closure strategies for turbulent and transitional flows

This page intentionally left blankCLOSURE STRATEGIES FORTURBULENT AND TRANSITIONAL FLOWSThe Isaac Newton Institute of Mathematical Sciences of the University ofCambridge exists to stimulate research in all branches of the mathematicalsciences, including pure mathematics, statistics, applied mathematics, theo-retical physics, theoretical computer science, mathematical biology and eco-nomics. The research programmes it runs each year bring together leadingmathematical scientists from all over the world to exchange ideas throughseminars, teaching and informal interaction.This book, which has grown out of a two-week instructional conference atthe Newton Institute in Cambridge, is designed to serve as a graduate-leveltextbook and, equally, as a reference book for research workers in industry oracademia.CLOSURE STRATEGIES FORTURBULENT AND TRANSITIONAL FLOWSedited byB.E. LaunderUMISTandN.D. SandhamUniversity of Southamptoncaxniioci uxiviisir\ iiissCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So PauloCambridge University PressThe Edinburgh Building, Cambridge cn: :iu, United KingdomFirst published in print format isbn-13 978-0-521-79208-0 hardbackisbn-13 978-0-511-06939-0 eBook (EBL) Cambridge University Press 20022002Information on this title: www.cambridge.org/9780521792080This book is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.isbn-10 0-511-06939-1 eBook (EBL)isbn-10 0-521-79208-8 hardbackCambridge University Press has no responsibility for the persistence or accuracy ofuiis for external or third-party internet websites referred to in this book, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgisnx-:,isnx-:cisnx-:,isnx-:c:ccCONTENTSContributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiIntroductionB.E. Launder and N.D. Sandham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Part A. Physical and Numerical Techniques1. Linear and Nonlinear Eddy Viscosity ModelsT.B. Gatski and C.L. Rumsey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92. Second-Moment Turbulence Closure ModellingK. Hanjalic and S. Jakirlic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473. Closure Modelling Near the Two-Component LimitT.J. Craft and B.E. Launder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024. The Elliptic Relaxation MethodP.A. Durbin and B.A. Pettersson-Reif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275. Numerical Aspects of Applying Second-Moment Closure to Complex FlowsM.A. Leschziner and F.-S. Lien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536. Modelling Heat Transfer in Near-Wall FlowsY. Nagano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887. Introduction to Direct Numerical SimulationN.D. Sandham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2488. Introduction to Large Eddy Simulation of Turbulent FlowsJ. Fr ohlich and W. Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679. Introduction to Two-Point ClosuresC. Cambon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29910. Reacting Flows and Probability Density Function MethodsD. Roekaerts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328vi ContentsPart B. Flow Types and Processes and Strategies for Mod-elling themComplex Strains and Geometries11. Modelling of Separating and Impinging FlowsT.J. Craft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34112. Large-Eddy Simulation of the Flow past Blu BodiesW. Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36113. Large Eddy Simulation of Industrial Flows?D. Laurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Free Surface and Buoyant Eects on Turbulence14. Application of TCL Modelling to Stratied FlowsT.J. Craft and B.E. Launder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40715. Higher Moment Diusion in Stable StraticationB.B. Ilyushin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424By-Pass Transition16. DNS of Bypass TransitionP.A. Durbin, R.G. Jacobs and X. Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917. By-Pass Transition using Conventional ClosuresA.M. Savill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46418. New Strategies in Modelling By-Pass TransitionA.M. Savill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Compressible Flows19. Compressible, High Speed FlowsS. Barre, J.-P. Bonnet, T.B. Gatski and N.D. Sandham . . . . . . . . . . . . . . . 522Combusting Flows20. The Joint Scalar Probability Density Function MethodW.P. Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58221. Joint Velocity-Scalar PDF MethodsH.A. Wouters, T.W.J. Peeters and D. Roekaerts . . . . . . . . . . . . . . . . . . . . . . 626Contents viiPart C. Future Directions22. Simulation of Coherent Eddy Structure in Buoyancy-Driven Flows withSingle-Point Turbulence Closure ModelsK. Hanjalic and S. Kenjeres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65923. Use of Higher Moments to Construct PDFs in Stratied FlowsB.B. Ilyushin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68524. Direct Numerical Simulations of Separation BubblesG.N. Coleman and N.D. Sandham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70225. Is LES Ready for Complex Flows?B.J. Geurts and A. Leonard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72026. Recent Developments in Two-Point ClosuresC. Cambon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740CONTRIBUTORSS. Barre, Universite de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex,[email protected]. Bonnet, Universite de Poitiers, 40 Avenue du Recteur Pineau, 86022 PoitiersCedex, [email protected]. Cambon, Laboratoire de Mecanique des Fluides et dAcoustique, Ecole Centralede Lyon, 36 avenue Guy de Collongue, BP 163, 69131 Ecully Cedex, [email protected]. Coleman, Aeronautics and Astronautics, School of Engineering Sciences, Uni-versity of Southampton, Southampton SO17 1BJ, [email protected]. Craft, Department of Mechanical, Aerospace and Manufacturing Engineering,UMIST, PO Box 88, Manchester M60 1QD, [email protected]. Durbin, Department of Mechanical Engineering, Stanford University, StanfordCA 94305-3030 [email protected]. Fr ohlich, Institut f ur Hydromechanik, Universit at Karlsruhe, Kaiserstr.12, D-76128Karlsruhe, [email protected]. Gatski, Computational Modeling & Simulation Branch, Mail Stop 128, NASALangley Research Center, Hampton VA 23681-2199, [email protected]. Guerts, Faculty of Mathematical Sciences, University of Twente, PO Box 217,7500 AE Enschede, The [email protected]. Hanjalic, Department of Applied Physics, Thermal and Fluids Sciences, Universityof Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, [email protected]. Ilyushin, Institute of Thermophysics SD RAS, Lavrentyev Avenue 1, 630090 Novo-sibirsk, [email protected]. Jacobs, TenFold Corporation, Draper, UT 84020, [email protected]. Jakarlic, Institute for Fluid Mechanics and Aerodynamics, Darmstadt Universityof Technology, Petersenstr. 30, D-64287, Darmstadt, [email protected] ixW.P. Jones, Department of Mechanical Engineering, Imperial College of Science Tech-nology and Medicine, University of London, Exhibition Road, London SW7 2BX,[email protected]. Kenjeres, Department of Applied Physics, Thermal and Fluids Sciences, Universityof Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, [email protected]. Launder, Department of Mechanical, Aerospace and Manufacturing Engineering,UMIST, PO Box 88, Manchester M60 1QD, [email protected]. Laurence, Department of Mechanical Engineering, UMIST, PO Box 88, Manch-ester M60 1QD, UK; and Electricite de [email protected]. Leonard, Graduate Aeronautical Laboratories, California Institute of Technology,Pasadena CA 91125, [email protected]. Leschziner, Imperial College of Science Technology and Medicine, Aeronautics,Department, Prince Consort Rd., London SW7 2BY, [email protected]. Lien, Department of Mechanical Engineering, University of Waterloo, Waterloo,Ontario N2L 3G1, [email protected]. Nagano, Department of Environmental Technology, Nagoya Institute of Technol-ogy, Gokiso-cho, Showa-ku, Nagoya 466-8555, [email protected]. Peeters, Corus Research, Development & Technology P.O. Box 10000, 1970CA IJmuiden, The [email protected]. Petterson-Reif, Norwegian Defence Research Establishment, N-2025 Kjeller, [email protected]. Rodi, Institut f ur Hydromechanik, Universit at Karlsruhe, Kaiserstr.12, D-76128Karlsruhe, [email protected]. Roekaerts, Department of Applied Physics, Thermal and Fluids Sciences, Univer-sity of Technology Delft, Lorentzweg 1, NL-2628 CJ Delft, [email protected]. Rumsey, Computational Modeling & Simulation Branch, Mail Stop 128, NASALangley Research Center, Hampton VA 23681-2199, [email protected]. Sandham, Aeronautics and Astronautics, School of Engineering Sciences, Uni-versity of Southampton, Southampton SO17 1BJ, [email protected] ContributorsA.M. Savill, Department of Engineering, University of Cambridge, Trumpington St.,Cambridge CB2 1PZ, [email protected]. Wouters, Corus Research, Development & Technology, P.O. Box 10000, 1970CA IJmuiden, The [email protected]. Wu, Department of Mechanical Engineering, Stanford University, Stanford CA94305-3030, USAxiaohua [email protected] for this volume rst began to be assembled in 1999 when a six-month Pro-gramme on Turbulence was held at the Isaac Newton Institute for MathematicalSciences in Cambridge. The Programme had its own origins in an Initiative on Tur-bulence by the UK Royal Academy of Engineering, which had identied the predictionof turbulent ow as a key technology across a range of industrial sectors. Researchersfrom dierent disciplines gathered together at the Newton Institute building in Cam-bridge to work on aspects of the turbulence problem, one of the most important ofwhich was closure of the averaged (or ltered) turbulent ow equations. An Instruc-tional Workshop on Closure Strategies for Modelling Turbulent and Transitional Flowswas held from April 6 to 16th. The aim of the workshop was for the experts on closureto present material leading up to the current state-of-the-art in a form suitable forresearch students and others requiring a broad overview of the eld, together withan