1. Introduction ................................................. 1
Part I. Fundamental Physical and Model Equations
2. The Fluid Flow Equations ..................................... 7
2.1. Mathematical Preliminaries ............................. 7
2.2. Kinematic Considerations ............................... 9
2.3. The Equations for Variable Density Flows .............. 10
2.3.1. The Continuity Equation ....................... 10
2.3.2. The Momentum Equations ........................ 11
2.3.3. The Energy Equation ........................... 14
2.4. Compressible Euler Equations .......................... 16
2.5. Low-Mach Number Scaling ............................... 20
2.6. Boussinesq Approximation .............................. 23
2.7. Variable Density Flow ................................. 23
2.8. Zero Mach Number Combustion ........................... 24
2.9. Initial and Boundary Conditions ....................... 25
3. The Viscous Fluid Flow Equations ............................ 27
3.1. The Stress and Strain Tensors for a Newtonian
Fluid ................................................. 27
3.2. The Navier-Stokes Equations for Constant Density
Flows ................................................. 31
3.3. Non-Newtonian Constitutive Equations for the
Shear-Stress Tensor ................................... 33
3.3.1. Generalized Newtonian Fluids .................. 33
3.3.2. Viscoelastic Fluids ........................... 34
3.3.3. Other Viscoelastic Models ..................... 37
3.4. Alternative Forms of the Advective and Viscous
Terms ................................................. 38
3.5. Nondimensionalization of the Governing Equations ...... 39
3.6. General Remarks on Turbulent Flow Simulations ......... 42
3.7. Reynolds-Averaged Navier-Stokes Equations (RANS) ...... 43
3.8. Large Eddy Simulation (LES) ........................... 47
3.9. Closing Remarks ....................................... 49
4. Curvilinear Coordinates and Transformed Equations ........... 51
4.1. Generalized Curvilinear Coordinates ................... 51
4.2. Calculation of Metrics ................................ 55
4.3. Transformation of the Fluid Flow Equations ............ 57
4.4. Viscous Terms ......................................... 60
4.5. Geometric Conservation Law ............................ 63
5. Overview of Various Formulations and Model Equations ........ 67
5.1. Overview of Various Formulations of the
Incompressible Flow Equations ......................... 67
5.1.1. Vorticity/Stream-Function Formulation ......... 67
5.1.2. The Vorticity/Vector-Potential Formulation .... 69
5.1.3. Vorticity-Velocity Formulation ................ 70
5.1.4. Pressure-Poisson Formulation .................. 70
5.1.5. Projection Formulation ........................ 71
5.1.6. Artificial-Compressibility Formulation ........ 71
5.1.7. Penalty Formulation ........................... 72
5.1.8. Hybrid Formulations ........................... 73
5.2. Model Equations ....................................... 75
5.2.1. Advection-Diffusion Equation .................. 75
5.2.2. Burgers' Equation ............................. 76
6. Basic Principles in Numerical Analysis ...................... 79
6.1. Stability, Consistency and Accuracy ................... 79
6.2. Fourier Analysis ...................................... 83
6.2.1. Fourier Analysis of First-Order Upwind ........ 85
6.2.2. Fourier Analysis of Second-Order Upwind ....... 86
6.3. Modified Equation Analysis ............................ 90
6.4. Verification via Sample Calculations .................. 94
7. Time Integration Methods .................................... 99
7.1. Time Integration of the Flow Equations ................ 99
7.2. Lax-Wendroff-Type Methods ............................ 100
7.3. Other Approaches to Time-Centering ................... 102
7.4. Runge-Kutta Methods .................................. 103
7.4.1. Second-Order Runge-Kutta ..................... 104
7.4.2. Third-Order Runge-Kutta ...................... 106
7.4.3. Fourth-Order Runge-Kutta ..................... 107
7.4.4. TVD Runge-Kutta Methods Applied to
Hyperbolic Conservation Laws ................. 109
7.5. Linear Multi-step Methods ............................ 113
7.5.1. Adams-Bashforth Method ....................... 113
