Turbulent Fluid Flow

Preface xiii

About the Companion Website xv

1 Introduction 1

1.1 What is Turbulent Flow? 1

1.2 Examples of Turbulent Flow 2

1.3 The Goals of a Turbulent Flow Study 7

1.4 Overview of the Methodologies Available to Predict Turbulence 9

1.4.1 Direct Numerical Simulation 9

1.4.2 Experimental Methods 10

1.4.3 Turbulence Modeling 11

1.5 The Plan for this Book 12

References 13

2 Describing Turbulence 15

2.1 Navier-Stokes Equation and Reynolds Number 15

2.2 What Needs to be Measured and Computed 16

2.2.1 Averaging 17

2.2.2 One-Point Statistics 19

2.2.3 Two-Point Correlations 21

2.2.4 Spatial Spectra 25

2.2.5 Time Spectra 28

Reference 29

3 Overview of Turbulent Flow Physics and Equations 31

3.1 The Reynolds Averaged Navier-Stokes Equation 31

3.2 Turbulent Kinetic Energy Equation 33

3.3 ¿¿¿¿ Equation 37

3.4 Reynolds Stress Equation 39

3.5 Vorticity Equation 40

3.5.1 Vortex Stretching and Reorientation 42

3.6 Enstrophy Equation 43

References 44

4 Turbulence at Small Scales 47

4.1 Spectral Representation of ¿¿¿¿ 48

4.2 Consequences of Isotropy 50

4.3 The Smallest Scales 54

4.4 Inertial Subrange 58

4.4.1 Relations Between 1D and 3D Spectra 58

4.4.2 1D Spatial and Time Series Spectra 61

4.5 Structure Functions 65

4.6 Chapter Summary 67

References 67

5 Energy Decay in Isotropic Turbulence 71

5.1 Energy Decay 71

5.1.1 Turbulent Reynolds Number 75

5.2 Modes of Isotropic Decay 76

5.3 Self-Similarity 77

5.3.1 Fixed Point Analysis 79

5.3.2 Final Period of Isotropic Decay 80

5.3.3 High Reynolds Number Equilibrium 84

5.4 Implications for Turbulence Modeling 87

5.5 Equation for Two-Point Correlations 88

5.6 Self-Preservation and the Kármán-Howarth Equation 92

5.7 Energy Spectrum Equation 94

5.8 Energy Spectrum Equation via Fourier Analysis of the Velocity Field 96

5.8.1 Fourier Analysis on a Cubic Region 97

5.8.2 Limit of Infinite Space 99

5.8.3 Applications to TurbulenceTheory 101

5.9 Chapter Summary 102

References 103

6 Turbulent Transport and its Modeling 107

6.1 Molecular Momentum Transport 107

6.2 Modeling Turbulent Transport by Analogy to Molecular Transport 110

6.3 Lagrangian Analysis of Turbulent Transport 112

6.4 Transport Producing Motions 115

6.5 Gradient Transport 119

6.6 Homogeneous Shear Flow 122

6.7 Vorticity Transport 128

6.7.1 Vorticity Transport in Channel Flow 130

6.8 Chapter Summary 132

References 133

7 Channel and Pipe Flows 135

7.1 Channel Flow 135

7.1.1 Reynolds Stress and Force Balance 138

7.1.2 Mean Flow Similarity 141

7.1.3 Viscous Sublayer 142

7.1.4 Intermediate Layer 143

7.1.5 Velocity Moments 145

7.1.6 Turbulent Kinetic Energy and Dissipation Rate Budgets 148

7.1.7 Reynolds Stress Budget 150

7.1.8 Enstrophy and its Budget 154

7.2 Pipe Flow 156

7.2.1 Mean Velocity 158

7.2.2 Power Law 160

7.2.3 Streamwise Normal Reynolds Stress 162

References 163

8 Boundary Layers 167

8.1 General Properties 169

8.2 Boundary Layer Growth 171

8.3 Log-Law Behavior of the Velocity Mean and Variance 174

8.4 Outer Layer 175

8.5 The Structure of Bounded Turbulent Flows 177

8.5.1 Development of Vortical Structure in Transition 177

8.5.2 Structure in Transition and in Turbulence 180

8.5.3 Vortical Structures 181

8.5.4 Origin of Structures 186

8.5.5 Fully Turbulent Region 192

8.6 Near-Wall Pressure Field 197

8.7 Chapter Summary 197

References 199

9 Turbulence Modeling 203

9.1 Types of RANS Models 204

9.2 Eddy Viscosity Models 207

9.2.1 Mixing Length Theory and its Generalizations 208

9.2.2 K-¿¿¿¿ Closure 211

9.2.2.1 K Equation 212

9.2.2.2 The ¿¿¿¿ Equation 212

9.2.2.3 Calibration of the K-¿¿¿¿ Closure 214

9.2.2.4 Near-Wall K-¿¿¿¿ Models 215

9.2.3 K-¿¿¿¿ Models 218

9.2.4 Menter Shear Stress Transport Closure 219

9.2.5 Spalart-Allmaras Model 221

9.3 Tools forModel Development 222

9.3.1 Invariance Properties of the Reynolds Stress Tensor 222

9.3.2 Realizability 226

9.3.3 Rapid Distortion Theory 226

9.4 Non-Linear Eddy Viscosity Models 227

9.5 Reynolds Stress Equation Models 229

9.5.1 Modeling of the Pressure-Strain Correlation 230

9.5.2 LRR Model 232

9.5.3 SSG Model 234

9.5.4 Transport Correlation 238

9.5.5 Complete Second Moment Closure 239

9.5.6 Near-Wall Reynolds Stress Equation Models 240

9.6 Algebraic Reynolds Stress Models 242

9.7 Urans 243

9.8 Chapter Summary 244

References 245

10 Large Eddy Simulations 251

10.1 Mathematical Basis of LES 252

10.2 Numerical Considerations 257

10.3 Subgrid-Scale Models 258

10.3.1 Smagorinsky Model 261

10.3.2 Wale Model 263

10.3.3 Alternative Eddy Viscosity Subgrid-Scale Models 265

10.3.4 Dynamic Models 266

10.4 Hybrid LES/RANS Models 270

10.4.1 Detached Eddy Simulation 271

10.4.2 A Hybrid LES/RANS Form of the Menter SST Model 272

10.4.3 Flow Simulation Methodology 273

10.4.4 Example of a Zonal LES/RANS Formulation 274

10.4.5 Partially Averaged Navier-Stokes 276

10.4.6 Scale-Adaptive Simulation 277

10.5 Chapter Summary 278

References 278

11 Properties of Turbulent Free Shear Flows 283

11.1 Thin Flow Approximation 283

11.2 Turbulent Wake 285

11.2.1 Self-Preserving FarWake 286

11.2.2 Mean Velocity 290

11.3 Turbulent Jet 292

11.3.1 Self-Preserving Jet 292

11.3.2 Mean Velocity 293

11.3.3 Reynolds Stresses 295

11.4 Turbulent Mixing Layer 298

11.4.1 Structure of Mixing Layers 298

11.4.2 Self-Preserving Mixing Layer 300

11.4.3 Mean Velocity 302

11.4.4 Reynolds Stresses 303

11.5 Chapter Summary 304

References 306

12 Calculation of Ground Vehicle Flows 309

12.1 Ahmed Body 309

12.2 Realistic Automotive Shapes 317

12.3 Truck Flows 324

12.4 Chapter Summary 326

References 327

Author Index 329

Subject Index 335