1 ε

−1

,

(4.20)

where C10 = 3.4, C11 = 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

or

Dτij τij Dk = , Dt k Dt

(4.21)

and

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

Gatski and Rumsey

or rewritten using matrix notation

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, b=

3

αn T(n) ,

(4.26)

n=1

with the same three-term tensor basis T(n) shown in equation (4.25). Equation (4.24) can be solved ` a la Galerkin by projecting this algebraic 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 (n) (m) (n) (m) (n) (m) αn − (T , T ) − 2a3 (T S, T ) + 2a2 (T W, T ) 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 

1 2  − a4 η 

   4 2 Anm =   2a2 η R    

−2a2 η 4 R2 −

2 4 2 η R a4

− 13 a3 η 4

0

(4.29)



− 13 a3 η 4 

    , 0     1 4  − η

(4.30)

6a4

which, when inverted, leads to the following expressions for the representation coeﬃcients a4 2 ({RS} + 2a a {RWS} − 2a a {RS } , (4.31) α1 = − 2 4 3 4 α0 η 2 {RWS} α2 = a4 a2 α1 + , (4.32) η 4 R2

6{RS2 } , α3 = −a4 2a3 α1 + η4

(4.33)

where α0 = 1 − 23 a23 a24 η 2 + 2a22 a24 η 2 R2 , and R2 (= − W2 / S2 ). The ﬂow parameter R2 is a dimensionless variable that is useful for characterizing the 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

γ02 α13

1 γ0 γ1 − 2 α12 4 2 γ12 − 2τ 2 γ0 {RS} − 2η 2 τ 2 η τ 4η τ

1 + 6 4η τ

γ1 {RS} + 2τ a2 {RWS} − a3 {RS } 2

a23 − R2 a22 3

α1

= 0.

(4.39)

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

νt ν+ σk

Dε ε ε2 ∂ = Cε1 P − fε Cε2 + Dt k k ∂xk

∂k , ∂xk

νt ν+ σε

(4.40)

∂ε , ∂xk

(4.41)

[1] Linear and nonlinear eddy viscosity models

35

where ν is the kinematic viscosity and νt = Cµ kτ is an ‘equilibrium’ eddy viscosity. Also,

Rek fε = 1 − exp − 10.8

σk = 1.0,

,

Rek =

k 1/2 d , ν

κ2 , κ = 0.41, Cµ (Cε2 − Cε1 ) Cε2 = 1.83, Cµ = 0.096,

(4.42)

σε =

Cε1 = 1.44,

(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 Cµ∗ = Cµ (near 0.09). On the other hand, this explicit algebraic stress solution 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 − 13 {S2 }I. 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 ¯bpq = Xi bij Xj , (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¯bpq Dbij j D j

D i

Xp bij Xjq + Xip bij X . = Xip Xq + 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¯bpq /Dt = 0 does hold so that equation (4.45) can be rewritten as Dbij = bik Ωkj − Ωik bkj , Dt

(4.46)

where

D j

X (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)) k

Ωij = Xi

bij 2 + a3 bik Skj + Sik bkj − bmn Snm δij a4 3 −a2

∗ bik Wkj

where Wij∗ = Wij +

−

∗ Wik bkj

1 Ωij . a2

(4.48)

+ a1 Sij = 0,

(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¯bij /Dt = 0 holds exactly. Since equation (4.48) 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 ¯bij 2¯ ¯ ¯ + a3 bik S kj + S ik bkj − bmn S mn δij a4 3

∗ −a2 ¯bik W kj

−

∗ W ik¯bkj

where now ∗ W ij

= W ij

(4.50)

+ a1 S ij = 0,

DXki j Ωij = Xk Dt

1 − ijr Ωr + Ωij , a2

(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 = −ijr Ωr so that

∗ W ij

= W ij

(4.52)

1 − 1+ ijr Ωr , a2

(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¯bij /Dt = 0 will hold. Spalart and Shur (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) ﬁxed (unbarred) system is a Cartesian system where ek = δkj , the unit vectors in the principal axes frame are simply given by i

(i)

ek = Xk .

(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 , (k)

Ωij = ei

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

0

−τ12

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∗4 , which is now given by 1 1 1 = + D a∗4 a4 k

(4.59)

or, using equation (3.30) for the turbulent kinetic energy, a∗4

=

1 Dk γ1 + + 1 − 2 (γ0 − 1) α1 η 2 τ ε Dt

−1

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∗4 = γ1∗ − 2γ0∗ α1 η 2 τ where

γ0∗ = γ0 − 1, γ1∗ = γ1 + 1 +

−1

τ,

Cε2 − Cε1 . Cε1 − 1

(4.65)

(4.66) (4.67)

[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

0.1 4

2 3

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

0.2

0.10 0.50

2 1.5 1.0 4 3

0.1 20

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

modified EASM original EASM

0.008

0.006

y, m 0.004

0.002

0 20

21

22

- m/s u,

23

24

Figure 4: Eﬀect of modiﬁed turbulent transport and viscous diﬀusion assumption on wake velocity proﬁle.

[1] Linear and nonlinear eddy viscosity models

5

43

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. (1991) ‘A one-equation turbulence transport model for high-Reynolds number wall-bounded ﬂows,’ NASA Technical Memorandum 102847. Baldwin, B.S., and Lomax, H. (1978). ‘Thin layer approximation and algebraic model for separated turbulent ﬂows,’ AIAA 16th Aerospace Sciences Meeting, Paper No. 78-257. Carlson, J.R., Duquesne, N., Rumsey, C.L. and Gatski, T.B. (2001) ‘Computation of turbulent wake ﬂows in variable pressure gradient,’ Comput. Fluids 30, 161–187. Cebeci, T., and Smith, A.M.O. (1974) Analysis of Turbulent Boundary Layers Academic Press, New York.

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Craft, T.J., Launder, B.E., and 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., and Suga, K. (1997) ‘Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model,’ Int. J. Heat and Fluid Flow 18, 15–28. Dacles-Mariani, J., Zilliac, G. G., Chow, J.S., and Bradshaw, P. (1995) ‘Numerical/experimental study of a wingtip vortex in the near ﬁeld,’ AIAA J. 33, 1561– 1568. Degani, D., and Schiﬀ, L.B. (1986). ‘Computation of turbulent supersonic ﬂows around pointed bodies having crossﬂow separation,’ J. Comput. Phys. 66, 173–196. Durbin, P.A. (1991) ‘Near-wall turbulence closure without ‘damping functions’,’ Theoret. Comput. Fluid Dynamics 3, 1–13. Durbin, P.A. (1993a) ‘A Reynolds stress model for near-wall turbulence,’ J. Fluid Mech. 249, 465–498. Durbin, P.A. (1993b) ‘Application of a near-wall turbulence model to boundary layers and heat transfer,’ Int. J. Heat and Fluid Flow 14, 316–323. Durbin, P.A. (1995) ‘Separated ﬂow computations with the k-ε − v 2 model,’ AIAA J. 33, 659–664. Durbin, P.A. (1996). ‘On the k-3 stagnation point anomaly,’ Int. J. Heat and Fluid Flow 17, 89–90. Fu, S., Huang, P.G., Launder, B.E., and Leschziner, M.A. (1988) ‘A comparison of algebraic and diﬀerential second-moment closures for axisymmetric turbulent shear ﬂows with and without swirl,’ ASME J. Fluids. Eng. 110, 216–211. Gatski, T.B. (1996) ‘Turbulent ﬂows – model equations and solution methodology,’ Handbook of Computational Fluid Mechanics, R. Peyret (ed.), Academic Press, 339–415. Gatski, T.B., and Jongen, T. (2000) ‘Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent ﬂows,’ Prog. Aerospace Sci. 36, 655–682. Gatski, T.B., and Speziale, C.G. (1993) ‘On explicit algebraic stress models for complex turbulent ﬂows,’ J. Fluid Mech. 254, 59–78. Girimaji, S.S. (1996) ‘Fully explicit and self-consistent algebraic Reynolds stress model,’ Theor. Comput. Fluid Dyn., 8, 387–402. Gulyaev, A.N., Kozlov, V.E., and Sekundov, A.N. (1993) ‘A universal one-equation model for turbulent viscosity,’ Fluid Dynamics 28, 485–494. Hanjalic, K. (1994) ‘Advanced turbulence closure models: a view of current status and future prospects,’ Int. J. Heat and Fluid Flow 15, 178–203. Johnson, D.A. (1987) ‘Transonic separated ﬂow predictions with an eddy-viscosity/Reynolds-stress closure model,’ AIAA J. 25, 252–259. Johnson, D.A., and Coakley, T J. (1990) ‘Improvements to a nonequilibrium algebraic turbulence model,’ AIAA J. 28, 2000–2003. Johnson, D.A., and King, L.S. (1985) ‘A mathematically simple turbulence closure model for attached and separated turbulent boundary layers,’ AIAA J. 23, 1684– 1692.

[1] Linear and nonlinear eddy viscosity models

45

Jones, W.P., and Launder, B.E. (1972) ‘The prediction of laminarization with a twoequation model of turbulence,’ Int. J. Heat Mass Transfer 15, 301–314. Jongen, T., and Gatski, T.B. (1998a) ‘A new approach to characterizing the equilibrium states of the Reynolds stress anisotropy in homogeneous turbulence,’ Theor. Comput. Fluid Dyn., 11, 31–47. Erratum: Theor. Comput. Fluid Dyn., 12, 71–72. Jongen, T., and Gatski, T.B. (1998b) ‘General explicit algebraic stress relations and best approximation for three-dimensional ﬂows,’ Int. J. Engr. Sci. 36, 739–763. Jongen, T., and Gatski, T.B. (1999) ‘A uniﬁed analysis of planar homogeneous turbulence using single-point closure equations,’ J. Fluid Mech. 399, 117–150. Kolmogorov, A.N. (1942) ‘Equations of turbulent motion in an incompressible ﬂuid,’ Izv. Akad. Nauk., SSSR; Ser. Fiz. 6, 56–58. Launder, B.E. (1982) ‘A generalized algebraic stress transport hypothesis,’ AIAA J. 20, 436–437. Launder, B.E., and Sharma, B.I. (1974) Lett. Heat Mass Transfer 1, 131–138. Menter, F. (1994) ‘Two-equation eddy-viscosity turbulence models for engineering applications,’ AIAA J. 32, 1598–1605. Mohammadi, B., and Pironneau, O. (1994) Analysis of the K–Epsilon Turbulence Model John Wiley & Sons, Chichester, UK. Monson, D.J., and Seegmiller, H.L. (1992) ‘An experimental investigation of subsonic ﬂow in a two-dimensional U-duct,’ NASA TM 103931. Patel, V.C., Rodi, W., Scheuerer, G. (1985) ‘Turbulence models for near-wall and low-Reynolds number ﬂows: a review,’ AIAA J. 23, 1308–1319. Pope, S.B. (1975) ‘A more general eﬀective-viscosity hypothesis,’ J. Fluid Mech. 72, 331–340. Prandtl, L. (1925) ‘Report on investigation of developed turbulence,’ ZAMM 5, 136– 139. Reynolds, O. (1895) ‘On the dynamical theory of incompressible viscous ﬂuids and the determination of the criterion,’ Phil. Trans. R. Soc. Lond. A 186, 123–164. Reynolds, W.C. (1987) ‘Fundamentals of turbulence for turbulence modeling and simulation,’ AGARD Report No. 755 1.1–1.66. Rodi, W. (1976) ‘A new algebraic relation for calculating the Reynolds stresses,’ ZAMM 56, T219–T221. Rodi, W. and Mansour, N.N. (1993) ‘Low-Reynolds number k − ε modelling with the aid of direct numerical simulation data,’ J. Fluid Mech. 250, 509–529. Rumsey, C.L., Gatski, T.B., and Morrison, J.H. (1999) ‘Turbulence model predictions of strongly-curved ﬂows in a U-duct,’ AIAA J. 38, 1394–1402. Sarkar, A., and So, R.M.C. (1997) ‘A critical evaluation of near-wall two-equation models against direct numerical simulation data,’ Int. J. Heat and Fluid Flow 18, 197–208. Savill, A.M., Gatski, T.B., Lindberg, P.-A. (1992) ‘A pseudo-3D extension to the Johnson–King model and its application to the EuroExpt S-duct,’ In Numerical

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Gatski and Rumsey Simulation of Unsteady Flows and Transition to Turbulence, (O. Pironneau, W. Rodi, I. L. Rhyming, A.M. Savill, and T.V. Truong, eds.), Cambridge, 158–165.

Schunk, P.R., and Scriven, L.E. (1990) ‘Constitutive equation for modeling mixed extension and shear in polymer solution processing,’ J. Rheology 34, 1085–1119. Sekundov, A.N. (1971) ‘Application of a diﬀerential equation for turbulent viscosity to the analysis of plane non-self-similar ﬂows,’ Fluid Dynamics 6, 828–840. Shih, T.-H., Zhu, J., and Lumley, J.L. (1995) ‘A new Reynolds stress algebraic equation model,’ Comput. Methods Appl. Mech. Engrg. 125, 287–302. Shur, M., Strelets, M., Zaikov, L., Gulyaev,A., Kozlov, V., and Secundov, A. (1995) ‘Comparative numerical testing of one- and two-equation turbulence models for ﬂows with separation and reattachment,’ AIAA 33rd Aerospace Sciences Meeting, Paper No. 95-0863. So, R.M.C., and Speziale, C.G. (1998) ‘Turbulence modeling and simulation,’ The Handbook of Fluid Dynamics, R.W. Johnson (ed.), CRC Press, 14.1–14.111. Souza Mendes, P.R., Padmanabhan, M., Scriven, L.E., and Macosko, C.W. (1995) ‘Inelastic constitutive equations for complex ﬂows,’ Rheol. Acta 34, 209–214. Spalart, P. R., and Allmaras, S. R. (1994) ‘A one-equation turbulence model for aerodynamic ﬂows,’ La Recherche A´erospatiale 1, 5–21. Spalart, P. R., and Shur, M. (1997) ‘On the sensitization of turbulence models to rotation and curvature,’ Aerosp. Sci. Technol. 5, 297–302. Spencer, A.J.M., and Rivlin, R.S. (1959) ‘The theory of matrix polynomials and its application to the mechanics of isotropic continua,’ Arch. Rational Mech. Anal. 2, 309–336. Speziale, C.G. (1979) ‘Invariance of turbulent closure models,’ Phys. Fluids 22, 1033– 1037. Speziale, C.G. (1987) ‘On nonlinear K-l and Kε models of turbulence,’ J. Fluid Mech. 178, 459–475. Speziale, C.G. (1991) ‘Analytical methods for the development of Reynolds-stress closures in turbulence,’ Annu. Rev. Fluid Mech. 23, 107–157. Speziale, C.G. (1998) ‘A review of material frame-indiﬀerence in mechanics,’ Appl. Mech. Rev. 51, 489–504. Speziale, C.G., Sarkar, S., and Gatski, T.B. (1991) ‘Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach,’ J. Fluid Mech. 227, 245–272. Speziale, C.G., Abid, R., and Anderson, E.C. (1992) ‘Critical evaluation of twoequation models for near-wall turbulence,’ AIAA J. 30, 324–331. Tennekes, H., and Lumley, J.L. (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˜ nada, California. 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´c and S. Jakirli´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´c and Jakirli´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

The Basic Linear Second-Moment Closure Model for High-Re-number Flows

2.1

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 + (fi uj + fj ui ) = Dt ∂xk ∂xk ∂xk ∂t Gij Lij

1

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

p −2Ωk (uj um ikm + ui um jkm ) + ρ

Rij

   ∂ui uj ν  ∂x  k  ν Dij

−

2ν

φij



∂ + ∂xk

∂ui ∂uj + ∂xj ∂xi

49

−

ui uj uk

t Dij

Dij

−

∂ui ∂uj ∂xk ∂xk

εij

    .   

p (ui δjk + uj δik ) ρ

p Dij

(2.1)

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 = 12 ijk Wkj associated with the rotation-rate (‘vorticity’) ∂Uj i tensor Wij = 12 ( ∂U ∂xj − ∂xi ), and the ﬂuid vorticity ωi = ijk Wkj . 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 ∂u ∂u εij = 2ν ∂xki ∂xjk — which is associated with smallest eddies — should reduce 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 ∂ul where ε = ν ∂x and δij is the Kronecker unit tensor. It should be menk ∂xk tioned that at high Reynolds numbers ε can be interpreted as the amount of

50

Hanjali´c and Jakirli´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 t and D p , need to be modelled. The most popular ﬂuctuating pressure, Dij ij t model for Dij is the generalized gradient diﬀusion (GGD), known also as the ∂φ . The application of GGD to the Daly–Harlow model: ϕuk = −Cφ τ uk ul ∂x l 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 = (−ui uj uk ) = Dij ∂xk ∂xk

k 2 ∂ui uj Cs 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 Dij and to adjust the coeﬃcient Cs . In many ﬂows the pressure transport is 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´enard convection) this is not the case and such models are not 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

(2.6)

which can be integrated to yield p at x: p 1 = ρ 4π

V

∂2 u ) + 2 ∂Ul ∂um − ∂fl dV (x ) , (u u − u l m m l ∂xl ∂xm ∂xm ∂xl ∂xl |r|

∂ui ∂uj Multiplication by + where r = ∂xj ∂xi the following exact expression for φij :

x − x.

p φij = ρ

∂ui ∂uj + ∂xj ∂xi

∂Ul +2 ∂xm

1 + 4π

∂um ∂xl

1 = 4π

1 ∂  p A r ∂n

V

∂xl ∂xm

∂ui ∂uj + ∂xj ∂xi

∂ui ∂uj + ∂xj ∂xi

(at x) and averaging yields

 2 ∂ u u ∂u ∂u m i j l  +

−

− p

∂xj

∂xi

φij,1

φij,2



(2.7)

∂fm ∂xm



φij,3

∂ui ∂uj + ∂xj ∂xi

∂ui ∂uj  dV (x ) + ∂xj ∂xi |x − x |

∂ ∂n



1  dA . r

φw ij

(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: w w φij = φij,1 + φij,2 + φij,3 + φw ij,1 + φij,2 + φij,3 ,

(2.9)

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.

52

Hanjali´c and Jakirli´c φij,3 is the ‘isotropization’ of stress production due to a body force; w w φw ij,1 , φij,2 , φij,3 is the wall blockage (‘eddy splatting’) and pressure reﬂection 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 ij into two 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 ij,2 modiﬁes the processes of stress production 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 w Based on the above reasoning several models of φw ij,1 and φij,2 have been proposed, the most frequently employed schemes being:

54

Hanjali´c and Jakirli´c • Shir (1973): w φw ij,1 = C1

ε 3 3 uk um nk nm δij − ui uk nk nj − uk uj nk ni · fw ; k 2 2

• Gibson and Launder (1978):

φw ij,2

=

C2w

(2.14)

3 3 φkm,2 nk nm δij − φik,2 nk nj − φjk,2 nk ni · fw ; 2 2

(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].

where C1w = 0.5, C2w = 0.3, fw =

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

∂Ui ∂xk

Pε1 +Pε2

−2νuk

∂ui ∂ 2 Ui ∂fi ∂ui + ν ∂xl ∂xk ∂xl ∂xl ∂xl

Pε3

Gε

−2ν

∂ui ∂ui ∂uk ∂ 2 ui − 2 ν ∂xk ∂xl ∂xl ∂xk ∂xl



+ 

Pε4

∂

ν

2

Y

∂ε

 ∂xk ∂xk   Dεν 



− uk ε

−

Dεt

Dε

2ν ∂p ∂uk  . (2.16) ρ ∂xi ∂xi 

Dεp

  

[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 ∂ ∂ Dεt = (−uk ε) = Cε uk ul . (2.17) ∂xk ∂xk ε ∂xl Hence, the model equation for ε has the form (Hanjali´c and Launder 1972): 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

2.3

C1 1.8

C2 0.6

C1w 0.5

C2w 0.3

Cε 0.18

Cε1 1.44

Cε2 1.92

Cε3 1.44

Cε4 −0.373

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

Hanjali´c and Jakirli´c The Model Equation for Scalar Flux θui

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 p ∂θ ∂Ui −uk θ −βgi θ2 + ∂xk ∂xk ρ ∂xi

Θ Piθ

U Piθ

∂θ ∂ui ∂ −(α + ν) − ∂xk ∂xk ∂xk

−εiθ

Giθ

φiθ

pθ ui uk θ + δik . ρ

t +D p Diθ iθ

(2.19)

The physical meaning of the various terms can be inferred by comparison with the Reynolds-stress equation: Θ is the ‘thermal’ production (nonuniform temperature ﬁeld interacting Piθ with turbulent stresses), U is the ‘mechanical’ production (mean ﬂow deformation interacting Piθ 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 − C3θ Giθ . − C2θ Piθ k

(2.20)

Θ is absent because the mean It is noted that the term corresponding to Piθ 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

2.3.2

∂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

= −ui uk

(2.21)

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)

2

∂θ εθ = 2α ∂x . A transport equation for εθ can be derived, resembling the ε j 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

εθ =

ε θ2 . R 2k

(2.22)

Hence, the model equation for θ2 is ∂ ∂Θ 1 θ2 Dθ2 − = −2ui θ ε + Cθ2 Dt ∂xi 2R 2k ∂xj 2.3.3

k ∂θ2 ui uj 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

2.4

Cθ 0.15

C1θ 3.5

C2θ 0.55

C3θ 0.55

R 0.5

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 2

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].)

58

Hanjali´c and Jakirli´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

(2.24)

ui uj 2 − δij leads to: k 3 ui uj Dk ui uj Dui uj − Dij = − Dk = (P + G − ε). Dt k Dt k

The expansion of aij =

(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´c and Launder 1972):

t Dij

∂uj uk ∂uk ui ∂ui uj ∂ ∂ k = (−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´c 1994, Jakirli´c 1997). Nagano 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, u3i , while evaluating the mixed triple moments 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´c and Jakirli´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

2 ∂ ∂ k = (−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 − 98 (A2 − A3 ). Speziale, Sarkar and Gatski (1991) (SSG) proposed 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

C1 1+ 0 C1

aij + C1 aik akj

1 − A2 δij 3

,

(3.4)

with C1 = 3.1(A2 A) 2 , C1 = 1.2 and C1 0 = 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 from pairs of DNS components of φij,1 , 11-22, 11-33, 22-33. Lines: 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´c and Jakirli´c

assuming local mean ﬂow homogeneity. The modelling task is reduced to expressing the fourth-order tensor bmi lj in terms of available second-order tensors (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¨ allback 1994), though cubic models have also been designed 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[a2mn (aik Wjk + ajk Wik ) + ami anj (amk Wnk + ank Wmk )], (3.7) 2

1 ∂Ui ∂Uj 1 ∂Ui ∂Uj + and Wij = − where Sij = 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 ij and was shown to 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. 3

The early model of Hanjali´c and Launder (1972) also contained the nonlinear term with the same value of the empirical coeﬃcient, though with opposite sign (Table 1).

C2 Paij

[2] Second-moment turbulence closure modelling

63

Table 1: Summary of coeﬃcients in pressure-strain models

φij,1 Authors

φij,2

Linear Quadratic

Linear

C1

C1

C2

C3

HL

2.8

0

−0.9

LRR IP

1.8

0

LRR QI

1.8

SSG

1.7

Quadratic Cubic C4

C5 C6

C7

C8

0.8

0.71 0.582 0

0

0

0

0.8

0.6

0

0

0

0

0

0.8

0.873 0.655 0

0

0

−1.05

0.9

0.8−

0.65

0

0

0.2

1.2

0.6

0.2

0

1/2 0.625A2

LT

1/2

6.3AA2

0.7C1

0

0.8

0.6 0.866 0.2

Abbreviations: HL: Hanjali´c and Launder (1972), LRR: Launder, Reece and Rodi (1975), SSG Speziale, Sarkar and Gatski (1991), LT: Launder and Tselepidakis (1994).

The model of the rapid pressure-strain term can alternatively be written

∂Uk k in terms of Sij , Pij and Dij = − ui uk ∂U + u u (Launder et al. 1975). j k ∂xi ∂xj 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 (Hadˇzi´c 1999): C2∗ = C2 , C3∗ = 4/3C4 − C3 , C4∗ = 1/2(C4 + C5 ), and C5∗ = 1/2(C4 − C5 ). It should be mentioned that the LRR-IP and LRR-QI models require the use of wall-echo terms φw ij deﬁned earlier, whereas the SSG model 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 Hadˇzi´c 1999), the SSG has some appeal as a compromise between the desired accuracy and computational economy: it is more practical than LRRQI with wall-correction term φw ij , and more accurate than both the IP and LRR-QI models if used without the φw ij .

64

Hanjali´c and Jakirli´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 = CµD 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 = −

φhij , k

(3.9)

where φhij is the ‘homogeneous’ (far from a wall) pressure-strain model, for which, in principle, any known model (without wall correction) can be used4 . The function fij is then used to obtain φD ij = k fij . With viscous eﬀects accounted 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 ij = (Πij − 1/3Πkk δij ) − (εij − ui uj /kε) instead of the convenp 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´c (2000b) have proposed a ‘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 h φ∗ij = (1 − kα)φw ij + kαφij

(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 φhij can be any known homogeneous model of φij , whereas the inhomogeneous part (satisfying the wall constraints) is deﬁned as

φw ij = −5

ε 1 ui uk nj nk + uj uk ni nk − uk ul nk nl (ni nj − δij ) . k 2

(3.13)

The unit normal vectors are obtained from n =

∇α . ||∇α||

(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´c and Jakirli´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´c 1996). The standard model 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´c 1996). 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 ). This modiﬁcation was shown to perform (which also requires modifying Cε1 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-

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ventional dissipation equation serves well. The term SΩ , deﬁned as: SΩ = Cε5 kWij Wij ,

(3.17)

was introduced long ago by Hanjali´c and Launder (1980) to enhance the effects 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´c et al. 1997). It is noted that Shimomura’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´c 1996).

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.

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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 + 2µt + µt + ∂x1 ∂x2 ∂x2 ∂x1 ∂U1 ∂U2 ∂U1 ∂U2 = −u21 − u22 − u1 u2 + . ∂x1 ∂x2 ∂x2 ∂x1

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 ∂U2 ∂U2 1 more complex ﬂows, ∂U ∂x1 , ∂x2 , ∂x1 , . . . can have signiﬁcant values and diﬀerent signs. Hence, P DSM and P EVM may be very diﬀerent. This becomes more 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, ∂U ∂x1 is much less than

∂U1 ∂x2 ,

but u21 is much greater than u22 near the airfoil surface,

2 ∂U2 are of importance (Bradshaw 1 hence both terms in P12 = − u22 ∂U ∂x2 + u1 ∂x1 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´c 1996). The standard k-ε EVM yields the same results for the same initial level of k and ε for both the compression and expansion. Other, industrially more relevant examples are stagnation regions, ﬂow impingement on a solid

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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. Hadˇzi´c et al. 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

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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.

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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 − ∂U2 1 W12 = − ∂U ∂x2 , since ∂x1 = 0). The ﬂuctuating Coriolis force ampliﬁes the 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 /∂x2 the system vorticity Ro = ωl /Ωl (here Ro = − ∂U2Ω ), or its reciprocal S = 3 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 −u22 dU dx2 Rij 4Ωu1 u2 −4Ωu1 u2 0 −2Ω(u21 − u22 ) 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 + 12 Rij (e.g. Launder et al. 1987). So 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

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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´c et al. 1998). It is interesting to note that this term resembles closely that proposed by Hanjali´c and Launder (1980) for nonrotating ﬂows to enhance the eﬀect of 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´c et 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 = 12 ωl + Ωl

[2] Second-moment turbulence closure modelling

a.

b.

75

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

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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´c et al. (1995). An overview of the performance of DSMs in ﬂows with diﬀerent pressure gradients involving separation is given in Hanjali´c et al. (1999). 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

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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

−

ω1

∂U1 ∂x1

vortex stretching

+

ω2

∂U1 ∂U1 + ω3 ∂x2 ∂x3

‘skew-induced’ ω1 (vortex bending)

∂ 2 u2 u3 ∂ 2 u2 u3 ∂2 + + (u2 − u23 ) . ∂x2 ∂x3 2 ∂x22 ∂x23

‘stress-induced’ generation of

(4.5)

ω1

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

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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´c (1994) and elsewhere.

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

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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).

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Figure 9: Mean velocity proﬁles at selected locations in the recovery region behind a backward-facing step ﬂow (Hanjali´c and Jakirli´c 1998) (above) and proﬁles of axial and tangential velocity in a spin-down ﬂow in an engine cylinder at the beginning of compression (below), (Jakirli´c et al. 2000).

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

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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´c and Launder (1976), Launder and Shima (1989), Hanjali´c and Jakirli´c (1993), Shima (1993), Hanjali´c et al. (1995) base their modiﬁcations on the 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; 6

Jakirli´c (1997, p. 44) has noted that the current ER proposal appears to be inconsistent: 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.

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Hanjali´c and Jakirli´c • Provision of a non-isotropic model for εij ; • 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´c et al. (1995) (see also Hanjali´c et al. 1997, Jakirli´c 1997, Hadˇzi´c 1998). This model has been used with reasonable success in a number 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].

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models for φw ij seems reasonable, despite some opposing views in the literature. 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´c et al. (1995) 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´c et al. 1997, 1999), Figure 10a. The sensitivity of stress invariants to 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

Ret 150

C2 = 0.8A1/2 ;

3/2

(

;1 ;

k 3/2 fw = min ; 1.4 2.5εxn

C1w = max(1−0.7C; 0.3);

C2w = min(A; 0.3),

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 =

ui uj 2 − δij k 3

eij =

εij 2 − δij . ε 3

εij , the stress dissipation rate model: 2 εij = fs ε∗ij + (1 − fs ) δij ε 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 + 32 pk q np nq fd √ fs = 1 − AE 2 ; fd = (1 + 0.1Ret )−1 .

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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 k +Cεν ν uj uk + SΩ + Sl , ε ∂xj ∂xl ∂xj ∂xl + Cε1 P + Cε3 G + Cε4 k

where

Ret Cε − 1.4 exp − fε = 1 − 2 Cε2 6

2

,

ε˜ = ε − 2ν

∂k 1/2 ∂xn

(5.1)

,

and SΩ and Sl have been deﬁned earlier, equations (3.17) and (3.18).

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6

85

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.

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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.

c.

d.

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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´c et 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´c et al. 1999). Figure 12a shows the evolution of friction 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

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b.

Figure 13: Oscillatingboundary 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´c et al. 1995). 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

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c.

Figure 13: Continued. in axial turbomachinery (Hanjali´c et al. 1994). Figure 14 shows the response 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´c 1997). Both the high- and low-Re-number DSM reproduce the 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

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Figure 14: Response of turbulent stress ﬁeld in a boundary layer encountering transverse wall movement (Hanjali´c et al. 1994). compression cylinder at two time instants (Jakirli´c et al. 2000). It is worth 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

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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´c 1997), compared 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 Hadˇzi´c and Hanjali´c 2000).

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DSM:

DNS:

Figure 16: Streamlines and axial velocity proﬁles computed with the low-Re DSM (Jakirli´c 1997) and DNS (Pascal 1998) at two time instants in a spindown operation of a rapid compression machine.

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Figure 17: Velocity vectors and proﬁles of swirl velocity in a single-shot compression engine: computations with the low-Re-number DSM (Jakirli´c et al. 2000).

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Figure 18: Pressure-induced separation on a ﬂat wall: above: friction factor; below: streamlines. Computations with the low-Re-number DSM (Hadˇzi´c 1999) 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´c and Jakirli´c 1998): the streamline patterns and 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

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Figure 19: Shear stress distribution in a boundary layer in adverse pressure gradient, computed with the standard DSM and wall functions (Jakirli´c 1977). 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´c and Jakirli´c 1998). 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´c 1999). It should be noted, however, that the application 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

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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

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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. Hadˇzi´c, from the Technical University Hamburg-Harburg, Germany. Thanks are also due to Mr. Bart Hoek from TU Delft for providing several drawings.

References Alam, M. and Sandham, N. (2000). ‘Direct numerical simulations of short laminar separation bubbles’, J. Fluid Mech. 403, 223–250. Anwer, M., So, R.M.C. and Lai, Y.G. (1989). ‘Perturbation by and recovery from bend curvature of a fully developed pipe ﬂow, Physics of Fluids A 1, (8), 1387–1397. Bardina, J., Ferziger, J.H. and Rogallo, R.S. (1987). ‘Eﬀect of rotation on isotropic turbulence; computation and modelling’, J. Fluid Mech., 154, 321–336.

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Basara, B., Hanjali´c, K. and Jakirli´c, S. (1997). ‘Numerical simulation of turbulent ﬂow in a car compartment with a second-moment turbulence closure’, Proc. ASME Fluid Engineering Division Summer Meeting, FEDSM97-3023. Bradshaw, P., Cebeci, T. and Whitelaw, J.H. (1981). Engineering Calculation Methods for Turbulent Flows, Academic Press. Chieng, C.C. and Launder, B.E. (1980). ‘On the calculation of turbulent heat transport downstream from an abrupt pipe expansion, Numerical Heat Transfer 3, 189– 207. Ciofalo, M. and Collins, M.W. (1989). ‘The k-ε predictions of heat transfer in turbulent recirculating ﬂows using an improved wall treatment’, Numer. Heat Transfer B 3, 21–47. Craft, T.J. (1998). ‘Developments in a low-Reynolds-number second-moment closure and its application to separating and reattaching ﬂows’, Int. J. Heat and Fluid Flow 19, 541–548. Craft, T.J., Gant, S.E., Iacovides, H. and Launder, B.E. (2001). ‘Development and application of a new wall function for complex turbulent ﬂows’, Proc. ECCOMAS 2001 Conference K. Morgan, (ed.). Craft, T.J., Gerasimov A., Iacovides, H., Launder, B.E. (2002). ‘Progress in the generalization of wall-function treatments’, Int. J. Heat and Fluid Flow (to appear). Craft, T.J. and Launder, B.E. (1991). ‘Computation of impinging ﬂows using secondmoment closures’, Proc. 8th Symp. Turb. Shear Flow, Munich, 8/1-8/5. Craft, T.J. and Launder, B.E. (1996). ‘A Reynolds stress closure designed for complex geometries’, Int. J. Heat and Fluid Flow 17, (3), 245–254. Craft, T.J. Launder, B.E. and Suga K. (1995). ‘A nonlinear eddy viscosity model including sensitivity to stress anisotropy’, Proc. 10th Symp. on Turbulent Shear Flows Pennsylvania State Univ., 23/19–23/25. Craft, T.J. Launder, B.E. and Suga K. (1997). ‘Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model’, Int. J. Heat and Fluid Flow 18, (1), 15–28. Demuren, A.O. and Rodi, W. (1984). ‘Calculation of turbulence driven secondary motion in non-circular ducts’, J. Fluid Mech. 103, 161–182. Durbin, P.A. (1991). ‘Near-wall turbulence closure modelling without “damping functions” ’, Theoret. Comput. Fluid Dynamics 3, 1–13. Durbin, P.A. (1993). ‘A Reynolds stress model for near-wall turbulence’, J. Fluid Mech. 249, 465–498. Dutzler, G.K., Pettersson-Reif, B.A. and Andersson, H.I. (2000) Laminarization of turbulent ﬂow in the entrance region of a rapidly rotating channel’, Int. J. Heat and Fluid Flow 21, 49–57. Fu, S., Launder, B.E. and Tselepidakis, D.P. (1987). ‘Accommodating the eﬀects of high strain rates in modelling the pressure-strain correlation, Thermoﬂuids report TFD/87/5, UMIST, Manchester. Fu, S., Leschziner, M. and Launder, B.E. (1987). ‘Modelling strongly swirling recirculating jet ﬂow with Reynolds-stress transport closures’, Proc. 6th Symp. Turbulent Shear Flows, Paper 17.6.

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Gibson, M.M. and Launder, B.E. (1978). ‘Ground eﬀects on pressure ﬂuctuations in the atmospheric boundary layer’, J. Fluid Mech. 86, 491. Gibson, M.M. and Rodi, W. (1981). ‘A Reynolds-stress closure model of turbulence applied to the calculation of a highly curved mixing layer’, J. Fluid Mech. 140, 189–222. Hadˇzi´c, I. (1999). ‘Second-moment closure modelling of transitional and unsteady turbulent ﬂows’, PhD thesis, Delft University of Techology, 1999. Hadˇzi´c, I. and Hanjali´c, K. (2000). ‘Separation-induced transition to turbulence: second-moment closure modeling’, Flow Turbulence and Combustion 63, 153–173. Hadˇzi´c, I., Hanjali´c, K. and Laurence, D. (2001). ‘Modelling the response of turbulence subjected to cyclic irrotational strain’, Physics of Fluids 13, (6), 1739–1747. Hanjali´c, K. (1994). ‘Advanced turbulence closure models: a view of current status and future prospects’, Int. J. Heat and Fluid Flow 15, (3), 178–203. Hanjali´c, K. (1996). ‘Some resolved and unresolved issues in modelling non-equilibrium and unsteady turbulent ﬂows’. In Engineering Turbulence Modelling and Experiments 3, W. Rodi and G. Bergeles, (eds.), Elsevier, 3–18. Hanjali´c, K. (1999). ‘Second-moment turbulence closures for CFD: needs and prospects’, Int. J. Computational Fluid Dynamics 12, 66–97. Hanjali´c, K. and Hadˇzi´c, I. (1996). ‘Modelling the transitional phenomena with statistical turbulence closure models’. In Transitional Boundary Layers in Aeronautics, R.A.W.M. Henkes and J.L. van Ingen, (eds.), North-Holland, 283–294. Hanjali´c, K., Hadˇzi´c, I. and Jakirli´c, S. (1999). ‘Modelling turbulent wall ﬂows subjected to strong pressure variations’, ASME J. of Fluid Engineering 121, 57–64. Hanjali´c, K. Jakirli´c S. and Durst, F. (1994). ‘A computational study of joint eﬀects of transverse shear and streamwise acceleration on three-dimensional boundary layers’, Int. J. Heat and Fluid Flow 15, (4), 260–282. Hanjali´c, K. Jakirli´c S. and Hadˇzi´c I. (1995). ‘Computation of oscillating turbulent ﬂows at transitional Reynolds numbers’, Turbulent Shear Flows, 9, F. Durst et al. (eds.), Springer, 323–342. Hanjali´c, K. Jakirli´c S. and Hadˇzi´c I. (1997). ‘Expanding the limits of “equilibrium” second-moment turbulence closures’, Fluid Dynamics Research 20, 25–41. Hanjali´c, K. and Jakirli´c, S. (1998). ‘Contribution towards the second-moment closure modelling of separated turbulent ﬂows’, Computers & Fluids 27, (2), 137–156. Hanjali´c, K. and Launder, B.E. (1972). ‘A Reynolds stress model of turbulence and its application to thin shear ﬂows’, J. Fluid Mech. 52, (4), 609-638. Hanjali´c, K. and Launder, B.E. (1976). ‘Contribution towards a Reynolds-stress closure for low-Reynolds number turbulence’, J. Fluid Mech. 74, (4), 593–610. Hanjali´c, K. and Launder, B.E. (1980). ‘Sensitizing the dissipation equation to irrotational strains’, J. Fluids Engineering 102, 34–40. Hogg, S. and Leschziner, M. (1989). ‘Computation of highly swirling conﬁned ﬂow with a Reynolds stress turbulence model’, AIAA J. 27, 57–63. Iacovides, H., Launder, B.E. and Li, H-Y. (1996). ‘The computation of ﬂow development through stationary and rotating U-ducts of strong curvature’, Int. J. Heat and Fluid Flow 17, (1), 22–33.

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Ishigaki, H. (1996). ‘Analogy between turbulent ﬂows in curved pipes and orthogonally rotating pipes’, J. Fluid Mech. 307, 1–10. Jakirli´c, S. (1997). ‘Reynolds-Spannungs-Modellierung komplexer turbulenter Str¨ omungen’, Doktor Arbeit, Univerit¨ at Erlangen-N¨ urnberg. Jakirli´c, S. and Hanjali´c, K. (1995). ‘A second-moment closure for non-equilibrium and separating high- and low-Re-number ﬂows, Proc. 10th Symp. Turbulent Shear Flows, Pennsylvania State University, USA. Jakirli´c, S., Tropea, C. and Hanjali´c, K. (1998). ‘Computations of rotating channel ﬂows with a low-Re-number second-moment closure model’, 7th ERCOFTAC/IAHR Workshop on Reﬁned Flow Modelling UMIST, Manchester, 28–29 May, 1998. Jakirli´c, S., Volkert, J., Pascal, H., Hanjali´c, K. and Tropea, C. (2000). ‘DNS, experimental and modelling study of axially compressed in-cylinder swirling ﬂow’, Int. J. Heat and Fluid Flow 21, (5), 627-639. Johansson, A.V. and H¨ allback, M. (1994). ‘Modelling of rapid pressure-strain in Reynolds-stress closures’, J. Fluid Mech. 269, 143–168. Jones, W.P. and Mussonge, P. (1988). ‘Closure of the Reynolds stress and scalar ﬂux equations’, Phys. Fluids 31, (12), 3589–3604. Johnson, R. W and Launder, B.E. (1982). ‘Discussion of “On the calculation of turbulent heat transport downstream from an abrupt pipe expansion” ’, Numerical Heat Transfer, Technical Note 5, 493–496. Kiel, R. and Vieth, D. (1995). ‘Experimental and theoretical investigation of the nearwall region in a turbulent separated and reattached ﬂow’, Experimental Thermal and Fluid Sciences 11, 243–256. Kim, S.E. and Choudhury, D. (1995). ‘A near-wall treatment using wall functions sensitized to pressure gradient’. In FED Vol. 217, Separated and Complex Flows, ASME 1995, 273–280. Launder. B.E. (1976). ‘Heat and mass transport’. In Turbulence, P. Bradshaw, (ed.), Topics in Applied Physics, 12, Springer, 232–287. Launder, B.E. and Morse, A.P. (1979). ‘Numerical prediction of axisymmetric free shear ﬂows with a Reynolds stress closure’. In Turbulent Shear Flows 1, L.J.S. Bradbury et al., (eds.), Springer, 279–294. Launder, B.E., Reece, G.J. and Rodi, W. (1975). ‘Progress in the development of Reynolds-stress turbulence closure’, J. Fluid Mech. 68, 537–566. Launder, B.E. and Samaraweera, D.S.A. (1979). ‘Application of a second-moment closure to heat and mass transport in thin shear ﬂows’, Int. J. Heat Mass Transfer 22, 1631–1643. Launder, B.E. and Tselepidakis, D.P. (1994) ‘Application of a new second-moment closure to turbulent channel ﬂow rotating in orthogonal mode’, Int. J. Heat and Fluid Flows 15, 2–10. Launder, B.E., Tselepidakis, D.P. and Younis, B.A. (1987). ‘A second-moment closure study of rotating channel ﬂow’, J. Fluid Mech. 183, 63–75. Laurence, D. (1997). Applications of Reynolds averaged Navier–Stokes equations to engineering problems, Von Karman Institute for Fluid Dynamics, Lecture Series, 1997-03.

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Lien, F.S. and Leschziner, M.A. (1993). ‘Second-moment modelling of recirculating ﬂow with a non-orthogonal collocated ﬁnite-volume algorithm’, Turbulent Shear Flows, 8, F. Durst et al, (eds.), Springer, 205–222. Lumley, J.L. (1978). ‘Computational modeling of turbulent ﬂows’, Adv. Appl. Mech. 18, 123–176. Luo, J. and Lakshminarayana, B. (1997). ‘Analysis of streamline curvature eﬀects on wall-bounded turbulent ﬂows’, AIAA J. 35, (8), 1273–1279. Manceau, R., Wang, M. and Laurence, D. (2001). ‘Inhomogeneity and anisotropy eﬀects on the redistribution term in RANS modelling’, J. Fluid Mech. 438, 307– 338. Manceau, R. and Hanjali´c, K. (2000a). ‘A new form of elliptic relaxation equation to account for wall eﬀects in RANS modelling’, Physics of Fluids 12, (9), 2345–2351. Manceau, R. and Hanjali´c, K. (2000b). ‘The elliptic blending model: a new near-wall Reynolds-stress closure’, (to appear in Physics of Fluids). Pettersson, B.A. and Andersson, H.I. (1997). ‘Near-wall Reynolds-stress modelling in noninertial frames of reference’, Fluid Dynamics Research 19, 251–276. Shih, T.H. and Lumley, J.L. (1993) ‘Critical comparison of second-order closures with direct numerical simulations of homogenous turbulence’, AIAAJ. 31, (4), 663–670 Shimomura, Y. (1993). ‘Turbulence modelling suggested by system rotation. In NearWall Turbulent Flows, So et al., (eds.), Elsevier Science Publishers B.V., 115–123. Spalart, P.R. and Coleman, G.N. (1997). ‘Numerical study of a separation bubble with heat transfer’, Eur. J. Mech. B 16, 169–189. Spalart, P.R. and Strelets, M.K. (2000). ‘Mechanism of transition and heat transfer in a separation bubble’, J. Fluid Mech. 403, 329–349. Speziale, C.G. (1991). ‘Analytical methods for the development of Reynolds-stress closures in turbulence’, Ann. Rev. Fluid Mech. 23, 107–157. Speziale, C.G. and Gatski, T.B. (1997). ‘Analysis and modeling anisotropies in the dissipation rate of turbulence’, J. Fluid Mech. 344, 155–180. Speziale, C.G., Sarkar S. and Gatski, T.B. (1991). ‘Modelling the pressure-strain correlation of turbulence: an invariant system dynamics approach’, J. Fluid Mech. 227, 245–272. Travin A., Shur, M., Strelets, M. and Spalart, P. (1999). ‘Detached-eddy simulations past circular cylinder’, Flow, Turbulence and Combustion 63, 293–313. Wenneberger, D. (1995). ‘Entwicklung eines vorhersagefa¨achigen Berechnungsmodells f¨ ur stark verdrallte Str¨ omungen mit Verbrenung’, Doktor Arbeit, Universit¨ at Erlangen-N¨ urnberg. Wizman, V., Laurence, D., Kanniche, M., Durbin, P. and Demuren, A. (1996). ‘Modelling near-wall eﬀects in second-moment closures by elliptic relaxation’, Int. J. Heat and Fluid Flow 17, (1), 255–266. Wallin, S. and Johansson, A.V. (2000). ‘A new explicit algebraic Reynolds stress model for incompressible and compressible turbulent ﬂows’, J. 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)

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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), ∂ui uj k ∂ dij = νδlk + cs uk ul , (2.2) ∂xk ε ∂xl 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 ∂uj p ∂ui φij ≡ + (2.3) ρ ∂xj ∂xi 3 ∂ 3 uk ul uj dV ∂ uk ul ui 1 + = − 4π V ol ∂rl ∂rk ∂rj ∂rl ∂rk ∂ri |r| φij1

1 − 2π

V ol

∂ 2 ul uj ∂ 2 ul ui + ∂rk ∂rj ∂rk ∂ri

∂Uk dV , ∂rl |r|

φij2

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 ∂Uk li + Xki , φij2 = Xkj ∂xl

(2.4)

li represents the integral of the two-point velocity-derivative where the tensor Xkj correlations: ∂ 2 ul ui dV 1 li Xkj = − . (2.5) 2π V ol ∂rk ∂rj |r|

This approach has been employed by Naot et al. (1973) and Launder et al. li is simply a linear function of the (1975) to derive models of φij2 in which Xkj 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 = 0 • Continuity: Xki li = 2u u . • Normalization: Xkk l i

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 γ+8 30γ − 2 ∂Ui ∂Uj 1 φij2 = − + (Pij − /3δij Pkk ) − k 11 55 ∂xj ∂xi 8γ − 2 − (2.7) (Dij − 1/3δij Dkk ), 11 where Dij ≡ −ui uk ∂Uk /∂xj − uj uk ∂Uk /∂xi .

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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 to be a nonlinear function of on the second-moments, and thus to allow Xkj 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 = 0), and making use of the Cayley– Applying the continuity constraint (Xki Hamilton theorem, leads to six equations: λ1 + 4λ2 = 0

(2.9a)

λ3 + λ4 + 5λ5 = 0

(2.9b)

λ6 + λ7 + λ8 + λ9 + 5λ10 = 0

(2.9c)

λ10 + λ11 + 4λ12 = 0

(2.9d)

λ13 + λ14 + 4λ15 + 2λ18 + 2λ19 + 10λ20 = 0

(2.9e)

λ16 + 4λ17 + 1/3(λ13 + λ14 + 2λ15 ) = 0

(2.9f)

Similarly, the normalization constraint leads to a further six equations: 3λ1 + 2λ2 = 4/3

(2.10a)

3λ3 + 4λ5 = 2

(2.10b)

2λ7 + 3λ8 + 4λ10 = 0

(2.10c)

λ9 + 3λ11 + 2λ12 = 0

(2.10d)

λ13 + 2λ15 + 3λ18 + 4λ20 = 0

(2.10e)

4λ15 + 9λ16 + 6λ17 = 0

(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 u22 is taken as the vanishing component, then the other two normal stresses can be written as u21 = (1 + δ)k and u23 = (1 − δ)k. The 2-component limit requires that ∂Uk = 0, ∂xl

l2 Xk2

(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

(2.12a) (2.12b)

3λ10 − 6(λ11 + λ12 ) − 2λ15 − 6(λ16 + λ17 ) + 4(λ18 + λ19 ) + 6λ20 = 0 (2.12c) 2λ20 = 0.

(2.12d)

These equations can be solved, leaving four undetermined parameters: λ1 =

8 15

2 λ2 = − 15

λ3 = λ4 = λ5 = λ6 = λ7 =

14 4 15 + 3 t 1 11 15 + 3 t 1 5 −t 1 10 + 8t − 2p 3 − 10 − 32 t + p

+ t − 2p

λ15 = 3s − 94 t

λ9 = − 15 2 t − 2p

λ16 = −2s + t

λ10 = p

λ17 = s

λ11 = 3t + p

λ18 =

λ8 =

1 5

λ12 =

− 34 t

−

λ13 = −6s − λ14 = −6s +

1 2p 27 4 t 33 4 t

15 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 2 +0.1 Pmj + Pmi − /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

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.

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(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 , (2.16) where c1 = 3.1(A2 A)1/2

2.2

c1 = 1.1

fA = A1/2 .

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)

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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 ∂ε εPkk ε2 k ∂ νδlk + cε uk ul , (2.18) = cε1 − cε2 + Dt 2k k ∂xk ε ∂xl 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

3

1/2

cε2 = 1.92/(1 + 0.7A2 A).

(2.19)

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

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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 u21 and u22 is rather too weak. It is, however, 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].

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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

u2c /(∆Uc2)

u2c /(∆Uc2)

0.4 Airfoil Expt. Pred. Solid Strip Expt. Pred.

0.3

0.2

0.2

0.1

0.1

0.0

Airfoil Expt. Pred. Solid Strip Expt. Pred.

0.3

0.0 0

500

1000

1500

x/θ 2000

600

0

500

1000

1500

x/θ 2000

(y1/2/θ )2

(y1/2/θ )2

600

Airfoil Expt. Pred. Solid Strip Expt. Pred.

450

Airfoil Expt. Pred. Solid Strip Expt. Pred.

450

300

300

150

150

0

0 0

500

1000

1500

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

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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

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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).

Expt. (Abrahamsson et al, 1997) Linear EVM Basic model TCL model TCL model at 70 diameters

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

z˙1/2 /y˙ 1/2 4.94 0.88 15.3 8.54 5.6

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

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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).

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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)

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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 : ∂ 3 uk ul θ dV ∂ 2 ul θ ∂Uk dV 1 1 − , (4.2) φiθ = − 4π V ol ∂rl ∂rk ∂ri |r| 2π V ol ∂rk ∂ri ∂rl |r| from which φiθ is traditionally modelled as φiθ = φiθ1 + φiθ2 ,

(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 = 2blki

∂Uk , ∂xl

where the tensor blki represents the integral: ∂ 2 ul θ dV 1 l bki = − . 4π V ol ∂rk ∂ri |r|

(4.4)

(4.5)

Adopting the same approach to modelling this tensor as was done for φij2 , blki can be modelled in terms of the Reynolds stresses and scalar ﬂuxes. However, the linearity principle (noting that the integral in equation (4.5) is linear in the scalar θ) requires that an expansion for blki , whilst possibly being nonlinear 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 blki = α1 ul θδik + α2 uk θδli + ui θδlk +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)

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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: bkki = 0 • Normalization: blkk = ul θ and applying them leads to the eight relations: α1 + 4α2 = 0

(4.7a)

α3 + α4 + α5 + 4α6 = 0

(4.7b)

α7 + α8 + α9 + α10 + α13 + 4α14 = 0

(4.7c)

α10 + α11 + 4α12 = 0

(4.7d)

3α1 + 2α2 = 1

(4.7e)

2α4 + 3α5 + 2α6 = 0

(4.7f)

2α8 + 2α10 + 3α13 + 2α14 = 0

(4.7g)

α9 + 3α11 + 2α12 = 0.

(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

≤ u2α θ2 ,

(4.9)

could not be violated. They did this by imposing the condition that the rate 2 of change of the diﬀerence uα θ − u2α θ2 should be zero when equality held or, mathematically, 2uα θ

Dθ2 Du2α Duα θ = u2α + θ2 , Dt Dt Dt

(4.10)

2 when uα θ = u2α θ2 . 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 blki , but the above constraint also led to both free coeﬃcients in the TCL model of φij2 being determined as zero.

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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 u22 is zero (Craft 1991). For φiθ2 , this condition translates to ∂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)

= 0

(4.12b)

= 0

(4.12c)

= 0

(4.12d)

= 0

(4.12e)

= 0.

(4.12f)

=

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

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written: φiθ2

∂Uk ∂Ui ∂Uk 1 ε P ∂Ul = 0.8uk θ − 0.2uk θ + /3 ui θ − 0.4uk θail + ∂xk ∂xi k ε ∂xl ∂xk ∂Um ∂Ul − 0.1uk θ (aim Pmk + 2amk Pim ) /k + +0.1uk θaik aml ∂xl ∂xm ∂Uk ∂Ul +0.15aml + amk ui θ − ami uk θ ∂xl ∂xk ∂Uk ∂Ui −0.05aml 7amk ui θ + uk θ ∂xl ∂xl ∂Ui ∂Ui , (4.13) + amk −uk θ aml ∂xk ∂xl

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 − c θ1 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 u22 = 0 regardless 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 + c , θ1 A2 ) + cθ1 aik uk θ + cθ1 aik akj uj θ − cθ1 rkaij k ∂xj (4.15)

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where + , cθ1 = −0.8, cθ1 = 1.7 1 + 1.2(A2 A)1/2 c∗θ1 = −0.2A1/2 , c θ1 = 0.6,

cθ1 = 1.1,

and the timescale ratio r is deﬁned2 as r = (2εθ /θ2 )(k/ε). In this case the factor A1/2 in the coeﬃcient c∗θ1 ensures that this part of the model also satisﬁes the 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].

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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.

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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.

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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`ese, Belgium. 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.

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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`ese, Belgium. 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¨ ur 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.

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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

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IMAGE VORTICITY

PRESSURE REFLECTION

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

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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 + ν∇2 ui uj = Pij Dt production redistribution dissipation transport viscous (1.3) the redistribution tensor is written φij = φhij + φw ij

(1.4)

where φhij is a homogeneous pressure-strain model, such as the simple IP model,

1 φhij = −c1 εaij − c2 Pij − Pkk δij . 3

(1.5)

Here aij ≡ ui uj /k − 23 δij is the anisotropy tensor. The term φw ij in (1.4) is 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 ij

=

ε cw 1

L 3 3 um ul nm nl δij − ui um nm nj − uj um nm ni + ··· k 2 2 d

(1.6)

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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 ij , has to 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: φhij 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 φhij models: IP, LRR (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 φhij .

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

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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: 1 p(x) = − 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 ∂p 1 ui j (x) = ∂x 4π

x) ui (x) ∂S( ∂xj d3 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

x) turbulence (Chou 1945) note at this point in the derivation that ui (x) ∂S( ∂xj is a function of x − x alone, so that the integral in (2.3) is a constant second order tensor, φhij :

ui

∂p 1 (x) = j ∂x 4π

F cnij (x − x ) 3 d x = φhij |x − x |

(2.4)

where φhij is not a function of x. The original form of the source in (1.1) motivates the standard practice of splitting φhij into slow and rapid parts, φhij = Nij + Mijkl

∂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 − 23 uk ul δij ). 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

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this end, an exponential function will be used as a device to introduce the cor

x) relation length of the turbulence into the formulation. Letting ui (x) ∂S( ∂xj =

Rij (x )e−|x−x |/L in (2.3) gives ∂p ui (x) = ∂xj

e−|x−x |/L 3 Rij (x ) d x. 4π|x − x |

(2.6)

The representation of non-locality in this formula can be described as both geometrical and statistical.

−|x−x |/L

(•) e

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 = φhij /kL2 . However, the present 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 φhij , evaluated in plane channel ﬂow with friction velocity Reynolds number of Rτ = 395. When this φhij is used as the 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 dramath ically when y + < ∼ 60. The magnitude and sign of φ12 are quite wrong; however 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

133 0.20

v2 component

uv component

0.1

0

-0.1

℘12 via elliptic relaxation h SSG formula φ 12

-0.2

0.15

0.10

0.05

DNS data

℘12 -0.3

0

0

100

200

0

100

y+

200

y+

u2 component

0

-0.1

-0.2

-0.3 0

100

200

y+

Figure 2: The eﬀect of elliptic relaxation on the SSG formula for φhij . Computation 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 φh12 near the wall, and brings φ12 into agreement with the DNS data. 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

∂ ∂uk ui uj ∂ = νTkl ui uj Tij ≡ − ∂xk ∂xk ∂xl

(3.1)

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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 ui uj + Ωl (ikl uk uj + jkl uk ui ) + ε Dt k ∂ ∂ ν ui uj + ν∇2 ui uj = Pij + φij + ∂xk Tkl ∂xl φhij aij L2 ∇2 fij − fij = − − k T

(3.4)

where

∂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 Pij = −ui uk

φij =

φhij

ui uj 2 +ε − 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 23 δij ε in the quasi-homogeneous limit. This is the behavior far from 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 ∂x2k

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/∂x2d 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 ∂x2d

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).

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The length scale in (3.4) is prescribed by analogy to (3.3) as  

L = max cL 

k 3/2 ε

, cη

ν3

1/4  

ε



.

(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 φhij in the elliptic equation; the only constraint is that the model relax to φij − εij = φhij − 23 εδij in the quasi-homogeneous limit. However, to date 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 Lφhij /k, making it and fij constant in the log-layer. This makes the Laplacian 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

∂v 2 ∂ νT + ν∇2 v 2 = kf + ∂xk ∂xk

L ∇ f −f

P = −c2 + c1 k

2

v2 Dv 2 +ε Dt k 2

∂U

v2 2 /T − k 3

(3.7)

∂U

j i where P = νT ∂xij + ∂U ∂xj ∂xi . These are solved in conjunction with the k-ε 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:

εv 2 ν 2v2 f = −5 lim = −20 lim d→0 k 2 d→0 εd4

(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

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6

10 3 Cf

4

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

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0.75

0.5

0.50

0.4

100Cf

0.25

0.3

Y

Cp 0

0.2 -0.25

0.1

-0.50

0 -0.5

0

0.5

1

1.5

U profiles

2

2.5

-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

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1.0 IP Launder & Shima Exp

U/Uw

R

ω

Uθ (r)

r

0.0 0.0

Schematic of ﬂow conﬁguration

0.2

(r-R)/R

0.4

Mean velocity distribution

0.4

uv/2k 0.3

IP Launder & Shima Exp

0.2

0.1

0.0 10-3

10-2

(r-R)/R

10-1

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

[4] The elliptic relaxation method

(a)

141

(b)

z x

Ω

y

Uwall

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

U/Ub

Durbin and Pettersson-Reif

-uv/u*02

Ro = 0.05

DNS RLA

1.0

1.0 Ro = 0.10 Ro = 0.20

Ro = 0.05

0.0

0.0 0.0 0.0

Ro = 0.10

Ro = 0.50

0.0 Ro = 0.20

0.0

0.0

DNS RLA

0.0 -1.0

0.0

y/h

1.0

Ro = 0.50

-1.0

Mean velocity distribution

0.0

y/h

1.0

Turbulent shear stress distribution

0.5 Ro = 0.05

0.0

-uv/k Ro = 0.10

0.0

0.0 Ro = 0.20 0.0

-1.0

DNS RLA

Ro = 0.50 0.0

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

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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 − 12 Ω2 (x2 + y 2 ) is the mean reduced pressure. The Coriolis acceleration 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

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1.0 DNS RLA

〈U z /Uw 〈

Ro = 0.0

0.5 Ro = 0.1

0.0

Ro = 0.2

0.0 0.0 0.0

1.0

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

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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

∂ ∂U ∂ ∂U 0 = 2ΩV + ν ν − uv + − uw ∂y ∂y ∂z ∂z ∂P ∗ ∂y ∂P ∗ ∂z

∂V ∂ ∂V ∂ ν ν = −2ΩU + − v2 + − vw ∂y ∂y ∂z ∂z

=

0 =

∂W ∂ ∂W ∂ ν ν − vw + − w2 ∂y ∂y ∂z ∂z

(4.2)

∂V ∂W + ∂y ∂z

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

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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-

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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) 2 ∂ Uθ 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 = −u2r

∂Uθ Uθ + u2θ ∂r r

and Crθ = (u2r − u2θ )

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, u2r = u2θ . 2

Editors’ note. Readers will note that a diﬀerent view is taken in [22].

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2.0

x

z/h

2h y

2h

1.0

z

Schematic of square duct

0.0 0.0

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.

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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 ∂Ωx ∂Ωx V +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

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∂V where Ωx = ∂W ∂y − ∂z . The two last terms are referred to as turbulent source 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¨ofdahl, L. and Nilsen, P.J. (1991). ‘Turbulence in 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. (1998). ‘Secondary ﬂow in rotating turbulent plane Couette ﬂow: direct simulation and second-moment modelling’. In Advances in Turbulence VII Frisch, U. (ed.), 301–304. Bech, K.H. and Andersson, H.I. (1997). ‘Turbulent plane Couette ﬂow subjected to strong system rotation’, J. Fluid Mech. 347, 289–314. Chou, P.Y. (1945). ‘On velocity correlations and the solution of equations of turbulent ﬂuctuations’, Quart. of Applied Math. 3, 38–54. Craft, T., Graham, L. and Launder, B.E. (1993). ‘Impinging jet studies for turbulence model assessment – II. An examination of the performance of four turbulence models’, Int. J. Heat Mass Transfer 36, 2685–2697. Daly, B.J. and Harlow, F.H. (1970). ‘Transport equations of turbulence’, Phys. Fluids 13, 2634–2649.

[4] The elliptic relaxation method

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Demuren, A.O. and Wilson, R.V. (1995). ‘On elliptic relaxation near wall models’. In Transition, Turbulence and Combustion II M.Y. Hussaini, T.B. Gatski, and T.L. Jackson (eds.), Kluwer, 61–71. Dreeben, T.D. and Pope, S.B. (1997). ‘Probability density function and Reynoldsstress modeling of near-wall turbulent ﬂows’, Phys. Fluids 9, 154–163. Durbin, P.A. (1991). ‘Near-wall closure modelling without “Damping Functions” ’, Theoret. Comput. Fluid Dynamics 3, 1–13. Durbin, P.A. (1993). ‘A Reynolds Stress model for near-wall turbulence’, J. Fluid Mech. 249, 465–498. Durbin, P.A. (1995). ‘Separated ﬂow computations with the k-ε-v 2 model’, AIAA J. 33, 659–664. Fu, S., Launder, B.E. and Tselepidakis, D.P. (1987). ‘Accommodating the eﬀects of high strain rates in modeling the pressure-strain correlation’, UMIST Technical Report TFD/87/5. Gibson, M.M. and Launder, B.E. (1978). ‘Ground eﬀects on pressure ﬂuctuations in the atmospheric boundary layer’, J. Fluid Mech. 86, 491–511. Hunt, J.C.R. and Graham, J.M.R. (1978). ‘Free-stream turbulence near plane boundaries’, J. Fluid Mech. 212, 497–532. Huser, A. and Biringen, S. (1993). ‘Direct numerical simulation of turbulent ﬂow in a square duct’, J. Fluid Mech. 257, 65–95. Huser, A., Biringen, S. and Hatay, F.F. (1994). ‘Direct simulation of turbulent ﬂow in a square duct: Reynolds stress budget’, J. Fluid Mech. 257, 65–95. Imao, S., Itoh, M. and Harada, T. (1996). ‘Turbulent characteristics of the ﬂow in an axially rotating pipe’, Int. J. Heat and Fluid Flow 17, 444–451. Johnston, J.P., Halleen, R.M. and Lezius, D.K. (1972). ‘Eﬀects of spanwise rotation on structure of two-dimensional fully developed turbulent channel ﬂow’, J. Fluid Mech. 56, 533–557. Kristoﬀersen, R. and Andersson, H.I. (1993). ‘Direct simulations of low-Reynoldsnumber turbulent ﬂow in a rotating channel’, J. Fluid Mech. 256, 163–197. Lai, Y. G. and So, R. M. C. (1990). ‘On near-wall turbulent ﬂow modelling’, AIAA J. 34, 2291–2298. Lamballais, E., Lesieur, M. and Metais, O. (1996). ‘Eﬀects of spanwise rotation on the vorticity stretching in transitional and turbulent channel ﬂow’, Int. J. Heat and Fluid Flow 17, 324–332. Launder, B.E. and Li, S.-P. (1994). ‘On the elimination of wall topography parameters from second-moment closure’, Phys. Fluids 6, 999–1006. Launder, B.E. and Shima, N. (1989). ‘Second-moment closure for the near-wall sublayer: development and application’, AIAA J. 27, 1319–1325. Launder, B.E., Reece, G.J. and Rodi, W. (1975). ‘Progress in the development of Reynolds stress turbulence closure’, J. Fluid Mech. 68, 537–566. Le, H., Moin, P. and Kim, J. (1997). ‘Direct numerical simulation of turbulent ﬂow over a backward-facing step’, J. Fluid Mech. 330, 349–374.

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Manceau, R., Wang, M. and Laurence, D. (2001). ‘Inhomogeneity and anisotropy eﬀects on the redistribution term in RANS modeling’, J. Fluid Mech. 438, 307– 338. Mansour, N.N., Kim, J. and Moin, P. (1988). ‘Reynolds-stress and dissipation budgets in a turbulent channel ﬂow’, J. Fluid Mech. 194, 15–44. Parneix, S., Durbin, P.A. and Behnia, M. (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

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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´c and Hanjali´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.

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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 ij ) − εδij , ∂xk 3

(1)

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where

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 34 34 w 4 Φij = Φkl nk nl δij − Φik nj nk − Φjk ni nk f, 2 2 4 ij = C1w ρ ε ui uj + C2w Φij,2 , Φ 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: 1 J

)

∂ dij = ∂xk

*

∂ 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 2 2 ϕ = u , v , 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, Cs ρk 2 2 u yη − 2uvxη yη + v 2 x2η , ε Cs ρk 2 2 = u yξ − 2uvxξ yξ + v 2 x2ξ , ε Cs ρk 2 = − u yξ yη − uv(xξ yη + xη yξ ) + v 2 xξ xη , ε

q11 =

(4)

q22

(5)

q12

(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

2ρ 2 v (Vη xξ − Vξ xη ) + uv(Vξ yη − Vη yξ ) J ρ 2 = − u (Vξ yη − Vη yξ ) + v 2 (Uη xξ − Uξ xη ) J

157

P22 = − P12

(9)

+uv(Uξ yη + Vη xξ − Uη yξ − Vξ xη ) ,

(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 = n21 f,

fy = n22 f,

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´c and Hanjali´c (1995) and 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

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Table 1: Coeﬃcients associated with the sources of the Reynolds-stress equations. ϕ = u2 α1

1 − C2 + 2C2 C2w fx

α5

−(C1 + 2C1w fx )

α2

−C2 C2w fy

α6

C1w fy

α3

C2 C2w fxy 2 (C2 − 2C2 C2w fx + C2 C2w fy ) 3

α7

−C1w fxy 2 (C1 − 1) 3

α1

−C2 C2w fy

α5

C1w fx

α2

1 − C2 + 2C2 C2w fx

α6

−(C1 + 2C1w fy )

α3

C2 C2w fxy 2 (C2 − 2C2 C2w fx + C2 C2w fy ) 3

α7

−C1w fxy 2 (C1 − 1) 3

α1

1.5C2 C2w fxy

α5

−1.5C1w fxy

α2

1.5C2 C2w fxy

α6

−1.5C1w fxy

α3

1 − C2 + 1.5C2 C2w (fx + fy )

α7

−[C1 + 1.5C1w (fx + fy )]

α4

−2C2 C2w fxy

α8

0

α4

α8

ϕ = v2

α4

α8

ϕ = uv

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.

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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:

u2P =

Am u2m + SC /AP + µP11

m=E,W,N,S

(Uw − Ue )P , ∆x

(13)

HP

where

µP11

=

4 8 2 2 − C2 + C2 C2w fx + C2 C2w fy ρu2 3 3 3 AP

∆x∆y P

,

(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 u2E and u2e are: u2E =

(Uw − Ue )E HE + µE , 11 AE ∆x

u2e =

(UP − UE ) He + µe11 , Ae ∆x

(15)

[5] Applying Second-Moment Closure to Complex Flows

161

with µe11 = (µP11 + µE 11 )/2. If the H/A terms are also linearly interpolated, the resulting form of u2e is: u2e =

1 2 uP + u2E 2 linear interpolation

(16)

1 P P E µ + µE + 11 (UP − UE ) − µ11 (Uw − Ue )P − µ11 (Uw − Ue )E /∆x, 2 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 u2P . (17) 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 µe11 (which equals 0.5[µP11 + µE 11 ]). This leads to 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

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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

(18)

with ρε (C1 + 2C1w fx ) u2 k ∂U 4 8 2 2 − C2 + C2 C2w fx + C2 C2w fy ρu2 B = , 3 3 3 ∂x

(19)

A =

(20)

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

C11 − D11 − (P11 + Φ11 − 2 ρε + A + B) k

3

ε

C1 + 2C1w fx

2 4 2 − C2 + C2 C2w (4fx + fy ) ρku2 3 3 = . C1 + 2C1w fx ε

,

(21)

(22)

Note that µ11 in (21) and (22) is associated with the diﬀerential gradient; in contrast, µP11 in (13) is applicable to the gradient approximation at grid-node P . The derivation of µ22 follows a path analogous to (18) to (22). Hence, only the end result is given, namely:

µ22

2 4 2 − C2 + C2 C2w (4fy + fx ) ρkv 2 3 3 = . C1 + 2C1w fy ε

(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 ∂U ∂V 3 2 2 − 1 − C2 + C2 C2w (fx + fy ) v . (24) +u 2 ∂y ∂x

Then (24) can be reformulated in either of the two forms:

∂V 3 C12 − D12 + 1 − C2 + (fx + fy ) ρu2 ∂U k 2 ∂x −ρuv = µ12 , + 3 ∂y ε C1 + C1w (fx + fy ) 2

(25)

or

−ρuv = µ21

∂V + ∂x

∂U 3 C12 − D12 + 1 − C2 + (fx + fy ) ρv 2

k ε

2 3 C1 + C1w (fx + fy ) 2

∂y

,

(26)

where

µ12 =

µ21 =

3 1 − C2 + C2 C2w (fx + fy ) ρkv 2 2 3 ε C1 + C1w (fx + fy ) 2 3 1 − C2 + C2 C2w (fx + fy ) ρku2 2 . 3 ε C1 + C1w (fx + fy ) 2

(27)

(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 ∂(P + ρv 2 )xξ ∂(ρuv)yξ ∂(ρuv)yη ∂(P + ρv )xη − − + . = ∂ξ ∂η ∂ξ ∂η

(30)

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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 = n22 (k 3/2 /ε)/(Cl ln ) (see equation (11)). Introduction of the apparent viscosity into (29) and (30) leads to:

∂ Uc ρϕ − ∂ξ

ϕ

q1 J

∂ϕ ∂ + Vc ρϕ − ∂ξ ∂η

ϕ

q2 J

∂ϕ = JSϕ ∂η

(31)

where for ϕ = U : q1u = µ11 yη2 + µ12 x2η ,

∂P = − ∂ξ

JSu

yη +

∂ρ(uv)uξ + ∂ξ with

∂P ∂η

yξ −

xη −

q2u = µ11 yξ2 + µ12 x2ξ ,

∂ρ(u2 )uξ ∂ξ

∂ρ(uv)uη ∂η

yη +

(32)

∂ρ(u2 )uη ∂η

yξ (33)

xξ ,

µ11 yξ ∂U µ11 yη ∂U , ρ(u2 )uη = ρu2 − , J ∂ξ J ∂η µ12 xξ ∂U µ12 xη ∂U uη = ρuv − , ρ(uv) = ρuv + , J ∂ξ J ∂η

ρ(u2 )uξ = ρu2 + ρ(uv)uξ

(34)

while for ϕ = V : q1v = µ21 yη2 + µ22 x2η ,

JSv =

∂P ∂ξ

xη −

∂P ∂η

∂ρ(uv)vξ − ∂ξ with

xξ +

yη +

q2v = µ21 yξ2 + µ22 x2ξ , ∂ρ(v 2 )vξ ∂ξ

∂ρ(uv)vη ∂η

xη −

(35)

∂ρ(v 2 )vη ∂η

xξ

yξ ,

µ22 xξ ∂V µ22 xη ∂V = − , ρ(v 2 )vη = ρv 2 + , J ∂ξ J ∂η µ21 yξ ∂V µ21 xη ∂V ρ(uv)vξ = ρuv + , ρ(uv)vη = ρuv − . J ∂ξ J ∂η ρ(v 2 )vξ

(36)

ρv 2

(37)

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3.3

165

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 q1ϕ + − ∂x ∂y ∂x ∂x

+

∂ϕ ∂ q2ϕ ∂x ∂y

= Sϕ

(38)

apparent diﬀusion

where, for ϕ = U , q1u = µ11 , Su = −

q2u = µ12 (39)

∂P ∂ρ(u2 )ux ∂ρ(uv)uy − − , ∂x ∂x ∂y

with ρ(u2 )ux = ρu2 + q1u

∂U , ∂x

ρ(uv)uy = ρuv + q2u

∂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 ∂ q1ϕ ∂x ∂x

←−

(q1u )e UE − [(q1u )e + (q1u )w ] UP + (q1u )w UW . (∆x)2

(41)

The normal-stress gradient on the RHS of equation (39) is also approximated by a central diﬀerence, yielding: ∂ ∂x

ρ(u2 )ux

←−

ρ(u2 )ux

=

E

− ρ(u2 )ux

2∆x

∂U q1u

∂x

−

∂U q1u ∂x

E

2∆x

W

W

∆ +

ρu2

∆x

(42)

.

With

∂U q1u ∂x

E

(q u ) (UEE − UP ) = 1 E , 2∆x

∂U q1u ∂x

= W

(q1u )W (UP − UW W ) , 2∆x

(43)

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Figure 3: Collocated cell storage arrangement. equation (42) becomes: ∂

(u2 )ux ←−

(q1u )E UEE

−

[(q1u )E

+ (q1u )W ] UP 4(∆x)2

+

(q1u )W UW W

∆ ρu2

+

. ∆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 one assumes that q1 is a constant, it is evident that the diﬀerence between the two approximations, DU , is proportional to: ∂x

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 =

Am ϕm + JSϕ .

(46)

m=E,W,N,S

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 )u2P − J (C1 + 2C1w fx ) u2P , k k A

(48)

B

with SC = max(A, 0),

SP =

min(A, 0) u2P + 10−15

+ 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 Fsshear acting on the cell’s southern face Areas is: Fsshear = τw Areas

(50)

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Figure 4: Boundary cell at a curved wall. where

1/2

τw =

ln E U =U−U , t

1/4

ρP kP Cµ κ

n

1/2 l k ∗ n P

Ut

(51)

ν

ln = (rP − rs ) · n.

(52)

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: (ηx , ηy , ηz ) (n1 , n2 , n3 ) = , ηx2 + ηy2 + ηz2 where

(53)

ηx2 + ηy2 + ηz2 = Areas .

(54)

Hence, the components of Ut in (52) can be written as: (Uxt , Uyt , Uzt ) = (U, V, W ) − (U n1 + V n2 + W n3 )n,

(55)

Uxt = (1 − n21 )U − n1 n2 V − n1 n3 W,

(56)

n22 )V − n1 n2 U − n2 n3 W, n23 )W − n1 n3 U − n2 n3 V.

(57)

with

Uyt Uzt

= (1 − = (1 −

(58)

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

169

for the x-momentum equation, 1/2

1/4

ρP kP Cµ κ

SC ← S C −

ln E

1/2 l k ∗ n P

Uxt Areas ;

(59)

Uyt Areas ;

(60)

Uzt Areas .

(61)

ν

for the y-momentum equation, 1/2

1/4

ρP kP Cµ κ

SC ← SC −

ln E

1/2 l k ∗ n P

ν

for the z-momentum equation, 1/2

S C ← SC −

1/4

ρP kP Cµ κ ln E

1/2 l k ∗ n P

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 ) 1/2

ρP κ∗ ln kP

(τw · τw ),

(62)

where lv is the viscous sublayer thickness. With ε = 2ν(k/ln2 ) in the viscous sublayer and ε = k 3/2 /κln in the log-law region, the average value of ε becomes: 3/2

k ε= P ln

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)

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while the shear stress itself is given by −ρuv τw .

(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 P = 1.098kP ,

(v 2 )w P = 0.247kP ,

(uv)w P = −0.255kP .

(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): u2P

2 w 2 w 2 = (u2 )w P t1 + (v )P n1 + 2(uv)P t1 n1

(68)

vP2

2 w 2 w 2 = (u2 )w P t2 + (v )P n2 + 2(uv)P t2 n2

(69)

w 2 w = (u2 )w P t1 t2 + (v )P n1 n2 + 2(uv)P (t1 n2 + t2 n1 ),

(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

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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

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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

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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)

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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

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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: +

5

,

J (ρϕ)n+1 − (ρϕ)n f n+1 ds = JSt + Sc , − ∆t

(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.

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Linearizing convective and diﬀusive ﬂuxes and splitting all sources into positive and negative contributions gives:

∂f J ∆(ρϕ) = f n + J St+ + St− + Sc+ + Sc− . − ∆t ∂(ρϕ)

(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:

∂f J JSt− + Sc− ∆(ρϕ) − (ρϕ)n+1 = f n + JSt+ + Sc+ . (75) − n ∆t ∂(ρϕ) (ρϕ) + δ

To retain the ‘∆’ form, the term

JSt− + Sc− (ρϕ)n (ρϕ)n + δ

is added to both sides of equation (75) to give:

∂f J JSt− + Sc− ∆(ρϕ) − − ∆t ∂(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 5 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

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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´c (1994) and Leschziner et al. (1999).

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 ij,2 to the rapid part of the pressure-strain term. This 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

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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

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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).

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SST

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´c–Hanjali´c (1995) linear Reynolds-stress model and the

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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-

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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´c–Hanjali´c model signiﬁcantly underestimates the interaction, thus 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

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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.

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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. Batten, P., Craft, T.J, Leschziner, M.A. and Loyau, H. (1999). ‘Reynolds-stresstransport modelling for compressible aerodynamic ﬂows’, J. AIAA 37 785–796. Batten, P., Leschziner, M.A. and Goldberg, U.C. (1997). ‘Average state Jacobians and implicit methods for compressible viscous and turbulent ﬂows’, J. Comp. Phys. 137 38–78. Barakos, G. and Drikakis, D. (1998). ‘Implicit unfactored implementation of twoequation turbulence models in compressible Navier–Stokes methods’, Int. J. Numer. Meth. Fluids 28 73–94. Brandt, A. (1977). ‘A multilevel adaptive solution of boundary value problems’, Math. Comput. 31 333–390. Chen, W.L., Lien, F.S. and Leschziner, M.A. (1998). ‘Non-linear eddy-viscosity modelling of transitional ﬂows pertinent to turbomachine cascade aerodynamics’, Int. J. Heat Fluid Flow 19 297–306. Craft, T.J. (1998). ‘Development in low-Reynolds-number second-moment closure and its application to separating and reattaching ﬂows’, Int. J. Heat and Fluid Flow 19 541–548. Craft, T.J. and Launder, B.E. (1996). ‘A Reynolds stress closure designed for complex geometries’, Int. J. Heat Fluid Flow 17 245–254. Daly, B.J. and Harlow, F.H. (1970). ‘Transport equations in turbulence’, Phys. Fluids 3 2634–2649. Gibson, M.M. and Launder, B.E. (1978). ‘Ground eﬀects on pressure ﬂuctuations in the atmospheric boundary layer’, J. Fluid Mech. 86 491–511. Haase, W., Chaput, E., Elsholz, E., Leschziner, M.A. and M¨ uller, U.R. (1996). ‘ECARP: European Computational Aerodynamics Research Project. II: Validation of CFD Codes and Assessment of Turbulence Models’, Notes on Numerical Fluid Mechanics 58. Hanjali´c, K. (1994). ‘Advanced turbulence closure models: a view of current status and future prospects’, Int. J. Heat Fluid Flow 15 178–203. Huang, P.G and Leschziner, M.A. (1985). ‘Stabilisation of recirculating ﬂow computations performed with second-moment closure and third-order discretisation’. Proc. 5th Turbulent Shear Flows, Cornell University, 20.7–20.12. Iacovides, H. and Launder, B.E. (1985). ‘ASM predictions of turbulent momentum and heat transfer in coils and U-bends’. Proc. 4th Int. Conf. on Numerical Methods in Laminar and Turbulent Flows, Swansea, 1023—1031.

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Jakirli´c, S. and Hanjali´c, K. (1995). ‘A second-moment closure for non-equilibrium and separating high- and low-Re-number ﬂows’. Proc. 10th Symp. on Turbulent Shear Flows, Pennsylvania State University, 23.25-23.30. Jameson, A., Schmidt, W. and Turkel, E. (1981). ‘Numerical solutions of the Euler equations by ﬁnite volume methods using Runge–Kutta time-stepping schemes’. AIAA-81–125. Kreplin, H.P., Vollmers, H. and Meier, H.U. (1985). ‘Wall shear stress measurements on an inclined prolate spheroid in the DFVLR 3 m × 3 m low-speed wind tunnel’. DFVLR Goettingen Report IB 222–84 A 33. Launder, B.E. (1989). ‘Second-moment closure: present . . . and future?’, Int. J. Heat Fluid Flow 10 282–299. Leschziner, M.A. (1990). ‘Modelling engineering ﬂows with Reynolds-stress turbulence closure’, J. Wind Engineering and Industrial Aerodynamics 35 21–47. Leschziner, M.A. (1994). ‘Reﬁned turbulence modelling for engineering ﬂow’. In Computational Fluid Dynamics ‘94, S. Wagner, J.Periaux and E.H. Hirschel (eds.), Wiley, 33–46. Leschziner M.A. (1995). ‘Computation of aerodynamic ﬂows with turbulence transport models based on second-moment closure’, Computers and Fluids 24 377–392. Leschziner, M.A., Batten, P. and Loyau, H. (1999). ‘Modelling shock-aﬀected nearwall ﬂows with anisotropy-resolving turbulence closures’. In Engineering Turbulence Modelling and Experiments 4, W. Rodi and D. Laurence (eds.), 19–36. Leschziner, M.A. and Ince, N.Z. (1995). ‘Computational modelling of three-dimensional impinging jets with and without cross ﬂow using second-moment closure’, Computers and Fluids 24 811–832. Leonard, B.P. (1979). ‘A stable and accurate convective modelling procedure based on quadratic upstream interpolation’, Comp. Meths. Appl. Mech Engrg. 19 59–67. Lin, C.A. and Leschziner, M.A. (1993). ‘Three-dimensional computation of transient interaction between radially injected jet and swirling cross-ﬂow using secondmoment closure’, J. Comp. Fluid Dynamics 1 419–428. Lien, F.S. and Leschziner, M.A. (1993a0. ‘Second-moment modelling of recirculating ﬂow with a non-orthogonal collocated ﬁnite-volume algorithm’. In Turbulent Shear Flows 8, Springer, 205–222. Lien, F.S. and Leschziner, M.A. (1993b). ‘A pressure-velocity solution strategy for compressible ﬂow and its application to shock/boundary-layer interaction using second-moment turbulence closure’, ASME J. Fluids Eng. 115 717–725. Lien, F.S. and Leschziner, M.A. (1994a). ‘A general non-orthogonal ﬁnite volume algorithm for turbulent ﬂow at all speeds incorporating second-moment closure, Part 1: numerical implementation’, Comp. Meths. Appl. Mech. Engrg. 114 123– 148. Lien, F.S. and Leschziner, M.A. (1994b). ‘Upstream monotonic interpolation for scalar transport with application to complex turbulent ﬂows’, Int. J. Num. Meths. in Fluids 19 527–548. Lien, F.S. and Leschziner, M.A. (1994c). ‘Multigrid acceleration for turbulent ﬂow with a non-orthogonal collocated scheme’, Comp. Meths. Appl. Mech. Engrg. 118 351–371.

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Lien, F.S. and Leschziner, M.A. (1995). ‘Second-moment closure for three-dimensional turbulent ﬂow around and within complex geometries’, Computers and Fluids 25 237–262. Loyau, H., Batten, P. and Leschziner, M.A. (1998). ‘Modelling shock/boundary-layer interaction with nonlinear eddy-viscosity closures’, J. Flow, Turbulence and Combustion 60 257–282. Meier, H.U., Kreplin, H.P., Landhauser, A. and Baumgarten D. (1984). ‘Mean velocity distribution in 3D boundary layers developing on a 1:6 prolate spheroid with artiﬁcial transition’. 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´erodynamique Numerique 3D Instationnaire avec Fermature BasReynolds au Second Ordre. Doctoral Thesis, Universit´e de Paris 6. 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

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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.

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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

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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

f2 f2

(1 + f2 )(1 − fw1 ){1 − 0.6 exp[−(Rt /45)1/2 ]} exp(−2 × 10−4 Rv13 )[1 − exp(−2.2Rv0.5 )]

Rv

(k/ε)[1/(1 + νt /ν)](1/Rt )fw1

fµ

(1 − fw2 ){1 + (60/Rt ) exp[−(Rt /55)1/2 ]}

1/2

3/4

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 ∂Ui + − ui uj , =− ν (2.2) Dt ρ ∂xi ∂xj ∂xj where D/Dt ≡ ∂/∂t + Uj ∂/∂xj .

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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 2 −ui uj = νt + (2.3) − δij k, ∂xj ∂xi 3 k2 νt = Cµ fµ , ε νt ∂k Dk ∂Ui ∂ ν+ − ui uj − ε, = Dt ∂xj σk ∂xj ∂xj νt ∂ε ε ∂Ui ε2 Dε ∂ ν+ − Cε1 ui uj − Cε2 fε . = Dt ∂xj σε ∂xj k ∂xj k

(2.4) (2.5) (2.6)

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 ∂Ui ∂ νt ∂k −ε = ν+ ∗ − ui uj Dt ∂xj σk ∂xj ∂xj k ∂ε ∂ (2.7) fw1 , 0 , + max −0.5ν ∂xj ε ∂xj Dε ∂ε ε ∂Ui ε2 ∂ νt − Cε2 f2 = ν+ ∗ − Cε1 ui uj Dt ∂xj σε ∂xj k ∂xj k 2 2 ∂ Ui k ∂k ∂Ui ∂ 2 Ui +fw2 ννt + Cε3 ν ∂xj ∂xk ε ∂xk ∂xj ∂xj ∂xk ∂ ε ∂k +Cε4 ν fw1 . (1 − fw1 ) (2.8) ∂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: y∗ 2 fw (ξ) = exp − , (2.9) ξ 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

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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

Two-equation model for heat transfer (DNS-based modelling)

3.1

Governing Equations

The energy equation may be written DΘ ∂ ∂Θ − uj θ . = α Dt ∂xj ∂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

(3.4)

Dεθ = Dεθ + Tεθ + Pε1θ + Pε2θ + Pε3θ + Pε4θ − Υεθ . Dt

(3.5)

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The terms on the right-hand sides in (3.4) and (3.5) are identiﬁed as   Molecular diﬀusion    2 2   ∂ kθ ∂ εθ   , Dεθ = α , Dkθ = α   ∂xj ∂xj ∂xj ∂xj      Turbulent diﬀusion      ∂uj kθ ∂uj εθ   , Tεθ = − , T kθ = −   ∂xj ∂xj      Mean gradient production      ∂u ∂U ∂θ ∂Θ ∂θ ∂θ ∂Θ j j 1 2  Pkθ = −uj θ , Pεθ = −2α , Pεθ = −2α , ∂xj ∂xk ∂xk ∂xj ∂xk ∂xj ∂xk  (3.6)  Gradient production     ∂θ ∂2Θ    Pε3θ = −2αuj ,   ∂xk ∂xj ∂xk      Turbulent production     ∂u ∂θ ∂θ  j 4   Pεθ = −2α ,   ∂xk ∂xk ∂xj      Destruction     2 2  ∂ θ  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 3  u = b1 y + b 2 y + b 3 y + · · ·     2 3  v = c1 y + c2 y + · · ·    2 3  w = d1 y + d2 y + d3 y + · · ·    2 3 2 2  k = ui ui /2 = [(b1 + d1 )/2]y + (b1 b2 + d1 d2 )y + · · ·     3 4  uv = b1 c1 y + (b1 c2 + c1 b2 )y + · · · (3.7)  εw = ν(∂ 2 k/∂y 2 )w = ν(b21 + d21 )    2 3  Θ+ = P ry + + g1 y + + g2 y + + · · ·     2  θ = θ w + h1 y + h2 y + · · ·     2 2 2 /2 + [(h + 2h θ )/2]y + · · ·  kθ = θ2 /2 = θw 2 w  1   2 3  vθ = c1 θw y + (c2 θw + c1 h1 )y + · · ·     2 2 2 εθw = α[∂ kθ /∂y ]w = α(h1 + 2h2 θw )

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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) (3.8) αt ∝ y 2 for p = 0 (with θw ﬂuctuations). 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)

ε4θ ε4θ Dε4θ ∂ 2 ε4θ + Tε4θ + CP 1 fP 1 Pkθ + CP 2 fP 2 Pk =α Dt ∂xj ∂xj kθ k −CD1 fD1

ε4θ 2 ε4θ ε4 − 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 Tε4θ term is also modelled in the same manner as in (3.11).

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Table 2: Existing εθ and ε4θ equation models.

Cλ Cs CP 1 CP 2 CD1 CD2 σφ τm fλ Aµ Aλ fR fP 1 fP 2 fD1 fD2 Additional term

Nagano-Kim(1988)

Nagano-Tagawa-Tsuji(1991)

0.11 — 0.9 0.72 1.1 0.8 1.0 4 1/2 (k/4 ε)(2R)

0.1 — 0.85 0.64 1.0 0.9 1.0

3/4 (k/ε) (2R)2 + 3.4(2R)1/2 /Rt 2

2

[1 − exp (−y + /Aλ )] √ — (30.5/ P r)(Cf /2St) — 1.0 1.0 1.0 1.0

[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)]

ααt (1 − fλ )(∂ 2 Θ/∂xj ∂xk )2 4 = (kθ /ε4θ )/(k/4 R ε)

— Cε2 = 1.9 fε = 1 − exp[−(Rt /6.5)2 ]

In (3.10), quantities ε4 and ε4θ , called the isotropic dissipation rates of k and kθ , are deﬁned by the following equations, respectively: ε4 = ε − ε7,

(3.12)

(3.13) ε4θ = εθ − ε7θ , √ where ε7 = 2ν(∂ k/∂y)2 , ε7θ = 2α(∂ ∆kθ /∂y)2 , and ∆kθ = kθ − kθw . √

3.3.2

Assessment procedure

In (3.9) and (3.10), εθ and ε4θ are the only unknown variables, and all turbulence quantities except εθ and ε4θ are supplied directly from the DNS data; i.e., Ui , ui uj , k, ε, ε4, 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.

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Table 2: (continued)

Cλ Cs CP 1 CP 2 CD1 CD2 σφ

Shikazono-Kasagi(1996)

Abe-Kondoh-Nagano(1995)

— 0.3 0.8 0.3 1.0 0.3 —

0.1 — 1.9 0.6 2.0 0.9 1.6

8 9 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 − ε4θ ε7θ /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 ε4θ 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 ε4 and ε4θ 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-

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Table 2: (continued) Sommer-So-Lai(1992)

Hattori-Nagano-Tagawa(1993)

Cλ Cs CP 1 CP 2 CD1 CD2 σφ τm

0.11 0.1 0.11 — 0.9 0.85 0.72 0.64 1.1 1.0 0.8 0.9 — 1.0 8

8 9 1/4 3/4 (k/ε) (2R)1/2 + 0.1(2R)1/2 /Rt (k/ε) (2R)1/2 + [7.9(2R)1/2 /Rt ]fd 9 ×(fεt /fλ ) + ,2 + , + , 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 ε4θ /εθ 1.0 fD1 ε4/ε (1/CD2 )(Cε2 fε − 1) fD2 ε/k)ε,θ Additional fεt [(CD1 − 2)(ε4θ /kθ )εθ + CD2 (4 ∗2 ∗ ααt (1 − fw )(∂ 2 Θ/∂xj ∂xk )2 −(ε /(2k ) + (1 − C )(ε /k P 1 θ θ θ )Pθ θ term fεt = exp[−(Rt /80)2 ] fd = exp[−(Rt /120)2 ] fw = [1 − exp{−y + /(30/P r1/3 )}]2 ε∗θ = εθ − 2α(kθ /y 2 ) Pθ∗ = uθ(∂Θ/∂x) P + = ν(dP/dx)/ρu3τ Cε2 = 1.9 fε = 1 − 0.3 exp[−Rt2 ]

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 ε4θ - and εθ -equation models at a two-equation level and those at a second-order closure level are shown in Figure 1, where in ε4θ -equation modelling εθ is obtained from εθ = ε4θ + ε7θ . The resultant characteristic time scale τθ is assessed in Figure 2. To assess the NK model, ε from the DNS, because the NK model the time scale τu in αt is given by k/4 is usually combined with the Nagano and Hishida model (1987) (4 ε-equation model).

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Figure 1: Assessment of ε4θ and εθ equations: (a) ε4θ equations (εθ = ε4θ + ε7θ ) at a two-equation level; (b) εθ equations at a two-equation level; (c) ε4θ - and εθ equations at a second-order closure level. As can be seen from Figures 1 and 2, the results of assessment for ε4θ 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

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Figure 2: Proﬁles of time scale τθ : (a) in ε4θ equations at a two-equation level; (b) in εθ equations at a two-equation level; (c) in ε4θ 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.

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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 ε4θ must be ε4θ ∝ y 2 . In the ε4θ equation of the NK and HNT models, however, the molecular diﬀusion term balances with CD1 ε4θ 2 /kθ term at y = 0. As a result, the wall-limiting behavior of ε4θ becomes ε4θ ∝ y 1 . Therefore, in the ε4θ equation, adding the extra term to reproduce the correct wall-limiting behavior of ε4θ 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 ε4θ equation at the wall, as suggested by Kawamura and Kawashima (1994). The budget data for the ε4θ and εθ equations are shown in Figure 3. The budget in√the ε4θ equation is represented by α(∂ 2 εθ /∂y 2 ) = α(∂ 2 ε4θ /∂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

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Figure 3: Budgets of modelled ε4θ equations (Pε14θ + Pε24θ + Pε34θ + Pε44θ + Tε4θ − Υε4θ ) and εθ equations (Pε1θ + Pε2θ + Pε3θ + Pε4θ + Tεθ − Υεθ ): (a) two-equation level ε4θ equations; (b) two-equation level εθ equations; (c) second-order closure level ε4θ - and εθ equations. √ number P rt or of a mixed time scale, e.g., τm = τu τθ or τm = τθ2 /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

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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

'

√ ( Rt 2 2R 2R 3 exp − + , 0.5 + R P r R3/4 200

(3.17)

t

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): ' √ ( k 2R 2R 26 Rh αt = Cλ fλ kτm = Cλ fλ k exp − + , ε 0.5 + R P r4/3 R3/4 220 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 ε4θ 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. (a) Modelling of Pε1θ , Pε2θ , Pε4θ and Υεθ The Pε1θ , Pε2θ , Pε4θ and Υεθ terms can be modelled in a way similar to the NK (Nagano and Kim 1988) and NTT models (Nagano et al. 1991): Pε1θ + Pε2θ + Pε4θ − 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)

The DNS data indicates that the Pε1θ and Pε2θ terms exert a great inﬂuence on the production of εθ near the wall. The modelling given by (3.20), however, is

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based on Pε4θ and Υεθ , so that the inﬂuence of other terms is not suﬃciently reﬂected. Therefore, we model the contributions from the Pε1θ and Pε2θ 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 the Pε1θ , Pε2θ and Pε4θ terms as √   k kθ λθ  G ,  Pε1θ = O   θ λ     √   kkθ 2 (3.21) S , Pεθ = O  θ         kk θ θ 4   , Pεθ = O 2 λ θ where G = [(∂Θ/∂xj )(∂Θ/∂xj )]1/2 represents the mean temperaturegradi1/2 is the mean strain rate, and λ = kν/ε ent, S = [(∂U i /∂xj )(∂Ui /∂xj )] and λθ = kθ α/εθ are the Taylor microscales for the velocity and√temperature ﬁelds, respectively. The above relations give Pε1θ /Pε4θ ≈ (λθ / kθ )G = √ G/(εθ /α)1/2 , Pε2θ /Pε4θ ≈ (λ/ k)S = S/(ε/ν)1/2 . Consequently, we deﬁne the parameters RT and RU as

1/2 (∂Θ/∂xj )2 G , (3.22) RT =

1/2 = 1/2 (ε /α) 2 θ (∂θ/∂xj ) RU =

(∂Ui /∂xj )2 (∂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 ﬂuctuating components. Apparently, the relations Pε1θ /Pε4θ ≈ RT and Pε2θ /Pε4θ ≈ 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)), contributions of Pε1θ and Pε2θ 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

RU =

fw (6),

u2τ fw (6). (ν + νt )(ε/ν)1/2

(3.24)

(3.25)

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The contributions of Pε1θ and Pε2θ can now be included in the model functions fP 1 and fP 2 as follows: ( fP 1 = (1 − fP 1 )fp , (3.26) fP 1 = exp(−7 × 10−5 RT 10 )[1 − exp(−2.2RT 1/2 )], ( fP 2 = (1 − fP 2 )fp , (3.27) fP 2 = exp(−7 × 10−5 RU 10 )[1 − exp(−2.2RU 1/2 )], 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. (b) Modelling of Pε3θ The Pε3θ term is negligibly small in comparison with Pε1θ , Pε2θ , Pε4θ and Υεθ . However, when compared with the sum of these terms, i.e., Pε1θ +Pε2θ +Pε4θ −Υεθ , the Pε3θ term becomes of the same order, so modelling Pε3θ is also important. In the present model, we adopt the following form similar to (2.8) in the k-ε model: 2 2 ∂k ∂Θ ∂ 2 Θ ∂ Θ k 3 Pεθ = ααt fw (12) + CP 3 α fR , (3.28) ∂xj ∂xk ε ∂xk ∂xj ∂xj ∂xk 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 * ) αt ∂εθ ∂ ∂ 3/2 ε ∂kθ Tεθ = fw (3) . (3.29) + Cεθ α [1 − fw (3)] ∂xj σφ ∂xj ∂xj k ∂xj (d) Modelled εθ -equation To sum up, the proposed εθ -equation can be written as αt ∂εθ εθ εθ ∂Ui Dεθ ∂Θ ∂ α+ − CP 1 fP 1 uj θ − CP 2 fP 2 ui uj = Dt ∂xj σφ ∂xj kθ ∂xj k ∂xj 2 ε2 ∂2Θ εθ ε −CD1 fD1 θ − CD2 fD2 + ααt fw (12) kθ k ∂xj ∂xk

206

Nagano ∂k ∂Θ ∂ 2 Θ k +CP 3 α fR ε ∂xk ∂xj ∂xj ∂xk * ) ∂ 3/2 ε ∂kθ +Cεθ α fw (3) . [1 − fw (3)] ∂xj k ∂xj

(3.30)

The wall reﬂection function fw (ξ) is given by (2.9). (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) (3.33) fD2 = (1/CD2 )(Cε2 fε − 1) [1 − fw (12)]

(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 /6.5)2 ].

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

(3.36) ε2 , k

= −εθ , = −CD1 fD1

(3.37) (3.38)

ε2θ εθ ε − CD2 fD2 , kθ k

(3.39)

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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 ε2θ ε2 kθ εθ ε εθ ε dεθ 1 ε2 kθ = − − C f − − (Cε2 fε − 1) U = . (3.40) ε2 ε 2 2 dx R k k k kθ k 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 Cλ ∂ k 2 ∂εθ εθ ∂Θ εθ ∂U εθ ε − CP 1 vθ fR − CP 2 uv − CD1 θ − CD2 = 0. σφ ∂y ε ∂y 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) CP 2 = (CD1 − CP 1 )/R + CD2 − (κ2 /P rt )/(σφ Cµ1/2 ),

(3.43)

where κ is the von K´ arm´ am constant. Equation (3.43) is similar to the wellknown relation in the k-ε model given by Cε1 = Cε2 − κ2 /(σε Cµ1/2 ).

(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.

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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

0.1 — 0.9 0.77 0.05 1.0 0.9 1.6 1.8

9 8 3/4 (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)] (= Pε3θ ) exp(−Rh /220) 1.9 1 − 0.3 exp[−(Rt /6.5)2 ]

fP 1

exp(−7 × 10−5 RT10 )[1 − exp(−2.2RT )]

fP 2 fp RT RU

10 )[1 − exp(−2.2R exp(−7 × 10−5 RU U )] 1/2 1 + 0.75 exp[−(Rh /40) ] fw (6)uτ θτ /[(α + αt )(εθ /α)1/2 ] fw (6)uτ 2 /[(ν + νt )(ε/ν)1/2 ]

1/2 1/2

(f ) Second-order closure modelling In second-order closure modelling, the turbulent diﬀusion term Tεθ and the production term Pε3θ 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

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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 proposed εθ -equation models (Pε1θ + Pε2θ + Pε3θ + Pε4θ + Tεθ − Υεθ ). RU =

u2τ + uv fw (6). (εν)1/2

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 )].

(3.46)

(3.47)

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Figure 5: Budget of temperature variance kθ .

The turbulent diﬀusion term, Tεθ , can be written as ) * k ∂εθ ∂ ∂ 3/2 ε ∂kθ Cs fR uj uk + Cεθ α [1 − fw (3)] Tεθ = fw (3) , ∂xk ε ∂xj ∂xj k ∂xj (3.48) where Cs = 0.11, and fR and Cεθ are exactly the same as in the two-equation heat-transfer model. The Pε3θ term may be written [see (3.28)] as k ∂2Θ ∂2Θ k ∂uj u ∂Θ ∂ 2 Θ + CP 3 α fR , (3.49) Pε3θ = 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

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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 9 √ αt ∂kθ ∂ 8 ∂ 1/2 1/2 T kθ = d n e [1 − f (28)] k k [f (28)] σ , +C u u j w w θ k θ k ∂xj σh∗ ∂xj ∂xj (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 ) 1 (x ≥ 0), σx = (3.51) −1 (x < 0). The ﬁnal formulation of the kθ -equation model is written as follows: Dkθ αt ∂kθ ∂ α+ ∗ = Dt ∂xj σh ∂xj 8 9 √ ∂ σuk u dk n ej [1 − fw (28)]1/2 k kθ [fw (28)]1/2 +Cθ ∂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

Nagano Numerical scheme

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

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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.

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(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

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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

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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).

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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

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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

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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 √ no temperature ﬂuctuations are created by the mean temperature 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

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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

Two-equation Model for heat transfer (eﬀects of Prandtl number)

4.1 4.1.1

Construction of kθ -ε4θ Model Eddy diﬀusivity αt for kθ -ε4θ model

As mentioned in the foregoing, the eddy diﬀusivity for heat, αt , is generally asfollows: given by (3.3). Features of αt in (3.3) with (3.17) are √ summarized 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 -ε4θ model (Nagano and Shimada 1996), we adopt a multipletime-scale similar to (3.17): : 2R 2R Bλ k (4.1) fη , τm = + ε Cm + R P r Rt θ 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 y4θ∗ n y4∗2−n = 1 − exp −A∗λ y4∗2 , where

√ A∗λ = Aλ (1 + Cη P r)n .

(4.2)

Here, Aλ and Cη denote model constants, and the dimensionless parameter y4θ∗ is deﬁned using the mixed length scale η4m as √ y4θ∗ = y/4 ηm = (1 + Cη P r)4 y∗, η4m =

1 Cη + η4 η4θ

−1

η4 √ . = 1 + Cη P r

(4.3)

222

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ε)1/4 represents the Kolmogorov microscale deﬁned The length scale η4 = (ν 3 /4 2 previously, and η4θ = (α ν/4 ε)1/4 is the Batchelor microscale (Batchelor 1959). y4∗ is deﬁned as y4∗ = y/4 η . From (4.3), the characteristic length scale η4m gives weight to the Kolmogorov microscale η4 for lower Prandtl number ﬂuids (P r < 1), while for higher Prandtl number ﬂuids the Batchelor microscale η4θ becomes dominant. [For low Prandtl number ﬂuids, the so-called Obukhov microscale (Obukhov 1949) η4o = (α3 /4 ε)1/4 = η4P r−3/4 could be used as the characteristic length scale of dissipation eddies. However, it is easily understood that η4 is always smaller than η4o because of the relation η4o /4 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 ε4 is represented by (3.12). We note that the fact that the sum of the power of y4θ∗ and y4∗ 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∗λ = Aλ (1 + Cη P r)n so that the maximum values of kθ at P r = 0.1, 0.71 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 y4θ∗ as fηθ = exp[−(4 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

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ratio of the respective time scales for dissipation eddies in the velocity and thermal ﬁelds becomes equal to √R/P r (Youssef et al. 1992; Abe et al. 1995), 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: ' ( : y4θ∗ 3/4 2R 120 k2 2R 1 √ , exp − + αt = Cλ fλ ε 0.2/P r1/4 + R P r (1 + 2 P r)1/4 Rt 25 fλ = 1 − exp(−7 × 10−4 y4θ∗ 1/4 y4∗7/4 ), √ y4θ∗ = (1 + 2 P r)4 y∗.

