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Dimitris Drikakis
William Rider
High Resolution Methods
for Incompressible and
Low Speed Flows
W ith 480 Figures a nd 32 Tables
4y Springer
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Contents
1.
I n t r o d u c t i o n
P a r t I . F u n d a m e n t a l P h y s i c a l a n d M o d e l E q u a t i o n s
2 .
T h e F l u i d F l o w E q u a t i o n s
7
2.1 M athem atical Preliminaries 7
2.2 Kine matic Con siderations 9
2.3 T he Eq ua tion s for Variable De nsity Flows 10
2.3.1 The Con tinuity Eq uatio n 10
2.3.2 The M om entum Eq uation s 11
2.3.3 The Energy Eq uatio n 14
2.4 Com pressible Euler Eq uatio ns 16
2.5 Low-Mach Nu m ber Scaling 20
2.6 Boussinesq A ppro xim ation 23
2.7 Variable Den sity Flow 23
2.8 Zero Mach Nu mbe r Com bustion 24
2.9 Init ial and Bo und ary Cond it ions 25
3 . T h e V i s c o u s F l u i d F l o w E q u a t i o n s
27
3.1 Th e Stress an d Strain Tensors for a Ne wto nian Fluid 27
3.2 Th e Navier-Stokes Eq uatio ns for Co nsta nt Density Flows . . . . 31
3.3 Non -New tonian Co nsti tutive Eq uation s for the Shear-Stress
Tensor 33
3.3.1 Generalized N ew tonian Fluid s 33
3.3.2 Visco elastic Flu ids 34
3.3.3 O th er Viscoelastic M odels 37
3.4 A lterna tive Forms of th e Adv ective and Viscous Term s 38
3.5 No ndim ensionalization of th e Gov erning Eq ua tion s 39
3.6 General Rem arks on Turbu lent Flow Simulations 42
3.7 Reyno lds-Averaged Navier-Stokes Eq uat ion s (RAN S) 43
3.8 Large Ed dy Simu lation (LES) 47
3.9 Closing Re m arks 49
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XIV Contents
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
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ontents XV
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
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XVI Contents
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. Project ion Methods: The Basic Theory and the Exact Pro
ject ion 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 A lgorithms for Incompressible Flow 223
11.5 Boundary Conditions 225
11.5.1 Solvability 225
11.5.2 Solid Wall Boundary Conditions 227
12. Approximate Project ion 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 Approxim ate 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
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Contents XVII
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 Tim e Integration 274
12.6.5 Circular Drop Problem 279
12.6.6 Results Using Various Filters 285
Part III . Modern High Resolution Methods
13 . Introduct ion to Modern High Resolut ion Methods
295
13.1 General Rem arks about High-Resolution Methods 295
13.2 The Concept of Nonoscillatory Methods and Total Variation . 301
13.3 Monotonicity 303
13.4 General Rem arks on Riem ann Solvers 305
14. High Resolut ion Go dunov Type M ethods for Project ion M eth
ods 309
14.1 First-O rder Algorithm 309
14.2 High-Resolution Algorithms 316
14.2.1 Piecewise Linear Methods (PLM ) 316
14.2.2 Piecewise Parabo lic 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 Extend ing Classical TVD Lim iters 336
14.5 Multidimensional Results 340
14.6 Viscous Terms 342
14.7 Stability 343
15. Centered High Resolut ion M ethods 347
15.1 Lax-Friedrichs Scheme 348
15.2 Lax-Wendroff Scheme 353
15.3 First-O rder 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 Th ird-O rder Centered Schemes 364
15.4.1 Nessyahu-Tadm or Second-Order Scheme 364
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XVIII Contents
15.4.2 Two-Dimensional Form ulation 367
15.4.3 Th ird-O rder 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 Lim iters 374
16.2.1 Flux Limiter for th e G odunov/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 Lim iter 384
16.4 The Characteristics-Based Scheme 384
16.4.1 Introductory Rem arks and Basic Formulation 384
16.4.2 Dimensional Sp litting 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 Un-
structu red Grids 404
16.7 The Weight Average Flux Method 406
16.7.1 Basic Form ulation 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 Estim ation of the W ave Speeds for the HLL and HLLC Rie-
mann Solvers : 420
16.14 HLLE Scheme 421
16.15 Com parison of CB and HLLE Schemes 421
16.16 Viscous TVD Limiters 424
17. Beyond Second Order M ethod s
429
17.1 General Rem arks on High-Order Methods 430
17.2 Essentially Nonoscillatory Schemes (ENO ) 433
17.3 ENO Schemes Using Fluxes 436
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Contents XIX
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 Meth-
ods 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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XX Contents
19.
High Resolution Methods and Turbulent
Flow Computation
529
19.1 Physical Considerations 529
19.2 Survey of Theory and Models 533
19.3 Relation of High-Resolution Methods and Flow Physics 536
19.3.1 Num erical Considerations 537
19.3.2 Relation of High-Resolution Methods to Weak Solu-
tions and Turbulence 538
19.4 Large Eddy Simulation: Standard and Implicit 539
19.5 Num erical Analysis of Subgrid Models 543
19.6 ILES Analysis 544
19.6.1 Explicit Modeling 544
19.6.2 Implicit Modeling 546
19.6.3 Lim iters 547
19.6.4 Energy Analysis 549
19.7 Computational Examples 552
19.7.1 Burgers' Turbulence (Burgulence) 552
19.7.2 Convective Planeta ry Boundary Layer 553
A. M ATH EM ATIC A Com mands for Num erical Analysis
557
A.I Fourier Analysis for First-O rder Upwind Methods 557
A.2 Fourier Analysis for Second-Order Upwind Methods 558
A.3 Modified Equation Analysis for First-O rder Upwind 559
B . Exam ple Com puter Imp lementations
563
B.I Appendix: Fortran Subroutine for the Characteristics-Based
Flux 563
B.2 Fifth-Order Weighted ENO Method 568
B.2.1 Subroutine for Fifth-Order WENO 568
B.2.2 Subroutine for Fifth-Order W EN O's Third-Order Based
Fluxes 570
B.2.3 Subroutine Fifth-Order WENO Smoothness Sensors . . 571
B.2.4 Subroutine Fifth-Order WENO Weights 572
C. Acknowledgements: Il lustrations Repro duced with Perm is
sion
575
References
577
Index
615
