Preface ....................................................... vii
Nomenclature ................................................. xvii
1. Introduction ................................................. 1
1.1. The basic equations ..................................... 1
1.1.1. The molecular dynamics ........................... 3
1.1.2. The basic equations .............................. 3
1.2. Turbulence models ....................................... 4
1.2.1. Stochastic models for large-scale turbulence ..... 5
1.2.2. Stochastic models for small-scale turbulence ..... 5
1.2.3. The unification of turbulence models ............. 6
Appendix 1A: Filter operations ............................... 6
1A.1. Spatial averages .................................. 6
1A.2. Ensemble averages ................................. 7
1A.3. The ergodic theorem ............................... 7
2. Stochastic variables ......................................... 9
2.1. PDFs of one variable .................................... 9
2.1.1. The need for the usage of probabilistic
concepts ......................................... 9
2.1.2. The definition of PDFs .......................... 10
2.1.3. General properties of PDFs ...................... 12
2.2. The characterization of PDFs by moments ................ 12
2.2.1. The calculation of moments by PDFs .............. 12
2.2.2. The calculation of PDFs by moments .............. 13
2.2.3. An example: the Gaussian PDF and its moments .... 15
2.3. PDFs of several variables .............................. 16
2.3.1. PDFs ............................................ 16
2.3.2. Correlations .................................... 17
2.3.3. Conditional PDFs ................................ 18
2.4. Statistically most-likely PDFs ......................... 19
2.4.1. The measurement of uncertainty .................. 20
2.4.2. Statistically most-likely PDFs .................. 21
2.5. Examples for statistically most-likely PDFs ............ 22
2.5.1. Second-order SML PDF: unbounded variables ....... 22
2.5.2. Fourth-order SML PDF: unbounded variables ....... 24
2.5.3. Second-order SML PDF: bounded variables ......... 25
2.6. Examples for other PDFs ................................ 26
2.6.1. Gamma and exponential PDFs ...................... 27
2.6.2. The beta PDF .................................... 28
Appendix 2A: Theta and delta functions ...................... 29
2A.1. A theta function for one variable ................ 29
2A.2. A delta function for one variable ................ 29
2A.3. The properties of delta functions ................ 30
2A.4. The extension to the case of several variables ... 31
3. Stochastic processes ........................................ 33
3.1. PDF transport equations ................................ 33
3.1.1. The Kramers-Moyal equation ...................... 33
3.1.2. Markov processes ................................ 34
3.1.3. Implications for PDF transport equations ........ 35
3.2. The Fokker-Planck equation ............................. 36
3.2.1. The Fokker-Planck equation ...................... 36
3.2.2. Transport equations for moments ................. 37
3.2.3. The limiting PDF ................................ 38
3.3. An exact solution to the Fokker-Planck equation ........ 40
3.3.1. The equation considered ......................... 40
3.3.2. The solution to the Fokker-Planck equation ...... 41
3.3.3. Means, variances and correlations ............... 42
3.4. Stochastic equations for realizations .................. 43
3.4.1. Stochastic differential equations ............... 44
3.4.2. The relationship to Fokker-Planck equations ..... 46
3.4.3. Monte Carlo simulation .......................... 47
3.5. Stochastic modeling .................................... 48
3.5.1. The set of variables considered ................. 48
3.5.2. The coefficients of stochastic equations ........ 48
Appendix 3А: The dynamics of relevant variables ............. 49
3А.1. The problem considered ........................... 49
3А.2. The projection operator .......................... 50
3A.3. An operator identity ............................. 51
3А.4. The dynamics of relevant variables ............... 52
3A.5. The equilibrium dynamics of relevant variables ... 53
3A.6. Colored Gaussian noise ........................... 54
3А.7. White Gaussian noise ............................. 56
4. The equations of fluid and thermodynamics ................... 57
4.1. The fluid dynamic variables ............................ 57
4.1.1. Means conditioned on the position ............... 57
4.1.2. The conditioned velocity PDF .................... 58
4.1.3. The fluid dynamic variables ..................... 59
4.2. From the molecular to fluid dynamics ................... 60
4.2.1. A model for the molecular motion ................ 60
4.2.2. The unclosed fluid dynamic equations ............ 61
4.3. The closure of the fluid dynamic equations ............. 63
4.3.1. The calculation of the deviatoric stress
tensor .......................................... 63
4.3.2. The heat flux calculation ....................... 65
4.3.3. Scaling parameters .............................. 67
4.3.4. The closure of the velocity and energy
equations ....................................... 68
4.3.5. The resulting basic equations ................... 70
4.4. The equations for multicomponent reacting systems ...... 70
4.4.1. The mass fraction equations ..................... 70
4.4.2. The caloric equation of state ................... 72
4.4.3. The thermal equation of state ................... 73
4.4.4. The equations for multicomponent reacting
systems ......................................... 75
4.4.5. Incompressible flows ............................ 76
4.4.6. The Boussinesq approximation .................... 77
4.5. Direct numerical simulation ............................ 79
4.5.1. The energy cascade .............................. 79
