Acknowledgements ............................................... xi
List of Symbols .............................................. xiii
Abbreviations ................................................. xix
1 Introduction ................................................. 1
2 Governing Equations .......................................... 5
2.1 The Flow and its Mathematical Description ............... 5
2.2 Conservation Laws ....................................... 8
2.2.1 The Continuity Equation .......................... 8
2.2.2 The Momentum Equation ............................ 8
2.2.3 The Energy Equation ............................. 10
2.3 Viscous Stresses ....................................... 13
2.4 Complete System of the Navier-Stokes Equations ......... 16
2.4.1 Formulation for a Perfect Gas ................... 18
2.4.2 Formulation for a Real Gas ...................... 19
2.4.3 Simplifications to the Navier-Stokes
Equations ....................................... 22
Bibliography ................................................ 26
3 Principles of Solution of the Governing Equations ........... 29
3.1 Spatial Discretisation ................................. 32
3.1.1 Finite Difference Method ........................ 36
3.1.2 Finite Volume Method ............................ 37
3.1.3 Finite Element Method ........................... 39
3.1.4 Other Discretisation Methods .................... 40
3.1.5 Central and Upwind Schemes ...................... 41
3.2 Temporal Discretisation ................................ 45
3.2.1 Explicit Schemes ................................ 46
3.2.2 Implicit Schemes ................................ 49
3.3 Turbulence Modelling ................................... 53
3.4 Initial and Boundary Conditions ........................ 56
Bibliography ........................................... 58
4 Structured Finite Volume Schemes ............................ 77
4.1 Geometrical Quantities of a Control Volume ............. 81
4.1.1 Two-Dimensional Case ............................ 81
4.1.2 Three-Dimensional Case .......................... 82
4.2 General Discretisation Methodologies ................... 85
4.2.1 Cell-Centred Scheme ............................. 85
4.2.2 Cell-Vertex Scheme: Overlapping Control
Volumes ......................................... 87
4.2.3 Cell-Vertex Scheme: Dual Control Volumes ........ 90
4.2.4 Cell-Centred versus Cell-Vertex Schemes ......... 93
4.3 Discretisation of the Convective Fluxes ................ 95
4.3.1 Central Scheme with Artificial Dissipation ...... 97
4.3.2 Flux-Vector Splitting Schemes .................. 100
4.3.3 Flux-Difference Splitting Schemes .............. 108
4.3.4 Total Variation Diminishing Schemes ............ 1ll
4.3.5 Limiter Functions .............................. 112
4.4 Discretisation of the Viscous Fluxes .................. 118
4.4.1 Cell-Centred Scheme ............................ 120
4.4.2 Cell-Vertex Scheme ............................. 121
Bibliography ............................................... 122
5 Unstructured Finite Volume Schemes ......................... 131
5.1 Geometrical Quantities of a Control Volume ............ 136
5.1.1 Two-Dimensional Case ........................... 136
5.1.2 Three-Dimensional Case ......................... 138
5.2 General Discretisation Methodologies .................. 141
5.2.1 Cell-Centred Scheme ............................ 142
5.2.2 Median-Dual Cell-Vertex Scheme ................. 145
5.2.3 Cell-Centred versus Median-Dual Scheme ......... 149
5.3 Discretisation of the Convective Fluxes ............... 153
5.3.1 Central Schemes with Artificial Dissipation .... 153
5.3.2 Upwind Schemes ................................. 157
5.3.3 Solution Reconstruction ........................ 157
5.3.4 Evaluation of the Gradients .................... 163
5.3.5 Limiter Functions .............................. 168
5.4 Discretisation of the Viscous Fluxes .................. 172
5.4.1 Element-Based Gradients ........................ 172
5.4.2 Average of Gradients ........................... 174
Bibliography ............................................... 177
6 Temporal Discretisation .................................... 183
6.1 Explicit Time-Stepping Schemes ........................ 184
6.1.1 Multistage Schemes (Runge-Kutta) ............... 184
6.1.2 Hybrid Multistage Schemes ...................... 186
6.1.3 Treatment of the Source Term ................... 187
6.1.4 Determination of the Maximum Time Step ......... 188
6.2 Implicit Time-Stepping Schemes ........................ 192
6.2.1 Matrix Form of the Implicit Operator ........... 193
6.2.2 Evaluation of the Flux Jacobian ................ 197
6.2.3 ADI Scheme ..................................... 201
6.2.4 LU-SGS Scheme .................................. 204
6.2.5 Newton-Krylov Method ........................... 210
6.3 Methodologies for Unsteady Flows ...................... 214
6.3.1 Dual Time-Stepping for Explicit Multistage
Schemes ........................................ 215
6.3.2 Dual Time-Stepping for Implicit Schemes ........ 217
Bibliography ............................................... 218
7 Turbulence Modelling ....................................... 227
7.1 Basic Equations of Turbulence ......................... 230
7.1.1 Reynolds Averaging ............................. 231
7.1.2 Favre (Mass) Averaging ......................... 232
7.1.3 Reynolds-Averaged Navier-Stokes Equations ...... 233
7.1.4 Favre- and Reynolds-Averaged Navier-Stokes
Equations ...................................... 234
7.1.5 Eddy-Viscosity Hypothesis ...................... 235
7.1.6 Non-Linear Eddy Viscosity ...................... 237
7.1.7 Reynolds-Stress Transport Equation ............. 238
7.2 First-Order Closures .................................. 240
7.2.1 Spalart-Allmaras One-Equation Model ............ 240
7.2.2 K-ε Two-Equation Model ......................... 243
7.2.3 SST Two-Equation Model of Menter ............... 247