appreciation of current issues. A sequence of 38 lectures by 19 dierent lecturersprovided background to techniques, examples of current applications, and reectionson possible future developments. Recognising that a gathering of so many expertsfrom a variety of backgrounds was somewhat unusual, it was felt that a more polishedversion of the lecture material would be of interest to a wider audience, as a reectionupon the current state of prediction methods for turbulent ows. The result has takenrather longer to appear than originally intended, but the opportunity has been takenfor contributions to be updated wherever possible, to take account of the most recentdevelopments.The Editors would like to thank the contributors for their eorts to improve thepapers, with a view to making them more accessible to the intended audience, whichremains that of the original workshop. Especial thanks are due to Dr T. Gatski whoshouldered the not inconsiderable task of shaping three separate contributions into asingle chapter on compressible ow. Thanks are also due to the Newton Institute forhosting the workshop, and the main sponsors of the Turbulence Programme, includingthe then British Aerospace (now BAE Systems), RollsRoyce, The Meteorological Of-ce, British Gas Technology, British Energy, and the UK Defence Evaluation ResearchAgency. Finally we record our appreciation to Mrs C. King who provided invaluablesecretarial support throughout the Workshop and in the editing of the present volume.ACRONYMSABL Atmospheric Boundary LayerAPG Adverse Pressure GradientASM Algebraic Stress Model [1]BL Binomial Lagrangian [21]C/D Coalescence/Dispersion [21]CDF Cumulative Distribution Function [20]CERT Centre dEtudes et de Recherches, ToulouseCE Constrained Equilibrium [21]CFD Computational Fluid DynamicsCMC Conditional Moment Closure [10]CPU Central Processing UnitDES Detached Eddy Simulation [8]DIA Direct Interaction ApproximationDNS Direct Numerical Simulation [7]DSM Dierential Second-moment-closure Model (also SMC) [2]EASM Explicit Algebraic Stress Model [1]ECL Ecole Centrale de LyonEDF Electricite de FranceEDQNM Eddy-Damped Quasi-Normal Markovian [9]EMST Euclidean Minimum Spanning Tree [20]ER Elliptic Relaxation [4]ERCOFTAC European Research Community On Flow, Turbulenceand CombustionESRA Extended SRA [19]EVM Eddy Viscosity Model [1]FFT Fast Fourier TransformGGD/GGDH Generalized Gradient Diusion (Hypothesis) [2]GLM Generalized Langevin Model [21]IEM Interaction-by-exchange-with-the-mean [21](also known as LMSE)ILDM Intrinsic Low-Dimensional Manifold [21]IP/IPM Isotropization of Production (Model) [2]ISAT In Situ Adaptive Tabulation [20,21]KH KelvinHelmholtzLDA Laser Doppler AnemometerLDV Laser Doppler VelocimetryLES Large Eddy Simulation [8]LHS Left Hand SideLIA Linear Interaction Approximation [19]LIPM Lagrangian IP Model [21]LMSE Linear Mean Square Estimation [20] (also known as IEM)LRR Launder, Reece and Rodi [2]LSES Large-Scale Eddy StructuresMCM Mapping Closure Model [21]MDF Mass Density Fraction [21]Acronyms xiiiMPP Massively Parallel ProcessingMUSCL Monotone Upwind Schemes for Scalar Conservation LawsNASA National Aeronautics and Space AdministrationNLEVM Nonlinear Eddy Viscosity Model [1]NSE NavierStokes EquationsONERA Oce National dEtudes et de Recherches AerospatialesPBL Planetary Bounday LayerPDF Probability Density Function [10]PIV Particle Image VelocimetryPOD Proper Orthogonal DecompositionPTM Production-Transition Modication [18]QI Quasi-IsotropicQUICK Quadratic Upstream-Interpolation for Convection Kinematics [5]RANS Reynolds Averaged NavierStokes [1]RDT Rapid Distortion Theory [9]RHS Right Hand SideRLA Ristorcelli, Lumley and Abid [4]RMS Root Mean SquareRNG Renormalization Group Theory [2]RSM/RSTM Reynolds Stress (Transport) Model [2]SDM Semi-deterministic Method [22]SGD Simple Gradient Diusion [2]SGS Sub-Grid Scale [8]SIG Special Interest GroupSIMPLE Semi-Implicit Method for Pressure-Linked Equation [5]SLM Simplied Langevin Model [21]SLY Savill, Launder and Younis (a second-moment transitionmodel due to Savill [17])SM Smagorinsky Model [8]SMC Second Moment Closure [2]SNECMA Societe Nationale dEtudes et de Conception de MoteursdAvionsSRA Strong Reynolds Analogy [19]SSG Sarkar, Speziale and Gatski [1]SSM Scale Similarity Model [8]SST Shear Stress Transport [1]TCL Two-Component Limit [3]TFM Test Field Model [9]TKE Turbulence Kinetic EnergyTPC Two-Point Closure [26]T-RANS Time-dependent RANS [22]T-S TollmienSchlichtingT3A, T3B . . . A series of transition test cases [17]TVD Total Variation DiminishingUMIST University of Manchester Institute of Science and TechnologyUMIST Upstream Monotonic Interpolation for Scalar Transport [5]UTS University of Technology, SydneyVLES Very Large Eddy SimulationIntroductionB.E. Launder and N.D. SandhamAlthough Computational Fluid Dynamics (CFD) has developed to a pointwhere it is a routine tool in many applications, several diculties remain.Numerical issues, such as grid generation, are often dicult and costly, in thesense that much time and eort has to be devoted to the task, but they aremanageable. The other main problem concerns the realistic physical modellingof turbulent and transitional ow, and is much less tractable.The aim of this volume is to provide a reasonably comprehensive, up-to-date and readable account of where the numerical computation of industriallyimportant, single-phase turbulent ows has reached. Turbulent ow appears insuch a diversity of guises that no single model used for engineering calculationscan expect to mimic all the observed phenomena to the level of approximationsought. Thus, dierent levels and types of modelling are adopted according tothe nature of the physical situation under study, the type of information to beextracted, and the accuracy required.The book has been organized within three main sections. In Part A the focusis on techniques (with applications serving to illustrate the appropriateness ofthe technique adopted) while Part B examines particular types of ow, usuallyadopting a single preferred modelling strategy. Finally, in Part C, some currentresearch approaches are introduced. Throughout, references to other articles inthe book are given by their chapter number in square brackets. The individualarticles themselves are sequenced broadly in terms of increasing complexity,at least within Parts A and B. The nomenclature undergoes some variationacross the chapters, reecting the dierences habitually adopted in the journalliterature over the dierent themes covered in the volume. Nomenclature forcore variables is dened at the start of Chapter [1] and this is essentiallycommon for Part A. Additional denitions or variants are provided in theindividual chapters as needed.Part A: Physical and Numerical TechniquesChapters [1][6] present a sequence of articles on single point closure. Theserepresent the core of what is usually understood by turbulence modelling.Chapter [1] by Gatski and Rumsey considers linear and non-linear modelsof eddy-viscosity type. It begins with algebraic variants of the mixing-lengthhypothesis and considers in turn various elaborations up to conventional two-equation models and the k--v2extension. The chapter closes with an extensivediscussion of non-linear eddy-viscosity models, a closure level which apprecia-bly enlarges the range of ows that may successfully be modelled, usually for12 Introductionlittle additional cost. However, when transport or force-eld eects on the tur-bulent uctuations are large, a formal second-moment closure is usually to bepreferred. Thus, one solves transport equations for all the second momentsi.e. the non-zero turbulent stresses and, in non-isothermal ows, the heat uxestoo. In Chapter [2] Hanjalic and Jakirlic provide an overview of the importantmodelling issues at this level and some of the modelling strategies adopted overthe last 25 years. The chapter concludes by presenting an impressive range oftest ows that have been computed with the form of closure adopted by theauthors group.Chapters [3] and [4] which follow, by Craft and Launder (CL) and Durbinand Petterson-Reif (DP) provide, in greater detail, particular modelling strate-gies in second-moment closure especially for the crucially important pressure-strain terms. Both are motivated by the aim of replacing the widely adopted,though limited, algebraic wall-reection scheme that attempts to account formodications to the pressure uctuations brought about by a wall. The DPchapter reviews the current form of the elliptic-relaxation method which re-places the algebraic scheme by a set of relatively simple partial dierentialequations. That by CL reviews the two-component-limit strategy; their aimis partly to remove the need for wall reection and partly to achieve a widerapplicability of the model in free ows by adopting a more elaborate treatmentfor the case where walls are absent.If one is going to adopt a model at second-moment closure level, onesoutput comprises point values of the stresses rather than the value of the eddyviscosity. The recommended strategies for incorporating such models into thecomputer code in order to achieve rapid convergence of the numerical solverare 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) diusion. Thediscussion covers both second-moment and eddy viscosity approaches withparticular focus being placed on an equation for the dissipation rate of meansquare temperature uctuations. A major requirement for heat transport mod-elling is that, besides gaseous ow where the molecular diusivities of heat andmomentum are of a similar magnitude, one also needs to cope with Prandtlnumbers both much less than (liquid metals) and much greater than (oils)unity.Sandham (Chapter [7]) and Fr ohlich and Rodi (Chapter [8]) introducesimulation-based approaches, dealing respectively with Direct Numerical Sim-ulation (DNS), where all scales of turbulence are resolved, and Large-EddySimulation (LES), where large scales are resolved and small scales modelled.These approaches are becoming increasingly realistic as computer performancecontinues to improve. DNS provides reference solutions for simple canonicalows, against