Turbulence Modeling for CFD

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Turbulence Modeling for CFD by David C. Wilcox Third Edition
Dedicated to my wife BARbARA my Children KINLEY and boB and my dad about the author Dr. David C. Wilcox, was born in Wilmington, Delaware. He did his under- graduate studies from 1963 to 1966 at the massachusetts Institute of Technology, graduating with a Bachelor of Science degree in Aeronautics and Astronautics From 1966 to 1967. he was employed by the mcDonnell douglas aircraft Di vision in Long Beach, California, and began his professional career under the guidance of A M.O. Smith. His experience with McDonnell Douglas focused on subsonic and transonic flow calculations. From 1967 to 1970. he attended the California Institute of Technology, graduating with a \h.D. in Aeronautics In 1970 he ioined TRW Systems, InC. in Redondo beach, California, where he performed studies of both high-and low-speed fuid-mechanical and heat-transfer problems, such as turbulent hypersonic flow and thermal radiation from a flame From 1972 to 1973, he was a staff scientist for Applied Theory, Inc, in Los Angeles, California. He participated in many research efforts involving numer ical computation and analysis of fluid flows such as separated turbulent flow, transitional flow and hypersonic plume-body interaction In 1973. he founded dcw industries. inc. a La canada, California firm en- gaged in engineering research and book publishing for which he is currently Pres ident. He has taught several fluid mechanics and applied mathematics courses at the University of Southen California and at the university of California, Los Angeles Dr wilcox has numerous publications on turbulence modeling, computational fluid dynamics, boundary-layer separation, boundary-layer transition, thermal ra diation, and rapidly rotating fluids. His book publications include texts entitled Elements of fluid Mechanics, Basic Fluid Mechanics and Perturbation Methods in the Computer Age He is an Associate Fellow of the American Institute of aeronautics and Astronautics (AIAA)and has served as an associate Editor for the AIAA Journal Contents Notation Preface VI 1 Introduction 1.1 Definition of an Ideal turbulence model 2 How Complex Must a Turbulence Model Be? 1.3 Comments on the Physies of Turbulence i.3. 1 Importance of Turbulence in Practical Situations 22335 3.2 General Properties of Turbulence 3. 3 The Smallest Scales of Turbulence 1.3.4 Spectral Representation and the Kolmogorov-5/3 Law 1.3.5 The law of the wall 15 1.4 A Brief History of Turbulence Modeling 1.3. 6 Power Laws 20 23 Problems .28 2 The closure Problem 33 2.1 Reynolds Averaging 34 2.2 Correlations 39 2.3 Reynolds-Averaged Equations 39 2.4 The Reynolds-Stress Equation 41 2.5 The Scales of Turbulence 43 2.5.1 Turbulence Intensity 44 2.5.2 Two-Point correlation tensors and related scales 45 Problems 50 3 Algebraic models 53 3.1 Molecular Transport of momentum 54 3.2 The Mixing- Length Hypothesis 57 3.3 Application to Free Shear Flows 60 CONTENTS 3.3. 1 The Far Wake 62 3.3.2 The Mixing Layer 67 3.3.3 The jet 70 3. 4 Modern Variants of the Mixing-Length Model 74 3.41 Cebeci-Smith model 79 3.4.2 Baldwin-Lomax Model 81 3.5 Application to Wall-Bounded Flows 84 3.5.1 Channel and Pipe Flow 84 3.5.2 Boundary layers 89 3.6 Separated Flows 4 3.7 The 1/2-equation model 96 3.8 Range of Applicability 100 Problems .102 4 One-Equation and Two-Equation Models 107 4.1 The Turbulence Energy Equation 108 4.2 One-Equation Models 4.3 Two-Equation Models ..122 4.3.1 The k-w model ,,124 4.3.2 The k-E Model 128 4.3.3 Other Two-Equation models 131 4.4 Closure Coefficients 133 4.5 Application to Free Shear Flows 136 4.5. 1 Developing the Similarity Solution .,137 4.5.2 Numerical Solution 143 4.5.3 Sensitivity to Finite Freestream Boundary Conditions . 147 4.5.4 Cross diffusion .151 4.5. 