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Statistical Theory and Modeling for Turbulent Flows

P.A. Durbin, B.A. Petterson Reif
Format
Book
Published
Chichester ; New York : John Wiley, c2001.
Language
English
ISBN
0471497363 (hbk.), 0471497444 (pbk.)
Contents
  • I Fundamentals of Turbulence xv
  • 1.1 Turbulence Problem 1
  • 1.2 Closure Modeling 6
  • 1.3 Categories of Turbulent Flow 8
  • 2 Mathematical and Statistical Background 13
  • 2.1 Dimensional Analysis 13
  • 2.1.1 Scales of Turbulence 16
  • 2.2 Statistical Tools 17
  • 2.2.1 Averages and P.D.F.'s 17
  • 2.2.2 Correlations 23
  • 2.3 Cartesian Tensors 30
  • 2.3.1 Isotropic Tensors 32
  • 2.3.2 Tensor Functions of Tensors; Cayley-Hamilton Theorem 33
  • 2.4 Transformation to Curvilinear Coordinates 38
  • 2.4.1 Covariant and Contravariant Tensor Quantities 38
  • 2.4.2 Differentiation of Tensors 40
  • 2.4.3 Physical Components 42
  • 3 Reynolds Averaged Navier-Stokes Equations 47
  • 3.1 Reynolds Averaged Equations 49
  • 3.2 Terms of the Kinetic Energy and Reynolds Stress Budgets 51
  • 3.3 Passive Contaminant Transport 55
  • 4 Parallel and Self-Similar Shear Flows 59
  • 4.1 Plane Channel Flow 59
  • 4.1.1 Logarithmic Layer 62
  • 4.1.2 Roughness 65
  • 4.2 Boundary Layer 66
  • 4.2.1 Entrainment 70
  • 4.3 Free Shear Layers 71
  • 4.3.1 Spreading Rates 76
  • 4.3.2 Remarks on Self-Similar Boundary Layers 77
  • 4.4 Heat and Mass Transfer 78
  • 4.4.1 Parallel Flow and Boundary Layers 78
  • 4.4.2 Dispersion from Elevated Sources 82
  • 5 Vorticity and Vortical Structures 89
  • 5.1 Structure 90
  • 5.1.1 Free Shear Layers 91
  • 5.1.2 Boundary Layers 95
  • 5.1.3 Non-Random Vortices 99
  • 5.2 Vorticity and Dissipation 99
  • 5.2.1 Vortex Stretching and Relative Dispersion 102
  • 5.2.2 Mean-Squared Vorticity Equation 103
  • II Single Point Closure Modeling 107
  • 6 Models with Scalar Variables 109
  • 6.1 Boundary Layer Methods 110
  • 6.1.1 Integral Boundary Layer Methods 111
  • 6.1.2 Mixing Length Model 114
  • 6.2 [Kappa]
  • [varepsilon] Model 118
  • 6.2.1 Analytical Solutions to the [kappa]
  • [varepsilon] Model 120
  • 6.2.2 Boundary Conditions and Near-wall Modifications 124
  • 6.2.3 Weak Solution at Edges of Free-Shear Flow; Free-Stream Sensitivity 131
  • 6.3 [Kappa]
  • [omega] Model 132
  • 6.4 Stagnation Point Anomaly 136
  • 6.5 Question of Transition 138
  • 6.6 Eddy Viscosity Transport Models 140
  • 7 Models with Tensor Variables 147
  • 7.1 Second Moment Transport 147
  • 7.1.1 A Simple Illustration 148
  • 7.1.2 Closing the Reynolds Stress Transport Equation 148
  • 7.1.3 Models for the Slow Part 150
  • 7.1.4 Models for the Rapid Part 153
  • 7.2 Analytic Solutions to SMC Models 158
  • 7.2.1 Homogeneous Shear Flow 160
  • 7.2.2 Curved Shear Flow 162
  • 7.3 Non-homogeneity 166
  • 7.3.1 Turbulent Transport 167
  • 7.3.2 Near-Wall Modeling 168
  • 7.3.3 No-Slip 169
  • 7.3.4 Non-Local Wall Effects 170
  • 7.4 Reynolds Averaged Computation 181
  • 7.4.1 Numerical Issues 181
  • 7.4.2 Examples of Reynolds Averaged Computation 185
  • 8 Advanced Topics 201
  • 8.1 Further Modeling Principles 201
  • 8.1.1 Galilean Invariance and Frame Rotation 202
  • 8.1.2 Realizability 205
  • 8.2 Moving Equilibrium Solutions of SMC 207
  • 8.2.1 Criterion for Steady Mean Flow 208
  • 8.2.2 Solution in Two-Dimensional Mean Flow 209