which turbulence closure assumptions can be checked, whilstLES is developing towards a practical method of prediction. Limitations onIntroduction 3Reynolds number, due to the range of turbulence scales that need to be re-solved, are emphasised in these contributions.An alternative perspective on turbulence closure is provided by Cambon inChapter [9]. Here the single-point approaches discussed in [1][5] are placedinto the context of multi-point and higher-order closures. Though mathemat-ically more demanding, such approaches contain more of the physics of tur-bulent ow and provide useful insight into fundamental phenomena, such asnonlinearity and non-locality. Emphasis is placed on two-point closures, withpractical examples of rotation and stratication used to illustrate the insightthat can be obtained with this approach.Part A concludes with Chapter [10], in which Roekaerts gives an intro-duction to the modelling of reacting ows. In this application mass-weightedaveraging is introduced for the rst time (this form of averaging is also usedin Chapter [19], when compressible, high-speed ows are discussed). To ac-count for chemistry eects, methods based on probability density functions(PDFs) are introduced. Applications of one-point scalar PDF methods andjoint velocity-scalar PDFs appear later, in Chapters [20] and [21].Part B: Flow Types and ProcessesPart B of the volume begins with a consideration of the capability of single-point closures and LES in tackling ows with separated ow regions and strongstreamline curvature. Craft in [11] examines the strengths and, all too often,the weaknesses of single-point closure when applied to separated and impingingows. This article, read in conjunction with the applications reported in [3][5],provides an overview of the performance achieved by the dierent modellinglevels. In Chapters [12] and [13], Rodi and Laurence discuss the capabilities ofLES, and we have our rst glimpse of a current debate concerning the extentto which LES will replace single-point closure approaches for practical prob-lems. 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 simulationof a laboratory experiment of ow around a blu-body, dominated by sep-aration and strong vortex shedding. Dierences between techniques requirefurther investigation, but the chapter ends on an optimistic note that LESwill soon become aordable and ready for practical applications. Laurencein [13], however, damps some of the high expectations, by suggesting that inmany industrial problems the increase in computer power will simply result inmore complete single-point predictions, and we may be waiting many years tosee LES widely used.The application of second- and third-moment closure to problems of hor-izontal shear ows aected by buoyancy is the theme of Chapters [14] and[15] by Craft and Launder, and Ilyushin. Stably stratied horizontal owsturn out to be far more dicult to capture than vertical mixed convection,4 Introductionwhere even a linear eddy viscosity model does fairly well in reproducing theobserved phenomena. The reason for this dierence in ease of predictability isthat, for vertical ow, buoyant eects in the mean momentum equation intro-duce additional shear which is usually the dominant feature of any change inturbulence structure. In the horizontal shear ow the only important eectsof stratication arise through the impact of buoyancy on the turbulent elditself. Because it is the vertical velocity uctuations that are mainly aectedby the stratication, second-moment closure is usually seen as the best start-ing point for closure. Yet, as both sets of authors point out, situations arisewhere second-moment closure is inadequate, though agreement with obser-vation may be restored if, instead, closure is eected at third-moment level.Looking ahead, an alternative route for dealing with this type of problem isdeveloped in [22] by Hanjalic and Kenjeres where the large-scale structures inRayleighBenard convection are resolved by employing a time-dependent solu-tion of the Reynolds equations using just a (highly) truncated second-momentclosure.The problem of by-pass transition has been the subject of single-pointturbulence modelling since the early 1970s. The rationale was originally pro-vided by the fact that at least some low-Reynolds-number two-equation eddy-viscosity models reproduced the reversion of a turbulent boundary layer backto (or towards) laminar when subjected to a severe acceleration. In view ofthat, it was conjectured that forward transition (from laminar to turbulentow) in the presence of a turbulent external stream could also be predictedby the same model . . . and so it proved. Since those early days the appre-ciation of the detailed processes taking place in by-pass transition has comea long way, progress being greatly assisted by DNS/LES studies of the typeprovided by Durbin, Jacobs and Wu in Chapter [16]. Savills survey of mod-elling approaches is divided into two parts, Chapter [17] dealing with the useof conventional closures that have been designed for fully turbulent ows whileChapter [18] considers special modelling features. A major aim of current re-search eorts is to drive down the level of external-stream turbulence at whichaccurate prediction can be made and it is this goal that has led to the use ofintermittency parameters and other devices discussed in [18].Compressible ows, which form the subject of Chapter [19], in fact containseveral dierent phenomena requiring the modellers attention. The rst isthe question of how one should perform the averaging process in a uid wherethe density is itself varying in time. From there, issues concerning the eectsof density uctuations on the dierent processes provide a major challenge.Finally structural changes to turbulence passing through a shock wave needto be considered. All these topics are addressed by Barre, Bonnet, Gatskiand Sandham. It is noted that questions of numerical solution, addressed inChapter [5], have taken account of the requirements of compressible ow.Indeed that chapter shows an application to a supersonic three-dimensionalIntroduction 5ow with a bow shock present.In [20] Jones presents a review of the one-point scalar PDF approach, appliedto ows with chemical reaction. It is argued that in the exact equation for ascalar PDF it is the term representing molecular mixing which presents thechief diculty, and various approaches are described. Applications to a jetdiusion ame illustrate the current state of the art. Extensions to a jointvelocity-scalar PDF, solved by means of a Monte Carlo method, are describedby Wouters, Peeters and Roekaerts in Chapter [21]. This approach has a uniedtreatment of all the terms in the averaged equations that must be closed,including conventional Reynolds stresses. The method is expensive, but resultsfor a blu-body stabilized diusion ame are promising.Part C: Future DirectionsIn this nal part of the volume, space is allocated to some of the strategiesthat 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. Ashas been signalled earlier, Hanjalic and Kenjeres [22] report that the prob-lem of RayleighBenard convection (in which a horizontal layer of uid, con-ned within the space between horizontal planes, is heated from below) iscaptured much better with a truncated second-moment closure if one adoptsa time-dependent rather than a steady-state numerical solution. Essentiallywhat results from the time-dependent simulation is a close replica of the largeeddy-simulation of the same ow. Put another way, the TRANS simulation iseectively a coarse-grid LES that uses a higher level of sub-grid model andis grid independent. Clearly, for the problem chosen it would seem that thisfusion of RANS and LES strategy is wholly satisfactory and superior (fromthe standpoint of accuracy or cost) to either a steady state RANS or a con-ventional LES. Ilyushin in Chapter [23] also develops inter-linkages betweentwo approaches to turbulence that are usually viewed as discrete. In this casehe shows, among other contributions, how a knowledge of just the second- andthird-order moments enable the probability density functions to be approxi-mated.Coleman and Sandham review the latest direct simulations of separationbubbles in Chapter [24]. Turbulent separation bubbles are at the current lim-its of computer power, with severe Reynolds number restrictions. However,DNS of transitional separation bubbles, an important phenomenon that insome cases controls the performance of aerofoils, are already at the Reynoldsnumbers encountered in applications. Guerts and Leonard consider, in Chap-ter [25], recent developments in LES and the issues facing LES that need to beaddressed for it to be developed into a reliable predictive tool. The guidelinesfor 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 LES6 Introductionfor complex ow problems. Closure methods will continue to require guidancefrom experiment and theory, and in Chapter [26] we conclude the volume witha review by Cambon of the potential for further insight coming from recentdevelopments in two-point closures.Part A.Physical and Numerical Techniques1Linear and Nonlinear Eddy ViscosityModelsT.B. Gatski and C.L. Rumsey1 IntroductionEven with the advent of a new generation of vector and now parallel processors,the direct simulation of complex turbulent ows is not possible and will notbe for the foreseeable future. The problem is simply the inability to resolve allthe component scales within the turbulent ow.In the context of scale modeling, the most direct approach is oered by thepartitioning of the ow eld into a mean and uctuating part (Reynolds 1895).This process, known as a Reynolds decomposition, leads to a set of Reynolds-averaged NavierStokes (RANS) equations. Although this process eliminatesthe need to completely resolve the turbulent motion, its drawback is thatunknown single-point, higher-order correlations appear in both the mean andturbulent equations. The need to model these correlations is the well-knownclosure problem. Nevertheless, the RANS approach is the engineering toolof choice for solving turbulent ow problems. It is a robust, easy