Handbook of computational fluid mechanics

Cover......Page 1
List of contributors......Page 2
Preface......Page 3
Contents......Page 5
I INTRODUCTION......Page 6
II CONTINUOUS MODELS......Page 7
B Finite-Difference Methods (FDM)......Page 9
C Finite-Volume Methods (FVM)......Page 11
D Finite-Element Methods (FEM)......Page 13
E Compact Finite-Difference Methods (CFDM)......Page 14
F Spectral Methods (SM)......Page 15
A Stability, dispersion, dissipation......Page 16
1 Stability......Page 17
C Space-time approximations......Page 18
V OSCILLATION CONTROL......Page 19
B Total Variation Diminishing property (TVD)......Page 20
C Flux-Corrected Transport (FCT)......Page 21
D Upwind and symmetric TVD principles......Page 22
E The MUSCL method......Page 24
F Essentially Non-Oscillatory method (ENO)......Page 25
VI CONCLUDING REMARKS......Page 26
REFERENCES......Page 27
I INTRODUCTION......Page 29
A Geometrical concepts......Page 33
B The different forms of the primitive formulation......Page 35
C The Reynolds-averaged Navier-Stokes equations......Page 39
A The finite-volume technique: cell-centred treatment......Page 40
1 Finite-difference techniques......Page 48
2 The closure stencil......Page 50
A The discrete form of the mass equation......Page 57
B The general form of the discrete momentum equation......Page 58
C Iterative algorithms for the coupled systems......Page 60
1 Approximate factorization techniques......Page 62
2 Conjugate-gradient type accelerations......Page 63
3 Block-correction accelerations......Page 66
4 Multigrid acceleration......Page 67
E Strongly coupled algorithms......Page 69
F Other flux reconstruction techniques......Page 75
G The non-linear problem......Page 76
H Perturbed continuity equation and false transient methods......Page 77
A Methods without interpolation......Page 81
1 The chequerboard problem in the collocated case......Page 84
2 The partially staggered grid: the ICED-ALE practice......Page 85
3 The fully staggered grid\" the MAC practice......Page 87
4 The TURF practice......Page 88
C Boundary conditions......Page 90
VI CLOSURE......Page 94
REFERENCES......Page 95
3 Navier-Stokes equations for incompressible flows: finite-element methods......Page 102
1 Differential equations......Page 103
3 Boundary conditions......Page 104
B Weak formulation of the steady-state Navier-Stokes syst......Page 105
1 Sobolev spaces......Page 106
2 Weak formulation......Page 107
1 The discrete problem......Page 108
2 The LBB condition......Page 109
3 Error estimates......Page 110
4 Verifying the LBB condition......Page 112
5 Ways in which the LBB condition fails......Page 114
1 Piecewise linear velocities......Page 116
2 Piecewise bilinear velocities......Page 119
4 Solenoidal elements......Page 120
6 Some other elements of interest......Page 121
E Inhomogeneous velocity boundary conditions......Page 122
1 Alternate weak formulations......Page 123
2 Pressure, stress and vorticity boundary conditions......Page 124
H The effects of numerical integration......Page 126
I Stabilized methods......Page 127
J Penalty methods......Page 128
1 Newton\'s method......Page 129
3 Continuation methods......Page 130
1 Weak formulation......Page 132
2 Spatial semi-discretization......Page 133
3 Single-step fully implicit schemes......Page 134
4 Single-step semi-implicit schemes......Page 135
5 Backward differentiation multistep schemes......Page 136
M Pressure Poisson equation formulation......Page 137
1 The streamfunction-vorticity formulation for plane flows......Page 138
2 The streamfunction formulation for plane flows......Page 139
2 Error estimates......Page 140
1 Recovery of the velocity......Page 141
2 Recovery of the pressure......Page 142
1 Boundary conditions......Page 144
2 Finite-element discretizations......Page 145
3 Solution algorithms......Page 146
IV LEAST-SQUARES FINITE-ELEMENT METHODS......Page 147
2 Least-squares principles......Page 148
2 The discrete equations......Page 149
3 Error estimates......Page 150
1 Equations derivable from the primitive variable formulation......Page 151
2 Additional boundary conditions......Page 152
1 The discrete equations......Page 153
2 Accuracy considerations......Page 154
VI SPECTRAL-ELEMENT METHODS......Page 155
3 Numerical integration......Page 156
REFERENCES......Page 157
4 Euler and Navier-Stokes equations for compressible flows: finite-volume methods......Page 161
I INTRODUCTION......Page 162
II GOVERNING EQUATIONS......Page 163
A Quasilinear conservative/non-conservative formulations of Euler equations......Page 165
III FINITE VOLUME FORMULATION......Page 174
A Control volume decomposition and surface vector definition......Page 176
A Basic concepts......Page 179
Courant-lsaacson-Rees (CIR)......Page 185
Godunov (G)......Page 186
2 Schemes for one-dimensional systems of conservation laws......Page 187
Conservative splitting......Page 190
Non-conservative splitting......Page 191
Numerical flux splitting......Page 192
Steger-Warming splitting......Page 193
Van Leer splitting......Page 195
Advection Upwind Splitting Method (AUSM)......Page 199