7.5.2. Adams-Moulton Method ......................... 116
7.5.3. Backward Differentiation Formulas ............ 119
8. Numerical Linear Algebra ................................... 121
8.1. Basic Numerical Linear Algebra ....................... 121
8.2. Basic Relaxation Methods ............................. 123
8.3. Conjugate Gradient and Krylov Subspace Methods ....... 126
8.4. Multigrid Algorithm for Elliptic Equations ........... 130
8.5. Multigrid Algorithm as a Preconditioner for
Krylov Subspace Methods .............................. 138
8.6. Newton's and Newton-Krylov Method .................... 139
8.7. A Multigrid Newton-Krylov Algorithm .................. 140
Part II. Solution Approaches
9. Compressible and Preconditioned-Compressible Solvers ....... 147
9.1. Reconstructing the Dependent Variables ............... 147
9.1.1. Riemann Solvers .............................. 148
9.1.2. Basic Predictor-Corrector .................... 152
9.1.3. Characteristic Direct Eulerian ............... 153
9.1.4. Lagrange-Remap Approach ...................... 155
9.2. Reconstructing the Fluxes ............................ 156
9.2.1. Flux Splitting ............................... 157
9.2.2. Flux Splitting Time Integration .............. 158
9.3. Preconditioning for Low Speed Flows .................. 160
9.3.1. Overview of Preconditioning Techniques ....... 160
9.3.2. Preconditioning Choices for Compressible
Flows ........................................ 161
9.3.3. Preconditioning of Numerical Dissipation ..... 167
9.3.4. Differential Preconditioners ................. 169
10. The Artificial Compressibility Method ..................... 173
10.1. Basic Formulation ................................... 173
10.2. Convergence to the Incompressible Limit ............. 174
10.3. Preconditioning and the Artificial
Compressibility Method .............................. 176
10.4. Eigenstructure of the Incompressible Equations ...... 177
10.5. Estimation of the Artificial Compressibility
Parameter ........................................... 180
10.6. Explicit Solvers for Artificial Compressibility ..... 183
10.7. Implicit Solvers for Artificial Compressibility ..... 184
10.7.1. Time-Linearized (Euler) Implicit Scheme ..... 184
10.7.2. Implicit Approximate Factorization
Method ...................................... 185
10.7.3. Implicit Unfactored Method .................. 186
10.8. Extension of the Artificial Compressibility to
Unsteady Flows ....................................... 188
10.9. Boundary Conditions .................................. 190
10.10.Local Time Step ...................................... 191
10.11.Multigrid for the Artificial-Compressibility
Formulation .......................................... 192
10.11.1. Rationale for Three-Grid Multigrid .......... 192
10.11.2. FMG-FAS Algorithm ........................... 193
10.11.3. Remarks on the Full Approximation
Storage (FAS) Procedure ..................... 196
10.11.4. Effects of Pre- and Post-Relaxation on
the Efficiency of FMG-FAS ................... 197
10.11.5. Transfer Operators .......................... 198
10.11.6. Adaptive Multigrid .......................... 201
11.Projection Methods: The Basic Theory and the Exact Pro
jection Method ............................................. 209
11.1. Grids - Variable Positioning ......................... 210
11.2. Continuous Projections for Incompressible Flow ....... 211
11.2.1. Continuous Projections for Constant
Density Incompressible Flow .................. 212
11.2.2. Continuous Projections for Variable
Density Incompressible Flow .................. 213
11.3. Exact Discrete Projections ........................... 213
11.3.1. Cell-Centered Exact Projections .............. 214
11.3.2. Vertex-Centered Exact Projections ............ 217
11.3.3. The MAC Projection ........................... 219
11.3.4. The MAC Projection Used with
Godunov-Type Methods ......................... 220
11.3.5. Other Exact Projections ...................... 223
11.4. Second-Order Projection Algorithms for
Incompressible Flow .................................. 223
11.5. Boundary Conditions .................................. 225
11.5.1. Solvability .................................. 225
11.5.2. Solid Wall Boundary Conditions ............... 227
12.Approximate Projection Methods ............................. 237