to use, andcost eective means of computing both the mean ow as well as the turbulentstresses and has been overall, a good ow-prediction technology.From a physical standpoint, the task is to characterize the turbulence. Oneobvious characterization is to adequately describe the evolution of represen-tative turbulent velocity and length scales, an idea that originated almost60 years ago (Kolmogorov 1942). The physical cornerstone behind the de-velopment of turbulent closure models is this ability to correctly model thecharacteristic scales associated with the turbulent ow. This chapter describesincompressible, turbulent closure models which (can) couple with the RANSequations through a turbulent eddy viscosity (velocity length scale). In thiscontext both linear and nonlinear eddy viscosity models are discussed. Thedescriptors linear and nonlinear refer to the tensor representation used forthe model. The linear models assume a Boussinesq relationship between theturbulent stresses or second-moments and the mean strain rate tensor throughan isotropic eddy viscosity. The nonlinear models assume a higher-order tensorrepresentation involving either powers of the mean velocity gradient tensor orcombinations of the mean strain rate and rotation rate tensors.Within the framework of linear eddy viscosity models (EVMs), a hierarchyof closure schemes exists, ranging from the zero-equation or algebraic models tothe two-equation models. At the zero-equation level, the turbulent velocity and910 Gatski and RumseyNomenclaturebij, b Reynolds stress anisotropy Wij mean rotation rate tensor intensor, (uiuj/2k) ij/3 transformed frameC, C eddy viscosity calibration X orthogonal transformationcoecient matrixD/Dt material derivative xi coordinate direction in(= /t +Uj/xj) inertial (Cartesian) frame (x, y, z)T, Tij represents the combined n tensorial expansion coecientseect of turbulent transport isotropic turbulent energyand viscous diusion dissipation ratek turbulent kinetic energy near-wall modied( ii/2) dissipation rateL characteristic length scale ij dissipation rate tensorin wall proximity boundary layer thicknessl mixing length displacement thicknessP, p mean pressure scalar invariant (SikSki)T turbulent kinetic energy von Karman constantproduction term densityR symmetric, traceless tensor ij viscous stress tensorin algebraic stress equation ij pressure strain rate2ow parameter correlation( W2/S2) kinematic viscositySij, S mean strain rate tensor t, t turbulent eddy viscosity( (Ui/xj +Uj/xi)/2) ti, to inner and outer eddy viscosityT characteristic time scale in turbulent time scale (= k/)wall proximity ij, Reynolds stress tensor ( uiuj)T(n)tensor basis element ij arbitrary time-independentUe edge velocity rotation rate of noninertialUi mean velocity component frameu friction velocity r rotation rate of noninertialWij, W mean rotation rate tensor framein noninertial frame dissipation rate per unit kinetic( (Ui/xj Uj/xi)/2) energyWij, W modied mean rotation ratetensor in inertial frame[1] Linear and nonlinear eddy viscosity models 11length scales are specied algebraically whereas, at the two-equation level, dif-ferential transport equations are used for both the velocity and length scales.Within the framework of nonlinear eddy viscosity models (NLEVMs), thecharacterizing feature is the (polynomial) tensor representation for the second-moments or Reynolds stresses. However, the method of determining the ex-pansion coecients diers among models. In some methods the expansioncoecients are determined through calibrations with experimental or numer-ical data and the imposition of dynamic constraints. In other methods, theexpansion coecients are related directly to the closure coecients used inthe full dierential Reynolds stress equations. The models derived using theselatter methods are sometimes referred to as explicit algebraic stress models.Over the years, there has been a multitude of models at the EVM andNLEVM levels proposed for the RANS equations. No attempt is made (sincewe would surely fail) to be all inclusive with the choice of models for eachlevel of closure discussed. Our goal, however, is to provide the reader witha broad perspective on the development of such models, so that, with thisbroader view, he or she will be better prepared to assess the viability of usinga particular closure scheme.2 Reynolds-averaged NavierStokes formulationAs a prelude to the discussion of the linear and nonlinear eddy viscosity mod-els, it is desirable to describe the Reynolds averaging procedure and the result-ing form of the mean momentum and continuity equations. In the Reynoldsdecomposition, the ow variables are decomposed into mean and uctuatingcomponents asf = f +f

. (2.1)The average of a uctuating quantity is zero f

= 0, and the mean quantityf can be extracted if a statistically steady or a statistically homogeneousturbulence is assumed. For example, if the turbulence is stationary,f(x) = limT 1T_ t0+Tt0f(x, t) dt, (2.2)and the average of the product of two quantities is f g = fg +f

g

.The velocity (ui) and pressure (p) elds can be decomposed into their mean(Ui, P) and uctuating parts (ui, p), and the resulting Reynolds-averagedNavierStokes (RANS) equations can be written asDUiDt = Uit +UjUixj= 1Pxi+ ijxj ijxj. (2.3)For an incompressible ow, the mass conservation equation reduces to themean continuity equation,Ujxj= 0. (2.4)12 Gatski and RumseyThe viscous stress tensor ij for a Newtonian uid and incompressible ow isgiven byij = 2Sij, (2.5)where is the kinematic viscosity, and Sij is the strain rate tensorSij = 12_Uixj+ Ujxi_. (2.6)As equation(2.3) shows, for closure the RANS formulation requires a modelfor the second-moment (or Reynolds stress) ij (= uiuj).3 Linear eddy viscosity modelsUsing continuity, equation(2.4), and the denition of the viscous stress, equa-tion(2.5), the RANS equation can be written in the formDUiDt + 1Pxi= ijxj+ xj_Uixj_. (3.1)For linear eddy viscosity models (linear EVMs), the equation is closed by usinga Boussinesq-type approximation between the turbulent Reynolds stress andthe mean strain rateij = 23kij2tSij, (3.2)where k (= ii/2) is the turbulent kinetic energy, and t is the turbulent eddyviscosity. In Section 3.4, it will be shown that such a closure model can beextracted from an analysis of a simple shear ow in local equilibrium. Whenequation(3.2) is used as the turbulent closure in linear EVMs, equation(3.1)can be rewritten asDUiDt + 1pxi= xj_( +t) Uixj_, (3.3)where the isotropic part of the closure model, 2k/3, is assimilated into thepressure term so that p = P + 2k/3. In the EVM formulation, the turbulenceeld is coupled to the mean eld only through the turbulent eddy viscosity,which appears as part of an eective viscosity ( +t) in the diusion term ofthe Reynolds-averaged NavierStokes equation. Since in general, t > , thisformulation of the problem can be rather robust numerically, especially whencompared to the alternative form of retaining the stress gradient ij/xjexplicitly in equation(3.1).In the remainder of this section, a hierarchy of linear eddy viscosity modelswill be presented ranging from the least complex (algebraic) to the most com-plex (dierential transport) means of specifying the turbulent eddy viscosityt.[1] Linear and nonlinear eddy viscosity models 133.1 Zero-equation modelsThe zero-equation model is so named because the eddy viscosity required inthe turbulent stress-strain relationship is dened from an algebraic relationshiprather than from a dierential one. The earliest example of such a closure isPrandtls mixing-length theory (Prandtl 1925). By analogy with the kinetictheory of gases, Prandtl assumed the form for the turbulent eddy viscosityin a plane shear ow with unidirectional mean ow U1(x2) = U(y) and shearstress 12 = xy = tdU/dy. The eddy viscosity was assumed to have theformt = l2dUdy, (3.4)where l is the mixing length that requires specication for each ow underconsideration. In a free shear ow, the mixing length would be a character-istic measure of the width of the shear layer. In a planar wall-bounded ow,the mixing length l in the near-wall region would be proportional to the dis-tance from the wall. These relationships, though simple, give rise to signicantinsights about the structure of turbulent ows. In the case of wall-boundedows, the law of the wall and the structure of the outer layer of the boundary-layer ow can be deduced. Several texts and reviews in the literature providean insightful description of the physical and mathematical basis for this typeof 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 CebeciSmith (see Cebeci and Smith 1974) and the BaldwinLomax (see Baldwin andLomax 1978) models. Even though the original development of these mod-els was motivated by application to compressible ows, no explicit accountwas taken of compressibility eects. Density eects are simply accounted forthrough a variable-mean-density extension of the incompressible formulation(t = t). These are two-layer mixing-length models that have an inner layereddy viscosity given byCebeciSmith: ti = l2_UixjUixj_12(3.5)BaldwinLomax: ti = l2_2WijWij, (3.6)where Wij is the rotation tensorWij = 12_Uixj Ujxi_, (3.7)and _2WijWij represents the magnitude of the vorticity. An outer layer eddyviscosity is given byCebeciSmith: to = 0.0168UeFK(y; ) (3.8)BaldwinLomax: to = 0.0269FwkFK(y; ym/0.3). (3.9)14 Gatski and RumseyThe mixing length is dened similarly in both models for zero-pressure-grad-ient ows. That is,l = y_1 ey+/A+_, (3.10)where = 0.41 is the von Karman constant, A+= 26 is the Van Driest damp-ing coecient, and y+is the distance from the wall in wall units (uy/).In the expressions for the outer layer eddy viscosity, is the boundary-layerthickness, is the displacement thickness, and Ue is the edge velocity. Ingeneral, the damping coecient A+can be a function of the pressure gradi-ent, but for present purposes it will be assumed to be constant. Throughoutthis subsection, attention will be focused on the form of the models for zero-pressure-gradient ows; extensions that include pressure gradient eects canbe found in the references cited for the particular