(4.4)

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)): αt ∂kθ ∂ √ ∂ + Cθ T kθ = k kθ fwθ σui uj eSi eN j e , ∂x σh ∂x ∂x (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

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Table 5: Model constants and functions of the proposed kθ -ε4θ model. Cλ

Cm

σh

∗ CD3

0.2 P r1/4 Cτ

1.6 fh fλ

0.025

3.0

(4.2)

0.10

Cθ

σφ

CP 1

CP 2

CD1

CD2 fD2

0.1

1.8

0.825

0.9

1.0

(4.15)

fh

fwθ

fP 1

fP 2

fD1

∗ fD3

(4.6)

(4.7)

1.0

1.0

1.0

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: ( ' ;t 5 Cm + R R 1+ , (4.6) fh = exp − 2R Pr 100 fwθ = (1 − fw )2 fw1/4

with fw = exp (−4 y ∗ /8) ,

(4.7)

;t = k 2 /(ν ε4) is the where fw is the wall reﬂection function similar to (2.9), R turbulent Reynolds number based on ε4, 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 + Pkθ − εθ . = α+ Dt ∂x σh ∂x (4.8) One may note that (4.8) is almost identical to (3.52). 4.1.3

Modelling the ε4θ -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 ε4θ -equation similar to that proposed by Nagano and Kim (1988): αt ∂ ε4θ ε4θ Dε4θ ∂ α+ + (CP 1 fP 1 Pkθ − CD1 fD1 εθ ) = Dt ∂xj σφ ∂xj kθ 2 2 ∂ Θ ε4θ + (CP 2 fP 2 Pk − CD2 fD2 ε) + ααt (1 − fλ ) , (4.9) k ∂xj ∂xk

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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 ε4θ is deﬁned by (3.13). In the proposed model, we use the above ε4θ -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 ε4θ × (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

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ﬁelds, the time scales contributing to the generation of turbulence should be directly expressed by using Pk and Pkθ as −ui uj ∂Ui −1 , k ∂xj −1 −uj θ ∂Θ 1 = . τΘ = Pkθ /kθ kθ ∂xj

1 = τU = Pk /k

(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 ε4θ -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)

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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, √ are strongly linked √ to the characteristic time scales of dissipation eddies 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 : : Pr 1 Cτ 1 1 . (4.14) 1 + Cτ = + = τηm τηu τηθ τηu R where Cτ is a weighting constant. Now, (4.14) is not directly incorporated into the present ε4θ -equation [(4.9)], but is indirectly reﬂected in the model constant CD2 and the model function fD2 as : Pr ∗ ∗ ∗ ∗ fD2 fD3 Rt 1 + Cτ , (4.15) CD2 fD2 = CD2 1 + CD3 R with

+ ∗2 , ∗ ∗ fD2 = (Cε2 f2 − 1) 1 − exp −4 y . CD2

(4.16)

∗ , C , and f ∗ are C ∗ = 0.025, C = 3.0, and f ∗ = f , respectively. Here, CD3 τ τ w D3 D3 D3 [We would like to emphasize that the proposed ε4θ -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 ε4θ /kθ , with the correct wall-limiting behavior of ε4θ ∝ 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 -ε4θ 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.

A

const

0 0

Case

Θw

kθw ε4θ w

α

∂Θ ∂y

0 0

w

B

=−

qw ρcp

Θw|inj on injection side 0 0

Θw|suc on suction side

C

0 0

const

D

0 0

const

E (Θw =const)

0 and ∂kθ /∂y = 0 0

E (qw =const) qw ∂Θ =− α ∂y w ρcp

α

∂Θ ∂y

Table 6: Boundary conditions of the proposed kθ -ε4θ model for various ﬂow ﬁelds.

0 0

w

F =−

qw ρcp

228 Nagano

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229

Case F. Backward-facing step ﬂow for various Prandtl number ﬂuids. The numerical scheme and the grid system are the same as for the k-4 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 -ε4θ 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

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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 particular, that the predicted εθ , which is given by εθ = ε4θ + 2α(∂ ∆kθ /∂y)2 , can

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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 −

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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 -ε4θ 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

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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

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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

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Nagano

(a)

(c)

(b)

(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

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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 1.21, Cλ Aλ Bλ 2R/P r y + → 0 (with R → P r).

(4.19)

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+ tematic variations with increasing P r: the constant heat ﬂux layer (vθ 1) 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

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(a)

239

(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

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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

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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 ,

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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

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(b)

(a)

(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

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ﬂows under arbitrary wall thermal boundary conditions show that the Prandtl number dependence identiﬁed by DNS is satisfactorily captured. The Reynolds and Prandtl (or Schmidt) number dependence of the proposed kθ -εθ model is thoroughly investigated in a pipe ﬂow, and we have demonstrated that the predicted heat (and mass) transfer rate coincides with reliable empirical formulas and experimental data with high accuracy. Of course, this agreement cannot be achieved without a high-performance k-ε model.

References Abe, K., Kondoh, T. and Nagano, Y. (1994). ‘A new turbulence model for predicting ﬂuid ﬂow and heat transfer in separating and reattaching ﬂows -I. Flow ﬁeld calculations’, Int. J. Heat Mass Transfer 37, 139–151. Abe, K., Kondoh, T. and Nagano, Y. (1995). ‘A new turbulence model for predicting ﬂuid ﬂow and heat transfer in separating and reattaching ﬂows -II. Thermal ﬁeld calculations’, Int. J. Heat Mass Transfer 38, 1467–1481. Antonia, R.A. (1980). ‘Behavior of the turbulent Prandtl number near the wall’, Int. J. Heat Mass Transfer 23, 906–908. Antonia, R.A., Danh, H.Q. and Prabhu, A. (1977). ‘Response of a turbulent boundary layer to a step change in surface heat ﬂux’, J. Fluid Mech. 80, 153–177. Azer, N.Z. and Chao, B.T. (1961). ‘Turbulent heat transfer in liquid metals fully developed pipe ﬂow with constant wall temperature’, Int. J. Heat Mass Transfer 3, 77–83. Batchelor, G.K. (1959). ‘Small-scale variation of convected quantities like temperature in turbulent ﬂuid, Part 1. General discussion and the case of small conductivity’, J. Fluid Mech. 5, 113–133. Bhatti, M.S. and Shah, R.K. (1987). ‘Turbulent and transition ﬂow convective heat transfer in ducts’. In Handbook of Single-Phase Convective Heat Transfer, S. Kaka¸c et al. (eds.), Sec. 4.1, Wiley. Borishansky, V.M., Gel’man, L.I., Zablotskaya, T.V., Ivashchenko, N.I. and Kopp, I.Z. (1964). ‘Investigation of heat transfer in mercury ﬂows in horizontal and vertical pipes’, Convective Heat Transfer in Two- and One Phase Flows, 340. Energiya, Moscow (Collected Papers). Cebeci, T. (1973). ‘A model for eddy conductivity and turbulent Prandtl number’, Trans. ASME, J. Heat Transfer 95, 227–234. Chieng, C.C. and Launder, B.E. (1980). ‘On the calculation of turbulent heat transport downstream from an abrupt pipe expansion’, Numerical Heat Transfer 3, 189–207. Gibson, M.M., Verriopoulos, C.A. and Nagano, Y. (1982). ‘Measurements in the heated turbulent boundary layer on a mildly curved convex surface’. In Turbulent Shear Flows 3, L.J.S. Bradbury et al. (eds.), Springer, 80–89. Harriott, P. and Hamilton, R.M. (1965). ‘Solid-liquid mass transfer in turbulent pipe ﬂow’, Chem. Eng. Sci. 20, 1073–1078. Hattori, H. and Nagano, Y. (1995). ‘Calculation of turbulent ﬂows with pressure gradients using a k-ε model’, JSME Int. J. II 38, 518–524. Hattori, H. and Nagano, Y. (1998). ‘Rigorous formulation of two-equation heat transfer model of turbulence using direct simulations’, Numerical Heat Transfer B 33, 153–180.

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Hattori, H., Nagano, Y. and Tagawa, M. (1993). ‘Analysis of turbulent heat transfer under various thermal conditions with two-equation models’. In Engineering Turbulence Modelling and Experiments 2, W. Rodi and F. Martelli (eds.), Elsevier, 43–52. Hinze, J.O. (1975). Turbulence, 2nd ed., McGraw-Hill. Hochreiter L.E. and Sesonske, A. (1974). ‘Turbulent structure of isothermal and nonisothermal liquid metal pipe ﬂow’, Int. J. Heat Mass Transfer 17, 113–123. Iida, O. and Kasagi, N. (1993). ‘Direct numerical simulation of homogeneous isotropic turbulence with heat transport (Prandtl number eﬀects)’, Trans. Jpn. Soc. Mech. Eng. B 59, 3359–3364. Itazu, Y. and Nagano, Y. (1998). ‘RNG modeling of turbulent heat ﬂux and its application to wall shear ﬂows’, JSME Int. J. B 41, 657–665. Jones, W.P. and Musonge, P. (1988). ‘Closure of the Reynolds stress and scalar ﬂux equations’, Phys. Fluids 31, 3589–3604. Kasagi, N. and Ohtsubo, Y. (1992). ‘Direct numerical simulation of low Prandtl number thermal ﬁeld in a turbulent channel ﬂow’. In Turbulent Shear Flows 8, F. Durst et al. (eds.), Springer, 97–119. Kasagi, N., Tomita, Y. and Kuroda, A. (1992). ‘Direct numerical simulation of passive scalar ﬁeld in a turbulent channel ﬂow’, Trans. ASME, J. Heat Transfer 114, 598– 606. Kawamura, H. and Kawashima, N. (1994). ‘A proposal of k-4 ε model with relevance to the near wall turbulence’. In Proc. Int. Symposium on Turbulence, Heat and Mass Transfer, Lisbon, Portugal, I.1.1–I.1.4. Kays, W.M. (1994). ‘The 1992 Max Jakob Memorial Award Lecture: Turbulent Prandtl number – where are we? Trans. ASME, J. Heat Transfer 116, 284–295. Kim, J. and Moin, P. (1989). ‘Transport of passive scalars in a turbulent channel ﬂow’. In Turbulent Shear Flows 6, J.-C. Andr´e et al. (eds.), Springer, 85–96. Kim, J., Moin, P. and Moser, R. (1987). ‘Turbulence statistics in fully developed channel ﬂow at low Reynolds number’, J. Fluid Mech. 177, 133–166. Kline, S.J. (1980–1981) AFOSR–HTTM–Stanford Conference Complex Turbulent Flows. Kondoh, T. Nagano, Y. and Tsuji, T. (1993). ‘Computational study of laminar heat transfer downstream of a backward-facing step’, Int. J. Heat Mass Transfer 36, 577–591. Launder, B.E. (1988). ‘On the computation of convective heat transfer in complex turbulent ﬂows’, Trans. ASME, J. Heat Transfer 110, 1112–1128. Moser, R-D, Kim, J. and Mansour, N.N. 1999 ‘Direct numerical simulation of turbulent channel ﬂow up to Reτ up to 590’ Phys. Fluids 11, 943–945 Myong, H.K. and Kasagi, N. (1990). ‘A new approach to the improvement of k-ε turbulence model for wall-bounded shear ﬂows’, JSME Int. J. II, 33, 63–72. Nagano, Y., Hattori, H. and Abe, K. (1997). ‘Modeling the turbulent heat and momentum transfer in ﬂows under diﬀerent thermal conditions’, Fluid Dynamic Research 20, 127–142. Nagano, Y. and Hishida, M. (1987). ‘Improved form of the k-ε model for wall turbulent shear ﬂows. Trans. ASME, J. Fluids Engng. 109, 156–160. Nagano, Y. and Itazu, Y. (1997). ‘Renormalization group theory for turbulence: eddyviscosity type model based on an iterative averaging method’, Phys. Fluids 9, 143– 153.

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Nagano, Y. and Kim, C. (1988). ‘A two-equation model for heat transport in wall turbulent shear ﬂows’, Trans. ASME, J. Heat Transfer 110, 583–589. Nagano, Y., Sato, H. and Tagawa, M. (1995). ‘Structure of heat transfer in the thermal layer growing in a fully developed turbulent ﬂow’. In Turbulent Shear Flows 9, F. Durst et al. (eds.), Springer, 343–364. Nagano, Y. and Shimada, M. (1994). ‘Critical assessment and reconstruction of dissipation-rate equations using direct simulations’. In Recent Developments in Turbulence Research, Z.S. Zhang and Y. Miyake (eds.), Int. Academic Publishers, 189–217. Nagano, Y. and Shimada, M. (1995a). ‘Rigorous modeling of dissipation-rate equation using direct simulations’, JSME Int. J. B 38, 51–59. Nagano, Y. and Shimada, M. (1995b). ‘Computational modeling and simulation of turbulent ﬂows’. In Computational Fluid Dynamics Review 1995, M. Hafez and K. Oshima (eds.), Wiley, 695-714. Nagano, Y. and Shimada, M. (1996). ‘Development of a two-equation heat transfer model based on direct numerical simulations of turbulent ﬂows with diﬀerent Prandtl numbers’, Phys. Fluids 8, 3379–3402. Nagano, Y., Shimada, M. and Youssef, M.S. (1994). ‘Progress in the development of a two-equation heat transfer model based on DNS databases’. In Proc. Int. Symposium on Turbulence, Heat and Mass Transfer, Lisbon, Portugal, 3.2.1–3.2.6. Nagano Y. and Tagawa, M. (1990a). ‘An improved k-ε model for boundary layer ﬂows’, Trans. ASME, J. Fluids Engng. 112, 33–39. Nagano Y. and Tagawa, M. (1990b). ‘A structural turbulence model for triple products of velocity and scalar’, J. Fluid Mech. 215, 639–657. Nagano, Y., Tagawa, M. and Tsuji, T. (1991). ‘An improved two-equation heat transfer model for wall turbulent shear ﬂows’. In Proc. ASME/JSME Thermal Engng. Joint Conference, Reno, USA, 3, 233–240. Newman G.R., Launder B.E. and Lumley J.L. (1981). ‘Modelling the behavior of homogeneous scalar turbulence’, J. Fluid Mech. 111, 217–232. Obukhov, A.M. (1949). ‘Structure of the temperature ﬁeld in turbulent ﬂows’, Izv. Akad. Nauk SSSR Geogr. Geophys. Ser. 13, 58. Reynolds, W.C., Kays, W.M. and Kline, S.J. (1958). ‘Heat transfer in the turbulent incompressible boundary layer III-arbitrary wall temperature and heat ﬂux’, NASA MEMO. 12-3-58W, Washington, DC. Robinson, S.K. (1991). ‘The kinematics of turbulent boundary layer structure’, NASA TM-103859. Rodi, W. (1984). ‘Turbulence models and their application in hydraulics – a state of the art review. International Association for Hydraulic Research-Publication (2nd edn.), Delft. Rodi W. and Mansour, N.N. (1993). ‘Low Reynolds number k-ε modelling with the aid of direct simulation data’, J. Fluid Mech. 250, 509–529. Sato, H., Shimada, M. and Nagano, Y. (1994). ‘A two-equation turbulence model for predicting heat transfer in various Prandtl number ﬂuids’. In Proc. 10th International Heat Transfer Conference, Brighton, 2, 443–448. Shikazono, N. and Kasagi, N. (1996). ‘Second-moment closure for turbulent scalar transport at various Prandtl numbers’, Int. J. Heat Mass Transfer 39, 2977–2987. Sideman, S. and Pinczewski, W.V. (1975). ‘Turbulent heat and mass transfer at interfaces: Transport models and mechanics’. In Topics in Transport Phenomena, C. Gutﬁnger (ed.), Hemisphere, 47–207.

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7 Introduction to Direct Numerical Simulation Neil D. 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

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Performance (Rmax) Gflops

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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

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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

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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.

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A corresponding Kolmogorov micro-timescale for the small eddies, τ , is given by 1/2 ν . (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

(4)

and the three components of the momentum equation ∂ui ∂ui 1 ∂p ∂ 2 ui + =ν . + uj ∂t ∂xj ρ ∂xi ∂xj ∂xj

(5)

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 .

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3

exact (spectral) 2nd 4th 6th (compact)

2.5

hk’

2

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 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.

f (x+h)+3f (x)+f (x−h) =

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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.

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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

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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

Sandham Y

–3h

+8.5h +3h

X

O

–7.5h

Z

–5h

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)

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(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

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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

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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

−

∂ u Jj + Jjp + Jjν , (12) ∂xj

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 terms. The triple moment term Jju = ui ui uj /2 is usually modelled as gradient diﬀusion with an eddy viscosity. The pressure transport term Jjp = p uj is usually ignored, while the viscous term Jjν = −ν∂k/∂xj can be included as is

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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

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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.

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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.

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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.

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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¨ohlich and W. Rodi 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¨ artel 1996, Piomelli 1998), and our 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

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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

(1)

∂ui ∂ (ui uj ) ∂Π ∂(ν 2Sij ) + = , + ∂t ∂xj ∂xi ∂xj

(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 turbulence modelling (RANS) as time averaging: u = limT →∞ T1 0T u dt. Reynolds averaging has the properties u = u,

uv = uv.

(3)

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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

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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/∆) 2

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 ˆ(ω) = u(x) e−iωx dx, since in Fourier space, where the spatial frequency ω is the independent variable, a convolution integral turns into a simple product. Equation (4) then reads 7 7 7(ω) . u(ω) = G(ω) u

(5)

Figure 4 illustrates the ﬁltering in Fourier space. Equation (5) allows the def7 F (ω) = 1 if |ω| ≤ π/∆ inition of another ﬁlter, the Fourier cutoﬀ ﬁlter with G and 0 elsewhere. From (5) it is obvious that only this ﬁlter yields u = u 7 F )2 = G 7 F . In all other cases the identity is not fulﬁlled. This can since (G

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G(x) 6

GG GB

-

∆

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 G F 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)

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u PPP q u

6

u

-

(b)

∆

x

u u H HH j 6

u

-

∆

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

(8)

where S ij and Π are deﬁned analogously to the unﬁltered case. The term τij = ui uj − ui uj

(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

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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

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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¨ ohlich et al. (1998, 2000).

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. 1

apart from some ﬁltering operations for the dynamic model, discussed below, which is of a somewhat diﬀerent nature

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275

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.

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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, τija , to the strain rate Sij of the resolved velocity ﬁeld: 1 τija = 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 + 13 δij τkk instead of Π, and ν + νt instead 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|

|S| =

2S ij S ij

(12)

which yields νt = (Cs ∆)2 |S|.

(13)

This amounts to assuming local equilibrium between the production of the SGS 3 /l. Introducing kinetic energy, P = −τija S ij and dissipation ε expressed by qSGS (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+

,

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)

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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 applied with any model τijmod (C, ∆, u) for τij or τija containing a parameter C 2 . The basic idea is to employ this model not only on the grid scale, or ﬁlter 7 as illustrated in Figure 5. This is the scale, ∆ but also on a coarser scale ∆ 7 so-called test scale with, e.g., ∆ = 2∆: sub-grid scale stresses (∆-level) :

τij = ui uj − ui uj ≈ τijmod (C, ∆, u) (16)

7 sub-test scale stresses (∆-level) :

mod 7 7 7 = Tij = u< i uj − ui uj ≈ τij (C, ∆, u). (17)

From the known resolved velocities ui the velocities u7i can be computed by 7 Similarly, the term . . to ui using an appropriate function G. applying the ﬁlter .= < 7 = 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 + τ= ij

(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): 7 u) 7 − τ mod (C, ∆, u). Lmod = τijmod (C, ∆, (19) ij 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 7 and G to obtain u 7 Lilly (1991). Principally, the consecutive application of G 2

In this subsection we distinguish between exact and modelled SGS stresses for clarity.

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Fr¨ ohlich and Rodi resolved scales

-

log E(ω)

unresolved scales

Tij

-

τij

-

resolved turbulent stresses HH HH H Lij 7 π/∆

π/∆

log ω

Figure 5: Illustration of the dynamic modelling idea as discussed in the text.

7 7 which generally is even of diﬀerent yields an eﬀective ﬁlter of width ∆ = ∆, 7 and G (e.g. when the box ﬁlter is used). This issue is generally type to G 7 7 with the Fourier = ∆, neglected in the literature. For that reason and since ∆ 7 7 instead of ∆ in this cutoﬀ ﬁlter presently used for illustration, we write ∆ 7 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 7 7 7 2 |S| Lmod = −2 C ∆ S ij + 2 C ∆ |S|S ij ij 2

(21)

with C = Cs2 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

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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 ≈ ui uj and ui uj ≈ ui uj , respectively, yields the model τija = Lm,a with ij Lm ij = ui uj − ui uj ,

(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 7 7 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 2

R,a τija = CL Lm,a ij + CB Lij − 2Cν ∆ | S | S ij

(25)

with LR ij = ui uj − ui uj , evaluated using (23). A further step is to combine (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);

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(c) CB and Cν with CL = 1 (Horiuti 1997). 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.

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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´etais and Lesieur 1992):

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´etais 1996). It can be shown that when implemented 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

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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¨ ohlich et al. (1998) where, for the particular case considered, adaptivity of 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),

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the exact spatial derivative of u formulated in Fourier space is <

∂u 7 7(ω). (ω) = iω G(ω) u ∂x

(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 ob7 vious that computing a derivative with ωeﬀ = ω amounts to replacing G(ω) 7 7 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

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(b)

(a)

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

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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 −1 to 7 7 with G(ω) Kleiser 1999). It uses an operation like multiplication of u(ω) ∗ 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

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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¨ artel 1996, Piomelli et al. 1996). This 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.

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287

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¨ ohlich (2001). In the technically relevant case of a 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¨ otzbach 1977).

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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)du/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.

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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

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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.

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Figure 8: Computation without wall function using ∆x+ = 62, ∆z + = + + + 30, ∆y1+ = 1.8. Left: u+ , right: u+ rms , vrms , wrms , −uv . Continuous lines are 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

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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

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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.

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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

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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. This type of feature is required for important ﬁelds of research such as ﬂuid-structure aerodynamic coupling and turbulence control. In the future, the applications of LES will turn from the present academic cases to more applied conﬁgurations.

References Baggett, J.S. (1998). ‘On the feasibility of merging LES with RANS for the nearwall region of attached turbulent ﬂows’. In Annual Research Briefs 1998, 267–277. Center for Turbulence Research. Balaras, E., Benocci, C. and Piomelli, U. (1996). ‘Two-layer approximate boundary conditions for large-eddy simulations’, AIAA Journal 34 1111–1119. Beaudan, P. and Moin, P. (1994). ‘Numerical experiments on the ﬂow past a circular cylinder at sub-critical Reynolds number’, Technical Report TF-62, Stanford University, 1994. Boris, J.P, Grinstein, F.F., Oran, E.S. and Kolbe, R.L. (1992). ‘New insights into large eddy simulation’, Fluid Dyn. Res. 10 199. Breuer, M. (1998). ‘Large eddy simulations for the ﬂow past a circular cylinder, numerical and modelling aspects’, Int. J. Numerical Methods in Fluids 28 1281–1302. Breuer, M. and Rodi, W. (1994) ‘Large eddy simulation of turbulent ﬂow through a straight square duct and a 180◦ bend’. In Fluid Mech. and its Appl., 26, P.R. Voke, R. Kleiser and J.P. Chollet (eds.), Kluwer. Crooke, J.W. and Hanratty, T.J. (1993). ‘Origin of turbulence-producing eddies in a channel ﬂow’, Phys. Fluids 5 1011–1022. Cabot, W. (1995). ‘Large-eddy simulations with wall models’. In Annual Research Briefs 1995, Center for Turbulence Research, 41–50. Cabot, W. (1996). ‘Near-wall models in large eddy simulations of ﬂow behind a backward facing step’. In Annual Research Briefs 1996, Center for Turbulence Research, 199–210. Chollet, J.P. and Lesieur, M. (1981). ‘Parameterization of small scales of three dimensional isotropic turbulence’, J. Atmos. Sci. 38 2747–2757, 1981. Comte, P. and Lesieur, M. (1981). ‘Large eddy simulation of compressible turbulence’. In Advances in Turbulence Modelling, Lecture Series 1998-05, Von Karman Institute for Fluid Dynamics, Rhode Saint Gen`ese, Belgium. Davidson, L. (1997). ‘Large Eddy Simulation: a dynamic one-equation subgrid model for three-dimensional recirculating ﬂow’. In 11th Symp. on Turbulent Shear Flows 3 26.1–26.6, Grenoble. Ferziger, J. and Peric, M. (1996). Computational Methods for Fluid Dynamics. Springer-Verlag.

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Ferziger, J.H. (1996). ‘Large eddy simulation’. In Simulation and Modelling of Turbulent Flows, T.B. Gatski, M.Y. Hussaini, and J.L. Lumley (eds.), Oxford University Press, 109–154. Fr¨ ohlich, J. (2001). ‘LES of vortex shedding past circular cylinders’. In Proceedings of ECCOMAS 2000, CIMNE, Universitat Polyt´ecnica de Catalunya, Barcelona, Spain. Fr¨ ohlich, J., Rodi, W., Kessler, Ph., Parpais, S., Bertoglio, J.P. and Laurence, D. (1998). ‘Large eddy simulation of ﬂow around circular cylinders on structured and unstructured grids’. In Notes on Numerical Fluid Mechanics 66, E.H. Hirschel, (ed.), Vieweg, 319–338. Fr¨ ohlich, J., Rodi, W., Bertoglio, J.P., Bieder, U. and Touil, H. (2000). ‘Large eddy simulation of ﬂow around circular cylinders on structured and unstructured grids, II’. In Notes on Numerical Fluid Mechanics, E.H. Hirschel (ed.), to appear. Fureby, C., Tabor, G., Weller, H.G. and Gosman, A.D. (1997). ‘Diﬀerential subgrid stress models in large eddy simulations’, Phys. Fluids 9 3578–3580. Germano, M. (1992) ‘Turbulence: the ﬁltering approach’, J. Fluid Mech. 238 325–336. Germano, M. (1999) ‘From RANS to DNS: towards a bridging model’. In Direct and Large-Eddy Simulation III, P.R. Voke, N.D. Sandham and L. Kleiser (eds.), Kluwer, 225–236. Germano, M., Piomelli, U., Moin, P. and Cabot, W.H. (1991). ‘A dynamic subgridscale eddy viscosity model’, Phys. Fluids A 3 1760–1765. Geurts, B.J. (1999). ‘Balancing errors in large-eddy simulation’. In Direct and LargeEddy Simulation III, P.R. Voke, N.D. Sandham, and L. Kleiser (eds.), Kluwer, 1–12 Ghosal, S. (1977). ‘An analysis of numerical errors in large-eddy simulations of turbulence’, J. Comput. Phys. 125 187–206. Gr¨ otzbach, G. (1977). ‘Direct numerical simulation of secondary currents in turbulent channel ﬂow’. In Lecture Notes in Physics 76, H. Fiedler (ed.), Springer-Verlag, 308–319. H¨artel, C. (1996). ‘Turbulent ﬂows: direct numerical simulation and large-eddy simulation’. In Handbook of Computational Fluid Mechanics, R. Peyret (ed.), Academic Press, 283–338. Hirt, C.W. (1968). ‘Heuristic stability theory for ﬁnite-diﬀerence equations’, J. Comp. Phys. 2 339–355. Horiuti, K. (1997). ‘A new dynamic two-parameter mixed model for large-eddy simulation’, Phys. Fluids 9 3443–3464. Kravchenko, A.G. and Moin, P. (1997). ‘On the eﬀect of numerical errors in large eddy simulation of turbulent ﬂows’, J. Comp. Phys. 131 310–322. Kravchenko, A.G., Moin, P. and Moser, R. (1997). ‘Zonal embedded grids for numerical simulations of wall-bounded turbulent ﬂows’, J. Comp. Phys. 127 412–423. Leonard, A. (1974). ‘Energy cascade in large eddy simulations of turbulent ﬂuid ﬂows’, Adv. Geophys. 18A 237. Lesieur, M. (1997). Turbulence in Fluids, 3rd edition Kluwer.

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Lesieur, M. and M´etais. O. (1996). ‘New trends in large-eddy simulations of turbulence’, Ann. Rev. Fluid. Mech. 28 45–82. Lilly, D.K. (1991). ‘A proposed modiﬁcation of the Germano subgrid-scale closure method’, Phys. Fluids A 4 633–635. Liu, S., Meneveau, C. and Katz, J. (1994). ‘On the properties of similarity subgridscale models as deduced from measurements in a turbulent jet’, J. Fluid Mech. 275 83–119. Lund, T.S. and H.-J. Kaltenbach, H.-J. (1995). ‘Experiments with explicit ﬁltering for LES using a ﬁnite-diﬀerence method’. In Annual Research Briefs 1995, Center for Turbulent Research, 91–105. Meinke, M., Rister, Th., R¨ utten, F. and Schvorak, A. (1998). ‘Simulation of internal and free turbulent ﬂows’. In High Performance Scientiﬁc and Engineering Computing, H.-J. Bungartz, F. Durst, and C. Zenger (eds.), Springer-Verlag, 61–79. M´etais. O. and Lesieur, M. (1992). ‘Spectral large-eddy simulation of isotropic and stable-stratiﬁed turbulence’, J. Fluid Mech. 239 157–194. Moin, P. (1997). ‘Numerical and physical issues in large eddy simulation of turbulent ﬂows’. In Proceedings of the International Conference on Fluid Engineering, Tokyo, July 13–16, 1997, 1, Japan Society of Mechanical Engineers, 91–100. Moser, R.D., Kim, J. and Mansour, N.N. (1999). ‘Direct numerical simulation of turbulent channel ﬂow up to Reτ = 590’, Phys. Fluids 11 943–946. Nikitin, N.V., Nicoud, F., Wasisto, B., Squires, K.D. and Spalart, P.R. (2000). ‘An approach to wall modelling in large-eddy simulations’, Phys. Fluids 12, 1629–1632. Piomelli, U. (1998). ‘Large eddy simulation turbulent ﬂows’. In Advances in Turbulence Modelling, Lecture Series, 1998–05. Von Karman Institute for Fluid Dynamics, Rhode Saint Gen`ese, Belgium. Piomelli, U., Cabot, W.H., Moin, P., and Lee, S. (1991). ‘Subgrid-scale backscatter in turbulent and transitional ﬂows’, Phys. Fluids A 3 1766–1771. Piomelli, U. and Liu, J. (1995). ‘Large eddy simulation of rotating channel ﬂows using a localized dynamic model’, Phys. Fluids 7 839–848. Piomelli, U., Moin, P., and Ferziger, J.H. (1988). ‘Model consistency in large eddy simulation of turbulent channel ﬂows’, Phys. Fluids 31 1884–1891. Piomelli, U., Yu, Y. and Adrian, R.J. (1996). ‘Subgrid-scale energy transfer and nearwall turbulence structure’, Phys. Fluids 8 215–224. Reynolds, W.C. (1989). ‘The potential and limitations of direct and large eddy simulations’. In Lecture Notes in Physics 357, J.L. Lumley (ed.), Springer-Verlag, 313–343. Rodi, W., Ferziger, J.H., Breuer, M., and Pourqui´e, M. (1997). ‘Status of large eddy simulation: results of a workshop’, J. Fluid Eng. 119 248–262. Rollet-Miet, P., Laurence, D., and Ferziger, J.H. (1999). ‘LES and RANS of turbulent ﬂow in tube bundles’, Int. J. Heat Fluid Flow 20 241–254. Salvetti, M.V. and Banerjee, S. (1995). ‘A priori tests of a new dynamic subgrid-scale model for ﬁnite-diﬀerence large-eddy simulations’, Phys. Fluids 7 2831.

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Salvetti, M.V. and Beux, F. (1998). ‘The eﬀect of the numerical scheme on the subgrid scale term in large-eddy simulation’, Phys. Fluids 10 3020–3022. Schumann, U. (1975). ‘Subgrid scale model for ﬁnite diﬀerence simulations of turbulent ﬂows in plane channels and annuli’, J. Comput. Phys. 18 376–404. Smagorinsky, J.S. (1963). ‘General circulation experiments with the primitive equations, I, the basic experiment’, Mon. Weather Rev. 91 99–164. Spalart, P.R. and S.R. Allmaras. ‘A one-equation turbulence model for aerodynamic ﬂows’, La Recherche A´erospatiale, 1994. Spalart, P.R., Jou, W.H., Strelets, M. and Allmaras, S.R. (1997). ‘Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach’. In Advances in DNS/LES, C. Liu and Z. Liu (eds.), Greyden Press. Speziale, C.G. (1998). ‘A combined large-eddy simulation and time-dependent RANS capability for high-speed compressible ﬂows’, J. Sci. Comput. 13 253–274. Stolz, S., Adams, N.A. and Kleiser, L. (1999). ‘The approximate deconvolution procedure applied to turbulent channel ﬂow’. In Direct and Large-Eddy Simulation III, P. Voke, N.D. Sandham and L. Kleiser (eds.), Kluwer, 163–174. Tamura, T., Ohta, I. and Kuwahara, K. (1990). ‘On the reliability of two-dimensional simulation for unsteady ﬂows around a cylider-type’, J. Wind Eng. Indust. Aerodyn. 35 275–298. Tennekes, H. and Lumley, J.L. (1972). A First Course in Turbulence, MIT Press. Piomelli, U. and Chasnov, J.R. (1996). ‘Large-Eddy Simulations: theory and applications’. In Turbulence and Transition Modelling, M. Hallb¨ ack et al. (eds.), Kluwer, 269–331. Vasilyev, O.V., Lund, T.S. and Moin, P. (1998). ‘A general class of commutative ﬁlters for LES in complex geometries’, J. Comput. Phys. 146 82–104. Vreman, B., Geurts, B. and Kuerten, H. (1994a). ‘Discretization error dominance over subgrid terms in Large Eddy Simulation of compressible shear layers in 2D’, Comm. Num. Meth. Engin. 10 785–790. Vreman, B., Geurts, B. and Kuerten, H. (1994b). ‘On the formulation of the dynamic mixed subgrid-scale model’, Phys. Fluids 6 4057–4059. Vreman, B., Geurts, B. and Kuerten, H. (1997). ‘Large-Eddy simulation of the turbulent mixing layer’, J. Fluid Mech. 339 357–390. Werner, H. (1991). Grobstruktursimulation der turbulenten Str¨ omung u ¨ber eine querliegende Rippe in einem Plattenkanal bei hoher Reynoldszahl, PhD thesis, Technische Universit¨at M¨ unchen, 1991. Werner, H and Wengle, H. (1993). ‘Large-Eddy Simulation of turbulent ﬂow over and around a cube in a plane channel’. In 8th Symp. on Turb. Shear Flows, F. Durst et al. (eds.), Springer-Verlag, 155–168. Zang, Y., Street, R.L. and Koseﬀ, J.R. (1993). ‘A dynamic mixed subgrid-scale model and its application to turbulent recirculating ﬂows’, Phys. Fluids 5 (12) 3186–3196.

9 Introduction to Two-Point Closures Claude Cambon Abstract An overview is given of nonlocal theories and models, ranging from linear to nonlinear. The background principles are presented and illustrated mainly for incompressible, homogeneous, anisotropic turbulence. In this case, which includes eﬀects of mean gradients and body forces and related structural effects, the complete rapid distortion theory (RDT) solution is shown to be a building block for constructing a full nonlinear closure theory. Firstly, a general overview of the closure problem is presented, which accounts for both the nonlinear problem and the nonlocal problem. 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

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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 uj ∂xj

(2.1)

Reynolds stress term

∂Ui =0 ∂xi

(2.2)

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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

and

−

∂p ∂xi

+ ν

Pressure term

∂ 2 ui ∂xj ∂xj

(2.3)

Viscous term

∂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

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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

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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

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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.

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3 3.1

305

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 − ij − (Dijk ) + Uk ∂t ∂xk ∂xk

(3.1)

R˙ ij ∂U

∂Ui Rkj − ∂xkj Rki is usually referred to as the production tensor where Pij = − ∂x k 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 ∂u

∂u

(r)

(s)

pressure and is given by Πij = p ( ∂xji + ∂xji ), consisting of one-point correlations 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 (w)

Πij = Πij + Πij + Πij

(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.

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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 (r)

Πij = 2

∂Um ∂xl

κi κm κj κm Φlj + Φil d3 κ 2 κ κ2

and (s) Πij

(3.3)

=

Tij d3 κ

(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 = νωi 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 ˙ with adjustable constants. basing the equation for / ˙ on the equation for k/k 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

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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.

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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 7∗i u 7j . The equation that governs the second-order speccovariance matrix u 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 7i (κ, t)u 7j (p, t)u 7l (q, t) u

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 −

1 uuuu = − 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 7i at diﬀerent wave vectors are actually the Fourier velocity components u 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.

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• 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 uuuu, + exact linearuuuterms + uuu = ∂t θ

5

the term uuuu in the left-hand side acts as a source term for increasing 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.

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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)

(4.3)

Combining elementary solutions of the form (4.3) via Fourier synthesis ui (x, t)

=

7i exp(ıκ · x) d3 κ u

(4.4)

the RDT solution is 7i [κ(t), t] = Gij (κ, t, t )u 7j [κ(t0 ), t0 ] u

(4.5)

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 7i (κ, t) on a diﬀerent wavevector K = κ(t0 ) at time t0 , a process of spectral u 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

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of velocity can be derived from a spectral tensor Φij (κ, t) (Batchelor 1953), 7i (k, t) in individual realisations by related to the Fourier transform u 7∗i (p, t)u 7j (κ, t) = u 7i (−p, t)u 7j (κ, t) = Φij (κ, t)δ(κ − p) u

(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

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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 = 7j = 0. Taken 0, obtained from (4.6) and the incompressibility condition κj u 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  Φ= 0 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) = 12 Φii is the energy density in κ-space, whereas H(κ, t) = ıκl lij Φij is the helicity spectrum and, along with Z, is zero in the

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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 7˙ i + Mij u 7j + νκ2 u 7i = si u

(4.12)

7˙ i = ∂ u 7i /∂t − λlm κl ∂ u 7i /∂κm corresponds to linear advection by the where u 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

7j (p, t)u 7k (q, t)d3 p u

(4.13)

in terms of a convolution integral, the usual expression of a quadratic nonlinearity, and Pijk = 12 (Pij κk + Pik κj ) which arises from the elimination of 7i (κ, t) = 0. pressure using the incompressibility condition κi u The evolution equation for Φij (Craya 1958), derived from (4.6) and (4.12), is Φ˙ ij + 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 Φ˙ ij , 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) = Pjkl (κ)

t −∞

κ+p+q=0

Gim (κ, t, t )Gkp (p, t, t )Glq (q, t, t )

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Cambon

1 Φqn (q , t ) Pmnr (κ )Φpr (p , t ) + Ppnr (p )Φmr (κ , t ) d3 p dt , (4.16) 2

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

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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’ (7), can be written as 1 κ 7=u 7+ı 7 v b . (5.1) N κ (A similar three-component term, denoted WK , is extensively used in Riley 7 ). The and Lelong (2000), p. 626, but it is not a true vector, in contrast to v scaling of the contribution from the buoyancy force allows one to deﬁne twice the total energy spectral density as 7∗i u 7i + N −27 v7i∗ v7i = u b∗7b.

(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. 7 is similar to (4.5), but with a constant wavevector The linear equation for v 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 ) =

Ni (κ)Nj− (κ) exp[ıσκ (t − t0 )],

(5.3)

=0,±1

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

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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 7∗i u 7j or v7i∗ v7j . They consist of spectral tensors, which are related to u 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 7∗i u 7j are Godeferd and Cambon 1994). The fact that some terms in u 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.

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5.2

317

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

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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 7 are perpendicular, these wavenumwavenumbers κ⊥ ∼ 0. Because κ and u bers 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

Cambon Ω

Energy g

containing

Energy

cone

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

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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

6

Cambon

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-

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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

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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.

References Bardina, J., Ferziger, J.M., Rogallo, R.S. (1985). ‘Eﬀect of rotation on isotropic turbulence: computation and modelling’, J. Fluid Mech. 154, 321–326. Batchelor, G.K. (1953). The Theory of Homogeneous Turbulence. Cambridge University Press. Batchelor, G.K., Proudman, I. (1954). ‘The eﬀect of rapid distortion in a ﬂuid in

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turbulent motion’, Q. J. Mech. Appl. Maths 7, 83–103. Bayly, B.J. (1986). ‘Three-dimensional instability of elliptical ﬂow’, Phys. Rev. Lett. 57, 2160. Benney, D.J., Saﬀman, P.G. (1966). ‘Nonlinear interactions of random waves in a dispersive medium’, Proc. R. Soc. London, Ser. A 289, 301–320. Bradshaw, P. (1969). ‘The analogy between streamline curvature and buoyancy in turbulent ﬂows’, J. Fluid Mech. 36, 177. Caillol, P., Zeitlin, W. (2000). ‘Kinetic equations and stationary energy spectra of weakly nonlinear internal gravity waves’, Dyn. Atm. Oceans 32, 81–112. Cambon, C. (1982). Etude spectrale d’un champ turbulent incompressible soumis ` a des eﬀets coupl´es de d´eformation et rotation impos´es ext´erieurement. Th`ese de ´ Doctorat d’Etat, Universit´e Lyon I, France. Cambon, C. (1989). ‘Spectral approach to axisymmetric turbulence in a stratiﬁed ﬂuid’. In Advances in Turbulence 2, H.H. Fernholtz and H.E. Fiedler (eds.) SpringerVerlag, 162–167. Cambon, C. (2001a). ‘Stability of vortex structures in a rotating frame’. In Turbulence Structure and Vortex Dynamics, J.C.R. Hunt and J.C. Vassilicos (eds.), Cambridge University Press, 244–268. Cambon, C. (2001b). ‘Recent developments in two-point closures’ (this volume [26]). Cambon, C. (2001c). ‘Turbulence and vortex structures in rotating and stratiﬁed ﬂows’, Euromech 396 and Eur. J. Mech. B (Fluids), to appear. Cambon, C., Buﬀat, M., and Bertoglio, J.P. (1997). ‘Research at the LMFA on turbulent and transitional ﬂows aﬀected by compressibility’, ERCOFTAC Bulletin 34. Cambon, C., Jacquin, L., Lubrano, J.L. (1992). ‘Towards a new Reynolds stress model for rotating turbulent ﬂows’, Phys. Fluids A 4, 812–824. Cambon, C., Mansour, N.N., Godeferd, F.S. (1997). ‘Energy transfer in rotating turbulence’, J. Fluid Mech. 337, 303–332. Cambon, C., Mao, Y., Jeandel, D. (1992). ‘On the application of time dependent scaling to the modelling of turbulence undergoing compression’, Eur. J. Mech. B (Fluids) 6, 683–703. Cambon, C. and Scott, J.F. (1999). ‘Linear and nonlinear models of anisotropic turbulence’, Ann. Rev. Fluid Mech. 31, 1–53. Cambon C., Teiss`edre C., Jeandel D. (1985). ‘Etude d’eﬀets coupl´es de d´eformation et de rotation sur une turbulence homog`ene’, J. M´ec. Th´eor. Appl. 4, 629–657. Carnevale, G.F., Martin, P.C. (1982). ‘Fields theoretical techniques in statistical ﬂuid dynamics, with application to nonlinear wave dynamics’, Geophys. Astrophys. Fluid Dyn. 20, 131. Craya, A. (1958). ‘Contribution a` l’analyse de la turbulence associ´ee `a des vitesses moyennes’, P.S.T. 345. Minist`ere de l’air. France Fauchet, G., Shao, L., Wunenberger, R., Bertoglio, J.P. (1997). ‘An improved twopoint closure for weakly compressible turbulence’, 11th Symp. Turb. Shear Flow, Grenoble, Sept. 8–10, 1997.

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Galmiche, M., Hunt, J.C.R., Thual, O., Bonneton, P. (2001). ‘Turbulence-mean ﬁeld interactions and layer formation in a stratiﬁed ﬂuid. Euromech 396’, Eur. J. Mech. B (Fluids), to appear. Gence, J.-N. (1983). ‘Homogeneous turbulence’, Ann. Rev. Fluid Mech. 15, 201–222. Gence, J.-N., Mathieu, J. (1979). ‘On the application of successive plane strains to grid-generated turbulence’, J. Fluid Mech. 93, 501–513. Godeferd, F.S., Cambon, C. (1994). ‘Detailed investigation of energy transfers in homogeneous turbulence’, Phys. Fluids 6, 2084–2100. Godeferd, F.S. and Staquet, C. (2000). ‘Statistical modelling and DNS of decaying stably-stratiﬁed turbulence: Part 2. Large and small scales anisotropy’, J. Fluid Mech., submitted. Goldstein, M.E. (1978). ‘Unsteady vortical and entropic distortions of potential ﬂows round arbitrary obstacles’, J. Fluid Mech. 89, 431. Greenspan, H.P. (1968). ‘The Theory of Rotating Fluids’, Cambridge University Press. Hanazaki, H., Hunt, J.C.R. (1996). ‘Linear processes in unsteady stably stratiﬁed turbulence’, J. Fluid Mech. 318, 303–337. van Haren, L. (1993). ‘Etude th´eorique et mod´elisation de la turbulence en pr´esence d’ondes internes’, Th`ese de Doctorat, Universit´e Lyon 1, France. van Haren, L., Staquet, C., Cambon C. (1996). ‘Decaying stratiﬁed turbulence: comparison between a two-point closure EDQNM model and DNS’, Dyn. Atmos. Oceans 23, 217–233. Hunt, J.C.R. (1973). ‘A theory of turbulent ﬂow around two-dimensional bluﬀ bodies’, J. Fluid Mech. 61, 625–706. Hunt, J.C.R., Carruthers, D.J. (1990). ‘Rapid distortion theory and the ‘problems’ of turbulence’, J. Fluid Mech. 212, 497–532. Hopf, E. (1952). ‘Statistical hydrodynamics and functional calculus’, J. Rat. Mech. Anal. 1, 87. Jacquin, L., Leuchter, O., Cambon, C., Mathieu, J. (1990). ‘Homogeneous turbulence in the presence of rotation’, J. Fluid Mech. 220, 1–52. Kraichnan, R.H. (1959). ‘The structure of turbulence at very high Reynolds numbers’, J. Fluid Mech. 5, 497–543. Leblanc, S. and Cambon, C. (1997). ‘On the three-dimensional instabilities of plane ﬂows subjected to Coriolis force’, Phys. Fluids 9(5), 1307–1316. Lee, J.M., Kim, J., Moin, P. (1990). ‘Structure of turbulence at high shear rate’, J. Fluid Mech. 216, 561–583. Lele, S.K. (1994). ‘Compressibility eﬀects on turbulence’, Ann. Rev. Fluid Mech. 26, 211–254. Leuchter, O., Benoit, J.-P., Cambon, C. (1992). ‘Homogeneous turbulence subjected to rotation-dominated plane distortion’, Fourth Turbulent Shear Flows. Delft University of Technology. Leuchter, O., Dupeuble, A. (1993). ‘Rotating homogeneous turbulence subjected to a axisymmetric contraction’, In Proc. Ninth Turbulent Shear Flows. Kyoto, 137, 1–6.

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Lumley J.L. (1967). ‘The structure of inhomogeneous turbulence ﬂows’, In Atmospheric Turbulence and Radio Wave Propagation, A.M. Yaglom, V.I. Tatarsky (eds.), 166–67, NAUKA. Lundgren, T.S. (1967). ‘Distribution function in the statistical theory of turbulence’, Phys. Fluids 10(5), 969–975. Mansour, N.N, Lundgren, T.S. (1990). ‘Three-dimensional instability of rotating ﬂows with oscillating axial strain’, Phys. Fluids A 2, 2089–2091. Monin A.S., Yaglom A.M. (1975). Statistical Fluid Mechanics I. MIT Press Orszag, S.A. (1970). ‘Analytical theories of turbulence’, J.Fluid Mech. 41, 363–386. Pedley, T.J. (1969). ‘On the stability of viscous ﬂow in a rapidly rotating pipe’, J. Fluid Mech. 35, 97. Pierrehumbert, R.T. (1986). ‘Universal short-wave instability of two-dimensional eddies in an inviscid ﬂuid’, Phys. Rev. Lett. 57, 2157–2159. Reynolds, W.C., Kassinos, S.C. (1995). ‘One-point modeling for rapidly deformed homogeneous turbulence’, Proc. Roy. Soc. Lond., A, 451–87. Riley, J.J., Lelong, M.-P. (2000). ‘Fluid motions in the presence of strong stable stratiﬁcation’, Ann. Rev. Fluid Mech. 32, 613–657. Riley, J.J., Metcalfe, R.W., Weisman, M.A. (1981). ‘DNS of homogeneous turbulence in density stratiﬁed ﬂuids’, Proc. AIP conf. on nonlinear properties of internal waves, B.J. West (ed.), AIP, 79–112. Rogers, M.M. (1991). ‘The structure of a passive scalar ﬁeld with a uniform gradient in rapidly sheared homogeneous turbulent ﬂow’, Phys. Fluids A 3, 144–154. Salhi, A., Cambon, C. (1997). ‘An analysis of rotating shear ﬂow using linear theory and DNS and LES results’, J. Fluid Mech. 347, 171-195. Salhi A., Cambon C., Speziale, C.G. (1997). ‘Linear stability analysis of plane quadratic ﬂows in a rotating frame’, Phys. Fluids 9(8), 2300–2309. Sarkar S. (1995). ‘The stabilizing eﬀect of compressibility in turbulent shear ﬂow’, J. Fluid Mech. 282, 163–286. Simone A., Coleman G.N., Cambon C. (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)

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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)

where Jjα 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 Jjα = −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

Basic statistical concepts

2.1

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

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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 }.

(12)

The joint PDF is deﬁned by f φ(ψ) =

∂n F φ(ψ) ∂ψ1 ∂ψ2 · · · ∂ψn

(13)

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

2.3

331

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 =

2.4

+∞ −∞

Q(ψ1 , ψ2 )fφ1 |φ2 (ψ1 |ψ2 )dψ1 .

(17)

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 4= Q

ρQ , ρ

(18)

and Favre-decomposition into mean and ﬂuctuation is deﬁned as 4 + 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

and in general,

; = 0 Q

Q = 0,

; = 0 Q

Q = 0.

It is also useful to deﬁne the Favre-probability density function by f4φ(ψ) = f φ(ψ)

ρ|φ = ψ ρ(ψ) = f φ(ψ) ρ ρ

(20)

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 4 Q(φ) =

+∞ −∞

Q(ψ)f4φ(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 ∂ ;i ∂ , = +U Dt ∂t ∂xi

(22)

and Favre-averaging the momentum and species equations (2) and (4) gives D 4 ∂p ∂ ∂ u Ui = − − Tij − ρu> j i Dt ∂xi ∂xj ∂xj ∂ ∂ D φ , Jjα + ρS;α − ρu> ρ φ4α = − j α 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 4k /∂xk , u or φ , which are zero in constant-density ﬂows. The full second∂U α i moment 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

4 4 D > u ∂ Uj − ρu> u ∂ Ui ui uj = −ρu> i k j k Dt ∂xk ∂xk

− ui

∂p ∂p ∂p 2 + uj − δij uk ∂xj ∂xi 3 ∂xk

−ρij ∂ 2 > − ρui uj uk + δij uk p ∂xk 3 ∂p ∂p −ui − uj ∂xj ∂xi ∂u 2 + δij p k . 3 ∂xk

(25a)

(25b) (25c) (25d) (25e) (25f)

Here the Reynolds average is denoted by an overbar and the combined use of 4 is used to denote the Favre average of a long an overbar and tilde, as in x, 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 ij = δij . (26) 3 For the dissipation of turbulent kinetic energy the standard dissipation equa>

tion 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 u k ∂ u> > i j ui uj uk = −Cs u> u , k l ∂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 φ and for scalar variance φ> φ respectively read or Reynolds ﬂux u> i ρ

4 4 D > φ ∂ Ui − ρu> u ∂ φ ui φ = −ρu> j i j Dt ∂xj ∂xj

−φ

∂p ∂xi

(28a) (28b)

∂ > ρuj ui φ ∂xj ∂p −φ ∂xi

−

>

+ρui S(φ),

(28c) (28d) (28e)

and, ρ

4 D > φ ∂ φ φ φ = −2ρu> j Dt ∂xj −ρφ ∂ > ρu φ φ − ∂xj j >

+2ρφ S(φ).

(29a) (29b) (29c) (29d)

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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 Πφi (28b), the viscous dissipation of scalar variance φ (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. φ is linked to the dissipation The dissipation rate of scalar variance g = φ> rate of mechanical energy by ωφ ≡

φ = Cφ ω ≡ Cφ . g 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 ∂ξ ∂ξ ∂ ρ = + ρUi ∂t ∂xi ∂xj

∂ξ ρID ∂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

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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¸cais du P´etrole, 51(2), 215–243. 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

2.1

343

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 3/2 k /ε k /ε 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´c (1996) proposed eliminating the explicit dependence on wall-distance by 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 ∂l ∂l 1/2 Dl = ∂xj ∂xj

(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 ε2 2 SN Y = max 0.83 F (F + 1) , 0 (2.5) k 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-

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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 ∂U ∂V ∂W Pk = − u 2 + v2 + w2 ∂x ∂y ∂z

∂V = − v 2 − u2 (3.1) ∂y

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 ∂V 2 Pk = 3νt (3.3) ∂y

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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 ∂U ∂xj can be written 2 as Pk = cµ εS where S is the non-dimensional strain rate ∂Ui ∂Uj k 1/2 1 (3.4) and Sij = 2 + S = [2Sij Sij ] ε ∂xj ∂xi To improve predictions they investigated replacing this form of generation by Pk = cµ εSΩ

(3.5)

where k Ω = [2Ωij Ωij ]1/2 ε

and

Ωij =

1 2

∂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.

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3.3

351

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

(3.7)

? 2 k k ,β ε 3cµ ε

3 4|(S)2 |

(3.8)

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

ε Pk = cµ ε (3.9) T S2 k

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.

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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

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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).

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(1978) wall-reﬂection terms: φij1 = −c1 ε (ui uj /k − 2/3δij )

(4.1)

φij2 = −c2 (Pij − 1/3Pkk δij ) ε 3 3 φw ij1 = c1w (ul uk nl nk δij − /2ui uk nj nk − /2uj uk ni nk ) fy k 3 3 φw ij2 = c2w (φlk2 nl nk δij − /2φik2 nj nk − /2φjk2 ni nk ) fy

(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 ij2 . In a simple shear ﬂow (with mean strain ∂U/∂y), energy is generated in u2 by P11 . A proportion is transferred into v 2 by φ222 = 1/3c2 P11 , whilst the wall-reﬂection term φw 222 opposes this transfer, thus leading to the desired 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 222 = −2cw φ222 fy , in opposing this redistribu2 tion, thus acts to increase v 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 ij2 . They 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: ∂Ul ul uk (nt nt δij − 3ni nj ) (l/2.5y) ∂xk ∂Uj ∂Uk ∂Ui 3 3 / / −0.1kalm nl nk δij − 2 nl nj − 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

φw ij2 = −0.08

Predictions of stresses and heat transfer using this model can be seen in Figures 8 and 9. Clearly the modiﬁed φw ij2 has a signiﬁcant eﬀect in the stagnation region, reducing turbulence energy levels, and thus predicted heat transfer rates, to values close to those measured experimentally.

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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.

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Figure 16: IsoMach contours, boattail Cp and internal nozzle p/pTjet for the Carson and Lee (1981) conﬁguration 1 afterbody (From Batten et al. 1999a). SST: model of Menter (1994); JH: model of Jakirli´c and Hanjali´c (1995); MCL: 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

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Jakirli´c and Hanjali´c (1995) models. In both of the above cases it might be 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.

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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´c, K. (1996), ‘Some resolved and unresolved issues in modelling nonequilibrium 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´c, T. Peeters, eds.), Delft. Jakirli´c, S., Hanjali´c, K. (1995), ‘A second-moment closure for non-equilibrium and 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

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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-` a-vis RANS methods, is assessed.

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

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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¨ ohlich and Rodi (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-`a-vis RANS calculations will be discussed.

2

LES Methods Used

The LES approach is introduced in the companion chapter of Fr¨ ohlich and 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

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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

3

Rodi

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´ arm´an vortex street for 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

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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

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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). St

CD

CD

CL

LR /D

grid Nx × Ny × Nz

KAWAMU, 3rd UW No SGS, NS UMIST2, CD Dyn, WF UKAHY2, CD Smag, WF TAMU2, Dyn, 3rd UW NS/WF

0.15

2.58

0.27

1.33

1.68

125 × 78 × 20

0.09

2.02

–

–

1.21

140 × 81 × 13

0.13

2.30

0.14

1.15

1.46

146 × 146 × 20

0.14

2.77

0.19

1.79

0.94

165 × 113 × 17

UK1, Smag, WF UK3, Smag, NS NT7, LDM, WF UOI, Dyn., NS IS3, Dyn. mix, NS TIT, Dyn, NS ST5, Smag, NS

CD

0.13

2.20

0.14

1.01

1.32

109 × 105 × 20

CD

0.13

2.23

0.13

1.02

1.44

146 × 146 × 20

CD

0.131 2.05

0.12

1.39

1.39

140 × 103 × 32

2.03(1) 0.18

1.29

1.20

192 × 160 × 48

Calculation method

LES RottachEgern Rodi et al. (1997)

LES Grenoble

5th UW 0.13

5th UW 0.133 2.79

0.36

1.68

1.36

112 × 104 × 32

3rd UW 0.131 2.62

0.23

1.39

1.23

121 × 113 × 127(2)

variable 0.161 2.78 3rd UW

0.28

1.38

1.02

107 × 103 × 20

0.126 2.21 0.125 2.04 0.129 2.25

0.16 0.20 0.20

1.47 1.22 1.50

≈ 1.0 185 × 105 × 25

0.134 1.64 0.142 1.79

≈ 0 0.305 2.8 0.012 0.614 2.04

100 × 76 100 × 76

0.137 1.72 0.143 2.0

≈0 0.07

0.426 2.4 1.17 1.25

170 × 170 170 × 170

RANS RMS, WF Franke and Two-layer RSM Rodi (1993)

0.136 2.15 0.159 2.43

0.27 0.06

1.49 1.3

70 × 64 186 × 156

Experiments

0.132 1.9 to 2.2

Voke (1997)

LES Sohankar (1998)

Smag, NS Dyn, NS OEDSM, NS

RANS Bosch (1995)

Std. k-ε, WF Kato–Launder k-ε, WF Two-layer k-ε Two-layer Kato–Launder k-ε

CD

0.98 1.0

1.38

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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

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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

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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).

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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

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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´e (1996) reported that the 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

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373

Figure 6: Turbulent component of kinetic energy along centre-line of square cylinder. (1) Pourqui´e (1996), (2) Franke and Rodi (1993), (3) Bosch (1995), (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¨ ohlich et al. (1998) using the same computer code. Further, DNS 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¨ ohlich 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

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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

Beaudan and Moin (1994)

5th order – .216 upwind 144 × 136 Smag. .209 ×48 Dyn. .203

Mittal and Moin (1997)

CDS

cD

Cpb

LR /D

θ

.96 .92 1.00

.89 .81 .95

1.56 1.74 1.36

85.3 84.8 85.8

401 × 120 Dyn. ×48

.207

1.00

.93

1.40

86.9

Kravchenko and B-spline 185 × 205 Dyn. Moin (2000) ×48

.210

1.04

.94

1.35

88.0

– 165 × 165 Smag. ×32/64 Dyn.

.22 .22 .21

Breuer (1998)

CDS

Experiments (origin Fr¨ ohlich et al. 1998, Kravchenko and Moin 2000)

.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

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¨ ohlich et al. (1998) with and without Smagorinsky model are not 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 u2 and v 2

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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 u2 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¨ ohlich et al. (1998) also report preliminary 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

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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¨ ohlich et al. (1998) obtained on a similar grid with the Smagorinsky model. Most global parameters and the mean velocity agree reasonably

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Figure 8: Total resolved Reynolds stresses u2 (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 u2 and u v in wake of circular cylinder (Re = 3900); from Kravchenko and Moin (2000). For key, see Figure 7b.

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Table 3: Global parameters for ﬂow past circular cylinder at Re = 140, 000. Author

Zmax

grid

SGS model

St

cD

−Cpback LR /D

θsep

Calculations Fr¨ ohlich et al. 1D (1998)

166×206×64

Breuer (1999)

165×165×64

2D

Smag.CS = .1

.217 1.157

1.33

.42

93.8

Dynamic .204 1.239 Smag.CS = .1 .217 1.218 Smag.CS = .065 .247 .707

1.398 1.411 0.677

.577 .416 .712

96.37 95.16 94.58

1.21 1.34

≈ 0.5

Experiments Cantwell and Coles (1983) Others (see Fr¨ ohlich et al. 1998, Breuer 1999)

.179 1.237 .2 ≈ 1.2

79

Figure 10: Flow past circular cylinder at Re = 140,000: (a) time-mean velocity u along centre-line; (b) total ﬂuctuations u2 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¨ ohlich et al. (1998). One clear conclusion of Breuer (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

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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

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Table 4: Global parameters for single cube calculation. Calculation method

xF 1

xT

xR1

grid Nx × Ny × Nz

LES Rodi et al. (1997)

UKAHY3, Smag. UKAHY4, Dyn. UBWM2, Smag.

1.29 1.00 0.81

– – 0.837

1.70 1.43 1.72

165 × 65 × 97 165 × 65 × 97 144 × 58 × 88

RANS Lakehal and Rodi (1997)

Std. k-ε WF KL-k-ε-WF Two-layer k-ε Two-layer KL-k-ε

0.65 0.64 0.95 0.95

0.43 – – –

2.18 2.73 2.68 3.40

110 × 32 × 32 110 × 32 × 32 142 × 84 × 64 142 × 84 × 64

1.04

–

1.61

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

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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

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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 calculated by Mathey et al. (1999) with the measurements of Meinders et al. (1997). Here calculations with a ﬁner 1403 grid are also included. There is generally good agreement between the LES calculations and the measurements except at the corner points at the bottom (A, D). The disagreement there is due to the lack of knowledge about temperature boundary conditions at the channel wall; for lack of any information an adiabatic condition has been used which is probably not entirely realistic. The calculations of heat transfer presented in Mathey et al. (1999) are of similar quality. The RANS calculations included in the workshop report are on the whole much poorer as was observed already for the single-cube case.

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Figure 13: Streamlines for case of matrix of cubes: (a) in horizontal plane at half-cube height (y/H = 0.5); (b) in vertical symmetry plane; (c) near the channel ﬂoor. LES calculations of Mathey et al. (1998, 1999), experiments of Meinders et al. (1998).

5

Conclusions

LES calculations have been reviewed for several basic bluﬀ body ﬂows with relatively simple body geometry but complex ﬂow behaviour. For the case of vortex-shedding ﬂow past a square cylinder many results were available, mainly from calculations submitted to two LES workshops. The calculations all show qualitatively the correct basic shedding features, but there are large quantitative diﬀerences in the results, except for the Strouhal number which appears not to be very sensitive to the details of the calculation. These large diﬀerences and the fact that no simulation is entirely satisfactory point to diﬃculties that LES methods still have with this ﬂow. The diﬃculties stem partly from the fact that the ﬂow is transitional and has a very thin reverse-

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Figure 14: Temperature distribution on surface of heated cube along paths ABCDE in the horizontal plane at half-cube height and along path ABCD in the vertical symmetry plane. Calculations of Mathey et al. (1999) on 1003 grid (SPRUNG1) and on 1403 grid (SPRUNG2); experiments of Meinders et al. (1998). ﬂow shear layer near the side walls, which could not be resolved properly in any of the calculations. The simulations had further diﬃculties to get both the near-wake region and the ﬂow recovery in the further downstream region predicted correctly; in the latter region numerical errors mainly associated with grid stretching may have had an adverse eﬀect. In a number of calculations the resolution in the spanwise direction was not suﬃcient and a larger extent of the computation domain in this direction may also be necessary. The more successful of the LES predictions are, however, better than any of the RANS calculations reported. In all RANS calculations, the stochastic turbulent ﬂuctuations are severely underpredicted, most likely because the high values in the experiment may stem from low-frequency variation of the shedding motion due to three-dimensional eﬀects which cannot be accounted for in two-dimensional RANS calculations. LES seems to pick up these motions and consequently give a better simulation of the details of the ﬂow. The price to be paid for this is a much larger computing time: the UKAHY2 LES calculations took 73 hours on a SNI S600/20 vector computer while the RANS

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calculations using wall functions took 2 hours and the ones using the two-layer approach 8 hours on the same computer. If the approach ﬂow is not uniform in the spanwise direction (e.g. cylinder in boundary layer) so that RANS calculations must be also three-dimensional, these will then be nearly as expensive as LES calculations. Finally, it should be added here that fairly successful LES calculations have also been carried out for the ﬂow around oscillating square cylinders (Murakami et al. 1995, Kobayashi et al. 1996). Calculations of the ﬂow around a circular cylinder at the fairly low Reynolds number of 3900 have shown that it is important to perform a three-dimensional simulation since two-dimensional simulations did not produce a recirculation zone in the mean. The subgrid-scale model was found to have little eﬀect at this low Reynolds number on the mean ﬂow and global parameters – for these quantities, already quite good agreement with the experiments was achieved with three-dimensional calculations without a model, but to get good agreement also for the ﬂuctuating quantities required good numerical accuracy and the use of a dynamic model. Calculations of the ﬂow around a circular cylinder in the upper subcritical regime (Re = 140, 000) proved to be much more diﬃcult and demanding. Here the subgrid-scale model was found to have considerably more inﬂuence. Most global parameters and the mean velocity could be predicted in reasonable agreement with the experiments, but in the far wake the use of an O-grid which was too coarse there caused deviations. The best results were obtained with the dynamic model, but this was not decisively better than the Smagorinsky model. Reﬁning the grid led on the whole to a deterioration of the results rather than to an improvement. The reasons for this behaviour are not understood and there are still problems with this case. The LES calculations obtained are, however, much better than the RANS calculations for this ﬂow. The ﬂow past surface-mounted cubes can be well predicted by LES, and there were much smaller diﬀerences between the various LES results for this fully turbulent ﬂow. Basically all the complex features of the three-dimensional ﬂow are simulated well by LES, even quantitatively, and the results are much better than those of RANS calculations using various versions of the k-ε model and also Reynolds-stress models. The better predictions cost, however, a high price: on the SNI S660/20 vector computer the LES calculations (UKAHY4) took 160 hours while the RANS two-layer k-ε model calculations took 6 hours and the RANS k-ε model calculations using wall functions only 15 minutes. From the calculations reviewed, no great superiority of one subgrid-scale model over another could be determined, but on balance the dynamic model seems to give the best results. There seems to be some superiority of the calculations using wall functions over those using the no-slip conditions for the cases with higher Reynolds numbers considered. Any numerical errors occurring in the various calculations are diﬃcult to judge, but it appears that the use of stretched grids in the downstream wake can introduce noticeable errors and

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that considerable numerical dissipation is often introduced in upwind schemes, even in those of ﬁfth order, and should be checked in future calculations. For complex bluﬀ body ﬂows many of the details can be calculated successfully by LES, even though the results are not yet entirely satisfactory for all cases considered. The comparison has shown, however, that LES is clearly more suited than RANS methods and has great potential for calculating these complex ﬂows, but that it is of course more expensive than the RANS method. However, this disadvantage will become less important in future as TeraFlops machines will be available soon (being several hundred times faster than the SNI S600/20 vector computer mentioned above) and also the computing power of workstations and PCs will keep growing rapidly. Further development and testing of LES methods is certainly necessary, but with these advances in computing power, LES will soon become aﬀordable and ready for practical applications.

Key to Workshop Submissions KAWAMU, ST5 = Kawamura/Kawashima (Science U. Tokyo), UMIST = University of Manchester Institute of Technology (Archambeau/ Leschziner), UKAHY2, UK1, UK3 = University of Karlsruhe (Breuer, Pourqui´e, Rodi), TAMU2 = Tamura/Itoh/Takakuwa (Tokyo Inst. Technology), NT7 = Niigata Inst. Technology (Mochida/Tominaga) and I.I.S., U. Tokyo (Murakami/Iizuka) UOI = U. of Illinois (Wang/Vanka), IS3 = Inst. Industrial Science, U. Tokyo (Kogaki/Kobayashi/Taniguchi), TIT = Tokyo Inst. Technology (Tamura/Nozuwa),

References Achenbach, E., (1968), ‘Distribution of Local Pressure and Skin Friction Around a Circular Cylinder in Cross-Flow up to Re = 5 × 106 ’, J. Fluid Mech. 34 625–639. Beaudan, P. and Moin, P., (1994), ‘Numerical Experiments on the Flow past a Circular Cylinder at Sub-Critical Reynolds Number’, Report No. TF-62, Mech. Eng. Dept., Stanford University, December 1994. Bosch, G., (1995), Experimentelle und theoretische Untersuchung der instation¨ aren Str¨ omung um zylindrische Strukturen, Dissertation, University of Karlsruhe.

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Breuer, M., (1998), ‘Large-Eddy Simulation of the Sub-Critical Flow Past a Circular Cylinder: Numerical and Modeling Aspects’, Int. J. Num. Meth. in Fluids 28 1281– 1302. Breuer, M., (1999), ‘A Challenging Test Case for Large-Eddy Simulation: HighReynolds-Number Circular-Cylinder Flow’. In Proc. Turbulence and Shear Flow Phenomena 1, S. Banerjee and J.K. Eaton (eds.), Begell House Inc., New York, 735–740; also in Int. J. Heat and Fluid Flow 21, 648–654 (2000). Breuer, M. and Rodi, W., (1996), ‘Large-Eddy Simulation of Complex Turbulent Flows of Practical Interest’. In Notes on Numerical Fluid Mechanics 52, E.H. Hirschel (ed.), Vieweg Verlag, 258–274. Cantwell, B. and Coles, D., (1983), ‘An Experimental Study of Entrainment and Transport in the Turbulent Near Wake of a Circular Cylinder’, J. Fluid Mech. 136 321–374. Davidson, L., (1997), ‘Large-Eddy Simulation: A Dynamic One-Equation Subgrid Model for Three-Dimensional Recirculating Flow’. In Proc. 11th Int. Symp. on Turbulent Shear Flows 3 26.1–26.6, Grenoble. Franke, R., (1991), Numerische Berechnung der instation¨ aren Wirbelabl¨ osung hinter zylindrischen K¨ orpern, Dissertation, University of Karlsruhe. Franke, R. and Rodi, W., (1993), ‘Calculation of Vortex Shedding past a Square Cylinder with Various Turbulence Models’. In Turbulent Shear Flows 8, U. Schumann et al. (eds.), Springer Verlag. Fr¨ ohlich, J. and Rodi, W., (2001), ‘Introduction to Large-Eddy Simulation of Turbulent Flows’. This volume. Fr¨ ohlich, J., Rodi, W., Kessler, Ph., Parpais, S., Bertoglio, J.B. and Laurence, D., (1998), ‘Large-Eddy Simulation of Flow Around Circular Cylinders on Structured and Unstructured Grids’. In Notes on Numerical Fluid Mechanics 66, E.H. Hirschel (ed.), Vieweg Verlag, 319–338. Germano, M., Piomelli, U., Moin, P., and Cabot, W.H., (1991), ‘A Dynamic SubgridScale Eddy-Viscosity Model’, Phys. Fluids A, 3 1760–1765. Hellsten, A. and Rautaheimo, P., (eds.), (1999), Proceedings 8th ERCOFTAC-IAHRCOST Workshop on Reﬁned Turbulence Modelling, June 1999, Rept. 127, Helsinki University of Technology. Kato, M. and Launder, B.E., (1993), ‘The Modelling of Turbulent Flow around Stationary and Vibrating Square Cylinders’, Proc. 9th Symp. Turbulent Shear Flows, Kyoto. Kobayashi, T., Taniguchi, N., Tsubokura, M., and Kogaki, T., (1996), ‘The Veriﬁcation of the SGS Model and Application of LES to a Practical Engineering Problem’. In Advances in Turbulence Research, Korea University Seoul, Korea, 1–20. Kogaki, T., Kobayashi, T. and Taniguchi, N., (1997), ‘LES of Flow around a Square Cylinder’. In Direct and Large-Eddy Simulation II, J.P. Chollet, P.R. Voke, L. Kleiser (eds.), ERCOFTAC Series, Vol. 5, Kluwer, 401–408. Krajnovic, S., M¨ uller, D. and Davidson, L., (1999), ‘Comparison of Two One-Equation Subgrid Models in Recirculating Flows’. In Direct and Large-Eddy Simulation III, P. Voke, N.D Sandham, L. Kleiser (eds.), Kluwer, 63–74.

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Kravchenko, A.G. and Moin, P., (2000), ‘Numerical Studies of Flow Over a Circular Cylinder at ReD = 3900’, Phys. Fluids 12 403–417. Lakehal, D. and Rodi, W., (1997), ‘Calculation of the Flow past a Surface-Mounted Cube with Two-Layer Turbulence Models’, J. Wind Eng. and Ind. Aerodyn. 67–68 65–78. 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. Lourenco, L.M. and Shih, C., (1993), ‘Characteristics of the Plane Turbulent Near Wake of a Circular Cylinder, a Particle Image Velocimetry Study’, Private Communication, cited in Beaudan and Moin (1994). Lyn, D.A. and Rodi, W., (1994), ‘The Flapping Shear Layer Formed by Flow Separation from the Forward Corner of a Square Cylinder’, J. Fluid Mech. 267 353–376. Lyn, D.A., Einav, S., Rodi, W. and Park, J.-H., (1995), ‘A Laser-Doppler Velocimetry Study of Ensemble-Averaged Characteristics of the Turbulent Near Wake of a Square Cylinder’, J. Fluid Mech. 304 285–319. Martinuzzi, R., (1992), Experimentelle Untersuchung der Umstr¨ omung wandgebundener, rechteckiger, prismatischer Hindernisse, Dissertation, Universit¨at ErlangenN¨ urnberg. Martinuzzi, R. and Tropea, C., (1993), ‘The Flow around a Surface-Mounted Prismatic Obstacle Placed in a Fully Developed Channel Flow’, J. Fluids Eng. 115 85–92. Mathey, M., Fr¨ ohlich, J. and Rodi, W., (1998), ‘Large-Eddy Simulation of the Flow Over a Matrix of Surface-Mounted Cubes’, Proc. Workshop on Industrial and Environmental Applications of Direct and Large-Eddy Simulations, August 5-7, 1998, Istanbul. Mathey, M., Fr¨ ohlich, J. and Rodi, W., (1999), ‘LES of Heat Transfer in Turbulent Flow Over a Wall-Mounted Matrix of Cubes’. In Direct and Large-Eddy Simulations III, P. Voke, N.D. Sandham and L. Kleiser (eds.), Kluwer, 51–62. Meinders, E.R. and Hanjalic’, K., (1998), ‘Vortex Structure and Heat Transfer in Turbulent Flow Over a Wall-Mounted Matrix of Cubes’, Proceedings of the Turbulent Heat Transfer II Conference, Manchester, UK, June 1998. Menon, S. and Kim, W.-W., (1996), ‘High-Reynolds-Number Flow Simulations Using the Localised Dynamic Subgrid-Scale Model’, AIAA paper 96–0425. Mittal, R. and Moin, P., (1997), ‘Suitability of Upwind-Biased Finite-Diﬀerence Schemes for Large-Eddy Simulation of Turbulent Flows’, AIAA J., 35 1415–1417. Murakami, S., Mochida, A., and Sakamoto, S., (1995), ‘CFD Analysis of WindStructure Interaction for Oscillating Square Cylinder’, Proc. 9th Ind. Conf. on Wind Engineering, New Delhi, India. Ong, L. and Wallace, J., (1996), ‘The Velocity Field of the Turbulent Very-Near Wake of a Circular Cylinder’, Experiments in Fluids 20 441. Pourqui´e, M., (1996), Private Communication. Rodi, W., (1993), ‘On the Simulation of Turbulent Flow past Bluﬀ Bodies’, J. Wind Eng. and Ind. Aerodyn. 46−47 3–9.

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Rodi, W., Ferziger, H.J., Breuer, M., Pourqui´e, M., (1995), Proc. Workshop on LargeEddy Simulation of Flows past Bluﬀ Bodies, Rottach-Egern, Germany, June 1995. Rodi, W., Ferziger, J.H., Breuer, M. and Pourqui´e, M., (1997), ‘Status of Large-Eddy Simulation: Results of a Workshop’, J. Fluids Eng. 119 248-262. Smagorinsky, J.S., (1963), ‘General Circulation Experiments with the Primitive Equations, Part 1, Basic Experiments’, Mon. Weather Rev., 91 99–164. Sohankar, A., (1998), Numerical Study of Laminar, Transitional and Turbulent Flows Past Rectangular Cylinders, PhD Thesis, Dept. of Thermo- and Fluid Dynamics, Chalmers University of Technology, Gothenburg. Sohankar, A., Davidson, L. and Norberg, C., (1999), ‘A Dynamic One-Equation Subgrid Model for Simulation of Flow Around a Square Cylinder’. In Engineering Turbulence Modelling and Experiments 4, W. Rodi and D. Laurence (eds.), Elsevier, 227–236. Tamura, T., Ohta, I. and Kuwahara, K., (1990), ‘On the Reliability of Two-Dimensional Simulation for Unsteady Flows around a Cylinder-Type Structure’, J. Wind Eng. and Ind. Aerodyn. 35 275–298. Voke, P.K., (1997), ‘Flow past a Square Cylinder: Test Case LES 2’. In Direct and Large-Eddy Simulation II, J.P. Chollet, P.R. Voke and L. Kleiser (eds.), ERCOFTAC Series, Vol. 5, Kluwer, 355–373. Werner, H. and Wengle, H., (1989), ‘Large-Eddy Simulation of the Flow over a Square Rib in a Channel’. In Proc. 7th Symp. Turbulent Shear Flows, Stanford University, 10.2.1–10.2.6. Xia, M., Karniadakis, G., Karamanous, G. and Sherwin, S.J. (1998), ‘Issues in LES of Wake Flows’, AIAA Paper 98-2893.

13 Large Eddy Simulation of Industrial Flows? D. Laurence Abstract This chapter develops the somewhat controversial opinion that, at present and in the near future, industry is not likely to use LES for actual engineering applications, despite, or rather because of, the daily use of (RANS) CFD. While the eﬀect of subgrid-scale models is small, there is a need for further improvement of numerical schemes in complex geometry, and some steps in this direction are presented. Finally some test-cases of industrial relevance are presented in the frame of acoustics and ﬂuid-structure interaction. These are among the few niches left vacant by RANS where industry might spend the large computing and staﬀ-time resources required for conducting a complex LES.

1

Introduction

Despite widespread academic use of LES, there are still very few ‘industrial’ LES calculations. In the year 2000, following a decade in which LES has been touted as the route for by-passing turbulence model limitations, it still needs to be shown that industry is ready to invest its own resources, manpower and computing, to resolve, via LES, a relevant engineering problem, without either the incentive of external funding, or for the sake of pure research investigations. We do not count here as ‘industrial’ the proactive government initiatives such as state- or EU-funded projects, or the ASCI project in the US. Industry has taken advantage of increased computer power to model more realistic geometries and complex physical processes rather than resorting to advanced turbulence modelling, either LES or RANS (Reynolds-averaged Navier– Stokes). Even second-moment closures are often regarded as too demanding. In power plants for instance, it has been realised that isolating a component for a numerical simulation always results in crude approximations concerning the inlet conditions. In most research experiments great eﬀorts are devoted to obtaining clean and well understood inlet conditions, but still on many occasions these are not well enough known in the computor’s opinion (ERCOFTAC workshops, Rodi et al. 1998). In practice no section of a plant contains honeycombs or careful contractions, or a straight pipe 100 diameters long, to insure fully developed ﬂow that would allow a component-by-component CFD analysis. The numerous bends, sudden expansions, obstacles, etc. always induce strong persistent secondary motions which need to be taken into account, even if attention is restricted to a limited sub-component of the plant (Archambeau et al. 1997). 392

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Thus, in practice, reliability of industrial simulations has been increased by extending them to entire systems, including interactions between components. While modelling is limited to eddy viscosity models, daily industrial calculations make use of the many advances of CFD such as unstructured grids, automatic mesh reﬁnement, non-conforming mapping, and the ability to cope with millions of nodes. The increases in computer power that make the LES of simple industrial problems possible today, have been used instead to analyse more and more complex geometries. Ironically, while academia seems to have unlimited access to supercomputers and is using them to show that fairly complex geometries can be tackled via LES, the same supercomputers available in industry are seldom used for such LES applications. At EDF R&D for instance, where LES has been investigated since 1980, CFD is now the main tool for thermal-hydraulics studies, replacing experiments, with numerous RANS calculations needing millions of nodes and weeks of CPU time naturally having priority over LES investigations. Demand for production simulations is steadily increasing, and whatever extra computer power is made available is immediately used up by RANS calculations for ﬁner meshes, extended geometries, and more complex phenomena (two-phase ﬂows, chemistry, combustion). Thus, one can wonder whether LES will not always remain a pure research tool, just like DNS, because industry’s priority is in other CFD areas. Spalart (1999) developed a similar argument in the aerospace industry, quoting unreachable numbers for the LES of airfoils, but we stress here that doubts concerning industrial applicability of LES is not limited to high Reynolds number industries. A ﬁnal argument is that cost of staﬀ time, as opposed to CPU time, is not continuously decreasing. Whereas signiﬁcant productivity gains in RANS-CFD applications can be achieved, thanks to commercial packages and user friendly environments that enable a simulation to be conducted from scratch within a week, an LES study is nowhere near a routine job. It requires at least 6 months work for staﬀ who are highly trained in the subtleties of turbulence and pitfalls of numerical analysis, or more often, 3 years of labour for a PhD student. The LES calculations reported herein, are of industrial relevance, but are still only research test cases. The two main obstacles preventing industrial LES becoming a reality are the poor energy conservation features of numerical schemes applicable to complex geometries, and the lack of suitable LES boundary conditions, not only at walls, but for inlet conditions other than periodic, and ideal fully developed pipe or boundary layer ﬂows.

2 Numerical schemes for LES in complex geometry Standard numerical schemes developed for the RANS equations are not suitable for LES. They are set up to conserve mass and momentum but not the kinetic energy of the resolved ﬁeld. To ensure stability, the truncation error

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is dissipative, resulting in an artiﬁcial energy dissipation that is often of the same order as that of a subgrid-scale model. It is then useless to seek improvements from advanced subgrid-scale models, such as the dynamic model. Indeed, in the buﬀer layer for instance, the latter naturally makes the SGS viscosity vanish at the appropriate rate, but the numerical dissipation remains. Breuer (1998), applied LES to the ﬂow over a circular cylinder with a number of convection schemes, centered second- and fourth-order, third-order, QUICK and blending schemes. Centered schemes performed best, with little diﬀerence between second- and fourth-order. He concluded that low numerical dissipation produced by a scheme is more crucial for LES than its formal accuracy, central second-order schemes are preferable to even higher-order hybrid schemes. This is presumably because dispersive errors cancel out through the statistical averaging process (the exact space-time location of an eddy is of little importance, but proper conservation of its intensity is essential), whereas even-order errors introduce a systematic bias. In the same way, lattice Boltzmann simulations can be thought of as extremely poor in terms of conventional numerical analysis (zero-order error on the instantaneous ﬁeld), yet they provide excellent statistical results. At EDF R&D, LES has been used since 1980 as a research tool, for academic ﬂows with complex physics, using pseudo-spectral methods and non-eddy viscosity types of SGS allowing backscatter etc. Starting in 1996 an attempt was made at EDF R&D to apply a general purpose ﬁnite element code with unstructured grids, N3S, to LES (Rollet-Miet et al. 1999). The standard convection scheme for N3S was the method of characteristics, a combination of curvilinear streamline upwinding and high-order interpolation. The scheme performed well for transient RANS calculations, but was found, as a result of tests on decaying grid turbulence, to be insuﬃcient for LES. The energy spectrum was well predicted, but the dynamic model yielded a Smagorinsky constant for the viscosity that was signiﬁcantly lower than usual to make up for the numerical dissipation. Interestingly the dynamic procedure seemed to adapt to the weaknesses of the numerics: the larger the timestep, the lower the viscosity constant. On average, the numerical scheme was responsible for 50% of the total dissipation. Energy spectra were correct but this level of numerical dissipation is not acceptable in regions of vanishing subgrid-scale dissipation such as the near-wall layers or transitional ﬂows. Finally the code was changed to the classical Adams–Bashforth Crank– Nicholson (AB–CN) scheme with a centred discretization in space, now entailing a severe time-step limitation for complex geometry (CFL of the order of 0.2). Another severe requirement for the numerical scheme is the need for higher accuracy with regard to pressure. Indeed in Fourier space, pressure acts as a limiter on the energy transfer created by the nonlinear term (when writing the Navier–Stokes equations in spectral space, pressure appears as a projection

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operator acting on the nonlinear term). In physical space, while in RANS the pressure is smoother than the velocity ﬁeld, in LES pressure variations occur down to the grid scale. This is obvious from considering vortex pairs in an array of Taylor vortices: the wavelength of the velocity spans 2 vortices, one clockwise and the other counter-clockwise, while the pressure wavelength is actually half that of the velocity; thus u = sin(κx) cos(κy), p = cos(2κx)+cos(2κy). To obtain equal accuracy for pressure as for velocity on such a vortex array, one should use more pressure nodes than velocity nodes. This is opposite from traditional ﬁnite element discretizations where pressure is given fewer degrees of freedom than the velocity! The classical P2–P1 triangular element for instance is parabolic in velocity and linear in pressure. The next improvement that was introduced in N3S–LES was to abandon the P2–P1 triangular element for a collocated discretization, heresy according to any ﬁnite element textbook. A number of LES calculations, sometimes quite coarse, have been successfully applied to bluﬀ body aerodynamics such as cubes, fences, or backsteps, whereas disappointing results are obtained for the LES of a simple channel ﬂow. Typically, wall-normal ﬂuctuations are underestimated and streamwise ﬂuctuations overestimated. In the RANS context one would point to the pressure-strain terms. i.e. not enough energy is transferred from the axial ﬂuctuations, directly generated by the mean velocity gradient, to the wallnormal ﬂuctuations which have no production. We believe this is due to insuﬃcient resolution of the pressure ﬁeld. Conﬁrmation of this can be found in the fact that underestimation of wall-normal ﬂuctuations is aggravated in the collocated approach which, compared to staggered arrangements, introduces pressure-gradient interpolation/smoothing operations. While the fully staggered arrangements seem to be ideal for LES, collocated arrangements have been developed on unstructured grids for reasons of simplicity, and currently seem to be the only option in commercial or industrial ﬁnite volume codes, a severe limitation for LES applications. We emphasise that industrial LES requires further work, mainly on numerical schemes for ﬁnite volumes on unstructured grids, and that in this new ﬁeld one should depart from the traditional know-how developed for unsteady RANS. Steps in this direction can be found, e.g. in Mahesh, Ruetsch and Moin (1999), who revert to the staggered-grid Harlow–Welch algorithm. An alternative for suppressing pressure oscillations on collocated grids without introducing damping was proposed by Dormy (1999), and a proven accurate, secondorder method on unstructured grids was suggested by Kobayashi, Pereira and Pereira (1999). Benhamadouche (2001) is currently developing at UMIST a numerical scheme, with staggered velocities and pressures on tetrahedra, that conserves energy by construction, by choosing space and time dicretisations such that the steps leading to the Bernoulli theorem are veriﬁed in the discrete sense. With this new scheme, and starting with a grid turbulence spectrum, an LES could be conducted, without any viscosity, until all energy is trans-

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ferred to the highest wave-number reproducing the theoretical result of a κ2 spectrum (starting from κ−5/3 , where κ is the wave number). Only a few words are needed on implementation of subgrid-scale-models or ﬁlters for unstructured grids, since the ingredients usually exist in general purpose codes already. For example the dynamic model (Germano et al. 1991) on an unstructured mesh, introduces no speciﬁc diﬃculties. A top-hat ﬁlter, G, can be constructed from a combination of identity (or mass matrix) and existing Laplacian operators. For instance, in one dimension, along the x-axis, the ﬁltered value of the velocity u at node i can be written:

∂2 1 1 1 G(ui ) = (ui−1 + ui + ui+1 ) = ui + (ui−1 − 2ui + ui+1 ) ≈ ui + δx2 u 3 3 3 ∂x2

. i

Better still, when a geometrical multigrid feature is present, the restrictionprojection operators can be used as ﬁlters, and it suﬃces to use the data stored at diﬀerent grid levels to apply the dynamic procedure. Finally the size, ∆, of the ﬁlter on an arbitrary control volume can be taken as the cube root, r, of the volume, i.e. ∆ = r. A geometric coeﬃcient, α, might be expected, but in the dynamic model framework this is of no importance because it can be considered that instead of dynamically tuning the Smagorinsky constant, CS , the procedure is tuning the product CS α, i.e. the SGS viscosity is rewritten as νt = (CS ∆)2 S 2 = (CS αr)2 S 2 , where S is the strain rate, and then the dynamic procedure is applied to CS = CS α, instead of CS .

3

Application to a Tube Bundle and Cylinder

The ﬂow in a portion of a large tube bundle, where periodic boundary conditions can be applied in all 3 directions, is an ideal case for LES (Rollet-Miet et al. 1999). Industrial interest lies in vibrations caused by ﬂuid-structure coupling or large temperature ﬂuctuations that eventually lead to thermal fatigue. The geometry is relatively simple, yet the ﬂow experiences complex strains, making this an attractive test case. The experiment provides data on mean velocities and Reynolds stresses for the ﬂow through an staggered array of tubes (see Figure 1). The Reynolds number based on the bulk velocity in the sub-channel between the tubes and the tube diameter is 40000. This ﬂow was considered during two ERCOFTAC/IAHR workshops on Turbulence Modelling and in both cases RANS models performed poorly. A ﬁrst run was computed with the Smagorinsky model and, after initialisation, statistics were accumulated over three domain-through-ﬂow times; this was found to be suﬃcient since span-wise averaging and symmetries are used in gathering statistics. For the second run, with the localised dynamic procedure of Piomelli and Liu (1994), statistics were computed over ﬁve domain-throughﬂow times.

[13] Large eddy simulation of industrial ﬂows?

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D

V0 x

L

y

Figure 1: Computation sub-domain. The value of the Smagorinsky constant was set to CS = 0.065, while the dynamic model yielded a somewhat higher average value of this coeﬃcient, with great spatial variability. Large values occurred in the separated shear layers, and slightly negative values appear near the wall on the upstream half of the tube. For the mean velocity, the agreement of the LES with all available data is excellent, and the results obtained from the two subgrid-scale models cannot be distinguished. For the Reynolds stresses, the overall agreement is fairly good. Figure 2 shows the results in the wake-to-impingement axis and on the cross section just behind the tube, a region where RANS models performed poorly. The striking and unusual feature observed on the wake axis is that the transverse velocity ﬂuctuations are far larger than the axial ones. Thus as the ﬂow approaches the stagnation point, u2 < v 2 , and since ∂U ∂x < 0, the production of the stresses and kinetic energy are ∂U ∂U > 0; Pvv = +2v 2 ∂x ∂x

∂U = − u2 − v 2 < 0, ∂x

Puu = −2u2 Pk

where U is the mean velocity. We notice from the experiment and LES that u2 is fairly constant (production balances dissipation) in the impingement region while v 2 decreases rapidly due to dissipation and negative production. Since u2 < v 2 , the kinetic energy is also decreasing, in contrast to what is found at a stagnation point on a single bluﬀ body, or an impinging jet. Very near the stagnation point, the kinematic blockage by the wall forces u2 ﬂuctuations to be converted into v 2 (wall echo eﬀect), and the dynamic model seems to capture the peak in v 2 better, although there is only one experimental point to suggest the existence of such a peak. On Figure 3 is shown the LES of a single cylinder at Re = 3900. The unstructured N3S–LES mesh of Kessler et al. uses only about one tenth of the nodes of the structured mesh adopted by Fr¨ ohlich et al., but the extra cost of using unstructured grids in this case just about balances the saving in nodes.

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Figure 2: LES of the ﬂow across a tube bundle. LES with Smagorinsky model (solid lines), LES with dynamic model (dashed lines), experiment (symbols).

As commented earlier, in an industrial environment, only a few tens of shedding cycles could be aﬀorded, while Fr¨ ohlich, with virtually unlimited access to academic supercomputers, was able to conduct the averaging processes over thousands of cycles. Still the mean ﬂow features of the structured and unstructured LES are similar and in good agreement with the results of Beaudan and Moin (1994).

4

Fluid-Structure Coupling

If industry decides to invest in LES, it is probably not for the higher accuracy of LES predictions in simple ﬂows, but rather for speciﬁc problems where it seems to be the only approach. For instance, in a nuclear power plant the control rods, which act as brakes, are very long and thin and they could be

[13] Large eddy simulation of industrial ﬂows?

399

Figure 3: LES of the ﬂow around a single Cylinder at Re = 3900 (Fr¨ olich et al. 1998). LESOCC ﬁnite volume code with 1,3 million nodes (left), and N3S–LES unstructured code with 176000 nodes (right). Streamlines for time averaged ﬂow showing secondary recirculations. subject to ﬂow induced vibrations; this is typically an area where the extra cost of LES is negligible in view of the massive investment in safety studies. The conﬁguration described in Figure 4, is a simpliﬁed mock-up of the actual geometry. A tube is submitted to a turbulent three-dimensional axial ﬂow crossing a perforated plate located just above the rod and generating high turbulent ﬂuctuations near the tube. Fluid-structure eﬀects are restricted to added mass eﬀects due to the ﬂuid (in the LES the tube is assumed to be a solid non-ﬁxed tube). The calculation by Longatte et al. (2001) yields reasonably good predictions of the spectrum of the ﬂuctuating ﬂuid-force on the control rod (Figure 5).

5

Acoustic Source Terms

The right-hand side of the acoustic equation for pressure contains derivatives of the instantaneous acceleration which are extremely diﬃcult to model in terms of RANS, but can be extracted from an LES and inserted into an acoustic propagation code such as EOLE (the LES cannot directly predict the noise at a distance from the source with suﬃcient accuracy since the acoustic pressure is a very small fraction of the calculated pressure). Figure 6 presents an LES of the CLARINETTE experiment at EDF (Lafon 1997). A diaphragm is inserted

400

Laurence φ150

φ254

φ150

400

948

Tube

φ

260

15

Figure 4: Mock up of a control rod. Geometry on the left and instantaneous velocity from LES on the right.

Figure 5: Turbulent force spectra ((N/m) 2/Hz) computed numerically (lines) and measured experimentally (points) at three diﬀerent locations along the tube at 0.3m, 0.6m and 0.9m from the plate. in a rectangular channel and strong noise sources appear on the edges of the jet. The noise predicted by N3S–LES and subsequent acoustic propagation calculation agree well with the experimentally established behaviour, namely proportional to the fourth power of the bulk velocity. Moreover, for the chosen ratio of diaphragm-to-channel width, a Coanda eﬀect appears, i.e. an early reattachment of the jet to one of the walls although the geometry is totally

[13] Large eddy simulation of industrial ﬂows?

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symmetrical. From the LES, an animation explains this phenomenon: once the jet attaches to one side, an intense recirculation appears, and feeds back perturbations into the lower edge of the jet (Figure 6). The mixing on this side is then stronger than on the free jet side. In other words the ‘friction’ of the jet on the recirculation side is larger than on the free side thus deﬂecting the jet towards the lower wall. This research is part of the PREDIT programme, a collaboration with the rail and car industry, and sponsored by the French government. Another geometry that has been studied consists of a wall-mounted rectangular box (Figure 7). Here the acoustic sources are mainly located in the shear layer separating from the leading edge, whereas contributions from the trailing corner and recirculation are signiﬁcantly weaker. This is valuable information since, while the overall noise is easily measured, identifying precisely the noise sources experimentally is much more challenging. A similar geometry also simulated using N3S–LES is the rib-roughened channel encountered in turbine blade cooling. RANS models tended to show some instability in this case, with the ﬂow structure oscillating between two and three recirculation bubbles. The time-averaged streamlines from the LES exhibit three recirculations, but this structure could never be found on the instantaneous ﬁelds (Figure 8). Owing to the high turbulence intensity developing on this inﬁnite series of ribs (periodicity is assumed), the ﬂow between ribs is extremely complex, with little relation to the time-averaged solution. Time-averaged plots sometimes only represent the artefact of taking a statistical average, signiﬁcantly diﬀerent from the actual ﬂow when the turbulence intensity is high.

6

Toward Industrial LES

The previous applications are very simpliﬁed mock-ups of actual problems. As explained in the introduction, for most engineering applications, it is usually not possible to isolate a sub-domain, where LES might be attempted, from the complex three-dimensional ﬂow in which it is embedded (a car mirror, a control rod in a nuclear vessel, an aircraft engine housing etc.). An LES used as a zoom eﬀect within a complex industrial apparatus would need to be embedded and coupled with a global RANS calculation. A promising solution is the Detached Eddy Simulation (DES) of Spalart (1999, 2000) whereby the RANS eddy viscosity is forced to reduce to a subgrid scale viscosity locally where a transition to LES is desired. The method is eﬃcient for detaching shear layers around bluﬀ bodies which exhibit a natural instability. A more general method would consist of generating instantaneous structures using all the information available from the RANS simulation, since, in general, it is not suﬃcient simply to reduce the viscosity and expect these structures to appear, without a considerable space- or time-lag to enable small instabilities to develop into actual turbulence. Such synthetic turbulence boundary conditions are currently being developed, for example by Cabot and Moin (1999) for

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Figure 6: LES of the ﬂow through a diaphragm (CLARINETTE mock-up), instantaneous velocity levels, and lower half of the unstructured mesh (before

[13] Large eddy simulation of industrial ﬂows?

403

Figure 7: LES of a wall mounted rectangular body.

Figure 8: LES of the ﬂow across over a ribbed wall. Time averaged streamlines (upper ﬁgure), and perspective view of the intensity of the instantaneous velocity ﬁeld; the iso-surface in white/light grey corresponds to a zero value of the streamwise velocity. On the near side, the ﬂow reattaches early, showing a hole in the iso-surface, while on the far side, the two recirculation bubbles are connected into a single one ﬁlling the cavity between the ribs.

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Figure 9: The 2D velocity ﬁeld in the inﬂow plane of the channel. wall parallel boundaries, or by Sergent et al. (2000) for inﬂow-like conditions (Figure 9). In the latter, a set of random vortices is generated that match the statistics given by the RANS calculation. Coherence of these synthetic structures in space and time is essential for them to be sustained in the LES calculation. Such a RANS-embedded LES calculation could then be used in a manner similar to the coastal engineering situation where it is customary to overlay a local calculation of, for example, a harbour, into a larger scale simulation of coastal currents and tides. Before it can be generalised and used on an industrial basis, the method would need to be made automatic via optimal control, that is, self-adjustment of the synthetic turbulence to match the statistics of the RANS calculation in an overlap region.

Conclusions It is now possible to apply LES to moderately complex geometry, at least beyond homogeneous and channel ﬂows, but, at the same time, industry has progressed into applying RANS to real-life problems and it is not certain that it will resort to LES in the near future, apart from speciﬁc problems such as acoustics or ﬂuid-structure interaction. Progress concerning conservative

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properties of numerical schemes for unstructured grids is however promising, as well as work on ‘synthetic turbulence’ for boundary conditions or embedded calculations that would allow LES to be used as a zoom eﬀect in a region of speciﬁc interest. CPU costs are decreasing, but staﬀ time to conduct and process the data of an LES is still a limiting factor, as well as turn-over time. The applications presented above still remain ‘academic’ test cases by industry standards, and will require some years before a panorama of industrial applications can be presented (without a question mark).

References Archambeau, F., Laurence, D., Martin, A., Maupu, V., Pot, G. (1997). ‘Reﬁned turbulence modelling for power generetion industry’ J. Hydraulic Research 35 (6), 749–772. Benhamadouche, S., Laurence, D. (2001). ‘An Energy conserving scheme for LES on unstructured grids’. Int. Conf. on Num. Methods in Fluid Dynamics, Oxford. Breuer, M. (1998). ‘LES of the subcritical ﬂow past a circular cylinder : numerical and modeling aspects’, Int. J. Numerical Methods in Fluids 28 1281-1302. Beaudan, P., Moin, P. (1994). ‘Numerical experiments on the ﬂow past a circular cylinder at sub critical Re number’. Reports TF-62 Stanford University, Mechanical Engineering Department. Bonnin, J.C., Buchal, T., Rodi, W. (1996). ‘Databases for testing of calculation methods for turbulent ﬂows’, ERCOFTAC Bulletin 28 48–54. Cabot, W., Moin, P. (1999). ‘Approximate wall boundary conditions in the LES of high Re number ﬂow’, Flow Turbulence and Combustion 63 269-291. Dormy, E. (1999). ‘An accurate compact treatment of pressure for colocated variables’, J. Computational Physics 151 676–683. Fr¨ olich, J., Rodi, W., Kessler, P., Parappis, S., Bertoglio, J.-P., Laurence, D. (1998). ‘Large eddy simulation of ﬂow around circular cylinders on structured and unstructured grids’. In Notes on Numerical Fluid Mechanics 66, E.H. Hirschel (ed.), Vieweg 319–338. Germano, M., Piomelli, U., Moin, P., Cabot, W. (1991). ‘A dynamic subgrid scale eddy viscosity model’, Phys. Fluids A. 3 1760–1765. Kobayashi, M.H., Pereira, J.M.C, Pereira, J.C.F. (1999). ‘A conservative ﬁnite-volume second-order accurate projection methods on hybrid unstructured grids’, J. Computational Physics 150 40–75. Lafon, P. (1997). ‘Noise of conﬁned ﬂows’, EDF R&D report HP 53/97/032. Longatte, E. et al. (2001). ‘Application of large eddy simulation to a ﬂow induced vibration problem’. Submitted to ASME PVP (Pressure Vessel & Piping Conference), Atlanta. Mahesh, K., Ruetsch, G. R. and Moin, P. (1999). ‘Towards LES in complex geometries’, Center for Turbulence Research Annual Briefs, 379-387.

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Martin, A., Alvarez, D., Cases, F. (1997). ‘3D calculations of 900Mw PWR Vessel, Accurate RCP start-up Flow’, 5th Int. Conference on Nuclear Engineering, Nice. Rodi, W., Ferziger, J., Breuer, M., Pourqui´e M. (1997). ‘Status of large eddy simulation; results of a workshop’, J. Fluid Eng. 119 248–262. Rollet-Miet, P., Laurence, D. (1995). ‘Large eddy simulation with unstructured grids and ﬁnite elements, Turbulent Shear Flow 10, Pennsylvania State University. Rollet-Miet, P., Laurence, D., Ferziger, J. (1999). ‘LES and RANS of turbulent ﬂow in tube bundles’, Int. J. Heat & Fluid Flow 20 (3), 241–254. Rodi W., Bonnin, J.C., Buchal, T., Laurence, D. (1998). ‘Testing of calculation methods for turbulent ﬂows: workshop results for 5 test cases’, Coll. Notes Internes DER EDF 98NB00004, ISSN 1161-0611. Sergent, E., Bertoglio, J.-P., Laurence, D. (2000). ‘Coupling between LES and RANS’, Euromech Coll. 412, Munich. Spalart, P.R. (1999). ‘Strategies for turbulence modelling and simulations’. In Eng. Turbulence Modelling and Experiments 4, W. Rodi, D. Laurence (eds.), Elsevier, 3–17. Spalart, P., Travin, A., Shur, M., Strelets, M. (2000). ‘Physical and numerical upgrades in the detached-eddy simulation of complex turbulent ﬂows’. Euromech Coll. 412, Munich.

14 Application of TCL Modelling to Stratiﬁed Flows T.J. Craft and B.E. Launder 1

Introduction

Chapter [3] has provided the rationale and the associated analysis for replacing the Basic Model of the pressure-strain process in second-moment closure by a more widely applicable approach – namely one satisfying the two-component limit (TCL). The present chapter will show that extending that model to include ﬂow problems where gravitational eﬀects are important also brings clear beneﬁts in terms of accuracy and, sometimes, reductions in computing times, too. However, not all closure problems in stably-stratiﬁed ﬂows can be resolved merely by ensuring that the non-dispersive pressure-containing correlations satisfy the two-component limit. If generation by mean shear is eﬀectively obliterated by the sink associated with the stable stratiﬁcation, the computed behaviour of the second moments becomes particularly sensitive to secondmoment transport processes. While convective transport is, of course, handled exactly at this closure level, the same is not true of diﬀusive transport. Usually a very simple gradient-diﬀusion model is adopted for these processes. Such a practice, however, is motivated by the desire to adopt a cheap and stable approximation for a process that is frequently of little importance. However, transport can be of great importance in strongly inhomogeneous, stably-stratiﬁed ﬂow – so something better must be provided. Section 2 focuses on modelling the extra pressure-containing correlations arising from buoyancy and provides a comparison of computational results from implementing this model both with experimental data and, where available, with the Basic Model. Then Section 3 goes on to consider a case of a stably-stratiﬁed mixing layer where a partial third-moment treatment is required. The analysis is ﬁrst developed before comparisons with experiments are discussed.

2 2.1

Closure Modelling for Stratiﬁed Flow The Second-Moment Equations

The exact second-moment transport equations, now containing buoyant terms, 407

408

Craft and Launder

may be written: Dui uj Dt

= − ui uk

∂Uj ∂Ui + uj uk ∂xk ∂xk

+

Pij

∂ui 1 ∂uj + p + ρ ∂xj ∂xi

φij

ui uj uk +

Gij

− 2v

∂ − ∂xk

1 ρ uj gi + ρ ui gj ρ

∂ui ∂uj ∂xk ∂xk

(2.1)

εij

puj pui ∂ui uj δki + δkj − ν ρ ρ ∂xk

.

dij

Usually the density ﬂuctuations, ρ , will be provoked by temperature or concentration ﬂuctuations, θ, and it is convenient to write: ρ = −α

ρθ , Θ

(2.2)

where α is the dimensionless volumetric expansion coeﬃcient:

Θ ∂ρ . α≡− ρ ∂Θ p (For an ideal gas, where Θ denotes temperature, α is of course equal to unity). The buoyant term may thus be rewritten: Gij ≡ −

α θuj gi + θui gj . Θ

The corresponding transport equation for θui then takes the form: Dθui Dt

∂Θ ∂Ui + uk ui = − uk θ ∂xk ∂xk

Piθ

p ∂θ + −εiθ + diθ , ρ ∂xi

−

αθ2 gi Θ Giθ

(2.3)

φiθ

while the mean square scalar variance is determined from the following transport equation: Dθ2 ∂Θ − 2εθ + dθ , (2.4) = −2uk θ Dt ∂xk where, as in (2.1), the terms in equations (2.3) and (2.4) denoted by ε and d signify dissipative and diﬀusive processes. The equation set (2.1)–(2.4) is, of course, unclosed, for the processes denoted φ, d and ε (representing the non-diﬀusive action of ﬂuctuating pressure,

[14] Application of TCL modelling to Stratiﬁed ﬂows

409

diﬀusion and dissipation) all require approximation. In this section, we focus especially on the modelling of φij and φiθ for, in the ﬂows to be considered, these are the vital processes to be approximated. As a preliminary, however, the following paragraphs indicate brieﬂy the practices adopted for the dissipation and diﬀusion processes.

2.2

Modelling the Dissipation and Diﬀusion Processes

For the dissipation terms, it is convenient, following Lumley (1978), to assume local isotropy, which implies: 2 εij = δij ε, 3

εiθ = 0;

(2.5)

where ε is the viscous dissipation rate of turbulent kinetic energy, k. Any shortcomings of this assumption (since, for example, such an approximation for εij clearly does not satisfy the TCL) may be considered to be absorbed in modelling the ‘turbulence’ or ‘slow’ parts of φij and φiθ . Diﬀusional transport, in the inhomogeneous examples to follow, is approximated by the rudimentary generalized gradient-diﬀusion hypothesis (GGDH) of Daly and Harlow (1970). dαβ

∂ = cαβ ∂xm

k ∂αβ um un ε ∂xn

,

(2.6)

where α and β may denote either a velocity or a scalar ﬂuctuation. As in unstratiﬁed ﬂows, the diﬀusive coeﬃcient cαβ is assigned a value of about 0.2. The dissipation rate of kinetic energy appearing above is, following conventional practice, determined from the transport equation: Dε ∂ = Dt ∂xk

k ∂ε cε uk ul ε ∂xl

+

ε2 cε1 ε (Pkk + Gkk ) − cε2 . 2 k k

(2.7)

As signalled in [3], however, for some years the UMIST group has adopted a diﬀerent strategy in choosing the coeﬃcients cε1 and cε2 from that usually adopted with either the k-ε EVM or with simpler (older) forms of secondmoment closure. In most of the calculations reported below we have taken: cε1 = 1.0;

cε2 =

1.92 1/2

,

(2.8)

1 + 0.7A2 A25

where A2 is the second invariant of the dimensionless anisotropic Reynolds stress, aij aji and A25 ≡ max(A, 0.25).1 The quantity A denotes the ﬂatness invariant: 1−9(A2 −A3 )/8 where the third invariant, A3 , is just aij ajk aki . 1

410

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The change in the coeﬃcients from their erstwhile ‘standard’ values (cε1 = 1.44; cε2 = 1.92) has brought a number of beneﬁts in uniform-density ﬂows which [3] has already mentioned. In stratiﬁed ﬂows a major additional beneﬁt is that we have been able to retain the same coeﬃcient (cε1 ) multiplying both Pkk and Gkk irrespective of whether the ﬂow is orientated vertically or horizontally and whether the stratiﬁcation is stable or unstable. This is far from the case when the Basic Model is adopted (see, for example, Hossain and Rodi 1982, p. 142). The remaining quantity for whose determination a route must be prescribed, is the scalar dissipation rate, εθ . Some workers, including on occasions the UMIST group, have provided a transport equation for this quantity. In dealing with stratiﬁed ﬂows, however, we have consistently adopted the simpler practice of determining εθ from: εθ = ε

rθ2 , 2k

(2.9)

where the time-scale ratio2 r is related to the heat-ﬂux correlation coeﬃcient as 3 ui θ ui θ r = (1 + A2θ ) ; A2θ ≡ . (2.10) 2 kθ2

2.3

TCL Modelling of Pressure Interaction in Buoyancy Affected Flows

The treatment here parallels that reported in [3]. Note ﬁrst that in buoyant ﬂows the pressure containing correlations may be written: p φij ≡ ρ

∂ui ∂uj + ∂xj ∂xi

1 = φij1 + φij2 + 4π

αgk ∂θ Θ ∂xk

∂ui ∂uj + ∂xj ∂xi

dVol , |r|

φij3

p ∂θ 1 = φiθ1 + φiθ2 + φiθ ≡ ρ ∂xi 4π

(2.11) αgk ∂θ ∂θ dVol . Θ ∂xk ∂xi |r|

(2.12)

φiθ3

The ﬁnal terms in these equations provide the additional eﬀects due to the gravitational ﬁeld. The primes applied to θ and Θ merely indicate that the quantities are evaluated at x which is located a distance r from the point where φij , φiθ are to be determined (see Figure 1). While the volumetric integrals, in principle, need to be carried out over all space, in practice, the time-average correlations between velocities and temperatures at x and x diminish rapidly with distance so, except in situations 2

r is the reciprocal of the time-scale ratio R introduced in Chapter 2.

