Statistical Theory and Modeling for Turbulent Flows

Cover......Page 1
Statistical Theory and Modeling for Turbulent Flows......Page 5
ISBN: 978-0-470-97206-9......Page 6
Dedication......Page 7
Contents......Page 9
Preface to first edition......Page 13
Motivation......Page 14
Acknowledgements......Page 15
Part I FUNDAMENTALS OF TURBULENCE......Page 17
1 Introduction......Page 19
1.1 The turbulence problem......Page 20
1.2 Closure modeling......Page 25
1.3 Categories of turbulent flow......Page 26
Exercises......Page 30
2.1 Dimensional analysis......Page 31
2.1.1 Scales of turbulence......Page 34
2.2.1 Averages and probability density functions......Page 35
2.2.2 Correlations......Page 41
2.3 Cartesian tensors......Page 50
2.3.1 Isotropic tensors......Page 52
2.3.2 Tensor functions of tensors; Cayley–Hamilton theorem......Page 53
Exercises......Page 58
3 Reynolds averaged Navier–Stokes equations......Page 61
3.1 Background to the equations......Page 62
3.2 Reynolds averaged equations......Page 64
3.3 Terms of kinetic energy and Reynolds stress budgets......Page 65
3.4 Passive contaminant transport......Page 70
Exercises......Page 72
4 Parallel and self-similar shear flows......Page 73
4.1 Plane channel flow......Page 74
4.1.1 Logarithmic layer......Page 77
4.1.2 Roughness......Page 79
4.2 Boundary layer......Page 81
4.2.1 Entrainment......Page 85
4.3 Free-shear layers......Page 86
4.3.2 Remarks on self-similar boundary layers......Page 92
4.4 Heat and mass transfer......Page 93
4.4.1 Parallel flow and boundary layers......Page 94
4.4.2 Dispersion from elevated sources......Page 98
Exercises......Page 102
5 Vorticity and vortical structures......Page 107
5.1.1 Free-shear layers......Page 109
5.1.2 Boundary layers......Page 113
5.2 Vorticity and dissipation......Page 118
5.2.1 Vortex stretching and relative dispersion......Page 120
5.2.2 Mean-squared vorticity equation......Page 122
Exercises......Page 124
Part II SINGLE-POINT CLOSURE MODELING......Page 125
6 Models with scalar variables......Page 127
6.1 Boundary-layer methods......Page 128
6.1.1 Integral boundary-layer methods......Page 129
6.1.2 Mixing length model......Page 131
6.2 The k–ε model......Page 137
6.2.1 Analytical solutions to the k–ε model......Page 139
6.2.2 Boundary conditions and near-wall modifications......Page 144
6.2.3 Weak solution at edges of free-shear flow; free-stream sensitivity......Page 151
6.3 The k–ω model......Page 152
6.4 Stagnation-point anomaly......Page 155
6.5 The question of transition......Page 157
6.5.1 Reliance on the turbulence model......Page 160
6.5.2 Intermittency equation......Page 161
6.5.3 Laminar fluctuations......Page 163
6.6 Eddy viscosity transport models......Page 164
Exercises......Page 168
7.1 Second-moment transport......Page 171
7.1.1 A simple illustration......Page 172
7.1.2 Closing the Reynolds stress transport equation......Page 173
7.1.3 Models for the slow part......Page 175
7.1.4 Models for the rapid part......Page 178
7.2.1 Homogeneous shear flow......Page 185
7.2.2 Curved shear flow......Page 188
7.2.3 Algebraic stress approximation and nonlinear eddy viscosity......Page 192
7.3 Non-homogeneity......Page 195
7.3.1 Turbulent transport......Page 196
7.3.2 Near-wall modeling......Page 197
7.3.3 No-slip condition......Page 198
7.3.4 Nonlocal wall effects......Page 200
7.4 Reynolds averaged computation......Page 210
7.4.1 Numerical issues......Page 211
7.4.2 Examples of Reynolds averaged computation......Page 214
Exercises......Page 229
8.1 Further modeling principles......Page 233
8.1.1 Galilean invariance and frame rotation......Page 235
8.1.2 Realizability......Page 237
8.2 Second-moment closure and Langevin equations......Page 240
8.3 Moving equilibrium solutions of SMC......Page 242
8.3.1 Criterion for steady mean flow......Page 243
8.3.2 Solution in two-dimensional mean flow......Page 244
8.3.3 Bifurcations......Page 247
8.4.1 Scalar diffusivity models......Page 251
8.4.2 Tensor diffusivity models......Page 252
8.4.3 Scalar flux transport......Page 254
8.4.4 Scalar variance......Page 257
8.5 Active scalar flux modeling: effects of buoyancy......Page 258
8.5.1 Second-moment transport models......Page 261
8.5.2 Stratified shear flow......Page 262
Exercises......Page 263
Part III THEORY OF HOMOGENEOUS TURBULENCE......Page 265
9 Mathematical representations......Page 267
9.1 Fourier transforms......Page 268
9.2 Three-dimensional energy spectrum of homogeneous turbulence......Page 270
9.2.1 Spectrum tensor and velocity covariances......Page 271
9.2.2 Modeling the energy spectrum......Page 273
Exercises......Page 282
10.1 Convolution integrals as triad interaction......Page 285
10.2.1 Small-k behavior and energy decay......Page 287
10.2.2 Energy cascade......Page 289
10.2.3 Final period of decay......Page 292
Exercises......Page 293
11 Rapid distortion theory......Page 298
11.1.1 Cauchy form of vorticity equation......Page 299
11.1.2 Distortion of a Fourier mode......Page 302
11.1.3 Calculation of covariances......Page 304
11.2 General homogeneous distortions......Page 308
11.2.1 Homogeneous shear......Page 310
11.2.2 Turbulence near a wall......Page 313
Exercises......Page 317
Part IV TURBULENCE SIMULATION......Page 320
12 Eddy-resolving simulation......Page 322
12.1.1 Grid requirements......Page 323
12.1.2 Numerical dissipation......Page 325
12.1.3 Energy-conserving schemes......Page 327
12.2 Illustrations......Page 330
12.3 Pseudo-spectral method......Page 335
Exercises......Page 339
13.1 Large eddy simulation......Page 342
13.1.1 Filtering......Page 343
13.1.2 Subgrid models......Page 347
13.2 Detached eddy simulation......Page 356
Exercises......Page 360
References......Page 362
Index......Page 370
TrUe LiAr......Page 374