algebraic models. The func-tions FK and Fwk are an intermittency and a wake function, respectively. TheKlebano intermittency function FK is given byFK(y; ) =_1 + 5.5_y_6_1, (3.11)and the wake function Fwk is given byFwk = min_ymFm; ymU2dif/Fm_ (3.12)withFm = 1_maxy(l_2WijWij)_. (3.13)In the above, ym is the distance from the body surface where Fm occurs, isthe boundary-layer thickness in the CebeciSmith model, and is ym/0.3in the BaldwinLomax model. The quantity Udif is the dierence betweenthe maximum and minimum total velocity in the prole. Unlike the CebeciSmith model, the BaldwinLomax model does not need to know the locationof the boundary-layer edge. As equation(3.13) suggests, the BaldwinLomaxmodel bases the outer layer length scale on the vorticity in the layer ratherthan on the displacement thickness, as in the CebeciSmith model. Extensionsand generalizations to more complex ows can be found in Cebeci and Smith(1974), Degani and Schi (1986), and Wilcox (1998).A disadvantage of the CebeciSmith and BaldwinLomax turbulence mod-els is that they possess an inherent dependency on the grid structure: quan-tities are evaluated and searched for along grid lines normal to walls. Thisdependency can be problematic for unstructured grids or for multiple-zonestructured grids. Also, it has been shown that these models, in their originalform, generally do not predict separated ows well. For example, when strongshock-induced separation is present, these models tend to predict the shockposition too far aft. However, the BaldwinLomax model with the Degani-Schi modication is often still used in industry for three-dimensional vortical[1] Linear and nonlinear eddy viscosity models 15ow applications because other models (including some of the one- and two-equation eld equation models) can diuse vortices excessively.3.2 Half-equation modelsThe motivation for the development of the JohnsonKing model (Johnson andKing 1985) was primarily the need to solve a particular class of ows turbu-lent boundary layer ows in strong adverse pressure gradients rather thanthe development of a universal model. The model was developed to accountfor strong history eects that were observed to be characteristic of turbulentboundary layers subjected to rapid changes in the streamwise pressure gra-dient. Johnson and King felt that the simple algebraic models (as outlinedin Section 3.1) could be modied suciently, without recourse to the moreelaborate dierential transport formulations (such as the two-equation formu-lation to be discussed in Section 3.4), to better predict ows with massiveseparation. Thus, advection eects were deemed essential, whereas turbulenttransport and diusion eects were assumed to have much less importance.This level of closure derives its name somewhat subjectively because anordinary dierential equation is solved instead of a partial dierential equa-tion. Nevertheless, this level of closure does generalize the algebraic modelsby specifying a smooth functional behavior for the eddy viscosity across theboundary layer and by accounting in a limited way for history (relaxation)eects by solving a transport equation for the maximum shear stress. Sincethe inception of the JohnsonKing model, it has undergone some modication(Johnson 1987, Johnson and Coakley 1990) to improve its predictive capabili-ties for a wider class of ows, and in particular, for compressible ows. For thepresent purpose, only the simpler incompressible formulation will be outlined.The JohnsonKing model is also a two-layer model; however, in this model,the eddy viscosity changes in a prescribed functional manner from the innerlayer form to the outer layer form. This functional form is given by (Johnsonand King 1985)t = to [1 exp (ti/to)] . (3.14)In the later form of the model (Johnson and Coakley 1990), which was alsoused in the solution of transonic ow problems, this functional dependencywas based on a hyperbolic tangent function. The inner layer eddy viscosity isgiven byti = l2_xy[my , (3.15)where l is the mixing length dened in equation(3.10) with A+= 15, and thesubscript m denotes maximum value along a grid coordinate line normal toa solid wall surface. In zero-pressure-gradient, two-dimensional ows in whichthe law of the wall holds, this expression for ti corresponds to the CebeciSmith inner layer eddy viscosity given in equation(3.5).16 Gatski and RumseyThe outer layer eddy viscosity is given byto = 0.0168UeFK(y; )(x), (3.16)which is the CebeciSmith form, equation (3.8), with the addition of the factor(x) that accounts for streamwise evolution of the ow. At each streamwisestation, (x) is adjusted so that the relationt[m = xy[mU/y[m(3.17)is satised. The remaining quantity that is needed is xy[m m, and thisis determined from a transport equation for the shear stress xy. Unlike con-ventional Reynolds-stress closures in which the transport equation for theturbulent shear stress contains modeled pressure-strain correlations and tur-bulent transport terms, this turbulent shear stress equation is extracted fromthe turbulent kinetic energy equation (cf. equation(3.30)) by assuming thatthe shear stress anisotropy b12 = xy/2k = 0.125 is constant at the point ofmaximum shear. The log-layer of an equilibrium turbulent boundary layer owis a constant stress layer; therefore, the assumption used here is not withoutmerit in an equilibrium ow. It is interesting to see that such an assumptiondoes not adversely impact the model performance for the class of separatedows for which it was developed. If the viscous diusion eects are neglected,the evolution equation for m isUmdmdx = b12_meqm_ mLmCdif3/2m(0.7 ym)_1 1/2(x)_, (3.18)where meq is the equilibrium value ((x) = 1) for the shear stress, Cdif = 0.5for (x) 1 and zero otherwise, and Lm is the dissipation length scale givenbyLm = y, ym/ 0.09/ (3.19)Lm = 0.09, ym/ > 0.09/. (3.20)Because (x) is not known a priori at each streamwise station, it is necessaryto iterate on the equation set at each station to determine its value.While the discussion here has focused on two-dimensional ows, extensionshave been proposed for three-dimensional ows (e.g., Savill et al. 1992) whichhave also yielded good ow eld predictions.The JohnsonKing model suers from the same disadvantage as the CebeciSmith and BaldwinLomax models: it relies on the grid structure becausequantities are evaluated and searched for along lines normal to walls. Forthis reason, the model has received less attention in the last decade with theincreased use of unstructured and multiple-zone structured grids, for whicheld-equation turbulence models are more ideally suited.[1] Linear and nonlinear eddy viscosity models 173.3 One-equation modelsUp to this point, both the zero- and half-equation models have focused onthe specication of an eddy viscosity (which is the underlying basis of thedevelopment of single-point closure schemes) rather than on a specication ofeither a turbulent velocity or length scale individually. At the one-equationlevel of closure, a transport equation is introduced, which in the earliest mod-els that date back to Prandtl was for the turbulent velocity scale (turbulentkinetic energy), with an algebraic prescription for the turbulent length scale.Modern-day approaches have evolved beyond this formulation to the solutionof transport equations for the turbulent Reynolds number or the turbulenteddy viscosity (velocity scale length scale). Some of these formulations willbe 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 primarilyon empiricism and on dimensional analysis arguments. Unlike the zero- andhalf-equation models discussed previously, this one-equation model is local;that is, the equation at one point does not depend on the solution at otherpoints. Therefore, it is easily usable with any type of grid: structured or un-structured, single block, or multiple blocks. The eddy viscosity relation is givenbyt = fv1, (3.21)where fv1 = 3/(3+ c3v1), and = /. The variable is determined byusing the transport equationD Dt = cb1(1 ft2)S + 1xk_( + ) xk_ + cb2_ xk_2(cw1fw cb12 ft2__ d_2(3.22)with auxiliary relationsS =_2WijWij + 2d2fv2 (3.23)fv2 = 1 1 +fv1(3.24)fw = g_ 1 +c6w3g6+c6w3_1/6=_g6+c6w31 +c6w3_1/6(3.25)g = r +cw2(r6r) (3.26)r = S2d2 (3.27)ft2 = ct3exp(ct42), (3.28)where d is the minimum distance to the nearest wall. The closure coecientsare given by: cb1 = 0.1355, cb2 = 0.622, = 2/3, = 0.41, cw1 = cb1/2+(1 +18 Gatski and Rumseycb2)/, cw2 = 0.3, cw3 = 2, ct3 = 1.2, ct4 = 0.5, and cv1 = 7.1. Although notdiscussed here, Spalart and Allmaras (1994) also developed an additional termthat is used to trip the solution from laminar to turbulent at a desired location.This feature may be important as the subsequent downstream predictions cancritically depend on the appropriate choice for the onset of turbulence.Over the years since its introduction, the SpalartAllmaras model has be-come popular among industrial users due to its ease of implementation andrelatively low cost. Even though this one-equation level of closure is based onempiricism and dimensional analysis, with characterizing ow features usuallyaccounted for on a term-by-term basis using phenomenological based models,it has tended to perform well for a wide variety of ows. As the study by Shuret al. (1995) has shown, the model can even outperform some two-equationmodels in separating and reattaching ows. Recently, Spalart and Shur (1997)have developed a rotation function, which multiplies the production term andsensitizes the SpalartAllmaras model to the eects of rotation and curvature.This function is based on the rate of change of the principal axes of the strainrate tensor.Other contemporary one-equation models using the eddy viscosity have beenproposed. One is the Gulyaev et al. (1993) model, which is an improved versionof the model developed by Sekundov (1971). It has been shown in the Russianliterature to solve a variety of incompressible and compressible ow problems(see Gulyaev et al. 1993 for selected references). Another is the model by Bald-win and Barth (1991), that has its origins in the k- two-equation formulationand was a precursor to the SpalartAllmaras model.In their