[14] Application of TCL modelling to Stratiﬁed ﬂows

O

411

Z ZZ r Z Z ~ Z 3 x x

Figure 1: Table 1: TCL models for φij3 and φiθ3 .

φij3 =

−

4 10

+

3 + 20

βi

− 14 βk

− 18

+ 18

3A2 80

1/ βi θ 2 3

(Gij − 1/3δij Gkk ) + 14 aij Gkk

um uj k

+ βj umkui um θ −

uk ui u θ + uk uj u θ + j i k k

um uj k

k2

um uk u θ 1 m 10 δij βk k

um un um uk u θ 1 n 20 δij βk k2

ui θ + umkui uj θ umkuk βk

uk ui um uj

3 − 40 βi

φiθ3 =

um uj k

+

uk uj um ui k2

+ βj umkui

βk um θ

um un u θ + 1 β um uk ui uj u θ n m k 4 k k2

− βk θ2 aik

where the ﬂow structure is undergoing very rapid change (as across the buﬀer layer), it seems reasonable to replace the mean ﬂow variable to be evaluated at x (in this case the scalar Θ ) by the value at x and thus remove it through the integral. Moreover, diﬀerentiation with respect to x and x is mutually independent. Consequently: φij3

βk = 4π

2 ∂ θ ui

∂ 2 θ uj + ∂rk ∂rj ∂rk ∂ri

dVol ≡ βk bikj + bjki . |r|

(2.13)

In the above βk is shorthand for αgk /Θ. Notice that the tensor bikj is precisely the same tensor as appeared earlier in φiθ2 , (see [3], equation (4.5)). Thus, by substituting the expressions for this tensor we obtain the model for φij3 given in Table 1, Craft (1991), Craft et al. (1996). Likewise: φiθ3 = βk bki .

(2.14)

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The quantity bki is simply: 1 4π

∂θ ∂θ dVol 1 =− ∂xk ∂xi |r| 4π

∂ 2 θ θ dVol . ∂rk ∂ri |r|

(2.15)

To devise an approximation for bki consistent with that obtained for the other two-point correlations, we suppose: bki = α1 θ2 δki + α2 θ2 aki + α3 θ2 amn anm δik + α4 θ2 aim akm .

(2.16)

By requiring that on contracting and performing the above integral, bkk = θ2 , one concludes that: 1 α1 = ; 3α3 + α4 = 0. 3 Then, on imposing the requirement that if u22 should fall to zero: G2θ + φ2θ3 = 0, one concludes that α2 = −1, α3 = α4 = 0. Thus ﬁnally: 1 φiθ3 = βi θ2 − βk θ2 aik . 3

(2.17)

It is to be noted that neither the expression for φij3 (Table 1) nor φiθ3 contain any empirical coeﬃcients. Thus, to an extent that is rarely found in closure modelling, comparisons with experimental or DNS data can be called predictions. In the case of isotropic turbulence, these formulae reduce to the quasi-isotropic forms (Launder 1975; Lumley 1975):

φij3 φiθ3

2.4

3 δij = − Gkk − Gkk 10 3 1 = − Giθ . 3

Some Applications of the Model to Buoyant Flows

In this section comparisons are made between the TCL and the Basic Models for a number of ﬂows. The TCL model was ﬁrst applied by Craft (1991) to consider homogeneous, horizontal, stably-stratiﬁed ﬂow that had been created by passing a uniform ﬂow past a screen of diﬀerentially heated horizontal rods, Webster (1964), Young (1975). The computations shown in Figure 2 have assumed local equilibrium (i.e. 12 (Pkk + Gkk ) = ε ) which is what has traditionally been adopted for this test case (though it may have been some way from the truth). Evidently, as the gradient Richardson number, Ri increases, the shear-stress correlation coeﬃcient decays, the turbulent Prandtl number increases and the horizontal heat-ﬂux correlation decreases moderately. The TCL model reproduces these measured responses at least as accurately as

[14] Application of TCL modelling to Stratiﬁed ﬂows

413

Figure 2: Second-moment sensitivity to buoyant stratiﬁcation in nominally horizontal, equilibrium, homogeneous ﬂow. (a) Shear-stress correlation coeﬃcient; (b) normalized turbulent Prandtl number; (c) Correlation coeﬃcient of horizontal heat ﬂux. — TCL model; – – – Basic model; symbols: experiments of Webster (1964), Young (1975). the Basic Model3 even though the two empirical coeﬃcients in the buoyant terms of the latter model were optimized by reference to these same data. As noted above, the TCL scheme, in contrast, has no adjustable coeﬃcients in the buoyant parts of φij and φiθ . A further application of the model, which focuses especially on the buoyant terms, has been reported by Van Haren (1992). The model is believed to be the same as that presented in Section 2.2 save that cε1 and cε2 took the constant 3

An extensive review of the capabilities of the Basic Model in buoyant ﬂow has been given by Launder (1989).

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Figure 3: Development of oscillatory waves in stably-stratiﬁed grid turbulence, from Van Haren (1992). In the right-hand graph, ‘realisable Rij -ε’ refers to the TCL model; in the left-hand graph ‘Rij -ε’ refers either to the Basic Model or a similar linear model. values 1.44 and 1.76 and a transport equation was solved for εθ . Van Haren considered the decay, in the absence of mean strain, of stably-stratiﬁed turbulence and, in particular, the oscillatory pattern that is known to be established during the decay due to reversals in sign of the vertical heat ﬂux. Van Haren generated 2-point EDQNM results of such a ﬂow and then tested how well various single-point closures did in reproducing the behaviour. Figure 3 compares

1/2

the time history of the normalized vertical heat ﬂux wθ/ w2 θ2 versus normalized time where N is the Brunt–V¨ ais¨al¨ a frequency. Quite clearly, the ‘extended’ k-ε model and a simple second-moment closure4 shown in the lefthand ﬁgure exhibit a signiﬁcantly too long period and a too rapid decay of the heat ﬂux compared with the EDQNM data. In contrast the new formulation shown in Figure 3b is rather successful at mimicking the EDQNM results. We turn now to inhomogeneous cases of self-preserving free shear ﬂows: the plane and axisymmetric vertical, buoyantly-driven plumes. Their spreading behaviour, obtained by Cresswell et al. (1989), is summarized in Table 2; for comparison the spreading rates of the plane and axisymmetric jets in stagnant surroundings (quoted in [3]) are also given. Comparisons are drawn both with experiments and with predictions obtained with the Basic Model. What is evident is that a far better overall agreement is achieved with the TCL invariant-dependent closure than with the Basic Model. Even where dis4 No details were provided of this scheme but it was presumably the Basic Model or the similar ‘quasi-isotropic’ model (Launder et al. 1975) frequently used by the group at the ECL.

[14] Application of TCL modelling to Stratiﬁed ﬂows

415

Table 2: Rate of growth of half-width for some self-preserving free shear ﬂows Flow

Basic Model

Plane plume Axisymmetric plume Plane jet Axisymmetric jet

0.078 0.088 0.100 0.105

Recommended experimental values 0.120 0.112 0.110 0.093

TCL Model

0.118 0.122 0.110 0.101

crepancies remain with experimental data, as in the case of the axisymmetric ﬂows, these are both smaller in magnitude than with the Basic Model and show a consistent behaviour across both the plume and the jet. The same workers extended their study to the case of a hot jet discharged vertically downwards into a cold water environment moving upwards at less than 2% of the jet velocity. Because of the net vertical force due to buoyancy on the less dense ﬂuid, the penetration length, of course, is crucially dependent on mixing. Figure 4a is a vector velocity plot in the vicinity of discharge showing the reversal in the jet direction while Figures 4b and 4c present respectively the shear stress and resultant mean velocity proﬁles. Evidently, the new model does signiﬁcantly better in capturing the eﬀects of buoyancy and shear in this quite complex recirculating ﬂow. Cresswell et al. (1989) were, moreover, agreeably surprised to report that, because the TCL model respected realizability, it led to a faster rate of numerical convergence and to a reduction of some 30% in computing time per run relative to the Basic Model, despite the greater algebraic complexity of the model itself.

3 3.1

Third-Moment Modelling and Application Modelling

As signalled in the Introduction, the presence of strongly stable stratiﬁcation may lead to a situation where the local state of turbulence is predominantly determined by diﬀusional transport from neighbouring, less stratiﬁed layers. In these circumstances the generalized gradient-diﬀusion hypothesis, usually adopted to represent diﬀusional transport of the second moments, is inadequate. The GGDH approximation can be regarded as a severe truncation of the third-moment equations for cases unaﬀected by buoyancy. Buoyant terms appear in the third-moment equations just as they do in the second-moment equations. Thus, it is possible to envisage a modest elaboration of the diﬀusion model merely by including buoyant terms; such a strategy was adopted

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Figure 4: Axisymmetric downward directed bouyancy jet from Cresswell et al. (1989). (a) Computed velocity vector plot; (b) turbulent shear stress one diameter downstream of the jet discharge; (c) mean velocity proﬁle one diameter downstream of the jet discharge. — TCL model; – – – Basic model; symbols: experiments.

[14] Application of TCL modelling to Stratiﬁed ﬂows

417

in Craft et al. (1996). However, as stable stratiﬁcation becomes increasingly dominant, it becomes more and more diﬃcult to make the system of equations, with their multiple intercouplings, converge. For that reason, it is now advocated that one should actually solve transport equations for some of the third-moments. But which ones? Even in a two-dimensional stratiﬁed shear ﬂow, there are twelve third-moments to be determined which represents a severe enlargement of the model. Accordingly, our current strategy has been to ﬁnd from transport equations simply those triple moments which contain density ﬂuctuations since these are the source of the gravitational eﬀects on the turbulence structure. Thus the elements of ui uj uk are obtained from the GGDH as in Section 2 while all other third moments are obtained from transport equations. Following Craft et al. (1997) these equations may be expressed in compact notation as: Duk uj θ Dt

1 2 + Pkjθ + Gkjθ + φkjθ − εkjθ + dkjθ = Pkjθ

(3.1)

Duk θ2 Dt

1 2 + Pkθθ + Gkθθ + φkθθ − εkθθ + dkθθ = Pkθθ

(3.2)

Dθ3 Dt

1 2 + Pθθθ + Gθθθ + φθθθ − εθθθ + dθθθ , = Pθθθ

(3.3)

where, in all equations, the P 1 s denote production by second-moment gradients, P 2 s arise from mean velocity or mean scalar gradients while the Gs are direct buoyant inﬂuences. The detailed form of these processes is given in Chapter [15], equation (4) but it is remarked that, within third-moment closure they can all be handled without modelling approximations. As before, the terms denoted φ, ε and d represent non-dispersive pressure interaction processes, dissipation and diﬀusion and all require closure approximations. Regarding the non-dispersive pressure interactions, for consistency with the modelling of the second-moments, one should logically apply the TCL strategy. However, the appropriate analysis does not appear to have been made. Accordingly, Craft et al. (1997, 2001) adopted precisely the same modelling strategy as is applied to the pressure-strain correlation when adopting the Basic Model; that is, in physical terms the pressure ﬂuctuations are regarded as an agent for promoting the return to isotropy of the turbulence ﬁeld. Thus:

ε 1 2 φkθα = −c1α θuk α − c2α Pkθα + Pkθα + Gkθα , k

(3.4)

where c1α = 1/0.075, c2α = 0.5, and α on the right-hand side denotes either uj or θ.5 5

Concerning the choice of coeﬃcients, c2α is taken as 0.5 to accord with the corresponding second-moment coeﬃcient while c1α is selected following Dekeyser and Launder (1985).

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Concerning dissipative processes, the proposals advocated by Dekeyser and Launder (1985), hereafter referred to as ‘DL’, are retained for εkjθ and εkθθ : k ∂ε δjk εjkθ = −c3ε θul ε ∂xl

(3.5)

k ∂εθ , εkθθ = −c3ε uk ul ε ∂xl

(3.6)

where c3ε = 0.1, following DL. For εθθθ (a process not considered by DL) the usual approximation, θ3 (3.7) εθθθ = 3 2r εθ , θ2 is adopted with the coeﬃcient r chosen as in equation (2.10). In earlier closure proposals for the diﬀusion of third moments, the usual route has been to express the fourth rank products in terms of products of the constituent quantities taken two at a time (Millionshtchikov 1941): αβγδ = αβ · γδ + αγ · βδ + αδ · βγ.

(3.8)

An argument against using such a form is that the Gaussian distribution of the ﬂuctuations on which the approximation, equation (3.8), rests, will be least accurate in regions where quadruple products are most inﬂuential, i.e. where the turbulence is strongly inhomogeneous. Possibly with such thoughts in mind, Kawamura et al. (1995) proposed instead that, to model ui uj uk um , departures from equation (3.8) should be accounted for by a gradient transport model. We apply this idea to all the quadruple products as follows:

∂ um uj uk θ ∂xm

∂ ≡ − um uj uk θ − um uj · uk θ + um uk · uj θ + um θ · uj uk ∂xm

∂ − um uj · uk θ + um uk · uj θ + um θ · uj uk . (3.9a) ∂xm

dkjθ ≡ −

The ﬁrst line of equation (3.9a) is then approximated by the GGDH, i.e,

djkθ = c3d1

∂ k ∂ um un uj uk θ − ∂xm ε ∂xn

∂ um uj · uk θ + um uk · uj θ + um θ · uj uk . ∂xm

(3.9b)

The other quadruple moments are modelled analogously as: dkθθ

∂ k ∂ ∂ 2 = c3d2 um un uk θ − um uk · θ2 + 2um θ · uk θ (3.10) ∂xm ε ∂xn ∂xm

dθθθ = c3d3

∂ k ∂ 3 ∂ um un θ − 3um θ · θ2 , ∂xm ε ∂xn ∂xm

(3.11)

[14] Application of TCL modelling to Stratiﬁed ﬂows

Splitter Plate

Fluid 1

Free Surface

0.560m y

Fluid 2

419

0.323m

x=0m

g

z x

Bottom Wall

x=5m

x=10m

x=40m

Figure 5: Geometry of the stably-stratiﬁed mixing layer of Uittenbogaard (1988). where c3d1 = c3d2 = c3d3 = 0.1. Introduction of the Kawamura modiﬁcation in modelling the above processes signiﬁcantly improved the stability behaviour and the quality of the computations.

3.2

Application

The test ﬂow chosen for examination was the salinity-stratiﬁed mixing layer measured at the Delft Hydraulics Laboratory by Uittenbogaard (1988), see Figure 5. The ﬂow is nominally two-dimensional, a light, fresh-water stream being injected above a more dense saline stream. The two streams are then allowed to mix together. A previous exploration of this ﬂow (Craft et al. 1996) had persuaded us that a more reﬁned treatment of the second-moment diﬀusion processes was needed though, on that occasion, as remarked above, just a fairly rudimentary extension of the GGDH concept was adopted. Figures 6–8 from Craft et al. (2001) (see also Kidger 1999) compare proﬁles of second-moment and partial third-moment closures for a range of measured quantities. Figure 6 shows the vertical proﬁle of relative density at a distance 40m downstream. We note from the experimental data (crosses) that the concentration has by no means equalized even at this most downstream position, with the mixing region being only about 0.2m in width. By contrast the linear k-ε EVM predicts that the mixture was approaching a fully mixed state, Craft et al. (1996). At second-moment level the TCL model clearly shows too much mixing while, even more serious, near the bottom and top, the concentration levels are respectively above the maximum and below the minimum levels in the entering streams. Such a development is impossible, of course. Figure 6b shows, however, that by including the 3rd-moment transport equations, satisfactory agreement with the experimental data is achieved. The reason for this dramatic improvement is the very diﬀerent proﬁles of the predicted salinity

420

Craft and Launder

0.6

0.6

TCL 2 0.5

TCL 3 0.5

Basic 2

TCL 2

k-ε

0.4

0.4

y (m)

y (m) 0.3

0.3

0.2

0.2

0.1

0.1

0.0 -0.2

0.0

0.2

0.4

0.6

ρ − ρ max ρ min − ρ max

0.8

1.0

1.2

0.0 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ρ − ρ max ρ min − ρ max

Figure 6: Relative density at x = 40m in the stably-stratiﬁed mixing layer. (a) Predictions using TCL and Basic second-moment closures and the k-ε model; (b) predictions using the TCL second-moment (TCL 2) and partial triplemoment (TCL 3) closures. Symbols: experiments of Uittenbogaard (1988). From Kidger (1999). ﬂux, shown in Figure 7. The second-moment closure exhibits strong reversals in scalar ﬂux near the bottom of the channel and near the free surface. It is that feature which causes the under- and over-shoots in the mean concentration. The 3rd moment closure, however, gives a similar variation of salinity ﬂux to the measured levels. We come now to the reason for this striking diﬀerence in behaviour between the two models. The most important feature is that the buoyant term in the salinity ﬂux equation (which is proportional to θ2 ) is conﬁned within a narrow inner band with the partial 3rd moment closure whereas it is dispersed over most of the channel depth when this is modelled by the GGDH, see Figure 7.6 It is the presence of this substantial buoyant sink in the vθ equation in regions where gradients of mean salinity are weak which produces the anomalous sign of vθ and the consequent over- and under-shoots in mean salinity. It may be inferred that the improved representation of the diﬀusive ﬂuxes, particularly of vθ2 which appears in the θ2 transport equation is thus crucial to getting the correct behaviour of the resultant mean concentration proﬁle. Finally, it is noted that while the Basic Model also beneﬁts from the 3rd-moment treatment, Kidger (1999), the improvement is by no means as great with the mean concentration proﬁle still exhibiting over- and under-shoots. 6

The experimental data conﬁrm the narrowness of spread but the magnitude of the ﬂuctuations is much lower possibly because a ﬁnite probe will ‘smear’ the variations.

[14] Application of TCL modelling to Stratiﬁed ﬂows

421

0.6

TCL 3 0.5

TCL 2 0.4

y (m) 0.3

0.2

0.1

0.0 -5.0

-2.5

0.0

2.5

5.0

7.5

10.0

vθ x 10 (kg/m s) 3

2

y (m)

Figure 7: Vertical density ﬂux at x = 10m in the stably-stratiﬁed mixing layer with TCL closures. — partial triple-moment closure; – – – second-moment closure; symbols: experiments of Uittenbogaard (1988). From Kidger (1999). 0.560

0.560

0.480

0.480

0.400

0.400

0.320

0.320

0.240

0.240

0.160

0.160

0.080

0.080

0.000 0.000

1.500

3.000

4.500

6.000

0.000 7.500

Scalar Variance (kg^2/m^6)

Figure 8: Scalar variance at x = 10m in the stably-stratiﬁed mixing layer with TCL closures. — partial triple-moment closure; – – – second-moment closure; symbols: experiments of Uittenbogaard (1988). From Kidger (1999).

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Acknowledgements The authors express their appreciation to Dr J.W. Kidger in preparing Figures 5–8. Mrs C. King wordprocessed the text both for this published version and the INI Conference.

References Craft, T.J. and Launder, B.E., (1990). ‘The decay of stably-stratiﬁed grid turbulence’, 4th Colloquium on CFD, UMIST, Manchester, 24–25 April 1990, Department of Mechanical Engineering, UMIST, Manchester, Paper 1.5. Craft, T.J., (1991). Second-Moment Modelling of Turbulent Scalar Transport, PhD Thesis, Faculty of Technology, University of Manchester. Craft, T.J., Ince, N.Z. and Launder, B.E., (1996). ‘Recent developments in secondmoment closure for buoyancy-aﬀected ﬂows’, Dyn. Atmos. Oceans 23, 99–114. Craft, T.J., Kidger, J.W. and Launder, B.E., (1997). ‘Importance of 3rd-moment modelling in horizontal, stably-stratiﬁed Flow’. Proc. 11th Symp. on Turbulent Shear Flows, Grenoble, 1997, 20.13–20.18. Craft, T.J., Kidger, J.W. and Launder, B.E., (2001). ‘Selective 3rd-moment closure for stably-stratiﬁed ﬂows’, to be published. Cresswell, R., Haroutunian, V., Ince, N.Z., Launder, B.E. and Szczepura, R.T., (1989). ‘Measurements and modelling of buoyancy-modiﬁed elliptic turbulent ﬂows’, Proc. 7th Symp. on Turbulent Shear Flows, Stanford University, 12.4.1–12.4.6. Daly, B.J. and Harlow, F.H., (1970). ‘Transport equations in turbulence’, Phys. Fluids 13, 2634. Dekeyser, I. and Launder, B.E., (1985). ‘A comparison of triple-moment temperaturevelocity correlations in the asymmetric heated jet with alternative closure models’. In Turb. Shear Flows 4, L.J.S. Bradbury et al. (eds.), Springer, 102–117. Hossain, M.S. and Rodi, W., (1982). ‘A turbulence model for buoyant ﬂows and its application to vertical buoyant jets’. In Turbulent Buoyant Jets and Plumes, W. Rodi (ed.) Pergamon. Itsweire, E.C., Holland, K.N. and Van Atta, C.W., (1986). ‘The evolution of gridgenerated turbulence in a stably-stratiﬁed ﬂuid’, J. Fluid Mech. 162, 299–338. Kawamura, H., Sasaki, J. and Kobayashi, K., (1995). ‘Budget and modelling of triplemoment velocity correlations in a turbulent channel ﬂow based on DNS’, Proc. 10th Symp. on Turbulent Shear Flows, Pennsylvania State University, 26.13–26.18. Kidger, J.W., (1999). Turbulence Modelling for Stably-Stratiﬁed Shear Flows and Free Surface Jets, PhD Thesis, UMIST, Manchester. Launder, B.E., (1975). ‘Prediction methods for turbulent ﬂows’, Von Karman Institute for Fluid Dynamics, Rhode St Gen`ese, Belgium. Launder, B.E., (1989). ‘The prediction of force-ﬁeld eﬀects on turbulent shear ﬂow via second-moment closure’. In Adv. in Turbulence 2, H.H. Fernholz and H.E. Fiedler (eds.), 338–358, Springer.

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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. Lumley, J.L, (1975). ‘Prediction methods for turbulent ﬂows’, Lecture Series 76, Von Karman Institute for Fluid Dynamics, Rhode-St-Gen`ese, Belgium. Lumley, J.L, (1978). ‘Computational modelling of turbulent ﬂows’, Adv. Appl. Mech. 18, 123. Millionshtchikov M.D., (1941). ‘On the role of the third noments in isotropic turbulence’, C.R. Acad. Sci. SSSR 32, 619. Uittenbogaard R.E., (1988). ‘Measurement of turbulence ﬂuxes in a steady, stratiﬁed, mixing layer’, 3rd Int. Sym. on Reﬁned Flow Modelling and Turb. Meas., Tokyo, 725–732. Van Haren L., (1992). ‘Comparison between one- and two-point closure models for freely decaying stratiﬁed ﬂow’. In Pole Europ´een Pilote pour la Turbulence (PEPIT), Conf. on Stratiﬁed Turbulence, Paris, France. See also Etude-Th´eorique et Mod´elisation de la Turbulence en Pr´esence d’Ondes Internes, Th`ese de Docteur-Ing´enieur, Ecole Centrale de Lyon, 1992. Webster C.A.G., (1964). ‘An experimental study of turbulence in a density stratiﬁed shear ﬂow’, J. Fluid Mech. 19, 221–245. Young S.T.B., (1975). ‘Turbulence measurements in a stably-stratiﬁed shear ﬂow’, Rep. QMC–EP6018, Queen Mary College, London.

15 Higher Moment Diﬀusion in Stable Stratiﬁcation B.B. Ilyushin 1

Introduction

In recent years there has been intensive development of turbulence models both at the 2-equation level of modelling as well as complete second-moment closure. These models combine (relative) ease of numerical solution with suﬃcient accuracy for many applications. However, the use of these models to predict turbulent transport in stratiﬁed ﬂows has given results which in some cases are even qualitatively incorrect, Lamb (1982). Both experimental and theoretical studies have identiﬁed the existence of large-scale eddy structures, LSES, in stratiﬁed ﬂows taking the form of turbulent ‘spots’ in stable stratiﬁcation and coherent structures in unstable stratiﬁcation. The long lifetime of LSES results in a ﬂuid pollutant particle being entrained by such an eddy and being transferred a considerable distance by the (LSES) without any appreciable change in its direction, see Figure 1. Thus feature does not accord with Euler’s concepts of turbulence diﬀusion in which turbulent transport is considered analogous to Brownian motion; that is, a random-walk process. In this case a particle can travel a large distance only from multiple motions of random directions. Thus, to model turbulent transport in stratiﬁed ﬂows where turbulent ﬂuctuations are strongly non-isotropic, one needs to take into account the contribution of LSES corresponding to the long-wave-length part of the spectrum.

(a)

(b)

Figure 1: Transfer of a ﬂuid (pollutant) particle in the ﬁeld of eddies by (a) the inertial spectrum range (b) under the action of LSES.

424

[15] Higher moment diﬀusion in stable stratiﬁcation

lg E(κ)

425

Ri < 0 Ri = 0 ~ κ –5/3 Ri > 0 lg κ

Figure 2: Qualitative dependence of the vertical velocity ﬂuctuations spectrum on a character of stratiﬁcation of the atmospheric boundary layer. As the ﬂow stability increases the turbulent ﬂuctuations decrease in intensity due to destruction of the LSES. The maximum of the energy spectrum moves to smaller scales (higher wave number), see Figure 2, and the upper limit of the interval range (κη ≈ 1/η) shifts to the region of larger scales (smaller κ)1 . This leads to a reduction in and ﬁnally to the disappearance of the inertial range (due to the merging of the dissipation and energy containing ranges). As a result, the inﬂuence of spatially non-local turbulent transport (large-distance transport by the action of LSES)2 in the ﬂow dynamics becomes weak while non-local (memory) eﬀects, which are aﬀected by the intermittency of turbulence, increase in importance. This means that gradient-diﬀusion models of turbulent transport (which correctly describe transport by eddies in the inertial range having a Gaussian PDF) become inaccurate since memory eﬀects can only be taken account of through diﬀerential equations for the turbulenttransport itself. For the case considered practically the whole energy spectrum exhibits a strongly non-Gaussian distribution. This feature has been informed by measurements in the atmospheric boundary layer (ABL). In the measurements of Pinus and Cherbakova (1966), among others, vertical ﬂuctuations developed a ‘spotty’ character, in which, for long periods, the level of velocity ﬂuctuations was below the detection level of the instrumentation, while at other times the magnitude of the upﬂow and downﬂow motions was very large, see Figure 3. Moreover, in stable stratiﬁcation the asymmetry of the PDF appears to be greatly reduced (Byzova et al. 1989). It is noted that in the convective (i.e. unstable) ABL, the PDF of the vertical velocity ﬂuctuations has a very marked asymmetry. From the above we may conclude that in stratiﬁed ﬂows (especially in stable stratiﬁcation) a model of the third moments that neglects buoyancy and which assumes the fourth-order cumulants 1 2

η ≈ 10−2 m for the convective ABL and η ≈ 10−1 m for the stably stratiﬁed ABL as well as apparently an eﬀect of two-point correlations

426

Ilyushin

%

2

1

–0.5

0

0.5

w, m/s

Figure 3: Vertical velocity probability in the stably stratiﬁed ABL. Lines are the Gaussian function and the PDF reconstructed by a Gram–Charlier series with S = 0, δ = 2.1, others of the high moments are equal zero. to be zero may well be inadequate, in particular for computing the behavior of the second-order moments. Chapter [14] (see also Craft et al. (1997)) provides support for this conclusion. In problems where detailed information is required about the distribution of higher moments in stably stratiﬁed ﬂows it is essential to use higher-order closure modelling without the assumption of zero fourth-order cumulants. To conclude this section, we note that the simultaneous presence of two types of ﬂuctuations, namely turbulent motions and internal waves, is an important characteristic of stably stratiﬁed ﬂows (including, in particular, the ABL) which makes the interpretation of experimental data diﬃcult. Spectral measurements in the atmosphere by Caughey (1977) have shown the presence of internal waves near the ground. The wave frequency is close to the Brunt–Vaisala frequency and clearly separated from the high-frequency region of turbulent ﬂuctuations. At larger heights above the ground the turbulent perturbations shift to lower frequencies resulting in a smoothing of the spectrum shape leading to a single-humped spectral proﬁle. Experiments by Finnigan et al. (1984) have shown that in the stably stratiﬁed ABL the destruction of internal waves can be an additional source of turbulent ﬂuctuations3 . The 3

Mechanisms of interaction of internal waves with turbulence were considered in a range of theoretical studies on simple models. However to fully reveal the mechanism of this interaction, it is necessary to carry out more complete measurements and analyses.

[15] Higher moment diﬀusion in stable stratiﬁcation

427

susceptibility of stably stratiﬁed ﬂows to external disturbance can result in the generation of ‘parasitic’ waves, as found by Moeng and Randall (1984) in a numerical simulation of such ﬂows with a third-moment closure assuming zero fourth-order cumulants. Andr´e et al. (1978) managed to suppress these by incorporating the ‘clipping approximation’ (see Andr´e et al. 1976) in the numerical algorithm. Alternatively, Moeng and Randall (1984) and others have added diﬀusion terms. The closure strategy presented in the next section leads to rationally-based turbulence models with diﬀusion items in the transport equations for the higher-order moments. It provides the correct direction of spectral ﬂux of turbulence energy (from large eddies to dissipative motions) and the necessary damping of non-physical ‘parasitic’ waves.

2

The closure strategy

Nowadays ﬁrst- and second-order closure methods are very widely used4 . In many cases such models provide all the essential information sought to the required level of accuracy. However, even second-order models are found to be inadequate in some cases. It is generally supposed that where ﬂow features are not adequately modelled with nth-order models, improvement in prediction will result from an (n+1)th-order model since such a model ﬁnds the (n+1)thorder moments from transport equations in which the generative agencies are handled exactly. This supposition is corroborated by practice. A general strategy for closure may be stated: cumulants of the orders 1, . . . , n are calculated from diﬀerential transport equations, the (n + 1)th-order cumulants (describing processes of turbulent diﬀusion in the nth-order cumulants equations) are calculated from approximate algebraic expressions derived from the corresponding diﬀerential transport equations in the stationary state in which (n + 2)th-order cumulants are treated as zero, see Figure 4. The present approach is believed to adopt suﬃciently general closure models. Its principle distinction is the use of cumulants as dependent variables rather than moments. The advantages of this approach have been brought out by analysing the predictive capability of turbulence models based on a hierarchy of equations for moments and cumulants (Monin and Yaglom 1967; Keller and Friedmann 1924). Obtaining algebraic models for the (n + 1)th-order cumulants is based on the idea that they exhibit a faster relaxation rate than the nth-order cumulants. In this case the (n + 1)th-order cumulants are determined by those of lower order. The diﬀerential equation for the (n+1)th-order cumulant may be written: ∂Cn+1 ∂Cn+1 Cn+1 = F (r, t, C1 , . . . , Cn+1 ) − , + Uj ∂t ∂xj τn+1

(1)

where the function F depends on space and time coordinates, cumulants of 4

We deﬁne the order of a turbulence model as the order of the highest moment determined from a diﬀerential transport equation.

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Ilyushin

Figure 4: Closure strategy for turbulence models. order no higher than (n + 1) (higher-order cumulants are assumed to be negligible) and a relaxation time scale, τn+1 , of the (n+1)th-order cumulant (which is assumed proportional to the overall turbulence time scale τ ≡ k/ε, where in this context ε may be thought of as the spectral ﬂux of kinetic energy): ˜ where Cn+1 ˜ is a coeﬃcient of proportionality). On expandτn+1 = τ /Cn+1 n+1 ing the cumulant C in a Taylor series about time t + τ and coordinate r + Uτn+1 , where t τn+1 , and retaining only the ﬁrst three terms, equation (1) can be expressed as: ∂Cn+1 ∂Cn+1 n+1 n+1 C (t + τn+1 , r + Uτn+1 ) = C (t, r) + τn+1 + Uj τn+1 ∂t t ∂xj r τ = F r, t, C1 , . . . , Cn+1 . (2) ˜ Cn+1 It should be noted that the right hand side of equation (2) involves the time moment t and coordinate r, whereas the cumulant is calculated at the time moment t + τn+1 at a point r + Uτn+1 . To deﬁne the cumulant Cn+1 at time

[15] Higher moment diﬀusion in stable stratiﬁcation

429

t and at a point with coordinate r we assume that changes in its value over the interval τn+1 in the region with size r + Uτn+1 are linear and can be taken into account by a change in the value of the relaxation coeﬃcient: τ F (r, t, C1 , . . . , Cn+1 ). (3) Cn+1 (t) = Cn+1 Evidently, the assumptions made to obtain the algebraic model (3) limit its use. Allowing for the fact that the typical time scale of relaxation of turbulent ﬂuctuations is larger if the corresponding wave vector is smaller (Monin and Yaglom 1967), we may conclude that the algebraic parameterization (3) of the cumulant will result in a larger error if the contribution of the long-wave ﬂuctuations (large-scale structures) to turbulent transfer mechanism is greater. Using the above approach for creating the nth-order models for n = 2 (see Figure 6) leads to a ‘standard’ second-order closure. In fact, using diﬀerential transport equations to calculate some of the second-order moments and algebraic relations for others is often suﬃcient. Of course, to close the second-order model it is not suﬃcient simply to express the triple correlation by means of known functions. Other correlations (pressure-containing and dissipative ones) also need to be approximated. There has recently been signiﬁcant progress in this ﬁeld and a large number of papers have been devoted to solving these problems. Our attention will be mainly directed at algebraic models for the triple correlations used in second-order models and at third-order closure models.

3

Algebraic models for triple correlations

The transport equations for the triple correlations in stratiﬁed ﬂows have the following form: ∂Cijkl ∂ uj uk D ui uj uk ∂Uk − ui uj ul + − ui ul = − Dt ∂xl ∂xl ∂xl ijk ui uj uk +βg ui uj θ δk3 − c3 , τ ∂Cijθl ∂Uj ∂ ui uj D ui uj θ ∂Θ − ui uj ul − ui ul θ − ul θ = Dt ∂xl ∂xl ∂xl ∂xl @ 2A 5 ui uj θ ∂ uj θ + ij − ui ul + βg ui θ δj3 − c3θ , (4) ∂xl τ @ A D ui θ 2 ∂Ciθθl ∂Θ @ 2 A ∂Ui ∂ ui θ − ui ul θ − ul θ − ul θ = − Dt ∂xl ∂xl ∂x ∂xl @ 2 Al @ A ui θ +βg θ3 δi3 − c3θθ , τ @ A @ A @ 2A ∂ θ2 D θ3 ui θ ∂Cθθθl @ 2 A ∂Θ − ul θ − ul θ − c3θθθ = − , Dt ∂xl ∂xl ∂xl τ where Ui and ui are the mean and ﬂuctuating components of the instantaneous velocity, β = 1/Θ is the coeﬃcient of volumetric expansion, Θ and θ are the

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Ilyushin

mean and ﬂuctuating components of the potential temperature, g is the gravity force acceleration, τ = k/ε is the time scale of turbulence, k = 12 ui ui is the turbulent kinetic energy, ε its dissipation rate, c∗∗∗ are the model coeﬃcients and C∗∗∗ the5fourth-order cumulants (see below). In equation (4) and below, the symbol means the sum of functions with a cyclic rearrangement ijk5 of indices i, j, k: ijk F (ui , uj , uk , ul ) = F (ui , uj , uk , ul ) + F (uk , ui , uj , ul ) + F (uj , uk , ui , ul ). Taking into account (1)-(3), the following algebraic models for the triple correlations may be derived: ∂ uj uk ∂Uk τ 5 − ui ul + βg ui uj θ δk3 , ui uj uk = c3 ijk − ui uj ul ∂xl ∂xl ) ∂ ui uj ∂Θ τ − ui θ ui uj θ = c3θ − ui uj ul ∂xl ∂xl * @ 2A 5 ∂Uj ∂ uj θ + ij − ui ul θ − ui ul + βg ui θ δj3 , ∂xl ∂xl ' @ A (5) @ 2A ∂ θ2 ∂Θ @ 2 A ∂Ui τ ui θ = c3θθ − ui ul θ − ul θ − ui ul ∂xl ∂xl ∂xl * @ A ∂ ui θ − ul θ + βg θ3 δi3 , ∂xl ' @ 2A ( @ A @ 3A ∂ θ ∂Θ τ − ul θ 2 . θ = c3θθθ − ul θ ∂xl ∂xl The complete set (5) of algebraic equations was applied by Canuto et al. (1994) to describe the turbulence structure in the case of unstable stratiﬁcation. It is evident that use of (5) in a second-order model (to model diﬀusion terms) requires the time-consuming inversion of twenty equations. Canuto et al. (1994) inverted the matrix by means of symbolic calculations. However, the triple-correlation algebraic models can be simpliﬁed signiﬁcantly for many of stratiﬁed ﬂows. In particular, Hanjali´c and Launder (1972) showed that the contribution of terms with mean velocity gradient to equations (5) was small and could be neglected. One can also, as adopted for example in Ilyushin and Kurbatskii (1997) ‘single out’ the vertical turbulent transport (the direction of the buoyancy force). In this way, more complete models are used for correlations containing the vertical velocity ﬂuctuations than those with a horizontal @ A velocity The algebraic models for the triple correlations w3 , @ 2 A ﬂuctuations. @ 2A w θ and wθ in the case of a horizontally homogeneous ﬂow take the

[15] Higher moment diﬀusion in stable stratiﬁcation form: @

w

A 3

@

3τ3 =− c3

@ 2A ∂ w βg˜ τ wθ w + c ∂z 3θ ∂ wθ 2βg˜ τ @ 2 A βgτ w + wθ + c3θ c3θθ ∂z @ 2A ∂ θ 2(gβ)2 τ τ˜ @ 2 A 3βgτ wθ , + w + c3θ c3θθ c3θθθ ∂z @

A

τ3θ w2 θ = − c3θ

@

431

2

A

(6a)

A τ3θθ wθ2 = − c3θθ

@ A 3τ @ 2 A ∂Θ ∂ w2 wθ − w c3 ∂z ∂z @ 2 A 2βgˆ ∂ wθ τ wθ +2 w + c3θθ ∂z @ 2A ∂ θ 2βgˆ τ @ 2 A 3βgτ , wθ + w + c3θθ c3θθθ ∂z @ A 3τ ∂Θ @ 2 A ∂ w2 2ˇ τ ∂Θ wθ − w − c3θ ∂z c3 ∂z ∂z 3ˇ τ ∂Θ @ 2 A ∂ wθ w +2 wθ − c3θ ∂z ∂z @ 2A @ A 3βgτ ∂ θ , wθ + w2 + c3θθθ ∂z

(6b)

(6c)

where τ3 =

τ 1+

τ3θ = τ3θθ =

3 2 2 c3 c3θ τ N

1+

+

4 4 c3 c3θ c3θθ c3θθθ τ N

3c3θθ +4c3 2 2 c3 c3θ c3θθ τ N

1+

9

τ +

9 c3 c3θ c3θθ c3θθθ

τ +

3c3θ +4c3θθθ 2 2 c3θ c3θθ c3θθθ τ N

,

τˇ =

τ 4N 4

9 c3 c3θ c3θθ c3θθθ

,

τ 4N 4

τˆ = , τ˜ =

τ

1+

, 3 2 2 c3 c3θ τ N

1+

, 3 2 2 c3θθ c3θθθ τ N

1+

3c3θ +4c3θθθ 2 2 , c3θ c3θθ c3θθθ τ N

τ

τ

@ A ∂Θ is the Brunt–Vaisala frequency (the correlation θ3 is ∂z not required for closing the second-order model). One can see that equation (6) incorporates the mechanism of triple-correlation damping in the case of a stably stratiﬁed ﬂow (τ 2 N 2 > 0) in contrast to the usually used tensorwhere N 2 = βg

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Ilyushin

invariant models of the form (see, for example, Hanjali´c and Launder 1972): @ A @ 3 A0 3τ @ 2 A ∂ w2 = − , w w c3 ∂z @ A @ 2 A0 @ 2 A ∂ wθ ∂ w2 τ , w θ wθ = − +2 w c3θ ∂z ∂z (7) @ A @ 2 A0 τ ∂ wθ @ 2 A ∂ θ2 wθ = − + w , 2 wθ c3θθ ∂z ∂z @ A @ 3 A0 ∂ θ2 3τ = − wθ . θ c3θθθ ∂z The triple-correlation algebraic models taking into account buoyancy eﬀects to a ﬁrst approximation (only terms linear in β are retained) may be derived from (6) taking into account equation (7): @

@

@

w

A 3

w2 θ

wθ2

A

A

@ =

w3

A0

@ 2 A0 + 3 βgτ c3 w θ

1+ @

=

w2 θ

A0

3 2 2 c3 c3θ τ N

+

τ c3θ

,

@ A A

2 @ 3 0 + 2βg wθ 2 0 w −N βg

3θθ +4c3 2 2 1 + 3c c3 c3θ c3θθ τ N @ A

A @ 2 A0 2 @ τ 2 θ 0 + βg θ 3 0 −2 N w + c3θθ wθ βg

=

1+

3c3θ +4c3θθθ 2 2 c3θ c3θθ c3θθθ τ N

,

(8)

,

or, for the case of a three-dimensional ﬂow: ui uj uk =

0 ui uj uk 0 + α βgτ c3 ui uj θ

1+

α 2 2 c3 c3θ τ N

,

@

A A @ 2 0 2 0 δ + u θ2 0 δ u u u u + βg θ −N i j 3 i j3 j i3 βg

, ui uj θ = 2ϕ 2N 2 1 + cϕ+1 + τ c3θ c3θθ 3 c3θ @ 3 A0

@ 2 A0 0 τ N2 θ + u u θ + βg θ δi3 −ψ u i i 3 @ 2A c3θθ βg

ui θ = , 1 + c3θψc3θθ + c3θθ3δci33θθθ τ 2 N 2 (9) α = δi3 + δj3 + δk3 ; ϕ = δi3 + δj3 ; ψ = 2(1 + δi3 ). ui uj θ0 +

τ c3 θ

The lack of experimental data for the distribution of the triple correlations in stably stratiﬁed ﬂows prevents any direct check on the adequacy of models (6), (9). However, it is known that the mechanism of damping the vertical velocity ﬂuctuations by buoyancy is similar to that of suppressing radial velocity ﬂuctuations in a swirling ﬂow in a straight circular-sectioned pipe. The

[15] Higher moment diﬀusion in stable stratiﬁcation

433

third-order moments distributions measured in Zaetz et al. (1985) for this ﬂow enable us to evaluate the adequacy of accounting for the mechanism of swirl @ A damping for the triple correlations v 3 @ 2 Ain the above approach. The models and u v for swirling ﬂow take the form5 : @ 3 A0 @ A0 v + 6 cτ3 Wr v 2 w

2 , τ W W ∂W 1 + 6 c3 r ∂r + r @ @ 2 A0 A0 u v + 2 cτ3 Wr u2 w @ 2 A u v = −

2 , τ W W ∂W 1 + 2 c3 r ∂r + r @ 3A v = −

(10)

where W is the circumferential velocity and the coeﬃcient c3 is taken as 4.75 (1/c3 ≈ 0.21) to give best agreement with measurements for the skewness factor Sv in non-rotating pipe ﬂow (see Figure 7a) with results of calculation by the model of Hanjali´c and Launder (1972): @ A @ 3 A0 @ 2A ∂ v2 τ vw2 v , 3 v =− −6 c3 ∂r r (11a) @ A @ 2 A0 τ @ 2 A ∂ u2 ∂ uv uw2 u v =− , v + 2 uv −2 c3 ∂r ∂r r @ A @ 2 A @ 2 A ∂ vw vw ∂ v2 w τ + vw , 2 v =− + −4 c3 ∂r r ∂r r @ A @ 2 A0 ∂ u2 ∂ uw uw τ u w =− + vw . 2 uv + c3 ∂r r ∂r (11b) One may note the similarity of accounting for swirl inﬂuence on the triple correlations in the model (10) with that of buoyancy inﬂuence in (8). 5 shows proﬁles of the radial skewness factor Sv and triple correlation @ Figure A 2 u v computed by including (and omitting) eﬀects of damping the third-order moments by swirl for a ﬂow in a cylindrical pipe by means of the algebraic models (10)6 . The results of calculations using the models of Daly and Harlow (1970) and Hanjali´c and Launder (1972), (11), as well as the measured data, are plotted in Figure 5. It can be seen that, taking into account the mechanism of swirl damping in (10), the calculated third-order moment proﬁles come closer to the measured ones. @

5

A0 v2w

In deriving equation (10), the contribution of the algebraic convective term including the swirl velocity (that is the term proportional to W/r) in the diﬀerential transport equation (4) for triple correlations has been also taken into account. 6 The work was carried out with Dr. S.N. Yakovenko.

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Ilyushin

0.0

Sv

0.0

/u*3

1

3 2

3

1

–0.2

2

1

–0.2

2,3

2,3

a) –0.4 0.0

0.5

r/R

b) 1.0

0.0

0.5

r/R

1.0

@ A @ A3/2 5: Radial (a) skewness factor Sv = w3 / w2 and (b) correlation A @Figure u2 v distributions in the rotated pipe ﬂow. Lines correspond to calculations without (dashed lines) and with (solid lines) rotation: 1 – by the model (Daly and Harlow 1970), 2 – by the model (14) (Hanjali´c and Launder 1972), 3 – by the model (13); symbols (• without rotation and with it) are the experimental data (Zaetz et al. 1985). The ﬂow that forms in the turbulent wake behind a self-propelled body in a stably stratiﬁed ﬂow is a well-known example of a turbulent stratiﬁed shear ﬂow. For comparatively weak stratiﬁcation, the wake develops at ﬁrst in practically the same way as in a homogeneous ﬂuid and expands symmetrically. However, buoyancy forces inhibit vertical turbulent diﬀusion. Consequently, at large distances from the body, the wake becomes ﬂattened out and then ﬁnally entirely ceases to grow in the vertical direction. Turbulent mixing causes a more uniform density distribution within the wake than outside it. Gravity forces tend to restore the earlier undisturbed state of stable stratiﬁcation. As a result, convective streams leading to the formation of internal waves in the surrounding ﬂuid appear in the plane normal to the wake axis. The vertical extent of the wake is an important characteristic illustrating the process of turbulent diﬀusion in a stably stratiﬁed ﬂuid. The results of numerical simulations of the turbulent wake dynamics in a linear stratiﬁed medium show that taking into account the buoyancy eﬀect in the triple-correlation algebraic models allows us to describe the behaviour of the wake height downstream of the body (this height is deﬁned by proﬁle of the @ A vertical component w2 of turbulence kinetic energy) in complete agreement

[15] Higher moment diﬀusion in stable stratiﬁcation

H 1

435

FD 565 3 1 4 Lin 103 6 5 Pao 65

1 2

0 .1 0 .0 0 1

0 .0 1

0 .1

1

t/ T 5

Figure 6: Evolution of wake height: lines are calculation results: 1 – model without buoyancy terms; 2 – model taking into account buoyancy eﬀects; symbols are experimental data (Lin and Pao 1979). with the measurements data (Lin and Pao 1979)7 in contrast to models which neglect this eﬀect (see Figure 6)8 .

4

Third-order closure models

There is a considerable number of references on the use of third-order turbulence models in the literature. However, as a rule, these models apply the Millionshikov quasi-normality hypothesis to model the diﬀusion processes in the equations for the triple correlations, that is, for n = 3 all cumulants of fourth and higher order are assumed to be zero (whereas, in the proposed approach, for n = 3 only cumulants of ﬁfth and higher order are regarded as zero). As a consequence, in the former case the triple-correlation equations are of ﬁrst order without the necessary mechanism of damping the triple correlations. Instead, in Andr´e et al. (1978) the physically incorrect procedure of ‘clipping’ the triple correlations values in accordance with the generalized Schwartz inequalities has been used. The approach proposed below allows us to overcome this diﬃculty. It has been corroborated by results of the paper by Hazen (1963) showing that to predict the early stages of turbulence the use of the quasi-normal hypothesis (i.e. the assumption of zero fourth-order cumulants) is limited to small ﬂuctuation magnitudes in which third moments are small. On the other hand, using the hypothesis assuming zero ﬁfth-order cumulants (for non-zero fourth-order cumulants) allows the width of applicability of the model to be markedly widened. The equations for second- and 7 The lack of measurement data for the proper triple correlation distributions prevents us making a direct comparison of computed third-order moments. 8 This work has been undertaken with Professor G.G. Chernykh and Dr. O.F. Voropaeva.

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Ilyushin

fourth-order moments of the velocity ﬁeld are as follows: ∂ ui uj ∂ ui uj + Uk ∂t ∂xk B ) C ∂ ui uj uk ∂Uj ∂p 1 ui ui uk =− − + βgk ui θ δjk + ∂xk ∂xk ρ ∂xj ijk C* B 2 ∂ uj , −ν ui ∂xk ∂xk (12) ∂ ui uj uk ul ∂ ui uj uk ul + Um ∂t ∂xm ) ∂ ui uj uk ul um ∂Ul ∂ ul um =− − + ui uj uk ui uj uk um ∂xm ∂xm ∂xm ijkl B C B C* ∂p ∂ 2 ul 1 − ν ui uj uk , +βgm ui uj uk θ δml + ρ ui uj uk ∂xl ∂xm ∂xm (13) where ρ is the density and ν is the viscosity of ﬂuid. To approximate fourthorder moments in equation (13), the ﬁfth-order cumulants are assumed to be zero: Cijklm = ui uj uk ul um − ui uj uk ul um − ui uk uj ul um − ui ul uk uj um − ui um uk ul uj − uj uk ui ul um − uj ul ui uk um − uj um ui uk ul − uk ul ui uj um (14) − uk um ui uj ul − ul um ui uj uk = 0. Taking into account (12), (13) and (14), one can derive the equation for the fourth-order cumulant Cijkl : ∂Cijkl ∂Ul − Cijkθ βgm δml = −Cijkm ∂t ∂xm ijkl )B C C C B B ∂p ∂p ∂p 1 ui uj uk − ui uj uk − ui uk uj − ρ ∂xl ∂xl ∂xl (15) B C* ∂p − ui ul uj ∂xk )B B C C 2 ∂ ul ∂ 2 ul +ν ui uj uk − ui uj uk ∂xm ∂xm ∂xm ∂xm B B C C* ∂ 2 ul ∂ 2 uk − ui uk uj − ui ul uj ∂xm ∂xm ∂xm ∂xm ∂ uj ul ∂ uj uk ul − ui um − ui uk um ∂xm ∂xm − ui uk um

∂ uj ul ∂ ui uk − uj ul um , ∂xm ∂xm

[15] Higher moment diﬀusion in stable stratiﬁcation

437

where Cijkθ = ui uj uk θ − ui uj uk θ − ui uk uj θ − uj uk ui θ is a mixed cumulant of velocity and potential temperature ﬂuctuations (see below). Equation (15) contains the unknown cumulant of velocity ﬂuctuations and ﬂuctuating pressure derivative as well as the cumulant including the viscosity (third group of terms on the right-hand side of the equation). To model these two cumulants we assume that their sum is represented by the relaxation term Cijkl /τ4 (where τ4 = τ /C˜4 with the coeﬃcient of proportionality C˜4 between the typical time scale of turbulence and that of the cumulant’s relaxation) and equation (15) transforms to: ∂Cijkl ∂Ul −Cijkm − Cijkθ βgm δml = ∂t ∂xm ijkl ∂ uj uk ul ∂ uk ul (16) − ui uj um − ui um ∂xm ∂xm Cijkl ∂ uj ul ∂ ui uk − uj ul um − C˜4 . − ui uk um ∂xm ∂xm τ Using (3), the cumulant algebraic model is derived from (16):   ∂Ul ∂ uk ul τ Cijkm + Cijkθ βgm δml + ui uj um Cijkl = − C4  ∂xm ∂xm ijkl ∂ uj uk ul + ui um ∂xm  ∂ uj ul ∂ ui uk  . + uj ul um + ui uk um ∂xm ∂xm 

(17)

The use of (17) for the cumulant Cijkl in determining the evolution of thirdorder moments from the diﬀerential transport equations implies a quicker relaxation of Cijkl than that of the third-order moments. This condition restricts the value of the coeﬃcient C4 ; C4 > C3 , where C3 is the coeﬃcient of proportionality between the typical time scale of turbulence and the typical time scale of the triple-correlation relaxation (the coeﬃcient of the model for the pressure-containing correlation in the moment equations). The Schwarz @ 3third A inequality for the triple correlation ui @

u2i

A@

A @ A2 u4i − u3i ≥ 0

(18)

is considered to ﬁx C4 . Equation (18) may be written more strongly as: @ 3 A2 u (19) Ciiii ≥ @ i2 A . ui

438

Ilyushin

Next we consider the decay of homogeneous turbulence according to the laws @ 2A @ 2A  w = w 0 exp − τt    @ 3A @ 3A t w = w 0 exp −C3 τ (20)    Ciiii = Ciiii0 exp −C4 τt . The condition C4 ≤ 2C3 − 1 for the upper limit of the coeﬃcient C4 follows from (19) taking into account (20). In using equation (17) for the cumulant Cijkl in the triple-correlation transport equation, the value of this coeﬃcient (C3 < C4 < 2C3 − 1) is taken to be equal to its upper limit: C4 = 2C3 − 1.

(21)

In the case of a stratiﬁed ﬂow the algebraic model (17) for the cumulant of the velocity ﬂuctuations contains the mixed cumulants of velocity and potential temperature ﬂuctuations. Using analogous arguments, the following equivalent algebraic models can be obtained for these cumulants:   ∂Uk ∂Θ τ Cijmθ Cijkθ = − C4θ + Cijkm + Cijθθ βgm δkm  ∂xm ∂xm ijkl ∂ uj uk ∂ uj uk θ ∂ uk θ + ui uj um + ui θum + ui um ∂xm  ∂xm ∂xm ∂ ui uj uk  + θum  ∂xm ijkl

Cijθθ = − Cτ4θ

  ∂Uj ∂Θ + 2Cijmθ + Ciθθθ βgm δjm Cimθθ  ∂xm ∂xm

@ A ∂ uj θ 2 ∂ ui uj θ ∂ uj θ + θum + 2 ui θum + ui um ∂xm ∂xm ∂xm  @ 2A @ 2 A ∂ ui uj ∂ θ  + θ um + ui uj um  ∂xm ∂xm ij

ijkl

'

Ciθθθ =

∂Ui ∂Θ + 3Cimθθ + Cθθθθ βgm δim ∂xm ∂xm @ A @ A @ 2 A ∂ ui θ ∂ θ2 ∂ ui θ 2 +3 θ um + 3 ui um θ + 3 θum ∂xm( ∂xm ∂xm @ 3A ∂ θ + ui um ∂xm ' @ 3A ( @ 2A @ A ∂ θ ∂ θ ∂Θ 4Cmθθθ . (22) + 6 um θ 2 + 4 um θ ∂xm ∂xm ∂xm

− Cτ4θ

τ Cθθθθ = − C4θθ

Cmθθθ

[15] Higher moment diﬀusion in stable stratiﬁcation

439

This algebraic model (22) of the cumulant contains the relaxation coeﬃcients which are not considered to be equal to each other because the mechanisms of decay of diﬀerent cumulants are determined by diﬀerent physical processes. The Schwarz inequalities for the triple correlations including temperature ﬂuctuations are @ 2 A @ 2 2 A @ 2 A2 @ 2 A @ 4 A @ 3 A2 θ − θ ui ui θ − ui θ ≥ 0; θ ≥ 0. (23) Equation (25) thus provides the following inequalities for the cumulants Cijθθ

@ 2 A2 u θ ≥ @ i 2A , ui

Cθθθθ

@ 3 A2 θ ≥ . θ2

(24)

These are used to deﬁne upper limits for the coeﬃcients C4θ and C4θθ of equation (22). For this, the decay of thermal inhomogeneity is considered for homogeneous turbulence according to the relations: + + , @ 2A @ 2A , ui θ = ui θ0 exp −C1θ τt , θ = θ 0 exp −r τt , + + @ 2 A @ 2 A , @ 3A @ 3A , ui θ = ui θ 0 exp −C3θ τt , θ = θ 0 exp C3θθθ τt , (25) , , + + Cθθθθ = Cθθθθ0 exp −C4θθ τt , Cijθθ = Cijθθ 0 exp −C4θ τt , where C1θ and r = τ /τθ are the relaxation coeﬃcients9 in the equations for the @ 2A second-order moments ui θ and θ , C@3θ and A C3θθθ @ Aare the corresponding coeﬃcients for the third-order moments u2i θ and θ3 . Applying analogous arguments to that used to ﬁx the value of C4 , and noting equation (25), we derive from (24) the following relations between the relaxation coeﬃcients of the second-, third- and fourth-order cumulants: C4θ = 2C3θ − C1θ ,

C4θθ = 2C3θθθ − r.

(26)

Finally it should be noted in this section that equations (21) and (26) between the relaxation coeﬃcients of the second-, third- and fourth-order cumulants have been obtained from rather sweeping assumptions. Further reﬁnements in their values may be made from numerical optimization.

5

Assessment of cumulant model

As mentioned above, the available data base for the statistical structure of turbulence in stably stratiﬁed ﬂows is limited to the distribution of second-order moments. It does not permit us to examine the capability of the cumulant model in these ﬂows by a direct comparison of third-order moments calculated from the equations using the cumulant models. Applying the cumulant 9

r is the reciprocal of the time-scale ratio R introduced in Chapter 2.

440

Ilyushin

z /zi 1

0 0.0

<wk’> /w*3

0.2

Figure 7: Proﬁles of vertical ﬂux of turbulent kinetic energy: the solid line is calculation by the model presented in Ilyushin and Kurbatskii (1996), the dashed line is calculation by the model in Andr´e J.C. et al. (1978); are laboratory experimental data (Willis and Deardorﬀ 1974), , , ×, , , ♦, , are observed data (Lenschow et al. 1980). model in the third-order closure model for the convective atmospheric boundary layer (Ilyushin and Kurbatskii 1996) enables the positive vertical ﬂux wk of turbulence energy to be predicted in complete agreement over the whole of the layer height including the near-ground layer (see Figure 7). The lack of detailed experimental data for the fourth-order moments in the ABL does not allow us to make a direct test of the cumulant models in this ﬂow. However, the eﬀect of stable stratiﬁcation on turbulence is seen by the suppression of vertical velocity ﬂuctuations. Noting the action of this suppression, turbulence in such ﬂows is characterized by strong anisotropy. Analogous mechanisms cause the large turbulence anisotropy in swirling pipe ﬂow and in turbulence near a wall. The distribution of higher-order moments has been measured in such ﬂows and this allows us to compare the fourthorder cumulants, measured and calculated, using the proposed model. Raupach (1981) measured distributions of the vertical velocity intensity, skewness @ A @ A3/2 @ A @ A2 Sw = w 3 / w 2 and ﬂatness factor Kw = w4 / w2 in the boundary layer on a ﬂat rough plate. However, data for the vertical distribution of the typical time scale of turbulence τ (or equivalently the spectral ﬂux of the

[15] Higher moment diﬀusion in stable stratiﬁcation

(a)

441

(b)

(c)

1

0

0

1

–1

1

2

4

6

8

10

@ A1/2 Figure@8: The distribution of turbulence intensity (σw = w2 (a), skewness A A @ 3 (b), and kurtosis K = w 4 /σ 4 (c) coeﬃcients of vertical Sw = w3 /σw w w variance: lines correspond to interpolated functions for σw and Sw ; the line on the plot (c) is the calculated proﬁle of kurtosis Kw ; symbols are the experimental data (Raupach 1981). turbulence kinetic energy) are lacking. It makes it impossible to deﬁne the Kw proﬁle by a direct substitution of measured values into (17). Nevertheless, taking into account the fact that for this ﬂow the algebraic@ gradient model A (7) adequately describes the behaviour of the third moment w3 , the cumulant value can be calculated from the algebraic expression obtained from (17). Figure 8c shows the proﬁle of the ﬂatness factor Kw calculated by substitut@ 2A ing @ 3 Ainto (17) analytic interpolating functions for the distribution of w and w (see the dashed lines in Figure 8 a,b) which were measured in Raupach (1981) (d is the displacement of the ﬂow thickness by the roughness and δ is the boundary layer thickness). It is seen that for the boundary layer on a ﬂat plate, the algebraic model (17) for the cumulant C3333 correctly describes the behaviour both within the boundary layer and near the plate where the structure of turbulent ﬂuctuations is characterized by substantial anisotropy caused by the rigid wall damping of the vertical velocity ﬂuctuations. The results of testing the cumulant model for a rotating ﬂow in a straight circular sectioned pipe are shown in Figure 9. Measurements (Zaetz et al. 1985) have been conducted in a rotating section of the pipe where the inﬂuence of transitional eﬀects on the turbulence is substantial. The experimental data10 in Figure 9 show that the eﬀect of pipe rotation does not inﬂuence the ﬂow structure for r/R > 0.6. The swirl ﬂow damps radial ﬂuctuations of velocity and decreases the ﬂatness factor. The calculations shown in Figure 9 qualita10

Near the axis the experimental points have a large error and the diﬀerence in the kurtosis values at r/R = 0.9 for both cases (with and without rotation) is apparently due to measurement errors.

442

Ilyushin

r/R 0.8

0.6

0.4

0.2

0.0 3

Kv 4

@ A @ A2 Figure 9: Proﬁles of the ﬂatness factor Kv = v 4 / v 2 in circular pipe ﬂow. Lines (solid – in swirling ﬂow, dashed – without swirl) are calculations; symbols ( – in swirling ﬂow; • – without swirl) are the experimental data (Zaetz et al. 1985). tively conﬁrm the eﬀects noted. However, agreement of the calculations with the measured data is worse than in the boundary layer on a plate (Figure 8). To ascertain the reasons for this disagreement between the measured ﬂatness factor and those calculated, more extensive studies of swirling ﬂow (both experimental and numerical) are required.

6

Conclusions 1. An increase in stable stratiﬁcation leads to a decrease in the inertial range of the turbulent ﬂuctuation spectrum. This decrease is caused mainly by the destruction of large-scale eddy structures. As a consequence, the inﬂuence of the spatially non-local turbulent transfer (i.e., the transfer to a large distance by the action of large-scale eddy structures) on the dynamics of a ﬂow becomes weaker whereas the inﬂuence of the non-local memory eﬀects conditioned by intermittency of turbulence increases. Gradient diﬀusion models which are applicable to turbulent transport on scales corresponding to the inertial range are inadequate for predicting such ﬂows.

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2. Large values of the ‘ﬂatness’ coeﬃcients of the velocity ﬂuctuation PDFs (fourth-order cumulants) for values of the skewness (third-order moments) close to zero in stably stratiﬁed ﬂows preclude the application of third-order closure models using the quasi-normality hypothesis of Millionshikov (the assumption of fourth-order cumulants) if one is to obtain an adequate description of the turbulence structure. 3. In many cases second-order closure models give all the required information about the statistical structure of the investigated ﬂow and it is then not useful to adopt higher-order closures to describe the turbulence. 4. Algebraic expressions for the triple correlations, taking into account buoyancy eﬀects, allow one to reproduce the suppression of vertical velocity ﬂuctuations by the action of stable stratiﬁcation. They can be used to model the diﬀusion terms in the transport equations of second-order closure models. 5. To calculate the distribution of higher-order moments in stably stratiﬁed ﬂows, one should apply higher-order closure models. The closure strategy adopted enables third-order turbulent transport models to be formulated accounting for non-zero values of the fourth-order cumulants.

Appendix For a complete statistical description of the hydrodynamic characteristic ﬁelds in a turbulent ﬂow, it is necessary to set the whole of the multi-dimensional joint distributions of probability for the values of these characteristics at every possible ensemble of space-time points. However the deﬁnition of such multi-dimensional distributions is a highly complex problem. Moreover these distributions are often inconvenient for applications because they are bulky. Thus, generally one is restricted to employing only some simpler statistical parameters describing particular properties of a stream. The most important parameters of probability distributions are the moments and some of their combinations (in this section the central moments of one-point distributions P (ui ) will be considered for simplicity): E D α β γ uα1 uβ2 uγ3 P (u) du moments, = u1 u2 u3 σi = Si = δi =

R A1/2 u2i dispersions, @ 3A ui skewness factors, σi3 @ 4A ui − 3; excess coeﬃcients, σi4

@

444

Ilyushin

where ui is the vector of turbulent ﬂuctuations of velocity (i, j, k = 1, 2, 3). The two latter coeﬃcients, dependent on values of higher-order moments, characterize the departure of the probability density function (PDF) from the normal (Gaussian) one (see Figure 10). The distribution with non-zero asymmetry (S > 0) corresponds to a ﬂow with intense but rare ﬂuxes directed positively alternating with slow but more probable ﬂuxes directed negatively (and vice versa for S < 0). A positive excess value indicates that the distribution is ‘less ﬂat’ than normal and a quantity is concentrated in regions of very large and very small values. Flows with such a distribution are characterized by a mixture of long periods of relative steadiness (when ﬂuctuations are small) with periods of increased activity (with turbulent ﬂuctuations). Such a statistical structure is termed ‘spotted’11 . The coeﬃcient of asymmetry Sδ of ﬂuctuating velocity derivatives along the coordinate has an important physical sense. Since mainly small-scale ﬂuctuations contribute to this quantity, its value reﬂects the statistical structure of small-scale turbulence. For scales in the inertial range of the turbulent spectrum, the Sδ value deﬁnes a single direction of energy transport through the spectrum: if it has a negative value this means that ﬂuctuations of the considered scale transfer energy on average to those of smaller scale and take energy from larger scale ﬂuctuations (Monin and Yaglom 1967)12 . Near zero, the logarithm of the characteristic function (Fourier transform of the PDF) can be presented as a Taylor series: r

ln χ(q) =

n,m,k=0

χ(q) =

Cn+m+k

(iq1 )n ) (iq2 )m (iq3 )k + O(qr ), n! m! k!

P (u) exp(iqu) du. R

Coeﬃcients of this expansion are termed cumulants (semi-invariants). They are more convenient for use as statistical characteristics of distributions than the moments because of the fact that cumulants of order higher than second describe how non-Gaussian (non-equilibrium) are distributions of ﬂuctuations and also because of their invariant properties. The ﬁrst four cumulants are: C1i = ui , C2ij C3ijk C4ijkl 11

= ui uj , = ui uj uk , = ui uj uk ul − ui uj uk ul − ui uk uj ul − ui ul uj uk ,

Such a statistical structure of ﬂuctuations is typical of small-scale turbulence (Monin and Yaglom 1967) and also for stably stratiﬁed turbulent ﬂows (see below). 12 It should be noted that in regions with strong intermittency (for example, near a jet boundary) such a unique connection of the sign of the velocity derivative asymmetry and the direction of the energy spectral ﬂux is not found because in this region the approximation of homogeneous and isotropic turbulence is not satisﬁed.

[15] Higher moment diﬀusion in stable stratiﬁcation

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0.5 δ> 0 S>0

0

−4

−2

0

2

Figure 10: Distributions with coeﬃcients of skewness and excess being significant in comparison with those of the equilibrium (Gaussian) distribution. It follows from the foregoing that to describe the turbulent state of a ﬂow it is suﬃcient to deﬁne cumulants of this expansion. However, following the theorem of Marcinkiewicz (1939), the expansion of the characteristic function can be a polynomial of second degree (normal distribution13 ) or an inﬁnite series. It seems to be impossible to deﬁne a complete (inﬁnite) set of coefﬁcients. The inﬁnite chain of transport equations for cumulants (moments) should be broken and closed by the application of models. It should be noted that such a breaking can lead to incorrect results (obtaining negative values of quantities which in practice, can only be positive: PDF, dispersions, energy, dissipation). The last circumstance is a consequence that by the deﬁnition of a ﬁnite number of cumulants (moments) the corresponding probability may not even exist. It is connected closely with the inadmissibility of arbitrarily clipping the Taylor series of the characteristic function logarithm.14 In the present chapter the assumption of a relaxation character of the PDF evolution to the equilibrium state is applied (by analogy with the τ approximation used in a range of problems in the kinetic theory of gases). This assumption means there 13

The normal distribution of the turbulent ﬂuctuations supposes their isotropy and homogeneity. Practically the only ﬂow that reasonably approximates these conditions is the turbulent ﬂow behind a grid. The overwhelming majority of applied problems have anisotropic external eﬀects (stratiﬁcation, rigid boundary). In such ﬂows experiment ﬁxes signiﬁcant values of the coeﬃcients of skewness and excess (characteristics reﬂected in the deviation of the PDF from the normal one). To describe the statistical structure of anisotropic turbulence the Gaussian PDF approximation appears to be insuﬃcient. 14 A signiﬁcant number of papers (see, for example Schumann (1977)), is devoted to the problem of the realizability of turbulence models.

446

Ilyushin

Figure 11: Spectrum of turbulent ﬂuctuations. will be no strong deviations of the PDF from the equilibrium one. Moreover, the Kolmogorov hypothesis concerning the locally isotropic structure of small-scale turbulence in the inertial range of the turbulent energy spectrum (see Figure 11) will be used as well. At present the correctness of this hypothesis is not in doubt. The statistical structure of this region of the spectrum is described with good accuracy by the equilibrium PDF and a motion of ﬂuid particles (or contaminant particles) in the ﬁeld of isotropic turbulence can be characterized as a Brownian motion (diﬀusion). The short-wave part of the spectrum corresponds to scales of dissipation of the turbulent kinetic energy into heat by viscous action. This region contains a comparatively small part of the turbulence energy. Corresponding to this region, ﬂuctuations have a complex statistical structure and are characterized by considerable values of the skewness and ﬂatness coeﬃcients (with a strongly

[15] Higher moment diﬀusion in stable stratiﬁcation

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non-Gaussian distribution). However, when modelling turbulent ﬂows in the frame of the method of statistical moments, their inﬂuence on the processes of turbulent transfer is believed to be negligible. The primary role of ﬂuctuations in this range – providing the sink of turbulent kinetic energy – is taken into account. In the case of developed turbulence, the rate of viscous dissipation is equal with good accuracy to the spectral ﬂux of turbulence energy and, for its deﬁnition, does not require the consideration of a complex statistical structure of small-scale turbulence. In contrast, the energy range corresponding to long-wave (large-scale) ﬂuctuations contains the main part of the turbulence energy and mainly determines the character of turbulent transport. Long-wave ﬂuctuations correspond to large-scale eddy structures (LSES). They are characterized by a comparatively large time of relaxation and contain information about the history and structure of the averaged ﬂow (memory eﬀects). Therefore, this region of the spectrum is usually anisotropic and the PDF (being considerably nonGaussian) is characterized by substantial values of skewness and ﬂatness coefﬁcients. Thus, ﬂuctuations of the energetic and inertial ranges of the spectrum are the main objects of study in problems of turbulence modelling15 .

References Andr´e J.C., De Moor G., Lacarr`ere P. and Du Vachat R. (1976). ‘Turbulence approximation for inhomogeneous ﬂow: Part 1. The clipping approximation’, J. Atmos. Sci. 33 476; ‘Part 2. The numerical simulation of penetrative convection experiment’, J. Atmos. Sci. 33 482. Andr´e J.C., De Moor G., Lacarr`ere P., Therry G., Du Vachat R. (1978). ‘Modelling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer’, J. Atmos. Sci. 35 1861. Byzova N.L., Ivanov V.N., Garger E.K. (1989). Turbulence in the boundary layer of the atmosphere. Gidrometeoizdat, Leningrad. (in Russian). Canuto V.M., Minotti F., Ronchi C., Ypma R.M., Zeman O. (1994). ‘Second-order closure PBL model with new third-order moments: comparison with LES data’, J. Atmos. Sci. 51 1605. Caughey, S.J. (1977). ‘Boundary layer turbulence spectra in stable conditions’, Boundary-Layer Meteorol. 11 3. Caughey S.G. (1982). ‘Observed characteristics of atmospheric boundary layer’. In Atmospheric Turbulence and Air Pollution Modelling, F.T.M. Nieuwstadt and H. van Dop (eds.), Reidel. 15

The complex statistical structure of large-scale eddy formations on the one hand, and their governing eﬀect on the dynamics of a turbulent ﬂow on the other, have led to the development of the method of describing turbulence based on the exact resolution of large eddies with the use of parameterizations to take into account small-scale ﬂuctuations (the LES method). See Chapters [8], [12], [13], [25].

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Craft T.J., Kidger J.W., Launder B.E., (1997). ‘Importance of third-moment modelling in horizontal, stably-stratiﬁed ﬂows’. Eleventh Symp. on Turbulent Shear Flows, Grenoble, France. Finnigan J.J., Einaudi F., Fua D. (1984). ‘The interaction between an internal gravity wave and turbulence in the stably stratiﬁed nocturnal boundary layer’, J. Atmos. Sci. 16 2409. Hanjali´c K., Launder B.E. (1972). ‘A Reynolds stress model of turbulence and its application to thin shear ﬂows’, J. Fluid Mech. 52 609. Hazen A.M. (1963). ‘Towards a non-linear theory of appearance of turbulence’, DAN USSR. 163 1282 (in Russian). Ilyushin B.B., Kurbatskii A.F. (1996). ‘New models for calculation of third-order moments in planetary boundary layer’, Izv. RAN. Phys. Atmos. Ocean. 34 772 (in Russian). Keller L., Friedmann A. (1924). ‘Die Diﬀerenzialgleichungen f¨ ur die turbulente Bewegung einer Kompressible Flussigkeit’. Proc. Int. Congr. Appl. Mech. (Delft). Lamb R.G. (1982). ‘Diﬀusion in convective boundary layer’. In Atmospheric Turbulence and Air Pollution Modelling, F.T.M. Nieuwstadt and H. van Dop (eds.), Reidel. Lenschow D.H., Wyngaard J.C., Pennel W.T. (1980). ‘Mean-ﬁeld and second-moment budgets in a baroclinic, convective boundary layer’, J. Atmos. Sci. 37 1313. Lin J.T., Pao Y.H. (1979). ‘Wakes in stratiﬁed ﬂuids’, Ann. Rev. Fluid Mech. 11 317. Marcinkiewicz, J. (1939). ‘Sur une propriet´e de la loi de Gauss’, Math. Zeitschr. 44 612. Moeng C.-H., Randall D.A. (1984). ‘Problems in simulating the stratocumulus-topped boundary layer with a third-order closure model’, J. Atmos. Sci. 41 1588. Monin A.S., Yaglom A.M. (1967). Statistical Fluid Mechanics (Vol 1,2), Nauka (in Russian) See also English translation (ed. J.L. Lumley) Vol 1 (1971) Vol 2 (1975) MIT Press. Pinus N.Z., Cherbakova E.F. (1966). ‘About wind velocity ﬁeld in stratiﬁed atmosphere’, Izv. Acad. Sci. SSSR, Phys. Atmos. Ocean. 2 1126. Raupach M.R. (1981). ‘Conditional statistics of Reynolds stress in rough-wall and smooth turbulent boundary layers’, J. Fluid Mech. 108 363. Schumann U. (1977). ‘Realizability of Reynolds-stress turbulence models’, Phys. Fluids. 20 721. Willis G.F., Deardorﬀ J.W. (1974). ‘A laboratory model of the unstable planetary boundary layer’, J. Atmos. Sci. 31 1297. Zaetz, P.G., Safarov N.A., Safarov R.A. (1985). ‘Experimental study of turbulent ﬂow characteristics behaviour by channel rotation around longitudinal axis’. In Modern Problems of Continuous Medium Mechanics, MFTI, Moscow, 136, (in Russian).

16 DNS of Bypass Transition P.A. Durbin, R.G. Jacobs and X. Wu Although direct numerical simulation (DNS) databases have had an impact on the development of models for fully turbulent ﬂows, computer simulations of bypass transition have only recently become available and their impact on modeling has been very much less. The conditions of bypass transition make direct simulation a natural resource; there is little doubt that its use in this ﬁeld will continue to grow. Direct simulation is restricted to low Reynolds number and to simple geometries because of its great demand on computer speed and memory. These restrictions may not be a debilitating handicap for many transition applications: by their very nature, transitional phenomena occur at Reynolds numbers that are low compared to fully turbulent ﬂow. Since focus is on the vicinity of the transition line, geometrical complexity is not a controlling factor. Simulation databases increasingly will be used to develop and test models, and to understand phenomena; for these purposes they provide highly resolved data, with well deﬁned inlet conditions. Computational data are often of higher quality than analogous laboratory experiments. The geometry and the nature of the external disturbances that provoke transition are prescribed analytically and can be reproduced exactly in Reynolds averaged simulations. Free-stream turbulence intensity and integral scale can be speciﬁed at the inlet. A wealth of instantaneous ﬂow ﬁelds and ensemble averaged data are available for basic and applied inquiries. Instantaneous ﬂow ﬁelds provide views of the development of perturbations in the transitional zone and elucidate the initial stages of coupling between free-stream eddies and boundary layer disturbances. These are the phenomological underpinnings of the averaged data. The ﬁgures contained herein provide a sampling of the phenomological and statical resources of DNS. Certain simulations have already begun to ﬁll a niche left open by laboratory experiments. The periodic, wake induced transition discussed in §3 is a case in point. In some applications the simulation codes themselves might be a tool to use for predictive purposes. For instance, bypass transition in low pressure turbines is critical to some designs. The need for accurate prediction might warrant direct simulation.

1

The role of bypass transition

Under a quiet free-stream, a laminar boundary layer becomes linearly unstable beyond a critical Reynolds number at which Tollmein–Schlichting waves start 449

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Figure 1: The ragged edge of the turbulent region is maintained by merging spots. Three side by side copies of the same image are displayed. to grow. The instability is via a subtle mechanism whereby viscosity destabilizes the waves (Drazin and Reid 1981). That subtle mechanism produces very slow growth. Transition to turbulence might occur 20 times farther downstream from the position of linear instability. Orderly transition occurs only after the waves have become nonlinear and inviscid mechanisms have come into play. Given the delicacy and slowness of the orderly transition route, it is not surprising that the boundary layer will forgo this process when it is subjected to relatively small disturbances. It is also not surprising that the orderly route is beyond the scope of statistical turbulence modeling. Statistical models are inherently restricted to chaotic processes. What, then, is the province of turbulence models in the ﬁeld of transition? When a laminar boundary layer is exposed to about 1% or more free-stream turbulence, the Tollmein–Schlichting route to transition is bypassed. A rather diﬀerent scenario is then encountered: highly localized, irregular motions occur inside the boundary layer. These develop into small patches of turbulence, called turbulent spots. Spots spread and intensify as they propagate downstream. Eventually they join onto a main turbulent region, thereby maintaining its upstream edge. As the spots join the main region they form interlaced zones of turbulent and laminar ﬂow, just upstream of a fully turbulent region. Figure 1 illustrates the ragged edge of the turbulent zone, as is maintained by merging spots. This ﬁgure is three copies of a single image; the periodicity is an artiﬁce. The streamwise length in which bypass transition occurs is quite short compared to the length of the orderly route; bypass transition is complete in a distance comparable to the distance to its onset (Simon and Ashpis 1996). In ﬁgure 3 the onset of transition is at about Rθ ≈ 250 (Rex ≈ 1.5 × 105 ) and it is complete by Rθ ≈ 600 (Rex ≈ 3 × 105 ). Figure 2 shows two turbulent spots in the transition zone, as extracted from a direct numerical simulation. This ﬁgure consists of contours of the ﬂuctuating u-component velocity in a plane near the wall. When portrayed in this

[16] DNS of bypass transition

451

0.20 z 0.15 0.10 0.05 x

0.00 0.75

0.875

1.0

1.125

1.25

1.375

1.5

flow Figure 2: Spots in the transition zone as seen via contours of u near the wall. manner, streamwise elongated structure within the oval patches of u contours becomes apparent. It is a generic property of strongly sheared perturbations to develop elongated contours of u-component velocity ﬂuctuations. Turbulent spots certainly can be detected in experiments, but high resolution space-time data are hard to obtain by laboratory measurements. Numerical simulation is quite the opposite. A plenitude of data are available; the diﬃculties lie in how to process and archive all that information. The occurence of spots and precursors to spots is random in space and time. Hence, bypass transition falls within the province of statistical ﬂuid dynamics. Whether conventional turbulence models are applicable to this phenomenon is an open question (Saville 1999), but the fact that it is stochastic is not. The application of turbulence modeling to bypass transition will be discussed in detail in other chapters of this volume. The present chapter is addressed to the phenomenon as revealed by direct numerical simulation (DNS) and to the potential for DNS to generate unique databases that can be used to develop and assess models. Under zero pressure gradient, when the level of turbulence is on the order of 3%, transition takes place near the critical Reynolds number of linear theory. This may well be a reﬂection of a global critical Reynolds number, rather than a relevance of linear stability per se. Upstream of global instability, even large perturbations of the boundary layer do not provoke transition. That upstream region can be described as a buﬀetted laminar boundary layer. The random nature of this region puts it into the province of Reynolds averaged (RANS) modeling. In most cases it is more important for the RANS model to capture transition than for it to predict the buﬀetted laminar region. The virtues of numerical simulation include the wealth of full-ﬁeld instantaneous and ensemble averaged data that is provided, the ability to prescribe precise inﬂow conditions and geometry, and the ability to perform ‘thought

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0.008

T3A DNS

Turbulent

Cf

0.006

0.004

0.002 Blasius

0 200

400

600

800

1000

Reθ

Figure 3: DNS and experimental Cf . experiments’. Thought experiments are simulations of real ﬂuid dynamical behavior that would be virtually impossible to achieve in a physical experiment. Such experiments have already elucidated the dominance of the normal component of free-stream velocity ﬂuctuations in provoking transition (Yang and Voke 1995). They have also shown that turbulent ﬂuctuations in impinging wakes exert far greater control of bypass transition than does the mean velocity deﬁcit (Wu et al. 1999). Drawbacks to DNS include its restriction to relatively low Reynolds numbers and to small, simple geometries. It imposes substantial CPU and computer memory requirements. The stringent requirements on computer resources are rapidly being alleviated by massively parallel computers. The simulations described herein were performed on computers with from 64 to 512 processors. On the order of 5 × 107 grid points were used. A single run might require ∼ 102 hours of CPU time and be performed in ∼ 4 weeks of wall clock time. Of course, post-processing the wealth of data takes place over a much larger time period. This all is analogous to laboratory experimentation: data are obtained with highly specialized equipment and are archived for subsequent analysis.

2

Transition induced by free-stream turbulence

The simplest case of bypass transition is that induced by free-stream turbulence passing above a ﬂat plate with no imposed pressure gradient. The Rolls–Royce T3a experiment has become the deﬁnitive data set for this conﬁguration (Roach and Brierly 1990). Although our DNS experiments were not designed to accurately reproduce the conditions of that experiment, a fairly

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453

u-component

v-component Figure 4: Streaks, spots and transition in a boundary layer beneath free-stream turbulence as evidenced in contours of ﬂuctuating u and v components of velocity. close approximation has been produced. Figure 3 compares ensemble averaged skin friction versus momentum thickness Reynolds number from the DNS of Jacobs and Durbin (2000) to the experiments of Roach and Brierly (1990). The smallest x-location at which dCf /dx = 0 can be deﬁned as the onset of transition. At this location data ﬁelds like those in ﬁgure 4 show the onset of turbulent spot precursors. These stand out more starkly in the v-contours than in those of u. The spots themselves are initially highly intermittent. They mature through the transition region. Downstream, where the skin friction in ﬁgure 3 has reached the fully turbulent level, the spots merge into the continuously turbulent boundary layer. Statistical ﬁelds are obtained by spanwise and temporal averaging of random ﬁelds like ﬁgure 2. The evolution of mean velocity proﬁles with downstream distance is displayed in ﬁgure 5 by log-linear plots. These show that the trends of experimental ensemble averaged data are closely mirrored in the DNS. The lower portion of ﬁgure 5 is a comparison of the root-mean-square streamwise velocity ﬂuctuation to measurements. The DNS ﬁelds are a very comprehensive data set. Only selected locations are plotted for comparison to experiment; further data are available in Jacobs and Durbin (2000). The contour plots in ﬁgure 6 illustrate the ability of DNS to provide full ﬁeld data – in this case contours of turbulent kinetic energy production are plotted; they show the increase in production as the boundary layer undergoes transition. In the present example of transition induced by free-stream turbulence, the comprehensiveness of the DNS ﬁelds might be seen as the primary merit of the statistical data. In section 3 below, on wake induced transition, the DNS dataset ﬁlls a gap left by a lack of experimental data. However, even in the present case, instantaneous ﬂow ﬁelds obtained from the numerical simulation provide views of the coupling between free-stream eddies and boundary layer disturbances that could not be obtained in the lab. These instantaneous pictures are the subject of the remainder of this section.

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Durbin, Jacobs and Wu 30 Reθ = 323 Reθ = 385 Reθ = 456 Reθ = 980 Log Law

U+

20

10

0

10-1

100

101

102

103

y+

U proﬁles

Reθ = 177 Reθ = 385 Reθ = 539 Reθ = 897

u+

3

2

1

0 0

50

100

150

200

y+

u proﬁles Figure 5: DNS and experimental proﬁles of U , u at a few streamwise location. Symbols are the data of Roach and Brierly (1990).

2.1

Streaks, spots and shear ﬁltering

An intriguing phenomenon occurs in the laminar boundary layer upstream of the onset of transition. Long streaks of u-component velocity perturbations are observed: see the u-component of ﬁgure 4. Streaks are a feature of zero pressure gradient laminar boundary layers buﬀeted by free-stream eddies. The streaks appear to be identical to what have been called ‘Klebanoﬀ modes’ (Kendall 1991). The latter refer to proﬁles of ﬂuctuating velocity that peak in the outer part of the boundary layer. (Ironically, these ‘modes’ are ampliﬁed by the mean shear through a process of that has recently been renamed ‘non-modal’ growth. The term ‘Klebanoﬀ mode’ is a misnomer.) Figure 7 shows proﬁles of ﬂuctuating velocity in the streak region. These proﬁles peak at y/δ∗ ≈ 1 and are in excellent accord with ‘Klebanoﬀ mode’ measurements.

[16] DNS of bypass transition

455

5

4

y / δ0

3

2

1

0

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

Re

Figure 6: Contour plot rate of turbulent kinetic energy production. The role of laminar streaks in the subsequent bypass transition is unclear. Some insights are currently emerging from examination of DNS ﬁelds. Although it has been proposed that the streaks might undergo a sinuous instability, leading to turbulent spots (Elofsson et al. 1999), no evidence of that was seen in any of our direct simulations. Indeed the streaks in ﬁgure 4 persist up to the turbulent zone with no indication of sinuous instability. Velocity vector plots show that the streaky structure consists of narrow, intense regions where u < 0 and broader, weaker regions of u > 0. The spots appear to be a localized instability of the strong negative jet, that is stimulated by a free-stream eddy. In the buﬀetted boundary layer, viscous stresses are able to withstand the free-stream perturbations and the boundary layer remains laminar, though highly perturbed. The elongated streaks initially arise through a shear ﬁltering mechanism, whereby low frequency components of the external turbulence penetrate the boundary layer, while higher frequencies are expelled by the shear (Jacobs and Durbin 1998; Hunt and Durbin 1999). Figure 8 consists of velocity ﬂuctuation contours in planes parallel to the wall: the lowest plan view is in the free-stream, the middle view is at the upper edge of the boundary layer, and the uppermost is near the surface. A qualitative change from nearly isotropic turbulence in the free-stream to highly elongated streaks near the wall is observed as the plane moves down through the boundary layer. The mean shear shelters the boundary layer from small scales of free-stream vortical perturbations. The rationale presented in Jacobs and Durbin (1998) for this phenomenon is that modes of the continuous spectrum of the Orr–Sommerfeld

456

Durbin, Jacobs and Wu 1 Reδ* = 337 Reδ* = 371 Reδ* = 438 Reδ* = 483 Reδ* = 537 Reδ* = 578 Reδ* = 621

u / umax

0.8

0.6

0.4

0.2

0 0

1

2

3

4

y / δ*

Figure 7: ‘Klebanoﬀ modes’ are an allusion to ﬂuctuating velocity proﬁles of this form. They are probably a signature of the streamwise streaks induced by free-stream turbulence in the laminar boundary layer. equation are practically zero inside the boundary layer, unless the frequency is low. Disturbances convected with the free-stream velocity do not couple to the slower ﬂuid within the boundary layer. In the wake-induced transition discussed in the next section the long streaks of ﬁgure 4 do not occur. In that case shorter ‘puﬀs’ are observed prior to the appearance of turbulent spots (Wu et al. 1999). The process by which they are created is not understood. Puﬀs clearly are transition precursors, within which the irregular motion leading to turbulent spots evolves. There appear to be generic elements to bypass transition in zero pressure gradient boundary layers: some form of streaks or puﬀs are precursors; they produce lifted, negative jets; the jets undergo local instability under free-stream forcing; and, ﬁnally, some form of turbulent spots emerge. Direct simulations have made a signiﬁcant impact on our understanding of the process of bypass transition by enabling the early stages, before turbulent spots appear, to be explored. By the time that spots appear, a local region has already broken down to turbulence. Prior to that stage laminar, highly sheared regions appear. They take the form of ‘backward jets’ relative to the averaged ﬂow. The total velocity in these jets is not reversed: they are regions of low speed ﬂow in which the perturbation velocity proﬁle has the form of a jet directed upstream. Irregularities similar to Kelvin–Helmholtz instability are triggered on these shear layers and at some late stage of development they become localized turbulent patches. Figure 10 illustrates the backward jets as they appear in the wake-induced transition. Similar spot precursors are observed beneath free-stream turbulence.

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This overview of bypass transition DNS gives a view of the phenomena that transition models must represent, albeit on a statistical level. There are three regions of concern: the buﬀetted laminar boundary layer, the intermittently turbulent region of spot formation, and the fully turbulent boundary layer. In the ﬁrst, shear ﬁltering and non-modal shear ampliﬁcation lead to elongated streaks in the u-component of velocity ﬂuctuations. The skin friction is only slightly elevated by the free-stream turbulence. The boundary layer remains stable to perturbations. In the intermittency region, localized perturbations trigger shear layer instabilities that evolve into turbulent spots. These spots spread laterally and grow longitudinally. They overtake the main turbulent region and thereby maintain its upstream edge. That edge is irregular, consisting of juxtaposed zones of laminar motion, in which laminar streaks are usually visible, and zones of fully turbulent motion. In the last region the boundary layer is turbulent across its entire span. Sometimes a RANS model is left to capture these processes on its own, if it can. However, explicit transition models can be invoked as well. Abu-Gannam and Shaw (1980) proposed the data correlation Rθtr = 163 + e6.91−T u

(2.1)

for the transition Reynolds number of zero pressure gradient boundary layers. T u is the turbulence intensity in percentage, 100u /U , measured in the freestream. In the intermittency region the eddy viscosity is increased from zero to its full value by a function, γ(x), that increases from zero to unity: the eddy viscosity is represented by γνT where νT is the fully turbulent value. If the transition has been predicted to occur at xtr , γ = 1 − e−(x−xtr )

2 /L2 tr

,

x ≥ xtr

is used to ramp the eddy viscosity. Ltr here is a transition length, which has been estimated as Ltr = 126θ in zero pressure gradient boundary layers.

3

Transition induced by periodic passing wakes

The previous section covered only a continuously turbulent free-stream. One very important instance of bypass transition arises in turbomachinery. In that case the boundary layer is perturbed by periodically passing turbulent wakes. To simulate periodic wake induced transition, laboratory experiments have been performed in which wakes are swept over a ﬂat plate (Liu and Rodi 1991). The wake generator is parallel to the leading edge of the plate and the wake centerline is traversed perpendicularly to the plate. Physical experiments are very diﬃcult to conﬁgure and it is an enormous task to acquire accurate data. Because the ﬂow is not statistically stationary, time-averaging is not equivalent to Reynolds averaging. The appropriate data are phase averages as a function

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free-stream

y=δ

y = 13 δ Figure 8: Horizontal sections through the free-stream and the boundary layer. of space and time; for instance the mean velocity is of the form U (x, y, φ) and the skin friction coeﬃcient is a function, Cf (x, φ), of downstream distance and phase. The phase angle is equivalent to time, modulo the passing period φ = 2π(t/T )mod 1 . In a lab experiment it is a challenge to accumulate these data. However, in DNS experiments averaging over samples both at constant phase and in the direction of homogeneity enables statistical ﬁelds to be accurately computed. A database of transition induced by period wakes has been created by Wu et al. (1999). Reynolds averaged computations of these data are discussed in Wu and Durbin (2000). The ﬂow conﬁguration is illustrated by ﬁgure 9. This shows wakes in the free-stream sweeping over the boundary layer. As time progresses the wakes convect from left to right. As they convect downstream, the centerline of a given wake traverses down a constant x section; indeed, the sloped wake was created by traversing a wake down the inlet plane. Figure 9 illustrates how the boundary layer thickens in the vicinity of x = 1.0, where it is highly perturbed by the impinging wake. Downstream of that it thins before thickening again beyond x = 1.5. In this furthest downstream region, the boundary layer has become fully turbulent, but upstream it is a highly buﬀetted, laminar layer. The upper frame of ﬁgure 9 shows instantaneous contours of the ﬂuctuating u-velocity. The central frame is a spanwise average at the same phase, while the lower frame is a RANS computation. All three show local thickening of the boundary layer beneath the wake within the laminar region and transition to a thicker, turbulent boundary layer farther downstream. It is clear from this ﬁgure that RANS models have the potential to represent, to a signiﬁcant degree, the buﬀeted laminar layer and the transition to turbulence. However, too much should not be read into the level of agreement between RANS and

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Figure 9: Contours of streamwise velocity component: (a) instantaneous u over one xy plane; (b) phase-averaged u; (c) unsteady RANS using v 2 -f model. From Wu and Durbin (2000). DNS in ﬁgures 9 and 11. For example, it is unlikely that pressure gradient and curvature eﬀects would be captured correctly at the level of modeling used in these ﬁgures. The RANS computations were done with a model developed and calibrated for fully turbulent ﬂow (Wu and Durbin 1999), with no modiﬁcation speciﬁcally for transition. There is probably a need to modify turbulence models if they are to be used in ﬂows in which transition must be computed with a high degree of accuracy. In other ﬂows, the crude ability of turbulence models to undergo a form of bypass transition might be suﬃcient. Prior to the fully turbulent zone, demarcated by boundary layer growth in ﬁgure 9, puﬀs are seen in contour plots of u near the wall (Wu et al. 1999). These sometimes develop into turbulent spots, depending on how they interact with the free-stream turbulence. The initial irregular motion that leads to turbulent spots is created by certain free-stream eddies that cause jetting motion near to the wall. The jet is in the upstream direction relative to the mean ﬂow. In ﬁgure 10 the phase averaged velocity has been subtracted from the instantaneous velocity to show the motions that are responsible for spot precursors. The ﬁgure shows a lifted shear layer that undergoes instabilities on a rapid time-scale. The three sections of this ﬁgure are three successive

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Figure 10: Development of a backward jet in the relative velocity ﬁeld and its breakdown to turbulence. The ﬁgures show the development of a spot precursor and its breakdown via inviscid instability. These are successive instants, with time increasing from top to bottom. instants in time, with time increasing from top to bottom. They are spaced by a time interval of 0.1 of the wake passing period, T . The time-step in the simulation was 10−3 T . The x range portrayed moves downstream with the wake. The topmost ﬁgure is suggestive of an eddy at x ∼ 1.0, y ∼ 0.01 triggering subsequent highly irregular motions. The instabilities that grow on the negative jet develop small-scale eddies that evolve into the turbulent spots.

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6π

φ

0

0

3.5

6π

φ

0

0

x

3.5

Figure 11: Contours of skin friction vs. x and φ. (a) phase-averaged u; (b) unsteady RANS using v 2 − f model. From Wu and Durbin (2000). A plan view of velocity contours would show a nascent spot at the last instant of ﬁgure 10. At a somewhat later stage a fully formed spot, as in ﬁgure 2, would emerge. Although the occurence of turbulent spots in transitional ﬂow has been known from experiments, an understanding of the precursors to developed spots has only recently begun to evolve from numerical simulations such at these. By the time turbulent spots are detected, the process has already become chaotic. Phase averaged data for periodic wake induced transition can be found in Wu and Durbin (2000). Figure 11 shows Cf (x, φ) contours for a ﬂat plate boundary layer beneath periodically passing wakes. The upper part of the ﬁgure consists of DNS data; the lower shows a RANS simulation of the same conﬁguration. These are contour plots. In a slice parallel to the x-axis, at constant φ, the contour elevations become a skin friction curve Cf (x) at that particular phase. Because the tongues between x ≈ 1 and x ≈ 2.25 in ﬁgure 11

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extend upstream at an angle to the axes, the constant phase section traverses from low, to high, to low and back to high Cf . The ﬁrst region of high Cf occurs where the external wake impinges on the laminar boundary layer. The periodic tongues in ﬁgure 11 are imprints upon the laminar boundary layer of the passing wake. The boundary layer is highly perturbed but the wakes do not cause transition until further downstream. Beyond x ≈ 2.5 the boundary layer is fully turbulent. The periodic buﬀetting by the passing wake is still seen in the turbulent region, but there it is a minor eﬀect. Single point closure models are meant for use in Reynolds averaged computational ﬂuid dynamics codes. They are routinely tested against a range of experiments. The same should be done for bypass transition models. DNS provides comprehensive databases to use in testing. In the present example of wake induced transition, quantitative phase averaged ﬁelds of velocity, skin friction and second order statistics are now available. Eﬀects of pressure gradients and curvature will be included in future simulations. These will greatly enhance the ability to tackle computational predictions of complex engineering ﬂows.

References Abu-Gannam, B.J. and Shaw, R. (1980). ‘Natural transition of boundary layers—the eﬀect of turbulence, pressure gradient and ﬂow history’, J. Mech. Eng. Sci. 22, 213–228. Andersson, P., Berggren, M. and Henningson, D. S. (1999). ‘Optimal disturbances and bypass transition in boundary layers’, Phys. Fluids 11, 134–150. Drazin, P. and Reid, W.H. (1981). Hydrodynamic Stability, Cambridge University Press. Hunt, J.C.R. and Durbin, P.A. (1999). ‘Perturbed shear layers’, Fluid Dyn. Res. 24, 375–404. Jacobs, R.G. and Durbin, P.A. (2000). ‘Bypass transition phenomena studied by computer simulation’, report TF-77, Department of Mechanical Engineering, Stanford University. Jacobs, R.G. and Durbin, P.A. (1998). ‘Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation’, Phys. Fluids 10, 2006–2011. Kendall, J.M. (1991). ‘Studies on laminar boundary layer receptivity to free stream turbulence near a leading edge’. In Boundary Layer Stability and Transition to Turbulence, D.C. Reda et al. (eds.), ASME-FED, 114, 23–30. Liu, X. and Rodi, W. (1991). ‘Experiments on transitional boundary layers with wake-induced unsteadiness’, J. Fluid Mech. 231, 229–256. Roach, P.E. and Brierly, D.H. (1990). ‘The inﬂuence of a turbulent freestream on zero pressure gradient transitional boundary layer development, part I: test cases T3A and T3b’. In Numerical Simulation of Unsteady Flows and Transition to Turbulence, O. Pironneau et al. (eds.), Cambridge University Press, 229–256.

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Savill, A.M. (1999). ‘One point closures applied to transition’. In Turbulence and Transition, M. Hallb¨ ack et al. (eds.), Kluwer, 233–268. Simon, F. and Ashpis, D. (1996). ‘Progress in modeling of laminar to turbulent transition on turbine vanes and blades’, NASA-TM 107180. Wu, X. and Durbin, P.A. (2000). ‘Boundary layer transition induced by periodically passing wakes’, J. Turbomachinery 122, 442–449. 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. Yang, Z. and Voke, P. (1995). ‘Numerical study of bypass transition’, Phys. Fluids 7, 2256–2264.

17 By-Pass Transition using Conventional Closures A.M. Savill 1

Introduction

In attempting to apply turbulence models to transitional ﬂows most practitioners either assume ‘point transition’ at an estimated transition location, and force a switch between laminar and turbulent computations at that point, or make use of experimentally-derived correlations to estimate the start of transition and then ‘ramp-up’ the turbulent eddy viscosity by scaling this with an empirical transition function until fully turbulent conditions are attained. These two approaches are generally used in conjunction with mixing length models or even simpler integral methods which remain the most common methods used today for everyday engineering design purposes. However Priddin (1975) showed that a low-Re k-ε model treatment (due to Jones and Launder (1972)) could be used to predict transitional ﬂow as well. Subsequently a few other successful low-Re k-ε model variant applications were reported by Arnal et al. (1980), Dutoya and Michard (1981) and then Liu (1989). Other encouraging results were obtained by Arad et al. (1982), using the alternative two-layer k-kl model scheme of Ng, (1971), by Wilcox (1992) using his alternative two-equation k-ω model, and even with a oneequation k-l model adapted by Manartonakis and Grundmann (1991). At the same time Rodi and Scheuerer (1984) used the low-Re k-ε model of Lam and Bremhorst – see Patel, Rodi and Scheuerer (1985) – to perform the ﬁrst extensive evaluation of such transition predictions for a range of simple test cases and a turbine blade. For free-stream turbulence intensities > 1% they found this model could quite accurately predict the observed by-pass transition (provided the initial conditions were carefully controlled). But early attempts to employ higher level Reynolds Stress Transport (RST) closures, by Donaldson (1969) and Finson (1975) failed to give quantitatively similar agreement with experiment. The ﬁrst direct inter-comparison of diﬀerent modelling techniques for predicting by-pass transition – ranging from the simplest correlation methods to direct numerical simulations – was reported by Pironneau, Rodi, Ryhming, Savill and Truong (1992). Two test cases were considered, based on data provided taken from a series of experimental studies conducted by researchers at Rolls-Royce. These involved transition on a sharp-leading-edge test plate 464

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(carefully controlled to avoid leading-edge separation and to maintain nominally zero pressure gradient) under the inﬂuence of precisely isotropic 3% (T3A Test Case) and 6% (T3B) free-stream turbulence (fst). A very wide range of results was obtained from the diﬀerent computational approaches adopted. Although industrial design methods incorporating the well-known Abu-Ghannam and Shaw (1980) correlations for the start and end of transition (which in fact ﬁtted the Rolls-Royce data well) provided adequate predictions, it was clear that the success of more academic, but potentially more predictive, transport model schemes was highly dependent on the exact closure approximations adopted. However the best of these produced excellent results, and the interest this generated led to the setting up of an ERCOFTAC Transition Modelling Special Interest Group (SIG) Project to evaluate a wider range of candidate turbulence models against a larger series of transition test cases – see Savill (1992, 1993a,b, 1994, 1996). The present contribution is based on the work of this Transition SIG and a number of similar projects to assess turbulence models for predicting transition. These include studies for natural convection ﬂows, reported by Henkes (1992); for pipe ﬂow, as reported by Jackson and He (1995); and for relaminarising ﬂow reported by Viala et al. (1995); as well as a similar AFOSR/NASAled evaluation exercise on by-pass transition modelling by USA research groups – see Simon (1994) and others, all of which have suggested that the LaunderSharma low Re k-ε model oﬀers generally the best predictions of any standard eddy viscosity approach. The ERCOFTAC Project results have proved particularly instructive because Test Case computations were initially done ‘blind’ (i.e. without access to, or indeed in some cases in advance of, Test Case data), using only speciﬁed initial/boundary conditions. In each case experimental data of high quality and detail, complemented by suﬃciently resolved numerical simulations, were provided to assess the predictions. The necessary industrial input was also secured to ensure relevance of the comparisons. In addition participants were subsequently encouraged to perform tests of the sensitivity of their results to the imposed conditions, model constants, and mesh resolution. This was particularly important since the very wide range of models being evaluated precluded the speciﬁcation of ﬁxed grids. Finally authors were asked to reﬁne or otherwise modify their model approaches before re-testing these on a wider range of cases listed in the table below.

2

Applying turbulence models to prediction of transitional ﬂows

In attempting to use (high-Re) turbulence models to predict transition one can take two diﬀerent approaches. One can either adopt the premise that the transition process can be modelled as a superposition of laminar and turbulent

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Table 1: ERCOFTAC Transition Modelling SIG Test Cases T3A-: T3A: T3A+: T3B: T3BLES: T3B+: T3C1-5:

T3D1-3: T3E: T3F : T3G : T3H : T3L1-6: T3K:

zero pressure gradient, 1% isotropic free-stream turbulence (theoretical or experimental initial conditions) zero pressure gradient, 3% isotropic free-stream turbulence (theoretical or experimental initial conditions) zero pressure gradient, 3% isotropic fst, but variable Lue (theoretical or experimental initial conditions) zero pressure gradient, 6% isotropic free-stream turbulence (theoretical or experimental initial conditions) zero pressure gradient, 4.5% weakly anisotropic free-stream turbulence (Simulated initial conditions) zero pressure gradient, 10% weakly anisotropic free-stream turbulence (experimental initial conditions) pressure gradient representative of aft-loaded turbine blade (expt. initial conditions; various Re): C1 (6%fst) & C2 (3%fst, same design Re); C3 & C4 (3%fst , lower Re without & with laminar separation) ; C5 (3%fst, higher Re) zero pressure gradient, 0.1% isotropic fst, following laminar separation (experimental conditions; various Re) strong favourable/adverse pressure gradient, 0.1%fst, relaminarisation/retransition (experimental initial conditions) weak & strong convex curvature, 0.7 & 2% free stream turbulence (experimental conditions) weak & strong concave curvature, ≈0.6% and 8.5% or 2.5% fst (experimental conditions) zero pressure gradient, 5% fst with heated surface (experimental conditions, variable Re etc.) semi-circular leading edge, 0.2% free-stream turbulence (experimental initial conditions; various Re, fst, etc) Low-Speed (LP Rotor) Turbine Cascade, ≈4% free-stream turbulence (experimental initial conditions)

NB. The T3A&B Test Cases have also been adopted for the NASA Project

ﬂow in changing proportions of time, γ, that the ﬂow is turbulent, i.e. – taking γ times a turbulent solution and (1 − γ) times laminar (where γ is an intermittency factor varying from 0 at the start of transition to 1 at the end) or introduce additional (low-Re) model approximations to handle the turbulence development through transition. The ﬁrst of these approaches could, in principle, be applied to any type of transition (natural, Taylor–G¨ ortler, by-pass), but requires some empirical input regarding the location of transition onset and the precise streamwise variation of γ. It is often used to improve point transition methods by including an allowance for the ‘length-of-transition’ over which γ varies from 0 to 1.