Approximate Riemann solver......Page 201
Lax-Wendroff (L W)......Page 203
Total Variation Diminishing (TVD) schemes......Page 204
Flux limiter method......Page 208
Slope limiter methods......Page 212
Lacor-Hirsch limiters......Page 215
Total variation diminishing schemes (TVD)......Page 216
Monotone upstream schemes for conservation laws (MUSCL)......Page 217
(i) Third-order upwind biased scheme......Page 218
(iv) Fromrn scheme (Fromm, 1968)......Page 219
Jameson flux function......Page 222
Essentially Non-Oscillatory schemes (ENO)......Page 223
C Multidimensional extension......Page 226
1 Van Leer splitting......Page 228
2 Advection upstream splitting......Page 229
4 Total variation diminishing methods......Page 230
5 Essentially Non-Oscillatory schemes (ENO)......Page 231
V DISCRETIZATION OF THE VISCOUS FLUX......Page 232
VI TREATMENT OF BOUNDARY CONDITIONS......Page 235
1 Euler equations......Page 237
Wall (v. n = 0)......Page 239
2 Navier-Stokes equations......Page 240
B Choice of suitable boundary conditions......Page 241
C Construction of numerical fluxes at boundaries......Page 243
For fully discretized formulation.\"......Page 245
For semidiscretizedformulation:......Page 246
Zero stability......Page 248
Lax-Richtmyer stability......Page 249
Godunov-Ryabenkii stability......Page 250
For the semidiscretizedf orm:......Page 251
4 Normal mode method......Page 256
For semidiscretizedformulation......Page 257
A Runge-Kutta algorithms......Page 260
B Implicit methods......Page 263
1 Alternating direction implicit method......Page 266
2 Lower-Upper factorization......Page 267
C Multigrid techniques......Page 271
Restriction phase......Page 272
Coarse-grid evolution......Page 273
Coarse-grid correction prolongation......Page 275
VIII APPLICATIONS......Page 277
IX CONCLUDING REMARKS......Page 278
REFERENCES......Page 279
I INTRODUCTION......Page 285
A Basic equations......Page 288
2 Compressible flow......Page 289
B Direct numerical simulation......Page 290
1 Resolution requirements for DNS......Page 292
C Large-eddy simulation......Page 295
1 GS and SGS turbulence......Page 297
2 Scale separation and filtering......Page 298
Incompressible flow......Page 303
Compressible flow......Page 304
4 Resolution requirements for LES......Page 306
III SGS MODELLING......Page 310
A Models for incompressible flows......Page 311
B Models for compressible flows......Page 317
C Near-wall modifications......Page 320
D Validation of SGS models......Page 322
IV NUMERICAL METHODS......Page 323
Spectral methods......Page 324
C Initial and boundary conditions......Page 328
V APPLICATIONS OF DNS AND LES......Page 330
REFERENCES......Page 333
I INTRODUCTION......Page 341
II INCOMPRESSIBLE TURBULENT MODELLING......Page 343
A Reynolds-averaged Navier-Stokes equations......Page 344
1 Reynolds stress models......Page 345
Pressure-strain rate correlation models......Page 348
Isotropization-of-production model......Page 349
Speziale, Sarkar and Gatski model......Page 350
Fu, Launder and Tselepidakis model......Page 351
Turbulent diffusion models......Page 356
2 Two-equation models......Page 359
Explicit algebraic stress models......Page 362
3 One-equation models......Page 365
4 Zero-and half-equation models......Page 366
C Wall functions for incompressible flows......Page 370
D Incompressible near-wall modelling......Page 373
III COMPRESSIBLE TURBULENT MODELLING......Page 378
A Mean conservation equations......Page 379
2 Total energy......Page 380
B Compressible Reynolds stress and two-equation models......Page 381
C Scalar flux and dilatation closure models......Page 386
1 Mass Flux......Page 387
2 Turbulent heat flux......Page 388
3 Pressure dilatation......Page 390
4 Dilatation dissipation......Page 391
D Wall functions for compressible flows......Page 392
E Compressible near-wall models......Page 396
IV NUMERICAL SOLUTION OF TURBULENT MODEL EQUATIONS......Page 397
A Pressure-velocity-based solution methodologies......Page 398
1 Pressure-velocity discretization techniques......Page 400
2 Pressure-velocity solution algorithm......Page 403
B Density-velocity-based solution methodologies......Page 404
1 Density-velocity discretization techniques......Page 407
2 Density-velocity solution algorithm......Page 409
ACKNOWLEDGEMENTS......Page 412
REFERENCES......Page 413
I INTRODUCTION......Page 418
A Numerical implications......Page 428
II GEOMETRY MODELLING AND SURFACE DEFINITION......Page 430
III STRUCTURED GRIDS......Page 433
A Algebraic methods......Page 434
B Elliptic mesh generation......Page 435
C Hyperbolic mesh generation......Page 436
D Adaptive meshing......Page 437
E Overset grids and blocking techniques......Page 438
A Advancing front methods......Page 439
B Delaunay triangulation methods......Page 441
C Edge- and face-swapping techniques......Page 444
D Other triangulation methods......Page 446
F Mixed element meshes......Page 447
F Adaptivity......Page 448
A Grid quality......Page 449
B Grid stretching for viscous flows......Page 450
C Grid-based strategies......Page 451
REFERENCES......Page 456
Index......Page 461