original forms, both the SpalartAllmaras and BaldwinBarth mod-els are known to cause excessive diusion in regions of three-dimensional vor-tical ow. Dacles-Mariani et al. (1995) proposed the use of a modied formof the production term; rather than basing it on the magnitude of vorticity(_2WijWij) alone, the following functional form is assumed:_2WijWij + 2 min(0,_2SijSij_2WijWij). (3.29)This method was shown to help for a particular application using the BaldwinBarth model, but it is not a universally accepted x. The problem of excessivediusion in some vortical ow applications by these models, in general, stillpersists.3.4 Two-equation modelsWhile the previous closure models discussed have focused on the specicationof 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 forboth the turbulent velocity and length scales of the ow. Many variations on[1] Linear and nonlinear eddy viscosity models 19this approach exist, but the most common approaches use the transport equa-tion for the turbulent kinetic energy for the turbulent velocity scale equation.On the other hand, the length scale equation has generally been the most con-troversial element of the two-equation formulation. For the present purposes,attention will be focused in this subsection on the k- and k- formulations,where is the turbulent energy dissipation rate and is the dissipation perunit turbulent kinetic energy.The turbulent kinetic energy equation k is easily derived from the uctu-ating momentum equation for ui by forming the transport equation for thescalar product uiui/2. The resulting equation can be written asDkDt = T +T, (3.30)where the right-hand side represents the transport of k by the turbulent pro-duction T = ikUi/xk, the isotropic turbulent dissipation rate, , and thecombined eects of turbulent transport and viscous diusion T. When equa-tion(3.2) is used, the turbulent production term can also be written in termsof the eddy viscosity asT = 2t (SikSki) = 2t2, (3.31)where the velocity gradient tensor is decomposed into the sum of the symmetricstrain rate tensor Sij and the antisymmetric rotation rate tensor Wij, 2=SikSki (or 2= S2 in matrix notation), and the trace SikWki = WS =0. In such a formulation, the behavior of the individual stress componentsis governed by the Boussinesq relation given in equation(3.2), which is anisotropic eddy viscosity relationship. In general, the evolution of the individualstress components is not isotropically partitioned among the components. Forthis eect to be accounted for, higher-order closures are required such as thenonlinear eddy viscosity models to be discussed later in this chapter or theReynolds stress formulation to be discussed in Chapter [2].Nevertheless, the Boussinesq relation is not without physical foundation.For example, in (thin) simple shear ow where an equilibrium layer exists, itis assumed thatxy =_C k (3.32)with C the model constant. In the region of local equilibrium, the energyproduction and dissipation rates are in balance, so that in a thin shear ow,the kinetic energy equation reduces toT = xyUy = =_ xy_Ck_2 (3.33)where equation(3.32) has been used. This yields the familiar closure model forthe turbulent shear stress,xy = Ck2Uy . (3.34)20 Gatski and RumseyDimensional analysis considerations dictate that the eddy viscosity t be givenby the product of a turbulent velocity scale and a turbulent length scale.With the velocity scale given by k1/2, the remaining task is the developmentof the scale variable. In this chapter, two such alternatives are considered.The rst is the turbulent energy dissipation rate which implies that k3/2/is proportional to the length scale, and the second is the specic dissipationrate1, , which implies that k1/2/ is proportional to length scale. Thus, theeddy viscosity t is given by the relationt = Ck2 = Ck, = k (3.35)for the k- two-equation model, andt = k (3.36)for the original k- two-equation model. The modeling coecient C usuallyassumes a value of 0.09. (Note that this value is slightly larger than the valueassumed in the derivation of the half-equation model in Section 3.2.) For thek- formulation, the kinetic energy equation (3.30) is suitably modied byusing the substitution = Ck (see Wilcox 1998). The coecient k inequation(3.37) is k = 1 for the k- model, whereas k = 2 for the k- model.(The reader should be aware that the most recent version of the k- model,as proposed by Wilcox (1998), is dierent from the original Wilcox version.The necessary references are provided in Wilcox 1998.)Consistent with the simplied form of a two-equation formulation, a gradi-ent-transport model for the turbulent transport is usually used in the kineticenergy equation,T = xj__ + tk_ kxj_, (3.37)where the rst term on the right is the viscous contribution, and the second isthe model for the turbulent transport. The coecient k is an eective Prandtlnumber for diusion, which is taken as a constant in incompressible ows. Thevalue of k is dependent on the particular scale variable used. The resultingsimple form of the modeled turbulent kinetic energy equation is an obviousappeal of the formulation.There are several variations to the modeled form of the transport equationfor the isotropic dissipation rate . A rather general expression (Jones andLaunder 1972) from which many of the forms can be derived and which canbe integrated to the wall is given byDDt = 1 (C1T C2) + xk__ + t_ xk_, (3.38)1i.e., dissipation rate of kinetic energy (k) per unit k.[1] Linear and nonlinear eddy viscosity models 21where C1 1.45 is usually xed from calibrations with homogeneous shearows, and C2 is usually determined from the decay rate of homogeneous,isotropic turbulence ( 1.90). The closure coecient acts like an eectivePrandtl number for dissipation diusion and is specied to ensure the correctlog-law slope of 1, = 2_C (C2C1). (3.39)During the late 1990s, the two-equation shear stress transport (SST) modelof Menter (1994), has gained increasing favor among industrial users, dueprimarily to its robust formulation and improved performance for separatedows over traditional two-equation models. One of the primary features ofMenters model is that it is a blend of Wilcoxs original k- formulation nearwalls and a k- formulation in the outer region and in free shear ows. Thus,the model does not have to contend with the problems often encountered byk- models near walls (see Section 3.5), while it still retains the k- predictivecapabilities in free shear ows.Since the transport equation for the turbulent kinetic energy k has beengiven previously in equation(3.30), only the transport equation for the specicdissipation rate of turbulence kinetic energy for the SST model is given here:DDt = tT 2+ xk__ + t_ xk_+ 21 F12kxkxk. (3.40)The function F1 is the blending function that is used to switch between thek- (F1 = 1) and the k- (F1 = 0) formulations,F1 = tanh(4), (3.41)where = min_max_ kCd; 500d2_; 42kCDkd2_, (3.42)and CDk represents the cross-diusion term (the last term in the equa-tion (3.40)), limited to be positive and greater than some very small arbitrarynumber. In the derivation of the modied equation, Menter neglects a set ofdiusion terms that are demonstrated to be small (Menter 1994) and also ne-glects the molecular viscosity in the cross-diusion term. The model constantsk, , , and model constants are evaluated from(k, , , )T= F1(k1, 1, 1, 1)T+ (1 F1)(k2, 2, 2, 2)T. (3.43)The other important feature of the SST model (which represents a departurefrom the component k- and k- models) is a modication to the denitionof the eddy viscosity to account for the eect of the transport of the principal22 Gatski and Rumseyturbulent shear stress. The denition of the eddy viscosity t in the model isaltered from the forms given previously in equation(3.36):t = 2b12kmax(2b12;_2WijWijF2), (3.44)where b12 (= 0.155) is the shear stress anisotropy (see Section 3.2). The blend-ing function F2 is given byF2 = tanh(22), (3.45)where2 = max_ 2kCd; 500d2_. (3.46)Without the modied form of equation(3.44), most k- and k- linear eddyviscosity models have been generally found to yield poor results for separatedows.The constants for Menters SST model are given by: k1 = 1.17647, 1 = 2,1 = 0.075, k2 = 1, 2 = 1.16822, and 2 = 0.0828. The constant 1 is afunction of 1 and 1 whereas 2 is a function of 2 and 2 as follows:1,2 = 1,2/C2/(1,2_C). (3.47)Notice that the value of k1 has been recalibrated by Menter from its original(Wilcox k- model) value of 2 to recover the correct at-plate log-law behaviorwhen using the modied eddy viscosity equation(3.44). The other coecients1, 1, and 1 are the same as those in Wilcoxs original model. The constantsk2, 2, 2, and 2 have a direct correspondence with the k- coecients:k2 = k 2 = (3.48)2 = C(C21) 2 = C11. (3.49)Menter uses the LaunderSharma (1974) coecients: C = 0.09, C1 = 1.44,C2 = 1.92, = 0.41, k = 1, with computed by way of equation(3.39).Although Menters SST model uses two heuristic blending functions (bothof which rely on distance to the nearest wall), they have held up well undera great number of applications and still remain in many production codes asthe same functions cited in the 1994 reference.It is worth mentioning that many two-equation models, including MentersSST model, are sometimes implemented by using an approximate productiontermT = t(2WijWij), (3.50)where 2WijWij is the square of the magnitude of vorticity. This form is aconvenient approximation because the magnitude of vorticity is often readily[1] Linear and nonlinear eddy viscosity models 23available in many CFD codes. However, this approximation, while often foundto be valid for thin-shear-dominated aerodynamic ows, may lead to seriouserrors in some ow situations. On the other hand, use of the full produc-tion term can sometimes cause problems such as overproduction of turbulenceand/or negative normal stresses near stagnation points or shocks. One methodcommonly used for alleviating this problem is through the use of a limiter suchasT = min(T, 20T) (3.51)on the production term in the k-equation. See Durbin (1996) for a more thor-ough discussion and alternate strategies.Much more could be discussed about