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The second approach requires that there should be some initial source of turbulence activity within, and/or outside the initial (pseudo-laminar) shear layer, and makes the inherent assumption that the transition is ‘diﬀusioncontrolled’ in the sense that it is triggered by diﬀusion of free-stream turbulence into the ﬂow. Such ‘by-pass transition’ only occurs for free-stream turbulence intensity levels > 1%. It was given this name by Morkovin because most of the development stages in natural transition are then by-passed and transition to turbulence then occurs at a point where breakdown of free-stream turbulence-induced streaks and secondary instability produces (in 2D) a line of turbulent spots across the span of the ﬂow. This is why most current industrial design methods still use a simple ‘point transition’ approach whereby an initial laminar boundary layer computation is triggered to turbulent at a point determined either empirically or via an established correlation of relevant experimental data. The present contribution concentrates on the latter approach, because this appears to oﬀer potentially the greatest generality in terms of predictability and range of applicability, but some discussion is included concerning the former, not least because recent work has shown the advantages to be gained from combining both approaches. It has to be stressed however that neither approach considers many of the individual mechanisms and stages of transition discussed for example by Hallb¨ ack et al. (1996). The low-Re modelling approach takes no speciﬁc account of receptivity, algebraic growth, secondary instability, or turbulent spots. Instead, the modelling implicitly assumes that diﬀusion of (generally isotropic) free-stream turbulence leads to a build-up of weakly correlated turbulence activity in the initial pseudo-laminar (essentially Blasius proﬁle) boundary layer and that transition is initiated once the local production of turbulence energy suﬃciently exceeds the local dissipation rate, ε. As in the case of fully turbulent ﬂows, all low-Re models are rather insensitive to the actual turbulent (spot) structure. Intermittency models do attempt to account for spot formation and growth rates, with varying degrees of sophistication based on an ever-expanding experimental database; but only a few of these make any allowance for receptivity and algebraic growth, and even those that do, lump these together with secondary instability eﬀects in so-called ‘sub-transition’ corrections.

2.1

Low-Reynolds Number Transport Models

A very large number of low-Re one and two-equation models (k-l, k-ε, k-τ and k-ω) have now been evaluated. These include most, if not all, of the best known variants including the models of Abid, Birch/Hassid–Poreh; Chien, Chang, Dutoya–Michard, McDonald and Fish, Hanjalic, Jones and Launder, Lam and Bremhorst, Launder–Sharma, Lien–Leschziner, Michelassi, Myong– Kasagi, Nagano–Hishida, Nagano–Tagawa, Kasagi–Sikasano, Biswas and Fuku-

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yama, Norris and Reynolds, Shih, To and Humphrey, Yang and Shih, Wilcox and Wolfshtein (for summary presentations and cross comparisons of formulations see Patel et al (1985), Pironneau et al (1992) and Henkes (1992)). Each version introduces diﬀerent low-Re extensions to the standard high-Re

tur1 2 2 2 bulence model equations for turbulence energy – k = 2 u + v + w and dissipation, ε, (or length scale) combinations – see for example Hallb¨ ack et al. (1996) – within the following basic set of low-Re equations for 2D shear ﬂow: ∂U 2 Dk ∂ νt ∂k + = νt ν+ Dt ∂y ∂ σk ∂y ADVECTION

PRODUCTION

VELOCITY DIFFUSION LAMINAR & TURBULENT

+

−

Πk PRESSURE

(1)

ε DISSIPATION

DIFFUSION

Dε ε = E + f1 Cε1 Dt k

∂U ∂y

2 +

∂ ∂y

νt ∂ε ε2 ν+ + Πε − f2 Cε2 σε ∂y k

(2)

(for numerical convenience often written in terms of ε˜ which equals 0 at the wall boundary) where ντ = fµ Cµ k 2 /ε and ε = ε˜ − D; D = εwall = 0; 2 E = Cε3 ννt ∂ 2 U /∂y 2 . The quantities E, fµ , f1 and f2 are low-Re damping factors, and Cε1 , Cε2 and Cε3 model constants, which vary for diﬀerent low-Re formulations. The associated eddy viscosity and dissipation damping factors (fµ and Fε ) are always functions of the Reynolds-number formulations adopted by the various model developers: Ry ≡

k 1/2 y uτ y y k2 k 3/2 νt lε ; R ≡ ≡ ; y+ ≡ ; Rε ≡ ∝ ; R = . t L 1/4 ν ν εν ν εη η (ν/ε)

It must be stressed that not all variants require inclusion of the full set of lowRe damping factors, and/or additional D and E factors in the representative velocity and length-scale transport equations, to ensure the modelling can be extended right through the buﬀer layer where high-Re models normally employ wall functions. It should also be noted that all current low-Re, eddy-viscosity models were originally developed to handle low-Re near-wall regions of turbulent ﬂows and that no models have so far been proposed primarily to model low-Re transition regions. This applies equally to low-Re Reynolds-stress transport (RST) schemes.

2.2

Alternative Correlation and Intermittency-Scaling Models

Intermittency weighting for transitional ﬂows has until recently only been employed in simple closure models, at least for design purposes. In particular

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a number of diﬀerent integral/correlation methods have been developed and used to some eﬀect. The ﬁrst moderately successful model was that of Dhawan and Narasimha (1958), which determined the onset of transition (at Rexs ) by extrapolating Narasimha’s own assumed universal intermittency distribution as a function of x, and proposed a correlation for the length of transition, λ, in terms of Rexs itself. Other well-known examples are the methods of Fraser and Milne (1986), which uses the Abu-Ghannam and Shaw (1980) correlation for the start of transition and their own correlation for transition length, and the Narasimha and Dey (1989) approach, which uses Narasimha’s alternative correlations for Rexs coupled with his later spot-formation model calibrated against a slightly modiﬁed version of Dhawan and Narasimha’s correlation for transition length. More recently Costa and Arts (1991) have modiﬁed Dhawan and Narasimha’s model to account also for the eﬀects of acoustic excitation at discrete frequencies for low fst (Tu = 0.4%). The same type of intermittency-weighting has now been utilised in a wide range of mixing-length models. Boyle (1990) has obtained some reasonable predictions for heat transfer on a range of turbine-blade test cases by introducing the standard correlations of Dunham (1972), Seyb (1971) and Abu-Ghannam and Shaw (1980) for Rexs , and Dhawan and Narasimha’s correlation for Reλ (with Abu-Ghannam and Shaw’s intermittency function), into a modiﬁed Baldwin–Lomax (1978) model. Singer et al. (1991) have tested the performance of both Narasimha and Dey’s linear-combination model and an alternative empirical ‘intermittency’ transition factor, proposed by Coustols and used extensively by Arnal and colleagues at CERT ONERA (see Arnal et al. (1980) and Arnal (1991)), which is a function of the momentum thickness at the start of transition, θxs , in both the widely used Baldwin–Lomax and Cebeci–Smith mixing-length models – see Hallb¨ ack et al. (1996). The results agreed closely with predictions obtained from Narasimha and Dey’s integral method for zero-pressure, gradient-low fst ﬂows. Reasonable comparison with experiment was also obtained for a range of simple test cases of by-pass transition, introducing eﬀects of variable adverse pressure gradient and concave stream-line curvature. However, some serious discrepancies were found for ﬂows inﬂuenced by favourable pressure gradients or 3D eﬀects. In addition it was concluded that a separate intermittency parameter might be needed to describe the development of any thermal ﬁeld. The McDonald and Fish (1973) ‘integrated’ one-equation model, which is used in other industrial design methods e.g. by Rolls-Royce and SNECMA, also utilises a transition weighting factor which is a function of θxs to scale the structure parameter a1 (≡ uv/k) and this produces quite good predictions for transition onset, at least at fst levels less than 1%. (Note that Costa and Arts (1991) found good agreement between their integral method results for

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Tu= 0.4%, in the absence of acoustic excitation, and those of McDonald and Fish for Tu= 0.25, 0.5 and 1.0% – nominally all in the ‘natural transition’ regime.) At the same time Young et al. (1993) have introduced the Dhawan and Narasimha intermittency description (with Rexs prescribed from experiment) into the one-equation model of Micheltree et al. (1993) and the ASM-l model of Gaﬀney et al. (1990). In both cases the mean ﬂow was separated into turbulent and non-turbulent components ‘conditionalised’ by γ. The only turbulencemodel modiﬁcation was to introduce a similar type of linear-combination allowance for the inﬂuence of Tollmien–Schlichting (T-S) waves on the mixinglength scale. Good agreement was obtained for the case of natural transition as studied originally by Schubauer and Klebanoﬀ (1955). Allowance for T-S waves had little eﬀect on the Cf predictions, but the predicted turbulence intensity proﬁles were considerably improved when this was included. Although the linear-combination concept inherent in the use of intermittency functions appears to work well in practice, experimental data indicate that transitional proﬁles are not a simple combination of purely laminar and fully turbulent forms. Therefore, in order to take better account of intermittency, zonal conditional averaging of the mean and turbulent equations is required. It should also be noted that only in those models which have a basis in the turbulent spot description of transition can the intermittency function clearly be identiﬁed as a discriminator between turbulent and non-turbulent ﬂow regions. In other cases the ‘intermittency factor’ is merely a transition weighting function for the variation of ﬂow parameters from laminar to turbulent states, and, as in the case of the ONERA approach, may exceed 1 in order to capture an observed overshoot of Cf near the end of transition. The theoretical analyses of Libby (1975, 1976), and subsequent work of Dopazo (1977) and Chevray and Tutu (1978) have laid the foundation for developing models based on true intermittency conditional equations. This is the approach adopted by Steelant and Dick (1995). They follow exactly the original work of Libby to derive separate equations for laminar and turbulent zone mean velocities, as well as equations for k and (simply by analogy) ε, which contain additional source terms relating to crossings of the turbulent/nonturbulent interface (and in the process this introduces a separate allowance for velocity and pressure-diﬀusion eﬀects which is now also known to be important to correctly model the ﬂow physics; see Savill [18]. Simplifying assumptions are introduced in order to model these, primarily by considering only an idealised purely 2D turbulent spot geometry, and closure is eﬀected using the Jones–Launder or the very similar Launder–Sharma version of the low-Re k-ε model equations – see Patel et al. (1985).

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The extra source terms added to equations (1) and (2) take the form: ∂γ U − Uo Sk = Ck ννt δ ∂x and Sε = Cε (ε/k)Sk , where Ck = 60, Cε = C1 Ck , δ denotes the local boundary layer thickness and U − Uo an entrainment velocity. However, in practice it was found that Cε should be a function of γ also. The intermittency variation adopted was that of Dhawan and Narasimha, with speciﬁed Reynolds numbers for the start and length of transition. Good agreement with experiment has been obtained for Tu= 0, 0.15% and 5% in zero pressure gradient, and 1.2%fst in favourable pressure gradient, although for the highest fst level the above additional source terms were neglected. Simon and Stephens (1991) have followed a similar approach, but introduced some further simplifying assumptions in order to derive a single global mean ﬂow equation and have combined this with an extended form of the Jones–Launder k-ε model in which again all the turbulence terms are conditionalised by an intermittency factor, this time using the prescription adopted by Narasimha and Dey (1989). Their method predicts transition onset in excellent agreement with the Abu-Ghannam and Shaw correlation over a range of fst from 1.4% to 6.2%. The predicted Stanton number distributions also agree closely with the data of Blair and Werle (1981). The use of prescribed intermittency variations is thus well established for transition prediction, but there have been few attempts thus far to develop model equations for the intermittency itself.

2.3

Typical Results For Simple Test Cases

During the last 10 years extensive parabolic model calculations have been performed for a series of sharp, leading-edge test cases subjected to variable free-stream turbulence, intensity, anisotropy, length-scale, and pressure gradient – e.g. see Table 1 – while elliptic computations have more recently been reported for some sharp and ﬁnite leading edge cases, thereby allowing some initial conclusions to be drawn regarding model performance. A large number of recent low-Re k-ε results obtained for zero pressure gradient, but variable external turbulence, boundary-layer ﬂows have now conﬁrmed earlier ﬁndings that the Launder–Sharma model variant (which employs Rt -dependent damping functions) is the best for predicting such by-pass transition; certainly better than other models employing Ry damping factors (see Figures 1 and 2). This is partly because the Launder–Sharma model incorporates fµ and D factors that produce the correct near-wall power law

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Figure 1: Comparison of predicted skin friction distributions from diﬀerent low Re k-ε models with Rolls Royce data for the T3A Test Case (3% free stream turbulence). dependence of uv and ε, whereas other models generally only satisfy the ﬁrst or second of these conditions. However Ry -dependent models, which have been modiﬁed to ﬁt exactly ‘data’ from direct numerical simulations of low-Re turbulent wall ﬂow, also failed to produce better predictions, while similarly poor predictions have also been obtained with the alternative Ry -dependent treatments of Nagano and Hishida (1987) and Myong and Kasagi (1990) – even after some modiﬁcation (see Pironneau et al. (1992)) – despite the fact that all these models provide better predictions for a wider range of turbulent ﬂows than many other older low-Re k-ε schemes. Encouragingly the ‘better’ models also showed the correct sensitivity to changes in free-stream boundary conditions, but were not particularly sensitive to the starting location or the initial conditions; except those for dissipation (free-stream length-scale). Other models showed greater sensitivity to initial conditions – again see Pironneau et al. Generally the best k-ε models were also better than the best k-l ones. For example, blind predictions performed for 3% and 6% (T3A & T3B) test cases by researchers at SNECMA, using a McDonald and Fish low-Re k-l design method, proved to be rather poor. Transition was predicted far too late in

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Figure 2: Shape factor predictions corresponding to Figure 1. both cases. Better results were obtained by Rolls-Royce using the alternative Birch/Hassid-Poreh Rolls-Royce k-l method, but like many other simple models this proved insuﬃciently sensitive to the inﬂuence of the pressure-gradients imposed in subsequent sharp-leading-edge (T3C) test cases. The Launder–Sharma k-ε model did correctly predict a delay in transition onset with acceleration, but, as expected, failed to capture the correct degree of sensitivity to pressure gradient variations representative of an aft-loaded turbine blade operating at its design condition unless an additional straindependent correction term was introduced. This model also proved to have two more generally important defects as exhibited by most alternative Rt -based schemes as well. Firstly, it underpredicts the extent, and hence the end, of transition and, secondly, it requires a large number of points across the boundary layer (> 80) for grid independent solutions. In fact it is apparently a little more sensitive to initial conditions and more demanding of grid resolution than some alternative low-Re formulations that work equally well or better in near-wall regions of turbulent ﬂows. A further numerical disadvantage is that it introduces both additional D and E terms into the transport equations for the k and ε respectively. Introduction of the latter involves higher order derivatives of the mean ﬂow which are inconvenient to evaluate in any 3D application. Successful extension to an elliptic framework also appears to require more precise discretisation of the near-wall D term, and careful treatment of both stream-wise and shear-wise derivatives, in addition to very ﬁne grid resolution. However, it has to be noted that the

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ﬁrst Launder–Sharma model results for a laminar backstep test case (T3D) compare favourably with experimental data and preliminary simulation results – see Savill (1994). There is of course no a priori reason why one should expect to predict low-Re transitional ﬂows successfully with any of the current low-Re models since, as stated above, these have all been developed to model only the lowRe near-wall regions of fully turbulent ﬂows. Fortunately it would seem that the turbulence Reynolds number is a suﬃciently general property of low-Re ﬂows that one can in fact obtain reasonable predictions, at least for integral properties, with such a gross assumption. Other advantages of Rt -dependent models are that they can easily be applied to non-planar wall geometries and incorporated into unstructured codes, and these provide powerful reasons for concentrating on them for practical ﬂow computations. However it must be regarded as at least partly fortuitous that the Launder– Sharma functions damping of uv with y happens to give roughly the correct functional change in uv with x through transition (as revealed by the ﬁrst by-pass transition simulations of Voke and Yang (1993)), since other Rt -based models, including the very similar Jones–Launder scheme, predict very diﬀerent transition locations – see Figures 1 and 2. In fact, the alternative Dawes (1992) Rt -based low-Re, k-ε model provides the best current agreement with the simulated variation of the principal (eddy viscosity) damping factor fµ through transition – see Figure 3. This helps to explain why the Dawes code employing this has been so successfully applied to real turbomachinery blading ﬂows. The Launder–Sharma scheme also fails to predict the correct development of the various turbulence proﬁles. Although it predicts a too rapid growth of k through transition, and hence a too short a transition length, and a Cf overshoot, the growth of the turbulence energy peak in the boundary layer is underpredicted as are the uv proﬁles. As any isotropic eddy viscosity model, it incorrectly predicts similar proﬁles for all the normal stresses as well. Comparisons of turbulence proﬁle developments for other models, which do not predict the location of transition or its extent correctly, are not meaningful.

3

Some simple model reﬁnements for improving predictions

One would generally expect that low-Re models which contain damping factors that are only a function of wall-proximity could be improved for use on transitional ﬂows by introducing additional x-dependent damping functions and reducing their Ry dependence in favour of Rt . This is indeed the case and a number of such improvements have already been employed in practice to overcome the deﬁciencies of progressively higher order closures.

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Figure 3: Comparison of fµ functional distributions extracted from LES (—): well before transition (x = 25mm from sharp leading edge), in pseudo-laminar boundary layer (x = 45mm), at start of transition (x = 95mm) and at end of transition (x = 195mm); versus models of Launder–Sharma (◦), Lam– Bremhorst (+), Chien (∆) and Dawes (– – –).

3.1

Integral Methods

Gostelow et al. (1994) have found that a new correlation for Re, based on a wide range of data from the University of Technology, Sydney, provides better prediction of Cf and H for the ERCOFTAC zero pressure gradient 3%fst test case, and a better transition length prediction for the higher 6%fst, than Narasimha and Dey’s model. Prihoda et al. (1998), at the Institute for Thermomechanics in Prague, have also produced equivalent results for the 3% and 6% test cases using a similar approach. Subsequently the UTS group have obtained better results for these two cases with a modiﬁed version of the Gostelow et al./Narasimha and Dey

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Figure 4: Eﬀect of introducing intermittency scaling to low-Re k-ε predictions for 3%fst (T3A) test case: - - original Nagano–Hishida model; – – – modiﬁed Nagano–Hishida model; versus original Launder-Sharma model —; symbols – experiment, see Pironneau et al. (1992). integral method which has been shown to produce good predictions for a lower (1%)fst and the on-design turbine-blade pressure-gradient test cases as well.

3.2

One-Equation Models

Manartonakis and Grundmann (1991) have obtained far better results for bypass transition with the McDonald and Fish approach by introducing alternative transition weighting factors, for both a1 and the turbulence length-scale l, which are functions of the external turbulence intensity and length scale. Very good results have now been reported for the 3% and 6% zero and variable pressure gradient test cases – see Savill (1993a, 1993b) – but necessary variations in the two additional model constants that have been introduced are not yet well deﬁned.

3.3

Two-Equation Models

There have been two recent attempts to improve transition prediction with low-Re k-ε models, that have only a wall-proximity dependence, by introducing additional streamwise directional dependence. Goulas and colleagues (Theodoridis et al. 1991) have used a modiﬁed Dhawan and Narasimha intermittency function to scale the eddy viscosity computed from the Nagano– Hishida model, and shown that results comparable to the Launder–Sharma model can then be obtained for zero pressure gradient 3 and 6%fst test cases – see Figures 4 and 5.

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Figure 5: Eﬀect of introducing intermittency scaling on predictions for 6%fst (T3B) test case: — · Original Yang–Shih model; — modiﬁed Yang–Shih model; standard Launder–Sharma model - -; symbols – experiment, see Pironneau et al. (1992).

Yang and Shih (1991), at NASA Ames, have adopted a slightly diﬀerent approach in attempting to improve the rather poor transition predictions produced by their Ry -based model derived from detailed analyses of DNS results. They used a weighting factor ‘γ’ (which was only linearly related to the true intermittency, although it was evaluated from the Abu-Ghannam and Shaw data, and used ‘γ’ = 1 in the turbulent free-stream, which is contrary to normal practice for intermittency descriminators which assume γ = 0 where boundary layer ﬂuctuations are negligible, even if the external turbulence intensity is non-zero) to scale all the turbulence terms in the momentum, k and ε equations, including their Launder–Sharma-type additional E factor. Excellent predictions were produced for both 3 and 6%fst test cases, slightly better even than those obtained with the Launder–Sharma model – see Figures 4 and 5. Abid (1993) has also demonstrated that the Myong–Kasagi model predictions can indeed be signiﬁcantly improved by reducing the dependence of the damping functions on wall proximity and substituting a dependence on Rt instead, but also by converting the original Myong–Kasagi model from a k-ε to an equivalent k-τ formulation (τ = k/ε) – see Figure 6. A diﬀerent variant of this model and the similar models of Nagano and colleagues, (Nagano and Hishida 1987; Nagano and Tagawa 1990) developed by Biswas and Fukuyama (1994), also removes almost all Ry dependence; avoids the use of both D and E factors; and has produced equally good predictions for

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Figure 6: Eﬀect of reducing wall-proximity dependence and switching from ε to τ in Myong–Kasagi k-ε model predictions for 3%fst test case: - - - with original y + -dependent damping functions; — converted to Rε -dependence; - – - converted to Rt dependence and; — · with τ in place of ε also. the zero and variable pressure gradient cases as the Launder–Sharma model – see Savill (1993a, 1993b). The value of switching from Ry to Rt damping factors has also been indicated by the success of the Dawes model which assumes Rt = 0.4Ry (that simulations show is the case for the very near wall region of turbulent ﬂows), in order to adapt a Reynolds (fµ )/Lam and Bremhorst (fε ) hybrid y-dependent low-Re model for use in an unstructured code. (Note that Rodi and colleagues have shown that it is also possible to obtain very good predictions with the original Ry -dependent Lam and Bremhorst model, at least in parabolic boundary layer computations, provided speciﬁc empirical adjustments are made to take account of free-stream turbulence level and pressure gradient variations again – see Pironneau et al. 1992). Regarding the key deﬁciencies noted for the Launder–Sharma model (and its closest equivalents): (a) Transition length prediction deﬁciencies may be rectiﬁed by introducing a ‘Production Transition Modiﬁcation’ (PTM), whereby the production of turbulence energy is artiﬁcially limited by an empirical scaling through transition to ensure a close ﬁt of the predicted end of transition with the correlation of Abu-Ghannam and Shaw, although the same result can be obtained by introducing an intermittency factor – see Simon (1994). (b) It had been thought that there would be even greater beneﬁts associated with the switch from ε to ω (≡ 1/τ ) as the length-scale dependent variable, since it is known that the k-ω model does not require a

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full low-Re implementation to handle turbulent near-wall ﬂows and is therefore considerably less demanding of grid resolution for boundary layers. However there are then problems with the imposition of a freestream cut-oﬀ boundary condition and low-Re damping factors are still required to handle transitional ﬂows. Although the Wilcox (1992) lowRe k-ω model employs solely Rt -dependent damping factors, predictions obtained at UMIST and elsewhere within the SIG, for the start and end of transition in the T3A & B cases have proved disappointing. (c) All k-ε models exhibit signiﬁcant discrepancies in their predictions for the eﬀect of variable pressure-gradient on transition. As indicated above, this is also a problem encountered with eddy viscosity models in general, and correction factors need to be introduced to sensitise them to individual strain rates in transitional and turbulent ﬂows. Unfortunately although several empirical connections have been tested, these have so far been found to lack generality for more complex ﬂows. The same is true of the strain-dependent RNG k-ε model which, although derived from a purely theoretical (Renormalisation Group Theory) analysis of the Navier–Stokes equations and diﬀerent simplifying assumptions, only provides an alternative, equally approximate, viscosity transport scheme, (albeit with directly implied tunable strain-dependent constants), and has not shown any advantages for transitional ﬂow predictions. Some progress has been made in increasing the sensitivity of eddy viscosity models to the irrotational straining encountered in stagnation regions, but the modiﬁcations employed are not suﬃciently general to also handle subsequent shear-layer curvature and pressure-gradient eﬀects. For variable pressure-gradient ﬂows, greater success has been obtained using a two layer kε/k-l treatment with the near-wall prescribed length scale providing a matching condition for the k-ε model employed in the outer boundary layer. However, this approach necessitates the inclusion of empirical correlations which are better deﬁned for adverse than favourable pressure gradients and therefore it is of limited predictive capability, particularly where strain history eﬀects are important.

3.4

Non-Linear Viscosity Models

To improve predictions with isotropic eddy-viscosity transport models requires better accounting for the local strain-ﬁeld eﬀects encountered even in simple attached variable-pressure-gradient boundary layers. To handle those subjected to convex and concave streamline curvature (and in addition swirl and rotational eﬀects on shear layers in general) it is clear that some kind of formal anisotropic extension to the Boussinesq stress-strain relation is necessary). This requirement has led to the development of a range of explicit non-linear

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Figure 7: Nonlinear k-ε model predictions (—) versus linear Launder–Sharma model (- -) and experimental results for 3%fst () and 6%fst (◦). k-ε (and even k-ω) models with terms up to (most commonly) quadratic, but sometimes cubic, order in both the rotational and shear strain invariants; as well as some alternative (implicit) algebraic-stress-model formulations derived from full Reynolds-stress-transport model closures. Careful analysis has shown that the cubic level treatment is required to capture both curvature and rotational eﬀects, but a suﬃciently general description can only be obtained by extending to at least quartic level, which has yet to be done. The UMIST group (Craft et al. 1997) and others have demonstrated that both a 2-equation (low Re k-ε with additional strain-dependent fµ model damping factor) and a 3-equation cubic non-linear approach (including an additional transport equation for the stress invariant itself derived from a truncated Reynolds-stress equation) can produce good predictions for the T3A and T3B test cases (see Figure 7). These are at least as good as those from the far simpler linear Launder–Sharma k-ε model and are obtained at only modest additional CPU expense (≈30%), although with some loss of numerical robustness. Unfortunately, attempts to apply a similar 2-equation (k-ω) cubic non-linear treatment to the same test cases have so far proved rather disappointing; with transition predicted either far too early or far too late. Results for variable pressure gradient transition test cases have also shown less of an improvement over the best linear k-ε models than expected, indicating a lack of generality of current non-linear extension calibrations.

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481

Reynolds Stress Transport Models

There are two major advantages in moving up to the RST level of closure to predict by-pass transition ﬂows, and indeed many more complex turbulent ﬂows. Stress transport models can account for eﬀects of (free-stream) turbulence anisotropy, and at the same time they automatically capture the main eﬀects of applied strain rates in both irrotational and curved ﬂows, while correctly modelling the production of uv due to the product of v 2 with ∂U/∂y (and also the production of v 2 from uv∂U/∂y). However, like nearly all simpler models they remain essentially local-equilibrium closures in the sense that all model approximations are made in terms of local quantities, without any speciﬁc allowance for strain history. In addition model constants are calibrated against a range of near equilibrium ﬂows and so do not introduce any additional (cross-spectral) length or time scales nor do they include a speciﬁc allowance for pressure-diﬀusion eﬀects. Only a very limited number of full diﬀerential stress-transport models have so far been extended to handle low-Re ﬂows and, as with low-Re k-ε models, all of these have been developed speciﬁcally to handle low-Re near-wall regions of fully turbulent ﬂows rather than low-Re transition regions. Furthermore, to the author’s knowledge, only four have so far been tested on the type of by-pass-transition test cases considered here. Of these the low-Re treatment employed in the Savill–Launder–Younis SLY model has so far proved superior to the alternative low-Re formulation adopted in the Launder–Shima (1989) scheme, and in two hybrid low-Re/TCL RST closure schemes developed by Launder and Cho (1994), and by Hanjalic, Jakirlic and Hadzic (1997) – see for example Dick (1995). The SLY model is in fact an extension of a low-Re diﬀerential secondmoment closure model devised earlier by Kebede, Launder and Younis (1985), KLY. Its further development for predicting by-pass transition ﬂows has been based on the knowledge gained from the evaluation of low-Re k-ε schemes for modelling near-wall turbulent, transitional, and re-laminarising ﬂows, and from previous work by the author in reﬁning high-Re ASM and RST schemes to predict the eﬀect of free-stream turbulence and wake-interactions on turbulent boundary layers for high-lift, multi-element aerofoil applications (Savill 1993a,b). The basic SLY model employs low-Re extensions to the high-Re RST model of Launder, Reece and Rodi (1975) – their simpler LRR2 model to be precise. Full details are given in Savill (1996) – see also Westin and Henkes (1997) for clariﬁcation of some speciﬁc implementation details. Essentially it extends the Launder–Sharma k-ε treatment to the higher RST level of closure. The two key elements in common are the retention of ε as the length-scale determining equation (avoiding the need to include any D term) and the inclusion of a fµ -damped turbulent eddy viscosity in the E term of the ε equation, which

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Figure 8: SLY model predictions without (- - -) and with non-local extensions (— ·) versus DNS (—) and experimental data (◦) for full range of T3A & B series of test cases, (1 to 10%fst) versus experimental data. turbulence simulations (DNS) show is physically required to correctly account for an explicit additional near-wall production of dissipation – see Rodi and Mansour (1993). The SLY model predictions are in fact particularly sensitive to the precise scaling of this term since it keeps the dissipation of turbulent ﬂuctuations in the initial pseudo laminar layer small until transition is wellestablished. The optimised value of Cε3 is in fact half the value applied in the original KLY model for fully turbulent ﬂow and this scaling of fµ ensures a very close ﬁt to the transition simulations of Voke and Yang (1993), within the transitional buﬀer layer to y + = 40 (which is approximately equivalent to δ + in the pre-transitional boundary layer). Outside this region the E term quickly becomes negligible. This SLY model can accurately predict the onset of transition under nominally zero pressure-gradient conditions for levels of free-stream turbulence in the range 1 to 10% (see Figure 8). Equally good predictions are obtained for Cf , H, and θ, and hence for the mean velocity proﬁle development in all cases. The growth of the initial u peak, the turbulence energy, and the development of the other Reynolds-stress proﬁles is again somewhat underpredicted, and the model only slowly asymptotes to fully turbulent conditions (although, as part of the 1995 Stanford Collaborative Testing of Turbulence Models co-ordinated by P. Bradshaw, it was shown accurately to predict Cf (1% high) for a fully turbulent boundary layer at Reθ = 10,000). Certainly for the 3 and 6%fst test cases, the length of transition and shape factor distribution, are predicted slightly better than with the Launder–Sharma

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Figure 9: Comparison of T3B test case experimental (◦) and DNS (—) steamwise growth of turbulence intensity with predictions using Launder–Sharma k-ε model (- - - -) and Savill–Launder–Younis (SLY) RST model with (— ·) and without (— —) non-local model extensions. k-ε model and the same is true of the turbulent proﬁle development although the maximum value of k is still underpredicted compared to both the experimental and simulation results as illustrated for the 6%fst case in Figure 9. Similar results have been obtained with the Launder–Shima low-Re RST treatment for this case – see Figure 10 – probably because the fµ damping introduced in the SLY scheme only has an indirect eﬀect through the E factor. However, it has been found that modifying the E factor to provide a better ﬁt to low-Re near-wall DNS ‘data’ for turbulent ﬂows as done in the Launder– Shima model does not improve transition predictions. In comparison, the SLY model exhibits exactly the correct sensitivity to pressure gradient for the on-design aft-loaded turbine-blade test case with 3%fst and also captures the correct trend of varying Reynolds number for the pressure gradient distributions representative of the same aft-loaded turbine oﬀ-design (T3C3-5 Test Cases), greatly improving on results obtained with both standard and corrected Launder–Sharma k-ε model treatments – see Figure 11. The SLY model has also been shown capable of handling both relaminarisation and retransition of a turbulent boundary layer – see Figure 12 – provided the same degree of mesh resolution is maintained as the boundary layer thick-

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Figure 10: Comparison of Launder–Shima RST (—) and Savill–Launder– Younis RST (– · ·) model predictions for 6%fst experimental test case T3B (). ness varies (dramatically in this case). In addition it has been found to predict the eﬀect of a three-fold variation in free-stream length scale for zero pressure gradient 3%fst and the separate inﬂuence of both weak and strong streamline curvature – see Savill (1995).

4

Towards practical computations for engineering ﬂows

Several of the k-ε models evaluated within the ERCOFTAC Transition SIG Project have now been applied to additional heat transfer and blading test cases – see Figures 13 and 14. In nearly all cases computations have been initiated near the leading edge for a 2D or mid-span computations. Again the Launder–Sharma model has been found to provide the best overall predictions of the un-modiﬁed models that have been evaluated, although even better re-

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Figure 11: Comparison Launder–Sharma k-ε model, with (—) and without (– ·) pressure gradient correction versus Savill–Launder–Younis RST ( – · ) model predictions for variable fst and pressure-gradient representative of an aft-loaded turbine blade operating at various experimental conditions (). sults have been obtained by introducing PTM scaling and similar alternative Rt -dependent low-Re treatments. Good results have also been reported with a multi-scale k-ε approach being developed by Crawford – see Simon (1994). In some cases superior predictions have been obtained with a PTM version of the Ry -dependent Lam and Bremhorst scheme, although only after introducing some empirical input γ. This last model has also been successfully applied in modiﬁed form to engine ﬂows by researchers at General Electric – see Dailey et al. (1994). However it should be noted that the very early successful predictions reported by Rodi and Scheuerer (1984) for heat transfer variation with increasing free stream turbulence were possibly misleading. It now appears that the reproduction of the experimentally observed shift to a monotonic decay of Stanton number may have been due to the Lam–Bremhorst model predicting a similar, but far too early Cf transition.1 1 A more recent application of non-linear and linear eddy viscosity models to compute heat-transfer coeﬃcients over a cascade of gas-turbine blades has been reported by Craft et al. (1997). This shows very clearly the superiority of the non-linear EVM which avoids the excessive heat-transfer coeﬃcients generated by the linear model both in the vicinity of the stagnation point and on the pressure surface towards the downstream end of the blade. (Ed)

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Figure 12: Comparison of SLY model predictions (—, - - -) and measured Cf results (⊕) from T3E test case (relaminarisation and retransition). Some 3D computations have recently been undertaken for the ERCOFTAC linear turbine cascade (T3K) test case, using a variety of closures including the Dawes low-Re k-ε model. However even with up to 1.2 million cells the predictions still appear to be limited by the degree of boundary-layer grid resolution that can be incorporated in the computations and the model treatment used to handle the leading edge stagnation region. At least as good results have been obtained by Gregory-Smith (1995) and colleagues at Durham, using a simple mixing length model and assuming laminar conditions throughout the blade passage up to 80% chord – their experimentally determined location of transition on the suction surface. RST models such as the SLY scheme need to be tested more extensively, particularly within an elliptic framework for ﬁnite leading edge ﬂows such as the T3L case, prior to 2D and 3D blading applications, but already exhibit the correct response to both streamline and in-plane curvature. It can therefore certainly be anticipated that the RST approach will eventually provide superior predictions to k-ε models for the T3L, T3K and other blading-type test cases, with ﬁnite leading edges, because such stress transport models correctly predict only a small growth in turbulence energy along stagnation streamlines. Already acceptable engineering results have been achieved for many turbomachinery ﬂows using the Dawes k-ε model combined with the Durbin treatment for irrotational strain and curved ﬂow regions.

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Figure 13: Model predictions for Daniel Turbine Blade test case: Launder– Sharma (LS), Lam–Bremhorst (LB) – both with Production Transition Modiﬁcation (PTM), Crawford Multiscale (MTS), and standard Launder– Sharma (LS). Successful prediction of attached-ﬂow by-pass transition can also be extended to laminar separation-induced transition by the inclusion of appropriate alternative transition correlation criteria or, with progressively less empiricism, intermittency presumptions and intermittency transport – see Savill [18].

Acknowledgements This contribution would not have been possible without the continued support of Rolls-Royce plc, DRA, the CEC & COST. Several developments owe their origins to discussions with close colleagues at UMIST (especially Professor Brian Launder), researchers at the University of Surrey (in particular Professor Peter Voke) and Dr John Coupland of Rolls-Royce Aerothermal Methods; with whom regular interaction has been maintained as part of an overall transition strategy. The assistance of Dr B.A. Younis of City University, who freely provided the original version of the low-Re RST code and assisted with the implementation of this, has been vital to the development of the SLY model, as have contacts with other researchers, too many to mention individually, whose help is also gratefully acknowledged. The Editors, organisers of the INI Programme and other colleagues contributed a number of constructive criticisms which I hope are reﬂected in this ﬁnal version of my notes.

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Figure 14: Model predictions for Daniel Nozzle Guide Vane test case (◦); computations: Launder–Sharma (+), Lam–Bremhorst (×), Nagano–Tagawa (), Kasagi–Sikasano (•), and Biswas–Fukuyama ().

References Abid, R. (1993) ‘Evaluation of two-equation turbulence models for predicting transitional ﬂows’, Int. J. Engineering Science 11 25. Abu-Ghannam, B.J. and Shaw, R. (1980) ‘Natural transition of boundary layers: the eﬀects of turbulence, pressure gradients and ﬂow history’, J. Mech. Eng. Sci. 22, 213. Arad, E., Berger, M. and Wolfshtein, M. (1982) ‘Numerical calculations of transitional boundary layers’, Int. J. Num. Methods Fluids 2 1–23. Arnal, D. (1991) ‘Transition description and prediction. Numerical simulation of unsteady ﬂows and transition to turbulence’. In Proc. 1st ERCOFTAC Workshop on Numerical Simulation of Unsteady Flows Transition to Turbulence and Combustion, Lausanne, O. Pironneau et al. (eds.), Cambridge University Press, 304–316. Arnal, D., Habiballah, M. and Delcourt, V. (1980) ‘Synth`ese sur les m´ethodes de calcul de la transition developp´ees `a DERAT’. ONERA, Rapp. Techn., OA No. 11/5018. Baldwin, B. and Lomax, H. (1978) ‘Thin-layer approximation and algebraic model for separated turbulent ﬂows’, AIAA Paper 78–257. Biswas, D. and Fukuyama, Y. (1994) ‘Calculation of transitional boundary layers with an improved low-Reynolds number version of the k-ε model’, ASME Journal Turbomachinery 25 11.

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Blair, M.F. and Werle, M.J. (1981) ‘Inﬂuence of free-stream turbulence on the zeropressure gradient, fully turbulent boundary layer’, AFOSR Technical Report 81– 0514. Boyle, R.J. (1990) ‘Navier–Stokes analysis of turbine heat transfer’. NASA-TM-102496. Cebeci, T. and Smith, A.M.O. (1974) Analysis of Turbulent Boundary Layers Academic Press. Chevray, R. and Tutu, N.K. (1978) ‘Intermittency and transport of heat in round jet’, J. Fluid Mech. 88 133. Costa, J. and Arts, T. (1991) ‘Boundary layer transition under the presence of discrete frequencies in the free-stream turbulence’, Proc. 10th ISABE 2 807. Craft, T.J., Launder, B.E. and Suga, K. (1997) ‘The prediction of turbulent transitional phenomena with a non-linear eddy-viscosity model’, Int. J. Heat Fluid Flow 18 15–28 Dailey, L. D., Jennions, I. K. and Orkwis, P. D. (1994) ‘Simulated laminar-turbulent transition with a low Reynolds number k-ε turbulence model in a Navier–Stokes ﬂow solver’, AIAA Paper 94-0189. Dawes, W.N. (1992) ‘The simulation of three-dimensional viscous ﬂow in turbomachinery geometries using a solution-adaptive unstructured mesh methodology’, J. Turbomachinery 114 (3) 528–537. Dhawan, D. and Narasimha, R. 1958) ‘Some properties of boundary layer ﬂow during transition from laminar to turbulent motion’, J. Fluid Mech. 3 418. Dick, E. (ed.) (1995) ‘Theme section on transition’, ERCOFTAC Bulletin 24. Donaldson, C. DuP. 1969) ‘A computer study of an analytical model of boundary layer transition’, AIAA J. 7 271–278. Dopazo, C. (1977) ‘On conditional averages for intermittent turbulent ﬂow’, J. Fluid Mech. 81 433. Dunham, J. (1972) ‘Prediction of boundary layer transition on turbomachinery blades’, AGARDO graph 164, 55. Dutoya, D and Michard, P (1981) ‘A program for calculating boundary layers along compressor and turbine blades’. In Numerical Methods in Heat Transfer (eds. R.W. Lewis, K. Morgan and O.C. Zienkiewicz), Wiley, 413–428. Finson, M.L. (1975) ‘A Reynolds stress model for boundary layer transition’. Phys. Sc. Inc., Rep.-No. TR-34. Fraser, C.J. and Milne, J.S. (1986) ‘The eﬀect of pressure gradient and free-stream turbulence intensity on the length of transitional boundary layers’, Proc. Inst. Mech. Eng. 200 (C3), 179. Gaﬀney, R.L., Salas, M.D. and Hassan, H.A. (1990) ‘An abbreviated Reynolds stress turbulence model for airfoil ﬂows’, AIAA Paper 90-1468. Gostelow, J.P., Hang, G., Walker, G.J. and Dey, J. (1994) ‘Modelling of boundary layer transition in turbulent ﬂows by linear combination integral method’, ASME94-G1-358. Gregory-Smith, D. G. (1995) ‘3D ﬂow simulation in turbomachinery’, Proc. VKI Turbomachinery Conference, Von Karman Inst. Rhode-St-Gen`ese Belgium.

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Hanjalic, K., Jakirlic, S. and Hadzic, I. (1997) ‘Expanding the limits of “equilibrium” second-moment closures’, Fluid Dynamics Research 20 25-41. Henkes, R.A.W.M. (1992) ‘Test case on natural convection’, Proc. EROCFTAC/IAHR Seminar on Assessment of Turbulence Models of Engineering Applications, EPF Lausanne. Hallb¨ ack, M., Henningson, D. S., Johansson, A. J. and Alfredsson, H. (eds.) (1996) Turbulence and Transition Modelling, Kluwer. Jackson, J.D. and He, S. (1995) ‘Simulation of transient turbulent ﬂow using various 2-equation low-Reynolds number turbulence models’, Proc. 10th Turbulent Shear Flows Symposium, Pennsylvania State University. Jones, W.P. and Launder, B.E. (1972) ‘The prediction of laminarization with a twoequation model of turbulence’, Int. J. Heat Mass Transfer 15 301–314. Kebede, W., Launder, B.E. and Younis, B.A. (1985) ‘Large amplitude periodic pipe ﬂow: a second-moment closure’, Proceedings 5th Symposium Turbulent Shear Flows, 16.23–16.29, Cornell Univ. Launder, B.E., and Cho, J.R. (1994) ‘The modelling of diﬀusion-controlled transition with the new model’, Proc. 6th Biennial Colloquium on Computational Fluid Dynamics, UMIST. 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. Launder, B.E. and Shima, N. (1989) ‘Second-moment closure for the near-wall sublayer: development and application’, AIAA J. 27 (10) 1319–1325. Libby, P.A. (1975) ‘On the prediction of intermittent turbulent ﬂows’, J. Fluid Mech. 68 273. Libby, P.A. (1976) ‘Prediction of the intermittent turbulent wake of a heated cylinder’, Phys. Fluids 19 494. Liu, S.L. (1989) ‘The prediction of boundary layer transition using low Reynolds number k-ε turbulence model’, Proc. 4th Asian Congress of Fluid Mechanics, Hong Kong, Vol. 1, A9. McDonald, H. and Fish, R.W. (1973) ‘Practical calculation of transitional boundary layers’, Int. J. Heat Mass Transfer 16 1729. Manartonakis, G. and Grundmann, R. (1991) ‘Transition in three-dimensional boundary layers with a one-equation model’, Proc. Boundary Layer Transition and Control Conference, Cambridge, UK, RAeS Publishers. Micheltree, R.A., Salas, M.D. and Hassan, H.A. (1990) ‘One-equation turbulence model fo transonic airfoil ﬂows’, AIAA J. 28 1625–1632. Myong, H. K. and Kasagi, N. (1990) ‘A new approach to the improvement of k-ε turbulence model for wall-bounded shear ﬂows’, JSME J. Series II 33 (1), 63. Nagano, Y. and Hishida, M. (1987) ‘An improved form of the k-ε model for wall turbulent shear ﬂows’, Trans. ASME J. Fluids Engineering 109 156. Nagano, Y. and Tagawa, M. (1990) ‘An improved k-ε model for boundary layer ﬂows’, Trans ASME J. Fluids Engineering 112 33.

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Narasimha, R. and Dey, J. (1989) ‘ Transition-zone models for two-dimensional boundary layers’, Indian Academy Proceedings in Engineering Sciences: Sadhana 14 (2), 93. Ng, K.-H. (1971) Predictions of Turbulent Boundary-Layer Developments using a Two-Equation Model of Turbulence. PhD thesis, Imperial College, London. Patel, V.C., Rodi, W. and Scheuerer, G. (1985) ‘Turbulence models for near-wall and low-Reynolds number ﬂows: A review’, AIAA J. 23 1308. Pironneau O., Rodi W., I.L. Ryhming, Savill A.M. and Truong T.V. (eds.) (1992) Proc. 1st ERCOFTAC Workshop on Numerical Simulation of Unsteady Flows Transition to Turbulence and Combustion, Lausanne, Cambridge University Press; see in particular A.M. Savill: A synthesis of T3 test case predictions. Priddin, C.H. (1975) The Behaviour of the Turbulent Boundary Layer on Curved Porous Walls. PhD thesis, Imperial College, London. Prihoda, J., Hlava, T. and Kozel, K. (1999) ‘Modelling of bypass transition including pseudolaminar part of the boundary layer’, ZAMM 79 S3, 5699–700. Rodi, W. and Mansour, N.N. (1993) ‘Low-Reynolds number k-ε modelling with the aid of direct simulation data’, J. Fluid Mech. 250 509–529. Rodi, W. and Scheuerer, G. (1984) ‘Calculation of laminar-turbulent boundary layer transition on turbine blades’, AGARD CP 390 on heat transfer and cooling in gas turbines, 18-1. Savill, A.M. (1992) ‘Evaluating turbulence model predictions of transition – an ERCOFTAC Special Interest Group project’, Appl. Scientiﬁc Research 51 555. Savill, A.M. (1993a) ‘Some recent progress in the turbulence modelling of by-pass transition’. In Near-Wall Turbulent Flows (eds. R.M.C. So and B.E. Launder) Elsevier, 829. Savill, A.M. (1993b) ‘Further progress in the turbulence modelling of by-pass transition’. In Engineering Turbulence Modelling and Experiments 2 (eds. W. Rodi and F. Martelli) Elsevier, 583. Savill, A.M. (1994) ‘Transition modelling for turbomachinery II’, Summary Proceedings of the 1st ERCOFTAC Transition SIG Workshop of the BRITE-EURAM AERO-CT92-0052 Project Workshop on Transition in Turbomachinery, VUB. Savill, A.M. (1995) ‘Predicting transition with turbulence models: the eﬀect of variable free-stream turbulence length scale’. In Advances in Turbulence – VI, S. Gavrilakis, L. Machiels and P.A. Monkewitz (eds.), Kluwer, 79–80. Savill, A.M. (1996) ‘Evaluation of turbulence models for predicting transition in turbomachinery ﬂows’, Minowbrook II Workshop on Boundary Layer Transition in Turbomachines, Sycracuse University, USA. Schubaer, G.B. and Klebanoﬀ, P.S. (1956) ‘Contributions on the mechanics of boundary layer transition’, NACA Technical Note 3489. Seyb, N.J. (1971) ‘Transitional boundary layers’, Rolls-Royce Report, Bristol Engine Division. Simon, F.F. (1994) Proceedings of the AFOSR Workshop on End-Stage Boundary Layer Transition, Minowbrook, Syracuse University, USA.

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Simon, F.F. and Stephens, C.A. (1991) ‘Modelling of the heat transfer in bypass transitional boundary layer ﬂows’. NASA-TP-3170. Singer, A., Dinavahi, S.P.G. and Iyer, V. (1991) ‘Testing of transition-region models: test cases and data’. NASA CR 4371 see also Trans. ASME JFE 114 73. Steelant, J. and Dick, E (1995) ‘Transition modelling by conditionally averaged equations’, ERCOFTAC Bulletin 24 62–64. Theodoridis, G., Prinos, P. and Goulas, A. (1991) ‘Prediction of transition boundarylayer ﬂows: a comparison of various low-Re turbulence models’, Proc. 10th ISABE 2, 16.3. Viala, S., Deniau, H., Hambruger, J. and Aupoix, B. (1995) ‘Prediction of boundary layer relaminarization using low-Reynolds number turbulence models’, Proc. 10th Turbulent Shear Flows symposium, Pennsylvania State University. Voke, P.R. and Yang, Z.Y. (1993) ‘Numerical solutions of the mechanisms of by-pass transition in the ﬂat plate boundary layer”, Proc. 9th Symp Turbulent Shear Flows, Kyoto, 21-2-1. Westin, K.J.A. and Henkes, R.A.W.M. (1997) ‘Application of turbulence models to bypass transition’, J. Fluids Eng. 119 (4), 859–866. Wilcox, D.C. (1992) ‘The remarkable ability of turbulence model equations to describe transition’, SAE Conference on Numerical and Physical Aspects of Aerodynamic Flows, Paper SAE-92, Long Beach, CA. Yang, Z. and Shih, T.H. (1991) ‘Extending turbulence modelling to bypass transition’, NASA CMOTT Research Briefs 83.

18 New Strategies in Modelling By-Pass Transition A.M. Savill 1

Introduction

The preceding chapter [17] has discussed the various earlier attempts to predict by-pass transition using conventional turbulence-model closure schemes across the full range of complexity from simple mixing-length/correlation approaches to Large Eddy Simulations (LES). A perspective has thus been provided on the state-of-the-art as evidenced by research journal publications and actual industrial applications, which accurately reﬂects developments in both academia and industry during the twenty-year period from the early 70s to mid 90s. However, over the last ﬁve years a number of existing established methodologies have been signiﬁcantly reﬁned in the light of improved physical understanding emerging from further experimental studies and, more especially, new numerical database developments. Moreover some additional alternative approaches have also now been proposed, researched and, in some cases, tested with greater success. It is these newer strategies for turbulence modelling of by-pass transition, which have not yet reached either the archival research literature or full industrial implementation, that form the subject of the present review.

2

Results from by-pass transition simulations

A number of (fully resolved) simulations have now been carried out for ﬂows undergoing by-pass transition. Some initial rather coarse mesh zero pressure gradient Direct Numerical Simulations performed by Yang and Voke for the 1st ERCOFTAC Workshop – see Pironneau et al. (1992) – in fact very accurately reproduced the experimental behaviour observed in experimental test cases with 3 and 6% free-stream turbulence (fst). Encouraged by this success, a large amount of the UK National Supercomputer resources (more than 10,000hr CPU on a Cray YMP) has been devoted to establishing a suﬃciently ﬁne mesh by-pass transition simulation database for a (T3BLES) Test Case similar to the Rolls-Royce T3B (6%fst) experiment. In order to avoid the sub-grid-scale modelling approximations as far as possible, this simulation has been run as a Direct Numerical Simulation so far as the low-Re transitional boundary layer is 493

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concerned, but a Smagorinsky-type SGS model has been employed to simulate the development of the higher Re free-stream turbulence which was extracted from a precursor simulation. In addition Yang and Voke have now also performed a number of coarser mesh Large-Eddy Simulations for the ERCOFTAC T3A, T3B, T3C1, T3D2 & T3L Test Cases – see Table 1 of [17] and Savill (1995). Each simulation appears to capture the essential features of the development of the mean ﬂow and turbulence ﬁelds, including, in the last case, the detailed features of the laminar separation region, and a second ﬁne resolution data base is now being created for this as part of a current European Thematic Network project – see Savill and Pittaluga (2000). In parallel the same researchers have carried out a series of LES numerical experiments for variations on the T3BLES case in order to investigate further the mechanisms of transition and the eﬀect of varying degrees of free-stream anisotropy.

2.1

Low-Reynolds Number Damping

As discussed in [17], the transition simulations provide some explanation for the success of particular low-Re k-ε models in that they show the damping eﬀect of the wall on the eddy viscosity hardly changes as the boundary layer develops from its pre-transitional pseudo-laminar state to its post-transitional fully turbulent form. Of the most widely used low-Re k-ε model approaches, the Dawes model formulation for fµ actually comes closest to describing the simulated damping – see Figure 1. The simulations also provide some very useful information on the relationship between Ry and Rt through transition. They indicate, perhaps rather surprisingly, that Ry and Rt , as in near wall turbulent ﬂows, remain linearly related over a wide range of Rt , Ry < 300; with Rt ≈ Ry in the pre-transitional layer and the constant of proportionality dropping from 0.8 to 0.5 between start and end of transition – see Figure 2 – compared to the fully turbulent very near wall value of 0.4 extracted from earlier Direct Numerical Simulations (see Patel et al. 1985).

2.2

Reynolds Stress Balances

Close examination of the simulated Reynolds stress balances for stations at x = 25mm and 45mm (pre-transitional/pseudo-laminar), x = 95mm (transition onset) and 195mm (end of transition) relative to the leading edge reveals the following: u-component balance. Proﬁles are well-established by x = 25mm, reﬂecting a rapid increase in level of production (largely balanced by dissipation

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Figure 1: Comparison of fµ functional distributions through transition with 6%fst as provided by DNS (—), k-ε models of Launder–Sharma (◦) and Dawes (- -) and used in SLY model E term (· · ·). and velocity-diﬀusion) from the leading edge, and production already exceeds velocity-diﬀusion at this initial station – Figure 3. Production then continues to increase slowly for succeeding stations, and the balance at the end of transition appears very similar to that seen in fully turbulent DNS, except that there is a pressure-strain peak near the wall; presumably due to a ‘stronger splatting’ eﬀect of imposed large-scale fst on the wall (an essentially non-local eﬀect). Pressure-diﬀusion, which would be zero in the absence of the streamwise variation in u, appears to be positive and signiﬁcant (only) in the very near-wall region; just before and within the transition region. v-component balance. Production is zero. All other balance terms are small (and the pressure-strain is initially negative), until near the onset of transition, when signiﬁcant redistribution of newly generated u into v starts to occur (and pressure-strain becomes positive, being largely balanced by velocitydiﬀusion and dissipation). Pressure-diﬀusion again appears to be positive and

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Figure 2: Variation in ratio of Ry to Rt through transition as extracted from the same LES stations as Figure 1; continuous lines upper to lower Rex = 16600, 29800, 62900 and 129100; (· − · Rt = Ry ; · · · , Rt = 0.4Ry ). signiﬁcant only near the wall where it is almost exactly opposed by a similar magnitude pressure-strain. However, the Reynolds stress balance at the end of transition is very diﬀerent from that seen in fully turbulent DNS, unless such a splitting of the pressure term is avoided (indicating that the pressure-strain is again enhanced by a non-local contribution due to the imposed large-scale fst). Even then the terms appear to peak further oﬀ the wall than in fully turbulent ﬂow. uv-component balance. Production and pressure-diﬀusion (also positive) and pressure-stain are similar in magnitude, but the latter two are largely in opposition to one another over the whole buﬀer layer – see Figure 4. All are smaller than equivalent u terms, and presumably the balance at x = 25mm is little altered from the leading edge. Production then increases more rapidly, particularly once the level of v in the boundary layer increases. The balance terms at the end of transition are similar to those seen in fully turbulent ﬂow DNS balances, but again these appear to peak further from the wall. w-component balance. Production and pressure-diﬀusion are zero. Again the other balance terms are all small, until signiﬁcant redistribution of u into

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Figure 3: (a)–(d). LES u-component Reynolds-stress balance compared with ‘local’ SLY RST predictions (bold) for same transition locations as Figure 1; (—) Production; ( - - ) Dissipation + Viscous Diﬀusion; (· − ·) Velocitydiﬀusion; (- - -) Pressure-strain and Pressure-diﬀusion; (· · ·) Convection + Error Term. w occurs due to the pressure-strain (balanced by dissipation) near the onset of transition, and the balance at the end of transition is very similar to that seen in fully turbulent DNS, with no clear evidence of any enhanced splatting eﬀect.

2.3

Deduced Transition Mechanisms

The picture of transition mechanisms that emerges from this analysis is shown in Figure 5. The wall boundary condition imposed on relatively large scale (≈ δ) fst results in damped initial proﬁles for u, v and w. The superposition of the imposed v proﬁle with the essentially Blasius mean velocity proﬁle results in the production of local uv within the pseudo-laminar layer; in addition to the large, but uncorrelated, externally imposed u, v and w ﬂuctuations. This local uv combines with the local mean shear to provide a source term for

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Figure 4: (a)–(d). LES uv Reynolds stress balance compared with ‘local’ SLY RST predictions (bold); notation as for Figure 3. local u ﬂuctuations. It appears that much of the energy associated with these ﬂuctuations is dissipated immediately, but the remainder is redistributed via pressure-strain into local w, where it is again rapidly dissipated (both u and w balances quickly attaining a fully turbulent form), and into additional local v, which combines with the local mean shear to produce more local uv and hence more local u. As a result it appears there is a lag in the development of the v and uv balances towards a fully turbulent form. Pressure-diﬀusion appears to play a signiﬁcant role in promoting high levels of u and uv near the wall. However, v remains small near the wall in the pre-transitional layer due to the dominating inﬂuence of enhanced wall-reﬂection eﬀects. When the steadily increasing pressure-strain redistribution from u to v overcomes this eﬀective negative oﬀset, v starts to grow and transition occurs. Also indicated is the possible additional interaction sequence if the imposed free-stream turbulence had been sheared, so that there was also an imposed uv. This would then have combined with the local mean shear to immediately produce local u ﬂuctuations, so that the development of all the balances would

[18] New strategies in modelling by-pass transition

∂

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∂ ∂

∂

∂

∂

Figure 5: Mechanism for inducing boundary layer turbulence as deduced from by-pass transition LES. probably have been more rapid, and any lag eﬀect relatively less important. This would be the case in practical blading situations where the free-stream turbulence is composed of chopped wakes from upstream blade rows so it may be that the simulation provides a rather stricter examination of the modelling than is strictly necessary. However, the eﬀect of the unsteady interaction with any leading edge or subsequent laminar separation may well then dominate and simulations for the T3L semi-circular leading edge test case then indicate transition is triggered primarily by vertical ﬂuctuations again – see Voke, Yang and Savill (1996).

2.4

Results from Numerical ‘Experiments’

The series of numerical ‘experiments’ performed by Yang and Voke have shed further light on the transition mechanisms, as follows: (a) Suppression of Free-Stream Diﬀusion and Interaction-at-a-Distance In their ﬁrst ‘experiment’ the free-stream turbulence was artiﬁcially re-

500

Savill moved from the portion of the inlet plane within and immediately above the initial laminar boundary layer. This should have had the eﬀect of both delaying gradient diﬀusion of external turbulence into the wall layer and reducing the strength of any ‘interaction-at-a-distance’ eﬀects through the pressure-ﬁeld associated with free-stream vortices. However, it was found that transition was only delayed when the inﬂow fst was displaced vertically by a distance larger than the free-stream length scale (of order δ), and that removing the fst over a larger region had a much smaller eﬀect. These results indicated that non-local pressure-interactions had at least as strong an inﬂuence on transition as velocity-diﬀusion.

(b) Suppression of Free-Stream Turbulence Diﬀusion In order to test the above conclusion, a second numerical experiment was performed in which a high viscosity layer was imposed between the free-stream turbulence and the laminar boundary layer, in order to prevent any gradient diﬀusion of external turbulence into the wall layer. Despite the imposition of such a ‘viscous-slab’, the transition location was hardly aﬀected, conﬁrming the important inﬂuence of non-local interactions through the global pressure ﬁeld. (c) Passive Temperature Marking Subsequent tagging of the original simulation with a passive temperature tracer did, however, show that transition occurred at the point where the free-stream turbulence penetrated the laminar shear-layer; indicating that a combination of local and non-local diﬀusion processes largely determine the location of transition. (d) Extreme Free-stream Turbulence Anisotropy A ﬁnal set of three numerical experiments was then conducted which involved artiﬁcially altering the anisotropy of the free-stream turbulence so that this was composed of just of u, v or w ﬂuctuations in turn. The results of these three simulations conﬁrmed the primary role of v ﬂuctuations because, under the inﬂuence of these alone, transition occurred at almost exactly the same location as for the original almost isotropic external turbulence conditions. By comparison, when the ‘turbulence; comprised only w or u ﬂuctuations, transition was progressively delayed further downstream.

2.5

Implications for Models in General

The ﬁndings from all the simulations serve to highlight the key physical weakness of most present approaches to modelling transition: that they assume the transition process is controlled solely by (gradient) diﬀusion of (isotropic) free-stream turbulence into the initial pseudo-laminar boundary layer. Thus it is clear, as might have been anticipated, that external vertical ﬂuctuations have the predominant inﬂuence on the initial pseudo-laminar boundary-

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Figure 6: Comparison of eddy viscosity model predictions with k and v as representative velocity scale for the T3A test case: (– –) two-layer k, (—) twolayer v; (- -) Launder–Sharma k-ε, (- - -) Lam–Bremhorst with production transition modiﬁcation (PTM). layer transition. This being the case, one would expect that simply changing the representative velocity scale variable from k to v would improve predictions and this has clearly been demonstrated to be the case for both two-layer k-ε/k- and k-ε models by Sieger (1999) – see Figure 6 – and by Janour (1997) respectively. A speciﬁc allowance should also clearly be included for pressure-diﬀusion of k and ε, since these can both be large; perhaps in the form of cross-diﬀusion terms for ε and k as proposed for turbulent ﬂow by Kawamura and Hada (1992) and Cotton (1989) respectively – see Savill (1996). A similar type of pressure-diﬀusion approximation for k has already been included in transition predictions by Shih, and by Abid, while an alternative (second-order meanﬂow derivative term) proposed by Rotta has been more successfully applied to near wake transition by Yao et al. (2000). There may be a case for scaling these by the Lu∞ /δ in order to capture the eﬀects of free-stream length-scale variations properly. Allowance for a damped inviscid interaction (including the eﬀect of variable free-stream length scale) has also been included in the initial/boundary conditions and modiﬁed model formulation of the one-equation eddy viscositytransport νt -92 model of Secundov and Vasiliev (see Dick 1995), resulting in equally good predictions for zero pressure gradient and the 1 to 6%fst test cases – Figure 7. While the multi-scale k-ε model of Crawford – see Simon (1994) (which actually just uses two separate velocity and length scales for the boundary layer and free-stream turbulence) eﬀectively both introduces an allowance for non-local interaction and reduces the reliance on pure velocity diﬀusion.

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The conditional k-ε scheme developed by Steelant and Dick (1995) also eﬀectively accounts for such non-local eﬀects across intermittency interfaces. In principle, a similar allowance for interaction-at-a-distance eﬀects should be included in all simpler models too, and indeed this has already been attempted in several cases. Thus, the integral method of Johnson and Ercan (1999) attempts to take some direct account of the pressure ﬁeld interaction between the external turbulence and the pseudo-laminar boundary layer by assuming that this eﬀectively provides an additional source of u ﬂuctuations. This model has produced surprisingly good predictions for the ERCOFTAC 3 and 6% test cases with zero- and ‘turbine-blade’ pressure gradient. It has recently also been shown to predict the simulated 5%fst test case just as accurately (see Figure 8), but predicts transition far too early for 10%fst, largely because it uses empirical information which is only valid up to 5 to 6%fst. The model is currently being modiﬁed to correct this deﬁciency and to introduce a pressure-gradient sensitive spot-generation model. The Durbin (1993) v-τ model, in principle, introduces all three allowances (v as transported velocity scale, as well as an elliptic relaxation equation for estimating the associated near-wall pressure-strain redistribution term which may also account for pressure-diﬀusion) and has apparently produced some good results for at least the T3A test case. However the wider applicability of this turbulence model has yet to be demonstrated and validated for other transitional ﬂow test cases.

3 3.1

RST model comparison with transition simulations Skin Friction Development

The Savill, Launder, Younis (SLY) low-Re model – see [17] – has been found to predict accurately the simulation test case with very weak free-stream anisotropy and it has also predicted the correct trends for the simulated extreme values of free-stream anisotropy – see Hallback et al. (1996). This is rather surprising in view of the conclusions reached from the simulations regarding the importance of non-local interaction eﬀects, since the model retains a purely local, gradient (velocity-only) diﬀusion approach to the treatment of by-pass transition. It would seem that the only explanation for the success of the basic SLY model must be that it over-estimates the eﬀect of diﬀusion by approximately the right amount to counteract the missing allowance for any pressure-ﬁeld interaction. Diﬀerences between the model and simulation results for the strong anisotropy case emphasise the fact that the model RST equations are intimately coupled with a single time (or length) scale, so that production occurs ﬁrst in the u component, but energy is then rapidly redistributed between components at the same rate. The simulation suggests the existence of important lag eﬀects.

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Figure 7: Results with the νt -92 model for the T3A and B series of test cases: (—) computations, (points) experimental data.

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Figure 8: Comparison of the Johnson and Ercan modiﬁed integral method (—) and (− ·) with the T3B experimental () and T3B LES (◦) results of Voke and Yang (). Both these and the missing pressure-ﬁeld eﬀects need to be introduced into the model, however, if this is to be truly predictive.

3.2

Predicted Reynolds Stress and Turbulence Energy Balances

The production is reasonably well predicted for the u and uv components, although the peak value for the latter moves too quickly towards the wall through transition – Figures 3 and 4. The dissipation is also fairly well predicted, at least in terms of the general shape and peak magnitude of the proﬁles. The velocity diﬀusion is only reasonably predicted for the u (and w) component very near the wall. Beyond this region it is predicted to have the wrong sign and the same is true for some locations for the uv (and v) component where the model again predicts the peak inﬂuence too close to the wall through transition. The pressure-strain is only well predicted for the w component. Otherwise the predictions are poor and indeed appear spectacularly incorrect for the uv (and v) components throughout transition. It can be seen that the discrepancies would in fact be far greater if it is assumed, as some have chosen to do, that the model approximation for diﬀusion also automatically accounts for pressure-diﬀusion as well (so the predicted term was therefore compared with the sum of the simulated velocity and pressure diﬀusion terms for each component).

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[NB. It should be noted that there are in fact three diﬀerent ways in which the total pressure term could have been split into deviatoric (pressure-strain) and non-deviatoric (pressure diﬀusion) parts: the classical separation assumed by most researchers; that suggested by Lumley (1978); and a third proposed more recently by Mansour, Kim and Moin (1988); but it is not clear which splitting of the simulated pressure term is the most appropriate] It is quite clear that gradient velocity-diﬀusion of external fst into the boundary layer plays far less of a role than assumed in the basic SLY model (and indeed by all other current low-Re model schemes). Allowance must be made for the equally important ‘interaction-at-a-distance’ mechanism acting through the pressure ﬁeld of the vortices in the external ﬂow. In particular the simulations suggested that RST modelling could be improved by introducing, into what must remain an essentially local, one-point closure model, allowances for (necessarily non-local) pressure-diﬀusion of v, uv and ε and nonlocal corrections to the turbulence/mean-strain part of the pressure-strainredistribution, thereby introducing an additional larger length- (and hence longer time-) scale, L, associated with the external v ﬁeld. The basic SLY model fails to capture the delay in the development of the v and uv balance proﬁles compared to their u and w component counterparts, because the stress equations are all coupled together with a single time (and length) scale and so the proﬁles all develop at the same rate. As a consequence they are all predicted to have a turbulent form by the end-of-transition, whereas the simulation indicates that only the u (and w) components have attained such a state at this location. The v and (hence) uv proﬁles still have a form more reminiscent of the u and w balances at the start-of-transition. An additional time-scale is required to capture this eﬀect. [Interestingly Henkes and Westin (personal communication) have shown that omitting the wall reﬂection part of the (local) rapid press-strain approximation has very little eﬀect on the predicted Cf distribution for 3 and 6%fst zero pressure gradient test cases, although the predicted development of the stress proﬁles and balances is not then so good]

3.3

Non-Local Pressure-Velocity Modelling Reﬁnements

Both the pressure-diﬀusion and pressure-strain must necessarily be non-local in eﬀect and there is certainly some evidence from the transition simulations that the imposition of relatively large scale (≈ δ) fst modiﬁes the pressurestrain at least near the wall. However it is probably reasonable to consider only non-local corrections to the rapid part of the pressure-strain (and any corresponding wall-reﬂection term) with separate allowance for pressure-diﬀusion. In fact, Rotta (1968) has proposed model expressions for non-local pressurestrain contributions to all the Reynolds stress components for idealised plane parallel inhomogeneous turbulent shear ﬂow, by expanding the pressure-velocity gradient about a point in a similar manner to that which he adopted for

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modelling non-local pressure-diﬀusion. This adds a third-order derivative term to the normal ﬁrst-order derivative terms for the (local) rapid pressure-strain. For the present application, the additional length scale, L, thereby introduced was again set equal to the fst length scale, and a variable multiplying constant CN Lij introduced, which previous computer optimisation for turbulent wake/boundary layer ﬂows by Savill (1993) suggested should be O(1) for u, O(0.1) for v and uv, and O(0.01) for w. Because the resulting non-local pressure-strain approximations again introduce higher-order derivatives of the mean ﬂow (which one would certainly prefer to avoid in any implementation for 3D complex geometry ﬂows), alternative expressions are now being developed in which these are replaced by ﬁrst and second derivatives of the mean and turbulence ﬁelds respectively, by following the strategy adopted for isotropic and wake fst interactions with turbulent boundary layers (Savill 1993, 1994). However the model constants have been adjusted to ensure the results from these are very similar to the original Rotta formulation. In both cases equivalent improvement in predictions is obtained as indicated by Figures 9 and 10 whilst the eﬀect of allowing for pressure diﬀusion by speciﬁcally including the Kawamura or Rotta term is successfully demonstrated by Figure 9.

4

Additional allowance for intermittency transport

Like other Rt -based low-Re models, the SLY RST model tends to underpredict transition length, particularly for the T3A− (1%fst) test case. Although the model is also particularly sensitive to the exact grid speciﬁcation at such a low fst level, this ﬁnding is not surprising since, as discussed previously in [17], 1% free-stream turbulence intensity is on the boundary between by-pass and natural transition conditions and so one might expect a stronger inﬂuence from intermittency eﬀects associated with spot generation in this case. In an attempt to introduce intermittency transport into the SLY low-Re RST model it was noted – see Savill (1996) – that a transport equation for intermittency has in fact already been proposed and included in both conditionally-averaged k-ε and RST models by Byggstoyl and Kollmann (B&K) and Janicka and Kollmann (1983) (J&K), for modelling ﬁrst turbulent freeshear ﬂows and then turbulent boundary layers. More recently Younis (personal communication) has extended the use of such an RST-γ model to adverse pressure gradient boundary layers with some success, while Cho and Chung (1992) (C&C) have introduced an alternative transport equation for γ into a conventional high-Re Reynolds-averaged k-ε scheme in which the eddy viscosity is conditionalised by γ, but the k and ε equations are not, although an extra intermittency source term is included in the latter. Their model has been applied successfully to a range of turbulent free shear ﬂows and is now being extended to low-Re to predict ﬁrst turbulent and later transitional wall

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Figure 9: Non-local SLY RST model predictions (bold) versus LES: Top: u-component, Middle: v-component and Bottom: uv component pressurevelocity and pressure-strain balance terms at onset and end of transition. ﬂows by Chung (personal communication). The two model equations for γ take similar forms and both also introduce an additional source term in the ε equation. In order to keep the most general formulation all the terms from both models have been retained in the ﬁnal SLY version and the various constants appropriately modiﬁed. However both the C&C and J&K models were formulated for use in turbulent ﬂows with a non-

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Figure 10: Non-local SLY RST model predictions for pressure-strain (bold - -) versus LES w-component balance (for which pressure-diﬀusion is zero); notation as for Figure 3. turbulent free-stream, and the C&C version in addition speciﬁcally assumes that ﬁne (dissipating) scales are generated by the entrainment of non-turbulent ﬂuid by large-scale eddies so that, when γ is high, the dissipation is increased. In the by-pass transition cases considered here not only is the initial boundary layer pseudo-laminar and the external ﬂow turbulent, but also the free-stream turbulence length-scale is greater than the boundary layer mixing-length so that Sε and Sγ are sink terms. Both modiﬁcations to the eddy viscosity expression were tested in the additional E factor of the low-Re ε equation (see [17] – equation (2)), but it was found that the Cho and Chung form had far too large an inﬂuence. The J&K/B&K form was therefore adopted: νt = fµ Cµ Cµγ γ

k2 ε

Cµγ = 0.25

(with and without the Launder–Sharma fµ comparable results were achieved in either case, indicating perhaps that it might be possible to use such an

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allowance for intermittency at the k- level to predict transitional ﬂows without employing a full low-Re treatment). This analysis has resulted in a model equation for γ adapted by Savill (1996) for inclusion in a simple 2D boundary layer code – see Dick (1995). In coupling the γ equation to the SLY RST equations additional source terms were introduced into both the uv and ε transport equations although Steelant and Dick have in fact indicated that the extra source terms appearing in the conditionalised stress equations due to crossings of turbulent/non-turbulent interfaces largely compensate one another. The production terms Puu in the u and Puv in the uv equations, (and hence also Pε ) are conditionalised by γ, and the initial and boundary condition for γ set to 0.01. However, because the intermittency is initially very low and does not always reach unity by the end of transition, conditionalising by γ itself results in transition being predicted too late, and Cf being predicted too low for the subsequent turbulent region. To avoid this the maximum value of (γ, 1 − γ) is used as the scaling factor instead. This has the eﬀect of limiting the production through transition in a similar manner to the empirical PTM scheme proposed by Schmidt and Patankar (1988 ) and Stephens and Crawford (1990), and Cε3 is then automatically reduced from its turbulent value of 0.5 to the transition optimised value of 0.25 in the middle of the transition region before recovering again to its fully turbulent level. Such a variation in the Launder–Sharma fµ also closely mimics the fµ variation extracted from the simulations. Limited optimisation for 1% and 3%fst (zero pressure gradient) test cases resulted in encouraging predictions for both γ and Cf for the T3A− and T3A test cases. After careful consideration of the original SLY intermittency model development and its subsequent reﬁnement and optimisation for by-pass transition ﬂows (as well as an open reassessment of the underlying physical modelling) by Barton and Savill (unpublished), a revised, more robust, intermittency transport model equation has now been adopted as follows: Dργ (4.1) = dγ + Pγ1 + Pγ2 − ρεγ + ρG Dt '' ( ( ∂ k ∂γ µ Diﬀusion: dγ = (1 − γ)3/2 cγ cµ ρ uv + (4.2) ∂xj ε σγ ∂xj Transport Equation:

1

Pkk k 2 2 ρk ∂γ Production (2): Pγ2 = cγ2 ε ∂xj ε Dissipation: εγ = cγ3 γ(1 − γ) k Production (1): Pγ1 = cγ1 ργ(1 − γ) 2

(4.3) (4.4) (4.5)

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Entrainment G = cγ4 γ(1 − γ)

ε k 1/2 ∂γ uv k ε ∂xj

(4.6)

ε ui uj k ε ∂γ . Dissipation Source Term: Sε = cε4 uv 1/2 ∂xj k Reynolds Stress Source: Sui uj = cγ5 γ(1 − γ)

(4.7) (4.8)

This set has ﬁrst been tested in a 2D boundary layer code on the 1% and 3% test cases. For each case the code was run with or without the intermittency equation coupled to the mean ﬂow and Reynolds stress transport equations. Normal manual optimisation of the model constants conﬁrmed that good agreement could be achieved with the experimentally measured intermittency and skin friction results – see Figure 11 – in uncoupled mode using a single set of constants: σγ = 1.0, cγ = 0.15, cγ1 = 2.2, cγ2 = 0.6, cγ3 = 0.1, cγ4 = 0.16, cγ5 = 0, cε4 = 0.1, in line with the previous ﬁndings of Savill (1996). Furthermore, for fully coupled solutions, even better overall agreement could be achieved with the same set, provided that then cγ5 = 0.1 and cγ1 becomes a linear function of the free-stream turbulence intensity: cγ1 = 1.0 for T3A− and 3.0 for T3A which implies the production of intermittency (turbulence) within the initial pseudolaminar boundary layer is directly proportional to the intensity of the external turbulence. Other work indicates this is reasonable for free-stream turbulence length scales of the order of the initial boundary layer thickness, as was the case in the ﬂows considered. A ﬁrst attempt was then made at automatic optimisation of model constants for diﬀerent initial and boundary conditions for intermittency (including vanishingly small intermittency as a prelude to attempting to predict ‘natural’ transition due to much weaker free-stream disturbances) by implementing a procedure whereby the uncoupled model code was run thousands of times for small incremental alterations in all of the model constants until least-squares diﬀerences between experiment and computation for the T3A− and T3A cases were minimised. Unexpectedly this resulted in a radically diﬀerent set of model constants: σγ = 1.0, cγ = 7 or 11.5, cγ1 = 0.13, cγ2 = 1.35 or 6.0, cγ3 = 0.1, cγ4 = 0.05, cγ5 = 0, cε4 = 0.1. Using the manually optimised set of constants it was found that transition predictions could successfully be extended down to a free-stream turbulence level of only 0.3% (below the limit for other turbulence models). However, switching to the second, automatically optimised set, allowed transition to be predicted at external turbulence levels an order of magnitude less (≈0.04%),

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Figure 11: Comparison of measured maximum intermittency growth rate (◦) and Cf (+) with SLY-(γ model predictions (− ·) for 3%fst test case T3A. closer to that achieved in the best laminar ﬂow research tunnels (≈0.008%) and on external aircraft surfaces under ﬂight conditions. The same intermittency-transport model equation has also now been coupled to the Dawes low-Re k-ε model employed within his unstructured Navier– Stokes code. Results presented for the Durham T3K Turbine test case by Biesinger and Savill (1997) reproduce the experimentally observed transition behaviour at midspan on the suction surface and both relaminarisation and retransition on the pressure surface. Subsequently a similar methodology has also been successfully applied to a multi-element aerofoil conﬁguration by Vicedo, Vilmin, Savill and Dawes (2000).

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Figure 12: en predicted n factors for the T3A (1%fst) test case.

5

Other possible approaches

The large eddy simulations performed by Voke and Yang (1993) have revealed evidence of wave-like behaviour, as well as free-stream-turbulence-induced streaks and isolated spots in the wall region ahead of transition onset (detected as the point of minimum Cf ). The division between natural and by-pass transition regimes is certainly not as clear as Morkovin and others have suggested, and the simulations certainly support ample evidence in the literature showing that the‘T-S mechanism’ can co-exist with other ‘strongly amplifying mechanisms’ over a wide range of fst intensities from well below to well above 1% – see also [17]. Interest is thus now developing in possible extensions of en and parabolised stability equation (PSE) approaches to variable fst by-pass transition cases, because there is already some evidence from the NASA Transition Project and elsewhere that both methods can be used successfully to predict the onset of transition for non-zero free-stream turbulence on both ﬂat and curved surfaces, although the ‘n’ factor then becomes an empirical function of the free-stream turbulence intensity. In particular Pereira and colleagues – see Sousa et al.

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Figure 13: PSE predictions (-) of Cf for the T3A test case versus experimental data (◦). (2000) – have demonstrated that an en method can predict transition onset for the ERCOFTAC Transition Modelling SIG T3A− test case with 1%fst – see Figure 12 – while Herbert has shown that use of the PSE can predict the T3A (3%fst) case well into the transition region with only a low number of modes representing the initial disturbance ﬁeld – see Figures 13 & 14. These ﬁndings led Savill (1997) to propose that such stability methods might usefully be directly coupled to an intermittency transport model approach with the former providing the necessary initial conditions for the latter at the onset of transition. Some progress has very recently been made towards such combined use of en and RANS methodologies by Sousa et al. (2000) and Hui (2000) in the context of a structured RANS scheme, while direct coupling of alternative PSE methods with an unstructured, adaptive mesh RANS code is being addressed by Vicedo, Dawes and Savill (2000). The last approach appears particularly attractive and work in this area could conceivably lead to a new design rationale for both external airframe (natural transition) and internal gas turbine (by-pass transition) applications.

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Figure 14: PSE predictions for the Reynolds stress proﬁles in the early stages of by-pass transition for the T3A test case.

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515

Concluding remarks concerning best choice current models

The following statements can be made based on a survey of the state-of-the-art as provided by the above and the preceding chapter, [17]: • Current industrial design models are insuﬃcient to predict transition truly. • All current computational methods can be improved by introducing allowances for pressure-ﬁeld eﬀects and intermittency variations. • Low-Re models which satisfy the wall-limiting conditions for uv + and ε+ , are better at predicting transition than those which satisfy either only one or neither of these. • Low-Re models which employ damping factors that are functions of a general property of low-Re ﬂows (in particular turbulent Reynolds number Rt ) are more appropriate for the prediction of low-Re transition regions than ones that only introduce a dependence on wall-proximity (e.g. via Ry , Rε ). • Many of the more recent low-Re turbulence models (which have been carefully calibrated against turbulence simulations to meet additional wall-limiting conditions and thus produce superior predictions for a wide range of fully turbulent ﬂows) contain damping factors that are functions of y and thus fail to predict transitional ﬂows correctly. • Such models can be improved for transition prediction by introducing extra x-dependent damping factors and/or replacing Ry , Rε by Rt ; eliminating the D and/or E factors; switching to v for the velocity scale; and introducing a nonlinear stress-strain relationship, a speciﬁc allowance for pressure-diﬀusion, and better sensitivity to straining (history). • Of all the low-Re model treatments examined thus far the use of Launder– Sharma-type damping factors (which are functions of Rt and satisfy the wall-limiting conditions for uv + and ε+ is recommended for most accurate prediction of mean ﬂow quantities. • However this model is demanding of grid resolution and requires speciﬁc sensitising to irrotational straining. It is better implemented in a low-Re RST model which can also capture some eﬀects of free-stream turbulence stress anisotropy and does not require excessive grid reﬁnement or speciﬁc strain-dependent corrections. • Unfortunately all models are sensitive to initial and free-stream boundary conditions for ε, which are usually not well-deﬁned (estimates based

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Savill on turbulent values for the mixing length or structure parameter distributions, and integral length scale, respectively are subject to considerable uncertainty). Consequently the eﬀects of variable and anisotropic free-stream length scales encountered in many practical ﬂows cannot yet be predicted. • All transport models require greater mesh resolution than can be accommodated in current 3D computations for complex engineering ﬂows on all but the very latest massively parallel computers.

Future research must focus on developing methods less demanding of CPU/ store, but oﬀering greater predictive abilities and applicability. The combination of PSE with intermittency transport modelling may oﬀer a novel route towards this goal, and, in view of the simulation ﬁndings, a good candidate transport model may be that of Durbin which adopts v as velocity scale and solves an elliptic relaxation equation to take speciﬁc account of the non-local damping eﬀect of walls. A new two-time-scale model of Yang and Shih, which uses a combination of Rt and a dimensionless strain rate parameter also merits investigation, particularly as this model has been shown to provide the best kε model predictions for ﬂow through rows of tube-bundle rods – see Watterson et al. (1999). Other studies suggest further consideration of models combining an Ry dependence for (low-Re) near wall regions and an Rt or perhaps RL -dependence for (low-Re) transition regions see [17]. In principle even better predictions should be feasible with the hybrid lowRe/TCL approach, since this allows a clearer distinction to be drawn between the modelling of low-Re transitional and (low-Re) near-wall eﬀects. However this approach clearly requires further development along the lines currently being revised by Hanjalic (1997, 1999), Hadzic (1997, 1999) and others. Continued work on a hierarchy of models, evaluated step-by-step against a larger series of progressively more complex test cases, is therefore recommended, but it is clear that input is also required from studies of much more complex engineering ﬂows. These last are needed both to assist the improvement of current physical modelling and ensure that the current ‘best’ models really will be able to improve predictions when many more complicating factors, including compressibility, rotation and unsteadiness, have to be taken into account. The ﬂow through turbomachinery and many other practical ﬂow geometries, such as tube bundles and heat exchangers, is of course subjected to the sequential eﬀects of convex and concave curvature as well as the combined eﬀect of streamline curvature and irrotational straining. Rather less attention has so far been paid in general to the modelling of such sequences and combinations of simple strain rates, although it has been known for some time that these may produce unexpected nonlinear eﬀects. Fortunately Savill and Tselepidakis (1994) have shown that the basic LRR RST model can qualitatively

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reproduce the eﬀects of sequences of convex and concave curvature including their unequal eﬀect on the shear stress (which is suppressed more by the stabilising eﬀect of convex curvature than it is increased by the destabilising eﬀect of concave curvature), but similar magnitude inﬂuence on the normal stress anisotropy. The same authors have also shown that the same model can also capture the manner in which the response to and recovery from each strain is modiﬁed by the opposing action of the other sense of curvature, at least when diﬀusion is negligible (as in wakes). However, in practice it is clear that signiﬁcant further lag eﬀects occur in such ﬂows. Alternative TCL modelling can provide better quantitative predictions than the LRR model for the variation of Reynolds stress anisotropy in homogeneous shear ﬂow subjected to both weak and strong convex or concave curvature. However the production term in the ε equation needs to be scaled by the nonequilibrium parameter Pk /ε to produce equally good results when applied to a shear ﬂow subjected to sequential convex-to-concave curvature. It is likely that similar non-equilibrium extensions will be required to most models to account correctly for lag and other strain-history eﬀects encountered when applying them to real engineering ﬂows. Further attention also needs to be directed to the handling of ﬁnite leading (and trailing) edges and associated problems posed by transition in leadingedge separation bubbles as well as unsteady interactions, including wakepassing transition with intermediate boundary-layer calming eﬀects. The T3L semi-circular and leading-edge test case has now become a new entry-level for model validation – see Savill and Pittaluga (2000) – and it has been clearly established that stagnation-region corrections (such as those proposed by Kato and Launder 1993, and Durbin 1983) are essential to maintain good predictions with even the Dawes and Launder–Sharma low-Re k-ε models. Somewhat unexpectedly, only poor results (too large a bubble and delayed transition) have so far been obtained with nonlinear k-ε schemes – perhaps reﬂecting insufﬁcient generality of model calibration. Very much better results have been reported by Papanicolau and Rodi (1999) using a two-layer approach with alternative Chen and Thyson-type correlations more representative of separated shear layer transition, and by Hadzic and Hanjalic (2000) using their TCL RST model approach. Moreover, Vicedo et al. (2000) have recently demonstrated that the addition of a simple prescribed-intermittency scaling (and subsequently intermittency transport) to the unmodiﬁed Dawes low-Re k-ε model can provide excellent predictions both up to and downstream of the leading edge. They have also now begun applying the same approach(es) to the prediction of unsteady transition in an LA turbine blade test case with equal success.

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Acknowledgements This contribution would not have been possible without the continued support of Rolls-Royce plc, DRA, the CEC and COST. Several developments owe their origins to discussions with close colleagues at UMIST (especially Professor Brian Launder), researchers at the University of Surrey (in particular Professor Peter Voke) and Dr. John Coupland of Rolls–Royce Aerothermal Methods; with whom regular interaction has been maintained as part of an overall Transition Strategy. The assistance of Dr. B.A. Younis of City University, who freely provided the original version of the low-Re RST code and assisted with the implementation of this, has been vital to the development of the SLY model, as have contacts with other researchers, too many to mention individually, whose help is also gratefully acknowledged. The Editors, organisers of the INI Programme and other colleagues contributed a number of constructive criticisms which I hope are reﬂected in this ﬁnal version.

References Abid, R. (1993). ‘Evaluation of two-equation turbulence models for predicting transitional ﬂows’, Int. J. Engineering Science 11 25. Biesinger, T.E. and Savill, A.M. (1998). ‘A contribution to the Durham Linear Turbine Cascade Test Case’. Proc. ERCOFTAC Turbomachinery SIG Workshop, Aussois. Byggstoyl, S. and Kollmann, W. (1986). ‘A closure model for conditioned stress equations and its application to turbulent shear ﬂows’, Phys. Fluids 29 1430. Byggstoyl, S. and Kollmann, W. (1981). ‘Closure model for intermittent turbulent ﬂows’, Int. J. Heat Mass Transfer 24 1811. Cho, J.R. and Chung, M.K. (1992). ‘A proposal of κ-ε-γ turbulence model’, J. Fluid Mech. 237 301. Cho, J.R. and Chung, M.K. (1990). ‘Intermittency modeling based on interaction between intermittency and mean velocity gradients’. In Engineering Turbulence Modelling and Experiments, W. Rodi and E. Ganic (eds.), Elsevier, 101. Cotton, M.A. (1989). Personal Communication. Dick, E. (Ed) (1995). ‘Theme section on transition’, ERCOFTAC Bulletin 24. Durbin, P.A. (1993). ‘Application of a near-wall turbulence model to boundary layers and heat transfer’, Int. J. Heat Fluid Flow 14 316. Hadzic, I. and Hanjalic, K. (2000). ‘Separation-induced transition to turbulence: second-moment closure modelling’, J. Flow Turb. and Comb., 63, 153–273. Hallback, M., Henningson, D. S., Johansson, A. J. and Alfredsson, H. (eds.) (1996). Turbulence and Transition Modelling, Kluwer Academic Publishers. Hanjalic, K., Jakirlic, S. and Hadzic, I. (1997). ‘Expanding the limits of ‘equilibrium’ second-moment closures’, Fluid Dynamics Research 20 25–41. Hui, J. (2000). Turbulence and Transition Predictions of Internal Flows. PhD Thesis, KTH, Stockholm

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Janicka, J. and Kollmann, W. (1983). ‘Reynolds stress closure model for conditional variables’. Proc. 4th Symp. Turbulent Shear Flows, 14.13–14.18 Karlsruhe, 14.13 Janour, P. (1997). ‘A new k-ε model for predicting transition’. Institute of Thermomechanics Report, Prague. Johnson, M.W. and Ercan, A.H. (1999). ‘A physical model for bypass transition’, Int. J. Heat Fluid Flow 20, 95–104. Kato, M. and Launder, B.E. (1993). ‘The modelling of turbulent ﬂows around stationary and vibrating square cylinders’. Proc. 9th Symp. Turbulent Shear Flows, 10-4.1–10-4.6 Kyoto. Kawamura, H. and Hada, K. (1992). ‘A k-ε two-equation model for fully developed channel ﬂow’. Submission to Stanford collaborative testing turbulence models, Science Univ. Tokyo. Kebede, W., Launder, B. E. and Younis, B.A. (1985). ‘Large amplitude periodic pipe ﬂow: A second-moment closure’. Proc. 5th Symp. Turbulent Shear Flows, 16.23– 16.28. Lumley, J.L. (1978). ‘Computational modeling of turbulent ﬂow’, Adv. Appl. Mech. 18 123–176. Mansour, N.M, Kim, J. and Moin, P. (1988). ‘Reynolds-stress and dissipation-rate budgets in a turbulent channel ﬂow’, J. Fluid Mech. 194 15–44 Papanicolau, E.L. and Rodi, W. (1999). ‘Computation of separated-ﬂow transition using a two-layer model of turbulence’, J. Turbomachinery 121 78. Patel, V.C., Rodi, W. and Scheuerer, G. (1985). ‘Turbulence models for near-wall and low-Reynolds number ﬂows: a review’, AIAA J. 23 1308. Pereira, J.C.F. (2000). ‘Coupling Navier–Stokes and stability calculations in the incompressible regime’. Proc. ECCOMAS Conference, Barcelona. Pironneau O., Rodi W., I.L. Ryhming, Savill A.M. and Truong T.V. (eds.) (1992). Proc. 1st ERCOFTAC Workshop on Numerical Simulation of Unsteady Flows Transition to Turbulence and Combustion, Lausanne, Cambridge University Press; see in particular A.M. Savill: A synthesis of T3 test case predictions. Rodi, W. and Mansour, N.N. (1993). ‘Low-Reynolds number k-ε modelling with the aid of direct simulation data’, J. Fluid Mech. 250 509–529. Rotta, J.C. (1968). ‘Statistical theory of non-homogeneous turbulence’. Papers 1 and 2. (translated by W. Rodi) Imperial College Reports TWF/TN/38–39. Savill, A.M. (1993a). ‘Some recent progress in the turbulence modelling of by-pass transition’. In Near-wall Turbulent Flows, R.M.C. So, C. Speziale and B.E. Launder (eds.), Elsevier, 829–848. Savill, A.M. (1993b). ‘Further progress in the turbulence modelling of by-pass transition’. In Engineering Turbulence Modelling and Experiments 2, W. Rodi and F Martelli (eds.), Elsevier, 583–592. Savill, A.M. (1994). ‘Transition modelling for turbomachinery II’. Summary Proc. of the 1st ERCOFTAC Transition SIG Workshop of the BRITE-EURAM AEROCT92-0052 Project Workshop on Transition in Turbomachinery, VUB, Brussels.

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Savill, A.M. (1995) ‘A summary report on the COST ERCOFTAC Transition SIG Project: Evaluating turbulence models for predicting transition’, ERCOFTAC Bulletin 24, 57–61. Savill, A.M. (1996). ‘Evaluation of turbulence models for predicting transition in turbomachinery ﬂows. Minowbrook II Workshop on Boundary Layer Transition in Turbomachines, Sycracuse University. Savill, A.M. (1999a). ‘Transition Prediction with turbulence models’. Proc. INI Workshop on Breakdown to Turbulence and its Control, Isaac Newton Institute, Cambridge. Savill, A.M. (1999b). ‘Non-local counter-gradient transport modelling’. Proc. INI Workshop on Mathematics of Closure, Isaac Newton Institute, Cambridge. Savill, A.M. and Pittaluga, F. (2000). ‘Implementation and further application of reﬁned transition prediction methods for turbomachinery and other aerodynamic ﬂows’, ERCOFTAC Bulletin 44, 45. Savill, A.M and Tselepidakis, D.P. (1994). ‘Adopting the new TCL Reynolds stress model for curved ﬂows’. Proc. 6th Biennial Colloquium on Computational Fluid Dynamics, UMIST. Sieger, K. (1999). Prediction of Transitional Boundary Layers with a New Two-layer Turbulence Model. PhD Thesis, University of Karlsruhe Simon, F.F. (1994). Proc. AFOSR Workshop on End-Stage Boundary Layer Transition, Minowbrook, Syracuse University. Sousa, J.M.M., Margues, N.P.C. and Pereira, J.C.F. (2000). ‘Coupling Navier–Stokes calculations with linear stability analysis in wing design’. Proc. ECCOMAS 2000 Special Session on Transition Prediction using Tubulence Models and PSE, Barcelona. Steelant, J. and Dick, E. (1995). ‘Transition modelling by conditionally averaged equations’, ERCOFTAC Bulletin 24, 62–64. Vicedo, J. Dawes, W.N. and Savill, A.M. (2000). ‘Coupling PSE to a low Re k-ε intermittency transport RANS approach’. Proc. ECCOMAS 2000 Special session on Transition Prediction using Turbulence Models and PSE, Barcelona. Vicedo, J., Vilmin, S., Dawes, W.N. and Savill, A.M. (2000). ‘Extension of intermittency transport modelling to natural transition in external aerodynamic applications’. Proc. RAeS Aerodynamics Research Conference 2000, Royal Aeronautical Society, London. Vicedo, J., Vilmin, S., Dawes, W.N., Hodson, H.P. and Savill, A.M. (2000). ‘The extension of CFD-friendly turbulence modelling to include transition’. Proc. 8th European Turbulence Conference, Barcelona. Voke, P.R. and Yang, Z.Y. (1993). ‘Numerical solutions of the mechanisms of by-pass transition in the ﬂat plate boundary layer’. Proc. 9th Symp Turbulent Shear Flows, Kyoto, 21-2-1–21.2-6. Voke, P.R., Yang, Z.Y. and Savill, A.M. (1996). ‘LES and modelling of transition following a leading-edge separation bubble’, In Engineering Turbulence Modelling and Experiments-3, W. Rodi and G. Bergeles (eds.) 601–610, Elsevier Science.

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Westin, K.J.A. and Henkes, R.A.W.M. (1997). ‘Application of turbulence models to bypass transition’, J. Fluids Eng. 119 859–866. Yang, Z. and Shih, T. H. (1991). ‘Extending turbulence modelling to by-pass transition’, NASA CMOTT Research Briefs 83. Yao, Y.F., Savill, A.M., Sandham, N.D. and Dawes, W.N. (2000). ‘Simulation of a turbulent trailing edge ﬂow using unsteady RANS and DNS’. Proc. International Heat and Mass Transfer Conference, Nagoya.

19 Compressible, High Speed Flows S. Barre, J.-P. Bonnet, T.B. Gatski and N.D. Sandham 1

Introduction

There is an ever-increasing need to be able to predict and control high subsonic to high supersonic speed ﬂows for the optimal design of aerospace vehicles. In this Mach number range, large variations in pressure can occur which can also lead to large density variations. Such variations in the state variables can have a signiﬁcant impact on both the mean and turbulence dynamics. Improving the prediction and control of compressible ﬂows requires an accurate description of the large scale (structure) dynamics and of the mean and other statistical properties of the ﬂow. Thus, a better understanding of the dynamics and improvements in the modeling of these ﬂow features is essential. Within the framework of the ensemble-averaged Navier–Stokes equations, compressible ﬂows have generally been computed using mass-weighted (Favreaveraged) variables. Higher-order correlations with incompressible counterparts have been modeled by variable density extensions to the incompressible form; meanwhile higher-order correlations unique to the compressible form have been isolated. Some of these compressibility terms, such as dilatation dissipation and the compressible heat ﬂux, have been widely used. Others, such as mass ﬂux or pressure-dilatation, have seen more limited application, as have explicit compressibility corrections to standard model terms. Many calculations have been performed neglecting (with or without justiﬁcation) extra compressible terms and using simple variable density extensions of the equations. The goal of this chapter is to provide the reader with an overall perspective of the experimental and numerical study of compressible, turbulent shear layers including the eﬀect of shock-turbulence interactions. Concepts and relevant modeling issues associated with turbulent boundary layers, mixing layers and wakes are discussed. In addition, the eﬀects of shock/turbulence interactions on ﬂow ﬁeld dynamics in both isotropic, homogeneous turbulence, and inhomogeneous (boundary-layer and mixing layer) ﬂows are discussed. Coupled with some fundamental theoretical aspects of compressible turbulent ﬂows, this should provide a summary of the current state of the art and a basis for further research.

522

[19] Compressible, high speed ﬂows

2

523

Background: Experimental and Simulation Data

Detailed and accurate data for mean and turbulent quantities in supersonic turbulent ﬂows are very important in order to increase our knowledge about compressible turbulent ﬂow dynamics and develop turbulence closures. Such data can conceivably be provided by both laboratory experiments and numerical simulations (DNS, and to a more limited extent LES). However for both experiments and simulation there are additional diﬃculties beyond those faced for low speed ﬂows. In this section we brieﬂy review the current status. The ﬁrst issue with experiments is to obtain suﬃciently high ﬂow quality. Great care must be taken concerning the initial turbulence levels. Parasitic acoustic perturbations can alter the external conditions of the ﬂow and make the results diﬃcult to generalize and not very easy to use as test cases for turbulence model validation. One example of this is in a supersonic mixing layer where it is diﬃcult to obtain, in a given wind tunnel, a wide range of convective Mach numbers Mc . Even in a classical air-air wind tunnel with good quality external conditions such as low external pressure gradients, low external noise and turbulence, and a suﬃciently large test section to avoid conﬁnement eﬀects, a variation in Mc from 0.4 to 1 can be considered large, but even this is not easy to obtain in a lot of experimental facilities. Another example is the case of a normal shock wave interacting with homogeneous and isotropic turbulence. In this case, it is very diﬃcult to generate an incoming turbulent ﬁeld free from parasitic eﬀects like pressure waves and low frequency pulsation. Another major diﬃculty with experiments is concerned with measurement techniques. The most popular techniques used to measure turbulent ﬁelds in supersonic ﬂows are hot-wire anemometry and Laser Doppler Velocimetry (LDV). While these techniques allow one to obtain a lot of turbulence data, they are not free of diﬃculty in these types of ﬂows. For the hot wire, the main problem lies with calibration, particularly when transonic Mach numbers and low Reynolds numbers are involved. It has been shown that neglecting transonic eﬀects may lead to about a 50% underestimation in root mean square velocity ﬂuctuations (Barre et al. 1992). LDV does not need calibration, but the ﬂow must be seeded with particles in order to generate Mie scattering light. In highly sheared supersonic compressible ﬂows, such as mixing layers at high Mc , seeding problems may corrupt the experimental results. Another problem with LDV seeding arises in the case of shock/turbulence interaction. The inertia associated with the seeding particles makes them unable to follow the local air velocity in the immediate downstream vicinity of the shock. In these cases, a particle drag correction must be applied to the data. This law must be computed accurately in order to keep experimental results representative of reality. Even more measuring diﬃculties arise when 3D measurements are necessary. Three-dimensional LDV in supersonic ﬂows is not an easy task, as discussed by Bonnet et al. (1996) and Gruber et al. (1993), so that it is some-

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times necessary to develop mixed measurements techniques including hot wire and LDV to deduce three-dimensional Reynolds stress behavior (Chambres 1997). Some new optical techniques, such as Filtered Rayleigh Scattering, Particle Image Velocimetry (PIV) and Collective Light Scattering have begun to be used (see Bonnet et al. (1998) for a complete description). These new techniques are very promising, but at this time no complete results for the Reynolds stress tensor in supersonic compressible ﬂows have been obtained. Hopefully, in the near future, they will be able to provide new information for variables like the thermodynamic turbulent ﬂuctuations which would be very useful in the ﬁeld of compressible turbulence modeling. Direct numerical simulations (DNS) of compressible ﬂows have mainly focused on simple homogeneous ﬂows, with and without shocks, to study the eﬀects of compressibility on turbulence. Studies of inhomogeneous turbulent shear ﬂows, such as boundary-layer and mixing-layer ﬂows, are less common but have been performed. Such ﬂows suﬀer from the same constraints as their incompressible counterparts in that grid resolution restrictions prevent highReynolds number computations. In the compressible, turbulent boundarylayer case, such constraints are somewhat mitigated by the fact that the viscosity increases near the wall, thereby decreasing the local eﬀective Reynolds number in a region where grid resolution is most critical. Large-eddy simulation for high-speed ﬂow is in its infancy. Extra sub-grid terms requiring modeling appear in the energy equation. Thus far attempts to model these terms have been based on extensions of standard techniques such as dynamic modeling from incompressible ﬂow, with some limited guidance from DNS. The interaction of sub-grid model and numerical method can be even more important for compressible ﬂow with shock waves, which require special schemes, often relying on extra dissipation. Since this is introduced near the grid scale it will have a strong eﬀect on the energy transfer from resolved to unresolved scales. This may well have implications of the choice of sub-grid model.

3

Compressible Navier–Stokes Equations

It is a useful prelude to both the experimental and numerical results to be presented in this chapter to detail the equations governing the motion in the compressible regime. In addition to the numerical simulations or the calculation of the mean statistics through a Reynolds-averaged approach, these equations provide a useful basis of analysis for the experimental results. It will be shown how an analysis of the energy-budgets within the context of the ensemble-averaged equations leads to a better understanding of the dynamics of compressible turbulent ﬂow.

[19] Compressible, high speed ﬂows

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The starting point is the mass, momentum, and (total) energy conservation equations. The conservation of mass equation is ∂ρ ∂(ρuj ) = 0, + ∂t ∂xj

(2.1)

and the conservation of momentum equation is ∂(ρui ) ∂(ρui uj ) ∂p ∂σij =− + , + ∂t ∂xj ∂xi ∂xj

(2.2)

with the viscous stress tensor σij deﬁned as

1 σij = 2µ Sij − Skk δij , 3

(2.3)

where ρ and ui are respectively the density and velocity components, µ is the molecular viscosity, and Sij is the strain rate 1 Sij = 2

∂ui ∂uj + ∂xj ∂xi

.

(2.4)

The total energy equation is given by ∂(ρE) ∂(uj ρH) ∂(ui σij ) ∂qj = − , + ∂t ∂xj ∂xj ∂xj

(2.5)

where the total energy and total enthalpy, ρE and ρH, respectively, are

ρE = ρ e +

ui ui 2

ρH

=ρ E+

p ρ

(2.6)

(2.7)

and e = cv T is the internal energy (cv is the speciﬁc heat at constant volume and is assumed constant), qj = −kT ∂T /∂xj is the heat ﬂux (kT is the thermal conductivity), and the equation of state is the perfect gas law p = ρRT . These equations, or reformulations thereof, can be used in a direct simulation approach. In a large-eddy simulation approach, the governing equations are ﬁltered to yield a set of equations for the large-scale motions of the ﬂow, and in a RANS approach the equations are Reynolds averaged to yield a set of equations for the statistical mean motion of the ﬂow.

2.1

Reynolds-averaged equations

While direct or large-eddy simulations of compressible turbulent ﬂows have previously focused on homogeneous and temporally developing ﬂows, with only

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recent applications to inhomogeneous ﬂows, numerical solutions of the compressible, Reynolds averaged Navier–Stokes (RANS) equations with suitable correlation closure models, have been available for more than three decades. In the compressible formulation, a Reynolds averaging approach is once again applied to the conservation equations (2.1), (2.2), and (2.5). It has been found that a rewriting of the equations using mass-weighted, or Favre (Favre 1965), variables is advantageous, since the equations take a more compact form relative to the Reynolds averaged variables and the terms in the equations can be shown to have analogous counterparts in the incompressible formulation. For a dependent variable f , the Favre ‘average’ is deﬁned as ρf f˜ = . ρ

(2.8)

The instantaneous value f can then be decomposed into either the usual Reynolds averaged variables or the Favre-averaged variables f = f + f = f˜ + f .

(2.9)

As might be expected, an extensive list of relations exist between the Reynoldsaveraged variables and the Favre-averaged variables. Some of these relations are provided in Appendix A. The equations for the mean density ρ and mean momentum ρ˜ ui are given by ∂ρ ∂ (ρ˜ uj ) = 0, (2.10) + ∂t ∂xj ρ

ui ) D˜ ui ∂p ∂σ ij ∂(ρτij ) ∂(ρ˜ ∂ (˜ uj ρ˜ ui ) = − + − , = + Dt ∂t ∂xj ∂xi ∂xj ∂xj

(2.11)

and the mean viscous stress tensor σ ij is

1 σ ij = 2µ Sij − Skk δij 3

1 2µ S4ij − S4kk δij

3

,

(2.12)

u is the Favre-averaged where µ is the mean molecular viscosity, and τij = u> i j correlation tensor. Equation (2.12) neglects contributions from µ , and assumes that Ui ≈ u ˜i . This assumed equality between the average velocities implies that the average ﬂuctuating velocity ui is small since ui − u ˜i = ui . Results from a DNS of a spatially evolving supersonic ﬂow (M∞ = 2.25) over a ﬂat plate with adiabatic wall boundary conditions (Rai et al. 1995) show a maximum variation of approximately 4.8 percent. This maximum occurs in the near-wall region at y + ≈ 18 (see Figure 1). Similar results were obtained by Huang et al. (1995) from simulations of a cold-walled channel ﬂow. The equation for the 4 is mean total energy ρE

4 ∂(ρE) ∂ ˜ = − ∂ q j + ρE u , u ˜j ρH + j ∂t ∂xj ∂xj

(2.13)

[19] Compressible, high speed ﬂows

527

1.0

0.000 _ u u~

0.8

−0.005

(a)

(b) −0.010

_ u, ~ u

_ u − ~u

0.6

−0.015

0.4 −0.020 0.2 0.0

−0.025 1

10

y

+

100

−0.030

1000

1

10

y

+

100

1000

Figure 1: Mean streamwise velocity proﬁle across supersonic boundary layer: (a) Favre and Reynolds mean, (b) mean velocity diﬀerence. (Velocities scaled with freestream velocity.) where

u ˜i u> u ˜i u + i i, 2 2 4 + p, ˜ =E H ρ

4 = cv T˜ + E

q j = −kT

(2.14) (2.15)

∂T ∂ T˜ − kT , ∂xj ∂xj

(2.16)

and T + u ρE uj = cp ρu> ˜i (ρτij − σ ij ) + j

ρui ui uj u . − σ ij ui − σij i 2

(2.17)

In the approximation of (2.16), ﬂuctuations in the thermal conductivity are neglected, and the Favre-averaged and Reynolds-averaged mean temperatures are taken as approximately equal. The equality of the two averages may once again be checked against DNS results. As Figure 2 shows from the results of Rai et al. (1995), the maximum variation within the boundary layer between the two temperature averages is approximately 1 percent occurring at y + ≈ 20. In the cold-walled case of Huang et al. (1995) a similar deviation level was found. The equation of state in mean variables is p = ρRT4,

(2.18)

4 or in terms of the mean total energy ρE

4 − 1ρ u 4 2 − ρk p = (γ − 1) ρE ˜2 + v˜2 + w

2

(2.19)

528

Barre et al. 2.0

0.020 _ T ~ T

(a)

(b)

0.015

_ ~ T, T

_ ~ T−T

1.5 0.010

1.0 0.005

0.5

1

10

+

100

1000

0.000

1

y

10

+

100

1000

y

Figure 2: Mean temperature proﬁle across supersonic boundary layer: (a) Favre and Reynolds mean, (b) mean temperature diﬀerence. (Temperatures scaled with freestream temperature.) where γ is the ratio of speciﬁc heats (cp /cv ), and R is the gas constant. The presence of the turbulent kinetic energy term k, ρk = ρ

u u> τii =ρ i i, 2 2

(2.20)

suggests a strong coupling between the mean equations and the turbulent transport equations. Use of the total energy as a primary variable facilitates the use of shock-capturing numerical methods. In the incompressible case, the only corresponding term requiring closure would be the Reynolds stresses ρτij ; however, the inclusion of the mean total T , the energy equation necessitates models for the turbulent heat ﬂux ρcp u> j turbulent mass ﬂux ui = −ρ u /ρ, and the turbulent transport or diﬀusion ρui ui uj . While some or all of these terms may not be signiﬁcant in a majority of compressible ﬂows, it is important to include them at the outset in order to recognize the assumptions invoked in obtaining more simple forms of the equations.

