Computational Models for Turbulent Reacting Flows

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 15
1.1 Introduction......Page 21
1.2 Chemical-reaction-engineering approach......Page 23
1.2.1 PFR and CSTR models......Page 25
1.2.2 RTD theory......Page 28
1.2.3 Zone models......Page 30
1.2.4 Micromixing models......Page 32
1.2.5 Micromixing time......Page 34
1.3 Fluid-mechanical approach......Page 35
1.3.1 Fundamental transport equations......Page 36
1.3.2 Turbulence models......Page 37
1.3.3 Chemical source term......Page 38
1.3.4 Molecular mixing......Page 43
1.4 Relationship between approaches......Page 44
1.5 A road map to Chapters 2–7......Page 45
2.1 Homogeneous turbulence......Page 47
2.1.1 One-point probability density function......Page 49
2.1.2 Spatial correlation functions......Page 52
2.1.3 Temporal correlation functions......Page 54
2.1.4 Turbulent energy spectrum......Page 56
2.1.5 Model velocity spectrum......Page 59
2.1.6 Spectral transport......Page 61
2.2 Inhomogeneous turbulence......Page 64
2.2.1 Expected values of derivatives......Page 65
2.2.2 Mean velocity......Page 67
2.2.3 Reynolds stresses......Page 68
2.2.4 Turbulent dissipation rate......Page 71
3.1 Phenomenology of turbulent mixing......Page 76
3.1.1 Length scales of turbulent mixing......Page 77
3.1.2 Phenomenological model for turbulent mixing......Page 78
3.2.1 One-point velocity, composition PDF......Page 82
3.2.2 Conditional velocity and scalar statistics......Page 87
3.2.3 Spatial correlation functions......Page 89
3.2.4 Scalar energy spectrum......Page 91
3.2.5 Model scalar spectrum......Page 93
3.2.6 Scalar spectral transport......Page 98
3.3 Inhomogeneous turbulent mixing......Page 100
3.3.1 Scalar mean......Page 101
3.3.2 Scalar flux......Page 102
3.3.3 Scalar variance......Page 104
3.3.4 Scalar dissipation rate......Page 106
3.3.5 Scalar covariance......Page 110
3.3.6 Joint scalar dissipation rate......Page 112
3.4 Differential diffusion......Page 116
3.4.1 Homogeneous turbulence......Page 117
3.4.3 Decaying scalars......Page 118
4.1 Direct numerical simulation......Page 120
4.1.1 Homogeneous turbulence......Page 121
4.1.2 Reacting flow......Page 122
4.2.1 Filtered Navier–Stokes equation......Page 124
4.2.2 LES velocity PDF......Page 126
4.2.3 Scalar transport......Page 128
4.2.4 Reacting flow......Page 129
4.3 Linear-eddy model......Page 130
4.3.1 Homogeneous flows......Page 131
4.3.2 Inhomogeneous flows......Page 133
4.4.1 Turbulent-viscosity-based models......Page 134
4.4.2 Reynolds-stress transport equation......Page 137
4.5 RANS models for scalar mixing......Page 140
4.5.1 Turbulent-diffusivity-based models......Page 141
4.5.2 Scalar-flux transport equation......Page 143
4.5.3 Scalar-variance transport equation......Page 145
4.5.4 Scalar-dissipation transport equation......Page 146
4.6 Non-equilibrium models for scalar dissipation......Page 147
4.6.1 Spectral relaxation model......Page 148
4.6.2 Spectral transfer rates......Page 152
4.7.1 Multi-variate SR model......Page 155
4.7.2 Mean scalar gradients......Page 157
4.7.3 Decaying scalars......Page 158
4.8 Transported PDF methods......Page 160
5.1 Overview of the closure problem......Page 161
5.1.1 Chemical source term......Page 162
5.1.2 Elementary reactions......Page 164
5.1.3 Non-elementary reactions......Page 166
5.1.4 Reynolds-averaged chemical source term......Page 170
5.1.5 Chemical time scales......Page 171
5.2.1 First-order moment closures......Page 173
5.2.2 Higher-order moment closures......Page 175
5.3 Mixture-fraction vector......Page 176
5.3.1 General formulation......Page 177
5.3.2 Definition of mixture fraction......Page 181
5.3.3 Example flows......Page 188
5.3.4 Mixture-fraction PDF......Page 194
5.4.1 Treatment of reacting scalars......Page 197
5.4.2 Application to turbulent reacting flows......Page 198
5.5 Simple chemistry......Page 200
5.5.1 General formulation: reaction-progress variables......Page 201
5.5.2 One-step reaction......Page 202
5.5.3 Competitive-consecutive reactions......Page 204
5.5.4 Parallel reactions......Page 209
5.6 Lagrangian micromixing models......Page 213
5.6.1 IEM model for a stirred reactor......Page 214
5.6.2 Age-based models......Page 215
5.6.3 Lagrangian models for the micromixing rate......Page 217
5.6.4 Mechanistic models......Page 218
5.6.5 Extension to inhomogeneous flows......Page 220
5.7.1 Definition of a flamelet......Page 221
5.7.2 Stationary laminar flamelet model......Page 224