two-equation models in general andthe k- and k- models in particular because of the ease with which they canbe applied and the widespread use they have enjoyed. The interested reader isreferred to the book by Mohammadi and Pironneau (1994), which is devotedentirely to the k- turbulence model, and the book by Wilcox (1998) which isprimarily focused on the k- model. Additional references are also providedin reviews by Hanjalic (1994), Gatski (1996), and So and Speziale (1998).3.5 Near-wall integrationIn the discussion of the lower order zero- and half-equation models, it wasclearly seen that the models were constructed for direct integration to thewall through the two-layer structure for the eddy viscosity. In addition, whileless explicit about its suitability for direct integration to the wall, the one-equation formulation, and specically the SpalartAllmaras model, was alsodeveloped with the capability of being used unaltered in wall-bounded ows.However, in the two-equation formulation, equations (3.35) and (3.38) fromthe k- model, in the high-Reynolds number form, do not provide an accu-rate representation in the near-wall, viscosity aected region. In addition, thedestruction-of-dissipation rate term C22/k, is singular at the wall since isnite, and the turbulent kinetic energy k = 0. There has been an extensive listof near-wall modications over the last two decades for both the eddy viscos-ity and the transport equation for the eddy viscosity. The near-wall turbulenteddy viscosity has taken the formt = fCk2 , (3.52)where f is a damping function. The transport equation for the dissipationrate has been generalized so thatD Dt = f1C1 kT f2C2 2k + xk__ + t_ xk_+E (3.53) = +D, (3.54)24 Gatski and Rumseywhere f1 is a damping function, f2 is used to ensure that the destructionterm is nite at the wall, and D and E are additional terms added to betterrepresent the near-wall behavior. A partial list of the various forms for thesefunctions can be found in Patel et al. (1985), Rodi and Mansour (1993), andSarkar and So (1997). While the list is not all inclusive, it does provide thefunctional forms which are used today for these near-wall functions.As seen in Section 3.4, the SST model solves the near-wall problem byswitching on the k- form near the wall. Another alternative to the in-troduction of damping functions is the elliptic relaxation approach that wasrst proposed by Durbin (1991). This approach has been extended to a fullReynolds stress closure Durbin (1993a); however, in the context of the currentchapter, the description of the approach will be limited to the two-equationk- formulation.In the context of a linear eddy viscosity framework, this is a three-equationmodel for the turbulent kinetic energy k, turbulent dissipation rate , and thenormal stress component 22. (The model has been referred to as the kv2or v2f model and is based on an elliptic relaxation approach. The notationused here will be slightly dierent, in keeping with our attempt to have aunied notation throughout the chapter.) A major assumption of the modelis that the eddy viscosity t should be given byt = C22T, (3.55)where T is the applicable characteristic timescale of the ow in the proximityof the wallT = max_, 6__1/2_ (3.56)and the coecient C 0.2. Since the timescale = k/ 0 as the wall isapproached, equation(3.56) reects the physical constraint that the charac-teristic time scale T should not be less than the Kolmogorov time scale _/.With this assumption, the dissipation rate equation (3.38) is rewritten asDDt = 1T (C1T C2) + xk__ + t_ xk_ (3.57)to account for the variability of the characteristic timescale. With the exceptionof the production-of-dissipation rate coecient C1, the closure coecients areassigned values close to the standard ones (Durbin 1993b, 1995), but C1 hasassumed dierent non-constant values (Durbin 1993b, 1995) to optimize thepredictive capability of the model.The modeled equation for 22 is approximated from the Reynolds stresstransport equation and is given byD22Dt = kf2222k + xk__ + tk_ 22xk_, (3.58)[1] Linear and nonlinear eddy viscosity models 25where the function f22 is, in general, obtained fromL22f22f22 = 22 (3.59)as discussed in Chapters [2] and [4]. The characteristic length scale L is denedin a manner analogous to the timescale so thatL = CLmax__k3/2 , C_3_1/4__, (3.60)where CL 0.25 and C 80. Since the length scale CLk3/2/ 0 as thewall is approached, equation(3.60) reects the physical constraint that thecharacteristic length scale L should not be less than the Kolmogorov lengthscale (3/)1/4.This approach to the near-wall integration problem clearly diers from thestandard damping function approach used in two-equation modeling. To datethere have been several applications of the methodology to a variety of owproblems. Continued application and renement may lead to a more extensiveadaptation of this technique for near-wall model integration. The interestedreader is encouraged to review the cited references for additional details andmotivation.4 Nonlinear eddy viscosity modelsThe linear eddy viscosity models just discussed have proven to be a valu-able tool in turbulent ow-eld predictions. However, inherent in the formu-lation are several deciencies which do not exist within the broader Reynoldsstress transport equation formulation. Two of the most notable decienciesare the isotropy of the eddy viscosity and the material-frame indierence ofthe models. The isotropic eddy viscosity is a consequence of the Boussinesqapproximation which assumes a direct proportionality between the turbulentReynolds stress and the mean strain rate eld. The material frame-indierenceis a consequence of the sole dependence on the (frame-indierent) strain ratetensor. These deciencies preclude, for example, the prediction of turbulentsecondary motions in ducts (isotropic eddy viscosity) and the insensitivityof the turbulence to noninertial eects such as imposed rotations (materialframe-indierence). Remedies for these deciencies can be made on a case-by-case or ad hoc basis; however, within the framework of a linear eddy viscosityformulation such defects cannot be xed in a rigorous manner.The category of nonlinear eddy viscosity models (NLEVMs) simply extendsin a rigorous or general manner the one-term tensor representation in terms ofthe strain rate (see equation(3.2)) used in the linear EVMs to the generalizedformij = 23kij +n

n=1

nT(n)ij . (4.1)26 Gatski and RumseySince one of the advantages of the NLEVMs is to be able to capture someeects of the stress anisotropies that occur at the dierential second-momentlevel of closure, it is helpful to recast some of the equations in terms of thestress anisotropy bij given bybij = ij2k ij3 . (4.2)For example, in terms of bij, equation(4.1) can then be rewritten asbij =n

n=1nT(n)ij , (4.3)where T(n)ij are the tensor bases and n are the expansion coecients whichneed to be determined.As was shown in Section 3, the linear EVMs, through the Boussinesq approx-imation, couple to the RANS equations through a simple additive modicationto the diusion term (see equation(3.3)). In the case of nonlinear EVMs, thiscoupling can be more complex. The coupling can be either through the directuse of equation(4.1) in equation(3.1) or through a modied form of equa-tion(3.3) given byDUiDt + 1pxi= xj_( +t) Uixj_+S.T. (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 de-pendent on both the number and form of the terms chosen for the tensorrepresentation.The choice of the proper tensor basis is, of course, dependent on the func-tional dependencies associated with the Reynolds stress ij or the correspond-ing anisotropy tensor bij. As seen from the transport equation for the Reynoldsstresses (e.g. Speziale 1991), the only dependency on the mean ow is throughthe mean velocity gradient. Thus, it has been generally assumed in develop-ing turbulent closure models for the Reynolds stresses, that, in addition tothe functional dependency on the turbulent velocity and length scales, thedependence on the mean velocity gradient be included as well. The turbu-lent velocity scale is usually based on the turbulent kinetic energy k and theturbulent length scale on the variable used in the corresponding scale equa-tion (which for the purposes here will be the isotropic turbulent dissipationrate ). The continuum mechanics community has dealt with such questionson tensor representations for several decades (e.g., Spencer and Rivlin 1959).Within this context, the stress anisotropy tensor is considered here with thefunctional dependenciesbij = bij(k, , Skl, Wkl), (4.5)[1] Linear and nonlinear eddy viscosity models 27where the dependence on the mean velocity gradient has been replaced by theequivalent dependence on the strain rate tensor (see equation(2.6)) and the ro-tation rate tensor (see equation(3.7)). (In dealing with tensor representations,it is sometimes better, for notational convenience, to use matrix notation toeliminate the cumbersome task of accounting for several tensor indices. Forthis reason, both tensor and matrix notation will be used in describing themodels in this and subsequent sections.) Equation (4.5) can be rewritten inmatrix notation asb = b(k, , S, W). (4.6)In the case of fully three-dimensional mean ow, a symmetric, traceless tensorfunction b of a symmetric tensor (S) and an antisymmetric tensor (W) canbe represented as an isotropic tensor function of the following ten (integrity)tensor bases T(n)(= T(n)ij ):T(1)= S T(6)= W2S +SW2 23SW2IT(2)= SWWS T(7)= WSW2W2SWT(3)= S2 13S2I T(8)= SWS2S2WST(4)= W2 13W2I T(9)= W2S2+S2W2 23S2W2IT(5)= WS2S2W T(10)= WS2W2W2S2W.