2.2

Morkovin’s Hypothesis and the Strong Reynolds Analogy

Before proceeding onto the development and analysis of models for the unknown correlations that arose in the development of the mean conservation equations, it is useful to examine one of the most important concepts associated with the study of compressible ﬂows. Morkovin’s hypothesis (Morkovin 1964) and the associated set of relations which constitute what is termed the strong Reynolds analogy (SRA) have proved useful in the analysis of compressible, supersonic ﬂows. (The reader is referred to the review by Brashaw (1977) and the study by Gaviglio (1987) as well as the book of Smits and Dussauge

[19] Compressible, high speed ﬂows

529

(1996) for further discussion on this topic.) These ideas were developed from an analysis of 2D supersonic boundary layer data valid for M < 5 boundary layer ﬂows at moderate values of wall heat transfer q w , but have been generally applied as a basic concept. In the discussion, the analysis will focus on a simple boundary layer ﬂow with velocity components (u, v) in the streamwise x and wall normal y directions, respectively. From an analysis of the data available at the time, Morkovin deduced that the . . . essential dynamics of these supersonic shear ﬂows will follow the incompressible form . . . This observation was quantiﬁed by the proposal that

ρ ρw

uv

and

u2τ

ρ ρw

2 u

u2τ

(2.21)

would depend little on Mach number . . . This hypothesis was deduced from Morkovin’s analysis of the solutions of the momentum and total enthalpy equations. These solutions, which will now be derived, are what Morkovin termed the strong Reynolds analogy (SRA). The derivations to follow are based on a linearization of the governing equations which is justiﬁed by the early success of Kovasznay’s categorization (Kovasznay 1953) of compressible turbulence into rotational, entropy, and acoustic modes. For moderate Mach number supersonic ﬂows, the pressure ﬂuctuations (acoustic mode) are neglected. From the perfect gas law (p = ρRT = ρ(γ − 1)cv T ) and for small ﬂuctuations, the pressure ﬂuctuations are related to both the density and temperature ﬂuctuations through p ρ T , = + p ρ T4

(2.22)

so that if the pressure (acoustic) ﬂuctuations are negligible, the density and temperature ﬂuctuations are related through ρ T =− . ρ T4

(2.23)

Since this is diﬀerent to the isentropic relation, the temperature ﬂuctuations must be considered to be non-isentropic. In the case of an adiabatic wall, the SRA as proposed by Morkovin (1964) (see also Young 1953) is given by T4t = T4tw

(2.24)

Tt

(2.25)

= 0,

u2 /(2cp )). The absence where the subscript t denotes total conditions (T˜t = T˜ +˜ of total temperature ﬂuctuations corresponds to neglecting ﬂuctuations due to the entropy mode. When these equations are coupled with the total enthalpy

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equation and terms which are quadratic in the ﬂuctuations are neglected, the temperature and velocity ﬂuctuations are related by uu . cp T ≈ −˜

(2.26)

(Note that it has additionally been assumed here that u ˜u v˜v .) This instantaneous relation between the temperature and velocity ﬂuctuations can be recast into more useful statistical relations. For example, by using the mean total enthalpy relation (neglecting quadratic ﬂuctuations) cp T4t = cp T4tw = cp T4 +

u ˜2 2

(2.27)

evaluated at the boundary layer edge, (2.26) becomes

T 2

u ˜ u2 ≈2 , u ˜e u ˜e T4tw − T4e

(2.28)

˜2 /γRT4), or, by using the local Mach number (M 2 = u

T 2

T4

≈ (γ − 1) M

u2 . u ˜

2

(2.29)

In terms of the correlation coeﬃcient Ru T , deﬁned as Ru T =

u T

u2 T 2

,

(2.30)

(2.29) yields Ru T ≈ −1

(2.31)

As was discussed at the beginning of this subsection, the pressure ﬂuctuations have been neglected in the analysis so that the density and temperature ﬂuctuations are related by (2.23). Thus, both (2.29) and (2.31) can be rewritten in terms of these density ﬂuctuations as

ρ2 ≈ (γ − 1) M 2 ρ

u2 u ˜

(2.32)

and Ru ρ ≈ 1,

(2.33)

respectively. One remaining expression associated with the SRA can be found from the turbulent Prandtl number deﬁned by P rt =

µt cp . kT

(2.34)

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531

A relation between the vertical turbulent heat ﬂux and the turbulent shear stress can be obtained directly from (2.26), so that u ˜=−

cp ρv T , ρu v

(2.35)

and a relation between the mean temperature and velocity gradients can be obtained from (2.27), so that u ˜=−

cp ∂ T4/∂y . ∂u ˜/∂y

(2.36)

Equations (2.35) and (2.36) can be combined to yield the ratio

ρu v ∂u ˜/∂y

or P rt =

∂ T4/∂y ρv T

≈1

µt cp ≈ 1. kT

(2.37)

(2.38)

Equations (2.26), (2.28), (2.29), (2.31), and (2.38) are the ﬁve relations obtained by Morkovin (1964) that constitute the strong Reynolds analogy. Additional relations have since been derived from this basic set. For example, a very strong Reynolds analogy (VSRA) has been proposed which is the instantaneous counterpart to (2.29), T T4

≈ − (γ − 1) M 2

u . u ˜

(2.39)

In order to account for heat ﬂux at the wall, an extended strong Reynolds analogy has been proposed (ESRA) (Gaviglio 1987, Rubesin 1990) T /T4 1 1

≈− 2 (γ − 1) M u /˜ u P rt 1 − ∂ T4t /∂ T4

(2.40)

where the proportionality to the turbulent Prandtl number P rt has been pointed out by Huang et al. (1995). Gaviglio (1987) assumed an equality between the characteristic length scales associated with both the velocity and temperature small-scale ﬂuctuations to arrive at the form given in (2.40), and Rubesin (1990) assumed a mixing length formulation to arrive at a relation for the static enthalpy ﬂuctuations which eventually led to (2.40). Both Morkovin’s hypothesis and the various forms of the Reynolds analogies can be used to validate both experimental and numerical simulation data as well as the predictive capability of RANS. In turn, experimental and numerical simulation data can be used to validate the assumptions used in the derivation of the relations comprising the Reynolds analogies.

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2.3

Turbulent transport equations

In the analysis of compressible, turbulent ﬂow ﬁelds through either physical or numerical experiments, the equations describing the transport of the turbulent Reynolds stresses will be needed. As was seen in the development of the mean ﬂow equations, the turbulent stresses are needed to close the equations. The turbulent stress equations lead directly to the turbulent kinetic energy equation which is especially useful in understanding the underlying ﬂow dynamics through an analysis of the component terms comprising the energy budget. The transport equations for the turbulent stresses in the compressible case are obtained in a manner analogous to the incompressible formulation although now the dependent variables are decomposed into Favre-mean and ﬂuctuating components and then Reynolds-averaging the equations (e.g. Gatski 1996). This procedure leads to the Reynolds-averaged equations for the Favre correlation tensor τij , ρ

∂u ˜j ∂u ˜i Dτij − ρτjk + ρΠij − ρij + ρMij + ρDij , = −ρτik Dt ∂xk ∂xk

where

ρΠij =

p

∂uj ∂ui + , ∂xj ∂xi

(2.42)

∂uj 2 ∂ui , ρij = ρδij + ρD ij = σik + σjk 3 ∂xk ∂xk

ρMij = 

ρDij = −

ui

(2.41)

∂σ jk ∂p − ∂xk ∂xj

+ uj

(2.43)

∂σ ik ∂p − , ∂xk ∂xi

(2.44) 

∂  u + σ u )  . ρu u u + p (ui δjk + uj δik ) − (σik j jk i ∂xk i j k turbulent transport

(2.45)

viscous diﬀusion

As the equation shows, the transport of the Reynolds stress is controlled by a balance among its production, redistribution ρΠij , destruction ρij , mass ﬂux contribution ρMij , transport and diﬀusion ρDij . A simpliﬁcation results when the turbulent kinetic energy equation is considered since it is simply the trace of the (2.41), ρ

∂u ˜i Dk + ρΠ − ρ + ρM + ρD, = −ρτik Dt ∂xk

where

ρΠii ∂u = p i , 2 ∂xi

(2.47)

ρii ∂ui , = σik 2 ∂xk

(2.48)

ρΠ = ρ =

(2.46)

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533

ρMii ∂σ ik ∂p ρM = − , = ρui 2 ∂xk ∂xi 

(2.49) 

ρDii ∂  ρui ui uk u  . ρD = =− + p ui δik − σik i 2 ∂xk 2

(2.50)

The terms on the right hand side of (2.46) are the pressure-strain correlation (which here appears as pressure dilatation), the compressible turbulent dissipation rate, the mass ﬂux contribution, and the turbulent transport, respectively. Theoretical models have been developed for all these terms, but with varying degrees of validation. Some of the experimental and numerical calibration and validation of models for these terms will be discussed in subsequent sections. As in the incompressible case, the deviatoric part of the tensor dissipation rate, (2.43) is usually assimilated into the pressure-strain rate correlation, which leaves the isotropic dissipation rate in the compressible Reynolds stress or turbulent kinetic energy formulations to be obtained from a modeled transport equation. In the compressible case, a partitioning of the isotropic dissipation rate can be invoked (Zeman 1990; Sarkar et al. 1991; see also Wilcox 1998) to explicitly account for compressibility eﬀects. The form of the partitioning is = ε + εd (2.51) where ε is the solenoidal (incompressible) dissipation and εd is the dilatation (compressible) dissipation. A common high-Reynolds-number form of the isotropic solenoidal dissipation rate equation that is used is given by ρ where

ε ∂u ˜i ε2 Dε − ρCε2 + ρDε == −ρCε1 τik Dt k ∂xk k ∂ ρDε = ∂xj

k µ + Cε ρτij ε

∂ε ∂xi

(2.52)

(2.53)

for a second-moment closure, or ∂ ρDε = ∂xj

µ µ+ t σε

∂ε ∂xj

(2.54)

for a two equation k-ε formulation. The ﬁrst and second terms on the right side of (2.52) are the production and destruction of dissipation, respectively, and the last term is the viscous diﬀusion. The closure coeﬃcients Cε1 , Cε2 , Cε , and σε often assume their incompressible values. In the form presented in (2.52), the dissipation rate equation is simply a variable density extension of the incompressible form. An additional term associated with the mean dilatation can be added to the right hand side of (2.52) to properly account for the behavior of compressed

534

Barre et al.

isotropic turbulence (Speziale and Sarkar 1991). Omission of the mean dilatation term causes the model to incorrectly predict the decrease of the integral length scale for isotropic expansion and the increase for isotropic compression. Another contribution to the right side, which is also generally neglected, arises from accounting for the variation of mean kinematic viscosity ν in the development of the dissipation rate transport equation (Coleman and Mansour 1991). This also leads to an additional term that is proportional to the mean dilatation. For the most part, however, these mean dilatation eﬀects can be neglected in ﬂows without shocks or in the absence of strong pressure gradients. El Baz and Launder (1993) have taken an alternate approach by simply using a solenoidal dissipation rate equation that has been sensitized to compressibility eﬀects through the turbulent Mach number. These eﬀects were accounted for through a modiﬁcation of the decay coeﬃcient Cε2 rather than the solenoidal and dilatational partitioning that has been discussed. They argue that this approach is more consistent with the single-scale framework in which the dissipation equation provides a model for the energy transfer to the small scales.

3

Turbulent Shear Layers

In this section, results from experimental and numerical calculations of turbulent boundary layers and turbulent mixing layers will be discussed. As was pointed out previously, the equations describing the behavior of turbulent compressible ﬂows as well as the fundamental relations derived from the SRA, have highlighted some turbulent correlations which are either unique to the compressible regime or which can be used to better characterize such ﬂows. These explicit compressible correlations include the mass ﬂux or average ﬂuctuating velocity, the dilatation dissipation, and the pressure-dilatation. In addition to these uniquely compressible terms, eﬀects of mean density gradients and turbulent heat ﬂux correlations need to also be considered. The modeling of such terms as well as their measurement are important in obtaining the proper understanding of the dynamics of high speed compressible ﬂows. In addition one may need to consider compressibility eﬀects on terms already in the equations, for example the pressure strain (Vreman et al. 1996). In supersonic turbulent boundary layer ﬂows without shocks (or eddy shocklets), both dilatation dissipation and pressure-dilatation eﬀects are minimal, and the two main eﬀects are due to the mean density variations and turbulent heat ﬂux. In free shear ﬂows other compressibility eﬀects play a more important role. This is consistent with the results of Sarkar (1995), who has shown that the relevant parameter to be used in assessing the eﬀect compressibility is the gradient Mach number

[19] Compressible, high speed ﬂows

535

Mg deﬁned by (d˜ u/dy)l , Mg = γRT4

(3.1)

where l is the length scale of the large eddies in the radial (or transverse) direction. In a supersonic boundary layer ﬂow, Mg is roughly constant (≈ 0.2); whereas, in a mixing layer ﬂow, for example, it is a linear function of the convective Mach number (Mg ≈ 2.2Mc ; see Section 3.3.1).

3.1

Turbulent boundary layers

Just as in the previous section where the consequences of the strong Reynolds analogy were presented, an equally important set of relations for the mean ﬂow behavior in turbulent boundary-layers can be derived. These are relations on the variation of both mean velocity and mean temperature in the log-layer of a compressible boundary-layer. This compressible law of the wall serves as a necessary validation of both computational and experimental results as well as the basis for compressible wall functions. Adaptations have been made for ﬂows other than ﬂat plate boundary-layers in zero pressure gradient; however, for the purposes here in the derivation of the fundamental forms, the simple case of planar ﬂows in zero pressure-gradient will be considered. 3.1.1

Law of the wall

Whilst the very existence of a logarithmic portion of the mean velocity proﬁle near a wall is still a subject of debate in some quarters, and there is no consensus on the exact values of the relevant constants, it is recognized that the logarithmic law of the wall is a suﬃciently good approximation to be useful for modelling purposes. The compressible law of the wall (Van Driest 1951) deﬁnes the behavior of both the mean velocity u ˜ and temperature T4, as well as the turbulent shear stress ρτxy in a region of the boundary layer. In zero pressure gradient planar ﬂow, the momentum and total energy equations (Cartesian coordinates) near the wall become σ tot = σ xy − ρτxy = σ w ,

(3.2)

where subscript w denotes conditions at the wall, and

T − u q tot = q y + ρv E q y + cp ρv> ˜σ w = q − u ˜σ w = q w

(3.3)

respectively. In extracting these equations from (2.11) and (2.13), advection eﬀects have been assumed to be small, and in the energy equation both mass ﬂux and turbulent and viscous transport have been neglected.

536

Barre et al.

In the log-law region the mean velocity and temperature proﬁles are given by ∂u ˜ (σ w /ρ)1/2 = , (3.4) ∂y κi y with κi the (incompressible) Karman constant, and (q/ρcp ) ∂ T4 , = −P rt ∂y (σ w /ρ)1/2 κi y

(3.5)

respectively. In the region of the boundary layer where (3.4) and (3.5) hold, the total heat ﬂux is taken to be q = −cp so that −cp

µt ∂ T4 P rt ∂y

(3.6)

µt ∂ T4 ˜σ w = qw + u P rt ∂y

where

µt = σ w

∂u ˜ ∂y

(3.7)

−1

(3.8)

The temperature distribution can then be (formally) extracted from (3.7) to give 2 q u ˜ P r u ˜ t w T4 T4w − + . (3.9) cp σw 2 Equation (3.9) should be viewed with caution since it is only an approximate relationship because the integration limit on the temperature gradient extends directly to the wall (the expression is not valid in the viscous sublayer). Equation (3.9), coupled with the compressible law of the wall (3.5), yields a relationship between the wall temperature and the wall heat ﬂux. Since the pressure is assumed constant in the wall layer, the compressible law of the wall for velocity (3.4) can be written in terms of the friction velocity u ˜τ (= σ w /ρw ) as F G

4 ∂u ˜ u ˜τ G HT . = ∂y κi y T4w

(3.10)

This velocity log-law can then be combined with the temperature relation (3.9) to obtain yu ˜τ u ˜τ u ˜vd = ln + Bc (3.11) κc νw or

u ˜vd

2cp T4w P rt

1/2

arcsin

qw + u ˜σ w σw D

− arcsin

qw σw D

,

(3.12)

[19] Compressible, high speed ﬂows

537

where the Van Driest velocity u ˜vd is deﬁned by 4 1/2 Tw

u ˜vd = and

D= κc = κc (Mτ , Bq , P rt ),

T4 q 2w 2cp T4w + 2 σw P rt

d˜ u,

(3.13)

1/2

(3.14)

Bc = Bc (Mτ , Bq , P rt ),

Bq = q w /ρw cp u ˜τ T4w . (3.15)

These functional dependencies generally have minimal eﬀect in the low supersonic regime so that the values for κc and Bc are close to their incompressible ones (e.g. Huang et al. 1993).

3.2

Compressibility eﬀects

In the evaluation of both experimental and numerical results of compressible boundary layer ﬂows under suitable conditions, compliance with the compressible law of the wall is an important test which needs to be applied to the results. 3.2.1

Mean density gradient

One of the best examples of how the log-law can be used to assess turbulent model performance can be found in the study of Huang et al. (1994), where it is shown how mean density gradients can signiﬁcantly alter two-equation turbulence model calibrations for boundary layer ﬂows. In order to ensure that the law of the wall is maintained, a unique relation among turbulence model closure constants must exist. For incompressible ﬂows, this constraint leads to the condition that σε =

κ2i . (Cε2 − Cε1 ) Cµ

(3.16)

Huang et al. 1994 showed that to recover the correct log-law in ﬂows with mean density gradients one should take σε = where

κ˜i

2

=

κ2i

κ˜i 2 Cε2 − C˜ε1 Cµ

y dρ y dρ 3 y 2 d2 ρ 1+ +3 − ρ dy 2 ρ dy 2 ρ dy

'

κ2 C˜ε1 = Cε1 1 + i Cµ σK

(3.17)

2

y dρ y 2 d2 ρ 3 − + ρ dy ρ dy 2 2

(3.18)

y dρ ρ dy

2 (

.

(3.19)

538

Barre et al.

If a k-ω formulation is used, the mean density eﬀect is present but does not have as severe an impact on the log-law. (See Viala (1995) for the impact of this eﬀect on a variety of two equation models.) 3.2.2

Turbulent heat ﬂux

As was shown in (2.17), the heat ﬂux plays a prominent role in the mean (total) energy equation. In the compressible formulation, it also directly aﬀects the pressure-velocity correlation p ui which appears in the turbulent transport term ρDij in (2.45) since

or

T p ui u> ρ ui + = i p ρ T4

(3.20)

T + RT 4ρ u . p ui = Rρui T + RT4ρ ui = Rρu> i i

(3.21)

A representative distribution of the heat ﬂux and mass ﬂux contributions (Rai et al. 1995) for the streamwise p u and cross-stream p v pressure-velocity correlations is shown in Figure 3. (Variables are scaled with free stream velocity and pressure. The spanwise correlation p w is an order of magnitude smaller and is not shown.) A wide range of models has been proposed and many of 0.0020

0.005

(b)

(a) 0.0015

0.000

______

p′v′

0.0010

_________

R ρv′′T′′

−0.005

~ ______

RT ρ′v′

0.0005

______

p′u′

_________

−0.010

R ρu′′T′′

0.0000

~ ______

RT ρ′u′

−0.015

1

10

+

y

100

1000

−0.0005

1

10

+

100

1000

y

Figure 3: Variation of pressure-velocity correlation across supersonic boundary layer: (a) p u , (b) p v . these are summarized in Gatski (1996). Gradient-diﬀusion models are the most popular and simplest closures for the turbulent heat ﬂuxes. However, more recent attempts have taken into account variable Prandtl number behavior. This has required the solution of transport equations for the temperature variance as well as models for the dissipation of temperature variance. In addition, analogous to the development of algebraic stress models in terms of tensor representations involving the mean strain rate and rotation rate

[19] Compressible, high speed ﬂows

539

tensors, explicit algebraic scalar ﬂux models have also been developed (e.g. Wikstr¨ om et al. 2000). While these models are not speciﬁcally developed for compressible ﬂows, their applicability to such ﬂows is based on the assumption that a variable mean density extension is suﬃcient. This approach can be partially validated from an analysis of direct numerical simulation results. Huang et al. (1995) (using the DNS data of Coleman et al. 1995) showed that in a cold wall, compressible boundary layer, at Mach numbers of 1.5 (Case A) and 3.0 (Case B), the (vertical) turbulent heat ﬂux is given by T ≈ ρv T + ρ v T . ρv T = ρv>

(3.22)

Huang et al. (1995) retained the triple ﬂuctuating product term in (3.22) since its eﬀect did extend outside the sublayer and had an eﬀect on the heat ﬂux proﬁle near its maximum value. Even then its contribution was only about 5% T . Figure 4 shows the variation of ρv T (= ρv> T ), of the total heat ﬂux ρv> ρv T , and ρ v T across the channel obtained by Huang et al. (1995). Case A had a lower Mach number and channel center to wall temperature ratio than the Case B. In the lower supersonic case, the contribution from ρ v T was not T and v T signiﬁcant indicating a near equality between the quantities v> 0.200

0.2000

_____ cp ρv″T″ _ ___ cp ρv′T′ _____ cp ρ′v′T′

(a) 0.150

0.1500

0.100

0.1000

0.050

0.0500

0.000

1

10

+

100

y

1000

_____ cp ρv″T″ _ ___ cp ρv′T′ _____ cp ρ′v′T′

0.0000

1

10

(b)

+

100

1000

y

Figure 4: Distribution of turbulent heat ﬂux quantities across channel. Results from Huang et al. (1995): (a) Case A, (b) Case B. (Quantities shown normalized with wall values.)

3.3

Mixing Layers

The compressible mixing layer is studied as a canonical ﬂow that is nevertheless representative of many practical applications, such as supersonic mixing in scramjet engines, infra-red and acoustic signature of hot jets, and afterbody ﬂows on aircraft. Studies have shown that plane mixing layers, annular mixing layers and, in some cases, jets can be assumed to exhibit very comparable behavior when dealing with mean or turbulent ﬂow characteristics.

540 3.3.1

Barre et al. Characteristic features

A schematic representation of a mixing layer is shown in Figure 5. Two parallel ﬂows (with corresponding mean velocities U1 , U2 ; Mach numbers M1 , M2 ; speeds of sound a1 , a2 ; and densities ρ1 , ρ2 ) merge together after the trailing edge of a splitter plate. As these ﬂows merge, the Kelvin-Helmholtz (KH) instability generates the structure shown in Figure 6 where shadowgraph visualizations of subsonic variable density plane mixing layers at diﬀerent Reynolds number from Brown and Roshko (1974) are shown. The Reynolds number in case (a) is four times the one in case (e). It is clear that increasing Reynolds number leads to more and more small scale turbulent structures, but the initial large scale structures resulting from the KH instability persist even at the highest Reynolds numbers. Large-scale structures are still present in supersonic

U1, M1, a1, ρ1

U2, M2, a2, ρ2 Figure 5: Schematic representation of a plane mixing layer. ﬂows as shown in Figure 7 where tomographic visualizations are presented for four plane mixing layer ﬂow conﬁgurations (from Debisshop 1993, and Chambres 1997) covering a convective Mach number range from 0.525 to 1. These visualizations are instantaneous pictures taken with an exposure time of 30ns. The supersonic ﬂow (upper ﬂow on Figure 7) is seeded with ether droplets and the picture is obtained by collecting Mie scattered light. These pictures can be viewed as instantaneous images of the ﬂow structure. In the Mc = 0.525 case, 2D large scale structures appear clearly and are almost as sharp as in the subsonic case of Brown and Roshko (1974). However, when the convective Mach number is increased up to 0.64, the ﬂow appears less organized. In particular, the upper edge of the shear layer seems to be more horizontal and less disturbed by the large scale structures than in the Mc = 0.525 case. The decrease in spreading rate with increasing Mc is also visible in these pictures. When the convective Mach number is increased up to Mc = 1, the visualization appears to show a more uniform mixing zone with a sharp boundary and a more homogeneous internal ﬂow. Two pictures are given for the Mc = 1 ﬂow in order to show the repeatability of the visualization process. Large scale structures are less organized here than in the low convective Mach number cases and are highly three-dimensional, as was clearly shown in sequence of experiments carried out by Clemens and Mungal (1992) and numerical simulations

[19] Compressible, high speed ﬂows

541

(a)

(b)

(c)

(d)

(e) Figure 6: Eﬀect of Reynolds number on the structure of low speed plane mixing layers. From (a), high Reynolds number to (e), low Reynolds number. (See Brown and Roshko (1974) for details.) by Sandham and Reynolds (1991). The trend towards three-dimensionality is reproduced in the linear stability characteristics of compressible mixing layers (Sandham and Reynolds 1990). Such calculations also reproduce the growth rate reduction of mixing layers indicating that mean ﬂow instabilities, large scale structures and shear layer spreading rate are coupled by the same underlying ﬂow physics. The persistence of large scale structure over a wide range of Mach and Reynolds numbers is an important feature of mixing layers, and needs to be taken into account when developing models of such ﬂows. When the Reynolds number is suﬃciently large the ﬂow reaches an asymptotic downstream state where self-similarity of the (properly normalized) mean and turbulent ﬁelds is observed. The approximate (error function) shape of the longitudinal mean velocity proﬁle for low speed ﬂow is shown in Figure 8. From this proﬁle one can deﬁne the thickness of the shear layer. It is, in fact, possible to deﬁne this thickness in many diﬀerent ways that are not completely

542

Barre et al. Y Mc = 0.525

Bord de fuite Mc = 0.64

Mc = 1

Mc = 1

X(cm) 10

20

30

Figure 7: Tomographic visualizations of three kind of compressible plane mixing layers at Mc = 0.525, 0.64, and 1 (from Chambres 1997). U1

δω

U2

Figure 8: Schematic shape of the mean streamwise velocity proﬁle of a mixing layer. equivalent. The most straightforward when dealing with turbulent ﬂows is the vorticity thickness given by δω =

∆U , ∂U ∂y max

(3.23)

where ∆U = (U1 − U2 ) is the velocity diﬀerence between the external ﬂows, and (∂U/∂y)max is the maximum mean velocity gradient of the ﬂow. When the self-similar state is reached, the mixing layer spreads linearly with a constant value of the spreading rate ∂δω /∂x. The spreading rate can be computed with a semi-empirical law proposed by Brown and Roshko (1974)

[19] Compressible, high speed ﬂows

543

as a function of the density and the velocity ratio of the external ﬂows, √ ∂δω (1 − r) (1 + s) √ = Cδ , (3.24) ∂x 0 2 (1 + r s) with r = U2 /U1 , s = ρ2 /ρ1 , and Cδ ≈ 0.181 is a constant representing the spreading rate of the mixing layer after a backward facing step at uniform density. The experimental database supporting this law is small, but it may be taken as broadly predicting the eﬀect of velocity and density ratio on mixing layer spreading rates. In supersonic ﬂows this law does not correctly describe the measured variation in spreading rate. This fact is illustrated in Figure 9 where measured spreading rates in a supersonic backward facing step are plotted versus the corresponding density ratio (the freestream Mach number is also shown). This evolution is compared to the Brown and Roshko law for r = 0 at diﬀerent s corresponding to the experimental conﬁguration tested. It is clear that the Brown and Roshko law does not take into account the compressibility eﬀects and that the observed decrease in spreading rate is not entirely due to density ratio evolution, but is instead a genuine compressibility eﬀect. At this stage 0.25 0.20 0.15

δω 0.10 0.05

M1 1

0 .1

.2

.4

.6 .8 1

2 2

3

4 5 6 4

6 8 10

ρ1/ρ2

Figure 9: Eﬀect of density ratio on spreading rate with U2 = 0: •, values for incompressible ﬂow. Other symbols are for compressible mixing layers: +, Maydew and Reed (1963); ∆, Ikawa (1973); ×, Siriex and Solignac (1966). (From Brown and Roshko 1974) it appears that an extra parameter must be introduced in order to quantify the compressibility eﬀect. The most useful parameter has been the convective Mach number originally named by Papamoschou and Roshko (1988) who drew upon linear stability analysis of a vortex sheet for justiﬁcation of the particular formulation. Figure 10 clearly shows the basis of the convective Mach number concept. In a frame of reference ﬁxed to large scale structures traveling with

544

Barre et al.

a constant convective velocity Uc , one can deﬁne two Mach numbers of the external streams Mc1 =

U1 − Uc a1

and Mc2 =

Uc − U2 . a2

(3.25)

In the convected frame of reference there exists a saddle point between two (a) M1, U1, ρ1

Uc M2, U2, ρ2

(b) U1 – Uc Mc =

U1 – Uc a1

1

Mc = 1

Uc – U2 a2

Uc – U2

Figure 10: Convective Mach number concept (Papamoschou and Roshko 1988): (a) laboratory frame of reference; (b) convective frame of reference. adjacent eddies. If it is assumed that the static pressure is constant across the mixing layer and that the compression of ﬂuid along particle paths leading to the saddle points is isentropic, then the mechanical equilibrium of this saddle point leads to the relation

(γ1 − 1) 2 1+ Mc 1 2

γ1 (γ1 −1)

(γ2 − 1) 2 = 1+ M c2 2

γ2 (γ2 −1)

,

(3.26)

where γ1 and γ2 are the ratios of speciﬁc heats for the two streams of ﬂuid. Using the deﬁnition of the two convective Mach numbers, an expression for Uc can be obtained as follows for the simpliﬁed case where γ1 = γ2 Uc =

a2 U1 + a1 U2 . a1 + a2

(3.27)

Papamoschou and Roshko (1988) thus obtained a new functional form for the mixing layer spreading rate, √ ∂δω (1 − r) (1 + s) ∂δω √ = Cδ Φ (Mc ) , (3.28) Φ (Mc ) = ∂x 2 (1 + r s) ∂x 0

[19] Compressible, high speed ﬂows

545

where Φ(Mc ) is a compressibility function which is the ratio between the observed spreading rate of a compressible mixing layer and the incompressible value at the same velocity and density ratio. Experimental data to support this are given in the next section. 3.3.2

Experimental and DNS results for RANS model validation

Many measurements of mean and turbulent quantities have been carried out using either hot-wires (Barre et al. 1994) or LDV (Elliott and Samimy 1990, Goebel and Dutton 1991, Gruber et al. 1993, Debisschop 1993, and Chambres 1997) in both plane mixing layers and supersonic jet conﬁgurations (Lau 1981, Bellaud 1999). The majority of these studies have been summarized in Lele (1994) and more recently in Smits and Dussauge (1996). Both experimental and numerical studies have conﬁrmed that the spreading rate decreases when the convective Mach number increases. Figure 11 shows the reduction in the normalized spreading rate Φ(Mc ) with convective Mach number for a wide range of experiments. Between Mc = 0.5 and Mc = 1 there 1.20

Plane Mixing Layer

1.10

Chambres (1997) Debisschop (1993) Elliot and Samimy (1990) Goebel and Dutton (1991) Papamoschou et al. (1998)

1.00 0.90 0.80

Φ(Mc) 0.70 0.60

Annular Mixing Layer

0.50

Lau (1981) Mistral (1993) Bellaud (1999)

0.40 0.30 0.20 0.00 0.25 0.50 0.75

1.00 1.25

1.50

Mc

Figure 11: Normalized growth rate versus convective Mach number for diﬀerent mixing layer studies. is a reduction by more than a factor of two. Above Mc = 1 the normalized spreading rate is relatively constant. The reduction in spreading rate appears to be also accompanied in a majority of the experiments by a corresponding reduction in the turbulent ﬂuctuations. Figures 12, 13, and 14 show results from several diﬀerent plane and annular mixing layer studies of the variation 2 of the normalized Reynolds stress components σu (= u ), σv (= v 2 ), and u v as a function of convective Mach number. As the ﬁgures show there are signiﬁcant diﬀerences between the various studies. Nevertheless, several of the experimental studies conﬁrm the behavior of decreasing ﬂuctuations

546

Barre et al. Plane Mixing Layer 0.24

0.18

Barre (1993) Chambres (1997) Debisschop (1993) Elliot and Samimy (1990) Goebel and Dutton (1991) Gruber et al. (1993)

0.16

Annular Mixing Layer

0.14

Bradshaw et al. (1964) Denis et al. (1998) Lepicovsky et al. (1987) Lau (1981) Mistral (1993) Bellaud (1999)

0.22 0.20

σu ∆U

max

0.12 0.10 0.00 0.20

0.40

0.60

0.80

1.00

1.20

Mc

Figure 12: Streamwise velocity ﬂuctuations as a function of the compressibility parameter Mc for plane and annular mixing layers. Plane Mixing Layer

1.16

Chambres (1997) Debisschop (1993) Elliot and Samimy (1990) Goebel and Dutton (1991) Gruber et al. (1993)

1.14 1.12

σv ∆U

0.10 max

Annular Mixing Layer

0.08

Bradshaw et al. (1964) Denis et al. (1998) Mistral (1993) Bellaud (1999)

0.06 0.04 0.00 0.20

0.40 0.60

0.80 1.00

1.20

Mc

Figure 13: Transverse velocity ﬂuctuations as a function of the compressibility parameter Mc for plane and annular mixing layers. with increasing Mc . This behavior has led to the conclusion by some that the turbulent anisotropy, deﬁned by

σu σv max

and

σu − σv , σu + σv max

(3.29)

exhibits a quasi-constant behavior over the convective Mach number range. Figure 15 shows experimental results obtained for (σu /σv )|max . For the majority of these studies, the anisotropy data is relatively constant; however, the studies of Goebel and Dutton (1991) and Gruber et al. (1993) actually show a

[19] Compressible, high speed ﬂows

547

0.18

Plane Mixing Layer

0.16

Barre (1993) Chambres (1997) Debisschop (1993) Elliot and Samimy (1990) Goebel and Dutton (1991) Gruber et al. (1993)

0.14 0.12

–u ′v ′ 0.10 2 (∆U ) max

Annular Mixing Layer

0.08

Bradshaw et al. (1964) Denis et al. (1998) Lau (1981) Mistral (1993) Bellaud (1999)

0.06 0.04 0.02 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Mc

Figure 14: Turbulent shear stress evolution as a function of the compressibility parameter Mc for plane and annular mixing layers.

growth with Mc . The Gruber et al. (1993) study is limited to a single convective Mach number, but the result is consistent with the Goebel and Dutton (1991) result. The reason for this growth in (σu /σv )|max is due to the fact that Goebel and Dutton (1991) show a relatively constant value in σu /∆U but a decrease in σv /∆U with Mc (see Figs. 12 and 13). The result is an effective increase in (σu /σv )|max with an increase in the compressibility eﬀects (see Figure 15). This contradiction in the experimental results is worrying if one wishes to obtain a physical interpretation of the eﬀects of compressibility and develop predictive models. Also the fundamental question still persists: What are the real mechanisms responsible for the decrease of the mixing and the turbulent activity at high Mach numbers? Some studies were carried out almost a decade ago to try to answer this question. From an analysis of numerical simulations, Zeman (1990) and Sarkar et al. (1991) proposed models for the extra-dissipation due to dilatational eﬀects in order to explain the reduction in the turbulence. The functional form for these models was given by = ε + εd = ε[1 + F(Mt )],

(3.30)

where the extra-dissipation was represented by the turbulent Mach number term εF(Mt ). This form of the partitioning allowed for the direct extension of the incompressible (solenoidal) form of the dissipation rate transport equation. Utilization of this type of model brought the calculations into agreement with the experimental results available. An example of the signiﬁcant inﬂuence this term had on the predictive behavior of a Reynolds stress model is shown in Figure 16 with the function F(Mt ) given simply by Mt2 (Mt is the turbulent

548

Barre et al. 3.6

Plane Mixing Layer 3.2

Chambres (1997) Debisschop (1993) Elliot and Samimy (1990) Goebel and Dutton (1991) Gruber et al. (1993)

2.8

σu σv

2.4 max

2.0

Annular Mixing Layer

1.6

Bradshaw et al. (1964) Denis et al. (1998) Mistral (1993) Bellaud (1999)

1.2 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Mc

Figure 15: Anisotropy evolution as a function of the compressibility parameter Mc for plane and annular mixing layers. 1.0

0.8 Without extra - dissipation With extra - dissipation "Langley Experimental Curve"

0.6

Φ(Mc) 0.4

0.2

0.0 0

1

2

3

4

5

Mc

Figure 16: Predicted compressibility function Φ(Mc ) from a RANS calculation using a Reynolds stress model. ‘Langley Experimental Curve’ from Kline et al. (1981). Mach number). The RANS calculation is from Sarkar and Lakshmanan (1991) who used a deﬁnition of Φ(Mc ) which diﬀered in detail (see their equation (25)) from that given in (3.28); nevertheless, the ﬁgure clearly shows the same qualitative trend of decreasing spreading rate with Mc displayed in Figure 11. Such results, as well as those by Zeman (1990) who used a slightly more complex form for F(Mt ), strongly suggested that the increase in compressible dissipation and the resultant decrease in the turbulence was the underlying dynamic reason for the reduction in spreading rate.

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0.07 M = 0.2 j M = 0.4 j M = 0.8 j M = 1.15 j M = 1.55 j M = 1.92 j M = 2.5 j M = 3.0 j M = 3.5 j

0.06 0.05

σp

0.04 2

ρjUj

0.03 0.02 0.01 0 0

1

2

3

r/ro Figure 17: Normalized pressure ﬂuctuations for diﬀerent Mach numbers. However, with the emergence of direct numerical simulations, additional features of such ﬂows became evident. Sarkar (1995) identiﬁed turbulence production as a key term for understanding the eﬀect of compressibility on homogeneous turbulence. For mixing layers Vreman et al. (1996) showed that compressible dissipation remains negligible even at large Mach numbers. According to these simulations, it would seem that diﬀerences in the level and distribution of the pressure ﬂuctuations are central to the reduction in the turbulent activity. With an increase in Mach number, one sees a reduction in the pressure ﬂuctuations, an observation which has been conﬁrmed in more recent simulations by Freund et al. (1997) and Pantano and Sarkar (2001). Vreman et al. use models of compressible vortices to conﬁrm the origin of the reduction of pressure ﬂuctuations. With this mechanism the reduction of growth rate is not due to extra explicit compressibility terms in the equations, but due to an implicit modiﬁcation of existing terms (at second moment closure level this would be the pressure strain). Since the pressure dilatation term remains small there are no terms in the k or equations that can be modiﬁed. Instead, if we follow the algebraic Reynolds stress modelling procedure to convert from second moment closure to a two equation model, it is the eddy viscosity relation that has to carry the compressibility eﬀect. Figure 17 (Freund et al. 1997) shows the pressure ﬂuctuation proﬁles from DNS of an annular mixing layer. Results for nine jet Mach numbers Mj ranging from 0.2 to 3.5 are given corresponding to convective Mach numbers ranging from 0.1 to 1.8. It is clear from these direct numerical simulations that a strong decrease in the maximum of σp /ρj Uj2 is observed with increasing Mach number. A ﬁrst consequence of an increase in Mach number across an eddy is to limit the inﬂuence of the pressure waves to a zone in space within which the speed diﬀerence is subsonic. This ‘sonic-eddy’ concept was ﬁrst presented by

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Breidenthal (1990) and later extended by Papamoschou and Lele (1993). Indeed, the inﬂuence of this lack of communication by the pressure can be quantiﬁed by the gradient Mach number (3.1) ﬁrst introduced by Sarkar (1995). This Mach number can be interpreted as the ratio of the shearing velocity across a large eddy to the sound speed. If this Mach number becomes too high then the shearing will become too fast relative to the sound speed. This will result in a decrease in the communication across the eddy and thus restrict the structure size. Figure 18 shows the variation of gradient Mach number Mg versus convective Mach number Mc in the center of an annular mixing layer. It is clear that, after a gradual increase, a saturation eﬀect is reached when Mc is close to unity. This trend is very similar to the one observed for spreading rate and turbulent correlations. While this interpretation of the role played by 2.5 2.0 1.5

Mg 1.0 0.5 0 0

0.5

1.0

1.5

2.0

Mc Figure 18: Gradient Mach number evolution versus convective Mach number at the center of an annular mixing layer (DNS results from Freund et al. 1997) the gradient Mach number as a measure of compressibility eﬀects is certainly plausible, most of the DNS results are not in agreement with the available experimental results. The principal diﬀerence is related to the structure of the Reynolds stress tensor. With the exception of the Pantano and Sarkar (2001) study, the numerical simulations (Sarkar 1995, Vreman et al. 1996, and Freund et al. 1997) predict strong degrees of anisotropy when the convective Mach number increases; whereas, the majority of the experimental results, excluding those of Goebel and Dutton (1991) and Gruber et al. (1993), show that the anisotropy of the Reynolds stress tensor is not strongly aﬀected by compressibility. Nevertheless, the changes in intensity of the pressure ﬂuctuations (and pressure-dilatation) in the compressible case as compared to the incompressible case is conﬁrmed by the available compressible direct simulations. All these results clearly suggest that the task of providing suitable modeling for compressibility eﬀects remains largely an open question. Recall that the

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551

ﬁrst models for dilatation-dissipation (Zeman 1990, and Sarkar et al.1991), postulated that the increase observed in the turbulent Mach number when the level of compressibility increased could explain the decrease in turbulent activity for the compressible cases. This model gave completely compatible results with experiments (Chambres 1997) for the turbulent intensities and anisotropy parameters. On the other hand, direct numerical simulations (Vreman et al. 1996, Freund et al. 1997, Pantano and Sarkar 2000) now suggest that this ﬁrst theory was inappropriate and that it must be replaced by a precise modeling of the pressure-strain correlation. It appears that the Reynolds stress anisotropy may be strongly aﬀected by the initial condition and run time of DNS. In this respect the results of Pantano and Sarkar (2001) now represent the closest approximation that we have to a self-similar compressible mixing layer simulation, which should be the starting point for physically-sound modelling work. Another deﬁciency is the restriction of DNS to ﬂow Reynolds numbers which are an order of magnitude lower than those of the experiments, thus making generalizations of such DNS results questionable. On the experimental side, several limitations still remain. For example, it is not possible at this time to obtain reliable experimental data for some compressibility terms, such as the pressure strain, which appears to play a principal role in the dynamics. However, by using the Reynolds analogies and related assumptions, it is now possible to establish budgets of turbulent kinetic energy in supersonic mixing layers (Chambres 1997, Bellaud 1999). These results are quite useful in validating closure models, and comparing the structure of the Reynolds stress tensor with the one obtained in incompressible ﬂows. Figure 19 shows the production and diﬀusion terms in the kinetic energy budgets of an annular mixing layer obtained from experiments of Bellaud (1999) and DNS of Freund et al. (1997) at Mc = 0.85. While these budgets are not completely equivalent, a qualitative agreement is clearly shown. The most important feature revealed by both these budgets is the asymmetry of the diﬀusion term between the subsonic and the supersonic edge of the ﬂow. At present, this type of compressibility eﬀect has no satisfactory explanation. Nevertheless, with this type of agreement one can assume that these budgets across the layer are a good approximation of the actual physical behavior which must be modeled.

4

Shock Wave/Turbulence Interactions

The presence of a shock wave in a turbulent ﬂow signiﬁcantly inﬂuences the turbulence energetics as well as the mean ﬁeld near and downstream of the shock. The compressibility eﬀects which inﬂuenced the dynamics of the shockfree ﬂows discussed in the previous section are also expected to be relevant in the presence of shocks. To make predictions, speciﬁc models have had to be built in order to properly take into account the drastic modiﬁcation of the

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1.00 0.80 0.60 0.40 0.20 0.00 –0.20 –0.40 –0.60 –1.20 –0.80 –0.40

0.00 •

0.40

0.80

1.20

Y

Figure 19: Comparison between experimental and DNS turbulent kinetic energy budgets across an annular mixing layer at Mc = 0.85.

Shock Reflection

Flap

Separated flow

Second shock reflection

Sonic line B.L. separation

Over-expanded Nozzle Transonic Flow

Figure 20: Shock wave/turbulence interaction conﬁgurations. turbulent ﬁeld subjected to strong pressure gradients. Some types of ﬂows of practical interest in which shocks are present are illustrated by the sketches shown in Figure 20. Such shock wave/boundary layer interactions have been extensively studied for many years both experimentally and numerically. Review articles like those of Green (1970), Korkegi (1974) and Fernholz and Finley (1981) summarize the main results for these types of ﬂow ﬁelds. In one set of experiments an oblique shock wave impinges

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on a boundary layer. It has been shown by Smits and Muck (1987) and Andreopoulos and Muck (1987) that, in these conditions, if the shock is strong enough the ﬂow separates near the interaction. As for the evolution of the turbulent quantities across the shock, it has been shown by Ardonceau (1984) and Smits and Muck (1987), for example, that the turbulence intensities and the anisotropy ratio are increased due to the shock interaction. The same kind of results are available for oblique shock wave interaction with a free shear layer. Data show a high ampliﬁcation of the turbulence intensities which depends on the shock strength (Dolling and Murphy 1982, Settles et al. 1982, Hayakawa et al. 1984, Samimy and Addy 1985, and Jacquin et al. 1991). Additional experimental studies have been performed either in a shock tube in order to study the (pure) interaction between a traveling normal shock wave reﬂected on the end wall of the tube and the subsequent interaction with the ﬂow induced by the incident shock (Troiler and Duﬀy 1985, Hartung and Duﬀy 1986) or by the wake of a perforated plate (Honkan and Andreopoulos 1992, Keller and Merzkirch 1990, Briassulis and Andreopoulos 1996, Briassulis et al. 1999, Poggi et al. 1998). All these experiments show a strong increase in density ﬂuctuations through the shock, but there is little data describing the behavior of the velocity ﬁeld. Unfortunately, in very complex conﬁgurations streamline curvature and unsteady separation problems both occur. This makes the study and the prediction of shock wave/boundary layer interaction problems diﬃcult. Thus, it is desirable to ﬁnd a simpler ﬂow problem to study which has at least some of the key dynamic features of the general problem, but without some of the complicating features. One such basic ﬂow problem is the interaction between a normal shock and an isotropic turbulent ﬁeld. The main advantage of investigating such a ﬂow is to isolate the net eﬀect of the strong gradients of mean quantities imposed by the shock wave. This basic conﬁguration has many advantages for both experiments and computations. For the experiments, it reduces the number of geometrical parameters, while, for numerical predictions, analytical theories as well as advanced simulation methods (LES, DNS, EDQNM) can be applied and compared. The remainder of this section will be divided in two parts describing the current understanding of the shock wave/homogeneous turbulence interaction and shock wave/boundary layer interaction problems.

4.1

Shock wave/homogeneous turbulence interaction

Early theoretical work on this problem was done in the 1950’s, based on linear analysis of a plane disturbance interacting with a shock wave. Ribner (1953, 1954, 1969), in particular, investigated how a shock wave is perturbed by an impinging single wave. The theory, later called Ribner’s theory, is based on the mode theory developed by Kovasznay (1953) and extended to the shock problem by focusing mainly on the acoustic ﬁeld generated by the interaction

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with the shock wave. More recently, numerical simulations of such ﬂows have emerged. The Rotman (1991), Lee et al. (1993, 1997), and Mahesh et al. (1995, 1997) simulations have shown, among other results, that the vorticity ampliﬁcations predicted are in good agreement with linear theories, and the DNS studies of Lee et al. (1993, 1997), Jamme (1998) and Hannapel and Friedrich (1992, 1995) have shown how isotropic turbulence becomes axisymmetric after the shock, which is also consistent with the linear analysis. In addition, the evolution of the turbulent spectra across the shock exhibits a bigger ampliﬁcation at large wave numbers compared to lower ones which means that the longitudinal integral length scales are decreased while the lateral ones seem unaﬀected by the shock interaction. In this section we consider in detail the predictions of linear theory based on the work of Jamme (1998), followed by comparisons with experiment. 4.1.1

Linear interaction analysis (LIA)

The problem under study is the interaction between a normal shock wave and an isotropic incoming turbulent ﬁeld. It is assumed that this interaction takes place in an inﬁnite medium with no boundaries. In this kind of ﬂow ﬁeld the mode theory ﬁrst proposed by Kovasznay (1953) can be applied. The theory results from a linearization of the Navier–Stokes equations for moderately supersonic ﬂows. In this case, if the turbulence level is suﬃciently small to keep the linear approximation valid, the ﬂuctuation ﬁeld can be reduced to a linear combination of three fundamental modes described by Vorticity Mode : Entropy Mode : Acoustic Mode :

u , u T , T

T = 0, T u = 0, u

p = 0, p p = 0, p

T p 1 ρ = = , ρ γp (γ − 1) T

with

ρ =0 ρ

(4.1)

ρ T =− ρ T

(4.2)

u =f u

p

p

.

(4.3)

Note that here we are using ﬂuctuations from conventionally averaged quantities, rather than the mass-weighted averages introduced earlier. In the linearized ﬂow the two formulations are equivalent. Any turbulent ﬁeld satisfying the initial assumptions can be reduced to a linear combination of these three independent modes. Furthermore, if there is a non-random phase relationship between these modes, one can then obtain (non-random) correlated turbulent ﬁelds for some physical quantities of interest. For the shock wave/turbulence interaction, the physics of the problem can be modeled by using this modal decomposition for both the incoming and the transmitted turbulence across the shock wave. This physical situation is sketched in Figure 21, where the incoming and refracted turbulent ﬁeld are

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shown. In the general case the incoming turbulent ﬂow is three-dimensional and consists of many waves with a continuous spectrum representative of a large Reynolds number isotropic turbulence. However, for clarity the linear interaction theory will be used in a simpliﬁed case.

Acoustic

Entropy

Vorticity Vorticity Entropy Acoustic Figure 21: Schematic of incoming and refracted turbulent ﬁeld with shock. First, consider the two-dimensional interaction of a simple vorticity/entropy wave with a two-dimensional normal shock wave. This simpliﬁed case is of great interest because the vorticity/entropy wave structure is representative of the classical turbulent ﬁeld encountered in many moderately supersonic turbulent ﬂows. The interaction between simple vorticity or entropy waves was ﬁrst done analytically by Ribner (1953) for a vorticity wave and by Chang (1957) for an entropy wave. The analysis presented here is adapted from Jamme (1998) who studied the problem shown in Figure 22. The shock wave is distorted by the incoming wave ﬁeld. The incoming wave can be expressed as  u1   = lAv eiκ(mx+ly−U1 mt)    u1

Vorticity Mode :

   v   1 = −mAv eiκ(mx+ly−U1 mt)

(4.4)

u1

 ρ1   = Ae eiκ(mx+ly−U1 mt)    ρ  1    

Entropy Mode :

T

ρ

1  = − 1   T ρ1  1       p1 = 0,

where l = sin ψ1

and

m = cos ψ1

(4.5)

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Figure 22: Sketch of a linear shock/wave interaction.

where κ = κ2x + κ2y is the wavenumber of the incoming wave, Av is the amplitude of the vorticity wave upstream of the shock, Ae is the amplitude of the entropy wave upstream of the shock, and the subscript 1 refers to upstream conditions relative to the shock. The Euler equations downstream of the shock can be written as

∂ρ ∂u2 ∂v2 ∂ρ2 + U2 2 = −ρ2 + ∂t ∂x ∂x ∂y

∂u2 ∂u 1 ∂p2 + U2 2 = − ∂t ∂x ρ2 ∂x ∂v ∂v2 1 ∂p2 + U2 2 = − ∂t ∂x ρ2 ∂y

(4.6)

∂s ∂s2 + U2 2 = 0 ∂t ∂x p2 ρ T = 2+ 2 p2 ρ2 T 2

and

s2 p ρ = 2 − 2. Cp γp2 ρ2

The initial conditions for the solution of Euler’s equations across the shock

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can be given by the linearized Rankine–Hugoniot equations u T 1 ∂ξ u2 (x = 0) = (1 − B1 ) + B1 1 + B2 1 U1 U1 ∂t U1 T1 v2 (x = 0) ∂ξ v = 1 + E1 U1 U1 ∂y

1 ρ2 (x = 0) ∂ξ = C1 u1 − ρ2 U1 ∂t p2 (x = 0) 1 ∂ξ = D1 u1 − p2 U1 ∂t

T + C2 1 T1 + D2

(4.7)

T1 , T1

where B1,2 , C1,2 , D1,2 , and E1 are functions of γ and M1 . With these starting conditions, the ﬁeld downstream of the shock wave can be written as u2 = F eiκx eiκ(ly−U1 mt) + Geiκ(mrx+ly−U1 mt) u1 v2 = Heiκx eiκ(ly−U1 mt) + Ieiκ(mrx+ly−U1 mt) u1 p2 = Keiκx eiκ(ly−U1 mt) p2

(4.8)

ρ2 K iκx iκ(ly−U1 mt) = + Qeiκ(mrx+ly−U1 mt) e e ρ2 γ T2 γ − 1 iκx iκ(ly−U1 mt) = − Qeiκ(mrx+ly−U1 mt) , Ke e γ T2 where r is ratio of the velocity upstream of the shock (U1 ) to the velocity downstream of the shock (U2 ). The shock wave deformation equation can then be given by 1 ∂ξ = Leiκ(ly−mU1 t) U1 ∂t (4.9) ∂ξ l = − Leiκ(ly−mU1 t) . ∂y m The seven coeﬃcients F , G, H, I, K, Q, and L appearing in (4.8) and (4.9) are the transfer coeﬃcients between the upstream and downstream shock ﬂows. They are obtained by solving a system of seven equations, composed of the three equations of motion downstream of the shock and the four equations resulting from the Rankine–Hugoniot conditions across the shock. The interested reader is referred to the reference of Jamme (1998) for details. This

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leads to two kinds of solutions for the pressure ﬂuctuations downstream of the shock wave. These two families of solutions depend on the angle ψ1 of the wave vector. There exists a critical angle ψc for the wave vector direction which separates the two kinds of pressure wave solutions. The value of this critical angle can be computed as a function of the upstream velocity U1 , the downstream velocity U2 , and the speed of sound a2 ,

cot2 (ψc ) =

a2 U1

2

−

U2 U1

2

.

(4.10)

Then, knowing ψc , the two families can be characterized as either (i) an evanescent mode (ψc < ψ1 < π2 ), where the acoustic wave after the shock decreases exponentially or (ii) a non-evanescent mode (0 < ψ1 < ψc ), where the acoustic wave after the shock propagates downstream without attenuation. The three-dimensional generalization of this theory can be obtained by integrating (over physical and Fourier space) the results from all the contributing waves included in the incoming ‘turbulent’ ﬁeld. This can include initially acoustic ﬁelds. An initial spectrum which gives a good representation of a real turbulent ﬁeld (see Jamme (1998) for details) is E(κ) = Cκ4 e−ακ , 2

(4.11)

In this section, LIA results are presented for four kinds of initial ﬁelds corresponding to typical cases which are most often encountered in real turbulence. These are: (1) Pure solenoidal turbulence (vorticity mode); (2) Pure entropic turbulence (entropy mode); (3) Mixed turbulence: in phase vorticity and entropy modes

ρ12

ρ

=

T12

T

φ=0

= (γ −

=⇒

u12

1) M12

U1

u T < 0

(φ is the phase shift between the vorticity and entropy waves); (4) Mixed turbulence: out of phase vorticity and entropy modes

ρ

ρ12

=

φ=π

T

T12

= (γ −

=⇒

1) M12

u T > 0.

u12

U1

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559

For these four cases, Figure 23 shows the streamwise evolution of the LIA computed turbulent kinetic energy (q 2 ) ampliﬁcation rate for a ﬂow with M1 = 1.5. The most striking result is that the nature of the incoming turbulence plays a crucial role in the evolution of the ampliﬁcation rate of q 2 . For Case (1) (pure solenoidal turbulence) a 40% ampliﬁcation of the initial turbulent kinetic energy (TKE) is observed through linear mechanisms. By contrast the pure entropy wave of Case (2) has its initial kinetic energy reduced by a factor of ten. When these two waves are coupled together, it is possible to obtain a larger TKE ampliﬁcation if u T < 0 (Case 3). Thus, the coupling of an entropy wave with a solenoidal wave can create up to a 100% increase in kinetic energy level if the two waves have no phase shift.

Figure 23: Turbulent kinetic energy ampliﬁcation for diﬀerent upstream conditions (M1 = 1.5) (from Jamme 1998). Figure 23 also shows that the TKE ampliﬁcation is dependent on the downstream distance from the shock wave. Up to half a wavelength after the shock there is a strong variation in the amplitude before reaching a stable value. These variations are due to the evolution of the evanescent part of the pressurevelocity correlation, and to the evanescent part of the pressure ﬂuctuations generated by the shock/turbulence interaction. Figure 24 shows the streamwise evolution of the (partitioned) kinetic energy ampliﬁcation rate for a Case (3) mixed turbulence ﬁeld with an incoming Mach number of M1 = 1.5. It is clear that the initial evolution of the ampliﬁcation rate is correlated with the decrease of the acoustic intensity of evanescent sound waves generated by the linear interaction process, and with the return to isotropy induced by the evanescent pressure-velocity correlation. These results show that it is necessary to deﬁne two kinds of post-shock turbulent ﬁelds when dealing with wave/shock interaction. The ﬁrst will be called the near ﬁeld and the second the far ﬁeld. Figures 25 and 26 show the

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Figure 24: Partitioned turbulent kinetic energy ampliﬁcation downstream of shock (mixed turbulence Case (3) with M1 = 1.5) (from Jamme 1998). 4.0

v2'2 v1'2 q22 q12

3.5 3.0 2.5 2.0 1.5

u2'2 u1'2

1.0 0.5 0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

M1 Figure 25: Turbulent velocity ﬂuctuations ampliﬁcation coeﬃcient downstream of the shock (near ﬁeld) for Case (3) (in phase mixed turbulence), from Jamme (1998). kinetic energy evolution of longitudinal and lateral velocity ﬂuctuations in the near and far ﬁeld as a function of Mach number. As Figure 25 shows for the near ﬁeld, ampliﬁcation rates for the longitudinal and lateral velocity ﬂuctuations exhibit either a growth or decay up to M1 = 2. Beyond this point, an asymptotic limit is reached with a kinetic energy ampliﬁcation ratio of about 2.3, and an anisotropy ratio of about 0.18 for the longitudinal to lateral ﬂuctuations energy ratio u22 /v22 (because of the assumed isotropy of the incoming ﬁeld u12 = v12 ). In contrast, the far ﬁeld (see Figure 26) does not exhibit such an asymptotic behavior for the anisotropy

[19] Compressible, high speed ﬂows

561 v2'2 v1'2 q22 q12

3.0 2.5

u2'2 u1'2

2.0 1.5 1.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

M1 Figure 26: Ampliﬁcation coeﬃcients for turbulent velocity ﬂuctuations downstream of the shock (far ﬁeld) for Case (3) (in phase mixed turbulence), from Jamme (1998).

(λ)

(λ1)upstream

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0

Mixed upstream turbulent field Case 3 Pure vorticity upstream turbulent field Mixed upstream turbulent field Case 4 1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

M1

Figure 27: Ampliﬁcation coeﬃcient for longitudinal Taylor microscale in the far ﬁeld (from Jamme 1998). coeﬃcient. The ampliﬁcation ratio of the turbulent kinetic energy reaches a value of about 2.6 beyond Mach 3. Another measure of interest is the Taylor microscale. the LIA predicts a decrease for both the longitudinal and lateral scales (see Figs. 27 and 28). For the longitudinal Taylor microscale, the ratio can reach 0.4 for M1 = 5 (Figure 27). LIA theory also predicts the evolution of the thermodynamic characteristics of the turbulent ﬁeld across the shock. The thermodynamic ﬂuctuations can be characterized by an analogy with a pure sound ﬁeld or a pure vorticity-entropy ﬁeld. As an example, consider the inﬂuence of the pressure ﬂuctuations on the density ﬂuctuation ﬁeld. The temperature and density ﬂuctuations are related

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(λ2) λ ( 2)upstream

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0

Mixed upstream turbulent field Case 4 Pure vorticity upstream turbulent field Mixed upstream turbulent field Case 3

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

M1

Figure 28: Lateral Taylor microscale ampliﬁcation coeﬃcient downstream (far ﬁeld) (from Jamme 1998). to the pressure and entropy ﬂuctuations through T 2 T

2

=

s2 (γ − 1)2 p2 + γ2 p2 Cp2

Acoustic 1 p2

Entropy

ρ2 s2 = 2 2 + γ p Cp2 ρ2 Acoustic

(4.12)

Entropy

ρ2 T 2 1 p2 (γ − 1)2 p2 (γ − 2) p2 − = − = − . (4.13) ρ2 γ 2 p2 γ2 p2 γ p2 T2 A ratio α between these quantities can be constructed and is given by

ρ2 − ρ

α=

T 2 T

ρ2

=−

(γ − 2) n2pρ γ nρT

(4.14)

ρ with

p2 /p

npρ = ρ2 /ρ

T 2 /T and nρT = 1 + , ρ2 /ρ

(4.15)

where npρ and nρT are polytropic coeﬃcients. For a pure isobaric turbulence only the vorticity and entropy modes are active and we have p = 0 =⇒ α = 0 =⇒ p

'

npρ = 0 nρT = 2.

(4.16)

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For a pure acoustic ﬁeld we may write: s = 0 =⇒ α = 0.6 =⇒ s

'

npρ = γ nρT = γ.

(4.17)

The consequence of this is that a given turbulent ﬂow will satisfy Morkovin’s hypothesis when α 0.6. Figure 29 shows the evolution of the LIA computed polytropic coeﬃcients (downstream far ﬁeld conditions) as a function of the incoming Mach number. The upstream turbulence was composed of a pure vorticity wave in this computation. For small Mach number (M1 < 1.2), the downstream turbulent ﬁeld is isentropic, but as the Mach number increases the polytropic coeﬃcients reach asymptotic values of nρT = 1.9 and npρ = 0.6. The resulting ﬁeld

2.0 npT

1.6 1.2 0.8

npρ

0.4 0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

M1 Figure 29: Evolution of polytropic coeﬃcients computed from LIA in far ﬁeld for pure vorticity upstream turbulence (from Jamme 1998). now exhibits non-negligible entropy ﬂuctuations which were not present in the incoming ﬁeld. In the case of composite upstream turbulence (vorticity and entropy mode with u1 T1 < 0) and for M1 = 1.5, the LIA results in the far ﬁeld downstream of the shock are npρ = 0.4 nρt = 1.96

(

=⇒ α = 0.035 =⇒

α = 0.058. 0.6

(4.18)

Thus, after the shock, the turbulent ﬁeld is a 5.8% pure sound ﬁeld. The turbulence then satisﬁes Morkovin hypothesis even though it is not isobaric. In this case p /p is not negligible since

p p

2

ρ 2 = 0.4 . ρ

(4.19)

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This example illustrates the fact that it is possible to obtain turbulent ﬁelds which satisfy Morkovin hypothesis without the necessity of having p /p = 0. 4.1.2

Comparison with experimental results

Only a few attempts have been made to study such a ﬂow. Blin (1993) and Jacquin et al. (1991) generated a supersonic quasi-homogeneous turbulent ﬂow using a grid as a sonic throat. The ﬂow downstream of the grid was supersonic with a relatively low Mach number (1.7), but with parasitic shock waves which signiﬁcantly deteriorated the quality of the turbulent ﬂow ﬁeld. It was necessary to ‘shock’ the ﬂow by means of a second throat creating a pure normal shock wave/free turbulence interaction. Unfortunately, experimental diﬃculties due to the very low turbulence level (close to the noise level of the anemometer) led to signiﬁcant diﬃculties in interpreting the results. On the other hand Debi´eve and Lacharme (1986) obtained a quasi-homogeneous turbulence by generating perturbations in the settling chamber of a supersonic wind tunnel at Mach 2.3. They showed that, after the expansion in the nozzle, the turbulent ﬁeld becomes strongly non-isotropic and the turbulence level is drastically decreased, which leads to experimental diﬃculties. A third kind of experiment was performed by Barre et al. (1996) who used a multi-nozzle system to generate a homogeneous and isotropic turbulent ﬁeld in a Mach 3 ﬂow. This turbulent ﬁeld then interacted with a normal shock wave generated at the center of the test section of a supersonic wind tunnel by means of a Mach eﬀect. The main advantage of such a device is to obtain a shock wave free from parasitic excitation, making this interaction quite close to an ideal interaction and in agreement with the assumptions for linear theories. However, some limitations still persist in this conﬁguration. First, the turbulent ﬁeld issued from the multi-nozzle contains some acoustic waves generated at the trailing edge of the multi-nozzle. These waves, even if they are damped when they reach the shock wave, make it increasingly diﬃcult to interpret the results because they have a large inﬂuence on the interaction with the initial turbulent ﬁeld. Second, the turbulence level is quite low immediately upstream of the shock wave which leads to experimental diﬃculties as well. However, as noted by Barre et al. (1996), this experimental conﬁguration yielded some interesting results relating to the behavior of the turbulence in such a shock/turbulence interaction. In order to obtain better quality results in the near future, important modiﬁcations to this experimental device are in progress to suppress these two main limitations just described. It is hoped that better quality for both the incoming ﬂow and the interaction itself can be obtained. For the remainder of this section we focus on the experimental results presented in Andreopoulos et al. (2000) for shock interaction with turbulence. Figure 30 shows the behavior of the ampliﬁcation ratio for the velocity ﬂuctu ations Gu2 = u 2 /u02 across the shock from experiments, LIA, rapid distortion theory (RDT, see Andreopoulos et al. for references), and DNS at several up-

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stream Mach numbers Ms . It is clear that the disagreement is very large due to the experimental diﬃculties with statistical convergence for the shock tube results, and due to the diﬀerences in Reynolds number compared to the DNS. Other results have been obtained by Barre et al. (1996) at Mach 3 in a supersonic wind tunnel, where the turbulence ampliﬁcation was measured by means of a hot-wire. In Figure 31 the measured results compared to LIA are shown. In the far ﬁeld agreement is quite good. However in the near ﬁeld there is disagreement, probably due to the fact that the pressure ﬂuctuations, which exist in the near ﬁeld turbulence, had to be omitted in the data reduction of the hot-wire results because no measurements of the pressure ﬂuctuations were available. 5 4 3

u1, 2x2 grid u1, 3x3 grid u2, 2x2 grid u2, 3x3 grid u3, 2x2 grid u3, 3x3 grid LIA, streamwise velocity RDT, streamwise velocity DNS, Lee et al (1993)

G u2 2 1 0 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20

Ms

Figure 30: Velocity ampliﬁcation ratio across a shock both from experiments, LIA and RDT, and DNS. (From Andreopoulos et al. 2000): u1 , longitudinal velocity; u2 , u3 , lateral velocities. There is only a limited amount of experimental data available for length scale evolution across such shock interactions. Figure 32 shows results obtained by Briassulis (1996) in a shock tube for diﬀerent upstream Mach numbers and diﬀerent values of the upstream integral scale. Values of the longitudinal integral scale ampliﬁcation rate GL1 (ξ1) across the shock are plotted at diﬀerent positions (x/∆) where ∆ is the mesh spacing of the grid acting as the turbulence generator. Results are given for three shock intensities (Ms = 1.26, 1.37, 1.50), and at several integral scale initial values (recall that the initial integral scale increases with x/∆). It is clear that in all cases, a decrease of the integral scale is found, but it appears to depend strongly on both the shock strength and the initial value of the upstream integral scale. At large x/∆, far from the grid, the initial integral scales are the largest and the scale reduction across the shock is much more important.

566

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2 2

3 2

LIA (near field) LIA (far field)

1 0 –20

–10

0

10

20

30

40

Xc (mm)

Figure 31: Longitudinal turbulence ampliﬁcation across a Mach 3 normal shock (from Barre et al. 1996): Xc corresponds to shock location. 1.0 0.9 0.8 0.7 0.6 GL (ξ1) 0.5 1 0.4 0.3 0.2 0.1 0.0

Mflow =0.36, Ms =1.26, Mr =1.18 Mflow =0.475, Ms =1.37, Mr =1.25 Mflow =0.6, Ms =1.5, Mr =1.39

0

10

20

30

40

50

60

70

80

90 100

x/∆

Figure 32: Ratio of the longitudinal integral length scale across the shock for diﬀerent upstream integral scale and Mach numbers (from Andreopoulos et al. 2000). The attenuation of the longitudinal integral scale is also dependent on the shock strength. For high Mach numbers the attenuation is larger for a ﬁxed upstream scale. Figure 33 from Barre et al. (1996) conﬁrms this fact. For a Mach 3 shock, the attenuation is about 0.14 which is roughly equivalent to the results obtained by Briassulis (1996) at Mach 1.5. Thus, it appears that beyond Mach 1.5 an asymptotic behavior is obtained for the reduction in longitudinal integral scale across the shock. However, it is clear that, at the present time, no systematic study concerning the eﬀect of the ratio be-

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1.0 0.8

Lf Lf0

0.6 0.4 0.2 0.0 –20

–10

0

10

20

30

40

Xc (mm)

Figure 33: Longitudinal integral scale attenuation across a Mach 3 normal shock (from Barre et al. 1996): Lf, downstream shock value; Lf0 , upstream shock value. tween the upstream integral scale and the shock thickness has been done. Concerning the evolution of the lateral integral scale, Barre et al. (1996) have shown experimentally that this scale is practically unaﬀected by the shock interaction. A direct consequence of these results for the turbulent scales is the evolution across a normal shock of the ﬂuctuating longitudinal mass ﬂow rate spectrum. Figure 34 shows the upstream and downstream hot-wire spectra obtained across a Mach 3 normal shock by Barre et al. (1996). (In Figure 34 spectra are represented in dimensional form. Their integral is the local velocity variance, thus this representation takes into account the turbulent kinetic energy ampliﬁcation across the shock). It is clear that the downstream spectrum is shifted towards the high frequencies. This fact conﬁrms the observed reduction in longitudinal integral scale across the shock (cf. Figure 33). The ampliﬁcation ratio of each turbulent scale resolved by the experiment can be obtained by computing the ratio of the two spectra. This ratio is presented on Figure 35. It is clear that the most ampliﬁed frequencies are higher than the ones corresponding to the incoming longitudinal integral scale. The shock then performs an energy transfer from the incoming integral scale towards smaller scales. This is conﬁrmed by DNS data from Lee et al. (1993), Hannappel and Friedrich (1995), and Jamme (1998). Unfortunately, it has not been possible up to this point to reach any general conclusions from the experimental studies on the following questions (among others): – What is (are) the physical parameter(s) that govern(s) all aspects of this interaction?

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10–5

10–6

Before Shock After Shock

10–7 100

1000

10000

k (m–1)

Figure 34: Upstream and downstream hot-wire signal spectra for a Mach 3 normal shock (from Barre et al. 1996). 6.0 5.0 4.0 PSD PSD0 3.0 2.0 1.0 0.0 0

500 1000 1500 2000 2500 3000

k (m–1)

Figure 35: Ratio of hot-wire signal spectra across a Mach 3 normal shock (from Barre et al. 1996): P SD, downstream shock power spectral density; P SD0 , upstream shock power spectral density. – How can the problem be parametrized? Should the shock thickness be taken into account? It is hoped that future experiments (some now in progress) can provide some answers to these questions. While progress has been slow, this does not diminish the importance of understanding the dynamics in this simpliﬁed situation before attempting to develop models for much more complex shock/turbulence interactions such as shock/boundary-layer interactions.

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569

Shock/boundary-layer interaction

It is only with recent improvements in computer hardware that direct numerical simulations of supersonic wall-bounded ﬂows with embedded shocks have become feasible. An accurate DNS of such a spatially evolving ﬂow ﬁeld is a diﬃcult task. Two obvious challenges are, ﬁrst, the large number of numerical grid points required in a full spatial simulation and, second, the ability to retain a high level of global numerical accuracy when shocks are present. The issue of global numerical accuracy for shock capturing in such ﬂows has been the focus of some fundamental studies (Adams and Shariﬀ 1996, Casper and Carpenter 1998). Casper and Carpenter analyzed the behavior of highorder schemes for unsteady ﬂows with shocks for aeroacoustic applications, and concluded that in the presence of moving shocks it is diﬃcult to achieve globally high-order accuracy. Nevertheless, Adams and Shariﬀ developed a hybrid scheme that uses a fourth order compact scheme in smooth regions and a fourth order ENO (Essentially Non Oscillatory) scheme near shocks. They showed that direct simulations of ﬂows with steady shocks were feasible provided a high-order scheme is used that satisﬁes certain shock capturing properties. The methodology has been applied to the direct numerical simulation of a compression ramp ﬂow at low Reynolds number (Adams 1998). Data relevant to turbulence modeling have yet to be extracted from this simulation, but clearly there is a large potential for future work on this theme. However, accurate computations at high Reynolds numbers comparable to experiments will remain outside the range of DNS for many years to come. The focus in this subsection will then be on the RANS predictions of supersonic boundary-layer ﬂows with shock interactions. Several examples of such ﬂows have been extensively documented and discussed (Fernholz and Finley 1976, 1980, 1981; Fernholz et al. 1989; Settles and Dodson 1991; Knight and Smits 1997). Since it is diﬃcult to measure ﬂow variables in close proximity to the wall, and to obtain highly accurate turbulence statistics in such ﬂows, detailed quantitative comparisons with RANS calculations are diﬃcult. Such comparisons mainly focus on distributions of mean variables across the boundary layers as well as pressure and skin friction distributions along the surface. Where available, comparisons with turbulence statistics provide a useful, though qualitative, guide to model performance. Shock/boundary-layer interactions can be classiﬁed on how the shock is generated (Smits and Dussauge 1996). One such category is an incident shock interaction in which the shock from an external source impinges on the layer. Another category is a compression surface in which the shock is generated by the same surface that generated the boundary layer. Examples of the former category include the reﬂection of an oblique shock at a turbulent boundary layer (see Haidinger and Friedrich 1995, also Figure 20), and the shock ﬁeld generated by a double-ﬁn geometry. For the doubleﬁn example, experimental results from Zheltovodov et al. (1994) have been

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used for comparison with computational results from turbulent models ranging from an algebraic Baldwin-Lomax model to a full Reynolds stress model (see Knight and Smits 1997, pp. 5.39–5.43). Comparisons with mean surface quantities such as wall pressure and temperature, and surface heat transfer have been made. While wall pressure variation has been successfully computed by the models, wall temperature and surface heat transfer predictions have not been as successful. As might be expected from such complex ﬂow ﬁelds, characterizing dynamic features such as shock/shock interactions as well as shock/boundary-layer interactions make it diﬃcult to assess and uniquely identify model deﬁciencies. Examples of the compression surface category include shocks generated on an airfoil, wing, bump, cone, (single) ﬁn, and ﬂap/ramp surfaces (see Figure 20). A multitude of studies of such ﬂow ﬁelds exist in the literature (e.g. Haase et al. 1993, 1997; Leschziner et al. 1999, Loyau et al. 1998) utilizing a variety of turbulence models ranging from simple algebraic models to full second-moment closures. As noted previously, the range of experimental data available for comparison with calculations may vary; however, wall pressure variation (or pressure coeﬃcient) and skin-friction coeﬃcient are fundamental and can provide important information about the general characteristics of the ﬂow ﬁeld as well as model performance. As an example, consider the ﬂow ﬁeld predictions of an RAE 2822 airfoil (Re = 6.2 × 106 , M∞ = 0.754 and angle-of-attack 2.57◦ ; see Case 10 of Cooke et al. 1979). Figure 36 shows the pressure coeﬃcient variation along the airfoil as well as predictions using a two-equation k-ε model and an explicit algebraic stress model (EASM). As is well known, linear eddy viscosity models are more dissipative than higher-order closures such as second-order models or (explicit) algebraic stress models, and this is shown through the downstream shift in shock location predicted by the k − ε model relative to both the experimental data and algebraic stress model. This downstream shift in shock location corresponds to a delayed separation zone on the airfoil. Bardina et al. (1997) compared diﬀerent linear eddy viscosity models including the Spalart-Allmaras one-equation model and the SST model (see [1]) and found that the SST model yielded the closest predictions to the experimental data. This showed that even though the SST model was a linear eddy viscosity model, its hybrid structure resulted in lower eddy viscosity levels than the simple k-ε model. Leschziner et al. (1999) used a second-moment closure model as well as the SST and k-ε models. Their study also showed improved predictions of the second-moment closure and the SST model over the the k-ε model. While this transonic airfoil example suggests that higher-order closures can provide improved ﬂow ﬁeld predictions, further validation tests on a more (geometrically) complex example show that predictive deﬁciencies may even exist at this level. Consider another case of a compression surface generated shock – the compression-corner ﬂow at free stream Mach numbers near 3. In

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–1.5 –1.0

Cp

–0.5 0.0 0.5

Experiment EASM K-ε

1.0

1.5 0.0

0.2

0.4

0.6

0.8

1.0

x/c

Figure 36: Surface pressure distributions for the RAE 2822 airfoil (Case 10). this case, the (mean) shock location is better deﬁned (although an inherent unsteadiness about some mean location usually exits). The two-dimensional compression ramp ﬂow is one of the most thoroughly measured wall-bounded supersonic ﬂows (see Fernholz et al. 1989 and Settles and Dodson 1991 for additional citations). Suﬃciently complete experimental data has been documented for diﬀerent ramp angles including 8◦ , 16◦ , 20◦ , 24◦ . Separation in the vicinity of the ﬂat-plate/ramp juncture occurs at 16◦ (incipient), 20◦ , and 24◦ . (Unsteady and three-dimensional eﬀects are present in the 24◦ case which are not completely quantiﬁed, and which limit its usefulness as a validation test case.) Figure 37a shows the measured (normalized) wall pressure variation along the ﬂat plate (s−s0 < 0) and ramp. The ﬁgure clearly shows how the inﬂuence of the ﬂat-plate/ramp juncture is seen to move upstream with increasing ramp angle. In addition, at a ramp angle of 20◦ the onset of separation is highlighted by the plateau in the pressure distribution along the surface. Also shown on the ﬁgure are computational results for the 16◦ case using the LRR (Launder et al. 1975) and SSG (Speziale et al. 1991) pressure-strain rate models in a full second-moment closure with wall functions (as an aside it should be noted that the use of wall functions may be problematic in calculations of separated ﬂow). Both models yield nearly identical results for the surface pressure variation, under-predicting the pressure rise (adverse pressure gradient) along the ramp surface. For the 16◦ case, the turbulence models incorrectly predict a slightly larger upstream inﬂuence than the experimental data as well as the less rapid rise in pressure along the ramp.

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4.0 3.5

3.0 8° ramp 16° ramp LRR SSG 20° ramp

(a) 2.5

(b)

Experiment LRR SSG

3.0

3

Cf x 10

_ _ pw/p∞

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−0.1

0.0

0.1

0.2

(s−s0)/s0

Figure 37: Streamwise distribution along surface of compression ramp: (a) normalized pressure for 8◦ , 16◦ , and 20◦ ramps; (b) normalized skin-friction coeﬃcient for 16◦ ramp. Another measure of the inﬂuence of the ﬂat-plate/ramp juncture on the ﬂow ﬁeld can be seen from the streamwise evolution of the skin-friction coeﬃcient. Figure 37b shows the distribution of the measured skin-friction coeﬃcient Cf along the surface for the 16◦ case. The computations are unable to replicate the incipient separation (possibly due to the use of wall functions). In addition, the models are also unable to correctly predict the downstream behavior of the ﬂow. This is consistent with the model wall pressure predictions shown in Figure 37 where the pressure rise is not as great as in the physical experiment. For the skin-friction coeﬃcient predictions, however, the diﬀerence in model predictions is greater than in the surface pressure case. An important aspect of shock/boundary layer interaction is turbulence ampliﬁcation. This feature has been a major focus of many of the direct simulation studies of boundary-free turbulence interacting with a normal shock (Lee et al. 1993, 1997, Mahesh et al. 1997). Such studies isolate this ampliﬁcation effect without the added complicating feature of the wall. The ability of two second-moment closures to predict turbulence ampliﬁcation in the 16◦ ramp case was examined by Abid et al. (1995). Figure 38 shows the evolution of the normalized second-moment τss along a streamline downstream through the shock. (The τss component is the streamwise component and the normalization is with the value at the inﬂow plane.) The initial streamline location was y + ≈ 2200 which is well away from any direct eﬀects of the wall functions used. Neither the LRR nor the SSG pressure-strain models were able to correctly predict the location or ampliﬁcation level of the experimental results. These examples were intended to show that simple mean properties of the ﬂow, such as surface pressure and skin friction, can be used as a guide to the

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8 Experiment LRR model SSG model

τss/τss0

6

4

2

0 −0.1

0

0.1

0.2

s Figure 38: Evolution of τss second-moment along streamline in compression ramp ﬂow. ﬂow dynamics, and that even well established closure models straightforwardly applied may not be able to replicate these mean properties. Extending such comparisons to the turbulence quantities (such as shown in Figure 38) becomes even more challenging and poor predictive performance may not be surprising. While the predictions of the compression ramp ﬂows are disappointing, several other shock/boundary layer studies have been conducted with more success. These include airfoils, blunt-ﬁns, bumps (plane, axisymmetric, skewed), and wings. Some examples are discussed in the recent papers by Leschziner and colleagues (Batten et al. 1999, Leschziner et al. 1999, Loyau et al. 1998) as well as reports on organized attempts at predicting a cross-section of aerodynamic ﬂows (Haase et al. 1993, 1997).

5

Concluding Remarks

The dynamics of compressible, turbulent ﬂows, with or without shocks, is quite complex. In this chapter some of the characterizing features of these ﬂows have been presented, as well as some of the methodologies used in the analysis and prediction of such ﬂows. These methodologies have included experiments, numerical simulations, and RANS calculations. From an analysis of the various experimental results from both shock tube and supersonic wind tunnels, it appears that the ampliﬁcation ratio of turbulent kinetic energy as turbulence passes through a shock wave is strongly

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inﬂuenced by the nature of the incoming turbulence. However, it is still diﬃcult to clearly parameterize the ampliﬁcation ratio behavior. Eﬀects of incoming Mach number, turbulent Reynolds number, compressibility level, turbulence intensity and integral scales have not been separated and quantiﬁed by the available experiments up to this point. The same conclusions also hold for the evolution of the turbulent scales and spectra across the shock. There have been several direct numerical simulations of homogeneous turbulence interacting with a shock wave during the last decade. These show that turbulence kinetic energy ampliﬁcation is consistent with the results of linear theories. Longitudinal length scales are reduced across the shock, although it is still unclear what the proper parameter or range of parameters are to describe this (upstream integral length scale, upstream turbulence level, turbulence Reynolds number, shock thickness etc.) Lateral scales are essentially unaﬀected by passage through the shock, while dissipation rate and dissipation scales both increase. For mainly shock-free turbulent ﬂows progress has been made in parameterizing the eﬀect of compressibility and sorting out the causes of the changes. For attached boundary ﬂows up to Mach 5 the strong Reynolds analogy holds and the essential turbulence dynamics follows that of incompressible ﬂow. The gradient Mach number may be used to distinguish these ﬂows (with low Mg ) from mixing layer and jet ﬂows where true compressibility eﬀects become important at lower absolute Mach numbers. For mixing layers the gradient Mach number is proportional to the convective Mach number, which has been found to collapse a wide range of experimental data. Direct numerical simulations are now showing that statistical terms such as the pressure strain are central to explaining the reduction of mixing layer growth rate with Mach number. Simulations are additionally providing data to form models of explicit compressibility terms such as dilatation dissipation and pressure dilatation. While simulation techniques (DNS and LES) are appealing on the grounds that they make few assumptions about the turbulence, practical calculations and predictions of complex compressible ﬂow ﬁelds are usually done by RANS calculations. As was outlined in this chapter, the mean conservation and turbulent transport equations written in Favre variables provides an eﬀective framework for developing models for the higher-order correlations which appear in these equations. In the compressible case, new correlations directly associated with the compressibility appear and these terms now require modeling. In the absence of shocks, or even in the presence of shocks under certain circumstances, the utilization of variable mean density extensions to the incompressible formulation is justiﬁed. When strong compressibility eﬀects arise, the modeling situation becomes less clear. As has been discussed, several questions about the underlying physics remain, and without this more complete picture it will not be possible to develop an accurate array of closure models for the compressible correlations. Currently, many RANS approaches revert

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to the simple variable density extensions just mentioned, and most attempts to include compressibility models are on a case-by-case basis. Clearly, more focused eﬀorts are required to develop a somewhat more general approach to predicting these supersonic ﬂows. It is hoped that this chapter has provided the reader with a more complete understanding of the broad scope of research that has been undertaken in an eﬀort to successfully predict and subsequently control supersonic turbulent ﬂows. Many challenges and opportunities remain in the study of the dynamics of such ﬂows.

Acknowledgements The authors thank Professor P.G. Huang for providing the direct numerical simulation data used in Figure 4, and Dr. S. Jamme for providing the LIA results.

A

Identities for Mass-Weighted Averages ρ f f˜ = f + , ρ

ρ = ρ + ρ

ρf = ρf˜ + ρf = ρf + ρf , f = f¯ + f = f˜ + f = f˜ + f ,

ρf = ρf + ρ f ,

(A.1)

ρf = 0

(A.2)

f = f − f˜

(A.3)

f = −

ρ f ρ

(A.4)

ρ f f = f − f˜ + f = f + f = f − ρ ρf = ρf − ρ f = 0,

=ρ

(A.6)

ρ f ρf =− = −f4 ρ ρ

f = −

ρf g

ρ f = ρ f

f

ρ f − ρ

g

g ) = ρf g − ρf g = ρ(f>

(A.7)

ρ g − ρ

ρ f

ρ

(A.5)

ρ g

(A.8)

(A.9)

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References Abid, R., Gatski, T.B., and Morrison, J.H. (1995). ‘Assessment of pressure-strain models for predicting compressible turbulent ramp ﬂows’, AIAA J. 33, 156–159. Adams, N.A. (1998). ‘Direct numerical dimulation of turbulent compression ramp ﬂow’, Theoret. Comput. Fluid Dynamics 12, 109–129. Adams, N.A., and Shariﬀ, K. (1996). ‘A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems’, J. Comput. Phys. 127, 27–51. Andreopoulos, J., and Muck, K.C., (1987). ‘Some new aspect of the shock wave boundary layer interaction in compression ramp corner’, J. Fluid Mech., 180, 405–428. Andreopoulos, J., Agui, J.H., Briassulis, G.K. (2000). ‘Shock wave-turbulence interactions’, Ann. Rev. Fluid Mech. 32, 309–345. Ardonceau, P. (1984). ‘The structure of turbulence in a supersonic shock wave/boundary layer interaction’, AIAA J. 22, 1254–1262. Bardina, J.E., Huang, P.G., and Coakley, T.J. (1997). ‘Turbulence modeling validation, testing, and development’, NASA TM 110446. Barre S. (1993). Action de la compressibilit´e sur la structure des couches de m´elange turbulentes supersoniques. Th`ese de Doctorat, E.N.S.A.E. Toulouse (France) – janvier. Barre, S., Alem, D., and Bonnet, J.P. (1996). ‘Experimental study of a normal shock/ homogeneous turbulence interaction’, AIAA J. 34, 968–974. Barre, S., Dupont, P., Dussauge, J.P. (1992). ‘Hot-wire measurements in turbulent transonic ﬂows’, Eur. J. Mech. B 11, 439–454. Barre S., Quine, C., and Dussauge, J.P. (1994). ‘Compressibility eﬀects on the structure of supersonic mixing layers: experimental results’, J. Fluid Mech. 259, 47–78. Batten, P., Leschziner, M.A., and Craft, T.J. (1999). ‘Reynolds-stress modeling of afterbody ﬂows’. In Turbulence and Shear Flow Phenomena – 1 S. Bannerjee and J.K. Eaton (eds.), Begell House, 215–220. Bellaud, S. (1999). Mesures et analyses d´etaill´ees des champs turbulents en couches de m’elange annulaire supersoniques. Th`ese de Doctorat Universit´e de Poitiers. Blin, E. (1993). Etude exp´erimentale de l’interaction entre une turbulence libre et une onde de choc. Th`ese de doctorat de l’Universit´e Paris VI. Bonnet, J.P., Chambres, O., Lammari, M., Barre, S., Braud, P. (1996). ‘Mesures 3D en couches de m´eange planes supersoniques’, Contrat DRET 94/2586 A., Octobre. Bonnet, J.P., Gresillon, D., Taran, J.P. (1998). ‘Nonintrusive measurements for highspeed, supersonic and hypersonic ﬂows’, Ann. Rev. Fluid Mech. 30, 231—273. Bradshaw, P. (1977). ‘Compressible Turbulent Shear Layers’, Ann. Rev. Fluid Mech. 9, 33–54. Bradshaw, P., Ferriss D.H., and Johnson, R.F. (1964). ‘Turbulence in the noiseproducing region of a circular jet’, J. Fluid Mech. 19, 591–624. Briassulis, G.K. (1996). Unsteady nonlinear interactions of turbulence with shock waves. PhD Thesis, City College, City University of New York.

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20 The Joint Scalar Probability Density Function W.P. Jones Abstract The purpose of this chapter is to review the joint probability density function, (PDF), evolution equation approach to calculating the properties of turbulent ﬂames. In this approach the closure of the micromixing term represents a central diﬃculty and for this, various proposals including the linear mean square estimation closure, coalescence-dispersion models and Langevin, binomial, mapping and Euclidean minimum spanning tree closures are appraised. Results obtained with the PDF equation using a modiﬁed coalescencedispersion model and a simple global reaction mechanism demonstrate that the method is capable of reproducing measured mixture fraction, fuel and CO2 proﬁles in jet diﬀusion ﬂames and counterﬂow premixed and partially premixed turbulent ﬂames. The levels of carbon monoxide are somewhat overpredicted in all the ﬂames considered but it is argued that this is primarily a consequence of limitations in the simpliﬁed chemical reaction mechanism used. The need for improved models to represent micromixing is also suggested.

1

Introduction

Calculation methods for predicting the properties of turbulent ﬂames in practical combustion systems invariably involve statistical methods in which the partial diﬀerential conservation equations are averaged to yield transport equations for the moments of the appropriate dependent variables. Alternative approaches such as direct solution of the exact instantaneous forms of the equations (DNS) for turbulent ﬂames are computationally prohibitively expensive for engineering applications. DNS is presently restricted to ﬂow at low Reynolds numbers and in simple geometries and is likely to remain so for the foreseeable future. Large Eddy Simulation (LES), although becoming a viable tool for engineering applications, does not alleviate the problems arising from combustion. In the case of LES, burning is likely to occur predominately within the unresolved sub-grid scales and the resulting closure problems are then similar to those arising in conventional moment closures. In moment closures, averaging leads to a loss of information with the consequent appearance of various unknown terms, dependent on the ﬂuctuating 582

[20] The joint scalar probability density function

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turbulence ﬁeld, and closure approximations are needed to represent them. In turbulent ﬂames the processes for which closure approximations are required include turbulent transport (diﬀusion) of heat, mass and momentum and turbulence-chemistry interactions. For turbulent transport a turbulence model is required but it is the turbulence-chemistry interactions which represent the central diﬃculty in turbulent ﬂames. More speciﬁcally, in moment closures a means must be available to evaluate the averaged net formation rates which appear in the transport equations for the mass fraction of each chemical species present. This in turn requires that the joint probability density function for the mass fractions of the necessary chemical species be known. If the assumption of ‘fast’ reaction can be invoked then the thermo-chemical state can often be determined uniquely in terms of a single independent scalar variable: for non-premixed combustion this is the mixture fraction and for premixed systems it is a reaction progress variable. In these circumstances it is possible to presume a suitable shape for the PDF in terms of the mean and variance of the appropriate scalar with the mean and variance being obtained from solutions of their respective transport equations. While this approach often yields good results for heat release and temperature it is clearly inappropriate where chemical reaction rates exert an important inﬂuence, for example in the emission of unburnt hydrocarbons and the formation and emission of carbon monoxide. In order to reproduce ﬁnite rate eﬀects, more independent scalars need to be incorporated and a multi-dimensional joint PDF is needed. In this circumstance the presumed shape approach becomes essentially intractable and the only viable method appears to be the PDF evolution equation approach. The joint PDF for a set of scalars, P (ψ; x, t), contains all single point, single time moments of the scalar ﬁeld. Consequently in the exact evolution equation for P (ψ; x, t) the chemical source terms associated with reaction appear in closed form though, as in all statistical approaches, the equation itself is unclosed. The unknown terms represent micromixing in the scalar space and turbulent transport in physical space. However, providing suﬃciently accurate approximations for these can be devised, the inclusion of chemical reaction should present little further diﬃculty. PDF evolution equation modelling has been investigated for well over two decades but it is only relatively recently that the methodology has become a potential tool for engineering calculations. While this is in part due to the successful development of Monte Carlo solution algorithms, it is mainly because of computer hardware developments which provide the increased resource necessary to obtain solutions to the PDF equation. Essentially, for the reasons above, there is currently considerable interest in the PDF evolution approach to turbulent combustion; reviews of the topic can be found in the works of O’Brien (1980), Pope (1985), Kollmann (1990, 1992), Kakhi (1994), Jones and Kakhi (1996) and Peters (2000). In the present work the exact PDF evolution equation is presented and discussed and various approaches to approximating the micromixing term are

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reviewed. Various stochastic or Monte Carlo methods are described for solving the equation for P (ψ; x, t) and these are to be combined with a ﬁnite diﬀerence scheme for obtaining the velocity ﬁeld using a conventional turbulence model. Applications of the method to the computation of turbulent non-premixed and premixed ﬂames, using a reduced chemical reaction scheme, demonstrates the capabilities and shortcomings of the formalism.

2

Derivation of the Scalar PDF Equation

The evolution equation for P (ψ) (the dependence on x and t is omitted for clarity) is an exact, unclosed equation and its derivation relies on the use of the conservation equations. Consequently the equation has a physical foundation. The most widely-used method of derivation is based on the concept of a ﬁnegrained density function pˆ(ψ, φ; x; t), see Lundgren (1967), and is deﬁned such that pˆ(ψ, φ)dψ is the probability that at position x and time t, ψα ≤ φα (x, t) ≤ ψα + dψα for all α = 1, . . . , N . Note that ψα denotes an independent variable (like x and t) and represents ‘position’ in the sample space of the dependent random variable φα (x, t); ψ denotes the set of ψα for α = 1, . . . , N . Clearly pˆ(ψ, φ) is a multi-dimensional quantity and it represents the PDF for a single realization of a turbulent ﬂow, so that if ψα ≤ φα (x, t) ≤ ψα + dψα is not satisﬁed for all α, then pˆ(ψ, φ) = 0, otherwise pˆ(ψ, φ) = 1. Consequently, the ﬁne-grained density can be expressed as pˆ(ψ, φ) ≡

N I

δ {φα (x, t) − ψα } .

(1)

α=1

In contrast, the conventional PDF P (ψ) is representative of an ensemble of realizations, such that its integral over a ﬁnite region of its sample space represents the likelihood of occurrence for an event. It is a straightforward matter to show, see Pope (1985), that P (ψ) = p(ψ, φ).

(2)

The details of the derivation of the evolution equation for P (ψ) can be found in O’Brien (1980) and Kollmann (1992). Kollmann (1989, 1990) derives the PDF equation in terms of the characteristic function, which is the Fourier transform of the PDF. Whatever the method of derivation the evolution equation for the

[20] The joint scalar probability density function

585

density weighted scalar PDF) is given by N 9 ∂ P˜ (ψ) ∂ P˜ (ψ) ∂ 8 + ρ¯u ˜k ρ¯ ω˙ α (ψ)P˜ (ψ) ρ¯ ∂t ∂x ∂ψα k α=1 I II III 9 ∂ 8 ρ¯uk |φ = ψP˜ (ψ) =− ∂x k IV * )B C N ∂Ji,α ∂ ˜ + φ = ψ P (ψ) ∂ψα ∂xi α=1 V

(3)

where ω˙ α and Ji,α represent the net formation rate per unit volume and molecular diﬀusive ﬂux, respectively, of species α and where ˜ indicates a density weighted quantity. The terms have the following signiﬁcance: • Term I: Rate of change in physical space; • Term II: Mean convection in physical space; • Term III: Chemical source production in compositional space; • Term IV: Turbulent transport in physical space; • Term V: Micromixing in compositional space. Assuming Fickian diﬀusion the ‘micromixing’ term can be expressed as −

N N α=1 β=1

∂2 ∂ψα ∂ψβ

)B

C * µ ∂φα ∂φβ ˜ φ = ψ P (ψ) , Sc ∂xk ∂xk

(4)

where Sc is the Schmidt number. With the assumption of constant µ it can also be shown, see Kollmann (1992), that the term can be expressed in terms of a two-point PDF ∂2 µ ∂ − lim Sc ∂ψβ x →x ∂xk ∂xk N

)

ψβ P (ψ, ψ ; x, x , t)dψ

β=1

where the primes denote terms based on conditions at x .

* ,

(5)

586

2.1

Jones

Terms Appearing in the PDF Equation

Inspection of equation (3) reveals that the chemical source production term appears in exact and closed form. In other words it is known in terms of ψ and P˜ (ψ). This feature of equation (3) has always constituted the main reason for using the PDF approach in reacting ﬂows. The implication is that realistic chemistry involving ﬁnite-rate kinetics can be incorporated explicitly in turbulent ﬂow calculations without an assumed ﬂame structure to characterize the thermochemistry. In comparison with other forms of thermochemical closure, this is clearly a novel, advantageous and unique feature. However, the micromixing term appears in unclosed form due to the appearance of the conditional expectation. To evaluate this term, the joint PDF of the scalars and their gradient (equation (4)) at the single-point level, or the two-point joint PDF of the scalars (equation (5)) is required. Needless to say, evolution equations for such higher-order PDFs contain further unknown terms and dramatically increase the dimensionality and/or computational cost of the problem. The micromixing term involves diﬀusion coeﬃcients and composition gradients, which stem directly from the molecular diﬀusion term in the instantaneous species conservation equations, and an order of magnitude argument demonstrates that it cannot be neglected, even at high Re. Since diﬀusion is important predominantly amongst the small scales of turbulent motion, an accurate representation of these scales is clearly essential to the determination of the evolution of the PDF and hence of the averaged scalar ﬁeld. In hydrocarbon combustion, this is particularly true for species such H2 and H which are formed and consumed by reaction (in the ﬁne scales of turbulent motion) and which possess much larger diﬀusivities than other scalars. As a consequence it can be argued that the classical closure problem associated with turbulent combustion, which manifests itself as a mean reaction rate in moment closures, is simply re-expressed in terms of the micromixing term in PDF closures. In fact the appearance of the chemical source term in exact form in the PDF formalism is merely a consequence of the mathematical manipulations employed to derive such an equation. As pointed out by Kollmann (1992), in PDF methods nonlinear terms are transformed into linear terms with variable coeﬃcients. This is achieved by converting the associated dependent variables (which make the nonlinear terms unclosed upon averaging) into independent variables of the PDF at the cost of increasing the dimensionality of the formalism. The results of such a manipulation do not overcome the basic physical problems inherent in any statistical approach to turbulence, whether it be Reynolds–Favre averaging, spatial averaging or transported PDF modelling. The PDF P˜ (ψ) is only a function of the scalar ﬁeld and includes no velocity information; consequently the turbulent diﬀusion of the PDF (term V in equation (3)) appears in unclosed form. The joint velocity-composition PDF allows this term to be expressed in exact form, but at the expense of introducing further unknowns and extra independent vari-

[20] The joint scalar probability density function

587

ables to the equation. Finally it is to be noted that, at the level of closure corresponding to equation (3), u ˜k is unknown. This quantity is conventionally supplied through the use of a turbulence model.

3

Solution Aspects of PDF Evolution Equations

At this stage it may appear somewhat premature to discuss the solution algorithm for the PDF evolution equation, particularly since the modelling of the unclosed terms has not yet been considered. However, the reality is that the method of numerical solution and the modelling under this formalism are strongly interconnected. For problems involving a single scalar dimension and in simple geometries, solutions of PDF equations using conventional ﬁnitediﬀerence techniques have been obtained (Janicka et al. 1978, 1979). However, in order to exploit the full potential of PDF equations, it is necessary to solve for a multi-dimensional PDF involving several non-conserved scalars. Pope (1981) estimated that the number of computer operations at each node and at each time step, if ﬁnite-diﬀerence equations were to be employed, was of the order of exp(6N ), where N is the number of scalar variables. In the same paper he described a solution algorithm for performing a Monte Carlo (particle-method) simulation of the scalar PDF equation. In this approach the computational cost rises only linearly with N and the PDF is simulated via an ensemble of stochastic particles. That a continuous PDF P˜ (ψ) has a discrete counterpart is a widely used principle. The histogram constructed from φ(i) (x, t), (where φ(i) represents the ith realisation or measurement) is an approximate discrete representation of the PDF. In the limit of an inﬁnite number of realizations (with the sampling ‘bin’ sizes tending to zero), the histogram tends to the PDF – this result is formalized by equation (2). With regard to the closures for the mixing term in equations (3) and (4), which we will discuss later, the implication is that, given a suitable approximation (written in PDF form), an equivalent particle model representation needs to be constructed. Thus in the section dealing with the mixing models, both interpretations will be given, usually for the case of a single scalar. For certain mixing models the extension to multi-scalar problems is simple; in other models it is still not clear how this can be achieved in a rigorous and/or tractable manner. However, the existence of a link between the particle method and the PDF approach is essential if the advantages of the latter method are to be exploited. Such a link and the necessary criteria for its use have been investigated by Pope (1979, 1981, 1985); it is argued that the fundamental basis for ‘equivalence’ of these two methodologies is that the resulting statistical moments (mean, variance, . . . ) arising from the particle-method approach should evolve in an identical manner to its deterministic counterpart in the limit of an inﬁnite number of notional particles.

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4

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Basic Modelling Requirements of the Molecular Mixing Term

Closure approximations constructed to represent the micromixing term must satisfy a number of properties of term V in equation (3), requirements which are discussed in Kakhi (1994) where various mixing models are also reviewed. The most basic requirements are that the approximation leaves the means of the scalars unchanged, i.e. ∆φ(x, t) ≡ φ(x, t + δt) − φ(x, t) = 0, and that the model gives rise to PDFs which correspond to a set of physically possible sequences of events. In addition to the general realizability condition, P˜ (ψ(x, t) ≥ 0, there are further conditions to be imposed depending on the properties of the actual scalars being considered. For example, if the PDF of a single scalar corresponding to a mass fraction is considered, then solutions of the modelled form of the PDF evolution equation must satisfy P˜ (ψ, x, t) = 0 for ψ < 0 or ψ > 1. In the case of a reacting mixture comprising N species with corresponding mass fractions Yα with sample space Yˆα then realizability constraints imply ˆ α < 0. ˆ α = 1 or Y (6) Y P˜ (Yˆα , x, t) = 0 for α

While the satisfaction of these requirements by any mixing model is essential if anything useful is to result, they are not exhaustive and there are many other conditions which could be considered and imposed. These include for example the linearity and independence properties of a set of passive scalar quantities, see Pope (1983), and the ‘localness’ concepts introduced by Subramanian and Pope (1998). However these have not so far found widespread use and will not be considered in detail here. Instead we turn to the constraints imposed on mixing models by the need to produce results in agreement with measurements in simple conﬁgurations involving passive scalar quantities. Of these requirements probably the most important are the scalar variance decay rates, i.e. the scalar dissipation rates, predicted by the model. All mixing models proposed in current use have utilised the assumption that the ratio of the scalar to mechanical turbulence integral time scale is constant so that D 2 ε φ = · · · − Cd φ2 , Dt k

(7)

where C2d is the time scale ratio, r, and where k/ε is used here to represent the integral time scale. For inert ﬂows and in situations where the mechanisms responsible for generating the velocity and scalar ﬂuctuations have roughly the same length scales, for example in thin shear layers where the generation and dissipation rates of turbulence energy and scalar variance are in approximate balance, then the constant Cd has a value around 2.0. While the assumption

[20] The joint scalar probability density function

589

has been very widely used in the context of RANS closures and gives reasonable results in many near equilibrium shear layer ﬂows, it is not of general applicability since the time scale ratio will be diﬀerent in diﬀerent ﬂows: values in the range 13 < r < 3 have been observed, see Warhaft and Lumley (1978). The real test of any mixing model lies in its ability to predict correctly the evolution of the scalar PDF and in discussing the statistical behaviour of the various models, reference will be made to the following parameters: skewness, ﬂatness and superskewness. These are the normalized moments of the PDF, deﬁned as follows:  n = 3 ⇒ skewness     n  ψ (ψ − φ) P (ψ)dψ n = 4 ⇒ ﬂatness (8) Ωn = 8 9n   2 P (ψ)dψ 2  (ψ − φ)  ψ  n = 6 ⇒ superskewness. For a reference Gaussian PDF all the odd moments are zero, because of symmetry, and the values of the normalized fourth and sixth moments are 3 and 15 respectively. A popular test case utilised by many workers is that based on isotropic homogeneous turbulence with initial conditions corresponding to segregated ﬂuid parcels (in composition space), with a PDF idealized as a double δ-function distribution. Here it is to be expected that the process of mixing/homogenization will result in the initial PDF relaxing to a smooth, bell-shaped distribution and, in the limit of t → ∞, converging to a δ-function at the mean composition. Though there is no formal proof that it should be so, it is conventional to assume that the bell-shaped distribution is Gaussian. Eswaran and Pope (1988) performed a DNS of decaying passive scalar ﬁelds in inert, homogeneous and isotropic turbulence. A statistically stationary velocity ﬁeld was maintained by ‘forcing’ the simulations i.e. energy was added to the velocity ﬁeld at low wave numbers, allowing the simulations to reach a quasi-equilibrium state where the rate at which energy was dissipated at the small scales was equal to the rate at which energy was added at the large scales. The results demonstrated that, as time progressed, P (ψ; t) evolved from a double δ-function distribution to an inverted parabola and subsequently to a bell-shaped distribution. The evolution of the PDF shapes appeared to be independent of initial conditions (such as the ratio of the mechanical and scalar turbulence length scales, which had a signiﬁcant eﬀect on the scalar variance decay rate). It was argued that this observation should considerably simplify the task of constructing a realistic mixing model and it was found that after about eight eddy turnover times (a relatively long period of time), the skewness, ﬂatness and superskewness of φ tended very closely to the Gaussian values of 0, 3 and 15 respectively. The tendency to Gaussianity was accompanied by an ever increasing lack of correlation between χ – the scalar dissipation rate – and φ(x, t). This is an important result since the micromixing term (equation (4)) is eﬀectively χ conditioned

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on φ(x, t). As a modelling recommendation, it is argued that the assumption of χ being independent of φ(x, t), although valid at long times, is not valid in the short-term range of mixing: this raises questions concerning the validity of the statistical independence assumption which is often invoked in combustion and turbulence modelling. Additional evidence of the tendency to Gaussian PDFs was provided by Tavoularis and Corrsin (1981) who measured the evolution of the temperature ﬁeld in a nearly homogeneous turbulent ﬂow with uniform mean velocity and mean temperature gradients. The overheat of the ﬂuid by heating rods was considered small enough to have negligible eﬀect on the turbulence, and hence the temperature can be viewed as a passive scalar. It was found that typical values of the skewness and ﬂatness factors of the temperature measured on the centre-line of the tunnel, far downstream, were indistinguishable from the values for a normally distributed random variable (which are 0 and 3, respectively); it was found that a Gaussian with the same mean and variance ﬁtted the PDFs nearly perfectly. The velocity-scalar joint PDF also closely resembled a jointly-normal distribution. Despite the above comments concerning the evolution of a decaying scalar ﬁeld to Gaussianity, it must be borne in mind that the scalars considered here are bounded quantities, e.g. species mass fractions, and so their PDFs are bounded and cannot be exactly Gaussian. The implication is that the PDF approaches a Gaussian distribution only in the limit of zero variance where most of the interacting ﬂuid ‘parcels’ are very close to the mean composition. However the distribution near the bounds of the scalar are unlikely to resemble Gaussian behaviour, but in the cases considered above this is immaterial given the degree of homogenization at that stage of the mixing process. The provision of a suitable model for micromixing is a central requirement in the PDF evolution equation approach. A number of proposals have been made and these are described and reviewed below. In discussing mixing models attention will be restricted to homogeneous turbulent ﬂows without reaction; this is usually the basis on which the models are derived. However in all cases the models are intended for application, without modiﬁcation, to inhomogeneous turbulent ﬂows both with and without reaction.