5 The Round-Jet/Plane-Jet Anomaly 154 4.6 Perturbation Analysis of the Boundary layer 156 4.6.1 The Log layer 156 4.6.2 The Defect Layer 161 4.6.3 The Viscous sublayer l75 4.7 Surface Boandary Conditions 180 4.7.1 Wall Functions 81 4.7.2 Surface roughness 182 4.7.3 Surface Mass Injection l86 4.8 Application to Wall-Bounded Flows 187 4.8.1 Channel and Pipe Flow 87 4.8.2 Boundary layers l89 4.9 Low-Reynolds-Number Effects 192 4.9.1 Asymptotic Consistency 193 49.2 Transition .200 CONTENTS 4.9.3 Channel and Pipe Flow 210 4.9.4 Boundary-Layer Applications 212 4.10 Application to Separated Flows 218 4.11 Range of Applicability 227 Problems 230 5 Effects of Compressibility 239 5. 1 Physical Considerations 239 5.2 Favre Averaging 241 5.3 Favre-Averaged Equations 243 5.4 Compressible-Flow Closure Approximations 249 5.4.1 Reynolds-Stress Tensor 250 5.4.2 Turbulent heat -Flux Vector ..250 5.4.3 Molecular Diffusion and Turbulent Transport 250 5.4.4 Dilatation Dissipation 251 5.4.5 Pressure Diffusion and pressure Dilatation 253 5.4.6 Pressure Work 254 5.4.7 k-w Model Equations for Compressible Flows 255 5.5 Mixing-Layer Compressibility Corrections 257 5.5.1 The Sarkar/Zeman/Wilcox Compressibility Corrections 258 5.5.2 Applications 259 5.6 Compressible Law of the Wall ..262 5.6.1 Derivation 262 5. 6.2 The effect of cross Diffusion 268 5.7 Compressible Boundary layers 269 5.8 Shock-Induced Boundary-Layer Separation 275 5.8.1 The Earliest Applications 275 5.8.2 The Use of Wall Functions for Shock-Separated Flows. 277 5.8.3 The Next Two Decades of"Progress 279 5.8.4 Effect of the Stress Limiter on Shock-Separated Flows 280 5.8.5 Transonic Flow Over an Axisymmetric Bump 283 5.8.6 Mach 2 Flow Past a Backward-Facing Step 284 5.8.7 Mach 3 Compression Corners and reflecting Shocks 285 5.8.8 Mach 11 Reflecting-Shock 289 5.8.9 The Reattachment Point Heat- Transfer Anomaly 290 5. 8.10 Three-Dimensional Applications .292 5.9 Summary 295 Problems 297 V11 CONTENTS 6 Beyond the Boussinesq Approximation 303 6.1 Boussinesq-Approximation Deficiencies 303 6.2 Nonlinear constitutive relations 308 6.2. 1 The Earliest Formulations 308 6.2.2 Algebraic Stress Models 311 6.2.3 Relation to the Stress limiter 317 6.2. 4 Lag model ,,320 6.3 Stress-Transport Models 322 6.3. 1 Closure Approximations .323 6.3.2 Launder -Reece-Rodi model 330 63. 3 Wilcox Stress-w Model 332 6.4 Application to Homogeneous Turbulent Flows 334 6.5 Application to Free Shear Flows 340 6.6 Application to Wall-Bounded Flows 343 6.6. 1 Surface Boundary Conditions/Viscous Modifications... 343 6.6.2 Channel and Pipe Flow 348 6.6.3 Rotating Channel Flow .351 6.6.4 Boundary layers ..352 6.7 Application to Separated Flows ,,361 6.7.1 Incompressible Backward-Facing Step 361 6.7.2 Transonic Flow Over an Axisymmetric Bump .,365 6.7.3 Mach 3 Compression Corners and Reflecting Shocks 366 6.7.4 Hypersonic Shock -Separated Flows .370 6.8 Range of Applicability 371 Problems 373 7 Numerical Considerations 381 7. 1 Multiple Time Scales and Stiffness 381 7.2 Numerical accuracy Near Boundaries 383 7.2.1 Solid Surfaces 383 7.2.2 Turbulent/Nonturbulent Interfaces 387 7.2.3 Sensitivity to Freestream Boundary Conditions 395 7. 2. 4 Viscous-Interface layer 397 7.3 Parabolic Marching Methods 399 7.4 Elementary Time-Marching Methods 403 7.5 Block-Implicit Methods 409 7.6 Solution Convergence and Grid Sensitivity 414 7.6.1 Iteration Convergence and Grid Convergence 414 7.6.2 Richardson Extrapolation 416 7.6.3 Grid Convergence Index 417 7.6. Near-Wall Grid-Point Spacing 418 Problems 420 CONTENTS IX g New orizons 427 8. 1 Background Information 428 8.2 Direct Numerical simulation .431 8.3 Large Eddy simulation 436 8.3.1 Filtering .437 8.3.2 Subgrid-Scale(SGS) Modeling 440 8.3.3"Off the Wall Boundary Conditions 442 8.3.4 Applications 444 8.4 Detached Eddy simulation 446 8.4.1 DES-Blending Functions 446 8.4.2 Applications 451 8.5 Chaos ..,..452 8.6 Further Reading 455 Problems 456 a Cartesian Tensor Analysis 459 Problems 464 b Rudiments of perturbation methods 465 Problems 475 C Companion Software 477 Bibliography 479 Index 509 CONTENTS

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