(4.7)The expansion coecients n associated with this representation can, ingeneral, be functions of the invariants of the own = n(S2, W2, k, , Ret), (4.8)where S2 = SijSji, W2 = WijWji are the strain rate and rotation rateinvariants, respectively, and Ret = k2/ is the turbulent Reynolds number. Ofcourse, a smaller number of terms could be used for the representation, such asin three-dimensions where the minimum number is ve to have an independentbasis; however, the expansion coecients will then be more complex (Jongenand Gatski 1998) and could possibly be singular. An advantage of using thefull integrity basis is that the expansion coecients will not be singular.The linear term T(1)is the strain rate S, as in the linear EVM case, andits coecient 1 is now the turbulent eddy viscosity t, which is used in equa-tion(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 becategorized based on the methodology used to determine the expansion coef-cients n. First, what is usually termed nonlinear eddy viscosity models arediscussed. These are models in which a polynomial expansion is assumed thatis a subset of equation(4.7), and the expansion coecients are determinedbased on calibrations with experimental or numerical data and physical con-straints. Second, what is usually termed algebraic stress models or algebraic28 Gatski and RumseyReynolds stress models are discussed. These are models in which a polyno-mial expansion is, once again, assumed from equation(4.7), but the expansioncoecients are derived in a mathematically consistent fashion from the fulldierential Reynolds stress equation. In both cases, an explicit tensor repre-sentation for b is obtained in terms of S and W.4.1 Quadratic and cubic tensor representationsIn this subsection, a few examples of nonlinear eddy viscosity models repre-sented by the tensor expansions in equations (4.1) or (4.3) are discussed. Themodels examined include expansions where both terms quadratic (n = 2) andcubic (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 varietyof assumptions required in identifying the expansion coecients n needed inthe algebraic representation of the Reynolds stresses.As a rst example of a nonlinear eddy viscosity model using an explicitrepresentation for the Reynolds stress anisotropy, consider the quadratic modelproposed by Speziale (1987). Speziales approach, while also motivated by theneed to include Reynolds stress anisotropy eects into a linear eddy viscositytype of formulation, diered in its development of the tensor representation forthe Reynolds stress anisotropy. Speziale assumed that the anisotropy tensorbij was of the formbij = bij_k, , Skl, Sklt_, (4.9)whereSklt = DSklDt UkxmSml UlxmSmk (4.10)is the Oldroyd convective derivative (e.g., Aris 1989, p. 185). This dependencyon the convective derivative was used to ensure that the nonlinear polyno-mial approximation would be frame-indierent, in keeping with the frame-indierent properties of the anisotropy tensor itself. Calibrations were basedon fully developed channel ow predictions using both k-l and k- two-equationmodels. The resulting tensor representation (using the strain rate and rotationrate tensor notation) wasb = 1S +2 (SWWS) +3_S2 13S2I_+DDSDt , (4.11)where D(k, ) was a closure coecient determined from the calibration. Writ-ten in this form, it can be seen that the introduction of the frame-indifferentconvective derivative simply modies two of the tensor bases given in equa-tion(4.7). Preliminary validation studies were done for a rectangular duct owand for a backstep ow to highlight the improved predictive capabilities of thenonlinear model over the corresponding linear eddy viscosity k-l and k- forms.[1] Linear and nonlinear eddy viscosity models 29Next consider the quadratic model proposed by Shih et al. (1995),b = 1S +2 (SWWS) . (4.12)The i coecients were determined by applying the rapid distortion theoryconstraint to rapidly rotating isotropic turbulence, and the realizability con-straints 0, no sum (4.13)and2 , Schwarz inequality (4.14)to the limiting cases of axisymmetric expansion and contraction. The coe-cients were optimized by further comparison with experiment and numericalsimulation of homogeneous shear ow and the inertial sublayer. Initial vali-dation studies were run on rotating homogeneous shear ow, backward-facingstep ows, and conned jets with overall improved predictions over the lineareddy viscosity models. It was also found that the standard wall function ap-proach yielded better predictions than any of the low-Reynolds number k-models. The algebraic representation given in equation(4.12) for the Reynoldsstresses was coupled with a standard k- two-equation model given in equa-tions (3.30) and (3.38). The values used for the coecients and other detailsof 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 andthat higher-order terms are needed to be able to predict ows with complexstrain elds. Craft et al. (1996) considered a model of the formb = 1S +2_S2 13S2I_+3 (SWWS) +4_W2 13W2I_ (4.15)+5_W2S +SW2 23SW2I_+6_WS2S2W_.(The form given here diers slightly from the form presented in Craft et al.,although the two representations can be shown to be equivalent.) Calibrationof the closure coecients was based on an optimization over a wide range ofows. These included plane channel ow, circular pipe ow, axially rotatingpipe ow, fully developed curved channel ow, and impinging jet ows. Thisalgebraic representation for the Reynolds stresses was then coupled with low-Reynolds number forms for the kinetic energy and dissipation rate equations(see equations (3.30) and (3.52)(3.54)). Attempts at extending the model toows far from equilibrium have been undertaken and are discussed in Craftet al. (1997). Other cubic models have been proposed, for example, by Apsleyand Leschziner (1998) and Wallin and Johansson (2000), and the interestedreader is referred to these papers for further details on their development andapplication.30 Gatski and Rumsey4.2 Algebraic stress modelsThe identifying feature of algebraic stress models (ASMs) is the techniqueused to obtain the expansion coecients n. As noted previously, these co-ecients have a direct relation to the Reynolds stress model used, or morespecically, to the pressure-strain rate correlation model. The algebraic stressmodel used here is based on the model originally developed by Pope (1975)for two-dimensional ows, and later extended by Gatski and Speziale (1993),to three-dimensional ows. The implementation has since been rened, andthe formulation to be presented is based on recent work by Jongen and Gatski(1998b).4.2.1 Implicit algebraic stress modelThe starting point for the development of ASMs is the modeled transportequation for the Reynolds stress anisotropy tensor bij (see Gatski and Speziale1993) given byDbijDt = 12k_DijDt ijkDkDt_= bij_Tk _ 23Sij_bikSkj +Sikbkj 23bmnSmnij_(4.16)+(bikWkjWikbkj) + ij2k + 12k_Tij ijk T_.where ij is the pressure-strain rate correlation, and Tij is the combinedeect of turbulent transport and viscous diusion (T = Tii/2). While it isoutside the scope of this chapter to discuss the modeling of the pressure-straincorrelation ij, it is necessary for the development of the algebraic stressmodel to specify a form for the pressure-strain rate model. For the purposeshere, the SSG model (Speziale, Sarkar, and Gatski 1991) will be used, and canbe written in the formij = _C01 +C11T_bij +C2kSij (4.17)+ C3k_bikSjk +bjkSik 23bmnSmnij_C4k (bikWkjWikbkj) ,where the closure coecients can, in general, be functions of the invariantsof the stress anisotropy. It should be noted that the functional form givenin equation (4.17) is representative of any linear pressure-strain rate modelwhich could be used as well. Substituting equation(4.17) into equation(4.16)[1] Linear and nonlinear eddy viscosity models 31and rewriting yieldsDbijDt 12k_Tij ijk T_= _bija4+a3_bikSkj +Sikbkj 23bmnSmnij_ (4.18) a2 (bikWkjWikbkj) +a1Sij_.The coecients ai are directly related to the pressure-strain correlation modelbya1 = 12_43 C2_, a2 = 12(2 C4) ,(4.19)a3 = 12(2 C3) , a4 = g,andg =__C112 + 1_T + C012 1_1=_0T +1_1, (4.20)where C01 = 3.4, C11 = 1.8, C2 = 0.36, C3 = 1.25, and C4 = 0.4. An implicit al-gebraic stress relation is obtained from the modeled transport equation for theReynolds stress anisotropy equation(4.18) when the following two assumptionsrst proposed by Rodi (1976) are made:DbijDt = 0, or DijDt = ijkDkDt, (4.21)andTij = ijk T. (4.22)Equation(4.21) is equivalent to requiring that the turbulence has reached anequilibrium state, Db/Dt = 0, and equation(4.22) invokes the assumptionthat any anisotropy of the turbulent transport and viscous diusion is pro-portional to the anisotropy of the Reynolds stresses. Both these assumptionsimpose limitations on the range of applicability of the algebraic stress model.Later in this section, some alternative assumptions will be proposed that willimprove the range of applicability of the ASM.With these assumptions, the left side of equation(4.18) vanishes, and theequation becomes algebraic:bija4+a3_bikSkj + Sikbkj 23bmnSmnij_ (4.23)a2 (bikWkjWikbkj) +a1Sij = 0,32 Gatski and Rumseyor rewritten using matrix notation 1a4b a3_bS +Sb 23bSI_+a2(bWWb) = R. (4.24)For linear pressure-strain rate models and an isotropic dissipation rate, itfollows that R = a1S. However, the generalization implied by using R isintended to indicate that the right-hand side of equation(4.24) can contain anyknown 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 besolved numerically in an iterative fashion. Unfortunately, such procedures canbe numerically sti, depending on the complexity of the ow to be solved. Itis desirable to obtain an explicit solution to this equation which still retainsits algebraic character. The rst attempt at this was by Pope (1975) whoobtained an explicit solution of equation(4.24) using a three-term basis (cf.equations (4.3) and (4.7)) for two-dimensional mean owsb = 1S +2(SWWS) +3_S2 13S2I_, (4.25)where the i are scalar coecient functions of the invariants _S2_ and _W2_.Gatski and Speziale (1993) derived a corresponding expression for three-di-mensional mean ows which required all ten terms from the integrity basisgiven in equation(4.7). A general methodology will now be presented thatallows for the systematic identication of the coecients i from an implicitalgebraic equation such as that given in equation(4.24).4.2.2 Explicit solutionWhile it is possible to implement the following methodology by using anynumber of terms in the tensor representation T(n), it is dicult to obtainclosed form analytic

closure strategies for turbulent and transitional flows

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