High-Resolution Methods for Incompressible and Low-Speed Flows (eBook)

Preface by Frank Harlow 7
Acknowledgements 9
Contents 11
1. Introduction 19
Fundamental Physical and Model Equations 22
2. The Fluid Flow Equations 23
2.1 Mathematical Preliminaries 23
2.2 Kinematic Considerations 25
2.3 The Equations for Variable Density Flows 26
2.4 Compressible Euler Equations 32
2.5 Low-Mach Number Scaling 36
2.6 Boussinesq Approximation 39
2.7 Variable Density Flow 39
2.8 Zero Mach Number Combustion 40
2.9 Initial and Boundary Conditions 41
3. The Viscous Fluid Flow Equations 42
3.1 The Stress and Strain Tensors for a Newtonian Fluid 42
3.2 The Navier-Stokes Equations for Constant Density Flows 46
3.3 Non-Newtonian Constitutive Equations for the Shear- Stress Tensor 48
3.4 Alternative Forms of the Advective and Viscous Terms 53
3.5 Nondimensionalization of the Governing Equations 54
3.6 General Remarks on Turbulent Flow Simulations 57
3.7 Reynolds-Averaged Navier-Stokes Equations ( RANS) 58
3.8 Large Eddy Simulation (LES) 62
3.9 Closing Remarks 64
4. Curvilinear Coordinates and Transformed Equations 66
4.1 Generalized Curvilinear Coordinates 66
4.2 Calculation of Metrics 70
4.3 Transformation of the Fluid Flow Equations 72
4.4 Viscous Terms 75
4.5 Geometric Conservation Law 78
5. Overview of Various Formulations and Model Equations 81
5.1 Overview of Various Formulations of the Incompressible Flow Equations 81
5.2 Model Equations 89
6. Basic Principles in Numerical Analysis 93
6.1 Stability, Consistency and Accuracy 93
6.2 Fourier Analysis 97
6.3 Modified Equation Analysis 104
6.4 Verification via Sample Calculations 108
7. Time Integration Methods 112
7.1 Time Integration of the Flow Equations 112
7.2 Lax-Wendroff-Type Methods 113
7.3 Other Approaches to Time-Centering 115
7.4 Runge-Kutta Methods 116
7.5 Linear Multi-step Methods 126
8. Numerical Linear Algebra 133
8.1 Basic Numerical Linear Algebra 133
8.2 Basic Relaxation Methods 135
8.3 Conjugate Gradient and Krylov Subspace Methods 138
8.4 Multigrid Algorithm for Elliptic Equations 142
8.5 Multigrid Algorithm as a Preconditioner for Krylov Subspace Methods 150
8.6 Newton’s and Newton-Krylov Method 151
8.7 A Multigrid Newton-Krylov Algorithm 152
Solution Approaches 156
9. Compressible and Preconditioned- Compressible Solvers 157
9.1 Reconstructing the Dependent Variables 157
9.2 Reconstructing the Fluxes 166
9.3 Preconditioning for Low Speed Flows 170
10. The Artificial Compressibility Method 182
10.1 Basic Formulation 182
10.2 Convergence to the Incompressible Limit 183
10.3 Preconditioning and the Artificial Compressibility Method 185
10.4 Eigenstructure of the Incompressible Equations 186
10.5 Estimation of the Artificial Compressibility Parameter 189
10.6 Explicit Solvers for Artificial Compressibility 192
10.7 Implicit Solvers for Artificial Compressibility 193
10.8 Extension of the Artificial Compressibility to Unsteady Flows 197
10.9 Boundary Conditions 199
10.10 Local Time Step 200
10.11 Multigrid for the Artificial-Compressibility Formulation 201
11. Projection Methods: The Basic Theory and the Exact Projection Method 218
11.1 Grids – Variable Positioning 219
11.2 Continuous Projections for Incompressible Flow 220
11.3 Exact Discrete Projections 222
11.4 Second-Order Projection Algorithms for Incompressible Flow 232
11.5 Boundary Conditions 234
12. Approximate Projection Methods 245