12.1. Numerical Issues with Approximate Projection
Methods .............................................. 237
12.2. Projection Algorithms for Incompressible Flow ........ 243
12.3. Analysis of Projection Algorithms .................... 244
12.3.1. Basic Definitions for Analysis ............... 244
12.3.2. Analysis of Approximate Projection
Algorithms ................................... 245
12.3.3. Incremental Velocity Difference Projection ... 247
12.3.4. Pressure Velocity Difference Projection ...... 248
12.3.5. Incremental Velocity Projection .............. 248
12.3.6. Pressure Velocity Projection ................. 249
12.3.7. Discussion of Analysis Results ............... 249
12.4. Pressure Poisson Equation Methods .................... 250
12.4.1. SIMPLE-Type Methods .......................... 251
12.4.2. Implicit High-Resolution Advection ........... 254
12.4.3. Implicit Direct Methods ...................... 255
12.5. Filters .............................................. 256
12.5.1. Classification of Error Modes ................ 256
12.5.2. Projection Filters ........................... 258
12.5.3. Velocity Filters ............................. 263
12.6. Method Demonstration and Verification ................ 271
12.6.1. Vortex-in-a-Box .............................. 271
12.6.2. Inflow with Shear ............................ 272
12.6.3. Doubling Periodic Shear Layer ................ 273
12.6.4. Long Time Integration ........................ 274
12.6.5. Circular Drop Problem ........................ 279
12.6.6. Results Using Various Filters ................ 285
Part III. Modern High-Resolution Methods
13.Introduction to Modern High-Resolution Methods ............. 295
13.1. General Remarks about High-Resolution Methods ........ 295
13.2. The Concept of Nonoscillatory Methods and Total
Variation ............................................ 301
13.3. Monotonicity ......................................... 303
13.4. General Remarks on Riemann Solvers ................... 305
14.High-Resolution Godunov-Туре Methods for Projection Meth
ods ........................................................ 309
14.1. First-Order Algorithm ................................ 309
14.2. High-Resolution Algorithms ........................... 316
14.2.1. Piecewise Linear Methods (PLM) ............... 316
14.2.2. Piecewise Parabolic Methods (PPM) ............ 320
14.2.3. Algorithm Verification Tests ................. 323
14.3. Staggered Grid Spatial Differencing .................. 325
14.4. Unsplit Spatial Differencing ......................... 327
14.4.1. Least Squares Reconstruction ................. 329
14.4.2. Monotone Limiters and Extensions ............. 333
14.4.3. Monotonic Constrained Minimization ........... 334
14.4.4. Divergence-Free Reconstructions .............. 336
14.4.5. Extending Classical TVD Limiters ............. 336
14.5. Multidimensional Results ............................. 340
14.6. Viscous Terms ........................................ 342
14.7. Stability ............................................ 343
15.Centered High-Resolution Methods ........................... 347
15.1. Lax-Friedrichs Scheme ................................ 348
15.2. Lax-Wendroff Scheme .................................. 353
15.3. First-Order Centered Scheme .......................... 358
15.3.1. Random Choice Method ......................... 359
15.3.2. FORCE ........................................ 361
15.3.3. Variants of the FORCE Scheme ................. 363
15.4. Second- and Third-Order Centered Schemes ............. 364
15.4.1. Nessyahu-Tadmor Second-Order Scheme .......... 364
15.4.2. Two-Dimensional Formulation .................. 367
15.4.3. Third-Order Centered Scheme .................. 369
16.Riemann Solvers and TVD Methods in Strict Conservation
Form ....................................................... 373
16.1. The Flux Limiter Approach ............................ 373
16.2. Construction of Flux Limiters ........................ 374
16.2.1.Flux Limiter for the Godunov/Lax-Wendroff
TVD Scheme .................................... 375
16.2.2.Flux Limiter for the Characteristics-
Based/Lax-Friedrichs Scheme ................... 376
16.3. Other Approaches for Constructing Advective
Schemes .............................................. 382