4.1

Linear Mean Square Estimation (LMSE) closure

Using equation (5) and the deﬁnition of conditional probability, P (ψ, ψ ) = P (ψ |φ(x, t) = ψ)P (ψ), the PDF equation describing homogeneous turbulence can be written as ) * N ∂P (ψ; t) ∂2 µ ∂ ψβ P (ψβ |φ = ψ)dψβ P (ψ) lim ρ = − ∂t Sc ∂ψβ x →x ∂xk ∂xk β=1

= −

N , ∂2 + µ ∂ E(φ lim |φ = ψ)P (ψ) . β Sc ∂ψβ x →x ∂xk ∂xk β=1

(9)

[20] The joint scalar probability density function

591

Dopazo (1973, 1979) and Dopazo and O’Brien (1974) considered the case of the single scalar, where E(φ |φ = ψ) is the expectation of the random variable φ(x , t) at a point x , conditioned on the event that φ(x, t) = ψ at a point x. If it is assumed that P (ψ |φ = ψ) is conditionally Gaussian, the theory of stochastic processes implies that E(φ |φ = ψ) = φ + r(x, x , t) (ψ − φ) ,

(10)

where φ is the expectation (mean) of the random variable φ(x, t) and r(x, x , t) is the correlation coeﬃcient of φ(x, t) and φ(x , t): @+ ,+ ,A φ(x, t) − φ(x, t) φ(x , t) − φ(x , t) r(x, x , t) = , (11) σφ (x, t)σφ (x , t) where σφ denotes the square root of the variance of φ. Dopazo and O’Brien (1976) point out that the above assumption does not imply that φ(x, t) and φ(x , t) are normally distributed: rather it constrains the expected value of φ(x , t), given φ(x, t), to a certain prescribed behaviour. Later O’Brien (1980) reformulated the problem in terms of linear mean square estimation, which involves less restrictive assumptions but gives rise to the same result. The requirement of homogeneity and isotropy together with equations (10) and (11) leads to the following result for the RHS of equation (9): 2 ∂ r 6 −3 [ψ − φ(x, t)] ≡ 2 [ψ − φ(x, t)] . (12) ∂2 =0 λφ Here, λ2φ can be interpreted as a (Taylor) microscale for the scalar ﬁeld given −1 by −2 ∂ 2 r/∂2 =0 , where = |x − x|. Equating the result of equation (12) to the LHS of equation (9), the closure can be expressed as ∂P (ψ; t) ∂ =A [(ψ − φ) P (ψ; t)] , ∂t ∂ψ

(13)

where A = 6ν/(σ λ2φ ) and where σ is the Prandtl or Schmidt number as appropriate. Multiplying equation (13) by (ψ − φ)2 and integrating over ψ space leads to dφ 2 2 (14) = −2Aφ . dt Comparing this result with a conventionally modelled scalar variance equation in homogeneous, isotropic and inert turbulence implies the identity χφ ≡ Aφ . 2

(15)

Referring to equation (13) as the LMSE model, the particle method simulation corresponding to this closure becomes dφ(p) Cd ε (p) (16) =− φ − φ , dt 2 k

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where A has been replaced with Cd kε consistent with equation (7). The model constitutes a linear, deterministic simulation of the mixing phenomenon, continuous in time. Analytical solutions for equation (13) are discussed in Dopazo and O’Brien (1976), O’Brien (1980) and Pope (1985); the results of such analysis show that if the initial condition is a normal distribution, the model preserves the Gaussianity which is considered as a satisfactory result. However, the drawback is that the initial functional form of the PDF is never relaxed and consequently, in the case of an initial double δ-function speciﬁcation for the PDF, the subsequent time evolution remains as two δ-functions which approach each other in composition space. Pope (1985) points out that such behaviour is to be expected since the model contains no information about the shape of the distribution – only the mean appears in equation (16). Despite this drawback, Dopazo (1979) argues that the modelling is applicable provided the turbulence is ‘given a chance’ to induce randomness and smear out the initial discontinuous state. In other words it is applicable if the PDF is continuous. Predictions based on LMSE were compared against measurements in Dopazo (1979) for grid-generated homogeneous turbulence with an asymmetric mean temperature proﬁle. In the far downstream region good agreement was found for the conditional expected value of temperature versus temperature.

4.2 4.2.1

Coalescence-Dispersion Mixing Models Curl’s Mixing Model

The work of Curl (1963) was concerned with the analysis of the mixing of clouds of droplets in a two-liquid phase chemical reactor. It was assumed that all the droplets in the system were of the same size and that coalescence took place between two droplets having a solute concentration of c1 and c2 . Redispersion occurred immediately to produce two equal droplets of concentration 1 2 (c1 + c2 ). The total concentration range was divided into L intervals of concentration width δc and P (l)δc was deﬁned as the fraction of the droplets with concentrations within the lth, 1 ≤ l ≤ L), interval. The basic balance equation constructed for P (c) stated that the rate of change of the number of droplets in a particular interval is inﬂuenced by: • the rate of creation of new droplets in the interval (c, c + δc) due to collisions of droplets from concentration intervals located symmetrically about c; • the rate of loss of droplets by collision of droplets from the interval (c, c + δc) with those from all other concentration intervals.

[20] The joint scalar probability density function This led to

 c 

∂P (c) = 4Q  ∂t

0

  P (c + α)P (c − α)dα − P (c) . 

593

(17)

Curl deﬁned Q as twice the collision rate divided by the total number of droplets in the system and solved equation (17) for the case of inert and reacting ﬂows using conventional ﬁnite-diﬀerence techniques. Subsequently Spielman and Levenspiel (1965) proposed a particle (Monte Carlo) method for the solution of such an equation and used the methodology to investigate the ﬂow characteristics of an ideal stirred tank reactor involving a dispersed phase system. Curl’s mixing closure has also been used in the context of continuous systems, to describe the turbulent mixing of gases. The work of Flagan and Appleton (1974) and the references cited therein may be consulted as examples of such applications. As noted by Dopazo (1979) the direct extrapolation of Curl’s model to the mixing of a single-phase gaseous scalar ﬁeld is questionable, particularly since it was originally formulated to describe the interactions of droplets. An implication would be that two ﬂuid ‘parcels’ at diﬀerent temperatures, say, would mix instantaneously to form ﬂuid parcels with a temperature equal to the average of the two original temperatures. This feature of the model is clearly unrealistic and has also been highlighted by Pope (1982) for stochastic simulations; considering the test situation discussed in section 4, an initial condition corresponding to a double δ-function was speciﬁed and the mixture allowed to evolve. Of the Np particles in the system, Np /2 were assigned φ(p) = 1 and the remainder as φ(p) = −1. In any particular time step, pairs of particles were selected, and the number chosen was the nearest integer to ε Nc = Bδt Np . k

(18)

This expression was chosen to ensure a variance decay rate consistent with equation (7) and, for Curl’s model, B ≈ 2. The particles were then allowed to interact along the straight line connecting them in composition space, such that their new concentrations ‘jump’ from their initial values to their joint mean value. The values of particles not selected remain unaltered. The results demonstrated that, after the ﬁrst time step, φ took the values −1, 0 and 1; after the second step they were −1, −1/2, 0, 1/2 and 1; after the jth step they were all integer multiples of 21−j . Therefore a continuous distribution is never achieved since certain values of the scalar sample space (those not corresponding to multiples of 21−j ) cannot be reached. In spite of this, after a suﬃcient period of time, a bell-shaped (but non-Gaussian and discrete) distribution is obtained. In contrast to the LMSE closure however, the PDF does relax from its initial shape.

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4.2.2

Modiﬁed Curl

Dopazo (1979) and Janicka et al. (1979) independently proposed modiﬁcations to Curl’s model in order to avoid some of its shortcomings. Although the starting point motivating their modelling was diﬀerent, the end results were very similar indeed, largely because their approach followed the ideas behind the derivation of Curl’s model. Essentially the diﬀerence between Curl’s original model and the modiﬁed forms lies in the treatment of the dispersion of the material point properties. In discussing these diﬀerences it is useful to introduce the concept of the transition PDF. The transition PDF is deﬁned such that Tr (ϕ, Ψ|ψ)dψ is the probability that the interaction of a material point φ = ϕ with a point φ = Ψ produces ψ ≤ φ ≤ ψ + dψ and (ϕ + Ψ − ψ) ≤ φ ≤ (ϕ + Ψ − ψ) + dψ. It is clear from the deﬁnition that the transition PDF has to satisfy the following properties: • In coalescence-dispersion models the mixing process is eﬀectively represented by the pairwise interaction of material points. These material points are only allowed to mix along the line joining them in composition space and this is consistent with the notion of physical mixing. Therefore, if ψ is outside the interval (ϕ, Ψ), it follows that Tr (ϕ, Ψ|ψ)dψ=0. This implies that the interval (ϕ, Ψ) denotes a locally convex domain for ψ. • We require Tr (ϕ, Ψ|ψ) = Tr (ϕ, Ψ|ϕ + Ψ − ψ). This eﬀectively states that after pairwise interaction, the new scalar properties ψ and ϕ + Ψ − ψ are formed with the same likelihood. This condition, together with the previous requirement, ensures that the mean of the dependent scalar, φ, remains unchanged by the micromixing process. • Necessarily Tr (ϕ, Ψ|ψ) is a PDF with respect to ψ. Therefore the normalization condition follows; when the integration is performed over the domain of ψ consistent with the comments made above Tr (ϕ, Ψ|ψ)dψ = 1. ψ

In terms of the transition PDF, coalescence-dispersion models can be expressed in the general form (using a density-weighted PDF)     ˜ ∂ P (ψ) ε ρ¯ dϕ dΨP˜ (ϕ)P˜ (Ψ)Tr (ϕ, Ψ|ψ) − P˜ (ψ) , (19) ≈ ρ¯B  ∂t k ϕ

Ψ

where B is a constant which can be evaluated by multiplying equation (19) by (ψ − φ)2 , integrating over ψ space, and then comparing the result with equation (7). In terms of Curl’s original speciﬁcation, where material points/ particles coalesce and disperse with properties involving their mean value, the transition PDF takes the following form: + , Tr (ϕ, Ψ|ψ) = δ ψ − 12 (ϕ + Ψ) . (20)

[20] The joint scalar probability density function

595

In modiﬁed forms of Curl’s model the material points that interact are not constrained to mix to their common mean values. In fact the degree of mixing can vary between none and complete (up to the mean composition). If it is assumed that all composition states between these limits can be attained with equal likelihood then the transition PDF will then be a uniform distribution, see Janicka et al. (1979), over the range of the locally convex domain (ϕ, Ψ):  1  |ϕ−Ψ| ψ ∈ [ϕ, Ψ] Tr (ϕ, Ψ|ψ) = (21)  0 ψ ∈ [ϕ, Ψ]. In this case discontinuous PDFs cannot arise (in the sense that Curl’s original model produces) since all possible composition states can be reached. However it must be noted that mixing is still represented as a jump process and such closures are essentially discontinuous representations of the mixing phenomenon. Also, when applied to the isotropic homogeneous turbulence test problem with initial conditions corresponding to a PDF comprising a double δ-function distribution, the resulting ﬂatness (equation (8)) and higher even moments of the resulting PDF increase indeﬁnitely with time, whilst for a Gaussian PDF these values are constant. In the particle method analogy of the generalised coalescence-dispersion model, a random selection of a pair of particles φ∗(p) (t) and φ∗(q) (t) is followed by ‘mixing’ such that the new concentrations are given by

φ∗(p) (t + δt) = φ∗(p) (t) + 12 x φ∗(q) (t) − φ∗(q) (t)

(22) φ∗(q) (t + δt) = φ∗(q) (t) − 12 x φ∗(q) (t) − φ∗(q) (t) , where x is a uniformly distributed random number and where the number of particles selected for mixing is given by equation (18). There have been a number of studies aimed at deriving improved forms of coalescence-dispersion models. Various alternative transition probabilities have been examined, for example by Pope (1982), Kosaly (1986) and Kosaly and Givi (1987). Pope (1982) formulated an age-biased model such that continuous PDFs with ﬁnite ﬂatness and superskewness could be obtained. This was subsequently investigated by Chen and Kollmann (1991). Attempts have also been made to overcome the limitations arising from the discontinuous nature of coalescence-dispersion models and these include the reaction zone conditioning approach of Chen and Kollmann and the ordered pairing method of Norris and Pope (1991). However no signiﬁcant advantage appears to arise from these various approaches and so they will not be considered further here. Further details may be obtained from the original papers and from the review of Jones and Kakhi (1996).

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4.3

Jones

Langevin Models

Nonlinear integral models (such as coalescence-dispersion models) can be interpreted stochastically as Poisson processes which form the basis of particle interaction (jump) models, see Pope (1985). Random diﬀusion processes can also be used to simulate the behaviour of micromixing, and these form the basis of Langevin models. Pope (1985) proposed a model for the mixing term using a Langevin equation: 1 δt φ2 2 δφ(t) = φ(t + δt) − φ(t) = g [φ(t) − φ(t)] + b δWt . τ τ

(23)

The ﬁrst term on the RHS is a linear deterministic term followed by a random diﬀusive term, g and b are constants to be determined, and B is a non-negative diﬀusion coeﬃcient for this process. The term τ is a characteristic turbulent time scale and δWt is the increment of a Wiener process which is deﬁned by the properties δWt = 0 and δWt2 = δt where δWt =5Wt+δt − Wt . A function which satisﬁes these properties is Wt ≡ s (δt)1/2 N i=1 υi , where υi , i = 1, . . . , Ns , represents a set of independent standardized (i.e. zero mean, unit variance) normal random variables. The function Wt is a continuous and non-diﬀerentiable function of time. The mean is preserved in this formulation, and the requirement for the correct variance decay rate leads to a relation between g and b leaving one constant to be speciﬁed. Pope points out that from any initial condition and in the absence of any disturbing inﬂuences, the PDF relaxes to Gaussian with this model. The rate at which the PDF relaxes depends on the value of b: for b = 0, LMSE is recovered for which there is no relaxation. A drawback of the model is that the term involving b can lead to the violation of scalar bounds and, for this reason, the expression shown in equation (23) is not considered as useful for representing scalar mixing. However, for the velocity ﬁeld which is not bounded, such a Langevin model can be and has been employed.

4.4

Binomial Models

Vali˜ no and Dopazo (1990) proposed a sequential LMSE and random jump process model capable of producing a Gaussian relaxation for bounded scalars. During a time step δt, two additive subprocesses take place involving φ(n) (t): ﬁrst all Np particles modify their values according to an LMSE variation and then Np B δt/τ of the particles are randomly selected out of the Np stochastic particles (here τ is a characteristic time scale, and B is a positive dimensionless constant whose value strongly aﬀects the relaxation rate of the PDF and its value requires calibration against measurements). The remaining Np (1 − B δt/τ ) particles do not alter their values, whilst those that are

[20] The joint scalar probability density function

597

randomly sampled modify their values according to 1

φ(p) (t + δt) − φ(t) = ζ (p) (φ − φ)2 2 ,

(24)

where ζ (p) is either a standardized Gaussian random variable if φ is an unbounded scalar, or a standardized binomially distributed random variable if φ is bounded. Since the latter case is relevant to reactive scalars, we will concentrate on its representation: ζ (p) =

η − MP M P(1 − P)

.

(25)

Here η is a binomially distributed integer random variable with mean M P and variance M P(1 − P), P is a real number in [0, 1] and η can take integer values between 0 and M . To ensure boundedness, the minimum and maximum values of φ correspond to η equal to zero and M , respectively. With this requirement it can be shown that M

=

(φ − φmin ) (φmax − φ) φ 2

(26)

P

=

(φ − φmin ) (φmax − φmin )

(27)

In homogeneous turbulence φ 2 decreases with time and thus, for ﬁxed bounds, equation (26) implies that M increases. Furthermore as M → ∞, the de Moivre–Laplace theorem shows that ζ (p) tends to a normal continuous random variable; thus a distribution close to Gaussian at long times is implied. For the test case corresponding to an initial double δ-function with φmin = 0 and φmax = 1, diﬃculties arise in the early stages of mixing with regard to equation (26). Vali˜ no and Dopazo show that, if the result of equation (26) is truncated to the nearest integer (M is an integer), no relaxation of the PDF is observed until M ≥ 2. This diﬃculty was overcome by allowing for random samplings of stochastic particles around each of the initial δ-functions. As φ 2 decreases, these two samplings converge towards that originally proposed with a single M and P speciﬁcation, given by equations (26) and (27). Monte Carlo simulations of homogeneous turbulence demonstrated that the moments of the PDF evolved correctly: i.e., to within statistical error (based on ﬁnite sample size), the mean and skewness were invariant, the variance decayed exponentially and the ﬂatness tended asymptotically to 3.0 – the Gaussian value. However, the rates at which the moments evolved were strongly aﬀected by the value of B ; for B = 0.5, the PDFs evolved towards a bellshaped distribution, but with the appearance of two spikes corresponding to the initial δ-functions. With B = 2 qualitatively better PDF evolution was obtained.

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Subsequently Vali˜ no and Dopazo (1991a) combined an LMSE and a binomial diﬀusion process to obtain a modiﬁed Langevin model: 2

(1 δt 2 2 (p) ζ δφ(t) = K 1− 1 φ 2τ Random diﬀusion term δt φ 2 − 1+K 1− 2 (φ(t) − φ) , τ φ∗ Linear deterministic term '

φ (t) φ∗

(28) where φ∗ 2 is equal to φ2M whenever φ(p) (t) > 0, and is equal to φ2m for φ(p) (t) < 0; the terms φ2M and φ2m are the maximum (positive) and minimum (negative) ﬂuctuations allowed, respectively, K is a constant to be adjusted to yield the correct evolution of the PDF shape and to be consistent with the scalar variance decay rate, equation (7). It is worth noting that the binomial model described in Vali˜ no and Dopazo (1990) is sequential but is neither continuous nor diﬀerentiable; the present (non-sequential) model is continuous but non-diﬀerentiable, and is therefore a more physically realistic representation of mixing. Equation (28) is similar to the proposal of Pope (1985), namely equation (23), except that the Wiener process (which may generate unbounded scalar values) is now replaced by a standardized binomially distributed random variable, ζ (p) , and deﬁned to ensure boundedness as in the previous binomial model. As time increases φ 2 decreases, so that φ 2 /φ∗ 2 → 0, φ (t)/φ∗ → 0 and for large values of t, equation (28) asymptotically yields relaxation to a Gaussian PDF. Comparison of the PDF forms (using K = 2.1) with the DNS data of Eswaran and Pope (1988) demonstrated very good agreement. However, the sensitivity of the results to the value of K was not discussed. Binomial models have been demonstrated to work well for PDFs of single scalars with ﬁxed bounds. However, the ability of the models to handle multi-scalar PDFs is not clear. The bounds of any particular reactive scalar are not ﬁxed but depend on the values of the other scalars; each scalar property, e.g. the speciﬁc mole number of H2 , CO, . . . , has its own bounds ﬁxed by conservation principles and the allowable scalar domain is a complicated hypervolume in multi-dimensional composition space. Consequently the extension of binomial models to combustion applications, where a number of species and temperatures have to be included, is not straightforward. Ensuring satisfaction of the realizability condition, equation (6), while maintaining mean values unchanged by mixing appears problematic. However some progress in this direction has been made by Hulek and Lindstedt (1996).

[20] The joint scalar probability density function

4.5

599

Mapping Closures

In the literature dealing with mixing models, the quest for predicting Gaussian PDFs in homogeneous turbulence appears to have culminated in the proposal for mapping closures. This closure has received a considerable amount of attention over a relatively short period of time, partly because of its ability to predict (in analytical form in certain cases) the correct evolution and asymptotic behaviour of passive scalar mixing in homogeneous ﬁelds. Chen et al. (1989) addressed the problem of closure for the mixing term by constructing φ(x, t) as a time-dependent mapping of a Gaussian reference ﬁeld; the evolution of the mapping characterizes the evolution of the scalar ﬁeld for which the PDF is sought. The mapping technique is capable of handling strongly nonGaussian φ(x, t). For homogeneous turbulent scalar ﬁelds the PDF equation can be written as follows: , ∂P (ψ; t) ∂ + E D∇2 φ|ψ P (ψ; t) = 0, (29) + ∂t ∂ψ where D is the diﬀusivity and E D∇2 φ|ψ is the expectation of D∇2 φ conditioned on φ(x, t) = ψ. The cumulative distribution function (CDF) of P (ψ; t) is deﬁned as ψ ˆ t)dψ. ˆ P (ψ; (30) F (ψ; t) ≡ −∞

Equation (30) taken together with (29) leads to the evolution equation for the CDF: ∂F (ψ; t) ∂F (ψ; t) + E D∇2 φ|ψ = 0. ∂t ∂ψ

(31)

The closure establishes the following mapping relation, (Pope 1991, Kollmann 1992): φs (x, t) ≡ X[θ(z), t],

(32)

where φ(x, t) is the dependent scalar ﬁeld, θ(z) is a time-independent standardized Gaussian reference ﬁeld residing at a spatial location z, and φs (x, t) is referred to as the surrogate ﬁeld, see Pope (1991), and is eﬀectively deﬁned through equation (32). The mapping X does not determine the relation between the locations of φ(x, t) and φ(z). However, if the relation between the locations in physical and reference space is restricted to a stretching transformation uniform in physical space, we have ∂zβ = δαβ m(t), ∂xα

(33)

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where m(t) accounts for the average rescaling produced by the stretching and molecular diﬀusion, see Gao (1991b). The mapping structed as follows: the value of the CDF of a turbulent ﬁeld at (where η is the sample space of the Gaussian reference ﬁeld) is equal to the value of the standardized Gaussian CDF at θ(z) = η, F (X(η, x, t)) = FG (η),

turbulent X is conX(η, x, t) presumed (34)

implying the relation ψ = X(η, x, t). A direct consequence of the monotonicity of the CDFs is that the mapping also increases monotonically, i.e. X(η1 ) < X(η2 ) for η1 < η2 . Diﬀerentiating equation (34) with respect to η ∂F ∂X ∂FG = ∂ψ ∂η ∂η we ﬁnd

P (ψ; t) = PG

∂X ∂η

−1 .

(35)

Performing chain diﬀerentiation of equation (34) with respect to time yields ∂F ∂F ∂X + = 0. ∂t ∂ψ ∂t

(36)

This result employs the concept of time independence of FG , the reference CDF. Vali˜ no et al. (1991) consider the case of time-dependent statistics, i.e. φs (x, t) ≡ X[θ(z, t), t] (cf. equation (32)). In this case the equation for PG (η) satisﬁes B C ∂θ ∂PG (η, t) ∂ θ = η PG (η, t) . (37) =− ∂t ∂η ∂t A closure hypothesis for ∂θ ∂t θ = η was proposed which preserved the Gaussianity for the reference ﬁeld and involved a free parameter β adjusted to produce the correct variance decay rate (obtained from DNS results). For β = 0 the results of Chen et al. (1989) are recovered. However, it was also demonstrated that the shape of P (ψ; t) was largely independent of the value of β. Substitution of equation (31) into (36) coupled with the fundamental assumption, see Pope (1991) and Gao (1991b), that the unknown statistics of the turbulent ﬁeld are the same as the known statistics of the surrogate ﬁeld, leads to ∂ (38) X(η, x, t) = E D∇2 φs |ψs = X(η, x, t) . ∂t It is relatively straightforward to show that the Laplacian of the surrogate ﬁeld is 2 2 2 2 ∂θ ∂θ ∂ θ , (39) +X ∇ φs (x, t) = ∇ X(θ(z), t) = m X ∂zi ∂zi ∂zi ∂zi

[20] The joint scalar probability density function

601

where X ≡ ∂X ∂θ . The conditional expectation of equation (39) can be established provided the derivatives of the mapping are completely speciﬁed by the condition φs (x, t) = ψ. This implies, see Kollmann (1992), that the mapping must be local, i.e. the variation of the reference ﬁeld at locations z = z has no inﬂuence on X(θ(z), t). Thus, the condition φ(x, t) = ψ is the image of the condition θ(z) = η. This, taken with the results for Gaussian statistics of the reference ﬁeld, see Pope (1991), leads to the evolution equation for the mapping: B C ∂θ ∂θ ∂X η 2 =m D X − 2 X , (40) ∂t ∂zi ∂zi θ where θ2 = 1 and D is treated as spatially uniform. Since equation (29) makes use of the mixing term employing Fickian diﬀusion, this strictly implies that D cannot vary from one specie to another – thus it appears that this level of closure is constrained to an equal diﬀusivity assumption. The mapping closure described above relates to single point statistics. As in all other one∂θ ∂θ point, one-time closures, length scale information is not available; ∂z in i ∂zi equation (40) requires external information and is similar in form to a scalar dissipation. Vali˜ no and Gao (1992) suggest the use of a characteristic time k/ε, which is what is used in other mixing models. Another possibility is to model the joint scalar and its gradient, see Vali˜ no and Dopazo (1991b). In this 2 case τφ ≡ φ /D∇φ ∇φ can be obtained directly from the joint PDF. How easily the latter option can be implemented is still not clear. Gao (1991a) presented a closed form solution to equation (40). Two important features of the solution were that it preserved the boundedness of the scalar ﬁeld and also predicted relaxation to a Gaussian distribution. Pope (1991) and Vali˜ no and Dopazo (1991a) obtained analytical solutions to equation (40) for the case of P (ψ; 0) = 12 [δ(ψ + 1) + δ(ψ − 1)]. Excellent agreement was found with the relaxation of PDFs arising from DNS. Monte Carlo simulations of the mapping closure for this test case, Vali˜ no et al. (1991) and Vali˜ no and Gao (1992), led to equally good agreement. The basic assumption used to obtain equation (40) can be expressed as E ∇2 φs |φs = ψ = E ∇2 φ|φ = ψ Es E = E (∇φ)2 |φ = ψ . (41) or E (∇φs )2 |φs = ψ Gao (1991b) tested this assumption by comparing predictions of Es (φs )/Es ( 12 ), for which an analytical expression exists (using the initial double δ-function speciﬁcation) against DNS results. Excellent agreement was obtained suggesting that the underlying assumption behind equation (41) is sound. In section 4 it was pointed out that the DNS results of Eswaran and Pope (1988) predicted Gaussianity after a relatively long period of time, implying the existence of a

602

Jones

substantial interim period of non-Gaussianity. Gao (1991b) demonstrated that the evolution of Es was closely related to its initial value, which can be aﬀected by diﬀerent initial length scales, consistent with DNS results. The observed persistence of non-Gaussianity in the ﬂatness and superskewness is clearly related to the behaviour of the tails of the PDF. When near the mean, the behaviour is strongly Gaussian, while near the bounds it is not (but after substantial mixing, very little probability, i.e. area under the PDF, exists near the tails). Miller et al. (1993) used mapping closures together with other mapping relations (referred to as Johnson–Edgeworth translations) and DNS results to show that the level of agreement of the normalized conditional dissipation of the scalar at the bounds was rather poor. The deﬁciency was attributed to the inability of the closures to allow for variations in the scalar bounds as the inert mixing process evolved. The DNS results they employed for validation demonstrated that the scalar maxima and minima migrated towards the mean with time; these were used as an empirical input for calculations where the bounds were allowed to relax, and the subsequent improvement in the predictions at the scalar extremities was noted. How useful mapping closures will be for prediction methods in practical ﬂows is still an open question and the problems associated with incorporating extra scalars (inert and reactive) in a mathematically and computationally tractable manner need to be resolved. Furthermore, although such a technique yields excellent results in homogeneous turbulence, it is not clear whether it will necessarily yield better results in inhomogeneous ﬂows where the turbulent time scales can vary and are not known. The use of k/ε as the only characteristic time scale may limit its usefulness relative to simpler and cruder models previously discussed.

4.6

EMST Mixing Model

The Euclidean minimum spanning tree (EMST) mixing model formulated by Subramanian and Pope (1998) retains some of the properties of mapping closures and overcomes some of the problems related to their extension to multiple scalars. The model can also be viewed as a generalisation of the ordered pairing model of Norris and Pope (1991). It utilises the concept of ‘localness’, the satisfaction of which requires mixing models to be local in the sample space. In terms of particles it implies that only particles with adjacent properties can interact and mix; for example a particle with property φ can mix only with particles with properties φ ± δφ. Since all of the scalars considered are continuous functions of position and time, this is clearly a property of the exact scalar conservation equations. In the EMST model a Euclidean minimum spanning tree constructed in scalar space is used to deﬁne neighbouring particles and mixing is represented by a pairwise interaction of particles with adjacent properties. A straightforward application of the model to a simple test problem led to ‘stranding’ patterns in scatter plots which was eliminated

[20] The joint scalar probability density function

603

by the incorporation of an intermittency-like aspect through the introduction of a particle age property. This is then used to distinguish a subset of particles for mixing. An integral turbulence frequency ω ∝ kε essentially determines the rate at which mixing occurs. Although it complies with localness concepts, the model does not satisfy linearity and independence principles, probably a reﬂection of the fact that it seems to be impossible to satisfy all exact properties simultaneously while retaining a useful model at the single-point single-time level of closure. The model has been shown to yield encouraging results for inert scalar mixing and, in contrast to other binary interaction models, to be consistent with the ﬂame sheet approximation in diﬀusion ﬂames. More recently Xu and Pope (2000) and Tang et al. (2000) have applied the model in a joint velocity-scalar PDF formulation to obtain extremely impressive results for a piloted turbulent diﬀusion ﬂame displaying local extinction. However at the present time it is uncertain whether this arises as a consequence of the mixing model or from the relatively detailed and realistic chemical reaction mechanism used in their calculations.

4.7

Models for the Turbulent Transport of the PDF

Traditional modelling practice suggests that in many ﬂows turbulent transport can be adequately represented through a gradient transport hypothesis. For many of the applications of the PDF method to thin shear ﬂows it appears to yield acceptable results for quantities relating to gross mixing patterns (e.g. mixture fraction). Hence we can write 9 ˜ (ψ) ∂ µ ∂ P ∂ 8 t , (42) ρ¯uk |φ = ψP˜ (ψ) ≈ − ∂xk ∂xk σt ∂xk IV where σt = 0.7 and µt is given by the k-ε model. With a second moment closure being used for the velocity ﬁeld, Chen and Kollmann (1988) used the following approximation: 9 ˜ (ψ) ∂ P k ∂ 8 ∂ u − ρ¯uk |φ = ψP˜ (ψ) ≈ cs ρ¯ u , (43) ∂xk ∂xk ε k j ∂xj IV with cs = 0.3. In the past, other proposals for closing the turbulent diﬀusion term have been made. Dopazo (1975) tentatively proposed an LMSE-type approximation and such an expression can be expected to work well in nearly Gaussian cases. Janicka et al. (1978) used a modelled transport equation for uk |φ = ψ (i.e. Reynolds averaging and for a single conserved scalar) – though the modelling involved a formidable task. Certain terms involving the pressure were

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Jones

neglected, through ad hoc assumptions or order of magnitude analysis, and conditional Reynolds stresses were modelled using a gradient transport hypothesis. However, these methods have now been largely replaced by the gradient models referred to above.

5

Stochastic Methods for Solving the Scalar PDF Equation

Because of the large number of independent variables involved in the PDF equation, ﬁnite diﬀerence techniques are not feasible and stochastic (Monte Carlo) methods have to be resorted to. In the methods to be described it will be assumed that turbulent transport of the PDF is represented by the gradient transport model as discussed in section 4.7 but no restriction will be placed on the mixing models to be used, other than that a particle method can be constructed to represent it. In the present context it is important to note that the Monte Carlo solution method is not a statistical simulation of turbulence but rather of the closed form of the PDF evolution equation. Stochastic particle methods can be formulated in either an Eulerian or Lagrangian framework and both approaches will be outlined here.

5.1

Eulerian Particle Method

In the Eulerian particle method the PDF is represented by an ensemble of N particles located at each node point with each particle having N properties corresponding to the random variables of the PDF. To utilise these particles the closed form of the PDF equation is ﬁrst discretised using an explicit, i.e forward time, approximately factored ﬁnite diﬀerence approximation which may be written as P˜ (ψ; x, t + ∆t) = (I + Ω∆t) (I + Ξ∆t) (I + T∆t) P˜ (ψ; x, t), Source Mixing Transport

(44)

where I is the unit matrix and where T, Ξ and Ω are diﬀerence coeﬃcient matrices associated with transport (convection and diﬀusion), micromixing and chemical reaction. Equation (44) can now be solved via a sequence of fractional steps. First the eﬀect of transport is calculated from the relation P˜ (ψ; x, t + ∆tT ) = (I + T∆t) P˜ (ψ; x, t), where ∆tT is a notional time step indicating that transport has been applied. A similar procedure is applied, ﬁrst to represent mixing and then to account for chemical reaction. On completion of these operations an approximation to P˜ (ψ; x, t + ∆t) is obtained. Obviously the discretisation must be stable and consistent and the time step must be suﬃciently small for the factorisation errors to be negligible.

[20] The joint scalar probability density function

605

In general this requires the time step to be such that the Courant and diﬀusion numbers are small, (compared with unity). The transport contribution to equation (44) at position ‘p’ can be written as Aα (x, tn )Pα (ψ; x, tn ), (45) P˜p (ψ; x, t + ∆tT ) = δt α

where the summation is over the points involved in the discretisation. To simulate transport the ensemble of particles representing the PDF at position ‘p’ and t + ∆tT is now obtained by selecting at random δtAα N particles from the ensemble at each node α involved in the discretisation. Mixing is then simulated by allowing the particles within each ensemble to mix according to the speciﬁed model, and chemical reaction is described by allowing each particle to react according to the chemical mechanism being used. A more detailed description of the method is provided by Pope (1981), Kakhi (1994) and Jones and Kakhi (1996). The method oﬀers the advantage that, in a single step, the particles are exchanged only between neighbouring grid nodes to form a new ensemble. Hence the numerical implementation is fairly straightforward since the stochastic particle locations can be described by integer pointers. However it also suﬀers from a number of major disadvantages which severely restricts its use. First the diﬀerence coeﬃcients Aα , which depend on the velocity, eddy viscosity and density, etc. must be positive. In practice this means that the method can only be used with orthogonal coordinate systems; it does not appear possible to construct a consistent and stable approximation with positive coeﬃcients for convection-diﬀusion problems using non-orthogonal coordinates. For orthogonal coordinates, upwind or hybrid schemes are required and, as is well known, these can give rise to excessive numerical diﬀusion, particularly in recirculating ﬂows. There is also a slightly more subtle point related to the minimum number of particles which must be used. For example, if at any node the inequality max (δtAα N ) < 1,

(46)

is violated then the simulated PDF at that node will be completely unaﬀected by transport processes! Furthermore proper account of convection and diﬀusion requires that the number of particles at each node must satisfy, for all Aα = 0, (δtAα N ) > m

(47)

where m is a number at least greater than 1. Since the maximum value of δt is determined by stability considerations and the Aα (x, t) depend on the mesh dimensions and ﬂow properties, equation (47) eﬀectively determines the minimum number of particles which may be used at each node point. In recirculating ﬂows particularly, the number of particles required can be prohibitively

606

Jones

large. For this reason the method is eﬀectively restricted to simple jet ﬂames for which it has been used extensively; for examples see Jones and Kollmann (1987), Chen and Kollmann (1988), Chen et al. (1989), Sion and Chen (1992) Vervisch (1991, 1992), Kakhi (1994), Jones and Kakhi (1997, 1998).

5.2

Lagrangian Particle Method

In Lagrangian particle methods the PDF is also represented by an ensemble of stochastic particles but in contrast they are allowed to move through the solution domain and change their positions in a continuous manner. As a consequence the joint PDF at position x is given by P (ψ, x, t) =

N 1 ω (p) x(p) − x), δ(φ(p) − ψ)δ(ˆ N ω ¯

(48)

p=1

where N is the total number of particles, ω (p) is the weight of pth particle, x ˆ(p) is the position of pth particle and the mean weight is deﬁned as N 1 (p) ω ¯= ω . N

(49)

p=1

The discrete diﬀerence approximation is then simulated stochastically by allowing particle properties to evolve in a speciﬁed manner with their positions being tracked as they move through the solution domain. Each particle obeys certain equations which govern its transport. The particles move in physical space by convection due to the mean ﬂow, and by diﬀusion due to the molecular and eddy diﬀusivity. 5.2.1

Transport

In Lagrangian PDF formulations convection is represented deterministically by translation of particle positions, while gradient-diﬀusion may be represented by a classical random walk in physical space. Convection and gradient transport diﬀusion processes can be described by the following stochastic diﬀerential equation dˆ x(p) (t) = D(ˆ x(p) (t), t) δt + E(ˆ x(p) (t), t) dW,

(50)

where x ˆ is the position of a stochastic particle, D and E are the drift and diﬀusion coeﬃcient, respectively, and W is an isotropic Wiener process. The Wiener process (Chandrasekhar 1954, Cox and Miller 1965) is a very useful and widely used stochastic process for representing the random forces

[20] The joint scalar probability density function

607

corresponding to a diﬀusion process. The Wiener process involves the random variable ξi ; and is deﬁned by W(tn ) ≡ δt

1/2

n

ξi ,

(51)

i=1

where tn = n(δt) and ξi is the sequence of n independent normal random variables with zero mean and a unit variance. The increment of this process, written as ∆W(tn ) = W(tn ) − W(tn−1 ),

(52)

has the following properties: ∆W(tn ) = δt1/2 ξn = 0;

(53)

∆W2 (tn ) = δtξn2 = δt.

(54)

That is, the increment of the Wiener process is a Gaussian random variable with zero mean and a variance equal to the time step δt. Equations (51) and (52) show that the Wiener process is a continuous process, since ∆W(tn ) → 0, that is ∆W(tn ) → ∆W(tn−1 ), as δt goes to zero. It also shows that the Wiener process is not diﬀerentiable because, as δt becomes inﬁnitely small, the derivative of the process, ∆W(tn )/δt, becomes inﬁnite. Comparing the Fokker–Planck equation corresponding to equation (50) with the PDF evolution equation, we ﬁnd the drift and diﬀusion coeﬃcients become : µt (55) E≡ 2 ρ¯ 1 ˜ + ∇µt . D≡u ρ¯

(56)

Hence the stochastic diﬀerential equation which represents the spatial transport of the PDF is µt 1/2 1 p ˜ + ∇µt δt + 2δt dW, (57) dˆ x (t) = u ρ¯ ρ¯ and hence the spatial transport during interval time δt is accounted for by moving the stochastic particles according to µt 1/2 1 (p) (p) ˜ + ∇µt δt + 2δt ˆ (t) + u dξ, (58) x ˆ (t + δt) = x ρ¯ ρ¯ where ξ is a standard normal random vector and x ˆ(p) denotes the position of pth particle.

608 5.2.2

Jones Mixing and Reaction

The modelled PDF equation written in Lagrangian form, (3), can be discretised using an explicit forward-time, approximately-factored ﬁnite-diﬀerence scheme similar to that adopted in the Eulerian method. This is then used to simulate the inﬂuence of mixing and reaction on the evolution of the PDF by ﬁrst sorting particles into their respective ‘control volumes’ and then allowing the resulting ensemble of particles in each cell to mix according to the speciﬁed mixing model. Each particle is then allowed to react via the selected reaction mechanism. The variable particle weights introduced in equation (48) allow the number of particles in each cell to be maintained roughly constant by the splitting into two or removal of particles as appropriate. This removal and splitting appears to be essential, particularly if the method is to be applied to two- and three-dimensional ﬂows with recirculation, and must be conducted in such a manner that the statistical properties remain unchanged. For mixing models based upon binary interactions, e.g. coalescence-dispersion models, and some others, the particle representation must also be modiﬁed to account for the diﬀering particle weights – for further details see Wouters (1998). The Lagrangian particle method is not restricted to any particular coordinate system and is thus quite general, and is relatively free from numerical diﬀusive type errors. However the implementation of the method in a computationally eﬃcient manner, particularly for complex ﬂows, is non-trivial and to date it does not appear to have been widely used for the scalar PDF equation methods described here – but see Jones and Weerasinghe (2000), Weerasinghe (2000) and Lindstedt et al. (2000). It has been applied most extensively in the context of joint velocity-scalar evolution equation formulations.

5.3

Stochastic Field Methods

The recently proposed stochastic ﬁeld method of Vali˜ no (1998) and Vali˜ no et al. (2000) represents an extremely promising method for solving the joint scalar PDF evolution equation. In this formulation the PDF is represented by a set of stochastic ﬁelds rather than particles. The method is purely Eulerian. The evolution of the stochastic ﬁelds are obtained from solution of standard conservation-type partial diﬀerential equations to which a stochastic source term is added. The resulting ﬁelds are continuous and diﬀerentiable in space and continuous but non-diﬀerentiable in time. As a consequence the ﬁeld equations can be solved using the relatively standard techniques found in most current CFD codes. The convergence properties of the method also appear to be signiﬁcantly better than most particle-based methods. So far the method has been utilised only in conjunction with the LSME mixing model and further investigation is needed into its use with alternative mixing models.

[20] The joint scalar probability density function

609

The ease with which the method can be incorporated into any Eulerian CFD code makes it very attractive indeed.

5.4

Chemical Reaction

The chemical source terms appear in exact and closed form in the PDF equation and as a consequence this process can be simulated deterministically. In the solution methods described above this is implemented by allowing each particle or ﬁeld to react over a time interval δt according to the speciﬁed reaction mechanism. This in turn requires solution of a system of ordinary diﬀerential equations (ODEs): d (p) ω˙ α (φ) φ = dt α ρ(φ)

where

α = 1, . . . , N.

(59)

For heat releasing reactions the resulting system of ODEs is stiﬀ and their solution is non-trivial, see Hindmarsh (1980). Furthermore a PDF equation solution will involve a large number of particles, typically of order 106 , and for each of these the system of equations (59) will have to be integrated over a time interval corresponding to several mean ﬂow through times. Consequently limitations in computer speed and memory preclude the implementation of detailed kinetic mechanisms, with the possible exception of thin shear layers, i.e. simple jet ﬂames, and reduced mechanisms represent the only practical approach. A number of techniques are available for reducing detailed kinetic mechanisms and these include traditional methods involving combinations of steady state and partial equilibrium assumptions to determine ‘minor’ species as a function of ‘major’ species (see, for example, Peters and Rogg 1993, Sung et al. 1998), computational singular perturbation methods (Lam and Goussis 1988, Massias et al. 1999) and intrinsic low-dimensional manifold techniques (Maas and Pope 1992, Blasenbrey and Maas 2000). In all these approaches the number of independent species, for which equations of the type (59) have to be solved, is reduced to a small number, typically between 4 and 12. Generally speaking the more species retained, the wider is the range of applicability of the reduced mechanism. However a detailed review of reduced chemical mechanisms is beyond the scope of the present chapter. A further strategy for reducing computational costs is to tabulate the chemistry. A simple way of doing this is to calculate the integrals of the reaction rates for a speciﬁed time interval for all possible chemical states and then to store the results in a ‘look-up’ table, see Chen et al. (1989). The scalar bounds (these determine the allowable scalar space) for the interpolation tables are constructed using mass conservation principles and the assumption of constant carbon-to-hydrogen (C/H) and oxygen-to-nitrogen (O/N) atom ratios (see, for example, Chen et al. 1989, Sion and Chen 1992), consistent with the equal diﬀusivity assumption implicit in the described turbulence and mixing

610

Jones

models. During Monte Carlo simulations, the changes of the scalar properties as a result of the reaction are obtained from multilinear interpolation based on the previously generated look-up table, thereby replacing the repeated time integration of the stiﬀ ODEs. This strategy is computationally eﬃcient for a small number of species, particularly if only major stable species are involved in the table. However the overriding constraint becomes the size of the ‘lookup’ table. As the complexity of the reduced mechanism increases, the number of independent scalars required to specify the thermo-chemistry increases. For interpolation tables including more than six scalars, the demands on memory can become quite unmanageable if a suﬃciently ﬁne resolution of the chemistry is required; coarse grids may lead to solutions casting doubts on the accuracy of the results, see Sion and Chen (1992). A further limitation of the approach is that, in any ﬂame calculation, only a small part of the allowable space will be accessed. This accessible part of the space, though inﬂuenced to some degree by the mixing model, is essentially determined by the chemical reaction mechanism; for any given inlet and initial conditions there will be no path to certain parts of the allowable space. One extremely attractive method for overcoming the shortcomings of the tabulation method described above is the in situ adaptive tabulation method (ISAT) formulated by Pope (1997). Here the chemistry is tabulated on the ﬂy using a binary tree and an adaptive technique is used to maintain tabulation/interpolation errors below a speciﬁed tolerance. The method can result in up to three orders of magnitude reduction in computer time compared with direct integration of the equations (59) and with this approach it is possible to incorporate quite detailed chemistry into the PDF equation method. Recently Xu and Pope (2000) and Tang et al. (2000) have, with impressive results, utilised the reduced mechanism of Sung et al. (1998), which involves 16 species and 12 reaction steps, in a joint velocity-scalar PDF equation computation of a series of piloted methane-air jet ﬂames.

6

Applications

In order to illustrate the PDF equation approach, some results taken from Jones and Kakhi (1998) and Jones and Prasetyo (1996) will be presented. In both cases the combustion mechanism applied was the four-step global scheme of Jones and Lindstedt (1988). With the assumption of equal diﬀusivities this involves four independent scalars. The chemistry was tabulated and a stochastic particle method used to solve the PDF equation. The cases considered correspond to a turbulent non-premixed ﬂame involving a jet of propane burning in stagnant air surrounds, measured by Godoy (1982), and an opposed jet counterﬂow conﬁguration. The turbulent counter ﬂow ﬂame has been the subject of extensive experimental studies and premixed, partially-premixed and non-premixed cases have been measured, see for example, Mastorakos (1993),

[20] The joint scalar probability density function

611

Mastorakos et al. (1993), Abbas et al. (1992), Kostiuk et al. (1993a, b), and Mounaim-Rousselle and Gokalp (1993). In the present case the calculation method has been applied to the conﬁguration of Mastorakos et al. It consists of two opposed pipes of diameter 25.4mm, the exits of which are separated by 20mm. These are surrounded by concentric pipes of diameter 51.8mm. The results of two cases are presented here: a premixed ﬂame resulting from the burning of opposed jets of a methane-air mixture, and the ﬂame arising from opposed jets one of which comprised a methane-air mixture and the other, pure air. Further details of the calculation methods, etc., may be obtained from the original papers.

6.1

Propane-Air Jet Diﬀusion Flames

The calculated and measured radial proﬁles for the propane-air ﬂame are shown in ﬁgures 1 to 3. The general features of the ﬂame are quite well reproduced by the calculation and the measured and predicted proﬁles of mixture fraction and C3 H8 are in close accord. The level of agreement displayed in the proﬁles of CO2 and O2 is also reasonable though there are some discrepancies. The measurements indicate that O2 does not penetrate to the centreline at x/D = 20 and 40 whereas the calculations show small but appreciable levels and the maxima in the predicted CO2 proﬁles occur slightly further from the centreline compared with the measurements. However, ﬁgure 2 displays a signiﬁcant degree of over-prediction in the levels of CO. The primary reason for this is probably associated with the chemical reaction mechanism being used. This has been shown previously to yield accurate results for laminar ﬂames but in this case accurate transport properties were used. It now appears that the mechanism is unduly sensitive to the inﬂuence of diﬀusivity under fuel rich conditions compared to detailed reaction schemes. This, combined with the equal diﬀusivity assumption implicit in the mixing models, results in excessively high CO concentrations. A striking observation however is that the two mixing models, LMSE and modiﬁed Curl, yield results which are practically indistinguishable.

6.2

Counterﬂow Flames

The counterﬂow conﬁguration considered is illustrated in ﬁgure 4. A steady ﬂow solution of the non-reacting methane-air mixture was used to provide initial conditions for the burning PDF calculation and the ﬂame was then ignited by setting the mixture in the vicinity of the stagnation region to fully burnt conditions. The solution was then marched suﬃciently forward in time for steady ﬂow conditions to prevail. Computations of the ﬂame were carried out using both the coalescence-dispersion model and the LMSE closure. However only the coalescence-dispersion generated plausible results and the ﬂame extinguished when the LMSE model was used.

612

Jones

Figure 1: Radial proﬁles of the measurements, Godoy (1982), and predictions for ξ and C3 H8 in a propane-air jet diﬀusion ﬂame issuing into stagnant surrounds. The k-ε model and the scheme of Jones and Lindstedt (1988) are employed. Predictions using the LMSE and coalescence-dispersion closures are shown; Re ≈ 24000. The predicted and measured centreline proﬁles of the major species are shown in ﬁgures 5 to 7 for the two premixed jets, φ = 0.9, and in ﬁgures 8 to 10 for the single premixed jet, φ = 1.3. For both ﬂames, the measured proﬁles of CH4 are reproduced to a good accuracy and, in conformity with the measurements, the predicted methane concentrations reduce to zero in the air

[20] The joint scalar probability density function

613

Figure 2: Radial proﬁles of the measurements, Godoy (1982), and predictions for CO and CO2 in a propane-air jet diﬀusion ﬂame issuing into stagnant surrounds. The k-ε model and the scheme of Jones and Lindstedt (1988) are employed. Predictions using the LMSE and coalescence-dispersion closures are shown; Re ≈ 24000. stream for the single premixed jet and to zero in the region between the two ﬂames of the two premixed jets. In the latter case the minimum O2 concentrations in the burning region, ﬁgure 6, are also well reproduced. This behaviour is consistent with the observed complete combustion and suggests that the fuel breakdown has been properly

614

Jones

Figure 3: Radial proﬁles of the measurements, Godoy (1982), and predictions for O2 and N2 in a propane-air jet diﬀusion ﬂame issuing into stagnant surrounds. The k-ε model and the scheme of Jones and Lindstedt (1988) are employed. Predictions using the LMSE and coalescence-dispersion closures are shown; Re ≈ 24000. simulated. A reasonable level of agreement is also evident in the proﬁles of CO2 , see ﬁgures 7 and 9, where the proﬁle shapes and maximum values are reproduced to an acceptable accuracy. However in the case of CO, ﬁgure 10, the level of agreement with measured values is less satisfactory. For the single premixed jet the predicted maximum CO concentration exceeds by around 30%

[20] The joint scalar probability density function

615

Hp stream 2

H

z r

r

Hp

stream 1

D Do

Figure 4: Counterﬂow jet.

0.10

Mean CH4 mole fraction

Mastorakos Coalescence−Dispersion

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 5: Counter ﬂow ﬂame: mean methane mole fraction proﬁle.

616

Jones

0.50 Mastorakos Coalescence−Dispersion

Mean O2 mole fraction

0.40

0.30

0.20

0.10

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 6: Counter ﬂow ﬂame: mean oxygen mole fraction proﬁle.

0.50 Mastorakos Coalescence−Dispersion

Mean CO2 mole fraction (x 5)

0.40

0.30

0.20

0.10

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 7: Counter Flow Flame: mean carbon dioxide mole fraction proﬁle.

[20] The joint scalar probability density function

617

0.20

Mean CH4 mole fraction

Mastorakos Coalescence−Dispersion

0.10

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 8: Counter ﬂow ﬂame: mean methane mole fraction proﬁle.

0.50 Mastorakos Coalescence−Dispersion

Mean CO2 mole fraction (x 5)

0.40

0.30

0.20

0.10

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 9: Counter ﬂow ﬂame: mean carbon dioxide mole fraction proﬁle.

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0.20

Mean CO mole fraction (x 5)

Mastorakos Coalescence−Dispersion

0.10

0.00 0.0

5.0

10.0 z (mm)

15.0

20.0

Figure 10: Mean carbon monoxide mole fraction proﬁle. the measured value of approximately 2.8%. As was the case for the propane-air ﬂame the main reason for this is probably related to the limitations associated with the reaction mechanism and its interaction with the mixing model.

7

Concluding Remarks

In the preceding sections the joint scalar PDF evolution equation method for calculating the properties of combusting ﬂows was reviewed. Attention was restricted to the equation for the joint PDF of the set of scalars (at the singlepoint, single-time level) describing reacting ﬂows. This constitutes about the simplest level of closure in terms of the PDF formulation applied to combusting ﬂows, and has the merit that chemical source (reaction rate) terms appear in closed form. In the method, the mean velocity and turbulence characteristics are obtained from a conventional turbulence model approach, and accurate mean velocity and turbulence ﬁelds are a clear prerequisite for success. Also, if the method is to be applied successfully to turbulent combustion, then an accurate mechanism for describing chemical reaction must be provided. As in all statistical approaches, the PDF transport equation is unclosed and approximations must be provided for the unknown terms. These comprise terms representing turbulent transport in physical space and micromixing in scalar space. For the former, a simple gradient transport model will probably

[20] The joint scalar probability density function

619

suﬃce under many circumstances, provided of course that the mean velocity ﬁeld is accurately reproduced by the turbulence model. Micromixing on the other hand presents a closure problem distinctly diﬀerent from that arising in conventional moment closures. In fact the closure of the micromixing term poses a serious diﬃculty and, together with the chemical reaction mechanism, is central to turbulence-chemistry interactions primarily because it involves molecular diﬀusivities and scalar gradients. These scale with the dissipative motions in turbulent ﬂow and this is precisely where combustion occurs. The most widely used classes of mixing models are those based on the binary interaction of ﬂuid parcels (coalescence-dispersion and EMST closures) and on linear mean square estimation (LMSE). While these are all capable of reproducing measured statistics, including variance decay rates, etc., they all have limitations in terms of their ability to represent PDF evolution. For example, in homogeneous isotropic turbulence they are incapable of reproducing the observed relaxation of an arbitrary initial (passive) scalar ﬁeld to a Gaussian PDF. This deﬁciency has motivated the formulation of binomial Langevin and mapping closures, both of which allow a relaxation to a Gaussian PDF under the appropriate conditions. However initial eﬀorts have concentrated on the evolution of a single, usually passive, scalar and the extension of these improved models to the case of several reactive scalars appears diﬃcult and further eﬀort is needed. Also, in the case of reactive scalars (species mass fractions) the PDF is likely to be far from Gaussian – because the mass fractions must lie in [0, 1] and because the PDF will be strongly aﬀected by reaction – and under these circumstances the ability of mixing closures to describe relaxation to a Gaussian PDF, although desirable, may not be a primary factor in accurately predicting turbulent ﬂame properties. Also the deviation from Gaussian in homogeneous isotropic turbulence may not be large, particularly with binary interaction closures: the skewness, ﬂatness and superskewness, which are often used to provide a measure of PDF shape, emphasise the behaviour far from the mean where probabilities are inevitably small. A potentially more serious limitation may arise from the fact that the majority of current mixing models involve use of a single time scale, k/ε, which clearly implies the ratio of mechanical to scalar turbulence time scale is constant. As is well known, this assumption is of limited generality. As an alternative the solution of an equation for the joint PDF of the scalar and its gradient has often been suggested and this alleviates the problem of evaluating turbulent scalar time scales. However for a set of reactive scalars, the increase in complexity would then be considerable indeed. An interim step would be to introduce a single scalar time scale based on the solution of a modelled equation for the scalar dissipation rate, such as that formulated by Jones and Musonge (1988). The precise deﬁnition of the scalar would have to be chosen with care in each case but for non-premixed ﬂames the mixture fraction could be used as a quantity representative of scalar mixing. A further implication of the use of an integral time scale is that the models are based on the notion of

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an equilibrium turbulence spectrum whereby ‘scalar turbulence’ is created in the large scale motions, transferred through an inertial subrange to the ﬁne scales where diﬀusivity acts to smooth out the scalar ﬂuctuations, i.e. where scalar dissipation occurs. Thus, at least for an inert ﬂow, the rate of transfer of ‘scalar turbulence’ through the inertial subrange will equal the scalar dissipation rate. The mixing models in current use are in fact global models which represent the rate of transfer through the inertial subrange rather than the mixing process itself; they include no direct information on the molecular diffusion processes that occur within the ﬁne scale dissipative scalar ﬂuctuations; amongst the assumptions implied is that of equal diﬀusivities. On an instantaneous basis, burning is likely to occur in the ﬁne scales of turbulent motion. If burning is in the laminar ﬂamelet regime then (instantaneous) ﬂame thicknesses will be smaller than the smallest scale dissipative ﬂuctuations, i.e. the ﬂame thickness will be smaller than the Kolmogorov length scale. Even if the conditions for ﬂamelet burning are not satisﬁed then the fast reaction rates associated with heat release will ensure that burning takes place in layers which are thin compared with the (inertial) length scales representative of energy containing turbulent motions. In both situations molecular diﬀusion processes are likely to exert an important inﬂuence on burning. Now, species with low molar mass, such as H and H2 , diﬀuse at a much greater rate than others and this is likely to have important consequences. In hydrocarbonair ﬂames H and H2 are, essentially, produced and consumed in the burning region, where they have maximum concentration, and are intimately linked to OH and CO formation; high levels of H will give rise to high levels of OH. If no account is taken of diﬀusion processes in devising models for micromixing and the assumption of equal molecular diﬀusivities is consequently implicitly invoked, then the predicted implied instantaneous levels of H and H2 are likely to be higher than those observed. In reality the higher diﬀusivities will result in H and H2 diﬀusing away (on an instantaneous basis) from the burning region at a greater rate than other species with a consequence that maximum H, H2 and OH concentrations will be reduced and this will impact on CO levels. Some indication of the magnitude of this eﬀect can be obtained from laminar ﬂame computations. Recently Hasko (2000) has used GRI-Mech 2.11 to compute a range of laminar counterﬂow ﬂames using alternatively the equal diﬀusivity assumption and accurate transport properties. With the equal diﬀusivity assumption the computed mass fractions of H and OH were too high by a factor of 2 to 3 and 1.4 to 2 respectively, depending on the strain rate. The eﬀect on CO levels was less pronounced, with the errors being typically around 10%. If the above description of turbulent burning is correct then it is important to note that the inﬂuence of molecular transport properties on micromixing and burning will be present at all Reynolds numbers. If the eﬀects are signiﬁcant what will be needed is a mixing model which provides some description of the ﬁne scale mixing processes in the presence of reaction. However, exactly how this is to be constructed is at present unclear though the linear-eddy model

[20] The joint scalar probability density function

621

of Kerstein (1991) may provide the basis for doing so. The results of direct numerical simulations of turbulent ﬂames, which are becoming available, may also provide guidance here. In spite of the above remarks the PDF equation method incorporating the modiﬁed coalescence-dispersion model and a simple global reaction mechanism has been demonstrated to lead to results in general agreement with measured mixture fraction, fuel and CO2 proﬁles for both jet diﬀusion ﬂames and counterﬂow premixed and partially premixed turbulent ﬂames. In cases where burning is predicted, the LMSE and a modiﬁed coalescence-dispersion closure produce practically identical results. However for the counterﬂow ﬂames it was not possible to obtain a steady burning solution with the LMSE closure and the implication is that the model predicts too low a value of critical strain rate above which extinction occurs. The high predicted levels of CO compared with measured values in all the ﬂames considered are almost certainly a consequence of limitations in the global reaction mechanism used, particularly under fuel rich conditions. The recent results of Lindstedt et al. (2000) suggest that better chemistry will bring signiﬁcant improvement in this regard.

Acknowledgements It is a pleasure to acknowledge my indebtedness to Dr Maziar Kakhi, Mr Yudi Prasetyo and Dr Rohitha Weerasinghe for their contributions to this work.

References Abbas, N.S.T., Lockwood, F.C., Hussein, A.M.M., and Hassan, M.A. (1992). A study of extinction and NOx in stretched premixed ﬂames. J. Institute of Energy 65, 77. Blasenbrey, T., and Maas, U. (2000). ILDMs of Higher Hydrocarbons and the Hierarchy of Chemical Kinetics. In Proc. Combustion Institute 28, 1623–1630. Chandrasekhar, S. (1954). Stochastic problems in physics and astronomy. In N. Wax (ed.), Selected Papers on Noise and Stochastic Processes. Dover Publications, Inc., New York. Chen, H., Chen, S., and Kraichnan, R. (1989). Probability distribution of a stochastically advected scalar ﬁeld. Physical Review Letters 63(24), 2657–2660. Chen, J.-Y., and Kollmann, W. (1988). PDF modelling of chemical non-equilibrium eﬀects in turbulent non-premixed hydrocarbon ﬂames. In Proc. Combustion Institute 22, 645–653. Chen, J.-Y., and Kollmann, W. (1991). Mixing models for turbulent ﬂows with exothermic reactions. In Turbulent Shear Flows 7, Springer-Verlag, 277–292. Chen, J.-Y., Kollmann, W., and Dibble, R.W. (1989). PDF modelling of turbulent non-premixed methane jet ﬂames. Combustion Science and Technology 64, 315– 346. Cox, D.R., and Miller, H.D. (1965). The Theory of Stochastic Processes. Methuen.

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Curl, R. (1963). Dispersed phase mixing: 1. Theory and eﬀects in simple reactors. A.I.Ch.E. Journal 9(2), 175–181. Dopazo, C. (1973). Non-isothermal turbulent reactive ﬂows: stochastic approaches. PhD thesis, State University of New York at Stony Brook. Dopazo, C. (1975). Probability density function approach for a turbulent axisymmetric heated jet: centreline evolution. Physics of Fluids 18(4), 397–404. Dopazo, C. (1979). Relaxation of initial probability density functions in the turbulent convection of scalar ﬁelds. Physics of Fluids 22(1), 20–30. Dopazo, C., and O’Brien, E.E. (1974). An approach to the autoignition of a turbulent mixture. Acta Astronautica 1, 1239–1266. Dopazo, C., and O’Brien, E.E. (1976). Statistical treatment of non-isothermal chemical reactions in turbulence. Combustion Science and Technology 13, 99–122. Eswaran, V., and Pope, S.B. (1988). Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids A 31(3), 506–520. Flagan, R.C., and Appleton, J.P. (1974). A stochastic model of turbulent mixing with chemical reaction: nitric oxide formation in a plug-ﬂow burner. Combustion and Flame 23, 249–267. Gao, F. (1991a, April). An analytical solution for the scalar probability density function in homogeneous turbulence. Physics of Fluids A 3(4), 511–513. Gao, F. (1991b, October). Mapping closure and non-Gaussianity of the scalar probability density function in isotropic turbulence. Physics of Fluids A 3(10), 2438– 2444. Godoy, S. (1982). Turbulent diﬀusion ﬂames. PhD thesis, University of London. Hasko, S.M. (2000). Private Communication. Hindmarsh, A.C. (1980). LSODE and LSODI, two new initial value ordinary diﬀerential equation solvers. ACM SIGNUM Newsletter 15(4). Hulek, T., and Lindstedt, R.P. (1996). Computations of steady-state and transient premixed turbulent ﬂames using PDF methods. Combustion and Flame 104(4), 481–504. Janicka, J., Kolbe, W., and Kollmann, W. (1978). The solution of a PDF transport equation for turbulent diﬀusion ﬂames. In Proc. Heat Transfer and Fluid Mechanics Institute, Stanford University Press, 296–312. Janicka, J., Kolbe, W., and Kollmann, W. (1979). Closure of the equations for the probability density function of turbulent scalar ﬁelds. Journal of Non-Equilibrium Thermodynamics 4, 47–66. Jones, W.P., and Kakhi, M. (1996). Mathematical modelling of turbulent ﬂames. In Unsteady Combustion, M.H.F. Culick and J.H. Whitelaw (eds.), Kluwer Academic Publishers, 411–491. Jones, W.P., and Kakhi. M. (1997). Application of the transported PDF approach to hydrocarbon turbulent jet diﬀusion ﬂames. Combustion Science and Technology 129, 393–430. Jones, W.P., and Kakhi, M. (1998). PDF modelling of ﬁnite-rate chemistry eﬀects in turbulent non-premixed jet ﬂames. Combustion and Flame 115, 210–229.

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Jones, W., and Kollmann, W. (1987). Multi-scalar PDF transport equations for turbulent diﬀusion ﬂames. In F. Durst, B. Launder, J. Lumley, F. Schmidt, and J. Whitelaw (eds.), Turbulent Shear Flows 5, Springer-Verlag, 296–309. Jones, W.P., and Lindstedt, R.P. (1988). Global reaction schemes for hydrocarbon combustion. Combustion and Flame 73, 233–249. Jones, W.P., and Musonge, P. (1988). Closure of the Reynolds stress and scalar ﬂux equations. Physics of Fluids 31(12), 3589–3603. Jones, W.P., and Prasetyo, Y. (1996). Probability density function modelling of premixed turbulent opposed jet ﬂames. In Proc. Combustion Institute 26, 275–282. Jones, W.P., and Weerasinghe, W.M.S.R. (2000). Probability density function modelling of an axisymmetric combustion chamber. In Advances in Turbulence VII, C. Dopazo (ed.), CIMNE, Barcelona, 497–500. Kakhi, M. (1994). The transported probability density function approach for predicting turbulent combusting ﬂows. PhD thesis, University of London. Kerstein, A.R. (1991). Linear-eddy modelling of turbulent transport. Part 6. Microstructure of diﬀusive scalar mixing fields. Journal of Fluid Mechanics 231, 361–394. Kollmann, W. (1989). PDF transport equations for chemically reacting ﬂows. In R. Borghi and S. Murthy (eds.), Turbulent Reactive Flows, Spinger-Verlag, 715–730. Kollmann, W. (1990). The PDF approach to turbulent ﬂow. Theoretical and Computational Fluid Dynamics 1, 249–285. Kollmann, W. (1992). PDF transport modelling. Modelling of Combustion and Turbulence, Von Karman Institute for Fluid Dynamics Lecture Series. Kosaly, G. (1986). Theoretical remarks on a phenomenological model of turbulent mixing. Combustion Science and Technology 49, 227–234. Kosaly, G., and Givi, P. (1987). Modelling of turbulent molecular mixing. Combustion and Flame 70, 101–118. Kostiuk, L.W., Bray, K.N.C., and Cheng, R.K. (1993a). Experimental study of premixed turbulent combustion in opposed stream: Part I: Non-reacting ﬂow ﬁelds. Combustion and Flame 92, 233. Kostiuk, L.W., Bray, K.N.C., and Cheng, R.K. (1993b). Experimental study of premixed turbulent combustion in opposed stream: Part II: Reacting ﬂow ﬁelds and extinction. Combustion and Flame 92, 396. Lam, S.H., and Goussis, D.A. (1988). Understanding complex chemical kinetics with computational singular perturbation. In Proc. Combustion Institute 22, 931–941. Lindstedt, R.P., Louloudi, S.A., and V´ aos (2000). Joint scalar PDF modelling of pollutant formation in piloted turbulent jet diﬀusion flames with comprehensive chemistry. In Proc. Combustion Institute 28, 149–156. Lundgren, T.S. (1967). Distribution functions in the statistical theory of turbulence. Physics of Fluids 10(5), 969–975. Maas, U., and Pope, S.B. (1992). Simplifying chemical kinetics: intrinsic low dimensional manifolds in composition space. Combustion and Flame 88, 230–264.

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Massias, D., Diamantis, D., Mastorakos, E., and Goussis, D.A. (1999). An algorithm for the construction of global reduced mechanisms with csp data. Combustion and Flame 117, 685–708. Mastorakos, E. (1993). Turbulent combustion in opposed jet ﬂows. PhD thesis, University of London. Mastorakos, E., Taylor, A.M.K.P., and Whitelaw, J.H. (1993). Mixing in turbulent opposed jet ﬂows. In Proc. 9th Symp. Turbulent Shear Flows, Kyoto, Japan. Miller, R.S., Frankel, S.H., Madnia, C.K., and Givi, P. (1993). Johnson–Edgeworth translation for probability modelling of binary scalar mixing in turbulent ﬂows. Combustion Science and Technology 91, 21–52. Mounaim-Rousselle, Ch., and Gokalp, I. (1993). Turbulent premixed combustion in counterﬂow geometry: the inﬂuence of coﬂow. Norris, A.T., and Pope, S.B. (1991). Turbulent mixing model based on ordered pairing. Combustion and Flame 83, 27–42. O’Brien, E.E. (1980). The probability density function (PDF) approach to reacting turbulent ﬂows. In Turbulent Reacting Flows, P. Libby and F. Williams (eds.), Springer-Verlag. 185–218. Appeared under ‘Topics in Applied Physics’, vol.44. Peters, N. (2000). Turbulent Combustion. Cambridge University Press. Peters, N., and Rogg, B. (eds.) (1993). Reduced Kinetic Mechanisms for Applications in Combustion Systems. Lecture Notes in Physics M15. Springer-Verlag. Pope, S.B. (1979). The relationship between the probability approach and particle models for reaction in homogeneous turbulence. Combustion and Flame 35, 41–45. Pope, S.B. (1981). A Monte Carlo method for the PDF equations of turbulent reactive ﬂows. Combustion Science and Technology 25, 159–174. Pope, S.B. (1982). An improved mixing model. Combustion Science and Technology 28, 131–135. Pope, S.B. (1983). Consistent modelling of scalars in turbulent flows. Physics of Fluids 26, 404–408. Pope, S.B. (1985). PDF methods for turbulent reactive ﬂows. Progress in Energy and Combustion Science 11, 119–192. Pope, S.B. (1991). Mapping closures for turbulent mixing and reaction. Theoretical and Computational Fluid Dynamics 2, 255–270. Pope, S.B. (1997). Computationally eﬃcient implementation of combustion chemistry using in situ adaptive tabulation. Combustion Theory and Modelling 1(1), 41–63. Sion, M., and Chen, J.-Y. (1992). Scalar PDF modelling of turbulent non-premixed methanol-air ﬂames. Combustion Science and Technology 88, 89–114. Spielman, L.A., and Levenspiel, O. (1965). A Monte Carlo treatment for reacting and coalescing dispersed phase systems. Chemical Engineering Science 20, 247–254. Subramanian, S., and Pope, S.B. (1998). A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees. Combustion and Flame 115(4), 487– 514.

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Sung, C.J., Law, C.K., and Chen, J.-Y. (1998). An augmented reduced mechanism for methane oxidation with comprehensive global parametric validation. In Proc. Combustion Institute 27 295. Tang, Q., Xu, J and Pope, S.B. (2000). PDF calculations of local extinction and no production in piloted-jet turbulent methane/air flames. In Proc. Combustion Institute 28, 133–139. Tavoularis, S., and Corrsin, S. (1981). Experiments in nearly homogeneous turbulent shear ﬂow with a uniform mean temperature gradient. Part 1. Journal of Fluid Mechanics 104, 311–347. Vali˜ no, L. (1998). A ﬁeld Monte Carlo formulation for calculating the probability density function of a single scalar in a turbulent ﬂow. Flow, Turbulence and Combustion 60, 157–172. Vali˜ no, L., and Dopazo, C. (1990). A binomial sampling model for scalar turbulent mixing. Physics of Fluids A 2(7), 1204–1212. Vali˜ no, L., and Dopazo, C. (1991a). A binomial Langevin model for turbulent mixing. Physics of Fluids A 3(12), 3034–3037. Vali˜ no, L., and Dopazo, C. (1991b). Joint statistics of scalars and their gradients in nearly homogeneous turbulence. In Advances in Turbulence 3, A. Johansson and P. Alfredsson (eds.), Springer-Verlag, 312–323. Vali˜ no, L., and Gao, F. (1992, September). Monte Carlo implementation of a singlescalar mapping closure for diﬀusion in the presence of chemical reaction. Physics of Fluids A 4(9), 2062–2069. Vali˜ no, L., Larroya, J.C., and Cazalens, M. (2000). A ﬁeld Monte Carlo formulation for solving multiscalar probability density functions in turbulent ﬂows: Application to the study of a turbulent non-premixed methane ﬂame using detailed chemistry. submitted for publication. Vali˜ no, L., Ros, J., and Dopazo, C. (1991, September). Monte Carlo implementation and analytic solution of an inert-scalar turbulent mixing test problem using a mapping closure. Physics of Fluids A 3(9), 2195–2198. Vervisch, L. (1991). Prise en compte d’eﬀets de cin´etique chimique dans les ﬂammes de diﬀusion turbulentes par l’approche fonction densit´e de probabilit´e. PhD thesis, University of Rouen. Vervisch, L. (1992). Applications of PDF turbulent combustion models to real nonpremixed ﬂame calculations. Modelling of combustion and turbulence. Von Karman Institute Lecture Series. Warhaft, Z., and Lumley, J.L. (1978). An experimental study of the decay of temperature ﬂuctuations in grid-generated turbulence. Journal of Fluid Mechanics 88, 659–684. Weerasinghe, W.M.S.R. (2000). Application of Lagrangian probability density function approach to turbulent reacting ﬂows. PhD thesis, University of London. Wouters, H.A. (1998). Lagrangian models for turbulent reacting ﬂows. PhD thesis, Delft University of Technology. Xu, J., and Pope, S.B (2000). PDF Calculations of Turbulent Nonpremixed Flames with Local Extinction. Combustion and Flame 123(3), 281–307.

21 Joint Velocity-Scalar PDF Methods H.A. Wouters, T.W.J. Peeters and D. Roekaerts In this chapter the joint velocity-scalar PDF approach is described. This approach was mainly developed by S.B. Pope and includes from the start the complete one-point joint statistics of velocity and scalars. This is conceptually appealing because it delivers in one framework closure models for Reynolds stresses, Reynolds ﬂuxes and chemical source terms. The present text is based on the PhD thesis of H.A. Wouters (Wouters 1998), which can be consulted for further details and other applications. The outline of this chapter is as follows. First the exact transport equation for the velocity-scalar PDF is introduced and the closure problem is discussed. Next the Monte Carlo solution method that is used to model and solve the PDF transport equation is described. Modelling of the unclosed terms describing acceleration in the turbulent ﬂow are treated in some detail. The closure of the micromixing terms can be done along parallel lines as is the case in the scalar PDF method discussed in [20] and is not elaborated here. A description of some methods for handling complex chemistry is included. In the ﬁnal section results of test calculations are presented for a challenging test case, combining a complex ﬂow pattern with eﬀects of high mixing rates and chemical kinetics. For this bluﬀ-body-stabilized diﬀusion ﬂame, the relative importance of modeling of the velocity, mixing and chemistry terms is studied.

1

PDF transport equation

From the full conservation equations given in [10], the transport equation for the joint velocity-scalar PDF f Uφ(V , ψ; x, t) can be derived (Pope 1985). By integrating this quantity over a range of values of V and ψ the probability that U and φ take values in these ranges is obtained. It is also useful to consider the joint velocity-scalar mass density function (MDF) deﬁned as F Uφ(V , ψ, x; t) = ρ(ψ)f Uφ(V , ψ; x, t). The transport equation for the MDF reads ∂ ∂ Vi F = − F+ ∂t ∂xi

−

1 ∂ ∂p ∂Tij − − F ∂Vi ρ(ψ) ∂xi ∂xj 1 ∂ ∂Vi ρ(ψ)

J

(1.1a) K

∂Tij ∂p − − U = V , φ = ψ F (1.1b) ∂xi ∂xj 626

[21] Joint velocity-scalar PDF methods

1 ∂Jiα − + Sα (ψ) F ρ(ψ) ∂xi

−

∂ ∂ψα

−

1 ∂ ∂ψα ρ(ψ)

627

J

(1.1c) K

∂J α − i U = V ,φ = ψ F , ∂xi

(1.1d)

in which the ﬁrst two terms (1.1a) and (1.1b) on the right-hand side describe the evolution in velocity space and the last two terms (1.1c) and (1.1d) describe the evolution in scalar space. (The notation A| B denotes the conditional expectation value of A upon condition B). Terms (1.1a) and (1.1c) occur in closed form whereas the unclosed terms (1.1b) and (1.1d) contain conditional averages because these eﬀects cannot be expressed in terms of the one-point distribution of U and φ. The terms (1.1a) that describe the evolution in velocity space contain the mean pressure gradient and the mean viscous stress tensor. Both terms can be expressed in terms of the mean velocity ﬁeld and thus occur in closed form. Assuming high Reynolds number, eﬀects of mean shear Tij are usually neglected. The unclosed terms (1.1b) describe the mean eﬀects of the ﬂuctuating pressure gradient and the ﬂuctuating viscous stress tensor, conditional on the values of velocity and scalars. The terms (1.1c) that describe the evolution in scalar space contain the eﬀects of the mean molecular diﬀusion ﬂux and of reaction. The reaction rate can be expressed as a function of the scalar variables φ and therefore it is contained in closed form. The unclosed term (1.1d) describes the eﬀects of molecular mixing or micro-mixing given by the conditional mean eﬀects of the ﬂuctuating diﬀusion ﬂux. Note that the conditioning here is on both scalars and velocity. In practice the same micromixing models are usually used in the velocity-scalar PDF approach as in the scalar PDF approach (discussed in [20]) and we will not elaborate on micromixing models here. Starting from the transport equation for the joint velocity-scalar mass density function it is straightforward to derive the equation for the joint velocityscalar Favre PDF f4Uφ. Both formulations are equivalent. The Eulerian transport equations for the moments of f4Uφ can be derived by multiplying each term by a function of Vi and ψα (e.g., Vi Vj , Vi Vj Vk , ψα ψβ , or Vi ψα ) and integrating over the phase space (V , ψ). In this way also a transport equation for the Reynolds stresses can be derived and it is a good question to ask which Eulerian second-moment closure model is implied by a choice of closure of the PDF transport equation, or the other way round, to ask whether a proposed second moment closure model has a Lagrangian counterpart. Such questions are discussed in detail in Pope (1994b) and Wouters (1998).

628

2

Roekaerts

Monte Carlo solution method

The PDF transport equation (1.1a–d) is a partial diﬀerential equation in many dimensions. In the case of d spatial dimensions and n independent scalars, solving the stationary equation by means of a ﬁnite-diﬀerence technique would require a suﬃciently accurate discretization in 2d + n dimensions. Alternatively, in a Monte Carlo solution method, the 2d + n dimensional mass density function is represented by a large ensemble of N notional ‘ﬂuid’ particles. The particle properties evolve according to particle models such that the evolution of the statistics of the particle ensemble corresponds to the modeled PDF evolution. The Monte Carlo method is computationally more eﬃcient already for the case d = 2, n = 1. Furthermore it oﬀers additional opportunities for model development. The basics of the Monte Carlo method and several of the numerical algorithms involved are described in detail by Pope (1985), Haworth and El Tahry (1990), Correa and Pope (1992) and Wouters (1998). This involves the method of fractional steps, the estimation of mean ﬁelds and mean ﬁeld gradients, particle boundary conditions and the integration of particle evolution equations. The method used here is based on the hybrid Monte Carlo method of Correa and Pope (1992). We consider systems whose one-point statistics depend on two spatial coordinates only (d = 2). All Eulerian quantities, like mean velocity and scalar ﬁelds and the mass density function, are deﬁned on a two-dimensional grid. The Eulerian joint velocity-scalar mass density function is represented by a spatially equally distributed ensemble of Lagrangian notional ‘ﬂuid’ particles with properties like position, velocity and composition. Here the term ‘notional’ particles is used to distinguish from real physical ﬂuid elements. Each of the particles represents an amount of mass ∆m and the particles are distributed such that the distribution of mass in space corresponds to the actual density ﬁeld. An estimate of the Eulerian Favre joint velocity-scalar PDF or, equivalently, the joint velocity-scalar mass density function in a cell is given by the velocities and scalar properties of the particles present in that cell. Note that the mass density function is deﬁned on a grid but that the ‘ﬂuid’ particles are not. The particles each have their own position and velocity and are not bounded to a grid. The estimation of the Eulerian PDF from the ensemble of Lagrangian particles is illustrated in ﬁgure 1. The particle properties evolve according to a modeled Lagrangian system of the following general form dxi = Ui dt dUi = dφα =

−Kiu dt + Du dWiu −Kαφ dt + Dφ dWαφ ,

(2.1a) (2.1b) (2.1c)

in which K u and K φ are the drift vectors for respectively the velocity and scalars. In this study, the diﬀusion terms are assumed to be isotropic and are

[21] Joint velocity-scalar PDF methods

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Figure 1: Illustration of the Eulerian PDF estimate fφN (ψ; x) from an ensemble of Lagrangian particles φ(i) in a cell at position x. given by Du and Dφ . The Wiener increments dWiU and dWαφ are independent. The speciﬁc expressions for the drift and diﬀusion terms for velocities and scalars deﬁne the model. Examples of particle velocity evolutions are given below. Examples of scalar evolutions were given in [20]. The coeﬃcients of the particle models contain basically two diﬀerent kinds 4i and of mean quantities. First there are the Eulerian mean quantities, like U 4 φα . These can be estimated from the particle properties e.g. by the following expression Nc 1 φ4 = φ(i) , (2.2) Nc i=1 in which the sum is over all particles present in a cell. (In practice also smoothing algorithms are used) Second, the modeled terms contain a characteristic mean turbulent frequency that determines the rate at which the modeled processes take place. This frequency cannot be estimated from the particle properties in the joint velocity-scalar description and it has to be supplied externally. Another special term is the mean pressure gradient. The mean pressure ﬁeld can be obtained from the velocity and density ﬁelds and, in principle, it is closed. Direct calculation of the mean pressure ﬁeld would require a pressure solver that uses the particle velocity and mass distributions as input. In this approach, stochastic ﬂuctuations in the particle ensemble may cause spurious pressure ﬂuctuations. Here, the mean pressure ﬁeld is not calculated directly. A hybrid Monte Carlo method is employed in which both the pressure and turbulent frequency ﬁelds are solved by an external Eulerian ﬁnite-volume method according to the method of Correa and Pope (1992).

630

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EULERIAN SUBMODEL

LAGRANGIAN SUBMODEL

Lagrangian Algorithm

; ∂p k U ∂xi k

Finite Volume Method

solve for particle properties: x(i) , U (i) , φ(i)

-

ρ ν

solve for: mean ﬁelds u , k, ;, p, u> U i j

6 (i)

∆φ(i)

φ

ρ(i) T (i)

∆t ?

Chemistry Scheme

Figure 2: Sketch of the hybrid Monte Carlo method. Left side: Lagrangian submodel with chemistry solver. Right side: Eulerian submodel that solves for mean ﬂow ﬁeld.

3

Hybrid ﬂow ﬁeld model

The hybrid Monte Carlo method of Correa and Pope (1992) consists of an Eulerian ﬁnite-volume submodel and a Lagrangian Monte Carlo submodel. The hybrid algorithm is illustrated in ﬁgure 2. The Eulerian model solves for the mean velocity, pressure, turbulent kinetic energy and turbulent dissipation. Given these mean ﬁelds, the Lagrangian Monte Carlo method solves for the joint velocity-scalar mass density function. Eﬀects of thermo-chemistry are solved in the Lagrangian algorithm. Mean thermo-chemical properties that inﬂuence the ﬂow ﬁeld, like density and molecular viscosity ﬁelds, are then coupled back to the ﬁnite-volume method which solves for the new turbulent ﬂow ﬁeld. Subsequent submodel calculations are performed until a stable solution is reached. In this algorithm, conventional modeling of the Eulerian ﬂow ﬁeld equations is needed also, and closer inspection is needed of the relation between closure in the Lagrangian and the Eulerian framework. During the Monte Carlo calculations, by an overall shift and scaling of particle velocities in each cell, the Eulerian mean velocities and turbulent kinetic energy are imposed onto the Lagrangian ﬂuid particles. As a result of

[21] Joint velocity-scalar PDF methods

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the hybrid method, the mean velocity and pressure ﬁeld correspond to each other but both are based on the density ﬁeld of the previous Monte Carlo submodel calculation. Density changes seen during a Monte Carlo submodel calculation have their full impact only after coupling with the ﬁnite-diﬀerence submodel. This can cause slow convergence when many switches between the submodels are needed and because, in general, Monte Carlo calculations are very time consuming. In more recent algorithms, which also calculate the turbulent dissipation by a Lagrangian model (see section 5), the mean turbulent frequency is determined from the particle properties and the particle representation of the density ﬁeld is directly used to solve for the mean pressure. In ﬂows with a strong density-ﬂow-ﬁeld coupling, these new methods may converge faster even though these models are more complex. An intermediate method is to use only mean velocity ﬁelds from the ﬁnite diﬀerence code and obtain second moments from the Monte Carlo particles, see Muradoglu et al. (1999). A basic requirement for the Monte Carlo algorithm is that the particle ensemble gives a physically correct estimate of the mass density function. This is expressed mathematically by the requirement that the sum of particle masses divided by the cell volume, gives a correct representation of the density ﬁeld. Using a two-dimensional rectilinear grid, large cells correspond to a large volume in three dimensions. Moreover, in axisymmetrical conﬁgurations, cells near the axis represent a small volume in three dimensions whereas cells far from the axis correspond to a larger three-dimensional volume. If all particles are to represent the same amount of mass ∆m, the particle number in a cell must depend on the local density and on the three-dimensional cell volume. For example, in cells with a small volume or with low density the number of particles is lower and the statistical accuracy in these cells is lower. As a result, the statistical accuracy varies throughout the domain which is undesirable. To remedy this eﬀect an ensemble with variable particle weights w(i) ∆m is used allowing the use of a nearly constant number of particles per cell. The system of adjustable particle weights is described in detail by Haworth and El Tahry (1990). In particular the average in equation (2.2) has to be replaced by a weighted average. As explained, the particle density representation on the one hand, and the mean velocity and pressure ﬁeld on the other, are not coupled directly in the hybrid method. During the Lagrangian submodel calculation there is no physical mechanism, by means of changes in the mean pressure gradient, to ensure that the particle weight-to-volume ratio remains constant. Therefore, to enforce the correct weight to volume ratio, a correction algorithm is needed for particle weights (or positions). The correction algorithm used in the applications described below, transfers particles to adjacent cells if the total weight in the cell is too high.

632

4 4.1

Roekaerts

Modeling velocity evolution Langevin models

The unclosed terms in the joint velocity-scalar mass density function equation that describe the evolution in velocity space read ∂ − ∂Vi

J

K

∂Tij ∂p − − U = V , φ = ψ f4 , ∂xi ∂xj

(4.1)

here written in terms of the Favre joint PDF instead of the mass density function. To illustrate the closure procedure, we ﬁrst consider the Simpliﬁed Langevin Model (SLM) (Haworth and Pope 1986). A Langevin model is a stochastic model for ﬂuid particle velocities. The modeled particle velocity equation consists of a linear drift towards the local Favre mean and an added isotropic diﬀusion term, and reads dUi∗

1 3 4i )dt + (C0 )1/2 dWi , =− + C0 ω(Ui∗ − U 2 4

(4.2)

in which ω = /k is the mean turbulent frequency, Ui∗ is a stochastic velocity realization, C0 a constant, and dWi a stochastic Wiener increment. The physical meaning of C0 and the properties of the Wiener process are explained below, after equation (4.5). Using Ito calculus, the Langevin model then corresponds to an evolution of the velocity PDF given by (Risken 1984, Gardiner 1990) 24

∂ f4U 1 3 ∂ 4i )f4U + 1 C0 ∂ f U . (Vi − U (4.3) = + C0 ω ∂t 2 4 ∂Vi 2 ∂Vi2 The basic properties of the SLM are that it yields the correct evolution of turbulent kinetic energy and that it yields isotropization of the velocity distribution. To describe complex ﬂow ﬁelds an extended model is needed: the Generalized Langevin model (GLM) which is considered next. Earlier studies used Langevin equations to model turbulent dispersion and Langevin equations for particle velocities were developed by assuming an analogy between ﬂuid particles and particles undergoing a Brownian motion. In contrast, the GLM was derived taking the Navier–Stokes equations as a starting point. For an extensive review of the use of Langevin models for turbulence modeling see Haworth and Pope (1986), Pope (1983) or Dopazo (1994) and references therein. From the Navier–Stokes equations, the equations of motion for a Lagrangian ﬂuid element can be derived. The exact equations read dxi = Ui dt

dUi =

−

(4.4a)

1 ∂p ∂Tij − dt ρ ∂xi ∂xj

∂Tij 1 ∂p + − − dt, ρ ∂xi ∂xj

(4.4b)

[21] Joint velocity-scalar PDF methods

633

with the Eulerian mean ﬁelds p(x, t) and U (x, t) evaluated at the ﬂuid element position x(t). The ﬁrst term on the right-hand side of equation (4.4b) can be expressed in terms of the one-point Eulerian velocity statistics but the second is unknown. The velocity increments are modeled by a Langevin equation according to

dUi∗

=

1 ∂p ∂Tij − − dt ρ ∂xi ∂xj

(4.5)

4j )dt + (C0 )1/2 dWi , Gij (Uj∗ − U

in which C0 is a positive constant, W (t) an isotropic Wiener process, the dissipation of turbulent kinetic energy and Gij is a second-order tensor which is modeled as a function of local mean quantities. A ‘general’ form of Gij in terms of the local mean velocity gradients, Reynolds stresses and dissipation was derived by Haworth and Pope (1986). The Wiener process is a stochastic process with zero mean and covariance dWi dWj = dtδij .

(4.6)

The rationale of using the Langevin model to model turbulent velocities is found in the consistency of the model with Kolmogorov’s hypothesis of turbulence (see Monin and Yaglom 1971). For high Reynolds number there is a separation between the large energy containing scales and the small scales at which viscous dissipation takes place. The cascade of energy from the large to the small scales occurs in the inertial subrange where the energy dissipation rate is the only relevant parameter. The eﬀects modeled by the tensor Gij are characterized by the time-scales of mean deformation |∂U /∂x|−1 and energy dissipation T = k/. The Wiener process models eﬀects that take place on a shorter time-scale. The Wiener or diﬀusion process is continuous but not diﬀerentiable and has the properties of a Markov process. This means that the stochastic velocity increments only depend on the present state of the ﬂuid element and not on its history. The use of a diﬀusion process is justiﬁed because the modeled small-scale processes are not dynamically important as stated by Kolmogorov’s theory. The speciﬁc form of the diﬀusion coeﬃcient is determined by the modeled Lagrangian structure function which, for the Langevin model, reads ∆Ui ∆Uj L ≡ (Ui (t + s) − Ui (t))(Uj (t + s) − Uj (t))L = C0 sδij ,

(4.7)

with ∆Ui the velocity increment of a Lagrangian ﬂuid element. This is consistent with Kolmogorov’s inertial range scaling with C0 being a universal constant. A value C0 = 2.1 was determined experimentally from diﬀusion measurements in grid turbulence by Anand and Pope (1985). For a detailed treatment of dynamical properties of the Langevin model and consistency with

634

Roekaerts

Kolmogorov’s theory see Haworth and Pope (1986), Pope (1994a) and Girimaji and Pope (1990). An important feature of the Lagrangian models is that they describe the evolution of individual ﬂuid element velocities. This means that, for ﬁnite model coeﬃcients, the Langevin model always describes physically realizable distribution of velocities (e.g., positive normal stresses and turbulent kinetic energy). Consequently, the Langevin model is always realizable by construction as are all moment equations derived from the Langevin model. Because the Langevin model describes the evolution of the full joint velocity distribution it also provides a modeled evolution equation for the second moments of velocity. In particular, in the case of an isotropic dissipation tensor (ij = 23 δij ), the exact expression for the pressure strain correlation in the Reynolds stress equation, Π∗ij

1 =− ρ

ui

∂p ∂p + uj ∂xj ∂xi

,

(4.8)

is represented by u + G u> Π∗ij = Gik u> jk i uk + j k

2 + C0 δij . 3

(4.9)

In the approach followed here, following the lines of Pope (1994b), the model parameters in the expression for Gij are speciﬁed such that the modeled pressure-strain correlation equals that of several well-known second-moment closures. A basic second-moment closure model for Π∗ij is the combination of a linear return-to-isotropy or Rotta term and a linear isotropization-of-production (IP) term. The model (with ﬁxed model constants C1 = 1.8 and C2 = 0.6) is referred to as the Isotropization-of-Production Model (IPM) and the corresponding GLM is called the Lagrangian IP model (LIPM). The Simpliﬁed Langevin model (SLM) corresponds to the Rotta model (only linear return-to-isotropy C1 = 4.15, C2 = 0). For the (L)IPM and the SLM/Rotta model, hybrid Monte Carlo calculations were performed in a bluﬀ-body stabilized diﬀusion ﬂame using both the Eulerian second-moment model and the Lagrangian Langevin models (Wouters et al. 1996). Some results of these calculations are presented below.

4.2

Third-moment and turbulent scalar-ﬂux equations

Since the full joint probability of velocities and scalars is known the Lagrangian models also provide modeled evolution equations for the third-moments of velocity and the turbulent scalar ﬂuxes. The third-moment equations are governed by the Langevin model for velocities whereas the combination of Langevin model and micro-mixing model provides a model for the scalar ﬂuxes.

[21] Joint velocity-scalar PDF methods

635

The modeled equations were published by Pope (1994b) and are brieﬂy reviewed here. The modeled equations are looked upon to assess the validity of the hybrid Monte Carlo methods. Modeled triple velocity correlations The evolution of the triple velocity correlations as modeled by the GLM reads D > ∂ > u u u = − u u u u Dt q r s ∂xi i q r s ∂ > > u + δ u> + u u (δiq u> ir q us + δis uq ur ) r s ∂xj i j

4i ∂U + Gij − ∂xj

>

>

(4.10) >

(δiq uj ur us + δir uj uq us + δis uj uq ur ).

An algebraic expression for the triple moments can be derived from equation (4.10) by neglecting the mean convective term (the left-hand side) and approximating the fourth order term by >

> > u u> > > ui uq ur us = u> i q r us + ui ur uq us + ui us uq ur .

(4.11)

The resulting algebraic expression can be found in Pope (1994b) and is not repeated here. For small anisotropies and low mean shear rates the full expression simpliﬁes to the model of Daly and Harlow (1970) (see [10], equation (27)) with the constant Cs given by −1/(3α1 ). For the Simpliﬁed Langevin model, α1 = −(1/2 + 2/3C0 ) and Cs takes the value 0.16. Modeled turbulent scalar-ﬂux equations The evolution equation for the turbulent scalar ﬂux, as modeled in the Lagrangian method, is given by the combination of GLM and a micro-mixing model. The concept of micro-mixing was already introduced in [20]. Here the modeled scalar ﬂux equations are given assuming that the interactionby-exchange-with-the-mean (IEM or LSME [20]) mixing model is used. Other mixing models predict the same evolution of scalar mean and variance. Diﬀerences between micro-mixing models occur in the evolution of third- and higher moments. The eﬀects on the modeled scalar ﬂux equation are expected to be small however. Equating the exact scalar-ﬂux equations with the modeled equations, in the same way as was done for the Reynolds-stress equations, the modeled terms read 1 φ φ , Πi = Gij − Cφ δij u> (4.12) i 2 k with the exact term given by Πφi = −φ

∂p . ∂xi

(4.13)

636

Roekaerts

Here, the assumptions have been made that the mean pressure gradient term can be neglected and that local isotropy prevails. The assumption of local isotropy implies that the ﬂux dissipation term φi is zero. Alternatively, the micro-mixing term, which models the decay of scalar variance, can be interpreted to model the dissipation of the scalar ﬂux which would contradict the assumption of local isotropy (Pope 1994b). By replacing the local mean scalar value φ with the conditional mean φ|U the scalar ﬂux model agrees with the local isotropy assumption. More generally speaking, mixing models must be formulated such that particles interact only with particles that are within the same range of velocities. This concept agrees with observations in DNS data where it is seen that mixing occurs in lamella-like structures that move with the same velocity. For a more detailed treatment of velocity-conditioned mixing see Fox (1996). Returning to the scalar-ﬂux pressure-scrambling term, a standard Eulerian model for the pressure-scrambling term is given by 4 > ∂ Ui , Πφi = Cφ1 ρ u> i φ + Cφ2 ρuj φ k ∂xj

(4.14)

with standard values of the constants Cφ1 = 3.0 and Cφ2 = 0.5 (see Jones 1994). The Lagrangian SLM/IEM model, with C0 = 2.1 and Cφ = 2.0, corresponds to the slow part of this Eulerian model but with Cφ1 = 3.075. Examples of a scalar ﬂux model using the GLM, including rapid pressure eﬀects, are given in Pope (1994b) and Wouters et al. (1996) In the hybrid Monte Carlo method however, scalar transport is governed completely by the Lagrangian models and the relationship with standard Eulerian models is of no practical importance.

4.3

Consistency

When solving the modeled velocity PDF transport equation with the hybrid Monte Carlo method some caution is needed. In the hybrid method the turbulent ﬂow ﬁeld is deﬁned twofold: on the one hand by the Eulerian mean velocity and turbulent kinetic energy ﬁelds of the ﬁnite-volume submodel, and on the other hand by the velocity distribution of the Lagrangian Monte Carlo submodel. Although a unique deﬁnition of the turbulence model is not guaranteed, both submodels use the same solution for the mean velocity and turbulent kinetic energy ﬁelds. Turbulent transport of mean momentum and production of turbulent kinetic energy are governed by the Reynolds stresses and consistency between the two submodels on the level of mean velocities and kinetic energy inevitably requires a consistent treatment of the second moments. Several studies, using the hybrid method, have used the k- model in combination with the SLM (Correa et al. 1994, Nooren et al. 1997). In the Eulerian

[21] Joint velocity-scalar PDF methods

637

k- model, the Reynolds stresses are given by an algebraic relation and the second-moment equations are not modeled explicitly. No equivalent model in terms of a Langevin model exists. As a result, the k- model cannot be part of a consistent hybrid model. In the hybrid method employed here, a consistent hybrid turbulence model is obtained by using the same second-moment equation in both submodels with the Langevin model chosen equivalent to the ﬁnite-volume turbulence model for the pressure-strain correlation. The third-moments are still treated inconsistently in the two submodels. In the Lagrangian method the third-moments are exact whereas in the second moment closure models these are given by a generalized gradient diﬀusion model. Eﬀects of this inconsistency on the level of the third moments are assumed to be small. The inconsistency at the level of the second moments in the SLM/k- model and the performance of two consistent hybrid models, SLM/Rotta and LIPM/IPM, were studied in more detail in a bluﬀ-body-stabilized diﬀusion ﬂame and the corresponding inert ﬂow. Some results of the bluﬀ-body ﬂow calculation are presented in section 7. Summarizing, in reacting ﬂows where diﬀerences in scalar ﬁelds couple back to the ﬂow ﬁeld through the mean density, large diﬀerences can occur between the standard ﬁnite-volume method (using an assumed-shape PDF method), and the hybrid Monte Carlo method; even when using the same ﬂow model in the ﬁnite volume part of the algorithms. The point made here is that these diﬀerences do not necessarily stem from the diﬀerent treatment of scalar ﬂuctuations but can be caused by a simple diﬀerence of the turbulence models. In the hybrid model diﬀerent representations for one and the same physical quantity are used. Firm conclusions on speciﬁc model assumptions at the level of second moment closure can only be made if the two representations that are used are consistent at that level.

5

Developments in Lagrangian turbulence modeling

In the previous section modeling of the velocity PDF evolution equation has been treated. This approach to modeling of turbulent ﬂows has its limitations. First, the turbulent dissipation or turbulent frequency has to be provided. In the approach used here, the mean frequency is provided by solving the Eulerian transport equations for the turbulent kinetic energy and turbulent dissipation. Second, the eﬀects of molecular viscosity and ﬂuctuating pressure gradients on the velocity distribution occur in unclosed form and are modeled by means of a Langevin model. In this section two extensions to the velocity PDF approach are reviewed. First the joint velocity-dissipation method is discussed. Second, the extension of this model by inclusion of the joint statistics of the velocity wave vector is discussed. Modeling of homogeneous turbulent ﬂows in terms of the joint velocitydissipation PDF was proposed by Pope and Chen (1990). A Langevin equa-

638

Roekaerts

tion for the turbulent dissipation or turbulent frequency is solved. The model dissipation satisﬁes two properties: (1) the dissipation obtains a log-normal distribution, and (2) the mean dissipation evolves according to the standard -equation. The model was extended to inhomogeneous ﬂows by Pope (1991). The main advantage of the model is that it provides a turbulent time-scale to the modeled velocity evolution now that the joint PDF of velocities and dissipation is known. Second, in inhomogeneous ﬂows the model provides a way to describe small-scale intermittency eﬀects. In principle this set of modeled Lagrangian equations is self-contained in the sense that no additional Eulerian transport equations have to be solved for turbulent ﬂow quantities. In an actual Monte Carlo simulation the mean pressure ﬁeld is still solved by means of a pressure solver but the pressure ﬁeld is completely determined by the Lagrangian velocity distribution. A further development in Lagrangian turbulence modeling is the description of the ﬂow ﬁeld in terms of the joint PDF of velocity dissipation and wave vector by Van Slooten and Pope (1997). In the rapid distortion limit, where the eﬀects of mean strain are much larger than viscous dissipation eﬀects, an exact solution of the Navier–Stokes equations can be written in terms of Fourier modes of the velocity ﬁeld. To be more speciﬁc, an exact solution exists in terms of the velocity and its wave number vector. The wave number vector is deﬁned as the unit vector in the direction of the wave vector. In the predicted second-moment equations the rapid pressure-strain eﬀects now occur in closed form whereas the slow term has to be modeled. Future studies have to show if the new model will yield better results in complex ﬂows where current statistical models have their limitations (e.g. swirling ﬂows). In principle the PDF description can, like moment closures, be extended further and sensible choices of included quantities may allow for a better description of certain physical processes. Of course the demand of practical applicability narrows down the possible model choices. At this moment the practical application of the PDF method is restricted to the velocity PDF and joint velocity-dissipation PDF methods, the former method being a stable and reliable tool in turbulent reacting ﬂow modeling. The second method allows for a better ﬂow ﬁeld description but is more complicated and has yet to show its value in complex ﬂow conﬁgurations.

6

Chemical reaction

Although the Monte Carlo PDF method allows for an exact treatment of chemistry from the point of view of turbulence modeling, the use of a full reaction mechanism would require the integration of a stiﬀ system of many coupled equations. For example a methane-air or natural gas-air diﬀusion ﬂame requires equations for about 30 species obeying more than 200 chemical reactions. The inclusion of NO-formation or the use of fuels with higher hydrocarbon species complicates things even further. In the Monte Carlo Method

[21] Joint velocity-scalar PDF methods

639

chemical reaction has to be followed in many thousands of particles and to make the calculation feasible judicious simpliﬁcation is needed. This section deals with some basic modeling of chemistry for methane-air or natural gas-air diﬀusion ﬂames. A detailed treatment of turbulent combustion chemistry is given in standard textbooks (see Gardiner (1984), Williams (1985), Warnatz et al. (1996) and references therein). Making certain assumptions about the ﬂame conditions, full mechanisms can be simpliﬁed to so-called skeletal mechanisms containing about 15 species and 40 reactions. Usually, skeletal mechanisms are simpliﬁed further by making certain assumptions about some of the elementary reactions. By assuming that some reactions are in chemical equilibrium or assuming that certain species are in steady state, reaction can be simpliﬁed to several steps (e.g. four or ﬁve step schemes containing ﬁve or six species respectively). This way of reducing chemical kinetics is described in detail by Peters and Rogg (1993) and Seshadri and Williams (1994). For the speciﬁc case of methane-air combustion see Smooke (1991). In the rest of this section, conserved-scalar models for methane-air diffusion ﬂame chemistry are discussed. A constrained-equilibrium model, which is used in this study, is given in more detail. Then, a three-scalar simpliﬁed kinetics scheme, obtained from the Intrinsic Low-Dimensional Manifold method, is discussed. Constrained-equilibrium (CE) model One of the simplest models to describe diﬀusion ﬂame chemistry is the ﬂamesheet model of Burke and Schumann (1928). The model assumes an inﬁnitelyfast irreversible global reaction of fuel and oxidizer to products which results in piecewise linear relations between composition and mixture fraction. The ﬂame-sheet model however does not include the formation of intermediate species like CO and H2 or radical species like O, H, and OH. Another simple model that can be expressed in terms of mixture fraction is the chemical equilibrium model. This model assumes high Damkohler number so that the reactions are fast enough to reach full chemical equilibrium. For the case of methane-air combustion this assumption is valid in the high temperature regions of the ﬂame but in the low temperature regions on the rich side of the ﬂame the slow burn-out of CO does not reach equilibrium and the CO concentration is underpredicted. This eﬀect can be remedied by assuming that slow three-body reactions, involving three reactants and two products or vice versa, do not reach equilibrium. These so-called partial-equilibrium models perform well in the low temperature regions of the ﬂame. In diﬀusion ﬂames reaction occurs mainly in a thin reaction zone around stoichiometric mixture fraction and chemistry is frozen outside this zone. For hydro-carbon combustion the constrained-equilibrium model of Bilger and St˚ arner (1983) combines the partial-equilibrium assumptions with the assumption of a reaction zone around stoichiometric mixture fraction. The model assumes a fuel breakdown or pyrolysis sheet at ξ = ξig > ξst . There all fuel

640

Roekaerts

Figure 3: Relation between fuel mass fraction Yfu , intermediate species mass fraction Yint , oxidizer mass fraction Yox , and mixture fraction for the constrained-equilibrium model. In this example ξst = 0.2, ξig = 0.38 and the maximal intermediate species concentration Yint,max = 0.2. reacts by a one-step irreversible inﬁnitely fast reaction to some intermediate species. At the stoichiometric mixture fraction ξ = ξst all intermediate species are consumed. For methane combustion, Bilger and St˚ arner take C2 H4 as the intermediate species and the pyrolysis sheet is located at ξig = ξst + 0.018. The relationship between fuel, intermediate species, oxidizer and mixture fraction is depicted in ﬁgure 3. In the application described below a simpliﬁed version of the constrained equilibrium model is used. Basically, only the ﬂame zone assumption of the model is used. More details about this constrained-equilibrium model and its performance in natural-gas diﬀusion ﬂames can be found in Peeters et al. (1993a) and Peeters (1995). Intrinsic Low-Dimensional Manifold method In contrast to the traditional way of simplifying detailed kinetics schemes, the Intrinsic Low-Dimensional Manifold (ILDM) method (Maas and Pope 1992) provides a way of simplifying detailed schemes without making a-priori assumptions. Conventional methods assume that certain reactions reach equilibrium or that certain species are in steady state. This involves processes that occur on scales much shorter than the scales of turbulence and that can be decoupled from the rest of the system. The ILDM method performs a timescale analysis locally in composition space and automatically deﬁnes the ‘slow’ and ‘fast’ time-scales. Several time scales are denoted as ‘slow’ thereby specifying the dimension of the reduced kinetics scheme. The ILDM method then assumes that the remaining ‘fast’ scales are in equilibrium.