5.7.3 Joint mixture fraction, dissipation rate PDF......Page 225
5.7.4 Extension to inhomogeneous flows......Page 226
5.8.1 General formulation: conditional moments......Page 227
5.8.2 Closures based on presumed conditional moments......Page 229
5.8.3 Conditional scalar mean:homogeneous flows......Page 231
5.8.4 Conditional scalar dissipation rate......Page 232
5.8.5 Extension to inhomogeneous flows......Page 234
5.9.1 Single reaction-progress variable......Page 236
5.9.2 Multiple reaction-progress variables......Page 238
5.10 Multi-environment presumed PDF models......Page 241
5.10.1 General formulation......Page 242
5.10.2 Extension to inhomogeneous flows......Page 246
5.10.3 Multi-environment conditional PDF models......Page 253
5.10.4 Extension to LES......Page 257
5.11 Transported PDF methods......Page 259
6.1 Introduction......Page 261
6.1.1 Velocity, composition PDF......Page 262
6.2 Velocity, composition PDF transport equation......Page 264
6.2.1 Mean convected derivative: first form......Page 265
6.2.2 Mean convected derivative: second form......Page 266
6.2.4 Conditional fluxes: the unclosed terms......Page 268
6.3.1 Derivation of the transport equation......Page 269
6.3.3 Relationship to Lagrangian micromixing models......Page 271
6.4.1 RANS mean velocity transport equation......Page 272
6.5 Models for conditional acceleration......Page 274
6.5.2 Velocity PDF closures......Page 275
6.5.3 Corresponding Reynolds-stress models......Page 276
6.5.4 Generalized Langevin model......Page 277
6.5.5 Extension to velocity, composition PDF......Page 278
6.5.6 Coupling with mean pressure field......Page 279
6.5.8 Large-eddy PDF methods......Page 280
6.6 Models for conditional diffusion......Page 281
6.6.1 Some useful constraints......Page 282
6.6.2 Desirable properties for mixing models......Page 283
6.6.3 Physical basis for desirable properties......Page 284
(i) Why Gaussian?......Page 285
(iii) Why linearity?......Page 286
(iv) Why should mixing be local in composition space?......Page 288
(v) Why account for the scalar length-scale distribution?......Page 291
CD model......Page 293
IEM model......Page 294
FP model......Page 295
6.6.5 Prospects for mixing model improvements......Page 306
6.7.1 Lagrangian notional particles......Page 307
6.7.2 Lagrangian fluid particles......Page 309
6.7.4 Relationship to Eulerian PDF transport equation......Page 310
6.7.5 Stochastic differential equations for notional particles......Page 312
6.7.6 Lagrangian velocity PDF closures......Page 314
6.7.7 Lagrangian mixing models......Page 316
6.8.1 Notional particles......Page 318
6.8.2 Empirical PDF......Page 320
6.8.3 Errors in mean-field estimate......Page 322
6.8.4 PDF estimation......Page 327
6.9.1 Stiff kinetics......Page 328
6.9.2 Decoupling from transport terms......Page 329
6.9.3 Pre-computed lookup tables......Page 330
6.9.4 In situ adaptive tabulation......Page 332
Reaction mapping......Page 333
Linear interpolation......Page 334
Ellipsoid of accuracy......Page 336
Binary tree......Page 338
Further improvements......Page 339
6.10.1 Turbulence frequency......Page 341
6.10.2 Lagrangian SR model......Page 342
6.10.3 LSR model with differential diffusion......Page 345
6.10.4 LSR model with reacting scalars......Page 346
7 Transported PDF simulations......Page 348
7.1 Overview of simulation codes......Page 349
7.2 Eulerian composition PDF codes......Page 351
7.2.1 Particle transport processes......Page 352
7.2.2 Numerical diffusion......Page 356
7.2.3 Other considerations......Page 357
7.3.1 Notional-particle representation......Page 360
7.3.2 Monte-Carlo simulation......Page 364
7.3.3 Boundary conditions......Page 366
7.3.4 Particle-field estimation......Page 368
7.3.5 Other considerations......Page 372
7.4 Velocity, composition PDF codes......Page 374
7.4.1 Mean conservation equations......Page 375
7.4.2 Notional-particle representation......Page 376
7.4.3 Monte-Carlo simulation......Page 377
7.4.4 Particle-field estimation and consistency......Page 378
7.4.5 Other considerations......Page 379
7.5 Concluding remarks......Page 381
A.1 Scalar spectral transport equation......Page 383
A.2 Spectral relaxation model......Page 385
A.3 Scalar dissipation rate......Page 388
B.1 Quadrature method of moments......Page 392
B.2 Direct QMOM......Page 393
B.2.1 Uni-variate case......Page 394
B.2.2 Bi-variate case......Page 399
B.2.3 Multi-variate case......Page 402
B.3 DQMOM–IEM model......Page 404
References......Page 407
Index......Page 428