4.5.2. The simulation of the energy cascade ............ 82
4.6. Reynolds-averaged Navier-Stokes equations .............. 84
4.6.1. Ensemble-averaged equations ..................... 84
4.6.2. The calculation of variances .................... 86
4.6.3. The closure of source terms ..................... 86
Appendix 4A: Second- and higher-order RANS equations ........ 87
4A.1. Second-order equations ........................... 88
4A.2. Third-order equations ............................ 89
4A.3. Fourth-order equations ........................... 89
5. Stochastic models for large-scale turbulence ................ 91
5.1. A hierarchy of stochastic velocity models .............. 92
5.1.1. An acceleration model ........................... 92
5.1.2. A velocity model ................................ 94
5.1.3. A position model ................................ 96
5.2. The generalized Langevin model for velocities .......... 97
5.2.1. The generalized Langevin model .................. 97
5.2.2. The implied moment transport equations .......... 98
5.2.3. Specifications of the generalized Langevin
model ........................................... 99
5.3. A hierarchy of Langevin models ........................ 101
5.3.1. The extended Langevin model .................... 101
5.3.2. The Langevin model ............................. 102
5.3.3. The simplified Langevin model .................. 102
5.4. The Kolmogorov constant ............................... 104
5.4.1. C0 for an equilibrium turbulent boundary
layer .......................................... 104
5.4.2. An explanation for the variations of C0 ........ 106
5.4.3. Landau's objection to universality ............. 108
5.5. A hierarchy of stochastic models for scalars .......... 109
5.5.1. The scalar dynamics ............................ 109
5.5.2. A hierarchy of scalar equations ................ 110
5.5.3. The boundedness of scalars ..................... 112
5.5.4. Comparisons with DNS ........................... 114
5.6. Compressible reacting flow: velocity models ........... 119
5.6.1. Compressibility effects on velocity fields ..... 119
5.6.2. Stochastic velocity models ..................... 123
5.6.3. Deterministic velocity models: the k-ε model ... 125
5.6.4. The inclusion of buoyancy effects .............. 129
5.7. Compressible reacting flow: scalar models ............. 131
5.7.1. Stochastic velocity-scalar models .............. 131
5.7.2. Hybrid methods ................................. 133
5.7.3. Assumed-shape PDF methods ...................... 136
Appendix 5A: Stochastic models and basic equations ......... 136
5A.1. A nonlinear stochastic model .................... 137
5A.2. The consistency with basic equations ............ 138
5A.3. The determination of model coefficients ......... 139
Appendix 5B: Consistent turbulence models .................. 141
5B.1. The model considered ............................ 141
5B.2. Coefficient relations ........................... 142
5B.3. The determination of stochastic model
coefficients .................................... 143
5B.4. A consistent RANS model ......................... 143
Appendix 5C: Nonlinear stochastic models ................... 144
5C.1. The limitations of the applicability of linear
stochastic equations ............................ 144
5C.2. A cubic stochastic model ........................ 145
5C.3. Comparisons with other methods .................. 146
5C.4. An application to CBL turbulence simulations .... 147
6. Stochastic models for small-scale turbulence ............... 153
6.1. The generalization of LES by FDF methods .............. 155
6.1.1. The unclosed LES equations ..................... 155
6.1.2. The stochastic model considered ................ 156
6.1.3. The closure of LES equations ................... 158
6.1.4. Hybrid methods ................................. 159
6.2. The closure of the equation for filtered velocities ... 160
6.2.1. The transport equation for the SGS stress
tensor ......................................... 160
6.2.2. The general algebraic expression for the SGS
stress tensor .................................. 161
6.2.3. Linear and quadratic algebraic SGS stress
tensor models .................................. 163
6.2.4. Scaling analysis ............................... 166
6.2.5. The theoretical calculation of parameters ...... 167
6.2.6. Comparison with DNS data ....................... 169
6.3. The closure of the scalar FDF transport equation ...... 170
6.3.1. The scalar-conditioned convective flux ......... 170
6.3.2. The diffusion coefficient ...................... 171
6.3.3. The scalar mixing frequency .................... 173
6.4. The closure of LES and FDF equations .................. 175
6.4.1. The closure of LES equations ................... 175
6.4.1. The modeling of the dynamics of SGS
fluctuations ................................... 176
Appendix 6A: The dynamic eddy length scale calculation ..... 177
Appendix 6B: The scalar-conditioned convective flux ........ 178
Appendix 6C: An assumed-shape FDF method ................... 179
7. The unification of turbulence models ....................... 181
7.1. The need for the unification of turbulence models ..... 181
7.1.1. Industrial applications of turbulence models ... 181
7.1.2. Basic studies by DNS ........................... 182
7.2. Unified turbulence models ............................. 183
7.2.1. A unified stochastic model ..................... 184
7.2.2. A unified model for filtered variables ......... 185
7.3. Some unsolved questions ............................... 187
7.3.1. The structure of unified turbulence models ..... 187
7.3.2. The parameters of unified turbulence models .... 188
8. References ................................................. 189
9. Author index ............................................... 201
10. Subject index ............................................. 205
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