[21] Joint velocity-scalar PDF methods

641

Consider a detailed kinetics scheme involving ns species and, given the timescales of turbulence, the nf fastest time-scales can be decoupled. The number of describing variable or degrees of freedom is then given by ng = ns − nf . The ILDM method performs a time-scale analysis for every point in composition space based on the eigenvalue-eigenvector representation of the local Jacobian of the system. If the eigenvalues of the system are in descending order then the ﬁrst ng eigenvectors of the system deﬁne the low-dimensional manifold. The other eigenvalues of the system have a lower negative real part which causes a fast relaxation of the system to the manifold. More details about the application of the ILDM method to methane-air combustion and the validation of the technique in laminar ﬂames are given in Schmidt et al. (1996). In Situ Adaptive Tabulation method Another development in turbulent reacting ﬂow modeling is the In Situ Adaptive Tabulation (ISAT) method (Pope 1997). This new algorithm allows for the use of detailed chemistry schemes, like skeletal mechanisms with 15 or more species, in PDF simulations of turbulent ﬂames, keeping the computational demands within reasonable limits. The idea behind the ISAT algorithm is to integrate the reaction system directly and to store the outcome of the integration for later use. Other points in scalar space, with a small distance to points previously used, do not require a new integration but obtain the reaction increment by an interpolation in the table. Reaction increments that cannot be interpolated with suﬃcient accuracy require a new integration and are stored in the table. As a result, only the accessed region in scalar space is stored in the table. Details on the algorithm, its numerical implementation and performance can be found in Pope (1997) and references therein. Applications have been reported in Yang and Pope (1998) and Saxena and Pope (1998). A promising perspective is the combination of ISAT and the use of augmented reduced mechanisms (Sung et al. 1998, James et al. 1999). The computational advantage oﬀered by the eﬃcient storage and lookup algorithm can be exploited by doing less reduction, keeping suﬃciently detailed chemistry at an aﬀordable computing time.

7 Modeling of a bluﬀ-body-stabilized diﬀusion ﬂame 7.1

Introduction

To illustrate the theoretical developments described above, this section reports on the study of a bluﬀ-body-stabilized methane-air diﬀusion ﬂame using second-moment closures and multi-scalar chemistry. The bluﬀ-body conﬁguration serves as a model for industrial type low NOx burners. Here the bluﬀbody ﬂame is chosen as a test case because the ﬂow pattern exhibits complex phenomena, like recirculation regions and stagnation points, and because the coupling between turbulence and chemistry is strong. The ﬂow is very sensitive

642

Roekaerts

to density ﬂuctuations and the ﬂame shows eﬀects of ﬁnite-rate chemistry. The conﬁgurations studied here were subject of a 1994 ERCOFTAC-SIG workshop (First ASCF Workshop 1994a,b). It was shown that the k- model is insufﬁcient to model the bluﬀ-body ﬂow characteristics. Here two second-moment closure models are used besides the standard k- model. Chemistry is modeled with the conserved-scalar CE model and with the three-scalar ILDM scheme. The objective is to make a comprehensive study of the relative importance of the model used for convection, for micro-mixing, and for chemical kinetics. Monte Carlo predictions are compared to available experimental data of mean velocities, turbulent kinetic energy and temperature. The Monte Carlo model was implemented in a computer code PDF2DS provided by S.B. Pope. (The Delft version of this code is called PDFD). The ﬁnitevolume submodel calculations are performed with BIGMIX which is an in-house ﬁnite-volume code of TU Delft (see Peeters 1995).

7.2

Test case speciﬁcation

The ﬂow conﬁgurations studied are an axisymmetric bluﬀ-body-stabilized methane-air diﬀusion ﬂame and the corresponding inert ﬂow which were the subject of a 1994 ERCOFTAC-SIG workshop (First ASCF Workshop 1994a,b). Also other authors have reported their results in the literature (e.g. Fallot et al. 1997, Warnatz et al. 1996). The conﬁguration consists of a fuel jet 5.4 mm in diameter, surrounded by a 50-mm-diameter bluﬀ body. The coaxial air inlet has an outer diameter of 100 mm and a low-velocity coﬂow of air is present around the air inlet. The thickness of the coaxial air pipe is 1 mm. Figure 4 depicts the bluﬀ-body conﬁguration and sketches a typical ﬂow pattern. In the experiment, the burner is unconﬁned but in the simulation, a solid wall is assumed at a radius of 200 mm to prevent numerical problems with outﬂow of particles in the Monte Carlo method. Test calculations have shown that the presence of this conﬁnement does not inﬂuence the ﬂow in the regions of interest. The inlet velocities for the fuel, air and coﬂow are 21 m/s, 25 m/s and 1 m/s, respectively. The fuel inlet Reynolds number based on the fuel inlet diameter is Re = 7000 and for the air inlet, based on the bluﬀ-body diameter, Re = 80 000. Initially boundary conditions were the same as prescribed at the SIG workshop (First ASCF Workshop 1994a). However, the original inlet conditions are modiﬁed to improve the predictions with respect to the length of the recirculation zone, and to obtain a stable convergence of the secondmoment equations. Inlet proﬁle calculations are set up to approximate the experimental conﬁguration. The ﬁnite-volume calculations are performed on a rectangular nonuniform mesh of 150 × 150 points. On this mesh grid independent solutions are obtained and diﬀerences with solutions on an 80 × 80 mesh are small. The Monte Carlo simulations are performed on a 80 × 80 mesh using 150 particles per

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F F

S

A

B S D

41 U

40 U 5.4 mm 50 mm

100 mm

1 mm

Figure 4: Bluﬀ-body ﬂow conﬁguration and characteristic ﬂow pattern. Mean 40 = 25 m/s, uniform air inlet velocity U 41 = 21 m/s. Bold fuel jet velocity U faced letters indicate: S stagnation regions; D ‘diﬀusion ﬂame’ region; F main reaction or ﬂame region; A and B scatterplot positions of ﬁgures 8 and 9. cell. To reduce stochastic ﬂuctuations, mean properties are averaged over 500 iterations. Additional Monte Carlo calculations with 150 particles per cell on a grid of 150 × 150 and with 400 particles per cell on a grid of 80 × 80, are performed to test the inﬂuence of grid dependence and the statistical accuracy. Monte Carlo simulations require about 200 Mb internal memory and 100 hours of CPU time on a HP–735 workstation or 10 hours on 8 PE’s of a massively parallel CRAY–T3E supercomputer.

7.3

Choice of PDF model

This section summarizes the models used in this study and gives a motivation for the speciﬁc model choices. Hybrid turbulence model Three diﬀerent hybrid turbulence models have been used. The hybrid SLM/k model is used because it is a standard turbulence modeling approach using the hybrid Monte Carlo method (see Correa et al. 1994, Nooren et al. 1997). In this ﬂow the inconsistency of this model is apparent (see Wouters 1998). Two consistent hybrid models used are the SLM/Rotta and the LIPM/IPM models. The SLM/Rotta model is used because it is seen as the simplest consistent hybrid turbulence model. The consistent LIPM/IPM model is used

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because the second-moment closure model is seen as a basic Reynolds-stress model and it is expected to perform reasonably well for the bluﬀ-body conﬁguration. In the inert ﬂow, several other second-moment models were evaluated, without reaching signiﬁcant improvement of the predictions. The approach of tuning the turbulence model to the experimental ﬂow data separately for every chemistry and mixing model is not used here. Rather, the turbulence model constants are kept constant while investigating the relative performance of diﬀerent micromixing and chemistry models. In the reacting case, ﬂow and scalar ﬁelds are strongly coupled and changes in the scalar ﬁeld model will directly aﬀect the ﬂow ﬁeld. If the ﬁtting procedure is employed, then changes in the turbulence model constants will be large, provided an acceptable choice for the constants even exists. Chemistry modeling Initially, chemical reaction is simpliﬁed using the conserved-scalar constrainedequilibrium (CE) model. It is assumed that the reaction is fast such that it is limited only by the mixing of fuel and oxidizer. CE predictions however, show an overprediction of the mean temperature and, moreover, the ﬂame length is overpredicted severely. Because of the strong recirculation, burned products ﬂow back into the inlet region where the mixing rate of species is large. There, eﬀects of partial premixing and ﬁnite-rate kinetics are important. To capture these eﬀects, a three-scalar ILDM reduced kinetics scheme is used. The ILDM scheme can successfully describe ﬁniterate kinetics eﬀects and it is able to describe the pure diﬀusion limited case correctly. Here, for the case of non-premixed methane-air combustion the detailed mechanism is simpliﬁed to a three-dimensional manifold, parameterized by mixture fraction and the H2 O and CO2 mass fractions. The remaining thermochemical variables like the other mass fractions, temperature and density, and the reaction rates of H2 O and CO2 are tabulated as a function of these three parameters. In order to avoid a CPU-intensive integration of the rate equations for H2 O and CO2 mass fractions during the Monte Carlo simulations, the reaction rates are pre-integrated for the speciﬁc time-step of the simulation. For all simulations performed in this study, thermo-chemical variables are tabulated as a function of the describing variables in a locally reﬁned table as described in Peeters et al. (1993b) and Peeters (1995). Scalar micro-mixing Using the conserved-scalar CE chemistry model, modeling is required for micro-mixing of a single inert scalar. For this case, four mixing models are available: IEM (also called LSME), coalescence-dispersion (C/D), binomial Langevin (BL) and mapping closure (MCM), which were introduced in sections 4.1, 4.2, 4.4 and 4.5 of [20].

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With ILDM chemistry, scalar mixing involves mixture fraction and two additional reacting scalars, YCO2 and YH2 O . The C/D model performs very well for inert single-scalar mixing in jet ﬂows (see Wouters et al. 1998) and it has been shown to perform well in reacting jet diﬀusion ﬂames with CE and with ILDM chemistry (Nooren et al. 1997). For these reasons, the C/D model is selected as a standard model for mixing of multiple reacting scalars. A second model that is selected for scalar mixing in combination with ILDM chemistry is the modiﬁed multi-scalar BL model which was described in Wouters (1998). The model parameter K, which also enters the one-scalar case discussed in [20] takes a standard value of 0.3 but here it is varied between 0.1 and 2.0 to study its eﬀect on the overall reaction. The constant Cφ takes the standard value 2. Because other studies, using the same ILDM chemistry, have obtained good results with higher mixing rates, several calculations with Cφ = 4 are performed.

7.4

Results for reacting ﬂow

Results for the reacting ﬂow are given in the following order: ﬁrst a summary of ﬂow ﬁeld predictions is given, next results for thermo-chemical ﬁelds of the conserved-scalar constrained-equilibrium model are shown and the limitations of this model are discussed. Then, improvements using a three-scalar ILDM reduced kinetics scheme are presented. Finally results on the performance of micromixing models are given. A study of the inert ﬂow, including consistency tests can be found in Wouters (1998). Flow ﬁeld predictions Experimentally, the ﬂow exhibits one stagnation point, at x = 70 mm with a peak in turbulent kinetic energy of k = 44 m2 /s2 . All three hybrid models in combination with CE chemistry and C/D micromixing, predict a minimum in the axial velocity for x ≈ 70 mm with the minimal value close to zero. The peak value of turbulent kinetic energy is underpredicted by all models. With the IPM, predictions are reasonable for x < 70 mm. Here, the initial decay of the axial velocity is predicted correctly. At higher axial distances, the performance of the model is unclear. In this region the mean temperature is overpredicted severely (see ﬁgure 5) which clearly has its eﬀect on the ﬂow ﬁeld. Here, the limitations of the CE chemistry model play a role. Constrained-Equilibrium results Figure 5 depicts the axial proﬁle of the mean temperature. Moving along the axis in the axial direction, the experimental data show a sharp rise in temperature, up to a peak temperature of 1667 K at x = 80 mm. After the peak temperature is reached, the temperature drops rapidly. CE predictions are in reasonable agreement with the experiments up to x = 80 mm but the predicted temperature rises further and reaches a maximum of 2020 K at x = 145 mm. Moreover, the high temperature zone extends over a much larger region.

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Figure 5: Axial proﬁles of temperature. Predictions use IPM turbulence model and C/D micro-mixing model. Symbols: —— CE chemistry model (Tmax = 2020 K); – – – ILDM chemistry model (Tmax = 1980 K); measurements (Tmax = 1667 K).

Figure 6: Radial proﬁles of mean temperature. Predictions with IPM turbulence model and C/D micro-mixing model. (a) CE chemistry, (b) ILDM chemistry. Lines: predictions, —— x = 10mm, – – – x = 30mm, · · · · · x = 50mm. Symbols: measurements, ◦ ◦ ◦ x = 10mm, x = 30mm, x = 50mm. Radial proﬁles of temperature, which are depicted in ﬁgure 6a, show that predictions are reasonable for x ≤ 50 mm. At x = 10 and 30 mm the temperature is overpredicted by ≈ 250 K but the shape of the proﬁles is good. At r = 25 mm a peak indicates the presence of a reaction zone starting at the edge of the bluﬀ-body. There the assumption of a diﬀusion ﬂame, that can be described by a conserved-scalar model, seems

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to be reasonable. At x = 50 mm, the temperature maximum is overpredicted because the temperature at the centerline rises too early (see also ﬁgure 5). At higher axial distances near the stagnation zone, mixing rates are high and a well-mixed composition of fuel and air exists above the recirculation zone. Here, reaction is not limited by mixing and eﬀects of ﬁnite-rate kinetics are important. The conserved-scalar CE model is not able to capture these eﬀects and therefore it overpredicts the temperature in this region. ILDM results Final results of the ERCOFTAC-SIG workshop (First ASCF Workshop 1994b) show that models that do not employ the fast-chemistry assumption perform much better downstream of the recirculation zone, even when using k- turbulence modeling. To satisfactorily describe the eﬀects of partial premixing and ﬁnite-rate kinetics, a three-scalar ILDM reduced kinetics scheme is used. In ﬁgure 5, the axial proﬁle of mean temperature is depicted for CE and ILDM chemistry models and measurements, and the implications of the chemistry model are clearly seen. ILDM chemistry yields a faster axial decay of temperature after the peak temperature is reached. The CE model overpredicts the temperature in this region even though the mixture fraction ﬁelds in the CE and ILDM calculations are almost the same. At x ≈ 45 mm, the ILDM prediction shows a sharp rise in temperature which is not seen in the measurements or in the CE predictions. The faster ignition is caused by diﬀerences in the ﬂow ﬁeld. Because of the strong coupling between chemistry and ﬂow ﬁeld, small diﬀerences in the mean density result in a much stronger recirculation with a minimal axial velocity on the axis of −3.9 m/s at x = 50 mm. Grey-scale plots of the mean temperature for CE and ILDM predictions and the measurements are shown in ﬁgure 7. Looking at the overall picture, the ILDM ﬂame is more compact and it is located closer to the bluﬀ body than seen in the experiments. Compared to the CE predictions the ﬂame length is much shorter and the predictions are in better agreement with the experiments. However, the temperature is still overpredicted by approximately 200 K in the ﬂame region. The reason for the improvements obtained with the ILDM chemistry scheme over the CE model is that the fast-chemistry assumption, made in the CE model, is not valid in this ﬂame. To illustrate the diﬀerences between the two chemistry models, ﬁgure 8 shows scatter plots of mixture fraction and H2 O mass fraction at two positions in the ﬂame. Scatter plot positions are sketched in ﬁgure 4. Figure 8a shows a scatter plot at x = 37.5 mm and r = 7.86 mm (position A) in the region where the mixture fraction variance and mixing rate reach a maximum. A scatter plot at x = 38 mm and r = 15 mm (position B) is shown in ﬁgure 8b. This position is located in the shear layer above the edge of the bluﬀ body at an axial location where the source terms of YCO2 and YH2 O reach a local maximum. At

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(a)

(b) 300 K

(c) 1900 K

Figure 7: Greyscale plot of the mean temperature. (a) CE chemistry predictions (Tmax = 2020 K), (b) measurements (Tmax = 1667 K), (c) ILDM chemistry predictions (Tmax = 1980 K). Contour values at 300, 500, . . . 1900 K. For the calculations only part of the domain is shown. position A, the mixture fraction variance is large and almost the entire mixture fraction space is occupied. For ξ > 0.25 points are scattered around the equilibrium line but for lower mixture fraction values, many points are found below the equilibrium limit. At this position, the ILDM model predicts a mean temperature below the equilibrium value. Position B is located in the shear layer above the edge of the bluﬀ body where the CE results showed that the assumption of diﬀusion limited chemistry gives reasonable predictions of the mean temperature (see also ﬁgure 6a). The scatter plot shows that the ILDM results are far from equilibrium. The ﬁgure clearly shows that mixing rates are high which yields many points near stoichiometry but with low YH2 O values.

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Figure 8: Scatterplot of H2 O mass fraction versus mixture fraction. ILDM chemistry with C/D micro-mixing. (a) at x = 37.5 mm and r = 7.86 mm, ξ4 = 0.285, (b) at x = 28 mm and r = 22 mm, ξ4 = 0.084. Typical scatterplot positions are indicated in ﬁgure 4. Solid line denotes the constrained-equilibrium limit. Dashed vertical line denotes the value of the mean mixture fraction. Radial proﬁles of the mean temperature, up to one bluﬀ-body diameter downstream, are depicted in ﬁgure 6b. The proﬁles show no overprediction of temperature at the edge of the bluﬀ body (15 < r < 25 mm) as seen in the CE results. The proﬁle at x = 50 mm clearly shows the early rise of temperature which corresponds to a ﬂame closer to the bluﬀ body. The ILDM results fail to show a peak at r = 25 mm as is seen in the CE proﬁles. The scatter plot 8b indicates also that the equilibrium temperature in not reached in this region. Summarizing, the three-scalar ILDM reduced chemistry scheme is able to describe the eﬀects of partial premixing and ﬁnite-rate kinetics that occur in this ﬂame. Mean temperature ﬁelds are in better agreement with the experiments than the CE predictions which show a large overprediction of the ﬂame height. With the ILDM scheme, temperature is still overpredicted by approximately 200 K in the entire ﬂame region. Results: micro- and macro-mixing This section focuses on the importance of modeling of the micro-mixing term in this ﬂow. In the reacting case studied here, where the coupling between turbulence and chemistry is strong, the eﬀects of the mixing model on the reaction rate and ﬂow ﬁeld may be large. In combination with the conserved-scalar CE chemistry model, micro-mixing is modeled with the single-scalar IEM, C/D, MCM and BL models. The inﬂuence of the micro-mixing model on the mixture fraction PDF was studied for inert scalar mixing in jet ﬂows by Wouters et al. (1998). In the inert jet ﬂow, speciﬁc mixing models aﬀect details of the scalar distributions but have no effect on the mean quantities. In reacting jet ﬂows it was shown that diﬀerences

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Figure 9: Scatterplot of H2 O mass fraction versus mixture fraction. ILDM chemistry using BL micro-mixing with K0 = 0.3. (a) ξ4 = 0.275, (b) ξ4 = 0.090. Lines and position are the same as those of ﬁgure 8. in the scalar PDF’s aﬀect the mean temperature. As a result, the IEM model predicts higher mean temperatures than the C/D and MCM models (Nooren et al. 1997, Nooren 1998). For this bluﬀ-body ﬂame however, the inﬂuence of the micro-mixing model on the mean temperature is small and the diﬀerences in the density have a negligible eﬀect on the ﬂow ﬁeld. The ILDM chemistry model uses mixture fraction and two reacting scalar to describe the chemistry. Micro-mixing of these three reacting scalars is modeled by C/D and modiﬁed BL models. Figure 9 shows scatter plots of YH2 O versus mixture fraction using BL micro-mixing with the model parameter K0 = 0.3. Scatter plot positions are identical to those of ﬁgure 8. As a result, points are much more spread over the ξ − YH2 O plane than with the C/D model. Many points above the equilibrium limit are found. Nevertheless, the presence of super-equilibrium values of YH2 O and YCO2 has a small eﬀect on the mean temperature and density ﬁelds. We now discuss three aspects: the strength of turbulent dispersion (model constant K0 ), the mixing rate and the relative importance of micro- and macromixing. Turbulent dispersion Variation of the model parameter K0 from 0.1 to 0.6 yields only small diﬀerences in mean temperature peak values and the inﬂuence on the overall mean temperature and ﬂow ﬁelds is small. Using a higher value of K0 = 2.0, the diﬀusion term spreads scalar values even more over the allowed scalar domain. As a results the mean temperature increases with a peak temperature of 1980 K. The ﬂame height increases by 50%. With K0 in the normal range of 0.3 to 0.6, diﬀerences with the C/D model in mean ﬁelds are small.

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Mixing frequency Studies of methane-air and natural gas-air jet ﬂames using the same or similar ILDM chemistry (see Nooren et al. (1997) and references therein) have reported necessary changes of the mixing rate to obtain stabilization of the ﬂame. The mixing rate was modiﬁed by changing the constant Cφ and/or by using another deﬁnition of the mean turbulence frequency ω (Masri and Pope 1990). In this ﬂame, stabilization is obtained using the standard deﬁnition of the mixing rate (see [10]) where the constant Cφ takes the standard value 2. To investigate the eﬀects of the mixing rate, a calculation with the ILDM chemistry model and the C/D mixing model was performed using Cφ = 4. This particular value of Cφ was used by Nooren et al. (1997) for modeling of a jet diﬀusion ﬂame and yielded ﬂame stabilization and good predictions of the mean temperature. Here, compared to the calculations with Cφ = 2, scalar variances are lower and the high temperature region extends much further. The peak temperature remains almost the same with Tmax = 2079 K. Also above the edges of the bluﬀ-body, the temperature and YCO2 and YH2 O source terms increase. Over the entire ﬂame region, the agreement with the experiments is not as good as for the predictions with a standard mixing rate constant Cφ = 2. Summarizing, for this ﬂame, no increase of the mixing rate is needed to obtain ﬂame stabilization and the temperature overprediction is larger using a higher mixing rate. The eﬀects of the mixing rate on the predictions is not studied further here. Micro- and macromixing The remaining deﬁciencies of the present PDF model are attributed to modeling of momentum and scalar transport. Scalar micro-mixing and the coupling between micro-mixing and reaction are not crucial in this ﬂow. Apparently, macro-mixing by turbulent convection is the driving force in scalar mixing. The underprediction of the turbulent kinetic energy at the stagnation point positions and the low Reynolds numbers of the fuel and air inlets (Re = 7000 and Re = 48 000 respectively) indicate that this ﬂow exhibits unstable phenomena. A low-frequency oscillation in the ﬂow ﬁeld causes an overprediction of the measured turbulent kinetic energy. The instationary convective scalar transport cannot be described by the present stationary Favre-averaged PDF model.

7.5

Discussion

Although the ﬂame predictions with the ILDM chemistry scheme show a large improvement over the conserved-scalar CE predictions, results are still far from perfect. The remaining deﬁciencies of the present PDF model are attributed to the modeling of transport of scalars and momentum. Our study of scalar micro-mixing shows that this eﬀect is not crucial in this ﬂow. Apparently here turbulent transport or turbulent diﬀusion is the driving mechanism in scalar mixing. The large underprediction of turbulent kinetic energy at the stagnation point positions and the low Reynolds numbers of the fuel and air jets indicates

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that this ﬂow exhibits unstable phenomena which cannot be described by our stationary Favre-averaged PDF model. Since the calculation of an instationary ﬂow using Monte Carlo PDF methods is still limited by the available computer power this test case cannot be described accurately by the velocity-scalar hybrid PDF model. A Eulerian scalar-PDF transport method does not have these limitations. In cases where the turbulent scalar ﬂuxes follow simple gradient-diﬀusion laws, Eulerian scalar-PDF transport methods can be used to perform instationary calculations. Otherwise, the self-contained joint velocity-dissipation-scalar PDF method, or Large Eddy Simulation and ﬁltered-PDF techniques of Colucci et al. (1998), are needed to perform instationary PDF simulations.

References Anand, M.S. and Pope, S.B. (1985) ‘Diﬀusion behind a line source in grid turbulence’. In Turbulent Shear Flows 4, L.J.S. Bradbury et al (eds.), Springer-Verlag, 46–61. Bilger, R.W. and St˚ arner, S.H. (1983) ‘A simple model for carbon monoxide in laminar and turbulent hydrocarbon diﬀusion ﬂames’, Comb. Flame 51 155–176. Burke, S.P. and Schumann, T.E.W. (1928) ‘Diﬀusion ﬂames’, Indus. Engin. Chem. 20 999–1004. Colucci, P.J., Jaberi, F.A., Givi, P. and Pope, S.B. (1998) ‘Filtered density function for large eddy simulation of turbulent reacting ﬂows’, Phys. Fluids 10 499–515. Correa, S.M. and Pope, S.B. (1992) ‘Comparison of a Monte Carlo PDF/ﬁnitevolume mean ﬂow model with bluﬀ-body Raman data’, Proc. Combustion Institute 24 279–285. Correa, S.M., Gulati, A. and Pope, S.B. (1994) ‘Raman measurements and joint PDF modeling of a nonpremixed bluﬀ-body-stabilized methane ﬂame’, Proc. Combustion Institute 25 1167–1173. Daly, B.J. and Harlow, F.H. (1970) ‘Transport equations in turbulence’, Phys. Fluids 13(11), 2634–2649. Dopazo, C. (1994) ‘Recent developments in PDF methods’. In Turbulent Reacting Flows, P. Libby and F. Williams (eds.), Academic Press, 375–474. Fallot, L., Gonzalez, M., Elamraoui, R. and Obounou, M. (1997) ‘Modelling ﬁniterate chemistry eﬀects in non-premixed turbulent combustion: test on the bluﬀ-body stabilized ﬂame’, Comb. Flame 110 298–318. First ASCF Workshop (1994) ‘Steady-state combustion chambers and furnaces’. Test case speciﬁcations, ERCOFTAC-SIG. First ASCF Workshop (1994) ‘Steady-state combustion chambers and furnaces’. Final results, ERCOFTAC-SIG. Fox, R.O. (1996) ‘On velocity-conditioned scalar mixing in homogeneous turbulence’,

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Phys. Fluids 8(10), 2678–2691. Gardiner, W.C. (1984) ‘Introduction to combustion modeling’. In Combustion Chemistry, W. Gardiner (ed.), Springer-Verlag, 1–19. Gardiner, C.W. (1990) Handbook of Stochastic Methods, second edition. SpringerVerlag. Girimaji, S.S. and Pope, S.B. (1990) ‘A diﬀusion model for velocity gradients in turbulence’, Phys. Fluids A 2(2), 242–256. Haworth, D.C. and El Tahry, S.H. (1990) ‘Probability density function approach for multidimensional turbulent ﬂow calculations with application to in-cylinder ﬂows in reciprocating engines’, AIAA Journal 29(2), 208–218. Haworth, D.C. and Pope, S.B. (1986) ‘A generalized Langevin model for turbulent ﬂows’, Phys. Fluids 29 387–405. James, S., Anand, M.S., Razdan, M.K. and Pope, S.B. (1999) ‘In situ detailed chemistry calculations in combustor ﬂow analysis’. In 44th ASME Gas Turbine and Aeroengine Technical Congress’. Indianapolis. 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. Maas, U. and Pope, S.B. (1992) ‘Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space’, Comb. Flame 88(3), 239–264. Masri, A.R. and Pope, S.B. (1990) ‘PDF calculations of piloted turbulent nonpremixed ﬂames of methane’, Comb. Flame 81(1), 13–29. Monin, A.S. and Yaglom, A.M. (1971) Statistical Fluid Mechanics. MIT Press. Muradoglu, M., Jenny, P., Pope, S.B. and Caughey, D.A. (1999) ‘A consistent hybrid ﬁnite-volume/particle method for the PDF equations of turbulent reactive ﬂows’, J. Comp. Phys. 154 342–371. Nooren, P.A., Wouters, H.A., Peeters, T.W.J., Roekaerts, D., Maas, U. and Schmidt, D. (1997) ‘Monte Carlo PDF modelling of a turbulent natural-gas diﬀusion ﬂame’, Comb. Theor. Modelling 1(1), 79–96. Nooren, P.A. (1998) Stochastic modeling of turbulent natural-gas ﬂames. PhD thesis, Delft University of Technology. Peeters, T.W.J., Roekaerts, D. and Hoogendoorn, C.J. (1993) ‘Modelling of turbulent non-premixed ﬂames. Part 1. Conserved-scalar chemistry models’. Technical Report BCWT.93.1, Technische Universiteit Delft. Peeters, T.W.J., Roekaerts, D. and Hoogendoorn, C.J. (1993) ‘Modelling of turbulent non-premixed ﬂames. Part 4. Adaptive property tabulation’. Technical Report BCWT.93.4, Technische Universiteit Delft. Peeters, T.W.J. (1995) Numerical modeling of turbulent natural-gas diﬀusion ﬂames. PhD thesis, Delft Universitity of Technology.

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Peters, N. and Rogg, B. (1993) Reduced Kinetic Mechanisms for Applications in Combustion Systems, Lecture Notes in Physics m15. Springer-Verlag. Pope, S.B. and Chen, Y.L. (1990) ‘The velocity-dissipation probability density function model for turbulent ﬂows’, Phys. Fluids A 2(8), 1437–1449. Pope, S.B. (1983) ‘A Lagrangian two-time probability density function equation for inhomogeneous turbulent ﬂows’, Phys. Fluids 26 3448–3450. Pope, S.B. (1985) ‘PDF methods for turbulent reactive ﬂows’, Prog. Comb. Theory Comb. Sci. 11 119–192. Pope, S.B. (1991) ‘Application of the velocity-dissipation probability density function model to inhomogeneous turbulent ﬂows’, Phys. Fluids A 3(8), 1947–1957. Pope, S.B (1994) ‘Lagrangian PDF methods for turbulent ﬂows’, Ann. Rev. Fluid Mech. 26 23–63. Pope, S.B (1994) ‘On the relationship between stochastic Lagrangian models of turbulence and second-moment closures’, Phys. Fluids 6(2), 973–985. Pope, S.B. (1997) ‘Computationally eﬃcient implementation of combustion chemistry using in situ adaptive tabulation’, Comb. Theor. Modelling 1(1), 41–63. Risken, H. (1984) The Fokker–Planck Equation. Springer-Verlag. Saxena, V. and Pope, S.B. (1998) ‘PDF calculations of major and minor species in a turbulent piloted jet ﬂame’. In Proc. Combustion Institute 27 1081–1086. Schmidt, D, Riedel, U., Segatz, J., Warnatz, J. and Maas, U. (1996) ‘Simulation of laminar methane-air ﬂames using automatically simpliﬁed chemical kinetics’, Comb. Sci. Tech. 113-114 3–16. Seshadri, K. and Williams, F.A. (1994) ‘Reduced chemical systems and their application in turbulent combustion’. In Turbulent Reacting Flows, P. Libby and F. Williams (eds.), Academic Press, 153–210. Smooke, M.D. (1991) Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane Air Flames, Lecture Notes in Physics 384. Springer-Verlag. Sung, C.J., Law, C.K. and Chen, J.Y. (1998) ‘An augmented reduced mechanism for methane oxidation with comprehensive global parametric validation’, Proc. Combustion Institute 27 295–304. Van Slooten, P.R. and Pope, S.B. (1997) ‘PDF modeling of inhomogeneous turbulence with exact representation of rapid distortions’, Phys. Fluids 9 1085–1105. Warnatz, J., Maas, U. and Dibble, R.W. (1996) Combustion. Springer-Ferlag. Williams, F.A. (1985) Combustion Theory, second edition. Benjamin/Cummings. Wouters, H.A., Nooren, P.A., Peeters, T.W.J. and Roekaerts, D. (1996) ‘Simulation of a bluﬀ-body-stabilized diﬀusion ﬂame using second-moment closure and Monte Carlo methods’, Proc. Combustion Institute 26 177–185. Wouters, H.A., Nooren, P.A., Peeters, T.W.J. and Roekaerts, D. (1998) ‘Eﬀects of

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micro-mixing in gas-phase turbulent jets’, Int. J. Heat Fluid Flow 19 201–207. Wouters, H.A. (1998) Lagrangian models for turbulent reacting ﬂows. PhD thesis, Delft University of Technology. Yang, B. and Pope, S.B. (1998) ‘Treating chemistry in combustion with detailed mechanisms – in situ Adaptive Tabulation in Principle Directions – premixed combustion’, Comb. Flame 112 85–112.

Part C. Future Directions

22 Simulation of Coherent Eddy Structure in Buoyancy-Driven Flows with Single-Point Turbulence Closure Models K. Hanjali´c and S. Kenjereˇs 1

Introduction

A major deﬁciency of conventional single-point closure models that are used in conjunction with Reynolds-averaged Navier–Stokes methods (RANS) is in their inability to account for a wide range of turbulence time and length scales, which characterize various mechanisms in turbulence dynamics. By their nature, RANS methods conceal any spectral and structural information, and have been regarded as unsuitable for detecting any identiﬁable eddy structure. This, in turn, prevents accounting for any spectral features, interactions between eddies of diﬀerent scales and energy transfer through the spectrum. Yet, it is known that the spectral dynamics is diﬀerent in every ﬂow and inﬂuences the gross turbulence features. Single-point RANS closures compensate for these deﬁciencies partially by various additions to the basic model to account nominally for departure from the simple ﬂows in which they were tuned and thus, indirectly, for spectral non-equilibrium. A way to partially overcome the problem is to introduce one or more additional turbulence scales by which to represent distinct parts of the turbulence spectrum. Such multi-scale models have been proposed e.g. by Hanjali´c, Launder and Schiestel (1980), Schiestel (1987) and others. Here the turbulence energy spectrum is divided into two or more parts, and the transport equations for turbulent kinetic energy (or the full stress tensor) and for the energy dissipation rate are solved for each spectral slice, including also their mutual interaction. Other multiple-scale models have also been proposed in the literature, with moderate success. A major problem appears in the large number of unavoidable empirical coeﬃcients, which are diﬃcult to determine because of the lack of information on spectral dynamics in various ﬂows. More advanced spectral models, that account fully for spectral dynamics have also been proposed (e.g. Besnard et al. 1996, Touil et al. 2000). These models look promising, but have not yet been widely tested and it is uncertain how suitable and attractive they are for predicting real complex ﬂows. A particular problem arises when the ﬂow is dominated by large coherent vortical structures. These large-scale eddies can be discerned in many turbulent ﬂows whether in the laboratory, in nature or in industrial appliances. 659

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They act as a major carrier of momentum, heat and species. In wall bounded non-isothermal ﬂows, where the walls are at diﬀerent temperatures, these eddies often provide the major communication link between the bounding walls and thus control the exchange of heat between them. Often, as in the case of vortex shedding, in ﬂows driven by buoyancy, or subjected to a magnetic ﬁeld or system rotation, the large structure has a deterministic character. Such vortical structures also appear in laminar ﬂows, but in turbulent regimes it is not always clear whether these structures should be regarded as true turbulence (smooth spectrum and probability density functions), or whether they should be interpreted as a form of mean motion with inherent, but deterministic (organized) unsteadiness. While the form of secondary motion can be captured with advanced (e.g. second-moment) closures, the convective transport by large turbulent eddies, usually referred to as ‘turbulent diﬀusion’, is modelled by gradient hypotheses, which is unrealistic for representing such a process. Because of their importance in governing the turbulence dynamics, the only way to account fully for their eﬀects is to resolve the large-scale eddies in space and time. Large-eddy structures can be fully resolved by Large Eddy Simulation (LES) techniques and these have long been viewed as the future preferred computational method that should soon replace the RANS approach as an industrial tool. While conventional LES has proven to be a very useful method in studying turbulent ﬂows, over the years it has shown some serious limitations. The major problem is in the excessive computational costs which increase sharply with the Reynolds and Rayleigh numbers, associated with the need to resolve a larger and larger range of eddy scales. Handling complex geometries, the treatment of wall boundaries and the resolution of near-wall ﬂow regions in wall-bounded ﬂows are other challenges that have not yet been fully resolved. The latter issue is especially critical if wall friction and heat transfer are in focus. A proper resolving of the buﬀer region and capturing the streaky structures and small-scale eddies occupying this region require a very ﬁne computational mesh not only in the wall-normal direction, but also in the spanwise and streamwise directions, imposing formidable requirements on the mesh density, which becomes comparable with that for a complete direct numerical simulation (DNS). In fact, most LES reported in the literature have reproduced successfully the wall friction factor and heat transfer coeﬃcients by employing a very ﬁne mesh so that the subgrid-scale stresses are of the same order as the viscous stresses. The use of wall-functions to bridge the near-wall region, as is done in RANS methods, appears to be a possible compromise, but so far without much success. These features make the LES still inapplicable to solving technological ﬂows at very high Reynolds and Rayleigh numbers. To quote Speziale (1998) ‘the traditional LES has never really lived up to its promise’.

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VLES and Hybrid RANS/LES methods

In order to overcome the cost problems and still resolve large-scale structures – at least in ﬂows where they play a dominant role, several intermediate or hybrid methods have been proposed recently, aimed at utilizing the advantages of both approaches: the simplicity and computational eﬃciency of RANS and the potential of LES to fully resolve a large-scale part of turbulence spectrum. These methods can generally be classiﬁed as Very-Large Eddy Simulation (VLES), the name implying essentially a form of LES with a cut-oﬀ ﬁlter at much lower wave number. This means: resolve less and model more! Modeling a larger part of the spectrum requires a more sophisticated model than the standard sub-grid-scale model, i.e. a form of RANS model that is not related to the size of the numerical mesh. The solution of the resolved part of the spectrum can either follow the traditional LES practice using grid size as a basis for deﬁning the ﬁlter (hence the name hybrid RANS/LES), or solve ensemble- or conditionally-averaged equations which implicitly involve time ﬁltering. Ha Minh and Kourta (1993) (also Aubrun et al. 1999) proposed, what they called a ‘semi-deterministic’ method (SDM), based on the decomposition of physical variables into a coherent part, obtained by an ensembleaveraging, and an incoherent part which is the residue of the ensemble-average operator, see Fig. 1. The coherent part is resolved by time integration of the three-dimensional ensemble-averaged Navier–Stokes equations (having the same form as the Reynolds-averaged equations), in which the ensemble-averaged (‘incoherent’) stresses are provided from a conventional single-point low-Re-number k-ε model. However, in the ﬂow which they considered (the backward-facing step), such a model gave a too high turbulent viscosity that suppressed instabilities and unsteadiness and failing to resolve large eddy structure, resulted in a steady solution. For that reason, the eddy viscosity was arbitrarily reduced by decreasing the value of the coeﬃcient Cµ from the conventional value of 0.09 to 0.06. This problem may be associated with the particular turbulence model used, which may be too dissipative: Tatsumi et al. (1999) succeeded in obtaining unsteady solutions and captured large eddy structures behind a step using a more advanced non-linear eddy-viscosity model without any re-tuning of the model. Unsteady computations of ﬂows around bluﬀ bodies with eddy shedding, even in cases when this shedding is not periodic but stationary over long-term average (quasi-steady recirculation), using conventional single-point closure with a suﬃciently small time step, can also be classiﬁed as a form of SDM. Such an approach essentially implies a separation of time scales: those of the shedding eddies and those of the rest of turbulence. Such computations, with a proper accounting for the contribution of the shedding eddies to the secondmoments (‘apparent stresses’), which may even outweigh the modelled ones, have produced much better agreement with experiments than steady compu-

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Figure 1: Original sketch of spectrum decomposition from Ha Minh and Kourta (1993)

tations with the same model (e.g. Johansson et al. 1993, Durbin 1995, Bosch and Rodi 1998). More recently, Spalart (1999) (also Travin et al. 1999, Nikitin et al. 2000) proposed a detached-eddy simulation (DES) method as a hybrid RANS/LES technique in which a single turbulence model (of any level) is used as a subgrid-scale model in regions away from solid boundaries, and as a RANS model in the wall boundary layer. The transition from the RANS to LES regions is continuous and smooth, activated automatically when the distance from the wall becomes larger than the prescribed criterion – here the largest dimension of the grid cell. In the LES region, the method requires grid spacing in all directions to be ﬁne enough to allow traditional large eddy simulations, whereas in the RANS region (close to a wall) the grid reﬁnement is needed only in the wall-normal direction, as required by the turbulence model applied. This technique was originally aimed at external aerodynamics at high Reynolds numbers, with thin boundary layers and massive separation regions, but seems applicable to other types of ﬂow. Although it brings signiﬁcant cost saving by allowing near-wall spanwise and streamwise grid spacing to be much larger than in full LES (no resolution of streaks and near-wall eddy structure), the costs may still be excessive for ﬂows at very large Reynolds numbers. Deﬁning the distance from a wall of complex topology may also pose a problem and brings in some arbitrariness.

3

The Time-Dependent RANS (T-RANS)

We consider here the Time-dependent RANS (T-RANS) approach as a convenient method to compute ﬂows with coherent vortical structure at very high Reynolds and Rayleigh numbers, where DNS and LES are inapplicable, and where some hybrid methods, such as DES may also have diﬃculties. Specifically, we focus here on Rayleigh–B´enard convection in which a distinct ‘or-

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ganized’ large-scale structure is known to exist. The approach can also be applied to other ﬂows with a dominant large eddy structure (vortex shedding, internal separation and recirculation, longitudinal vortices, natural convection in enclosures, ﬂows with rotation and in a magnetic ﬁeld). In addition to predicting ﬂow features and heat transfer accurately we also demonstrate that the T-RANS approach can serve to identify the organized motion and its reorganization due to imposed ﬂow control methods, be it of a distributed type (extra body force, such as magnetic), or a boundary control (non-plane wall conﬁguration). The approach belongs to the class of VLES, and is close to the SDM method of Ha Minh and Kourta (1993)1 by which the large structure is fully resolved by solving three-dimensional ensemble-averaged Navier–Stokes equations in time and three-dimensional space, whereas the ‘rest’ of the turbulence is modelled by conventional turbulence closure models (‘subscale model’). As compared to LES, here both contributions to the turbulent ﬂuctuations and long-term statistical averages are of equal order of magnitude. Close to a solid wall, the unresolved (modelled) part is more dominant, and this imposes a special requirement to model accurately the wall phenomena. On the other hand, resolving the large-scale motion enables one to capture accurately the large scale transport, which in conventional RANS methods is usually modelled inadequately by gradient hypotheses. For buoyancy-driven ﬂows the unresolved random motion is modelled using a low-Re-number k-ε-θ2 algebraic stress/ﬂux closure model. The large-scale deterministic motion, which is the major mode of heat and momentum transfer in the central region, is fully resolved by the time solution. The considered T-RANS closure also brings a substantial computational advantage. The ‘sub-scale model’ – here RANS – is less dependent on the spatial grid. The time step can be larger, allowing implicit time marching, the numerical mesh away from a solid boundary does not need to be very ﬁne and is not directly related to the subscale model. Because only very large coherent eddies are resolved, good statistics can be obtained with a relatively small number of realizations. The problem of deﬁning inﬂow conditions at open boundaries is less restrictive than in LES. Because of the possibility of treating very high Ra and Re numbers the approach can be used for the computation of complex technological and environmental ﬂows of practical relevance. It is worth mentioning that the T-RANS approach can also be used for two-dimensional simulations of ﬂows with one homogeneous direction. In this case the simulation can still reproduce the coherent large-scale structure, but ensemble-averaged in the homogeneous direction, which is still more than a standard RANS can do. Such a simulation of Rayleigh–B´enard convection 1

At the time we started exploring this approach in early 1995, we were not aware of the work of Ha Minh’s group: we considered a very diﬀerent type of ﬂows, driven by buoyancy, and the subscale model was also diﬀerent, hence a diﬀerent name, T-RANS

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revealed the convective-roll-cell pattern and the wall Nusselt number in broad accord with the ensemble-averaged full 3-dimensional T-RANS simulations (Kenjereˇs and Hanjali´c 2000).

3.1

Equations

The instantaneous motion of an incompressible ﬂuid ﬂow driven by thermal buoyancy is described by the continuity, momentum and energy equation (using the Boussinesq approximation for density variation): ˜i ∂U =0 ∂xi ˜i ∂U ∂ = ∂t ∂xj

˜i ∂U ˜i U ˜j ν −U ∂xj

−

(1)

˜ 1 ∂ P − Pref ρ

∂xi

˜ ∂Θ ∂ = ∂t ∂xj

˜ − Θref + + βgi Θ

F˜i (2)

˜ ν ∂Θ ˜U ˜j . −Θ P r ∂xj

(3)

5 The term F˜i stands for other body forces if they appear (Coriolis force if the ﬂow is subjected to system rotation, Lorentz force in the case of an electrically conductive ﬂuid in a magnetic ﬁeld, etc.). Here we deal only with the thermal buoyancy and this force has been expressed separately, but the same approach has also been successfully used for ﬂows subjected to combined buoyancy and magnetic force (Hanjali´c and Kenjereˇs 2000). For the resolved (‘ﬁltered’) motion, the above equations can be written as (essentially the same form as for the LES):

∂Ui =0 ∂xi ∂Ui ∂Ui + Uj ∂t ∂xj ∂ = ∂xj

∂Ui ν − τij ∂xj

−

(4)

1 ∂ (P − Pref ) + βgi (Θ − Θref ) + Fi ρ ∂xi

∂Θ ∂Θ ∂ = + Uj ∂t ∂xj ∂xj

(5)

ν ∂Θ − τθj , P r ∂xj

(6)

where the stands for resolved (implicitly ﬁltered) quantities, and τij and τθj represent contributions due to unresolved scales to the momentum and temperature equations respectively, which are both provided by the subscale model.

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The subscale model

Because only the largest scales are fully resolved in time and space, care must be taken to provide an adequate ‘subscale’ model for the remaining, relatively large, unresolved part of turbulence spectrum. In the near-wall region the direct eﬀect of the large-scale structure is small and the subscale model provides a major contribution to the turbulent transport of momentum, heat and matter. Hence, the role of the subscale model is more important than, for example, the subgrid-scale model in LES. Besides, the subscale model should not be dependent on grid size, as it is in LES, simply because the unresolved motion spans a larger range of scales than deﬁned by the numerical mesh. It seems natural to use a single-point closure as practiced in the RANS approach. In some ﬂows where the coherent structure is not as dominant and the spectrum has a smooth shape, the single-point closure model may need modiﬁcation as mentioned earlier (e.g. Ha Minh and Kourta 1993). However, because in the ﬂows considered here the scale of the large structure is well separated from the rest of turbulence, there is no need for such a modiﬁcation. Furthermore, the turbulent transport by large-scale motion is fully resolved, and the importance of the subscale model is particularly signiﬁcant very close to the wall where the large-scale convection is usually negligible. Hence, there seems to be no need to solve diﬀerential transport equations for second-moments (turbulent stress, heat ﬂux), and simple algebraic models may suﬃce. The study reported here was performed using a ‘reduced’ algebraic expression for heat ﬂux τθi ≡ θui , derived by truncation of the modelled RANS diﬀerential transport equation for θui by assuming weak equilibrium, but retaining all major ﬂux production terms (all treated as time-dependent):

k ∂Θ ∂Ui τθi = −Cφ τij + ξτθj + η βgi θ2 + θfi , ε ∂xj ∂xj

(7)

5

where fi represents other ﬂuctuating body forces, and ξ = 1 − C2θ = 0.45 and η = 1 − C3θ = 0.45 are empirical coeﬃcients (see Section 2.3 in [2]). The turbulent stress tensor τij = ui uj and the correlations involving the other body forces should also be expressed in similar algebraic forms by truncation of the full transport equations for these quantities. However, in the present work we use simple eddy diﬀusivity expressions for these moments and account for the buoyancy by additional terms in the transport equations for the turbulent kinetic energy k and its dissipation rate ε, e.g.

τij = −νt

∂Ui ∂Uj + ∂xj ∂xi

2 + kδij . 3

(8)

The closure of the expressions for subscale quantities is achieved by solving the equations for turbulent kinetic energy k, its dissipation rate ε and temperature variance θ2 (all modiﬁed for low-Re-number and near-wall eﬀects),

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where (omitted for clarity in the equations that follow) indicates that all terms in these equations are computed from the instantaneous resolved quantities, resulting in the three-equation model k-ε-θ2 : Dk Dt Dθ2 Dt Dε Dt

= Dk + Pk + Ggk + Gfk − ε

(9)

= D θ + Pθ − ε θ

(10)

= Dε + Pε1 + Pε2 + Ggε + Gfε − Y

(11)

where D stands for diﬀusion, P is production by mean ﬁeld gradients, Gg is production by thermal buoyancy and Gf by other body forces, ε is turbulence energy dissipation, and Y is destruction of ε. All these terms are deﬁned by conventional expressions using standard values for the coeﬃcients and the thermal-to-mechanical turbulence time-scale ratio R = θ2 ε/kεθ = 0.5. For the implementation of Lorentz-force eﬀects in an electrically conductive ﬂuid in a magnetic ﬁeld, see Hanjali´c and Kenjereˇs (2000a, b).

3.3

Evaluation of second moments

Comparison of T-RANS and DNS statistics, which serves as a ﬁrst check if the T-RANS approach is meaningful, requires care in the interpretation of the statistics in order to account for the contribution of the resolved motion. The triple decomposition, by which the instantaneous ﬁeld is assumed to consist of long-term average (time-mean), ensemble-averaged (quasi-periodic) large-scale structure and random (stochastic) ﬂuctuations, provides a satisfactory tool, indicating that the long-term statistics can be well reproduced by accounting for both the random and ensemble-averaged contributions. For steady ﬂows with a distinct large-scale deterministic structure, any instantaneous ﬂuid ﬂow ˜ i , t) can be decomposed into time-mean Ψ(xi ), quasiproperty at a point Ψ(x ˆ i , t) and random ψ(xi , t): periodic (deterministic) Ψ(x ˆ (xi , t) + ψ (xi , t) = Ψ (xi , t) + ψ (xi , t) . ˜ (xi , t) = Ψ (xi ) + Ψ Ψ

(12)

By performing long-term time averaging at a point in space and assuming that because of the spectral gap between them, the deterministic and random motion are not directly interacting, the second moments are obtained as the sum of deterministic and modelled contributions. For example, it follows from applying long-term averaging, the second moment for two arbitrary variables Ψ and Υ is: ˜Υ ˜ =Ψ Υ+Ψ ˆΥ ˆ + ψγ. (13) Ψ In order to illustrate the implication of the approach proposed above, we consider the long-term averaged energy equation, which for steady Rayleigh–

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B´enard convection reduces to: ∂ ∂xj

ν ∂Θ − ΘUj − τθj P r ∂xj

= 0,

(14)

where τθi is provided from the subscale model presented above, i.e.:

k ∂Θ ∂Ui τθi = θui = −Cφ τij + ξτθj + η βgi θ2 + θfi . ε ∂xj ∂xj

(15)

Further averaging over homogeneous (horizontal) planes yields the expression for the total heat ﬂux in the vertical (z) direction W = Uz and U j = 0): ν ∂Θ ˆ ˆ − ΘW − τθw = const. P r ∂z

4

(16)

T-RANS Simulation of Rayleigh–B´ enard Convection

In order to demonstrate the capability of the T-RANS approach, we investigate several cases of steady Rayleigh–B´enard convection over ﬂat and wavy bottom walls. Rayleigh–B´enard convection is characterized by self-organized, large-scale convective roll cells, which ﬁll the vertical spacing between the two horizontal walls. This structure originates from plumes which rise from the outer edge of the boundary layer at the heated surface (updrafts) and sink downward from the upper cold boundary (downdrafts). The rise of plumes and their impingement on the opposite horizontal surface produce a horizontal motion in the wall boundary layer which governs the wall heat transfer. This in turn generates buoyancy which causes the rise of plumes. As the Ra number increases, the regularity of the cell pattern disappears, the plumes detach from the horizontal boundary layers and evolve into thermals. The large convective roll cells become unsteady and more disorderly, ﬂuctuating in amplitude and orientation with a frequency characterized by the Ra number. However, the small scale ﬂuctuations in velocity and temperature, originating primarily in the wall boundary layers, are transported by convective cells much as a passive scalar (e.g. Cioni et al. 1996, 1997). Both experiments (e.g. Chu and Goldstein 1973) and direct numerical simulations, DNS, (Gr¨ otzbach 1982, Cortese and Balachandar 1993) indicate that despite disorder, large coherent structures can be identiﬁed even at very high Ra numbers. A recent DNS by Kerr (1996) in the range of Ra numbers close to the fully turbulent regime (Ra ≈ 2 × 107 ) shows that the large structure governs the apparent chaotic behaviour of turbulent Rayleigh–B´enard convection. This evidence indicates the existence of two distinct scales of motion: large amplitudes associated with

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thermals, plumes and convective cells, and the small-scale turbulence generated mainly in the wall boundary layer and carried away by the large scale structure. The separation of the scales of the coherent convective cellular motion from the rest of turbulence in Rayleigh–B´enard convection (and other turbulent ﬂows with dominant large structures) makes these ﬂows very suitable for the VLES technique. By fully resolving the large-scale deterministic convective structure and associated momentum and heat transport (regarded as particularly diﬃcult to model with single-point closures), a simple eddy-diﬀusivity – or algebraic closure can be used to model the unresolved motion. For the conventional ﬂat-wall Rayleigh–B´enard convection a series of computations was performed covering Rayleigh numbers from 6.5×105 to 1012 (currently being extended to Ra = 1015 !). It is noted here that the present limit on DNS is Ra ≈ 2 × 107 (Kerr 1996), and that the highest LES simulated Ra number available in the literature is Ra = 108 (Eidson 1985, Peng and Davidson 2000). However, already at this Ra number the well resolved LES becomes very costly, at least when accurate heat transfer is required, because of the need to use a very dense grid in all three directions in the very thin thermal boundary layers on the walls. The T-RANS computations were ﬁrst performed for Ra = 6.5 × 105 which is the same as in the DNS of W¨ orner (1994), to enable a direct comparison. We also performed LES for the same Ra number, using the same mesh as for T-RANS, which enabled comparison of all three simulation approaches. The larger values of Ra number were then considered in order to demonstrate applicability of T-RANS to high Ra numbers where neither DNS nor LES can be applied.

4.1

Rayleigh–B´ enard convection over a ﬂat wall

Quantitative validation. We present ﬁrst the T-RANS computations of the long-term averaged temperature proﬁles and important second-moments: turbulent heat ﬂux and temperature variance in the vertical direction for classical Rayleigh–B´enard convection. The results were obtained by averaging over horizontal homogeneous planes and are compared with the DNS results of Gr¨ ozbach (1990) and W¨ orner (1994). The mean temperature proﬁles, (denoted in the ﬁgures as T rather than Θ) normalized with the wall temperature Tw and the domain height D for a range of Rayleigh numbers from 3.8 × 105 to 109 , see Fig. 2, show the characteristic uniform temperature in the core region with steep gradients in the wall thermal boundary layers that become progressively thinner as the Rayleigh number increases. Fig. 3 shows the predicted long-term averaged turbulent heat ﬂux in the vertical direction (a) and temperature variance (b) over the channel height for Ra = 107 . The contributions of both, the resolved and modelled parts are of the same order of magnitude, the resolved part dominating over most of the channel

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Figure 2: Long-term averaged vertical temperature proﬁles in Rayleigh–B´enard convection (Kenjereˇs and Hanjali´c 1999a) central region, and the modelled part dominating in the near-wall region. The molecular contribution to the heat ﬂux is also plotted, and the sum of all three a.

b.

Figure 3: The modelled and resolved parts of: (a) vertical turbulent heat ﬂux; (b) temperature variance (Kenjereˇs and Hanjali´c 1999a)

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a.

b. Figure 4: Normalized mean temperature proﬁles in Rayleigh–B´enard convection, (a.) near-wall blow-up, (b.) semi-logarithmic plot for T-RANS and DNS over range of Ra numbers (Hanjali´c and Kenjereˇs 2000a) contributions, normalized with the wall heat ﬂux, shows that the long-term averaged total heat ﬂux is constant over the entire channel height (here equal to 1). A near-wall blow-up of the same results, as well as semi-logarithmic plots over the entire channel height, normalized with the buoyancy temperature Tq = Q/Uq and length scale Zq = α/Uq , where Q is the wall heat ﬂux and Uq = (βα2 gQ/ν)1/4 is the buoyancy velocity, are shown in Figs. 4 and 5.

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a.

b. Figure 5: Distribution of normalized vertical heat ﬂux (near-wall blow up) (a.), and temperature variance (b.) in Rayleigh–B´enard convection: T-RANS and DNS (Hanjali´c and Kenjereˇs 2000a) Excellent agreement is obtained despite signiﬁcant diﬀerences in Rayleigh numbers, providing a suﬃcient proof that the T-RANS method is capable of reproducing faithfully the long-term averaged temperature ﬁeld. It is noted that although, in the long-term average over homogeneous horizontal planes, the ﬂow is steady and seems very simple with only one (vertical) inhomogeneous direction, the dominating large-scale structure in transporting

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momentum and heat is the major reason for the failure of a steady computations with conventional single-point closures to reproduce the mean ﬂow features and turbulence statistics in Rayleigh–B´enard convection. The main deﬁciency of eddy-viscosity/diﬀusivity models, if considered in a steady mode, is the gradient transport hypothesis for the momentum and heat ﬂux. Secondmoment closure, in which the turbulent ﬂux is determined from a diﬀerential equation, oﬀers no better prospects because the gradient transport model of triple moments and, especially, of pressure diﬀusion, seems to be totally inadequate for this type of ﬂows (e.g. W¨ orner 1994). It will be interesting to see whether the 3rd-moment treatments advocated in [14] and [15] are more successful in reproducing such ﬂows. Qualitative validation: structure morphology. A second test of the applicability of the T-RANS approach is the comparison of structure morphology in DNS, LES and T-RANS realizations. Various identiﬁcation criteria have been applied including numerical visualization of the coherent structure morphology, critical point theory and vortex dynamics. An illustration of the T-RANS performance and its relation to DNS and LES is given in Fig. 6, showing the instantaneous structure, captured by the kinematic vorticity number Nk = (ωk2 /2Sij Sij )1/2 for DNS, LES and T-RANS. A very similar picture is obtained if the second invariant ∆ of the velocity gradient tensor Aij = ∂Ui /∂xj is used instead of Nk (not shown here). All three computations were performed for the same Rayleigh number, Ra = 6.5 × 105 . The captured structure depends on the adopted threshold value of Nk , which, in this case, was chosen to be 2. In the background is the numerical mesh used in each computation. Note that the grid is uniform in the homogeneous direction and clustered close to the walls in the wall-normal direction. The ﬁgures illustrate the eﬀect of ﬁltering: while the structure in all cases is irregular but evenly distributed in the homogeneous planes, it is clear that both the LES and the T-RANS ﬁlter out the small-scales, and that the implicit cut-oﬀ wave number in the T-RANS is smaller than in the LES, the former displaying only the very large structures. A three-dimensional snap-shot of the plumes and thermals structure is given in Fig. 7a, showing two isothermal surfaces of the instantaneous nondimensional temperature Θ∗ = Θref ± 0.1, where the + sign corresponds to plumes rising from the bottom wall and the − sign to those descending from the upper wall. Note that the two surfaces in Fig. 7a for the selected values of Θ∗ correspond to temperature contours more in the plume cores, hence they exhibit a deviation from the expected mushroom shapes. Such types of structure with ﬁnger-like plumes and with horizontal (‘planform’) structures in between have been detected earlier both by experiments and by DNS. Temperature ﬁelds in the vertical and horizontal midplanes, given by diﬀerent intensities of grey scale, for the same instant of realization as in Fig. 7a, are given in Figs. 7b

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orner (1994) DNS: 201×201×51, W¨

LES: 82×82×32

T-RANS:

Figure 6: Qualitative comparison of captured vortical structures for DNS, LES 1/2 = and T-RANS realization: Ra = 6.5×105 , P r = 0.71, Nk = |ωi |2 /2Sij Sij 2; top view, with underlying computational mesh (Hanjali´c and Kenjereˇs 2000b) and 7c. The network of polygonal cells with ﬁngerlike plumes in between, usually associated with laminar Rayleigh–B´enard convection, is clearly visible in the horizontal plane despite the fact that the Ra number (107 ) is well into the turbulent regime. These plumes are actually the main carriers of heat, and the plume locations correspond to those where the local Nusselt numbers reach their maximum values, as seen in Fig. 12a (to be discussed below). The plumes move randomly and interact with each other causing a strong vertical motion. Again, very similar pictures were obtained from DNS (not shown here)

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a.

b.

c.

Figure 7: Thermal plumes (a) and temperature ﬁelds in vertical (b) and horizontal (c) midplanes in Rayleigh–B´enard convection, Ra = 107 , P r = 0.71 (Hanjali´c and Kenjereˇs 2000b)

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Figure 8: Instantaneous trajectories of massless particles in DNS at Ra = 6.5 × 105 (left) and T-RANS at Ra = 107 (right) indicating spiraling updrafts (above) and evolution from three line sources (below) (Kenjereˇs and Hanjali´c 1999b) and T-RANS results, except for a diﬀerence in scales. The identical scenario of planform structures has been observed in experimental studies of Theertan and Arakeri (1997). Next we compare directly some results of visualization of the instantaneous structure patterns (randomly selected realizations) in DNS and T-RANS computations for the same or similar Rayleigh numbers. Of particular interest is the identiﬁcation of the ﬁnger-like regions with intense spiraling updrafts extending in the vertical direction. Cortese and Balachandar (1993) detected such a structure in their DNS arguing that the origin of vorticity is in horizontal ﬂow, induced by ﬂuid motion towards irregularly spaced sites of plume or thermal release (or away from irregularly spaced stagnation points). The tilting and stretching of rising plumes by buoyancy in the vertical direction produce spiral structures. Fig. 8a, obtained by releasing a number of massless particles in one frozen realization of DNS of W¨ orner (1994) and one T-RANS realization, indeed supports the above ﬁnding: the trajectories show a spiraling tendency. A release of particles from line sources placed at the vertical midplane (Fig. 8b) showed also the spiraling motion around randomly oriented horizontal axes. A better demonstration of the existence of three-dimensional vortical structures is obtained by plotting the projection of instantaneous trajectories on the central horizontal plane, z ∗ = 0.5 and on several vertical

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DNS

T-RANS

Figure 9: Instantaneous trajectories of massless particles in central horizontal and ﬁve vertical planes for DNS, W¨ orner (1994) and T-RANS realization both for Ra = 6.5 × 105 , P r = 0.71 (Hanjali´c and Kenjereˇs 2000a) planes, see Fig. 9. These trajectories are plotted by releasing 1500 massless particles from uniformly distributed origins over the sampling planes of the instantaneous ﬁelds, and their distributions were calculated by applying a second-order, Runge-Kutta time-advection method. Although the streamline pictures portray very complex ﬂow, three primary and distinctive regimes can be easily observed: the regions with strong and well deﬁned plane circulation (roll structure), the regions with one-dimensional movements (dark lines) and divergent stagnation regions (unstable focus points). As seen, the DNS and T-RANS results show qualitatively very similar patterns. The only diﬀerence is in the size of the rolls – DNS shows smaller roll patterns but this is to be expected because the T-RANS can per se capture only the very large structure while the smaller ones are ﬁltered out.

4.2

Eﬀects of Wall Topology: Rayleigh–B´ enard Convection over Wavy Walls

In order to further illustrate the potential of the T-RANS approach, we consider high Ra number (up to Ra = 109 ) Rayleigh–B´enard convection over heated wavy walls with diﬀerent wave lengths and amplitudes. The two-dimensional (2D) and three-dimensional (3D) topologies are deﬁned by sinusoidal surface variation in one and two directions, SB (x) = 0.1 cos(xπ), and SB (x, y) = 0.1 cos(xπ) cos(yπ), respectively. Krettenauer and Schumann (1992) performed DNS and LES of the same 2D conﬁguration, though at a much lower Rayleigh number, 5.5 × 104 , and observed that the gross features of the ﬂow statistics,

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Figure 10: Eﬀects of bottom wall topology on large coherent structures (Nk = 2) at τ ∗ = 50, and contours of resolved vertical velocity W , temperature Θ and of subscale (modelled) temperature variance θ2 at τ ∗ = 200 (Hanjali´c and Kenjereˇs 2000a). Left ﬁgures 2D wavy wall; right ﬁgures 3D wavy wall such as the proﬁles of turbulence variance and ﬂuxes were not very sensitive to the variations of the bottom-wall topology. On the other hand, the motion’s structure persisted considerably longer over the wavy terrain than over ﬂat surfaces. An insight into the eﬀect of wall topology on the spatial organization of the large coherent structures can be gained from Fig. 10. Here we show the instantaneous structures over the 2D and 3D bottom wavy walls, with topology deﬁned above. The structures are identiﬁed again by the kinematic vorticity number Nk and show the situation in the early stage of ﬂow development (τ ∗ = 50) after the onset of heating in an initially uniform stagnant ﬁeld, Fig. 10a. The large 2D structures extending in the y-direction and located in the centre of the cavity are observed for the imposed 2D topology. Contrary to this, for the 3D wave topology, the coherent structures are located in the near-wall regions and are signiﬁcantly smaller in size. At a later stage the structure loses the initial wall-topology-aﬀected pattern and becomes similar in shape and size for both conﬁgurations. However, a signiﬁcantly diﬀerent ﬂow reorganization can still be observed (diagonally oriented for the 3D topology and around the central y-axis for the 2D topology). In both cases, a close correlation between thermal plumes and large structures can be observed.

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Typical patterns of the structural organization at a later stage (τ ∗ = 200) are presented in Figs. 10b, 10c and 10d, where the eﬀect of the bottom wall conﬁguration is shown on the mean properties (W, Θ) and second moments (θ2 ) respectively for each of the two wall topologies. In the initial stage of heating, τ ∗ = 50 (corresponding to Figs. 10a), the contours of the vertical velocity and temperature show a regular ﬂow pattern determined by the wall conﬁguration and extending in the vertical direction (not shown here), with the sites of the plume generation located at the surface wave crests. However, for the 2D topology at τ ∗ = 200 the locations of the plume realization are not ﬁxed any more, the 2D structure orientation is lost, and the plume’s movement produces a strong horizontal motion. The second moments (the modelled part) show a similar behaviour: the contours of θ2 in the vertical planes indicate that the largest temperature variance is concentrated in the near-wall regions. This is what we expected, since the main role of the subscale model is to produce the correct near-wall behaviour, while in the outer region the largescale-dominated motion is fully resolved. A similar trend appears in the 3D surface wave conﬁguration. Initially, the plumes rise from the surface peaks and sink into the surface valleys. At τ ∗ = 200 the initial organization of the ﬂow cannot be observed anymore, Fig. 10b. The thermal plumes occupy a signiﬁcantly larger space and not simply the regions close to the bottom surface peaks, as found in the initial phase of ﬂow development. Figure 11 shows the inﬂuence of the bottom-wall topology on the instantaneous distribution of the Nusselt number for three diﬀerent bottom-wall conﬁgurations: a ﬂat bottom surface, the 2D and 3D wavy surface topologies, the latter two with the same wave lengths. It was observed that the 3D wave conﬁguration promotes a fully developed regime earlier than others. The wave length in the 2D conﬁguration retains the initial ﬂow organization for a very long period (almost 200 dimensionless time units). When the fully developed stage was reached, the integral Nusselt number approached the value for the ﬂat wall, indicating that the imposed waviness of the bottom wall aﬀects the integral heat transfer in the fully developed state only marginally. The Nusselt numbers shown are for the upper ﬂat cold walls rather than for the bottom wavy walls in order to illustrate the eﬀect of the bottom wall topology on the heat transfer at the opposite ﬂat wall. Both cases show a strong organization in the Nusselt number distribution that reﬂects closely the wall conﬁgurations. The organization in the initial stage (τ ∗ = 50) is more orderly, but the wall-topology eﬀect is also visible much later, (τ ∗ = 300), particularly for 2D waviness. It is also interesting to note that the 3D waviness tends to smooth the Nusselt number distribution in the later stage more eﬃciently than the 2D topology.

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Figure 11: Inﬂuence of bottom wall topology on the instantaneous local Nusselt number: (a) ﬂat bottom wall, (b) 2D waviness, SB = 0.1 cos (xπ), (c) 3D waviness SB = 0.1 cos (xπ) cos (yπ) (Hanjali´c and Kenjereˇs 2000a)

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Figure 12: N u–Ra correlation obtained by 2D and 3D T-RANS compared with the available DNS and experiments (Kenjereˇs and Hanjali´c 2000a)

4.3

The Nusselt–Rayleigh Number Correlation

We now turn back to classical Rayleigh–B´enard convection over a ﬂat wall and discuss the T-RANS prediction of integral heat transfer. A number of experimental, DNS and LES results are available in the literature. The DNS results can serve as a good reference for T-RANS validation at low Ra numbers (up to 2×107 ), where they show a general trend N u ∝ Ra2/7±0.01 in accord with some earlier experiments. For moderate Ra numbers, 107 < Ra < 1010 , several experimental correlations seem to follow the same N u–Ra correlation. However, the high Ra number range is more uncertain: several recent experiments for Ra > 1010 disagree considerably over the Ra-number exponent. While Chavanne et al. (1997) observed a continuous increase in the Ra number exponent, approaching 0.4 at Ra = 5 × 1012 , recent experiments by Niemela et al. (2000) extending to Ra numbers of 1017 found no such trend. The latter authors reported that their measurements are correlated remarkably well by a single power law N u = 0.124Ra0.309 over eleven decades of Ra number, from 106 to 1017 ! It should be noted that early experiments at low Ra numbers were performed in geometries with a low aspect ratio, hence they could be contaminated by the side-wall eﬀects (friction, blockage, heat losses). More recent experiments at very high Ra numbers have been performed with liquid helium at very low temperatures, which can yield only the long-term averaged

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integral heat transfer coeﬃcients, and hardly any information about the ﬂow structure. This brief outline of current uncertainties in establishing the N u–Ra correlation for very high Ra number, even for a very simple geometry such as Rayleigh–B´enard convection, is included here to emphasize the potential of the T-RANS and other VLES techniques, which at present oﬀer the only viable method both to predict heat transfer and to gain insight into the convective roll-cell pattern and other coherent structures and their role in ﬂows driven by buoyancy or other body forces. Fig. 12 shows the N u–Ra correlation obtained by both the 2-dimensional and 3-dimensional T-RANS for a range of Ra numbers from 105 to 1012 . As mentioned earlier, we are extending currently these simulations to still higher Ra numbers. The T-RANS results are in excellent agreement with the available DNS results for lower Ra numbers, as well as with most experimental data, including the most recent correlation of Niemela et al. (2000) for high Ra numbers.

5

Conclusions

A time-dependent Reynolds-averaged Navier–Stokes (T-RANS) approach has been presented and its potential as a simulation technique for solving turbulent ﬂows at very high Reynolds and Rayleigh numbers discussed. The particular strategy adopted is restricted to ﬂows with distinct coherent large-eddy structures. The approach belongs to the class of Very Large Eddy Simulations, and is similar to the semi-deterministic model of Ha Minh and Kourta (1993). The ensemble-averaged Navier–Stokes equations are solved in time and space, with a single-point closure playing the role of the ‘subscale model’ for the unresolved motion. In comparison with conventional LES, the model of the unresolved motion covers a much larger part of the turbulence spectrum, in fact, almost the complete stochastic turbulence, whereas the large deterministic structure is fully resolved. Unlike LES, both the resolved and unresolved contributions are of the same order of magnitude, and in the near-wall regions the contribution of the unresolved part is dominant. This places especial importance on the subscale model, which needs to provide a good reproduction of near-wall turbulence transport. The method was applied to analyze the Rayleigh–B´enard convection over both ﬂat and wavy walls, over a range of Rayleigh numbers. The large-scale convective roll cells are numerically fully resolved in time and space, whereas the second-moments (heat ﬂux, stress, scalar variance) associated with the remaining unresolved turbulence spectrum for velocity and scalar variables are provided by a subscale model for which a single-point, three-equation k-εθ2 algebraic stress/ﬂux closure was used. The T-RANS simulations of the classical Rayleigh–B´enard convection reproduce distributions of the mean ﬂow properties, wall heat transfer and second-

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moment turbulence statistics in agreement with the DNS results for the same Rayleigh numbers. Moreover, when scaled with the buoyancy velocity deﬁned in terms of the wall heat ﬂux, the proﬁles of the ﬁrst and second moments in the near-wall regions collapse almost onto one curve over a range of Rayleigh numbers spanning four decades. The computations also captured the largescale deterministic structure in accord with the DNS, LES and experimental ﬁndings, as demonstrated by comparing qualitatively the structure morphology identiﬁed by several criteria, primarily the kinematic vorticity number and second invariant of the velocity gradient tensor. Comparisons of velocity vectors, instantaneous trajectories of massless particles, planform and plume structures, (Kenjereˇs and Hanjali´c 1999a, 1999b) provided additional illustration of a striking similarity between selected T-RANS and DNS realizations, thus supporting the claim that the T-RANS approach can be used to study coherent structures and their organization in various types of Rayleigh–B´enard and similar ﬂows. The method has also been shown to be useful in predicting the eﬀect of various means to control the ﬂow and turbulence. As an example of boundary control, two types of bottom-wall non-planar topologies, 2D and 3D wavy walls, were considered, showing the expected structural reorganization and its eﬀect on mean ﬂow properties, heat transfer and turbulence statistics. The method has also been extended to simulate ﬂows of electrically conductive ﬂuid subjected simultaneously to thermal buoyancy and a magnetic ﬁeld (‘magnetic Rayleigh–B´enard convection’), (Hanjali´c and Kenjereˇs 2000b), the latter being regarded as an example of distributed (body force) ﬂow control. The method demonstrated its ability to predict the eﬀect of the magnetic ﬁeld on the reorganziation of the large eddy structure, as well as the resultant modiﬁcation of the heat transfer. Finally, the method was shown to be useful for predicting the ﬂow structure and heat transfer at very high Rayleigh numbers (at present Ra = 1012 ), which are inaccessible to any other numerical simulation technique, and are also still challenging for experiments. The method can also be used for the computation of technological and environmental ﬂows with realistic complex geometries.

References Aubrun, S., Kao, P.L., Ha Minh, H. and Boisson, H. (1999). ‘The semi-deterministic approach as way to study coherent structures: case of a turbulent ﬂow behind a backward-facing step’. In Engineering Turbulence Modelling and Experiments, edited by W. Rodi and D. Laurence, Elsevier, 491–499. Balachandar, S. (1992). ‘Structure in turbulent thermal convection’, Phys. Fluids A, 4 2715–2726. Besnard, D., Harlow, F., Rauenzahn, R and Zemach, C. (1996). ‘Spectral transport model of turbulence’, Theor. Comput. Fluid Dyn. 8 1.

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Bosch, G. and Rodi, W. (1998). ‘Simulation of vortex shedding past a square cylinder with diﬀerent turbulence models’, Int. J. Num. Meth. Fluids 28 601–611. Chavanne, X., Chill´ a, B., Castaign, B., H´ebral, B., Chabaud, B. and Chaussy, J. (1997). ‘Observation of ultimate regime in Rayleigh–B´enard convection’, Phys. Rev. Lett. 79 3648–3651. Chu, T. and Goldstein, R.J. (1973). ‘Turbulent natural convection in a horizontal layer of water’, J. Fluid Mech. 60 141–159. Cortese, T. and Balachandar, S. (1993). ‘Vortical nature of thermal plumes in turbulent convection’, Phys. Fluids A 5 (12) 3226–3232. Durbin, P.A. (1995). ‘Separated ﬂow computations with the k-ε-v 2 model’, AIAA Journal 33 (4) 659–664. Gr¨ otzbach, G. (1982). ‘Direct numerical simulation of laminar and turbulent B´enard convection’, J. Fluid Mech. 119 27–53. Gr¨ otzbach, G. (1983). ‘Spatial resolution requirement for direct numerical simulation of Rayleigh–B´enard convection’, J. Comput. Phys. 9 241–264. Ha Minh, H. and Kourta, A. (1993). ‘Semi-deterministic turbulence modelling for ﬂows dominated by strong organized structures’. In Proc. 9th Int. Symp. on Turbulent Shear Flows, Kyoto, Japan 10.5-1–10.5-6. Hanjali´c, K. and Kenjereˇs, S. (2000a). ‘T-RANS simulation of deterministic eddy structure in ﬂows driven by thermal buoyancy and Lorentz force’. Submitted for publication in Flow Turbulence and Combustion. Hanjali´c, K. and Kenjereˇs, S. (2000b). ‘Reorganization of turbulence structure in magnetic Rayleigh–B´enard convection: a T-RANS study’, J. Turbulence 1 (8) 1– 22. Hanjali´c, K., Launder, B.E. and Schiestel, R. (1980). ‘Multiple-time-scale concepts in turbulent transport modelling’. In Turbulent Shear Flows 2, edited by L.J.S. Bradbury et al., Springer, 36–49. Johansson, S. H., Davidson, L. and Olsson, E. (1993). ‘Numerical simulation of vortex shedding past triangular cylinders at high Reynolds number using a k-ε turbulence model’, Int. J. Num. Meth. in Fluids 16 859. Kenjereˇs, S. and Hanjali´c, K. (1995). ‘Prediction of turbulent thermal convection in concentric and eccentric annuli’, Int. J. Heat and Fluid Flow 16 (5) 428–439. Kenjereˇs, S. and Hanjali´c, K. (1999a). ‘Transient analysis of Rayleigh–B´enard convection with a RANS Model’, Int. J. Heat and Fluid Flow 20 329–340. Kenjereˇs, S. and Hanjali´c, K. (1999b). ‘Identiﬁcation and visualization of coherent structures in Rayleigh–B´enard convection with a time-dependent RANS’, J. Visualization 2 (2) 169–176. Kenjereˇs, S. and Hanjali´c, K. (2000). ‘Convective rolls and heat transfer in ﬁnitelength Rayleigh–B´enard convection: a two-dimensional study’, Phys. Rev. E 62 (6A), 7987–7998. Kerr, M.R. (1996). ‘Rayleigh number scaling in numerical convection’, J. Fluid Mech. 310 139–179.

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Kretenauer, K. and Schumann, U. (1992). ‘Numerical simulation of turbulent convection over wavy terrain’, J. Fluid Mech., 237 261–299. Niemela, J.J., Skrbek, L., Sreenivasan, K.R. and Donnelly, R.J. (2000). ‘Turbulent convection at very high Rayleigh numbers’, Nature 404 837–840. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D. and Spalart, P.R. (2000). ‘An approach to wall modeling in large-eddy simulations’, Phys. Fluids 12 (7) 1629– 1632. Peng, S.-H. and Davidson, L. (private communication) Schiestel, R. (1987). ‘Multiple time scale modeling of turbulent ﬂows in one-point closures’, Phys. Fluids 30 722–731. Spalart, P.R. (1999). ‘Strategies for turbulence modelling and simulations’. In Engineering Turbulence Modelling and Experiments 4, edited by W. Rodi and D. Laurence, Elsevier, 3–17. Speziale, C.G. (1998). ‘Turbulence modeling for time-dependent RANS and VLES: a review’, AIAA Journal 36 (2) 173–184. Tatsumi, K., Iwai, E., Neo, E.C., Inaoka, K. and Suzuki, K. (1999). ‘Prediction of time-mean characteristics and periodical ﬂuctuation of velocity and thermal ﬁelds of a backward-facing step’. In Turbulence and Shear Flow Phenomena 1, edited by S. Banerjee and J.K. Eaton, Begell House, Inc., New York, 1167–1172. Theerthan, S.A. and Arakeri, J.H. (1997). ‘Planform structure of turbulent free convection on horizontal surfaces’. In 2nd Int. Symposium on Turbulence, Heat and Mass Transfer 573–580. Touil, H., Bertoglio, J.P. and Parpais, D. (2000). ‘A spectral closure applied to anisotropic inhomogeneous turbulence’. In Advances in Turbulence VIII, edited by C. Dopazo et al., CIMNE Barcelona, 689–692. Travin, A., Shur, M., Strelets, M. and Spalart, P. (1999). ‘Detached-eddy simulations past a circular cylinder’, Flow, Turbulence and Combustion 63 293–313. W¨ orner, M. (1994). Direkte Simulation turbulenter Rayleigh–B´enard Konvektion in ﬂussigem Natrium. Dissertation, Univ. of Karlsruhe, KfK 5228, Kernforschungszentrum Karlsruhe.

23 Use of Higher Moments to Construct PDFs in Stratiﬁed Flows B.B. Ilyushin 1

Introduction

The distributions of moments calculated using turbulence models of diﬀerent closure levels contain information about the statistical structure of turbulent ﬂuctuations. This information is suﬃcient for a wide range of practical problems. However, as was noted in [15], for a complete statistical description of the ﬂow-ﬁeld characteristics of a turbulent ﬂow one should deﬁne all the multidimensional joint probability distributions for values of these characteristics at every possible ensemble of points in space and time. The active use of this approach (the use of PDFs in the study of turbulence) began with the work of Monin (1967) and Lundgren (1967). These papers include the set of equations for the PDF of a random velocity ﬁeld for developed turbulence. The advantage of using the PDF method for studying turbulence begins with the fact that every PDF from the family of ﬁnite-dimensional PDFs of the velocity ﬁeld contains information about the whole associated system of statistical moments. However, the set of equations for the PDFs of the turbulence ﬁeld is not closed. The complexity of solving this problem can be judged by noting that a closed form equation for the one-point joint PDF for developed inhomogeneous turbulence can only be obtained by applying additional hypotheses and the one-point joint PDF is itself multi-dimensional in character. In the papers of Lundgren (1967) and Onufriev (1970, 1977b) the model equations for the one-point PDF of the velocity and scalar ﬁelds have been derived on the basis of a number of phenomenological assumptions. Use of these equations opens the possibility of extending the semi-empirical methods for turbulent transfer, Onufriev (1977a). One can ﬁnd an example of the numerical simulation of turbulent diﬀusion processes through the application of the model of Onufriev (1977b) for the one-point PDF in the paper by Belotserkovskii (1987) and the use of the PDF method for describing turbulent diﬀusion of a scalar in Pope (1980, 1983). Numerical integration of the model equation for the PDF usually requires powerful computer facilities and is rather expensive. However, the inherent advantages of the PDF method lead the writer to foresee the extensive use of this approach for studying the structure of turbulent ﬂows in the near future. In this chapter another useful aspect of the use of the PDF for representing the turbulent transport structure will be considered. As is known, when large-scale eddy structures with long lifetimes (coherent structures) are formed in a turbulent ﬂow, the turbulent transfer mecha685

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nism exhibits non-local properties which cannot be described adequately with gradient-diﬀusion models. For a satisfactory description it is necessary to apply higher-order closure models. Chapter [15] is devoted to the use of such models for calculating the turbulence structure of the velocity and temperature ﬁelds in stably stratiﬁed ﬂows. Application of the higher-order closure models to describe the turbulent transfer of a passive scalar requires the use of model equations (or approximations) for mixed covariances (for example, the concentration and temperature-ﬂuctuation covariances). Information about their behaviour is absent so a direct check of the adequacy of such models is diﬃcult. However there is an alternative way to solve this problem. The long lifetime of coherent structures (compared with the typical time scale of turbulence) presuppose their weak statistical dependence on eddies in the inertial range of the spectrum. This allows us to consider the transfer by eddies in the inertial range of the turbulent ﬂuctuation spectrum (‘background turbulence’) to be represented by gradient models of turbulent diﬀusion and the transfer under the action of coherent structures as statistically independent processes. The PDF of the background turbulence has a Gaussian distribution and can be easily distinguished from the PDF of the complete velocity ﬂuctuations. The remaining part of the PDF corresponds to eddies of the energy-containing range. From a representation of this part one can reconstruct the velocity ﬁeld of the coherent structures u ˜i which can then be used to take into account their contribution to turbulent transfer as a correction to the mean velocity ﬁeld Ui . This approach thus allows for mass transfer by coherent structures directly in the convection terms of transport equation for species concentration as well as for turbulent transfer through the eﬀect of background turbulence in the diﬀusion term using a ‘standard’ gradient type model: ∂C ∂C ˜i ) + (Ui + u ∂t ∂xi coherent structures

=

∂ ∂C Cµ τ ui uj ∂xj ∂xi

background turbulence

To adopt this strategy one needs to reconstruct the velocity-ﬂuctuations PDF with known (calculated) moments of the PDF distribution.

2

Use of higher moments to construct PDFs in stratiﬁed ﬂows

The theory allows us to reconstruct the single and exact PDF from a knowledge of the distribution of the moments. However, it requires a knowledge of all the moments of the considered PDF distribution. In practice only some lower-order moments are usually known. Thus the inﬁnite series is clipped after a small number of terms. As a consequence the PDF thus obtained does not always satisfy the necessary requirement of positive values within the whole domain

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of deﬁnition. Indeed, reconstructing the PDF for example as a Gram–Charlier series '

(

1 1 1 + Sw ξ 3 − 3ξ + δw ξ 4 − 6ξ 2 + 3 + · · · , 3! 4! (1) (where ξ = w/σ, σ is the dispersion, Sw is the skewness of the PDF distribution of velocity w and δw is its excess) using ﬁrst-, second- and third-order moments, we do not satisfy the positivity condition. The PDFs calculated from (1) are shown in Figure 1 as well as a graph of the polynomial in square brackets in (1). It is seen that using the assumption of zero cumulants of order higher than third order (corresponding to curve 2 in Figure 1), the third degree polynomial remaining in square brackets (and also the PDF, as a consequence) has a region of negative values at any value of skewness (see Figure 1b). This approximation corresponds to the quasi-normal hypothesis of Millionshtchikov (1941). When clipping the series (1) at the fourth term (that is taking into account nonzero values of the fourth-order cumulant), the distribution obtained becomes positive everywhere for some values of Sw and δw , in particular for the stably stratiﬁed planetary boundary layer (PBL). For example, for Sw = −0.2 (Chiba 1978) and with δw = 2.1 (see Figure 3 of [15]) the PDF reconstructed from the ﬁrst four moments with the help of the series (1) is everywhere positive. Thus, the use of the quasi-normal hypothesis of Millionshtchikov (1941) does not allow one to reconstruct a positive PDF by means of (1), whereas in some cases the assumption of zero cumulants of order higher than fourth does permit the use of the Gram–Charlier series (1) for reconstructing the PDF. However, the PDF created in this a way is not necessarily positive over the whole domain in ﬂows where there is a signiﬁcant asymmetry of turbulent ﬂuctuations (for example, for values Sw and δw corresponding to the convective PBL). This fact restricts the use of the Gram–Charlier series (as well as other exact formulations of the PDF as an inﬁnite series based on moments or its compositions) for PDF-reconstruction to the ﬁrst three or four moments of the distribution. This restriction is connected closely with the inadmissibility of arbitrary clipping of the Taylor series of the characteristic function logarithm. Thus, in this chapter another way of reconstructing the PDF of the vertical velocity is applied and considered for the case of a horizontally homogeneous convective PBL.

ξ2 1 P (w) = √ exp − 2 2π

3

Constructing PDFs in the convective PBL

The ‘convective boundary layer’ is usually taken to mean the atmospheric layer where there is a direct inﬂuence from the underlying surface heated by the sun’s rays (Figure 2). For conditions where the Rayleigh number in the lower regions of the PBL becomes larger than some critical value, large-eddy convective motions, similar to Benard cells, appear and these play the main role in the vertical transfer of momentum, heat and matter. Circulation in such

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P(w) Gram–Charlier polynomial

( b)

( a)

3

3

2 –5

2

w

0

5

1

0 .3

3 1 2 4

–5

0

8

5w

Figure 1: (a) PDF reconstructed with use of the Gram–Charlier series: (1): Gaussian function (Sw=0 , δw = 0); (2): Sw=−0.2 , δw = 0; (3): Sw=−0.2 , δw = 2.1; and Gram–Charlier polynomial (b). cells is shown schematically in Figure 3b on the x-y plane. Non-symmetric boundary conditions (heating below, stable stratiﬁcation at the upper boundary) cause an asymmetric distribution of the PBL vertical velocity amplitudes: steep rising motions and gentler descending ones (see Figure 3a). Such a character of the motion has been conﬁrmed by ﬁeld experiments (Byzova et al. 1991). The curve in Figure 3c represents a typical realization of the vertical component of velocity for unstable stratiﬁcation obtained at an altitude of 120m and smoothed over a period of about one minute. This curve is not exactly periodic because it represents a cross-section of a quasi-regular system of cells. However, noting the character of this curve, one can recognize that such a realization corresponds to the velocity ﬁeld along a line passing through the cell centers. The horizontal size of the cells λ is between three and ﬁve kilometers. The cells occupy, in the vertical direction, practically the whole PBL. The distributions of statistical characteristics of the velocity and temperature ﬁelds (second- and third-order moments) reﬂecting such a structure of the convective PBL have been calculated by Ilyushin (1998)

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Figure 2: Structure of the convective atmospheric boundary layer.

Figure 3: Vertical motion shape in the convective large scale eddies (a), (b); and the measured realization of vertical velocity smoothed with the period 1min. (c). and are shown in Figure 4 together with the data of measurements in the convective PBL taken from Caughey (1982) and Lenschow et al. (1980). One can see that the calculated proﬁles are in good agreement with the data of observations and can be used to reconstruct the PDF of the vertical velocity ﬂuctuation in the convective PBL. The PDF of the turbulent ﬂuctuations of the vertical velocity is represented as a superposition of two independent distributions: the inertial range of the turbulent spectrum (background turbulence) Pb (u) and a large-wave region PDF spectrum Pc (v) (u and v are the vertical velocity ﬂuctuations of back-

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z /zi

1 Q

0

1

1 <w2 >/ w*2

1

e h /w *2

0

1

1

2

2

< θ >/( Q /w ) o *

<wθ>/Q0

zi /w *3

z /zi 1

0.0

0.2 <w >/w*3

0.0

3

0.2

0

4 <wθ

<Ew>/w* 3

2

>w* /Qo2

0.0 0.8 2 * <w θ >/ Qow

Figure 4: Results of simulation of the convective ABL evolution: solid lines are calculated proﬁles of second- and third-order correlations (diﬀerent lines correspond to times from 10 am to 5 pm); symbols are the experimental data. ground turbulence and the vertical velocity ﬁeld of coherent structures, respectively) P (u, v) = Pb (u)Pc (v) = 

'

1 u2 exp − 2πσb2 2σb2



( ' ( ' (  a+ (m+ − v)2 a− (m− − v)2   , (2) + − exp −  σ + exp − 2(σ + )2  σc 2(σc− )2 c  c 

background turbulence PDF

upﬂow

downﬂow

Large Scale Eddy Formation PDF

where σb is the dispersion of background turbulence, a+ and a− are weighting

[23] Use of higher moments to construct PDFs in stratiﬁed ﬂows

691

coeﬃcients, σc+ and σc− are the dispersions, and m+ and m− are the distribution maximums (the average velocities) of upﬂow and downﬂow of coherent structures. Considering the total turbulent velocity ﬂuctuation w as the sum of u and v, we obtain the total PDF:

P (w) = R

=

P (u, v)δ(w − u − v) du dv = '

a+ (m+ − w)2 exp − 2 2πσ+ 2σ+

(

'

a− (m− − w)2 + exp − 2 2πσ− 2σ−

(

, (3)

2 = (σ + )2 + σ 2 and σ 2 = (σ − )2 + σ 2 . From the conditions where σ+ c − c b b R P (w) dw Rw

2 P (w) dw

= 1,

= w2 ,

R wP (w) dw Rw

= 0,

3 P (w) dw

= Sw σ 3

(4)

(here σ = w2 1/2 is the dispersion and Sw = w3 /w2 3/2 is the skewness 2 , σ 2 , m+ and m− are: factor), the connections among a+ , a− , σ+ − a+ + a− a++ m+ + a− m−, + + 2 , + 2 + 2 a (m ) + σ+ ++ a (m+ )2 + σ− + , , 2 + a− (m− )3 + 3m− σ 2 a+ (m+ )3 + 3m+ σ+ −

= = + =

1, 0, σ2 Sw σ 3 .

(5)

2 and σ 2 are found from the assumption (De Baas et The conditions for σ+ − 2 and σ 2 should be equal to al. 1986) that the square of the dispersions σ+ − the square of the average velocities of upﬂow and downﬂow (m+ )2 and (m− )2 correspondingly: 2 2 σ+ = (m+ )2 , σ− = (m− )2 . (6)

This assumption means that the ﬂux directed positively (negatively) remains mainly positive (negative) taking into account possible scatter. The necessary conditions for closure of the equation set (for σb ) is found from the wavelet model (Tennekes and Lumley 1972). Here we suppose that the main part of the energy-containing range of the ﬂuctuation spectrum is deﬁned by a single main wavelet with a typical wave number corresponding to the spectrum maximum (see Figure 5). The simpliﬁed eddy with a typical wave number κv is considered in the form of a localized perturbation of energy in wave number space (wavelet) with energy Ev = E(κv )κv . This consideration ensures the cascade transfer of turbulence energy from large-scale eddies to dissipative eddies. In the present work the coherent structure in the convective PBL is −2/3 assumed to be the wavelet containing the energy Ec = aε2/3 κmax (where a = 1.6 ± 0.02 is the Kolmogoroﬀ constant, Andreas (1983), and κmax is the maximum of the spectrum of the turbulent energy – see Figure 5). The total turbulence energy is equal to E=

κ max κmin

Ev (κv ) = Ev (κmax )

∞ i=0

3−2/3

i

≈ 2Ec .

(7)

692

Ilyushin

Figure 5: Presentation of the long-wave part of the turbulent ﬂuctuations spectrum as the wavelet. A similar result can be obtained by integration of the inertial range (without accounting for the departure from the Kolmogoroﬀ spectrum in the dissipation range) from κmax to inﬁnity and taking into account also the energy of half of the wavelet (which is outside of the inertial range, see Figure 5): E≈

∞

1 aε2/3 κ−5/3 dκ + Ec = 2Ec , 2 κmax

(8)

or σb2 = E (recall (σb2 + σc2 )/2 = E). Taking account of this result, κmax is expressed as κmax = (2a/E)3/2 ε and then λmax = 2π/κmax . The calculated proﬁles of λmax and the ratio σb2 /w2 in the convective atmospheric boundary layer are shown in Figure 6. Here one can seen that in the mixing layer the calculated distribution of λmax is in agreement with the observed data (Caughey 1982). The linear function λmax = 5.88z/zi corresponds to conditions of free convection in the near-ground layer. The calculated ratio σb2 /w2 is equal to 1/3 in the mixing layer of the PBL. Analysis of the experimental spectrum of w2 in the atmosphere (Monin and Yaglom 1967) gives the same result σb2 /w2 ≈ 1/3. The necessary condition for the dispersion of the background turbulence σb is 1 (9) σb2 = w2 . 3 Taking account of the variation in sign of m+ > 0 and m− < 0 (which corre-

[23] Use of higher moments to construct PDFs in stratiﬁed ﬂows

z /zi

693

z /zi

1

5.88z/z

0 0

i

λmax

2

0 1/ 3

1

σb2/<w2>

Figure 6: The calculated and measured (Caughey 1982) proﬁles of λmax = 2π/(εE/2a)3/2 and the ratio σb2 /w2 . sponds to upﬂow and downﬂow), equations (7), (9) have unique solutions:

√ √ m− = σ4 S − S 2 + 8 ; m+ = σ4 S + S 2 + 8 ; √ √ S − S2 + 8 S − S2 + 8 − + √ a =− √ ; a = ; 2 S2 + 8 2 S2 + 8 (10)

2 √ 2 (σc+ )2 = σ16 S + S 2 + 8 − 13 w2 ; (σc− )2 =

σ2 16

S−

√

2

S2 + 8

− 13 w2 .

The results of reconstructing the PDF using the calculated distributions of the second- and third-order moments are shown in Figure 7. The picture shows that the results of reconstruction correspond to the observed data: an upﬂow within a coherent structure with larger turbulent energy (σc+ ) and larger velocity (m+ ) occupying a smaller region (of size ≈ Pc (m+ ) < Pc (m− )) and a downﬂow with smaller energy (σc− < σc+ ) and smaller velocity (m− < m+ ). Such a structure for the convective PBL arises from the vertical asymmetry in the generation mechanism for the turbulent ﬂuctuations: in the near-ground layer a growth of turbulence energy is caused by the mean velocity shear whereas, in the upper (stably stratiﬁed) part of the PBL, turbulent ﬂuctuations are strongly suppressed. Therefore the intensity of the vertical velocity ﬂuctuations in ‘upﬂows’ appears to be larger than that in ‘downﬂows’. Figure 8 shows the PDF proﬁles obtained in the measurements (Byzova et al. 1991) in the atmosphere compared with calculations using the model of Deardorﬀ. As is obvious the reconstructed PDF in the present work P (w) (see Figure 7)

694

Ilyushin

2 2 2

0

Figure 7: The reconstructed proﬁle of PDF of the vertical velocity ﬂuctuations for z/zi = 0.5. is in qualitative agreement with those shown in Figure 8: it has the maximum in the region of negative velocities and the more convex part in that of positive velocities. To obtain the distribution of vertical velocity ﬂuctuations from individual realizations, it is necessary to apply a procedure of ensemble averaging. For horizontally homogeneous conditions, ensemble averaging and averaging over horizontal coordinates can be regarded as equivalent procedures. And ﬁnally, taking into account Taylor’s hypothesis of frozen turbulence, averaging over horizontal coordinates can be replaced by time averaging for ﬁxed horizontal coordinates if the time of this averaging is larger than the drift (under the action of averaged ﬂow) time D/U of the largest turbulent eddies. This allows us to decrease signiﬁcantly the time required to obtain the measurements. Proﬁles of PDF shown in Figure 8 have been obtained using this method. The following assumptions have been made. For the horizontally homogeneous PBL, the probability, P (w) dw, of the velocity falling within the range [w; w + dw] is proportional to the horizontal size of the region with such velocity, P (w) dw ≈ dx. The continuity condition for velocity applied to a region between the large-scale eddies requires that the adjoining eddies have opposite directions of swirling (see Figure 9). A pair of such eddies we denote as a ‘simple cell’. For a statistical analysis of the sequence of such identical cells carried by the wind it suﬃces to study just one cell. This cell is represented as a coherent structure which (within the adopted assumption) is a wavelet centered at the location κmax and of horizontal size λmax . It is evident that the iterative character of the structures in the PBL makes the averaging within

[23] Use of higher moments to construct PDFs in stratiﬁed ﬂows

695

Figure 8: Distributions of vertical velocity variance in the convective PBL: (a) observed data and (b) calculation using Deardorﬀ’s model (from Byzova et al. 1991). the time interval λmax /(2U ) and the averaging over an inﬁnite time equivalent. Thus, we may consider that the part, Pc (w), of the reconstructed PDF that has to do with coherent structures, characterizes the local vertical ﬂuxes up and down, that is one eddy (half of the cell). This allows us to reconstruct a single ﬁeld of the vertical velocity (and then of the horizontal one) for this cell. As mentioned above, the size of the horizontal region for the coherent structures (where the vertical velocity is equal to w) ˜ is taken to be proportional to the probability Pc (w): ˜ dx Pc (w) ˜ dw ˜= . (λmax /2) The vertical velocity ﬁeld of the coherent structures w(x, ˜ z) can be found from this diﬀerential equation. The horizontal velocity ﬁeld u ˜(x, z) is deter-

696

Ilyushin

z/zi 1

0

x/λmax

1

Figure 9: The reconstructed velocity ﬁeld of the coherent structure in the convective PBL.

Figure 10: Conditionally averaged (a) updraughts and (b) downdraughts in terms of (i) u and (ii) w (from Schmidt and Schumann (1989) calculated by the LES-method. mined from the equation of mass conservation. The result of reconstructing the coherent-structure velocity ﬁeld (˜ u, w) ˜ in the convective PBL is shown in Figure 9. The conditionally averaged (a) updraughts and (b) downdraughts in terms of (i) u and (ii) w (from Schmidt and Schumann 1989) are shown in Figure 10 (solid lines are positive values and dashed lines are negative ones). One can see qualitative agreement for the reconstructed ﬁeld of the

[23] Use of higher moments to construct PDFs in stratiﬁed ﬂows

697

coherent-structure velocities with that calculated in this latter paper by an LES simulation.

4

Modeling the turbulent transport of substance

To describe the process of species dispersion in the convective PBL, we use a model taking directly into account the eﬀect of mass transfer by coherent structures (in the advection terms of the equation for the crosswind integrated concentration, see Ilyushin and Kurbatskii (1996)). To account for the turbulent diﬀusion of matter by background turbulence, a ‘standard’ gradientdiﬀusion model is applied:

∂Cy ∂Cy ∂ ∂Cy ∂Cy 4 Cs τ w2 , + (U + u ˜) = +w ∂t ∂x ∂z ∂z ∂z

(11)

where Cs is the model coeﬃcient of the model whose value is determined from the requirement that the coeﬃcient of turbulent diﬀusion near the surface should equal κu∗ z (Monin and Yaglom 1967): Cs τ w2 −→ z→0

2 E02 2 ≈ 4.52 Cs κu∗ z ⇒ Cs ≈ 0.07. Cs 3 ε0 3

(12)

The results of a simulation of a pollutant jet spreading from sources placed both near ground level and in the middle of the mixed layer are presented. The crosswind integrated concentration ﬁelds averaged oven one period, λmax /Umax , are shown in Figures 11 and 12. A near-ground source was realized at zs /zi = 0.07. The maximum concentration centerline (i.e the locus of maximum concentration) ﬁrst moves parallel to the surface and then starts to rise rapidly at a downwind distance x∗ = 0.5 (here x∗ = xw∗ /(zi Ux )), creating a local maximum of concentration near the inversion layer. This maximum is located at a height of z = 1.1zi at about x∗ = 2.0. This is the same distance downwind as that observed in the laboratory experiments of Willis and Deardorﬀ (1976) (there, x∗ = 1.75), but the height is greater (z = 0.75zi in the Willis and Deardorﬀ (1976) experiment). This diﬀerence between the calculation and the experimental data may be connected to the larger size of the coherent structures in the calculations (see Figure 6, λmax (z)) near the inversion layer than in the observed data in the PBL. At about x∗ = 2.5, the centerline begins to descend back into the middle of the PBL in a manner similar to the laboratory results. The plume centerline for the mid-level elevated source descends rapidly, impinging on the ground at x∗ = 0.9, with a maximum concentration there of Cy = 1.8q0 /(zi Ux ) (see Figure 12). Both these features are in good agreement with the experiments (Willis and Deardorﬀ 1981). The plume then rebounds from the surface, producing a second line of high concentration, which rises near the inversion layer. The calculated and measured ground-level concentrations for these cases are shown in Figures 13 and 14. The calculated behaviour

698

Ilyushin

Z/Zi

(a)

1.0

0.5

0

1

2

z/zi

0.1 0.5 0.2 0.75 1

1 1.25

1.25

3

x*

1

1.5 3

0.75

2 5

0

1

2

X*

3

Figure 11: Dimensionless crosswind integrated concentration Cy zi Ux /q0 for point sources of height z/zi = 0.067: (a) calculated and (b) measured (Willis and Deardorﬀ 1976). of the plume centerline for both cases corresponds closely with Willis and Deardorﬀ’s tank experiments. It should be noted that the laboratory experiments did not include wind-shear and Coriolis eﬀects in contrast with the natural PBL observation and the simulation. To conclude this section, we note that, in VLES, the inﬂuence of coherent structures is accounted for by separation of the periodic properties of motions. In this chapter we have accounted for the inﬂuence of the coherent structures by highlighting the periodic properties of motions at the PDF level (in contrast with the VLES method). This approach is just one of the possible applications of the constructed PDF approach.

5

Conclusions 1. The distributions of ﬂuctuating velocity moments of order higher than second mainly characterize statistical properties of the large-scale eddy structures corresponding to the long-wavelength part of the turbulent

[23] Use of higher moments to construct PDFs in stratiﬁed ﬂows

z/zi

(b)

0.05

0.1

0.2 0.4 0.6 0.8 1.0

1 0.9

4

699

1.1

1.2 3 2 1.4 1.6

1.0 0.8

1.4

1

2

3

X*

4

Figure 12: Dimensionless crosswind integrated concentration Cy zi Ux /q0 for point sources of height z/zi = 0.5: (a) calculated and (b) measured (Willis and Deardorﬀ 1981). ﬂuctuation spectrum which gives rise to non-local features of turbulent transfer. 2. For a range of ﬂows characterized by a slight asymmetry of the turbulent ﬂuctuations, the representation of the PDF as a series with zero cumulants of order higher than fourth meets the necessary condition of positivity over the whole domain of deﬁnition and can be applied, for example, in stably stratiﬁed turbulent ﬂows. 3. The transport by coherent structures and the transfer caused by background turbulence (by eddies of the inertial range) can be considered as statistically independent processes. 4. The PDF of the vertical velocity ﬂuctuations can be represented as the superposition (product) of two independent distributions. The ﬁrst corresponds to the PDF of the velocity ﬂuctuations from the inertial range

700

Ilyushin

C0

C0

2 6

4

1

2

0

0 0

2

x*

Figure 13: The calculated groundlevel concentration (solid line) compared with the corresponding value from the laboratory experiment (solid circles) for source height at 0.067zi .

0

1

2

x*

Figure 14: The calculated groundlevel concentration for source height at 0.5zi . Line and symbols are as in Figure 13.

and has the form of a Gaussian distribution with zero average. The second corresponds to velocity ﬂuctuations of the coherent structures and has the form of a sum of two Gaussian distributions whose averages represent ‘upﬂows’ and ‘downﬂows’. The PDF created in this way can be reconstructed from a knowledge of ﬁrst-, second- and third-order statistical moments. 5. The velocity ﬁeld of coherent structures can be constructed from the PDF proﬁle of the vertical velocity in this structure for the horizontally homogeneous PBL. 6. The method of modelling turbulent mass transfer, which takes into account the eﬀects of the coherent structures by means of additional convection terms in the equation for concentration and which uses standard gradient models for parameterizing turbulent diﬀusion processes corresponding to eddies in the inertial range, allows us to describe non-local properties of turbulent mass transfer without the application of higherorder closure models.

References Andreas, E.L. (1983). ‘Spectral measurements in a disturbed boundary layer over snow’, J. Atmos. Sci. 44 1912.