  • 8.2.3 Bifurcations 212
  • 8.3 Passive Scalar Flux Modeling 215
  • 8.3.1 Scalar Diffusivity Models 215
  • 8.3.2 Tensor Diffusivity Models 216
  • 8.3.3 Scalar Flux Transport 218
  • 8.3.4 Scalar Variance 221
  • 8.4 Active Scalar Flux Modeling: Effects of Buoyancy 222
  • 8.4.1 Second Moment Transport Models 224
  • 8.4.2 Stratified Shear Flow 226
  • III Theory of Homogeneous Turbulence 229
  • 9 Mathematical Representations 231
  • 9.1 Fourier Transforms 232
  • 9.2 3-D Energy Spectrum of Homogeneous Turbulence 233
  • 9.2.1 Spectrum Tensor and Velocity Covariances 234
  • 9.2.2 Modeling the Energy Spectrum 236
  • 10 Navier-Stokes Equations in Spectral Space 247
  • 10.1 Convolution Integralss as Triad Interaction 247
  • 10.2 Evolution of Spectra 249
  • 10.2.1 Small [kappa]-Behavior and Energy Decay 249
  • 10.2.2 Energy Cascade 250
  • 10.2.3 Final Period of Decay 254
  • 11 Rapid Distortion Theory 257
  • 11.1 Irrotational Mean Flow 258
  • 11.1.1 Cauchy Form of the Vorticity Equation 258
  • 11.1.2 Distortion of a Fourier Mode 261
  • 11.1.3 Calculation of Covariances 262
  • 11.2 General Homogeneous Distortions 267
  • 11.2.1 Homogeneous Shear 268
  • 11.2.2 Turbulence Near a Wall 271.
Description
xiii, 285 p. : ill. ; 25 cm.
Notes
Includes bibliographical references (p. [277]-282) and index.
Technical Details
  • Access in Virgo Classic
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    g| I t| Fundamentals of Turbulence g| xv -- g| 1.1 t| Turbulence Problem g| 1 -- g| 1.2 t| Closure Modeling g| 6 -- g| 1.3 t| Categories of Turbulent Flow g| 8 -- g| 2 t| Mathematical and Statistical Background g| 13 -- g| 2.1 t| Dimensional Analysis g| 13 -- g| 2.1.1 t| Scales of Turbulence g| 16 -- g| 2.2 t| Statistical Tools g| 17 -- g| 2.2.1 t| Averages and P.D.F.'s g| 17 -- g| 2.2.2 t| Correlations g| 23 -- g| 2.3 t| Cartesian Tensors g| 30 -- g| 2.3.1 t| Isotropic Tensors g| 32 -- g| 2.3.2 t| Tensor Functions of Tensors; Cayley-Hamilton Theorem g| 33 -- g| 2.4 t| Transformation to Curvilinear Coordinates g| 38 -- g| 2.4.1 t| Covariant and Contravariant Tensor Quantities g| 38 -- g| 2.4.2 t| Differentiation of Tensors g| 40 -- g| 2.4.3 t| Physical Components g| 42 -- g| 3 t| Reynolds Averaged Navier-Stokes Equations g| 47 -- g| 3.1 t| Reynolds Averaged Equations g| 49 -- g| 3.2 t| Terms of the Kinetic Energy and Reynolds Stress Budgets g| 51 -- g| 3.3 t| Passive Contaminant Transport g| 55 -- g| 4 t| Parallel and Self-Similar Shear Flows g| 59 -- g| 4.1 t| Plane Channel Flow g| 59 -- g| 4.1.1 t| Logarithmic Layer g| 62 -- g| 4.1.2 t| Roughness g| 65 -- g| 4.2 t| Boundary Layer g| 66 -- g| 4.2.1 t| Entrainment g| 70 -- g| 4.3 t| Free Shear Layers g| 71 -- g| 4.3.1 t| Spreading Rates g| 76 -- g| 4.3.2 t| Remarks on Self-Similar Boundary Layers g| 77 -- g| 4.4 t| Heat and Mass Transfer g| 78 -- g| 4.4.1 t| Parallel Flow and Boundary Layers g| 78 -- g| 4.4.2 t| Dispersion from Elevated Sources g| 82 -- g| 5 t| Vorticity and Vortical Structures g| 89 -- g| 5.1 t| Structure g| 90 -- g| 5.1.1 t| Free Shear Layers g| 91 -- g| 5.1.2 t| Boundary Layers g| 95 -- g| 5.1.3 t| Non-Random Vortices g| 99 -- g| 5.2 t| Vorticity and Dissipation g| 99 -- g| 5.2.1 t| Vortex Stretching and Relative Dispersion g| 102 -- g| 5.2.2 t| Mean-Squared Vorticity Equation g| 103 -- g| II t| Single Point Closure Modeling g| 107 -- g| 6 t| Models with