12.1 Numerical Issues with Approximate Projection Methods 245
12.2 Projection Algorithms for Incompressible Flow 251
12.3 Analysis of Projection Algorithms 252
12.4 Pressure Poisson Equation Methods 258
12.5 Filters 264
12.6 Method Demonstration and Verification 279
Modern High-Resolution Methods 301
13. Introduction to Modern High-Resolution Methods 302
13.1 General Remarks about High-Resolution Methods 302
13.2 The Concept of Nonoscillatory Methods and Total Variation 308
13.3 Monotonicity 310
13.4 General Remarks on Riemann Solvers 312
14. High-Resolution Godunov-Type Methods for Projection Methods 315
14.1 First-Order Algorithm 315
14.2 High-Resolution Algorithms 322
14.3 Staggered Grid Spatial Differencing 331
14.4 Unsplit Spatial Differencing 333
14.5 Multidimensional Results 346
14.6 Viscous Terms 348
14.7 Stability 349
15. Centered High-Resolution Methods 352
15.1 Lax-Friedrichs Scheme 353
15.2 Lax-Wendroff Scheme 358
15.3 First-Order Centered Scheme 363
15.4 Second- and Third-Order Centered Schemes 369
16. Riemann Solvers and TVD Methods in Strict Conservation Form 378
16.1 The Flux Limiter Approach 378
16.2 Construction of Flux Limiters 379
16.3 Other Approaches for Constructing Advective Schemes 387
16.4 The Characteristics-Based Scheme 389
16.5 Flux Limiting Version of the CB Scheme 409
16.6 Implementation of the Characteristics-Based Method in Unstructured Grids 409
16.7 The Weight Average Flux Method 411
16.8 Roe’s Method 414
16.9 Osher’s Method 417
16.10 Chakravarthy-Osher TVD Scheme 419
16.11 Harten, Lax and van Leer (HLL) Scheme 421
16.12 HLLC Scheme 424
16.13 Estimation of the Wave Speeds for the HLL and HLLC Riemann Solvers 425
16.14 HLLE Scheme 426
16.15 Comparison of CB and HLLE Schemes 426
16.16 Viscous” TVD Limiters 429
17. Beyond Second-Order Methods 434
17.1 General Remarks on High-Order Methods 435
17.2 Essentially Nonoscillatory Schemes (ENO) 438
17.3 ENO Schemes Using Fluxes 441
17.4 Weighted ENO Schemes 444
17.5 A Flux-Based Version of the WENO Scheme 449
17.6 Artificial Compression Method for ENO and WENO 452
17.7 The ADER Approach 453
17.8 Extending and Relaxing Monotonicity in Godunov- Type Methods 460
17.9 Discontinuous Galerkin Methods 472
17.10 Uniformly High-Order Scheme for Godunov- Type Fluxes 474
17.11 Flux-Corrected Transport 477
17.12 MPDATA 480
Applications 482
18. Variable Density Flows and Volume Tracking Methods 483
18.1 Multimaterial Mixing Flows 483
18.2 Volume Tracking 494
18.3 The History of Volume Tracking 499
18.4 A Geometrically Based Method of Solution 503
18.5 Results For Vortical Flows 523
19. High-Resolution Methods and Turbulent Flow Computation 533
19.1 Physical Considerations 533
19.2 Survey of Theory and Models 537
19.3 Relation of High-Resolution Methods and Flow Physics 540
19.4 Large Eddy Simulation: Standard and Implicit 543
19.5 Numerical Analysis of Subgrid Models 547
19.6 ILES Analysis 548
19.7 Computational Examples 556
A. MATHEMATICA Commands for Numerical Analysis 560
A.1 Fourier Analysis for First-Order Upwind Methods 560
A.2 Fourier Analysis for Second-Order Upwind Methods 561
A.3 Modified Equation Analysis for First-Order Upwind 562
B. Example Computer Implementations 566
B.1 Appendix: Fortran Subroutine for the Characteristics- Based Flux 566
B.2 Fifth-Order Weighted ENO Method 571
C. Acknowledgements: Illustrations Reproduced with Permission 577
References 579
Index 616