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De Baas, A.F., van Dop, H. and Nieuwstadt F.T.M. (1986). ‘An application of the Langevin equation for inhomogeneous conditions to dispersion in a convective boundary layer’, Quart. J. R. Met. Soc. 112 165. Belotserkovskii O.M. (1987). ‘Numerical models of failure’, Problems of turbulent ﬂows. Nauka (in Russian). Byzova N.L., Ivanov V.N. and Garger E.K. (1991). Experimental investigations of atmospheric diﬀusion and pollution dispersion calculations. Gidrometeoizdat (in Russian). Caughey S.G. (1982). ‘Observed characteristics of atmospheric boundary layer’. In Atmospheric Turbulence and Air Pollution Modelling, edited by F.T.M. Nieuwstadt and H. van Dop, Reidel. Ilyushin B.B. and Kurbatskii A.F. (1996). ‘Modeling the pollutant spreading in the convective ABL’, Izv. RAN. Phys. Atmos. and Ocean. 32 307 (in Russian). Ilyushin B.B. (1998). ‘Modelling the non-local turbulent transport of momentum, heat and substance in the convective PBL’. In Proc. 2nd Engineering Foundation Conference on Turbulent Heat Transfer, Manchester. Lenschow D.H., Wyngaard J.C. and Pennel W.T. (1980). ‘Mean-ﬁeld and secondmoment budgets in a baroclinic, convective boundary layer’, J. Atmos. Sci. 37 1313. Lundgren T.S. (1967). ‘Distribution functions in statistical theory of turbulence’, Phys. Fluids 10 969. Millionshtchikov M.D., (1941). ‘On the role of the third noments in isotropic turbulence’, C.R. Acad. Sci. SSSR 32, 619. Monin A.S. and Yaglom A.M. (1967). Statistical Fluid Mechanics (Vol. 1 & 2). Nauka (Moscow). See also English translation (ed J.L. Lumley), Vol. 1 (1971), Vol. 2 (1975), MIT Press. Onufriev A.T. (1970). ‘About equations of semiempirical turbulent transport theory’, J. Appl. Mech. & Tech. Phys. 2 62–71. Onufriev A.T. (1977a). ‘Phenomenological models of turbulent theory’. Aerogasdynamics and physical kinetics. Nauka (in Russian). Onufriev A.T. (1977b). ‘On modeling the equation for the PDF in semiempirical turbulence theory’. Turbulent ﬂows. Nauka (in Russian). Pope S.B. (1980). ‘Probability distributions of scalars in turbulent shear ﬂows’. In Turbulent Shear Flows 2, edited by L.J.S. Bradbury, F. Durst, B.E. Launder, F.W. Schmidt and J.H. Whitelaw, Springer-Verlag. Pope S.B. (1983). ‘Consistent modeling of scalars in turbulent ﬂows’, Phys. Fluids 26 404. Schmidt H. and Schumann U. (1989). ‘Coherent structure of the convective boundary layer derived from large-eddy simulations’, J. Fluid Mech 200 511. Tennekes H. and Lumley J.L. (1972). A First Course in Turbulence. MIT Press. Willis, G.E. and Deardorﬀ, J.W. (1976). ‘A laboratory model of diﬀusion into the convective boundary layer’, Quart. J. Roy. Meteor. Soc. 102 427. Willis, G.E. and Deardorﬀ, J.W. (1981). ‘A laboratory study of dispersion from a source in the middle of the convective boundary layer’, Atmos. Environ. 15 109.

24 Direct numerical simulations of separation bubbles G.N. Coleman and N.D. Sandham Abstract In this chapter recent simulations of separation bubble ﬂows are reviewed. This class of simulations contain a range of physical phenomena with laminar or turbulent separation and turbulent reattachment. Streamline curvature and extra rates of strain are present, compared to simple parallel shear ﬂows. Simulations with initially laminar ﬂow also contain a laminar–turbulent transition process. All this occurs within a simple geometry, providing good test cases for comparison with other predictive methods such as LES and RANS.

1

Introduction

Our discussion of separated ﬂow DNS will be limited to separation of an incompressible laminar or turbulent boundary layer from a smooth ﬂat surface. Either the boundary layer cannot follow a discontinuous change in the surface geometry, under nominally zero-pressure-gradient conditions (and the ‘surface separates from the ﬂow’), or it is decelerated by an adverse pressure gradient (APG), with no change in surface curvature (and the ‘ﬂow separates from the surface’). These cases are represented by, respectively, the ﬂow over a backward-facing step, and the transpiration-induced detachment and reattachment of a ﬂat-plate boundary layer. Both conﬁgurations include reattachment, leading to a closed separation bubble. The present focus is upon three-dimensional simulations in two-dimensional geometries, for Reynolds numbers low enough that all (or nearly all) relevant scales of motion can be explicitly resolved. Two-dimensional computations of transitional separation bubbles (e.g. Pauley et al. 1990), and higher-Reynolds-number LES of the backward-facing step are not considered here. Neither are the very low Reynolds number DNS of bluﬀ-body ﬂows (circular and elliptic cylinders, spheres, normal ﬂat plates, and ﬁnite-thickness aerofoils), which often involve interaction of surface curvature and APG eﬀects (Karniadakis and Triantafyllou 1992; Zhang et al. 1995; Najjar and Balachandar 1998; and Mittal and Balachandar 1995). Application of DNS to separated ﬂows is a challenging task. The heart of the challenge lies in the numerical issues introduced by the streamwise variation inherent to any separated ﬂow. These include: (1) specifying realistic inﬂow and 702

[24] Direct numerical simulations of separation bubbles

703

outﬂow boundary conditions; (2) deﬁning a domain large enough to both accommodate the thickening boundary layers and detached shear layers induced by the separation, and to allow suﬃcient development length upstream of the separation (often needed to recover from the inﬂow treatment) and downstream of the re-attachment (to minimise the eﬀect of the outﬂow boundary condition); and (3) providing the spatial resolution needed to simultaneously capture the attached boundary layers and detached shear layers throughout the domain, for a ﬂow at a reasonably large Reynolds number. As we shall see below, DNS is just beginning to rise successfully to each of these challenges.

2

Backward-facing step

The Le et al. (1997) backward-facing step study is noteworthy for a number of reasons. It applied DNS to a geometry more complex than any yet considered, and pioneered use of turbulent inﬂow boundary conditions for spatially developing ﬂows. (Previous DNS of fully developed wall-bounded turbulence were able to use periodic streamwise conditions – either by simulating strictly parallel cases such as the plane channel (Kim et al. 1987) or by adding ‘growth terms’ to the governing equations to account for divergence of the mean streamlines (Spalart 1988).) Another unique feature of the Le et al. work is the ability to compare directly with experimental results: the DNS greatly beneﬁts from Jovic and Driver’s (1994) backward-facing step measurements at the identical conditions (the experiment was in fact commissioned to address issues raised by the DNS). The computation uses a staggered-grid, fractional-step method (Le and Moin 1991; Kim and Moin 1985), in conjunction with a ‘capacitance matrix’ Poisson-equation solver for pressure; the spatial discretization is second-order accurate, as is the compact-storage mixed implicit–explicit time-advance algorithm (Le et al. 1997). As discussed in section 1.8, such second-order accurate schemes will require more grid points than spectral methods to provide the same resolution of smallest-scale structures. The inﬂow ﬁeld consists of random (‘structure-free’) disturbances superimposed upon a mean proﬁle from Spalart’s (1988) Reθ = 670 zero-pressure-gradient DNS. After entering the domain through a plane of width 4h and height 5h (using h to denote the step height), the ﬂow develops over a distance of 10h before encountering the step, and experiencing a 5-to-6 (20%) expansion. This rather long development length is needed for the ﬂow to evolve physically realistic turbulence structure before encountering the step. After another 20 step heights, the ﬂow exits the domain, where a convective boundary condition (with a uniform convection velocity Uc of 0.8 the maximum mean streamwise velocity at the outﬂow plane) is applied to each of the velocity components. Roughly a third of the domain is thus ‘overhead’ needed to recover from the inﬂow treatment. A no-stress wall is applied at the upper boundary, located 6h above the lower wall. When

704

Coleman and Sandham

0

5

10

15

x/h

Figure 1: Instantaneous spanwise vorticity contours with negative values shown with dashed lines. (From Le et al. (1997).)

Figure 2: Comparison of pressure coeﬃcient from the simulation of Le et al. (1997) (solid line) with the experiment of Jovic and Driver (1995) (symbols). (From Le et al. (1997).) it reaches the edge of the step (at x = 0) the boundary layer has thickness 1.2h, and Reynolds number Reh (based on step height and maximum mean velocity) of 5100, and second-order statistics that agree well with those of the Reθ = 670 target (Spalart 1988). Over 9 million grid points are needed to resolve this ﬂow, requiring 13 Mwords of central memory. A uniform grid spacing is applied in the streamwise and spanwise directions, with points clustered in the vertical direction near the horizontal walls upstream and downstream of the step. (The spanwise resolution is slightly too coarse very near the wall, but this does not lead to a statistically signiﬁcant error.) Results were generated on a Cray C90, requiring on the order of 1000 CPU hours to obtain well-converged time-averaged statistics. An indication of the spatial resolution of the Le et al. simulation is given by the instantaneous vorticity contours shown in ﬁgure 1; the overall ﬁdelity of their results is reﬂected in the good agreement with Jovic and Driver’s (1994) experimental mean pressure and skin friction distributions (the sym-

[24] Direct numerical simulations of separation bubbles

705

0.004

0.002

Cf

0

– 0.002

– 0.004 0

5

10

15

20

x/h

Figure 3: Comparison of skin friction coeﬃcient from the simulation of Le et al. (1997) (solid line) with the experiment of Jovic and Driver (1995) (symbols). (From Le et al. (1997).) bols in ﬁgures 2 and 3). This agreement was especially signiﬁcant, since at this Reynolds number the magnitude of the negative mean skin friction in the recirculation region is much larger than that found in experiments done at larger Re. After demonstrating close correspondence between the DNS results and other statistics measured by Jovic and Driver, including the location of the mean re-attachment point (which agreed within 3%) and the downstream evolution of the mean velocity and Reynolds-stress proﬁles, Le and Moin turn their attention to quantities only available from the simulation. For example, they investigate the instantaneous structure of the detached shear layer (and the associated temporal and spanwise variations of the re-attachment region) and its connection to mean and instantaneous pressure ﬂuctuations throughout the ﬂow. Turbulent kinetic energy budgets are also examined, which reveal the strong similarity of the region above the recirculation with two-dimensional plane mixing layers, as well as the slow development of a conventional (nominally) zero-pressure-gradient boundary layer downstream of re-attachment. Even twenty step-heights downstream of the step, evidence of the sudden expansion and re-attachment can still be seen in the mean velocity proﬁle. By documenting the unexpectedly slow recovery, especially in terms of deviations from the standard zero-pressure-gradient log law, Le and Moin’s ﬁndings indirectly exposed an inaccuracy in many of the previous backward-facing-step skin-friction measurements. Since they were based on Clauser chart readings (and thus implicitly assumed the log law was valid where it was not), the experiments tended to report skin friction coeﬃcients that were signiﬁcantly smaller than the actual value. Le et al. and Jovic and Driver illustrate the manner in which DNS and laboratory experiments can complement each other, especially

706

Coleman and Sandham

when they are done simultaneously. Data from the Le et al. simulation have already been widely used for turbulence closure. The diﬀerential a priori analysis by Parneix et al. (1998) evaluated a second-moment model and showed that the main problem was closure of the u v equation. Modiﬁcations to the model used for the transport terms led to improvement in the prediction of the magnitude of the backﬂow velocity. Other studies have developed novel modelling approaches using the data. Peng and Davidson (2000) propose a two-equation scheme by which transport equations for the eddy viscosity and turbulence kinetic energy are solved, while Perot (1999) uses a strategy based on body-force potentials.

3

Flat-plate separation bubble with turbulent detachment

Two DNS studies, Spalart and Coleman (1997) and Na and Moin (1998), have been made of the rapid separation and re-attachment of a fully turbulent ﬂat-plate boundary layer (see also Coleman and Spalart 1993). The ﬂows are essentially equivalent: an incoming zero-pressure-gradient turbulent boundary layer at Reθ ≈ 300 is subjected to prescribed transpiration (suction followed by blowing) at a virtual wall opposite the ﬂat plate, which creates a short separation bubble (cf. ﬁgure 4 below, from Spalart and Coleman (1997), with Na and Moin’s (1998) ﬁgure 17). Although it is well deﬁned, there is no exact companion experiment for this ﬂow, and thus the DNS represents a ‘stand alone’ ‘numerical’ experiment. As such, it had to be indirectly validated, for example by examination of spectra and two-point correlations, and by contrasting DNS statistics from regions where the local conditions correspond to available canonical results. The mean streamlines from Spalart and Coleman are shown in ﬁgure 5 (the bubble is slightly larger in Na and Moin; see their ﬁgure 16). Note the ﬂattening of the pressure distribution in ﬁgure 4, due to the blockage from the bubble in ﬁgure 5a. The Spalart and Coleman simulation has the advantage of including heat transfer, through an isothermal wall (ﬁgure 5b); on the other hand, Na and Moin’s inﬂow turbulence (for reasons given below) is in a slightly more realistic state when it encounters the APG. Statistical data from both investigations are available for public distribution (Jim`enez et al. 1998). Both simulations assume periodic spanwise variations, use a second-order compact-storage semi-implicit time-advance scheme, and impose at an upper virtual wall a transportation proﬁle Vtop (x) that changes sign (ﬁgure 4) while maintaining zero vorticity (i.e. ∂u/∂y = dVtop /dx, v = Vtop and ∂w/∂y = 0; this condition, rather than the commonly used no-stress condition, is needed to avoid injecting vorticity downstream of the suction–blowing transition.) The major diﬀerences between the studies are in the numerical methods. Spalart and Coleman employ a Fourier/Jacobi spectral code, similar to that

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Figure 4: Turbulent separation bubble pressure coeﬃcient (left scale), showing distribution with separation (solid line) compared with the inviscid result (dashed line), and transpiration proﬁle (chain dotted, right scale). The shaded areas show the fringe zones. (From Spalart and Coleman (1997).)

Figure 5: (a) Mean ﬂow streamlines with dots showing the edge of the vortical region. Also shown along the axis are the locations where the probability of reverse shear crosses 1% and 50%. (b) Temperature contours: 0 (wall), 0.05, 0.1, . . . , 0.9, 0.95, 0.99, 1 (free stream). (From Spalart and Coleman (1997).) used by Spalart and Watmuﬀ (1993) for their DNS of a non-separating boundary layer. Eﬀective inﬂow/outﬂow conditions are set by a variation of the ‘fringe’ method (Spalart and Watmuﬀ 1993), which allows a periodic Fourier discretization to capture mean streamwise variations. This involves adding source terms to the governing equations that are active only in the ‘fringes’ of the domain (the cross-hatched regions in ﬁgure 5); these sources convert the exiting turbulent velocity and temperature ﬁelds into appropriate inﬂow

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conditions, before they re-enter the periodic domain. The favourable pressure gradient downstream of the separation bubble also helps ‘reprocess’ the ﬂow before it exits and re-enters the domain (the fringe method would not be effective for a separated boundary layer that did not re-attach well upstream of the exit). The initial portion of the ‘useful region’ (where the fringe terms are absent) consists of a zero-pressure-gradient (ZPG) ‘settling zone’, which allows the inﬂow to recover from any anomalies obtained within the fringes, and to develop a well-deﬁned standard ZPG turbulent boundary layer before the APG is applied. This objective was not completely met: while fully turbulent, the ﬂow entering the APG region is only partially developed (it possesses a reasonable shape factor, H ≈ 1.7, but the skin-friction is 16% smaller than the usual Reθ = 300 value). Spalart and Coleman recommend that a favourable-pressure-gradient settling zone be considered for any future separation-bubble DNS (based on experience from the Spalart and Watmuﬀ non-separating APG study). Another indication of the challenge associated with accurately capturing this ﬂow is the small but ﬁnite vorticity ﬂuctuation magnitude found above the bubble at the height of the virtual wall (see ﬁgure 6 of Spalart and Coleman). Weak temperature oscillations also appear and are responsible for the T = 1 contours just below the virtual wall, shown in ﬁgure 5b. These are present even though the spectral method uses 200 eﬀective grid (quadrature) points in the wall-normal direction (640 streamwise and 258 spanwise, for a total of 32.7 million, points are used). While these do not seriously corrupt the results, they do represent a numerical error and point to the disadvantage of the monotonically expanding wall-normal grid deﬁned by the spectral method. For this ﬂow with its rapidly changing thickness (indicated by the solid symbols in ﬁgure 5a) at this resolution, the accuracy of a spectral method is oﬀset by the inﬂexibility of the grid distribution. But even uniform grid spacing (cf. Na and Moin 1997) or local clustering near the transpiration plane cannot be expected to fully remedy this diﬃculty, given the rapid downstream thickening of the layer aﬀected by the separation and the sharp (streamwise, wall-normal, and spanwise) gradients at the ‘inviscid superlayer’, the viscous–inviscid interface with the freestream (see the turbulent region of ﬁgure 9 below, from Spalart and Strelets’ laminar-separation DNS). Future numerical studies of separated boundary layers would do well to monitor the level of freestream vorticity ﬂuctuations, when assessing the quality of the results. Spalart and Coleman observed instantaneous ﬂow reversals well upstream of the mean separation, almost immediately after the APG is experienced (contrast ﬁgures 4 and 6). (These local ﬂow reversals have also been seen in boundary layers with zero pressure gradient (Spalart 1988).) The probability of reversed shear increases monotonically within the APG region to a maximum of about 70%, and then falls rather more rapidly under the inﬂuence of the favourable pressure gradient. (The vertical arrows in ﬁgure 5a indicate the locations at which 1% and 50% ﬂow reversal occurs.) They ﬁnd no

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Figure 6: Skin friction (solid line) and heat transfer (shown as twice the Stanton number, chain dashed line) distributions (left scale), together with probability of reverse wall shear stress (dashed line, right scale). (From Spalart and Coleman (1997).) fundamental diﬀerence in the instantaneous structure of the APG boundary layer upstream and downstream of the mean separation. The locations of the mean separation deﬁned by the 50% probability and zero mean skin friction are nearly identical, implying there is little skewness in the wall-shear probability distribution, which agrees with the Dengel and Fernholz (1990) and Alving and Fernholz (1995) experiments, and Na and Moin’s DNS. The actual mean separation occurs well before the nominal location inferred by the mean streamline bounding the bubble (ﬁgure 5a), which implies that the region of mean backﬂow contains a thin upstream sliver adjacent to the wall; the same feature also appears in the Na and Moin ﬂow. Besides defeating the boundary-layer assumptions, the ﬂow near the bubble violates several convenient turbulence modelling assumptions: Spalart and Coleman ﬁnd negative production of turbulent kinetic energy, local countergradient heat transfer, and breakdown of the Reynolds velocity–temperature analogy. The latter (also revealed in the experiment of River, Johnston and Eaton 1994) can be seen by the opposite behaviour of the skin-friction and heat-transfer coeﬃcients in ﬁgure 6. While the skin friction drops approaching separation, the wall heat transfer increases – despite the upward orientation of the mean velocity vectors upstream of the bubble (ﬁgure 5a), the mean temperature gradient at the wall becomes steeper. Na and Moin (1998) reproduce many of the features found by Spalart and Coleman, using a ﬁnite-diﬀerence method. They utilise the second-order staggered-grid fractional-step algorithm developed by Le et al. (1997) (for the backward-facing step work described above), but improve upon Le et al.’s in-

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ﬂow treatment by feeding the instantaneous velocity ﬁeld from a previously generated zero-pressure-gradient boundary layer DNS (Spalart 1988) into the inﬂow one plane at a time, after randomising the amplitudes (but maintaining the phases). This approach deﬁnes an inﬂow requiring a much shorter development length, about one quarter of that needed by the previous scheme, and also produces a better developed ZPG boundary layer than in Spalart and Coleman. The convective boundary condition of Le et al. (1997) is used at the exit. Another diﬀerence relative to Spalart and Coleman’s approach was the use of a uniformly spaced wall-normal grid adjacent to the virtual wall at which the transpiration is applied. By doing so, they presumably were able to reduce the level of spurious freestream vorticity ﬂuctuations (although this diagnostic is not presented). Even though the ﬁnite-diﬀerence scheme is formally much less accurate than the spectral method, its ability to more arbitrarily distribute the spatial resolution within the domain is a deﬁnite advantage for this ﬂow. Reasonable results were obtained using only 12.8 million points, less than 40% of the number needed by Spalart and Coleman. A central part of Na and Moin’s analysis is investigation of turbulent pressure ﬂuctuations, and their relationship to the instantaneous vortical structures that convect up and over the bubble (ﬁgure 7). They propose a relationship between passage of these vortical structures and the magnitude of the pressure ﬂuctuations in various regions (largest in the middle of the shear layer above the bubble), as well as ﬂuctuations of the location of the mean re-attachment line. Turbulent kinetic energy budgets and mean velocity proﬁles reveal that the ﬂow qualitatively resembles a plane mixing layer far downstream of the bubble. As is the case in the backward-facing step, the boundary layer recovers very slowly after it re-attaches, even though here it is subjected to a strong favourable pressure gradient. The turbulent separation bubble of Spalart and Coleman (1997) has been modelled by Hanjalic et al. (1999). They used a low Reynolds number version of a second-moment closure and added an extra source term to the dissipation equation. The same transpiration proﬁle was used, but there was some arbitrariness about the inﬂow condition. They carried out an inﬂow-outﬂow calculation, rather than using a fringe method, which entailed making some assumptions about the dissipation proﬁle at the inﬂow. They ﬁnd generally good agreement, though the bubbles are thinner than in the DNS.

4

Flat-plate separation bubble with laminar detachment

DNS of bubbles involving laminar separation (ﬁgure 8) can be grouped into two categories – those that do not include fully developed turbulent boundary layers downstream of re-attachment and those that do. The former (Dovgal

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Figure 7: Iso-surfaces of instantaneous pressure ﬂuctuation. Iso-values: (a) ∗ = 3238, (b) in 3350, (c) 3463, and (d) 3575. (From Na and Moin tU0 /δin (1998).) et al. 1994; Maucher et al. 1994, 1999) focus primarily upon the growth of discrete two- and three-dimensional instability waves introduced upstream of the separation (which may or may not become turbulent and close into a bubble, depending upon the APG and disturbances imposed); the latter are concerned with separated laminar boundary layers, and their subsequent turbulent reattachment.

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Figure 8: The mean ﬂow structure of a laminar separation bubble. (Horton 1968). This second category has been the subject of two independent investigations, Alam and Sandham (2000) and Spalart and Strelets (2000). Both study separation of a laminar ﬂat-plate boundary layer, induced by aspiration through the opposite boundary. In both cases the detached shear layer quickly becomes turbulent, and its rapid spreading leads to re-attachment a short distance downstream of the separation line. As a result, a closed bubble again forms, but unlike the turbulent-detachment, here the re-attachment is not forced by a favourable pressure gradient caused by blowing at the opposite wall, but rather by the turbulence that develops in a detached free shear layer. After the re-attachment, the ﬂow relaxes towards a normal (undistorted) turbulent boundary layer. Another diﬀerence between this and the turbulent-separation case is that the Reynolds numbers (based on bubble length) at which practical laminar separation bubbles occur (e.g. over aerofoils) can be accommodated by DNS. This rare and important advantage implies that all of these results apply ‘as is’ (without, for example, needing to appeal to Reynolds-number similarity) to realistic and relevant engineering ﬂows. Both studies use spectral methods with a ‘fringe’ zone near the downstream boundary to allow the use of periodic boundary conditions in what is in reality a spatially evolving ﬂow. In both cases it is recognised that some low level disturbances from the outﬂow do recycle to the inﬂow boundary. The eﬀect of this is swamped by applied forcing in Alam and Sandham (2000), while in Spalart and Strelets (2000) these disturbances are not implicated in the observed transition mechanism (visualised in ﬁgure 9). Alam and Sandham’s simulation is of a shorter separation bubble, and uses a computational grid of up to 384 × 128 × 160 compared with Spalart and Strelets’ 1000 × 120 × 120. Diﬀerences in the suction proﬁle applied to the upper boundary give rise to two subtly diﬀerent transition processes. In Spalart and Strelets the bubble is longer, with signiﬁcant reverse ﬂow within the bubble. This leads in their case to a ‘transition by contact’ where the recirculating turbulent ﬂow from

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Figure 9: Visualisation of the vorticity magnitude and temperature. Mean separation is at x = 2.25 and mean reattachment at x = 4.25. (From Spalart and Strelets (2000).)

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Figure 10: Iso-surfaces of (a) wall-normal vorticity, showing (locations A and B) Λ-shaped structures at diﬀerent stages of evolution, and (b) spanwise vorticity, showing (locations A and B) shear layers forming above the Λ-structures shown in (a). (From Alam and Sandham (2000).) the back of the bubble triggers transition in the shear layer above the front of the bubble. As such, no upstream disturbances are required and the transition is not due to a convective instability. By contrast Alam and Sandham have a shorter bubble with smaller peak reverse ﬂow and a favourable pressure gradient downstream of reattachment (which most likely accounts for their observation of 100% forward ﬂow towards the end of the computational domain, in contrast to Spalart and Strelets who always observed some reversed ﬂow regions). In the Alam and Sandham simulation upstream disturbances, applied via a transition strip on the wall upstream of the separation point, are required for transition. When these were turned oﬀ (Alam, private communication) the transition process moved downstream. Alam and Sandham also observe the

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Figure 11: Turbulence kinetic energy budgets at three locations (a) inside the bubble, (b) just downstream of reattachment and (c) near end of computational box. The terms in the budgets are production (P ), dissipation (), triple u ), pressure transport (J p ), viscous transport (J ν ) and moment transport (Ji,i i,i i,i convection (Ru ). (From Alam and Sandham (2000).)

Λ-shaped vortices (ﬁgure 10) characteristic of convectively unstable boundary layer transition. The change in transition behaviour may be linked to a change in local stability characteristics from convective to absolute, which occurs for maximum backﬂows of the order of 15% (Alam and Sandham). This analysis may be relevant to the phenomenon of transition by contact described by Spalart and Strelets. Connections to large scale vortex shedding and eventual failure of the separated ﬂow to re-attach (‘bubble bursting’) are at present unclear. Results from both simulations have already begun to be analysed for their implications on closure methods. Budgets of turbulence kinetic energy at different downstream locations are shown in ﬁgure 11. Near reattachment these resemble one half of a mixing layer (cf. Le and Moin, Na and Moin), with signiﬁcant contributions from production, advection and the triple-moment transport term, and smaller but still important contributions from pressure transport. Further downstream the budgets illustrate the emergence of characteristic near-wall turbulence. The proﬁles of mean velocity in the recovery region shown in ﬁgure 12 are typical of the simulation results. The recovery is very gradual, occupying a distance up to 25 times the boundary layer thickness at reattachment.

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Figure 12: Streamwise proﬁles in the relaxing boundary layer downstream of reattachment. The locations are expressed in the legend according to number of reattachment boundary layer (99.5%) thicknesses δR downstream of reattachment and (in brackets) from simulation inﬂow in units of inﬂow displacement thickness. (From Alam and Sandham (2000).) Spalart and Strelets included in their paper a direct comparison with results from the one-equation Spalart–Allmaras model (see Chapter [1]). The model successfully captured the transition by contact process without any special modiﬁcation, and boundary layer thickness variations along the bubble were well predicted. Skin friction distributions in the bubble and in the recovery region were generally poorly predicted, along with heat transfer. The latter is believed to be due to a violation of Reynolds analogy, as was also seen in Spalart and Coleman (1997) described above. Another comparison of RANS with this DNS has been carried out by Hadzic and Hanjalic (2000). They applied low Reynolds number second-moment closure and also correctly captured the transition process without special treatments. The peak reverse ﬂow skin friction was a factor of two diﬀerent from the DNS, but they obtained improved behaviour in the recovery region downstream of reattachment compared with simpler eddy viscosity models. The Alam and Sandham (2000) simulation contains an additional challenge for models as the transition process is dependent on small amplitude upstream disturbances. Several two-equation models were tested for this ﬂow by Howard et al. (2000). In this study the model calculation replicated the DNS in the use of the same fringe zone and periodic streamwise boundary conditions.

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Thus there was no need to specify inﬂow proﬁles. The transition strip was represented by a non-zero k distribution upstream of separation, with the k distribution taken from the DNS. With standard models it was nevertheless found to be necessary to ﬁx the transition by turning on the production of turbulence term at some prescribed streamwise location. The only case where this wasn’t required was with the Wilcox (1994) k − ω model for transition, in which model constants are sensitised to the turbulence This model gave the best overall results, but still signiﬁcantly underpredicted the peak negative skin friction in the bubble. The two-equation models all gave very poor predictions of skin friction and integral boundary layer properties in the recovery region downstream of reattachment.

5

Conclusions and future directions

DNS of separating and reattaching ﬂows, possibly including transition to turbulence are now clearly feasible. Transitional separation bubble simulations are already at realistic Reynolds numbers, comparable to laboratory studies and practical applications. Turbulent separation simulations are still at relatively low Reynolds numbers, but are nevertheless beginning to provide data for model development. In the future it is hoped that databases covering a range of Reynolds numbers can be produced so that Reynolds number trends can be unambiguously known and predicted. The backward-facing step DNS highlighted problems with experimental measurements which have been remedied by new experiments. Current models generally perform only adequately for these ﬂows. A common error is under-prediction of the magnitude of the reverse ﬂow inside the separation bubbles. From the limited studies so far it does seem that second-moment approaches do better than standard two-equation models. Nonlinear eddy viscosity models remain to be tested. There has also been little published work to date on validation of subgrid models, wall functions and other fundamental aspects of LES against the separated ﬂow DNS. Now that reliable DNS databases are appearing, one can expect over the next few years to see a signiﬁcant expansion of such work.

References Alam, M. and Sandham, N.D. (2000) Direct numerical simulation of ‘short’ laminar separation bubbles with turbulent reattachment J. Fluid Mech. 410, 1–28. Coleman, G.N. and Spalart, P.R. (1993) Direct numerical simulation of a small separation bubble. Near-Wall Turbulent Flows (ed. C.G. Speziale and B.E. Launder), pp. 277–286. Elsevier. Dovgal, A.V., Kozlov, V.V. and Michalke, A. (1994) Laminar boundary layer separation: instability and associated phenomena Prog. Aerospace Sci. 30, 61–94. Hadzic, I. and Hanjalic, K. (2000) Separation-induced transition to turbulence: second moment closure modelling. J. of Flow, Turb. and Comb. 63, 153–173.

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Hanjalic, K., Hadzic, I. and Jakirlic, S. (1999) Modeling turbulent wall ﬂows subjected to strong pressure variations. J. Fluids Engineering 121, 57–64. Horton, H.P. (1968) Laminar separation in two- and three-dimensional incompressible ﬂow. PhD Dissertation. University of London. Howard, R.J.A., Alam, M.A. and Sandham, N.D. (2000) Two-equation turbulence modelling of a transitional separation bubble. J. of Flow, Turb. and Comb. 63, 175–191. Jim´enez, J. et al. (1998) A selection of test cases for the validation of large-eddy simulations of turbulent ﬂows. AGARD-AR-345. Jovic, S. and Driver, D.M. (1994) Backward-facing step measurement at low Reynolds number, Reh = 5000 NASA Tech. Mem. 10887. Jovic, S. and Driver, D.M. (1995) Reynolds number eﬀects on the skin friction in separated ﬂow behind a backward facing step. Exps. Fluids 18, 464–467. Karniadakis, G.E. and Triantafyllou, G.S. (1992) Three-dimensional dynamics and transition to turbulence in the wake of bluﬀ objects. J. Fluid Mech. 238, 1–30. Kim, J., Moin, P. and Moser, R.M. (1987) Turbulence statistics in fully-developed channel ﬂow at low Reynolds number. J. Fluid Mech. 177, 133–166. Le, H. and Moin, P. (1991) An improvement of fractional step methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 92, 369–379. Le, H., Moin, P. and Kim, J. (1997) Direct numerical simulation of turbulent ﬂow over a backward-facing step. J. Fluid Mech. 330, 349–374. Maucher, U., Rist, U. and Wagner, S. (1994) Direct numerical simulaiton of airfoil separation bubbles. Proc. Second European Computational Fluid Dynamics Conference, Stuttgart, 5–8 Sept., 1994. (ed. S. Wagner et al.), 471–477. Wiley. Maucher, U., Rist, U. and Wagner, S. (1999) Transitional structures in a laminar separation bubble. Proc. STAB-Symposium, Berlin, 10–12 Nov., 1998. Notes on Num. Fluid Mech., Vieweg. Mittal, R. and Balachandar, S. (1995) Eﬀect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders. Phys. Fluids 7, 1841–1865. Na, Y. and Moin, P. (1998) Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 370, 175–201. Najjar, F.M. and Balachandar, S. (1998) Low-frequency unsteadiness in the wake of a normal ﬂat plate J. Fluid Mech. 370, 101–147. Parneix, S., Laurence, D. and Durbin, P.A. (1998) A procedure for using DNS databases. J. Fluids Eng. 120, 40–47. Pauley, L.L., Moin, P. and Reynolds, W.C. (1990) The structure of two-dimensional steady separation. J. Fluid Mech. 220, 397–411. Peng, S-H. and Davidson, L. (2000) New two-equation eddy viscosity transport model for turbulent ﬂow computation. AIAA Journal 38, 1196–1205. Perot, B. (1999) Turbulence modelling using body force potentials. Phys. Fluids 11, 2645–2656.

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River, R.B., Johnston, J.P. and Eaton, J.K. (1994) Heat transfer on a ﬂat surface under a region of turbulent separation. J. Turbomachinery 116, 57–62. Spalart, P. (1988) Direct simulation of a turbulent boundary-layer up to Rθ = 1410 J. Fluid Mech. 187, 61–98. Spalart, P.R. and Coleman, G.N. (1997) Numerical study of a separation bubble with heat transfer. Eur. J. Mech. B Fluids 16, 169–189. Spalart, P.R. and Strelets, M.Kh. (2000) Mechanisms of transition and heat transfer in a separation bubble. J. Fluid Mech. 403, 329–349. Spalart, P.R. and Watmuﬀ, J.H. (1993) Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371. Zhang, K-Q., Fey, U., Noack, B.R., K¨ onig, M. and Eckelmann (1995) On the transition of the cylinder wake Phys. Fluids 7, 779–794.

25 Is LES Ready for Complex Flows? Bernard J. Geurts and Anthony Leonard Abstract Recent developments in LES modelling, combined with further advances in capabilities of computers and numerical methods, provide a strong incentive to apply LES to complex ﬂows. This brings to focus a number of issues in the development of LES that are not yet fully resolved. These include modelling and numerical elements, their respective errors, and the potential for interaction between these two sources of error. We discuss the relative importance of some of the errors that can arise in simple as well as complex ﬂows and give global criteria and guidelines that can be helpful in order to arrive at a form of LES that is robust and accurate.

1

Introduction

The intricate nature of high Reynolds number turbulent ﬂow has to date deﬁed detailed rigorous or direct numerical analysis and, consequently, has given rise to a number of modelling strategies. Such strategies are aimed at reducing the complexity of the underlying system of equations while retaining suﬃcient information to reliably predict the ﬂow phenomena of interest in an application. These two conﬂicting requirements are prominent in large-eddy simulation. In recent years there has been a signiﬁcant interest and associated development in large-eddy simulation (LES) which raises the question whether the LES approach can already be applied to ﬂows of engineering interest and consequently move away from academic problems. To address this question objectively in this generality is quite diﬃcult. Instead, we will focus on some outstanding open problems that need to be confronted in order to develop LES, in particular for complex ﬂows. While there has undeniably been considerable progress in LES concerning modelling and numerical issues which has resulted in a number of large-scale complex ﬂow predictions by a large number of diﬀerent research groups using LES, the signiﬁcance of some of the not yet resolved problems suggests that both a ‘yes’ and a ‘no’ answer are still possible for the question raised. In particular, as we discuss below, the answer strongly depends on the type of information one wants to predict. One can choose between the ‘implicit’ and the ‘explicit’ ﬁltering approach to LES (Ghosal 1999). We adopt here the explicit ﬁltering approach because 720

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it is more amenable to analysis and allows for a separation of issues related to modelling and numerical treatment. In this approach a spatial ﬁlter is applied to the Navier–Stokes equations. The reduction of the ﬂow complexity and information content that is achieved in this way depends strongly on the type and the width of the adopted ﬁlter. At one extreme, the width of the applied ﬁlter may be so large that virtually all information contained in the solution is removed while, at the other extreme, a very small ﬁlter-width may be adopted which does not reduce the complexity at all. Subsequently, the ﬁltered equations need to be closed by the introduction of a model for the sub-grid stresses and ﬁnally the resulting system of equations is treated numerically. These elements of LES, i.e. the information retained in the ﬁltered ﬁeld, the suitability of and the need for accurate subgrid modelling, and the role of and contamination arising from the numerical treatment will be focused upon. In view of the desired strong reduction in computational eﬀort compared to direct numerical simulation (DNS), these various sources of error can be quite signiﬁcant (Ghosal 1996; Geurts 1999) and can lead to intricate interactions with some unexpected consequences. The large-eddy modelling of incompressible ﬂow includes ﬁltering of the convective terms which leads to the turbulent stress tensor τ τij = ui uj − ui uj

; i, j = 1, 2, 3

(1)

as the major so-called subgrid term. This expression contains contributions from the ﬁltered (ui ) and the unﬁltered (ui ) velocity ﬁelds and cannot, in practice, be expressed in terms of the ﬁltered solution alone. Various modelling strategies have been proposed, some of which ﬁnd their origin in physical arguments while others start from rigorous information about properties of the stress tensor. We consider some recent modelling strategies which aim at optimising the use of the scales that are available in an LES in order to arrive at improvements in the subgrid model. This ‘inverse modelling’ (Geurts 1997) or ‘approximate deconvolution’ (Domaradzki and Saiki 1997; Stolz and Adams 1999) can give rise to models that combine high correlation with suitable transfer of energy from the resolved to the unresolved scales. For geometrically complicated ﬂows the use of a convolution ﬁlter, i.e. the use of a constant ﬁlter-width, may not be desirable. In such ﬂows one may observe regions of high turbulence intensity with many small-scale contributions next to regions of weakly turbulent ﬂow with predominantly large-scale components in the solution or next to regions of little interest such as the far wake of a bluﬀ-body ﬂow. An eﬃcient LES of such situations calls for spatial ﬁlters with nonuniform width. This, however, gives rise to additional terms that contribute to inter-scale energy-transfer in a speciﬁc way depending directly on the non-uniformity of the ﬁlter-width. This type of contribution has largely been left unstudied in literature even though its order of magnitude may be comparable to contributions from the turbulent stress tensor.

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In order to arrive at a slightly more quantitative approach to the question of adequacy of LES it is important to incorporate the type of information one wants to extract from the simulation. Since this diﬀers widely in the various application areas the capabilities of present-day LES can already be of some or even suﬃcient use in certain areas while the same LES capabilities provide insuﬃcient accuracy in other applications, e.g. due to restricted computational resources and numerical capabilities. To quantify this somewhat, we consider the prediction of a quantity q, which has an exact value qe . If we want to predict q and allow an error ε then accurate predictions of only a certain range of scales in the ﬂow are required. We denote q(n) the value of q evaluated with the ﬁrst n modes of the largest scales. The requirement |q(n)−qe | < ε speciﬁes the necessary number of modes nq (ε) for the quantity q. If in an LES one can provide nq (ε) modes with suﬃcient accuracy then the quantity q is adequately predicted. Clearly, a decrease in ε will lead to a (strong) increase in nq (ε) and, moreover, diﬀerent quantities q will require possibly very diﬀerent values of nq (ε) in order to yield the desired accuracy. However, the number of modes given by nq (ε) is ideal assuming modelling errors and numerical errors are negligible. Thus a central question is how LES should be designed to arrive at an accurate prediction of at least these ﬁrst nq (ε) modes. Because of these two types of errors any LES aiming at an adequate prediction of the quantity q should involve NLES > nq (ε) modes. The success of LES clearly depends on how much larger NLES needs to be, compared to the absolute minimum nq (ε). One might expect ‘good’ subgrid models as well as ‘good’ numerics to lead to a (slight) decrease in NLES . The latter will be illustrated in this chapter. Other factors may, however, lead to an increase in the required NLES . For example, a somewhat higher ‘numerical’ value for the minimal resolution may exist, e.g., in order to provide a stable treatment of ﬂow near walls or interfaces or to provide enough resolution to generate a stable solution in case inﬂow and outﬂow boundaries are introduced such as in spatially developing ﬂows. It is clear that the lower limit nq (ε) needs to be respected in any LES and this may well explain some of the reported failures of LES in predicting certain quantities in complex ﬂows. A number of test-cases have been studied with LES by a large number of groups in recent years (see e.g. Rodi et al. 1997 and [12]) and several appear very illustrative in relation to the failure to reach the lower value nq (ε). For a strict DNS the number of modes NDNS that needs to be incorporated is related to the reciprocal Kolmogorov length. Of course the main virtue of LES is that NLES is related to the reciprocal ﬁlter-width, i.e. 1/∆ so that NLES can be much lower than NDNS . At the same time the desire to predict q with an accuracy ε implies that NLES > nq (ε) which therefore provides a strict upper-bound for ∆. Stated diﬀerently, there is an obvious limit to the amount of information that can be ‘ﬁltered away’ if one insists in maintaining a minimal accuracy for the prediction of some ﬂow property. In order to have

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some independent control over the interaction between numerical and modelling errors in the subsequent LES the resolution should be ﬁne enough. In particular this implies that the mesh-size h has to be suﬃciently smaller than ∆. Appropriate values for the ratio ∆/h depend on the spatial discretization scheme that is used but should be at least larger than 2 or 4 for fourth- or second-order methods, respectively. We will illustrate this below. In total, the desired accuracy with which q needs to be predicted fully controls an upperbound for the mesh-size. Actual LES is ﬂawed and constrained in many ways compared to the ‘optimal eﬃciency LES’ which would need only nq (ε) modes. Thus the actual number of modes used, NLES , must be larger than nq (ε) and the corresponding ﬁlter-width and resolution, h, need to be smaller. Better numerics and modelling can help to increase acceptable values of ∆ and h and vice versa, but never beyond bounds set by nq (ε). The organisation of this chapter is as follows. In section 2 we brieﬂy formulate the ﬁltering approach to LES and identify some basic properties of the LES modelling problem that deserve to be incorporated into any modelling attempt. Recent developments in subgrid modelling which aim at incorporating information from the scales that are directly available in an LES will be considered. Section 3 describes additional complications to the LES equations that arise from extension to complex ﬂows. This involves the use of nonuniform ﬁlters and gives rise to additional terms in the ﬁltered equations which have a speciﬁc contribution to the inter-scale energy transfer that will be interpreted and estimated. The numerical treatment of the LES equations is another source of unavoidable and sometimes surprising errors. Section 4 is devoted to some unexpected consequences and paradoxes associated with the interaction of numerical and modelling errors. All sources of error in LES can be controlled to some degree at the expense of adding to the computational eﬀort. Some guidelines for LES which aim at keeping the mixture of errors within reasonable bounds will be suggested in section 5 where we also collect some concluding remarks.

2

The ﬁltering and inversion approach to LES

In this section we brieﬂy introduce the ﬁltering of the Navier–Stokes equations to derive the governing equations for large-eddy simulation. Some properties of the ﬁltered equations will be mentioned which all have consequences for the underlying modelling problem. As an illustration, algebraic properties of the turbulent stress tensor will be considered in more detail as well as the use of approximate inversion techniques and dynamic modelling. The starting point in the ﬁltering approach is the introduction of the ﬁlter operator L which is used to ﬁlter the Navier–Stokes equations. The ﬁlter considered in this section is a convolution ﬁlter and in one dimension this is

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deﬁned by

∞

u(x, t) = L(u) =

−∞

G(x − ξ)u(ξ, t)dξ

(2)

where G denotes the normalised ﬁlter-kernel. In three spatial dimensions we consider the application of ‘product ﬁlters’ which are deﬁned by u(x, t) ≡ L(u) = L1 (L2 (L3 (u))) where each of the one-dimensional ﬁlter-operations Lj corresponds to one of the Cartesian coordinates and can be written in a way similar to (2). The ﬁlter-kernel G used in LES typically has most of its ‘weight’ concentrated around the origin in a bounded domain of size ∆ which we refer to as the ﬁlter-width. It can be shown that the application of this convolution ﬁlter commutes with partial derivatives that occur in the Navier–Stokes equations, i.e. L(∂t u) = ∂t (L(u)) and similarly L(∂j u) = ∂j (L(u)) where ∂t and ∂j denote partial diﬀerentiation with respect to time t and spatial coordinate xj respectively. The ﬁlter operator does not commute with the product operator S(ui , uj ) = ui uj and as is well known the ﬁltering of the nonlinear terms in the convective ﬂux gives rise to the turbulent stress tensor τijL where we explicitly use the label ‘L’ to emphasise the role of the ﬁlter. In incompressible ﬂows and in case convolution ﬁlters are used, this is the only new term that arises in the ﬁltered equations and can be expressed in the following way: τijL = ui uj − ui uj = L(S(ui , uj )) − S(L(ui ), L(uj )) = [L, S](ui , uj ).

(3)

Here we introduced the commutator of the ﬁlter L with the product operator S for later convenience. This relation shows that the basic modelling problem in LES is completely identiﬁed with properties of the commutator [L, S]. The ﬁltered Navier–Stokes equations take on the same form as the unﬁltered equations with the exception that the divergence of the turbulent stress tensor appears as an extra term in the ﬁltered equations. With the unﬁltered equations written as N S(u) = 0 with N S a symbolic notation for the ‘Navier–Stokes’ operator, the ﬁltered equations can be written as N S(u) = −∂j τijL . In this way the divergence of the turbulent stress tensor appears as a ‘source-term’ for the evolution of the ﬁltered solution. In practice it cannot be expressed in terms of the ﬁltered solution alone and ideally would require full knowledge of the unﬁltered solution. Thus the closure problem in LES is to ﬁnd suitable expressions for τ L in terms of u alone. This modelling process is central in LES and can be guided by incorporating any sound physical properties of smallscale phenomena in turbulent ﬂow or by taking into account the mathematical structure of the ﬁltered equations. We illustrate some of the latter possibilities next. The ﬁltered equations have a number of rigorous properties which can be used to assist in the modelling process. There are several symmetries of the ﬁltered equations known, such as translational and rotational symmetry,

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Gallilean invariance and scale invariance (Ghosal 1999) involving an equivalence transformation. Similarly the realisability requirements for the turbulent stress tensor (Vreman et al. 1994) can be used to restrict the multitude of possibilities for modelling τ L . We will not consider these properties here but instead turn to algebraic properties of τ L and their use in subgrid modelling. The commutator deﬁning the turbulent stress tensor τ L shares a number of properties with the Poisson-bracket in classical mechanics. An important property of Poisson-brackets is in the context of LES known as Germano’s identity (Germano 1992) [L1 L2 , S] = [L1 , S]L2 + L1 [L2 , S] i.e.

τ L1 L2 = τ L1 L2 + L1 τ L2

(4)

where L1 and L2 denote any two ﬁlter operators and τ K = [K, S] is the turbulent stress tensor associated with a ﬁlter K. Similarly Jacobi’s identity holds for S, L1 and L2 : [L1 , [L2 , S]] + [L2 , [S, L1 ]] = −[S, [L1 , L2 ]] i.e.

[L1 , τ L2 ] − [L2 , τ L1 ] = τ [L1 ,L2 ] .

(5)

This formulation of the Jacobi identity holds for general ﬁlters. In case convolution ﬁlters are considered the right hand side in (5) is zero. The expressions in (4) and (5) provide relations between the turbulent stress tensor corresponding to diﬀerent ﬁlters and can be used to dynamically model τ L . The success of models incorporating (4) is by now well established and applied in many diﬀerent ﬂows. In the traditional formulation one selects L1 = H and L2 = L where H is the so called test-ﬁlter. In this case one can specify Germano’s identity as

τ HL (u) = τ H (L(u)) + H τ L (u) . (6) The ﬁrst term on the right hand side involves the operator τ H acting on the resolved LES ﬁeld L(u) and during an LES this is known explicitly. The remaining terms need to be replaced by a model. In the dynamic modelling approach the next step is to assume a base-model mK corresponding to the ﬁlter-level K and optimise any coeﬃcients in it in accordance with e.g. an optimal compliance with the Germano identity in a least squares sense (Lilly 1992). Several choices for the base model have been used varying from the Smagorinsky eddy-viscosity model to mixed versions consisting of similarity models, e.g. Bardina’s model or the tensor-diﬀusivity model, combined with an eddy-viscosity term. The ﬁrst base model gives rise to the dynamic model and the second option to what is known as the dynamic mixed models. In actual simulations this approach has proven to be very successful, mainly because these models avoid excessive dissipation in relatively quiescent regions

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of the ﬂow whereas appropriately high values of eddy-viscosity arise in regions with large turbulent intensity. In implementations of the dynamic procedure shortcomings of the assumed base model can require some technical adjustments. As an example, the use of dynamic eddy viscosity is not guaranteed to yield positive and relatively smoothly varying dynamic coeﬃcients. This could lead to numerical instabilities and for that reason ‘clipping’ and averaging over suitable parts of the ﬂow domain are introduced. The self-adjusting property of the model-parameters proceeds dynamically in accordance with the local instantaneous ﬂow properties and does not require ad hoc parameters other than in specifying the test-ﬁlter H. In addition, the dynamic modelling is appealing in many applications because it displays a self-restoring feedback mechanism. In fact, an under-prediction of the dynamic eddy viscosity typically tends to lead to a slight increase of small scale components of sizes comparable to the ﬁlter-width which in turn will increase the eddy viscosity and thus remove some of the newly arisen small scale components. This feedback has several appealing consequences for applications of LES. A quite complete comparison of a large number of subgrid models, combining the ideas of energy-dissipation, similarity and Germano’s identity for turbulent ﬂow in a temporal mixing layer can be found in Vreman et al. (1997). Recently, the use of inverse modelling approaches has been developed which gives rise to a further development of dynamic mixed subgrid models. We give a brief illustration of this next. The operator formulation allows to readily identify ‘generalised’ similarity models which involve approximate inversion deﬁned by L−1 (L(xk )) = xk for 0 ≤ k ≤ N (Geurts 1997). With this operation it is possible to partially reconstruct the unﬁltered solution u from the ﬁltered solution u and use this information in the deﬁnition of a subgrid model. Without any inversion the original similarity model by Bardina et al. (1983) can be written in a shorthand notation, as mB = [L, S](L(u)), i.e. applying the deﬁnition of the turbulent stress tensor directly to the available ﬁltered ﬁeld. A direct generalisation of −1 this arises from mGB = [L, S] L (L(u)) using the approximate inversion. This model was analysed in a kinematic simulation as well as for single Fourier modes. Compared to the original Bardina similarity model the generalised model showed to combine high correlation with improvements in dissipative properties while retaining the possibility to represent backscatter of energy. More recently the approximate inversion was combined with dynamic modelling (Kuerten and Geurts 1999). This was based on the choice L1 = H and L2 = H−1 L for which Germano’s identity can be speciﬁed to

τ L (u) = τ H H−1 L(u) + H τ H

−1 L

(u) .

(7)

Compared to the traditional formulation which involves the modelling of terms which correspond to length scales ∆L and ∆HL this extension which incorporates the (approximate) inverse of the test-ﬁlter H−1 requires modelling of terms on the scale of ∆L as before and ∆H−1 L . Since ∆H−1 L ≤ ∆L ≤ ∆HL the

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terms that require modelling are smaller and at the same time it is easier to maintain modelling assumptions, e.g. involving properties of an inertial range. Dynamic models based on the above have been applied successfully. A further extension involving repeated application of H−1 is also possible formally. However, since (approximate) inversion is not a very well behaved operation for the smaller scales, in actual applications one faces the risk of reconstructing small scale contributions which have been contaminated with possible numerical artifacts. Therefore there is a clear practical upper-bound to the number of times H−1 can be used beneﬁcially and from recent experience 3 or 4 appears a deﬁnite upper-bound. Another way of optimising the use of the information contained in an LES which is more implicit gives rise to the ‘tensor-diﬀusivity’ or ‘gradient’ model. Again, the basis for this model is, essentially, approximate deconvolution. Consider, for example, the Gaussian ﬁlter G(z) =

exp(−z 2 /σ 2 ) √ . πσ

(8)

For such a ﬁlter we ﬁnd that the commutator τ L is given by uv − u v =

∞

k=1

σ2 2

k

1 ∂ku ∂kv . k! ∂xk ∂xk

(9)

The full inﬁnite series above is equivalent to deconvolving u and v (clearly a singular operation) then forming the product uv and then applying the ﬁlter operation. Taking only the ﬁrst term in the series above as a subgrid model we have in d dimensions uv − u v ≈

σ 2 ∂u ∂v 2 ∂x ∂x

(10)

where repeated indicies are summed and = 1, 2, . . . , d. Use of this approximation on the ﬁltered, constant density, incompressible momentum equation for ui yields the following subgrid force on the RHS ∂ 2 ui −σ 2 S jk 2 ∂xj ∂xk

(11)

where S jk is the strain-rate tensor of the ﬁltered velocity ﬁeld. Hence the term ‘tensor diﬀusivity’ model which dates back to Leonard (1974). Similarly one could arrive at this model using a Taylor expansion on the Bardina model. A direct application of this model in LES can lead to an ill-behaved system of equations as was analysed in Vreman et al. (1996b). However, the appealing property of being able to represent backscatter, without the need of an additional ﬁltering, can be retained if this base model is combined with a dynamic eddy-viscosity. In that case a computationally eﬃcient and competitive subgrid model is arrived at (Vreman et al. 1997; Leonard and Winckelmans 1999).

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Non-uniform ﬁlters and LES of complex ﬂows

The desire to extend LES to complex ﬂows in an eﬃcient way implies that one typically encounters situations in which the turbulence intensities vary considerably within the ﬂow domain. In certain regions of the ﬂow a nearly laminar, smoothly evolving ﬂow may arise while a lively, ﬁne scale turbulent ﬂow can be present in another region. This calls for a ﬁltering approach involving a ﬁlter with a non-uniform ﬁlter-width. Here we will study some consequences of applying such ﬁlters and in particular identify and estimate the additional terms that arise from using a variable ﬁlter-width. We consider the eﬀect of applying a general, compact support ﬁlter which is deﬁned by u → u: u(x, t) ≡ L(u) =

x+∆+ (x) H(x, ξ) x−∆− (x)

∆(x)

u(ξ, t)dξ

(12)

where ∆+ , ∆− ≥ 0 denote the x-dependent upper and lower ‘bounding functions’ of the ﬁlter. The ﬁlter domain can also be represented by the ﬁlter-width ∆ = ∆+ + ∆− and the ‘skewness’ σ = ∆+ − ∆− which together with the ‘normalised kernel’ H specify the properties of the ﬁlter. We can derive the LES equations for nonuniform ﬁlters and identify a ‘mean’ term associated with the Navier–Stokes operator acting on the ﬁltered solution and several new terms which are related to commutators containing the ﬁlter L as was sketched in the previous section. Incompressible ﬂow is governed by the Navier–Stokes equations subject to the constraint of divergence free velocity ﬁelds. In dimensionless form this system of equations can be written in conservation form as ∂j uj = 0

;

∂t ui + ∂j (ui uj ) + ∂i p −

1 ∂jj ui = 0 Re

; i = 1, 2, 3

(13)

where p denotes the ‘pressure’, Re the Reynolds number and the summation convention is adopted. If we apply the ﬁlter to the system of equations in (13) commutators of L with partial derivatives and multiplication arise. After some manipulation we ﬁnd: ∂j uj = −[L, ∂j ](uj ). (14) This shows that application of a non-convolution ﬁlter to the continuity equation gives rise to terms which in general violate the local conservation form. Filtering the Navier–Stokes equations yields ∂t ui + ∂j (ui uj ) + ∂i p −

1 ∂jj ui = Re

1 − [L, ∂i ](p) − [L, ∂jj ](ui ) + [L, ∂j ](S(ui , uj )) + ∂j ([L, S](ui , uj )) . (15) Re in which the ‘Navier–Stokes’ operator applied to the ﬁltered ﬁeld is identiﬁed on the left hand side. The ﬁrst three terms on the right hand side are related to

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commutators of L and partial derivatives ∂j which as before implies violation of the conservation property in general. The turbulent stress tensor arises in the last term on the right hand side. It is the only ﬁlter-term in case convolution ﬁlters are adopted while more general ﬁlters yield the full system of equations (14) and (15). The central modelling problem for the continuous formulation and general ﬁlters is now extended to approximations modelling commutators like [L, ∂x ], [L, ∂xx ] and [L, S] in terms of operations on u. A mathematically consistent modelling can be arrived at using approximate inversion (Geurts 1997) which can be extended to non-uniform ﬁlters in a consistent way with the help of symbolic manipulation software such as maple. The new terms that have arisen can be shown to obey the same algebraic identities as put forward in relation to [L, S]. These identities can be used e.g. in a dynamic modelling of the commutators of non-uniform ﬁltering and partial derivatives which need to be taken into account for complex ﬂows. In order to establish the importance of these new commutators in relation to the regular ﬁlter terms [L, S] and to interpret in what way these terms contribute to the inter-scale energy transfer we analyse the commutators for general high order ﬁlters acting on suﬃciently smooth signals (Geurts et al. 1997). Such N th-order ﬁlters are deﬁned by requiring the ﬁrst N moments to be invariant, i.e. L(xk ) = xk for k = 0, 1, . . . , N − 1. In the following we will apply such ﬁlters and retain only the leading order terms assuming suﬃciently smooth signals for the moment. In one dimension one may readily show that

u(x) = u(x) + ∆N (x)MN (x) u(N ) (x) + · · ·

(16)

for N th-order ﬁlters. Here MN ∼ L(xN ) is the N th-order moment. If we turn to the decomposition of a typical term: ∂x (u2 ) = ∂x (u2 ) + ∂x ([L, S](u)) + [L, ∂x ](S(u))

(17)

one observes the commutators [L, S] and [L, ∂x ] to arise naturally. As derived in detail e.g. in Geurts et al. (1997) and Geurts (1999) one ﬁnds corresponding expressions for these commutators given by

[L, ∂x ](S(u)) = A(x) ∆N −1 ∆ MN + B(x) ∆N MN + ···

∂x ([L, S](u)) = a(x) ∆N −1 ∆ MN + b(x) ∆N MN

+c(x) ∆N MN + · · · where A, B, a, b and c are smooth functions containing combinations of derivatives of the solution u. From this we infer that a constant ﬁlter-width implies [L, ∂x ] = 0 and the leading order term of the turbulent stress tensor equals [L, S] ∼ ∆N for N th-order ﬁlters. However, non-uniform ﬁlters clearly give rise to contributions to the commutators which are a priori of equal order of magnitude. From this we infer, unlike ﬁndings in Van der Ven (1995) and

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Vasilyev et al. (1998) that it is not possible to remove the commutators [L, ∂x ] by a careful selection of the ﬁlter. In fact, all ﬁlters that would reduce this commutator are of higher order and consequently will also reduce the usual term [L, S]. The only possibility to control [L, ∂x ] independently is by reducing non-uniformity of the ﬁlter-width, e.g. by keeping grid-nonuniformity, which usually deﬁnes local ﬁlter-widths, small. We will quantify this to some extent next and illustrate the dynamic eﬀect associated with the new commutators. A detailed analysis of the new commutators can be obtained e.g. in a single-wave analysis in which we assume a solution u = sin(κx). For illustration purposes we consider a symmetric top-hat ﬁlter for which one has [L, ∂x ](sin(κx)) = −A sin(κx). Since A = sin(κ∆(x)/2)/(κ∆(x)/2) depends on x through ∆ we observe the importance of maintaining smooth and comparably slow spatial variations in ∆. In particular

cos(κ∆/2) − 2 sin(κ∆/2) ∆ 1 1 A = = − ((κ∆)2 ) + ((κ∆)4 ) + · · · . (κ∆/2) ∆ 24 1920 (18) In order to appreciate the magnitude of this commutator compared to [L, S] in a dynamic context we recall that [L, ∂x ](S(u)) needs to be compared to ∂x ([L, S](u)). After some manipulation we ﬁnd

∆ (x) (cos(2κx) − 1)} ∆(x) (19) where the characteristic ﬂux function C for this ﬁlter is given by [L, ∂x ](S(u)) + ∂x ([L, S](u)) = C(κ∆(x)){κ sin(2κx) +

C(z) =

1 1 4 z sin(z) − 2 + 2 cos(z) = − z2 + z + O(z 6 ). 2 z 12 180

(20)

The two contributions to the ﬂux have a ‘weight’ κ and (∆ /∆) respectively from which we infer that if variations in ∆ are suﬃciently slow, i.e. |∆ | |κ∆| then ﬁlter-width non-uniformity can be disregarded. We infer that the dynamic eﬀect of the new commutator is related to the sign of ∆ . One may interpret this as follows. A decreasing ﬁlter-width contributes to the backscatter of energy; in particular it appears that subgrid contributions tend to become resolved and thus shift to the grid-scale modes. Conversely an increase in ﬁlter-width is associated with extra dissipation since resolved scales which are convected into such regions tend to become subgrid contributions. In a priori estimates of these terms based on DNS of temporal boundary layer ﬂow (Geurts et al. 1994) it appeared that close to solid walls the ﬂux contribution from the commutators [L, ∂j ] is about half as large as that arising from [L, S] and hence one cannot avoid modelling the new commutators near walls since at high Reynolds numbers these are automatically associated with strong grid clustering. Likewise, a high correlation of the new commutators and (generalised) similarity models was observed which suggests eﬃcient and accurate ways to model these contributions.

[25] Is LES ready for complex ﬂows?

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731

Interaction between numerical and modelling errors

From the ﬁltering of the equations in the previous sections it has become clear that a large number of subgrid terms arise which need to be modelled and subsequently treated numerically. If the ratio of the ﬁlter-width to the grid-spacing of the LES-grid is too small then signiﬁcant numerical errors can occur and interact with the modelling errors discussed above. We will illustrate some consequences of the implicit ﬁltering approach in which the ﬁlter-width ∆ and grid-spacing h are identiﬁed and confront this with the explicit ﬁltering approach in which the ratio ∆/h is chosen larger than 1. The implicit ﬁltering approach has the beneﬁt of computational eﬃciency in relation to the amount of information contained in the solution but this beneﬁt can be obscured completely by an adverse interaction between the diﬀerent errors which can contaminate much larger scales. Such an interaction between errors can be controlled in the explicit ﬁltering approach but leads to an increase in computational eﬀort. In actual LES a suitable balance, expressed partly by an appropriate ratio between ∆ and h should therefore be used. We consider the numerical eﬀects by tracing the operations on a representative contribution for a convolution ﬁlter. This allows the comparison of different spatial discretization methods, ﬁlter-widths and ﬁlter implementations which are the main sources of local error. We focus on ﬁltering ∂j (ui uj ) + ∂i p in the Navier–Stokes equations and ﬁnd ∂j (ui uj ) + ∂i p = [δj (ui uj ) + δi p + Di ] + ∂j τij = [δj (ui uj ) + δi p + Di ] + [∂j mij + Ri ] (m)

= δj (ui uj ) + δi p + δj mij + [Di + Di

+ Ri ]

(21)

where Di denotes the discretization error arising from application of a spa(m) tial discretization method δj to the convective terms, Di is the error when implementing the model mij , e.g. ﬁltering as well as discretization errors and Ri = ∂j (τij − mij ) is the total ‘model-residue’ associated with mij . This term can only be determined in a priori evaluations and is of course unknown during an actual LES. So, whereas formally ∂j (ui uj ) + ∂i p is needed in an LES strictly speaking only δj (ui uj ) + δi p is directly available and two main sources of discrepancy can be identiﬁed. Whereas the subgrid-term ∂j τij is usually modelled with a subgrid-model, the discretization error Di is not taken into account. The ﬁrst question is whether this is justiﬁed and for this purpose an a priori comparison of diﬀerent spatial discretization methods and ﬁlter-widths was made for turbulent ﬂow in a mixing layer (Vreman et al. 1994, 1996a). The magnitude and ratio of the discretization error Di and the ﬂux due to the turbulent ﬂux ∂j τij determines in large part the reliability of LES predictions. We evaluated these terms for a well developed ﬂow and evaluated the errors

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associated with a second- and a fourth-order ﬁnite volume discretization operator δj . It was shown that if ∆ = h, the discretization error Di is larger than the subgrid term for both methods and in this case LES predictions would not be reliable even with a perfect subgrid-model for τ . If ∆ is suﬃciently larger than h, i.e. smoother ﬁelds are represented on the same grid, the contribution of ∂j τij is considerably larger than Di . In this regime the second-order method shows only a relatively small decrease of Di for increasing ∆, whereas the fourth-order method shows a rapid decrease of Di . This was observed for both ﬁne and very coarse LES-grids and can serve as a guidance in selecting ∆ for a speciﬁc discretization method on a given grid. An interesting related study can be found in Mason and Callen (1986) in which the Smagorinsky constant was varied at ﬁxed grid resolution. When ∆ is large the ﬁltered ﬁelds become smoother which reduces the discretization error at the expense of containing only little information about the smaller scales. A good compromise appears the choice ∆ = 2h for the fourth-order method while a second-order method requires a higher value of ∆/h. In that case fourth-order discretizations are more eﬃcient than second-order ones. (m) The second question we address is how the diﬀerent errors Di , Di and Ri interact dynamically. We consider the dynamic mixed model in combination with the discretization schemes used above as well as a pseudo-spectral method. Discrepancies between LES and ﬁltered DNS results arise mainly from shortcomings of the model and from numerical discretization on a relatively coarse grid. In LES these sources of error interact which complicates testing, since separation of subgrid-modelling and numerical eﬀects is diﬃcult (Meneveau 1994). We propose approximate separation of the eﬀects of modelling and discretization error by incorporating LES at higher resolution. We consider the evolution of the total kinetic energy E:

E= Ω

1 ui ui dx 2

(22)

where Ω is the ﬂow domain. As a function of time E displays a gradual decrease. Variations in spatial discretization method show variations in the predictions for E whose magnitude is of the same order as would arise when changing from a dynamic subgrid model to e.g. the Bardina similarity model. These eﬀects of the errors increase considerably if we use ∆ = h instead of ∆ = 2h. We can approximately separate the modelling and discretization eﬀects and focus on their interaction by incorporating a ﬁne-grid LES. The discretization error in LES will become smaller if the resolution is increased at constant ∆. The discretization error in such a ‘ﬁne-grid LES’ will be considerably smaller and we can obtain LES predictions with negligible discretization error eﬀects. The diﬀerence between these two large-eddy simulations can then give an indication of the eﬀect of the discretization error: εd = ELES − Eﬁne-grid LES whereas the diﬀerence between the ﬁne-grid LES and the ﬁltered DNS measures the eﬀect of the modelling error: εm = Eﬁne-grid LES − Eﬁltered DNS .

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2000

4000

1000

3000 errors in the total kinetic energy

errors in the total kinetic energy

[25] Is LES ready for complex ﬂows?

0

–1000

–2000

–3000

–4000 0

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–1000

20

40

60 time

80

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–2000 0

20

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Figure 1: Error decomposition with modelling error (solid) and discretizations error eﬀect for the three discretization methods: dashed (second-order), dashdotted (spectral) and dotted (fourth-order). Left ﬁgure ∆ = h and right ﬁgure ∆ = 2h. We calculated these errors for the turbulent ﬂow in a mixing layer on a representative grid which is about six times coarser in each direction than required for DNS. The corresponding ﬁne-grid LES has been performed with the ratio between ∆ and h in the ﬁne-grid LES suﬃciently large, i.e. the ﬁelds are quite smooth on grid-scale and discretization errors will be considerably reduced. The quantities εd and εm are shown in ﬁgure 1. The discretization error eﬀects are smaller than the modelling error only if ∆ ≥ 2h whereas with implicit ﬁltering εd is even larger than εm . The second-order scheme is observed to give the smallest discretization error eﬀect for this quantity. This does not imply that the discretization error itself is small, but only its eﬀect on the evolution of the total kinetic energy. For the fourth-order and pseudospectral methods the discretization error and modelling error eﬀect have opposite sign, which implies that the discretization error assists the subgrid-model in the representation of this quantity: the total error is considerably smaller than the modelling error. These observations suggest that e.g. for the spectral scheme, improvement of the subgrid-model (decrease of the modelling error) is expected to give worse results, since the total error will increase. Likewise one can infer that an increase in the resolution may result in worse predictions. All these errors and their interactions can be extremely disturbing for more complex ﬂows for which no proper separation is available and one has to rely on intuition and previous experience in order to judge and justify the outcome of a particular simulation. Since in the past this has proven to be a very underdeveloped area, we list some (hopefully) useful guidelines in the next section.

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Some guidelines for predictable LES

In this section we will formulate some general guidelines which can enhance the credibility of a ﬂow simulation within the LES approach. Since there has been a large development in the capabilities of computers and numerical methods, it has become possible to use some of these capabilities to systematically vary certain numerical elements within a ‘reference’ LES and monitor the sensitivity of the predictions. Since LES is in many respects close to a direct numerical simulation, several of the guidelines are of relevance for numerical reasons only whereas certain suggestions are more speciﬁc to the LES context. The following list which compiles the guidelines is necessarily incomplete and somewhat biased given the fact that LES is still a lively and rapidly developing ﬁeld of research. Moreover, we have put forward some guidelines which may add too much to the computational cost; however, we have taken the liberty to formulate an LES approach focusing more on reliability than on strict saving of computational eﬀort. Eventually, LES can oﬀer an expensive and reliable answer irrespective of the quality of the subgrid model and in part also quite independent of the quality of the numerical methods involved, provided the subgrid contributions are suﬃciently reduced. This ‘escape-route’, however, would be virtually identical to a well resolved DNS and be not very practical in most cases. However, the fact that LES has this limit built into it can be used also to infer about the reliability of any given ‘reference’ LES. The list of guidelines presented below has been split into mainly numerical, mainly modelling, and interaction issues. There can be a strong interaction and interdependence between modelling and numerics and for that reason some points are described under more than one topic. As a whole, an LES is as strong as its weakest element (as are many other approaches). • Numerical guidelines: — Use smooth grids with low stretching and skewness and ‘equal’ resolution in each direction. This is of importance since the formal as well as the attained accuracy of spatial discretization schemes can be considerably aﬀected by either shortcoming of the grid. Moreover, the resolution should be ‘comparable’ in each of the coordinate directions in order to avoid contamination of the solution in an under-resolved direction through ‘folding back’ of energy contained in modes which are well resolved in another coordinate direction. This obviously is ﬂow-dependent and is associated with the degree of nonhomogeneity in the ﬂow locally. — Avoid numerical dissipation. In particular for turbulent ﬂow the presence of some numerical dissipation can cover shortcomings e.g. in resolution, grid properties or modelling used in the approach. Although this may appear helpful in cases in which the resolution is too low anyway, it adds to the unreliability of LES and can seriously aﬀect the predictions, in

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particular if the same approach would be used for other ﬂows, at other resolutions or for other ﬂow conditions. — Validate your code. In order to eliminate as much as possible numerical artifacts and remaining uncertainties regarding resolution, inﬂow and outﬂow conditions, geometrical description of the ﬂow domain etc. validation is essential. This could incorporate comparison with simpler theories, e.g. using linear stability theory, checking whether basic symmetries of the equations are also contained in the numerical formulation, using experimental data if available and comparison with available ﬁltered DNS data in case validation for simple ﬂows is included. It is sensible to have some discipline of version management of the software. — Vary numerical parameters. In any numerical study a certain number of relevant numerical parameters appear and basically no physically relevant prediction should depend on any of these parameters. Variations in resolution, deﬁnition of geometry, inﬂow/outﬂow boundaries, numerical method and method of evaluation of the simulation results should be considered for LES as well as DNS. — Incorporate LES predictions at diﬀerent ﬁlter-width to mesh-size ratios into a ﬂow analysis. As was illustrated in the previous section a carefully selected set of LES predictions can be used to appreciate the inﬂuence of some of the errors involved and to some extent one could estimate these errors from the combined LES predictions. With present day computers it is now feasible to conduct several LES investigations of one ﬂow and from it extract a quantitative appreciation of the reliability of the predictions. • Modelling guidelines: — Use dynamic modelling. Dynamic modelling is appealing since it does not add any ad hoc parameters to the modelling other than properties of the test-ﬁlter. The approach has proven to be quite robust and possesses a self-restoring property if the resolution is suﬃciently high. Moreover, at suitable resolution it avoids the introduction of special wall treatment. Both properties give rise to a prediction of the large-scale ﬂow which is quite robust, suitable for transitional and turbulent ﬂow and can be used in quite general inhomogeneous ﬂows. — Use explicit ﬁltering approach. In contrast to the implicit ﬁltering setting of LES the explicit ﬁltering oﬀers independent control and treatment of the various steps relevant within LES. Using all available resolution aimed only at predicting small scale properties of the ﬂow can lead to considerable unreliability and contamination of much of the predictions

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Geurts and Leonard of all the other scales. In the explicit ﬁltering some of the resolution can be used to increase the reliability with which the smaller scales are predicted.

— Incorporate similarity and dissipation into modelling. Both these properties arise naturally from spectral considerations of LES; similarity is an inertial range property of the turbulent stress tensor itself and the required energy transfer to smaller scales is eﬃciently represented by dissipation although more reﬁned ways of energy transfer may be required if other elements in the LES approach become more reﬁned. — Use smooth grids with low stretching and skewness and ‘equal’ resolution in each direction. For modelling this point is relevant since high stretching and skewness give rise to signiﬁcant additional terms in the equations which require to be modelled. Similarly, unevenness in the grid can lead to sizeable eﬀects in the implementation and evaluation of the actual model and obscure much of the model’s potential. — Incorporate LES predictions at diﬀerent ﬁlter-width to mesh-size ratios into a ﬂow analysis. From a carefully designed set of LES predictions one could infer errors arising from the subgrid modelling. — Vary numerical parameters. The inﬂuence of the subgrid model can be controlled to some extent by a suitable change in the numerical parameters. Moreover, several models require additional ﬁltering and diﬀerentiation and the number of points available to do this is usually quite restricted and hence can make the implemented model appear to have diﬀerent properties compared to the continuum formulation. — Optimise use of scales available in LES. The options for LES modelling oﬀered by inverse modelling, approximate deconvolution and/or subgrid estimation have not been fully exploited and can be beneﬁcial to LES. • Restricted interaction guidelines: — Choose ∆/h appropriately. The ratio of the ﬁlter-width ∆ to (local) mesh-size h is an important parameter in LES. If it is large then the LES prediction will appear smooth on the grid-scale and the quality of the prediction will be mainly restricted by the quality of the subgrid model. Conversely, if this ratio is small then the eﬀects of numerics will be large. In practice a ratio ∆/h ≥ 2 appears adequate when fourthorder methods are employed but this ratio should be increased to 4 if second-order methods are used. — Incorporate LES predictions at diﬀerent ﬁlter-width to mesh-size ratios into a ﬂow analysis.

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In order to develop LES for general complex ﬂows the treatment of the near wall region is crucial and not yet well developed in LES. Similarly, if shocks or detailed capturing of chemistry is required also including multi-phase ﬂows then a proper modelling of this ‘near interface’ region is vital. Finally, an appreciation and possibly an estimate of the error in the LES predictions should be aimed at. For this purpose the use of approximate inversion of the ﬁltering, the compliance with algebraic properties and other rigorous characteristics of the subgrid terms and a systematic variation of e.g. the resolution, independent of the ﬁlter width, should be developed. The ﬁlter-width ∆ (with suitable h) and a certain subgrid model imply a certain number of modes NLES to be involved which should at any rate be suﬃciently larger than the number of modes nq (ε) needed to predict a quantity q with a desired level of accuracy. It appears relevant to quantify suitable numbers nq and corresponding NLES for several geometrically simple ﬂows which are well documented and allow for a fully resolved DNS as well in order to provide a well controlled point of reference. From an estimate of NLES for these ﬂows and a number of subgrid models and numerical methods it would be possible to formulate more general selection and design rules for reliable LES in the future. For the modelling process it is advisable to respect rigorous guidelines (e.g. symmetries, realisability, algebraic properties, inequalities) and to formulate some error monitoring and control aiming at the prediction of an ‘error-bar’ during a simulation.

Acknowledgements Stimulating and inspiring discussion with several participants in the turbulence programme of the Isaac Newton Institute has been much appreciated and helpful. One of us (BJG) would like to thank the European Science Foundation for supporting part of this work.

References Bardina, J., Ferziger, J.H., Reynolds, W.C. (1983). ‘Improved turbulence models based on large eddy simulations of homogeneous incompressible turbulence’, Stanford University, Report TF-19. Domaradzki, J.A., Saiki, E.M. (1997). ‘A subgrid-scale model based on the estimation of unresolved scales of turbulence’ Phys. Fluids 9, 1. Germano, M. (1992). ‘Turbulence: the ﬁltering approach’, J. Fluid Mech. 238, 325. Germano, M., Piomelli U., Moin P., Cabot W.H. (1991). ‘A dynamic subgrid-scale eddy viscosity model’, Phys. Fluids 3, 1760. Geurts, B.J. (1997). ‘Inverse modeling for large-eddy simulation’, Phys. Fluids 9, 3585. Geurts B.J., Vreman A.W., Kuerten J.G.M., Van Buuren R. (1997). ‘Non-commuting ﬁlters and dynamic modelling for LES of turbulent compressible ﬂow in 3D shear

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layers’. In Direct and Large-Eddy Simulation II: Grenoble, P.R. Voke, L. Kleiser, J.P. Chollet (eds.), Elsevier, 47. Geurts B.J., Vreman A.W., Kuerten J.G.M. (1994). ‘Comparison of DNS and LES of transitional and turbulent compressible ﬂow: ﬂat plate and mixing layer’. In 74th Fluid Dynamics Panel and Symposium on Application of DNS and LES to transition and turbulence, Crete, AGARD Conf. Proceedings 551:51. Geurts B.J. (1999). ‘Balancing errors in LES’. In Direct and Large-Eddy Simulation III, Proceedings of the Isaac Newton Institute Symposium/ERCOFTAC Workshop, Cambridge UK, 12–14 May, 1999, N.D. Sandham, P.R. Voke, L. Kleiser (eds.), Elsevier, 1–12. Ghosal, S. (1996). ‘An analysis of numerical errors in large-eddy simulations of turbulence’, J. Comp. Phys. 125, 187. Ghosal, S. (1999). ‘Mathematical and physical constraints on large-eddy simulation of turbulence’, AIAA J. 37, 425. Kuerten J.G.M., Geurts B.J. (1999). ‘Dynamic inverse modelling in les of the temporal mixing layer’, Proceedings Second AFOSR Conference on DNS and LES. Leonard, A. (1974). ‘Energy cascade in large-eddy simulations of turbulent ﬂuid ﬂows,’ Adv. Geophys., 18, 237. Leonard, A., Winckelmans, G.S. (1999). ‘A tensor-diﬀusivity model for Large-eddy Simulation’. In Direct and Large-Eddy simulation III: Cambridge, Proceedings of the Isaac Newton Institute Symposium/ERCOFTAC Workshop, Cambridge UK, 12–14 May, 1999, N.D. Sandham, P.R. Voke, L. Kleiser (eds.), Elsevier, 147–162. Lilly, D.K. (1992). ‘A proposed modiﬁcation of the Germano subgrid-scale closure method’, Phys. Fluids A 4, 633. Mason, P.J., Callen, N.S. (1986). ‘On the magnitude of the subgrid-scale eddy coeﬃcient in large-eddy simulations of turbulent channel ﬂow’, J. Fluid Mech. 162, 439. Meneveau, C. (1994). ‘Statistics of turbulence subgrid-scale stresses: Necessary conditions and experimental tests’, Phys. Fluids 6, 815. Piomelli, C., Moin P., Ferziger J.H. (1998). ‘Model consistency in large eddy simulation of turbulent channel ﬂows’, Phys. Fluids 31, 1884. Rodi, W., Ferziger J.H., Breuer M., Pourqui´e M. (1997). ‘Status of large-eddy simulation: results of a workshop’, Trans. ASME 119, 284. Stolz, S., Adams, N.A. (1999). ‘An approximate deconvolution procedure for largeeddy simulation’, Phys. Fluids, 11, 1699. Vasilyev, O.V., Lund, T.S., Moin, P. (1998). ‘A general class of commutative ﬁlters for LES in complex geometries’, J. Comput. Phys. 146, 82. Van der Ven, H. (1995). ‘A family of large eddy simulation ﬁlters with nonuniform ﬁlter widths’, Phys. of Fluids 7, 1171. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1994). ‘Realisability conditions for the turbulent stress tensor in large eddy simulation’, J. Fluid Mech. 278, 351. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1995). ‘A priori tests of Large Eddy

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Simulation of the compressible plane mixing layer’, J. Eng. Math. 29, 299. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1994). ‘Discretization error dominance over subgrid-terms in large eddy simulations of compressible shear layers’, Comm. Num. Meth. Eng. Math. 10, 785. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1996a). ‘Comparison of numerical schemes in Large Eddy Simulation of the temporal mixing layer’, Int. J. Num. Meth. in Fluids 22, 297. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1996b). ‘Large eddy simulation of the temporal mixing layer using the Clark model’, Theor. Comp. Fluid Dyn. 8, 309. Vreman A.W., Geurts B.J., Kuerten J.G.M. (1997). ‘Large-eddy simulation of the turbulent mixing layer’, J. Fluid Mech. 339, 357

26 Recent Developments in Two-Point Closures Claude Cambon Abstract Extensions of the frontiers of rapid distortion theory (RDT) and multi-point closures are discussed, especially developments leading towards inhomogeneous turbulence. Recent works related to zonal RDT and stability analyses for wavepacket disturbances to non-parallel rotational base ﬂows are presented. Application of linear theories to compressible ﬂows are touched upon. Homogeneous turbulence is revisited in the presence of dispersive waves, taking advantage of the close relationship between recent theories of weakly nonlinear interactions, or ‘wave-turbulence’, and classical two-point closure theories. Among various approaches to multi-point description and modelling, a review is given of multi-scale or multi-tensor transport models, which use, more or less explicitly, a spectral formulation.

1

Inhomogeneous turbulence

Multi-point formulations are not nearly as well-developed for inhomogeneous as for homogeneous turbulence. An assumption of weak inhomogeneity, in which variations of the ﬂow statistics take place over distances greater than O(), the size of the large turbulent eddies, allows some progress to be made, as, to a lesser extent, does the RDT limit of weak turbulence.

1.1

Linear theories

The solution of RDT with a known mean ﬂow Ui , arbitrarily varying in space, is a diﬃcult problem in general, but becomes somewhat simpler if the mean ﬂow is irrotational, as in the classical case of high Reynolds number ﬂow past a body, outside the wake and boundary layer. Consider a particle convected by the ﬂow, having position x at time t and x at time t. The deformation of ﬂuid elements is characterised by the Cauchy tensor Fij (x, t, t ) =

∂xi ∂xj

(1)

and the evolution of vorticity for inviscid incompressible ﬂow is then described by the Cauchy solution ωi (x, t) = Fij (x, t, t )ωj (x , t ) 740

(2)

[26] Recent developments in two-point closures

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The above formulation is exact and expresses the classical theory of inviscid vortical dynamics. However, in the context of RDT for irrotational mean ﬂows, one can neglect the ﬂuctuating part of the velocity compared with the mean part, x and x are related by mean ﬂow convection and Fij becomes the deformation tensor of the mean ﬂow alone. Since the mean ﬂow is assumed irrotational, there is no mean vorticity, and (2) describes the ﬂuctuating vorticity giving the curl of the ﬂuctuating velocity ui . The solution for the latter can be shown to be ui (x, t) = Fji−1 (x, t, t )uj (x , t ) +

∂φ ∂xi

(3)

where the velocity potential φ can, in principle, be obtained using the incompressibility requirement that ∇.u = 0, leading to a Poisson equation for φ whose source term derives from the divergence of the ﬁrst term on the right of (3) and which introduces nonlocal dependence of ui (x, t) on the entire velocity ﬁeld at time t , not just that at x . Given Ui , the above formulation provides a means of calculating the RDT solution for ui , with its Green’s function in physical space Gij (x, x , t, t ), and related development of statistical moments. It is, however, clear that detailed implementation of the above procedure is quite complicated, even for the simplest case of RDT with an irrotational mean ﬂow. In this case, equation (3) can also be used in compressible turbulence, without assuming a divergence-free velocity ﬁeld (Goldstein 1978). Calculation of mean trajectories is essential and one may follow particles convected by the mean velocity ﬁeld according to x˙ i = Ui (x, t)

(4)

with the associated Fij evolving as ∂Ui F˙ij = Fkj ∂xk

(5)

with initial conditions Fij (x, t , t ) = δij . In addition to irrotational mean ﬂows, only the case of pure plane shear attracted signiﬁcant interest in classical RDT (Townsend 1976). The latter case yields solutions for which it is no longer possible to separate pure distortion from the potential term, as in (3), but advanced semi-analytical solutions exist in the presence of simple boundary conditions (see Hunt and Carlotti 2000, for promising new developments). Assuming weak inhomogeneity, considerably more progress can be made without the need for irrotationality of the mean ﬂow, although simpliﬁcations occur in the latter case. As discussed earlier in [9], turbulence which is ﬁnescale compared with the overall dimensions of the ﬂow can be treated under RDT by following a notional particle moving with the mean velocity. Thus, the results obtained for strictly homogeneous turbulence can be extended to the

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weakly inhomogeneous case, but with a mean velocity gradient matrix λij (t) which reﬂects the ∂Ui /∂xj seen by the moving particle (Hunt 1973, Durbin and Hunt 1980). This idea has been formalised in the context of ﬂow stability (see the ‘geometric optics’ of Lifschitz and Hameiri 1991) using an asymptotic approach based on the classical WKB method, which is traditionally used to analyse the theoretical ray limit (i.e. short waves) in wave problems (see e.g. Lighthill 1978). The solution is written as ui (x, t) = ai (x, t) exp[ıψ(x, t)/δ]

(6)

with a similar expression for the ﬂuctuating pressure, where ψ is a real phase function and δ is a small parameter expressing the small scale of the ‘waves’ represented by (6), while ai (x, t) is a complex amplitude which is expanded in powers of δ according to the WKB technique. Over distances of O(δ), one can use a spatial Taylor’s series representation for ψ, up to the linear term, and approximate ai as constant. It is then apparent that (6) is locally a plane-wave Fourier component of wavenumber κi (x, t) = δ −1

∂ψ ∂xi

(7)

The amplitude ai (x, t) in (6) and the corresponding equation for the ﬂuctuating pressure are expanded as an asymptotic series in powers of δ and the result inserted into the linearised equations without viscosity. At leading order, ∂ψ one ﬁnds that ψ˙ = ∂ψ ∂t + Uj ∂xj = 0, i.e. the wave crests of (6) are convected by the mean ﬂow, with its trajectories given by (4). The spatial derivatives of ψ˙ = 0 yield κ˙ i = −λji (t)κj (8) where, as before, λij = ∂Ui /∂xj and the dot represents the mean-ﬂow material derivative ∂/∂t + Ui ∂/∂xi . At the next order, one obtains (0)

a˙ i

(0)

= −Mij (t)aj

(9)

after elimination of the pressure using the leading-order incompressibility con(0) (0) dition κi ai = 0, where ai is the leading-order term in the expansion of ai . Equations (8) and (9) have exactly the same form as the basic equations of homogeneous RDT (Townsend’s equations) and therefore, together with (4), describe the weakly inhomogeneous case at leading order. The only diﬀerence is that, rather than being simple time derivatives, the dots represent mean-ﬂow material derivatives, implying that one should follow mean ﬂow trajectories which diﬀer from one to another. The stability context is not discussed here for the sake of brevity. One should just point out that the ‘geometric optics’ for short wave disturbances can

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provide real insight into the nature of instabilities – e.g. elliptical, hyperbolic, and centrifugal – that occur in non-parallel ﬂow, with and without system rotation. Classical massive eigenvalue problems provide little or no such insight (see Cambon 2001; Godeferd, Cambon and Leblanc 2001). Continuing with pure linear theories, promising applications to compressible turbulence follow from RDT and ‘geometric optics’, as well as from Ribner’s (1954) theory for shock-wave turbulence interaction. It seems interesting to explore the common background of these approaches, diﬀerent though their applications are. In both RDT and stability analysis, distortion of the disturbance ﬁeld is mediated by a large-scale base ﬂow, whereas it is induced by linearised jump relationship across an inﬁnitely thin shock wave in the Ribner’s theory. The latter ‘Linear Interaction Approximation’ (see Lele 1994 and [19]), was recently revisited by Fabre et al. (2001) in order to complement analytical calculation of the linear transfer matrix, which connects any upstream disturbance mode to its downstream counterpart, when the disturbance ﬁeld is passing through a shock wave. Any kind of upstream disturbance (input) can be constructed, and its output calculated, with an arbitrary combination of the three modes, solenoidal, acoustic and entropic, which are uncoupled in the absence of shock waves or mean distortion in the linear limit (e.g. Kovasznay 1953). Previous LIA applications have dealt with homogeneous quasiisotropic and incompressible upstream turbulence. Recent work (Fabre et al. 2001) shows the behaviour of an upstream spot of temperature, which turns into a pair of vortices, as shown in ﬁgure 1 by results from LIA and DNS. This shows how powerful the method is, and illustrates the fact that Fourier synthesis for disturbances with a linear transfer function (as in homogeneous RDT!) is not an impediment to address strong inhomogeneity. Homogeneous RDT was generalized by Simone et al. (1997) towards isentropic compressible turbulence, using Fourier modes for disturbances to a mean ﬂow with spatiallyuniform velocity gradients, with emphasis on the stabilising-destabilising effects of compressibility on shear ﬂows. This technique gave access to linear coupling of pressure, solenoidal and dilatational disturbance modes by strong mean velocity gradients, possibly including an acoustic mode. On the other hand, ‘geometric optics’ can also be applied to weakly compressible turbulence, for wavepackets that are convected along mean streamlines (Eckhoﬀ and Storesletten 1978), but the short-wave WKB development discards the acoustic mode (long waves). An entropic disturbance mode, however, can be called into play in a nontrivial way by the latter technique. Hence, homogeneous RDT and ‘geometric optics’ are no longer governed by similar disturbance equations in the weakly compressible case. Simple explanations for diﬀerent types of compressible modes and their coupling, in the three theories, can be proposed as follows. In both LIA and homogeneous RDT, there is no physical cutoﬀ due to the external distortion, because the mean velocity gradients are spatially-uniform in the former, and the shock-wave is inﬁnitely thin in the latter. Hence, disturbances can be broadband or not, and long waves, such as

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Cambon 6

4

Y/a

2

0

-2

-4

-6

Figure 1: Isolines of pressure downstream of a shock wave, for the case where a Gaussian entropy spot s /Cp = exp(−r2 /a2 ) ( is a small amplitude) has passed through the shock wave. The upstream Mach number is 4 and the elapsed time corresponds to t = 4a/C2 , with C2 the upstream sound speed. The linear solution for p /(p2 ) with 0.02 step between isovalues is shown, with its counterpart from DNS superimposed ( = 0.01 in DNS, and has not to be speciﬁed in linear theory) (courtesy Fabre et al. 2000). acoustic waves, are not a priori forbidden. This is no longer valid for ‘geometric optics’, in which long waves are not allowed by the WKB development, even if the small parameter δ in (6) does not need to be related to the physically relevant length scales of the problem. In turn, the homogeneous assumption is not consistent with the presence of the entropic disturbance mode in RDT, since it actually consists of thermal inhomogeneities. It is not necessary to delve further into this area to be convinced that exciting issues remain for investigation. A more general theory has to be built in order to account for all the linear interactions between the three (solenoidal, acoustic, entropic) or four (solenoidal, dilatational, pressure and entropic) modes, for various external distortions.

1.2

Transport models for statistical quantities

Only incompressible turbulence is considered in what follows. In turbulent ﬂows, the ﬂuctuating ﬁeld is not the single component (6), but instead con-

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sists of a random superposition of such components. The two-point velocity correlations can be expressed as ui (x, t)uj (x + r, t), but the spectral tensor Φij (k, x, t) now varies (slowly) with position. As one might expect, given the behaviour of the underlying local Fourier components described above, it can be shown that, at leading order, weakly inhomogeneous turbulence evolves according to Φ˙ ij + Mik Φkj + Mjk Φik = 0 (10) where the dot now represents the operator ∂ ∂ ∂ − λji κj + Ui ∂t ∂xi ∂κi

(11)

and expresses both convection by the mean ﬂow and evolution of the wavenumber of individual Fourier components according to (9). The spectral evolution equation (10) corresponds to the RDT limit of its homogeneous equivalent, Craya’s equation, provided the dot operator is interpreted appropriately. Thus, following the mean ﬂow, the leading-order, local spectral tensor Φij (k, x, t) behaves as in homogeneous RDT, being given in terms of its initial values and the RDT Green’s function. The obvious way to incorporate nonlinearity and viscosity into this description is by employing Φ˙ ij + Mik Φkj + Mjk Φik = Tij − Dij − 2νκ2 Φij

(12)

rather than (10) to describe spectral evolution, where Tij could be modelled using a homogeneous spectral closure, as in [9]. For the sake of completeness, the tensor Dij would typically represent inhomogeneous diﬀusion across the mean streamline. In practice, problems with the above approach include the assumption that the turbulence is only weakly inhomogeneous, which may be a reasonable approximation for ﬁne-scale turbulence incident on a body, outside the boundary layer and wake, but is not appropriate in general. A second problem is the considerable computer resources required for anisotropic spectral calculations, of which one needs to conduct one per mean streamline even in the case of a steady mean ﬂow. Even for homogeneous turbulence, going beyond the isotropic case entails a high computational cost for two-point simulations using classical closures, a cost which is not insigniﬁcant compared with that of direct numerical simulation. Thus, it is currently unattractive to solve the full set of equations resulting from closures such as DIA, TFM or EDQNM in the inhomogeneous case without simpliﬁcations. In this context, two types of approach seem particularly promising. The ﬁrst type of approach takes inhomogeneity into account via the basis set of modes used to express the ﬂuctuations, while, as far as possible, maintaining the structure of equations of the correlation matrix similar to that of

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0.6

0.85

0.6

Figure 2: Application of the SCIT model to the ﬂow over an F2A airfoil. The contours show the ratio of the ﬂux dissipation (from the largest scale) to the dissipation rate f /, as an indicator of the unsteadiness of some regions (courtesy Hatem Touil, PhD Thesis). the homogeneous case. The modes which substitute for Fourier components may, for instance, be chosen to satisfy the boundary and incompressibility conditions. Accordingly, strong inhomogeneity due to solid boundaries can be accommodated by the very deﬁnition of the ﬂuctuation modes (see also Cambon 1982, chapter V). This approach is illustrated by the recent work of Turner (1999), who considered the problem of channel ﬂow using suitably chosen modes whose amplitude equations are analogous to those of Fourier modes in the homogeneous case and which were closed via a random phase approximation. The normal modes of the linear problem might well be good candidates in this type of approach. The second type of approach uses a dual physical/spectral space representation of the two-point correlations ui (x, t)uj (x , t) in which Fourier decomposition with respect to the separation variable, r = x − x, and a position variable such as (x + x) /2 are employed. The remaining necessary assumption is the separation of spectral and physical space dependencies of the correlations, for example by treating the inhomogeneity as weak. The ‘geometric optics’ method could lend support to this approach, along the lines touched upon in the previous subsection. Similar mathematically tractable techniques include using weighted spectral transforms (the Gabor transform, as in Nazarenko et al. 1999, for RDT) and WKB developments again. The main interest, however, of the latter approach does not reside in recovering equations similar to homogeneous RDT (the ‘geometric optics’ does the job in a more general and elegant way), but instead in reconstructing from RDT solutions a slowly space-varying Reynolds-stress tensor, so that a feed-back from the ﬂuc-

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tuating to the mean ﬂow can be allowed. Outside quasi-linear theory, this approach is mainly illustrated by semi-empirical transport models, discussed below, which treat the dependency with respect to the position variable by analogy with one-point modelling. These models cannot incorporate all the information from the general equation (12), but they retain some element of its structure. Transport models for the joint physical/spectral space energy spectrum E(κ, x) have been developed, which describe inhomogeneity in a similar way to the diﬀusive terms in the k − ε model, but allow a better treatment of dissipation, calculated using a quasi-isotropic spectral model to describe the energy cascade. Such models lie somewhere between one-point and full two-point modelling in both cost and realism. Examples include the SCIT (Simpliﬁed Closure for Inhomogeneous Turbulence) model developed at Lyon (Parpais and Bertoglio 1996; Touil et al. 2000) and the LWN (Local Wave Number) model developed at Los Alamos (Besnard et al. 1995; Clark and Zemach 1995). The SCIT model allows for an element of spectral description, but involving some heuristic closures similar to those used in one-point formulations. Diﬀusive terms in the spectral equations are accounted for by closure hypotheses like those employed in one-point modelling, a feature intended to extend the regime of applicability of the model to more strongly inhomogeneous ﬂows. The computer requirements are reduced by employing an isotropic spectral model obtained by integration of the anisotropic energy distribution over spherical shells in κ-space. This entails considerable loss of spectral information, which must subsequently be made good by additional closure hypotheses. Nonetheless, the model contains more of the underlying physics of turbulence, in particular the cascade, than do one-point models and ﬁrst results for complex ﬂows are promising (see ﬁgure 2).

2

Anisotropic turbulence with dispersive waves

This case was illustrated by stratiﬁed and rotating turbulence in [9], using two-point closure (TPC). More generally, mathematical developments in the area of wave-turbulence theory (WT) have recently renewed interest in ﬂows that consist of superimposed dispersive waves, in which nonlinear interactions drive the long time behaviour. Individual modes are of the kind ui (x, t) = ai (t) exp[ı(κ · x − σκ t)]

(13)

with a known analytical dispersion law for σκ = σ(κ). Similar averaged nonlinear amplitude equations can be found using either WT or TPC, the advantages and drawbacks of which are brieﬂy discussed below. In the case of wave-turbulence, statistical homogeneity has an equivalent counterpart, obtained by assuming random phases for the wave ﬁelds. As an interesting exception, phase randomisation is not needed in some mathematical developments of WT (Babin et al. 1998). On the other hand, one should

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perhaps recall that a random phase approximation does not necessarily imply statistical homogeneity, if the basic modes diﬀer from simple Fourier modes (Turner 1999). In addition, isotropic dispersion laws such as σκ = |κ|α in (13) are almost exclusively treated in WT for deriving Kolmogorov spectra, with the key hypothesis of constant and isotropic energy ﬂuxes across diﬀerent scales associated with a wavenumber |κ| (Zakharov et al. 1992). By contrast, in geophysical ﬂows, dispersion laws are anisotropic, with for instance σ = ±βκx /κ2 in the case of Rossby waves, σ = ±2Ωκ /κ for inertial waves and σ = ±N κ⊥ /κ for gravity waves (κx , κ and κ⊥ are the components of the associated wavevector respectively in the zonal direction, and the directions parallel or perpendicular to the rotation/gravity vectors). In the latter two 3D cases, this anisotropy is reﬂected by the strange conical — ‘St Andrew’s cross’ — shape of iso-phase surfaces in typical experiments with a localized point forcing (see views of this type in McEwan 1970; Mowbray and Rarity 1967; Godeferd and Lollini 1999), and by angular-dependent energy drains when looking at nonlinear interactions (as illustrated by Cambon et al. 1997; Godeferd and Cambon 1994; Godeferd et al. 2000). The TPC and WT theories share in general an important background. Kinetic equations for mean spectral energy densities of waves are found in WT, similar to homogeneous TPC. Their slow evolution is governed by similar energy transfer terms, which are cubic in terms of wave amplitudes (triads). There is also a possibility that these transfers involve fourth-order interactions (quartets) in WT when triple resonances are forbidden by the dispersion laws and/or by geometric constraints (e.g. shallow waters, plasma physics). When triple resonances are allowed, for instance in cases of rotating, stably stratiﬁed and MHD turbulence (Galtier et al. 2000), WT kinetic equations have exactly the same structure as their counterpart in elaborated TPC (e.g. Canevale and Martin 1982; Godeferd and Cambon 1994; Cambon et al. 1997), with the advantage of justifying the quasi-normal assumption by a rigorous asymptotic development at leading order (Benney and Saﬀman 1966). Hence, WT can give a theoretical reference to TPC in the limit of small interaction parameter (e.g. Rossby number, Froude number, magnetic number in MHD). The shape of the typical eddy damping parameter, which remains the heuristic correction to quasi-normal transfer in EDQNM, is unimportant in this limit (Scott, private communication). Its only role is to regularise the resonance operators, which reduce to Dirac delta functions in WT. Moreover, the ‘Markovianization’ (i.e. the drastic shortening of time-memory in cubic transfer terms with respect to second-order correlations) in EDQNM may have similar consequences to the averaging procedure over the slow time in WT equations. Beyond the weak nonlinearity assumption, the eddy damping, or more generally the nonlinear contribution to Kraichnan’s response function, can regain some importance for moderate interaction parameters, in allowing extrapolation from WT through TPC towards a larger domain, until the case of strong interactions is reached (e.g.

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pure isotropic turbulence without external or wave eﬀects, for which classic TPC models work satisfactorily). Proposals for renormalising such generalized eddy damping were oﬀered by Carnevale and Rubinstein (private communications).

3

Concluding remarks

With respect to most present-day closure strategies, RDT and multi-point closures give a better access to two particular aspects of turbulence. The ﬁrst is structural anisotropy, which is induced by mean velocity gradients, body forces, or the direct blocking eﬀect of solid boundaries. The second is the energy cascade involving complex multi-scale interactions, which have to be accounted for in a more or less explicit way in the case of spectral imbalance. As an example of the former in the context of homogeneous turbulence subjected to space-uniform mean velocity gradients, RDT solutions exhibit strong anisotropy. In this case RDT has the same starting point as an initial value, linear stability analysis, and some mechanisms of instability (e.g. the elliptic ﬂow instability) cannot be accounted for by classical one-point closure methods, since the instability would aﬀect the rapid pressure-strain term in a full Reynolds-stress models through a very complex angular distribution in wavespace. The challenge of reproducing RDT behaviour, by transporting extra tensors, was addressed by Cambon et al. (1992) and Reynolds and Kassinos (1995), with some limited success. Without instabilities induced by additional mean strain or mean shear, turbulence subjected to rotation, stable stratiﬁcation or simple MHD coupling, can be seen as a superposition of dispersive waves. In such cases, the RDT solutions for statistical quantities are relatively simple, and it is the nonlinear coupling which can build a complex structural anisotropy, for instance via resonant triadic interactions. As stressed earlier, the nonlinear problem of closure is better tackled by decomposing the background velocity and temperature ﬁelds in terms of normal, or eigen, modes of the linear operator. Accordingly, nonlinear equations are written for the intensities, or amplitudes, of such modes, and then for their statistical correlations. The latter quantities are kept constant in the inviscid RDT limit. Related correlation matrices have to be anisotropic in close connection with the spatial symmetries of the background dynamical equations. When divergence-free velocity ﬁelds are considered, the Craya–Herring frame of reference in 3D Fourier space (Craya 1958), or equivalently a poloidal-toroidal decomposition in physical space, yields a basis of solenoidal modes, which is useful for eventually deriving complete bases of eigenmodes (Cambon 2001). Though RDT is tractable with an assumption of strict homogeneity (provided it is permitted by dynamical equations), anisotropic equations resulting from TPC or from wave-turbulence may be complicated and computationally demanding. The main additional diﬃculty

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lies not in the number of independent spectra or co-spectra (multi-modal aspect), but in the extra angular dependency in Fourier space of these quantities. This dependency can be parameterized using angular harmonics, as used by Herring (1974), Teiss`edre (1984), Zemach (private communication) and Sanderson et al. (1986), but the number of necessary relevant harmonics can become larger and larger as anisotropy develops. Such complex anisotropy, created by nonlinear transfer and ‘slow’ pressure terms, is outside the scope of single-point closure models, even if improved versions are shown to work satisfactory when the dynamics is dominated by RDT (see Cambon et al. 1992 and Reynolds and Kassinos 1995 for rotating turbulence, and Craft and Launder 1990 for stably-stratiﬁed turbulence). More generally, eﬃcient numerical procedures have to be developed to render fully anisotropic TPC tractable, even in the homogeneous case, and interesting procedures are in progress, such as the adaptation of pseudo-spectral schemes or Monte Carlo techniques (Kaneda 1992). Another approach is to try to derive analytically asymptotic models (Godeferd et al. 2000) and scaling laws (Caillol and Zeitlin 2000; Galtier et al. 2000), but simple arguments such as constant isotropic ﬂux have to be discarded altogether as being totally irrelevant to this context. Inhomogeneity and anisotropy resulting from solid boundaries may be accounted for in a ‘rational’ way, as illustrated by Turner (1999) (TPC for channel ﬂows), and Carlotti (2000) (RDT for bounded shear ﬂow). On the other hand, weak inhomogeneity far from boundaries can be approached by the WKB techniques presented in section 1. It is worthwhile to explore how to perform the extension, towards spectral turbulent transport equations, of such asymptotic linear stability analyses, in which disturbances in terms of short wavepackets are convected and distorted following mean ﬂow trajectories. In the presence of dispersive waves, it is also possible to advect weakly inhomogeneous turbulence spectra following group velocities cg = ∇σκ . This approach would lead to replacing Ui by Ui ± cg in the equation for characteristic curves (4). Such applications of ray methods to turbulence transport have received preliminary applications in stably stratiﬁed (Riley, Godeferd, private communication) and rotating (Le Penven et al. 1993) weak turbulence. Another application would be to try to reconcile the ‘geometric optics’ mentioned above (Eckhoﬀ and Storesletten 1978) to the geometric optics extensively used in aeroacoustics, which deals with actual acoustic rays. Looking at nonlinear TPC, a last example, of limited but rational application to inhomogeneous turbulence, is the generalization of EDQNM with a related model for unidirectional ﬂuxes (gradient in physical space) of triple velocity and pressure-velocity correlations by Laporta (1995). Ultimately, one wishes to derive engineering models using elements from TPC. Unfortunately, a full angular dependence in Fourier space is presently too complicated to account for in a weakly inhomogeneous transport model for the

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spectral Reynolds stress tensor components Φij (κ, x, t). At most, sphericallyaveraged spectra φij (κ, x, t) are considered, such as for the kinetic energy spectrum E(κ, x, t) = ϕii /2. Accordingly, one has to forget the idea of recovering exactly the asymptotic RDT limit, and there is a need to model ‘rapid’ terms connected to wave-distortion, as in (11) and (10), as well as pressure-strain correlations. The latter closure problem arises from spherically averaging linear algebraic terms in (10), (12), because of the angular-dependent factor in Mij , which reﬂects contributions from pressure ﬂuctuation in the RDT dynamics. A model for the angular dependence of Φij , was proposed by Cambon et al. (1981), in order to derive a consistent closure of a spherically averaged equation (12) in homogeneous turbulence, and some interesting proposals were made by Laporta (1995) to extend this towards some inhomogeneous cases. For the sake of simplicity, such closure problems are treated in a similar manner as in single-point closure modelling, both in the LWN approximation (Los Alamos group) and in the SCIT models (Touil et al. 2000). In the same way, only heuristic corrections have been proposed in the vicinity of walls. Accordingly, the main advantage with respect to classic singlepoint closure models is the access to radial energy transfer and the related ‘classic’ cascade, with a better prediction of the dissipation rate. For instance, the imbalance between radial energy ﬂux from the largest scales εf and the dissipation rate ε can be exploited, as also illustrated in ﬁgure 2. Models from the proper orthogonal decomposition (POD) of Lumley (1967) are good candidates for extending some two-point stochastic models in terms of amplitudes, obtained by Galerkin projection of background dynamical equations, towards strongly inhomogeneous ﬂows. Since a small number of modes are suﬃcient to represent the energetic domain, low dimension dynamical models can be derived. These procedures cannot, of course, model the smallest scales which include high vorticity and are involved in the cascade process near the dissipative range. There remains work to do matching low dimension, very large scale dynamic models, for instance using POD modes, with quasi-homogeneous TPC and subgrid-scale models for LES. Another domain of interest is Lagrangian statistics and dispersion of particles (Godeferd et al. 1997; Nicolleau and Vassilicos 2000; Kaneda and Ishida 1999), with important previous work with TPC including the Lagrangian renormalised approximation (LRA) of Kaneda (1992). Even if the starting point of kinematic simulation (Nicolleau and Vassilicos 1999) is similar to that used for initialising pseudo-spectral DNS, the computation of trajectories from frozen velocity ﬁeld yields interesting results for one- and two-particle dispersion. Linear wave dynamics, as in RDT, can be incorporated very easily in any realization of the velocity ﬁeld (Godeferd et al. 1997), including random phases and random orientation of wavevectors, yielding realistic anisotropic dispersion in the presence of stratiﬁcation, rotation, and probably other organizing eﬀects such as mean shear. It should be noted that the RDT procedure

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cannot be used directly to provide Lagrangian correlations, due to the intrinsic nonlinearity of the ﬂuctuating trajectory equations, even if the velocity ﬁeld is governed by a linear equation. However, it can still be used as an essential ingredient to derive Lagrangian nonlocal models (see also Kaneda and Ishida 2000), which include much more of the real dynamics than classic ‘local’ Lagrangian stochastic models.

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