16.3.1. Positive Schemes ............................. 382
16.3.2. Universal Limiter ............................ 384
16.4. The Characteristics-Based Scheme ..................... 384
16.4.1. Introductory Remarks and Basic Formulation ... 384
16.4.2. Dimensional Splitting ........................ 386
16.4.3. Characteristics-Based Reconstruction
in Three Dimensions .......................... 389
16.4.4. Reconstructed Characteristics-Based
Variables in Two Dimensions .................. 392
16.4.5. High-Order Interpolation ..................... 393
16.4.6. Advective Flux Calculation ................... 396
16.4.7. Results ...................................... 397
16.5. Flux Limiting Version of the CB Scheme ............... 404
16.6. Implementation of the Characteristics-Based
Method in Unstructured Grids ......................... 404
16.7. The Weight Average Flux Method ....................... 406
16.7.1. Basic Formulation ............................ 406
16.7.2. TVD Version of the WAF Schemes ............... 408
16.8. Roe's Method ......................................... 409
16.9. Osher's Method ....................................... 412
16.10.Chakravarthy-Osher TVD Scheme ........................ 414
16.11.Harten, Lax and van Leer (HLL) Scheme ................ 416
16.12.HLLC Scheme .......................................... 419
16.13.Estimation of the Wave Speeds for the HLL
and HLLC Riemann Solvers ............................. 420
16.14.HLLE Scheme .......................................... 421
16.15.Comparison of CB and HLLE Schemes .................... 421
16.16."Viscous" TVD Limiters ............................... 424
17.Beyond Second-Order Methods ................................ 429
17.1. General Remarks on High-Order Methods ................ 430
17.2. Essentially Nonoscillatory Schemes (ENO) ............. 433
17.3. ENO Schemes Using Fluxes ............................. 436
17.4. Weighted ENO Schemes ................................. 439
17.4.1. Third-Order WENO Reconstruction .............. 441
17.4.2. Fourth-Order WENO Reconstruction ............. 442
17.5. A Flux-Based Version of the WENO Scheme .............. 444
17.6. Artificial Compression Method for ENO and WENO ....... 447
17.7. The ADER Approach .................................... 448
17.7.1. Linear Scalar Case ........................... 449
17.7.2. Multiple Dimensions: Scalar Case ............. 451
17.7.3. Extension to Nonlinear Hyperbolic Systems .... 453
17.8. Extending and Relaxing Monotonicity in
Godunov-Type Methods ................................. 455
17.8.1. Accuracy and Monotonicity Preserving
Limiters ..................................... 455
17.8.2. Extrema and Monotonicity Preserving
Methods ...................................... 460
17.8.3. Steepened Transport Methods .................. 465
17.9. Discontinuous Galerkin Methods ....................... 467
17.10.Uniformly High-Order Scheme for Godunov-Type
Fluxes ............................................... 469
17.11.Flux-Corrected Transport ............................. 472
17.12.MPDATA ............................................... 475
Part IV. Applications
18.Variable Density Flows and Volume Tracking Methods ......... 479
18.1. Multimaterial Mixing Flows ........................... 479
18.1.1. Shear Flows .................................. 480
18.1.2. Rising Bubbles ............................... 482
18.1.3. Rayleigh-Taylor Instability .................. 483
18.2. Volume Tracking ...................................... 490
18.2.1. Fluid Volume Evolution Equations ............. 492
18.2.2. Basic Features of Volume Tracking
Methods ...................................... 493
18.3. The History of Volume Tracking ....................... 495
18.4. A Geometrically Based Method of Solution ............. 499
18.4.1. A Geometric Toolbox .......................... 500
18.4.2. Reconstructing the Interface ................. 502
18.4.3. Material Volume Fluxes ....................... 510
18.4.4. Time Integration ............................. 513
18.4.5. Translation and Rotation Tests ............... 515
18.5. Results For Vortical Flows ........................... 519
18.5.1. Single Vortex ................................ 521
18.5.2. Deformation Field ............................ 525
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