Scalar Variables g| 109 -- g| 6.1 t| Boundary Layer Methods g| 110 -- g| 6.1.1 t| Integral Boundary Layer Methods g| 111 -- g| 6.1.2 t| Mixing Length Model g| 114 -- g| 6.2 t| [Kappa]--[varepsilon] Model g| 118 -- g| 6.2.1 t| Analytical Solutions to the [kappa]--[varepsilon] Model g| 120 -- g| 6.2.2 t| Boundary Conditions and Near-wall Modifications g| 124 -- g| 6.2.3 t| Weak Solution at Edges of Free-Shear Flow; Free-Stream Sensitivity g| 131 -- g| 6.3 t| [Kappa]--[omega] Model g| 132 -- g| 6.4 t| Stagnation Point Anomaly g| 136 -- g| 6.5 t| Question of Transition g| 138 -- g| 6.6 t| Eddy Viscosity Transport Models g| 140 -- g| 7 t| Models with Tensor Variables g| 147 -- g| 7.1 t| Second Moment Transport g| 147 -- g| 7.1.1 t| A Simple Illustration g| 148 -- g| 7.1.2 t| Closing the Reynolds Stress Transport Equation g| 148 -- g| 7.1.3 t| Models for the Slow Part g| 150 -- g| 7.1.4 t| Models for the Rapid Part g| 153 -- g| 7.2 t| Analytic Solutions to SMC Models g| 158 -- g| 7.2.1 t| Homogeneous Shear Flow g| 160 -- g| 7.2.2 t| Curved Shear Flow g| 162 -- g| 7.3 t| Non-homogeneity g| 166 -- g| 7.3.1 t| Turbulent Transport g| 167 -- g| 7.3.2 t| Near-Wall Modeling g| 168 -- g| 7.3.3 t| No-Slip g| 169 -- g| 7.3.4 t| Non-Local Wall Effects g| 170 -- g| 7.4 t| Reynolds Averaged Computation g| 181 -- g| 7.4.1 t| Numerical Issues g| 181 -- g| 7.4.2 t| Examples of Reynolds Averaged Computation g| 185 -- g| 8 t| Advanced Topics g| 201 -- g| 8.1 t| Further Modeling Principles g| 201 -- g| 8.1.1 t| Galilean Invariance and Frame Rotation g| 202 -- g| 8.1.2 t| Realizability g| 205 -- g| 8.2 t| Moving Equilibrium Solutions of SMC g| 207 -- g| 8.2.1 t| Criterion for Steady Mean Flow g| 208 -- g| 8.2.2 t| Solution in Two-Dimensional Mean Flow g| 209 -- g| 8.2.3 t| Bifurcations g| 212 -- g| 8.3 t| Passive Scalar Flux Modeling g| 215 -- g| 8.3.1 t| Scalar Diffusivity Models g| 215 -- g| 8.3.2 t| Tensor Diffusivity Models g| 216 -- g| 8.3.3 t| Scalar Flux Transport g| 218 -- g| 8.3.4 t| Scalar Variance g| 221 -- g| 8.4 t| Active Scalar Flux Modeling: Effects of Buoyancy g| 222 -- g| 8.4.1 t| Second Moment Transport Models g| 224 -- g| 8.4.2 t| Stratified Shear Flow g| 226 -- g| III t| Theory of Homogeneous Turbulence g| 229 -- g| 9 t| Mathematical Representations g| 231 -- g| 9.1 t| Fourier Transforms g| 232 -- g| 9.2 t| 3-D Energy Spectrum of Homogeneous Turbulence g| 233 -- g| 9.2.1 t| Spectrum Tensor and Velocity Covariances g| 234 -- g| 9.2.2 t| Modeling the Energy Spectrum g| 236 -- g| 10 t| Navier-Stokes Equations in Spectral Space g| 247 -- g| 10.1 t| Convolution Integralss as Triad Interaction g| 247 -- g| 10.2 t| Evolution of Spectra g| 249 -- g| 10.2.1 t| Small [kappa]-Behavior and Energy Decay g| 249 -- g| 10.2.2 t| Energy Cascade g| 250 -- g| 10.2.3 t| Final Period of Decay g| 254 -- g| 11 t| Rapid Distortion Theory g| 257 -- g| 11.1 t| Irrotational Mean Flow g| 258 -- g| 11.1.1 t| Cauchy Form of the Vorticity Equation g| 258 -- g| 11.1.2 t| Distortion of a Fourier Mode g| 261 -- g| 11.1.3 t| Calculation of Covariances g| 262 -- g| 11.2 t| General Homogeneous Distortions g| 267 -- g| 11.2.1 t| Homogeneous Shear g| 268 -- g| 11.2.2 t| Turbulence Near a Wall g| 271.
    596
      
      
    a| 5
    650
      
    0
    a| Turbulence x| Mathematical models.
    700
    1
      
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