7.3 Large-Eddy Simulation ................................. 250
7.3.1 Spatial Filtering .............................. 251
7.3.2 Filtered Governing Equations ................... 252
7.3.3 Subgrid-Scale Modelling ........................ 254
7.3.4 Wall Models .................................... 257
7.3.5 Detached Eddy Simulation ....................... 258
Bibliography ............................................... 259
8 Boundary Conditions ........................................ 271
8.1 Concept of Dummy Cells ................................ 272
8.2 Solid Wall ............................................ 274
8.2.1 Inviscid Flow .................................. 274
8.2.2 Viscous Flow ................................... 279
8.3 Farfield .............................................. 281
8.3.1 Concept of Characteristic Variables ............ 281
8.3.2 Modifications for Lifting Bodies ............... 283
8.4 Inlet/Outlet Boundary ................................. 287
8.5 Injection Boundary .................................... 289
8.6 Symmetry Plane ........................................ 290
8.7 Coordinate Cut ........................................ 291
8.8 Periodic Boundaries ................................... 292
8.9 Interface Between Grid Blocks ......................... 295
8.10 Flow Gradients at Boundaries of Unstructured Grids .... 298
Bibliography ............................................... 299
9 Acceleration Techniques .................................... 303
9.1 Local Time-Stepping ................................... 304
9.2 Enthalpy Damping ...................................... 305
9.3 Residual Smoothing .................................... 306
9.3.1 Central IRS on Structured Grids ................ 306
9.3.2 Central IRS on Unstructured Grids .............. 309
9.3.3 Upwind IRS on Structured Grids ................. 309
9.4 Multigrid ............................................. 312
9.4.1 Basic Multigrid Cycle .......................... 313
9.4.2 Multigrid Strategies ........................... 315
9.4.3 Implementation on Structured Grids ............. 316
9.4.4 Implementation on Unstructured Grids ........... 322
9.5 Preconditioning for Low Mach Numbers .................. 327
9.5.1 Derivation of Preconditioned Equations ......... 328
9.5.2 Implementation ................................. 330
9.5.3 Form of the Matrices ........................... 331
Bibliography .......................................... 342
10 Consistency, Accuracy and Stability ........................ 351
10.1 Consistency Requirements .............................. 352
10.2 Accuracy of Discretisation ............................ 353
10.3 Von Neumann Stability Analysis ........................ 354
10.3.1 Fourier Symbol and Amplification Factor ........ 354
10.3.2 Convection Model Equation ...................... 355
10.3.3 Convection-Diffusion Model Equation ............ 356
10.3.4 Explicit Time-Stepping ......................... 357
10.3.5 Implicit Time-Stepping ......................... 363
10.3.6 Derivation of the CFL Condition ................ 367
Bibliography ............................................... 370
11 Principles of Grid Generation .............................. 373
11.1 Structured Grids ...................................... 376
11.1.1 C-, H-, and O-Grid Topology .................... 377
11.1.2 Algebraic Grid Generation ...................... 379
11.1.3 Elliptic Grid Generation ....................... 383
11.1.4 Hyperbolic Grid Generation ..................... 385
11.2 Unstructured Grids .................................... 388
11.2.1 Delaunay Triangulation ......................... 389
11.2.2 Advancing-Front Method ......................... 394
11.2.3 Generation of Anisotropic Grids ................ 395
11.2.4 Mixed-Element/Hybrid Grids ..................... 400
11.2.5 Assessment and Improvement of Grid Quality ..... 402
Bibliography .......................................... 405
12 Description of the Source Codes ............................ 415
12.1 Programs for Stability Analysis ....................... 417
12.2 Structured 1-D Grid Generator ......................... 417
12.3 Structured 2-D Grid Generators ........................ 418
12.4 Structured to Unstructured Grid Converter ............. 419
12.5 Quasi 1-D Euler Solver ................................ 419
12.6 Structured 2-D Euler/Navier-Stokes Solver ............. 420
12.7 Unstructured 2-D Euler/Navier-Stokes Solver ........... 421
12.8 Visualisation Tool .................................... 423
Bibliography ............................................... 423
A Appendix ................................................... 427
A.l Governing Equations in Differential Form .............. 427
A.2 Quasilinear Form of the Euler Equations ............... 433
A.3 Mathematical Character of the Governing Equations ..... 434
A.3.1 Hyperbolic Equations ........................... 434
A.3.2 Parabolic Equations ............................ 436
A.3.3 Elliptic Equations ............................. 436
A.4 Navier-Stokes Equations in Rotating Frame of
Reference ............................................. 438
A.5 Navier-Stokes Equations Formulated for Moving Grids ... 441
A.6 Thin Shear Layer Approximation ........................ 445
A.7 Parabolised Navier-Stokes Equations ................... 447
A.8 Axisymmetric Form of the Navier-Stokes Equations ...... 448
A.9 Convective Flux Jacobian .............................. 450
A.10 Viscous Flux Jacobian ................................. 452
A.11 Transformation from Conservative to Characteristic
Variables ............................................. 455
A.12 GMRES Algorithm ....................................... 458
A.13 Tensor Notation ....................................... 462
Bibliography ............................................... 463
Index ......................................................... 465
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