Acknowledgments ................................................ ix
List of Symbols ................................................ xi
Abbreviations ................................................ xvii
1 Introduction ................................................. 1
2 Governing Equations .......................................... 7
2.1 The Flow and Its Mathematical Description ............... 7
2.2 Conservation Laws ...................................... 10
2.3 Viscous Stresses ....................................... 14
2.4 Complete System ofthe Navier-Stokes Equations .......... 16
References .................................................. 26
3 Principles of Solution of the Governing Equations ........... 29
3.1 Spatial Discretization ................................. 32
3.2 Temporal Discretization ................................ 46
3.3 Turbulence Modeling .................................... 55
3.4 Initial and Boundary Conditions ........................ 58
References .................................................. 59
4 Structured Finite-Volume Schemes ............................ 73
4.1 Geometrical Quantities of a Control Volume ............. 77
4.2 General Discretization Methodologies ................... 80
4.3 Discretization ofthe Convective Fluxes ................. 89
4.4 Discretization ofthe Viscous Fluxes ................... 113
References ................................................. 116
5 Unstructured Finite-Volume Schemes ......................... 121
5.1 Geometrical Quantities of a Control Volume ............ 126
5.2 General Discretization Methodologies .................. 130
5.3 Discretization of the Convective Fluxes ............... 139
5.4 Discretization of the Viscous Fluxes .................. 157
References ................................................. 162
6 Temporal Discretization .................................... 167
6.1 Explicit Time-Stepping Schemes ........................ 168
6.2 Implicit Time-Stepping Schemes ........................ 176
6.3 Methodologies for Unsteady Flows ...................... 202
References ................................................. 206
7 Turbulence Modeling ........................................ 213
7.1 Basic Equations of Turbulence ......................... 215
7.2 First-Order Closures .................................. 225
7.3 Large-Eddy Simulation ................................. 235
References ................................................. 244
8 Boundary Conditions ........................................ 253
8.1 Concept of Dummy Cells ................................ 254
8.2 Solid Wall ............................................ 255
8.3 Far-Field ............................................. 262
8.4 Inlet/Outlet Boundary ................................. 268
8.5 Injection Boundary .................................... 270
8.6 Symmetry Plane ........................................ 271
8.7 Coordinate Cut ........................................ 272
8.8 Periodic Boundaries ................................... 273
8.9 Interface Between Grid Blocks ......................... 275
8.10Flow Gradients at Boundaries of Unstructured Grids ..... 278
References ................................................. 279
9 Acceleration Techniques .................................... 283
9.1 Local Time-Stepping ................................... 284
9.2 Enthalpy Damping ...................................... 284
9.3 Residual Smoothing .................................... 286
9.4 Multigrid ............................................. 293
9.5 Preconditioning for Low Mach Numbers .................. 307
9.6 Parallelization ....................................... 320
References ................................................. 329
10 Consistency, Accuracy, and Stability ....................... 337
10.1 Consistency Requirements ............................. 338
10.2 Accuracy of Discretization Scheme .................... 339
10.3 Von Neumann Stability Analysis ....................... 340
References ................................................. 355
11 Principles of Grid Generation .............................. 357
11.1 Structured Grids ...................................... 360
11.2 Unstructured Grids .................................... 373
References ................................................. 388
12 Software Applications ...................................... 395
12.1 Programs for Stability Analysis ....................... 397
12.2 Structured 1-D Grid Generator ......................... 397
12.3 Structured 2-D Grid Generators ........................ 398
12.4 Structured to Unstructured Grid Converter ............. 399
12.5 Quasi 1-D Euler Solver ................................ 399
12.6 Structured 2-D Euler/Navier-Stokes Solver ............. 400
12.7 Unstructured 2-D Euler/Navier-Stokes Solver ........... 405
12.8 Parallelization ....................................... 406
References ................................................. 407
Appendix ...................................................... 409
A.1 Governing Equations in Differential Form .............. 409
A.2 Quasilinear Form of the Euler Equations ............... 415
A.3 Mathematical Character of the Governing Equations ..... 416
A.4 Navier-Stokes Equations in Rotating Frame of
Reference ............................................. 419
A.5 Navier-Stokes Equations Formulated for Moving Grids ... 421
A.6 Thin Shear Layer Approximation ........................ 425
A.7 PNS Equations ......................................... 427
A.8 Axisymmetric Form of the Navier-Stokes Equations ...... 427
A.9 Convective Flux Jacobian .............................. 429
A.10 Viscous Flux Jacobian ................................. 431
A.11 Transformation from Conservative to Characteristic
Variables ............................................. 433
A.12 GMRES Algorithm ....................................... 437
A.13 Tensor Notation ....................................... 440
References ................................................. 441
Index ......................................................... 443
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