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Turbulent Fluid Flow

Peter S. Bernard,Kenneth Kiger|2019.03.01

1 Introduction 1

1.1 What is Turbulent Flow? 1

1.2 Examples of Turbulent Flow 3

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 12

1.5 The Plan For This Book 13

2 Describing Turbulence 15

2.1 Navier-Stokes Equation and Reynolds Number 15

2.2 What Needs to be Measured and Computed 17

2.2.1 Averaging 17

2.2.2 One-Point Statistics 19

2.2.3 Two-Point Correlations 22

2.2.4 Spatial Spectra 26

2.2.5 Time Spectra 30

Problems 31

3 Overview of Turbulent Flow Physics and Equations 33

3.1 The Reynolds Averaged Navier-Stokes (RANS) Equation 33

3.2 Turbulent Kinetic Energy Equation 35

3.3  Equation 40

3.4 Reynolds Stress Equation 42

3.5 Vorticity Equation 43

3.5.1 Vortex Stretching and Reorientation 45

3.6 Enstrophy Equation 47

Problems 48

4 Turbulence at Small Scales 51

4.1 Spectral Representation of ǫ 52

4.2 Consequences of Isotropy 54

4.3 The Smallest Scales 59

4.4 Inertial Subrange 63

4.4.1 Relations Between 1D and 3D Spectra 63

4.4.2 1D Spatial and Time Series Spectra 67

4.5 Structure Functions 70

4.6 Chapter Summary 73

Problems 75

5 Energy Decay in Isotropic Turbulence 77

5.1 Energy Decay 77

5.1.1 Turbulent Reynolds Number 81

5.2 Modes of Isotropic Decay 83

5.3 Self-Similarity 84

5.3.1 Fixed Point Analysis 86

5.3.2 Final Period of Isotropic Decay. 87

5.3.3 High Reynolds Number Equilibrium 91

5.4 Implications for Turbulence Modeling 94

5.5 Equation for Two-Point Correlations 97

5.6 Self-preservation and the Kármán-Howarth Equation 100

5.7 Energy Spectrum Equation 103

5.8 Energy Spectrum Equation via Fourier Analysis of the Velocity Field 105

5.8.1 Fourier Analysis on a Cubic Region 105

5.8.2 Limit of Infinite Space 108

5.8.3 Applications to Turbulence Theory 110

5.9 Chapter Summary 111

Problems 112

6 Turbulent Transport and Its Modeling 115

6.1 Molecular Momentum Transport 115

6.2 Modeling Turbulent Transport by Analogy to Molecular Transport 118

6.3 Lagrangian Analysis of Turbulent Transport 120

6.4 Transport Producing Motions 124

6.6 Homogeneous Shear Flow 132

6.7 Vorticity Transport 138

6.7.1 Vorticity Transport in Channel Flow. 140

6.8 Chapter Summary. 143

Problems 144

7 Channel and Pipe Flows 147

7.1 Channel Flow 148

7.1.1 Reynolds Stress and Force Balance 150

7.1.2 Mean Flow Similarity 153

7.1.3 Viscous Sublayer 154

7.1.4 Intermediate Layer 156

7.1.5 Velocity Moments 157

7.1.6 Turbulent Kinetic Energy and Dissipation Rate Budgets 161

7.1.7 Reynolds Stress Budget 163

7.1.8 Enstrophy and Its Budget. 167

7.2 Pipe Flow 170

7.2.1 Mean Velocity 172

7.2.2 Power Law 174

7.2.3 Normal Streamwise Reynolds Stress 177

Problems 180

8 Boundary Layers 181

8.1 General Properties 183

8.2 Boundary Layer Growth 185

8.3 Log-Law Behavior of the Velocity Mean and Variance 188

8.4 Outer layer 190

8.5 The Structure of Bounded Turbulent Flows 192

8.5.1 Development of Vortical Structure in Transition 192

8.5.2 Structure in Transition and in Turbulence 195

8.5.3 Vortical Structures 197

8.5.4 Origin of Structures 203

8.5.5 Fully Turbulent Region 208

8.6 Near-Wall Pressure Field. 212

8.7 Chapter Summary 212

Problems 217

9 Turbulence Modeling 219

9.1 Types of RANS Models 221

9.2 Eddy Viscosity Models 224

9.2.1 Mixing Length Theory and Its Generalizations 224

9.2.2 K −  Closure 227

9.2.3 K − ω Models 236

9.2.4 Menter Shear Stress Transport Closure 237

9.2.5 Spalart-Allmaras Model 239

9.3 Tools for Model Development 241

9.3.1 Invariance Properties of the Reynolds Stress Tensor 241

9.3.2 Realizability 244

9.3.3 Rapid Distortion Theory 245

9.4 Non-Linear Eddy Viscosity Models 246

9.5 Reynolds Stress Equation Models 249

9.5.1 Modeling of the Pressure-Strain Correlation 249

9.5.2 LRR Model 251

9.5.3 SSG Model 254

9.5.4 Transport Correlation 259

9.5.5 Complete Second Moment Closure 260

9.5.6 Near-Wall Reynolds Stress Equation Models. 261

9.6 Algebraic Reynolds Stress Models 263

9.7 URANS 264

9.8 Chapter Summary 265

Problems 268

10 Large Eddy Simulations 271

10.1 Mathematical Basis of LES 272

10.2 Numerical Considerations 278

10.3 Subgrid-Scale Models 279

10.3.1 SmagorinskyModel 282

10.3.2 WALE Model 284

10.3.3 Alternative Eddy Viscosity Subgrid-Scale Models 286

10.3.4 Dynamic Models 287

10.4 Hybrid LES/RANS Models 292

10.4.1 Detached Eddy Simulation 293

10.4.2 A Hybrid LES/RANS Form of the Menter SST Model 295

10.4.3 Flow Simulation Methodology 296

10.4.4 Example of A Zonal LES/RANS Formulation 297

10.4.5 Partial-Averaging Navier-Stokes 298

10.5 Chapter Summary 301

Problems 304

11 Properties of Turbulent Free Shear Flows 305

11.1 Thin Flow Approximation 305

11.2 Turbulent Wake 308

11.2.1 Self-Preserving Far Wake 309

11.2.2 Mean Velocity 313

11.3 Turbulent Jet 314

11.3.1 Self-Preserving Jet 316

11.3.2 Mean Velocity 318

11.3.3 Reynolds Stresses 319

11.4 Turbulent Mixing Layer 322

11.4.1 Structure of Mixing Layers 323

11.4.2 Self-Preserving Mixing Layer 325

11.4.3 Mean Velocity 328

11.4.4 Reynolds Stresses 330

11.5 Chapter Summary 332

Problems 332

12 Calculation of Ground Vehicle Flows 335

12.1 Ahmed Body 336

12.2 Realistic Automotive Shapes 344

12.3 Truck Flows 350

12.4 Chapter Summary 351

Author Index 354

Subject Index 360