Drikakis / Rider High-Resolution Methods for Incompressible and Low-Speed Flows

1. Auflage 2005
ISBN: 978-3-540-26454-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 622 Seiten

Reihe: Computational Fluid and Solid Mechanics

The study of incompressible ?ows is vital to many areas of science and te- nology. This includes most of the ?uid dynamics that one ?nds in everyday life from the ?ow of air in a room to most weather phenomena. Inundertakingthesimulationofincompressible?uid?ows,oneoftentakes many issues for granted. As these ?ows become more realistic, the problems encountered become more vexing from a computational point-of-view. These range from the benign to the profound. At once, one must contend with the basic character of incompressible ?ows where sound waves have been analytically removed from the ?ow. As a consequence vortical ?ows have been analytically 'preconditioned,' but the ?ow has a certain non-physical character (sound waves of in?nite velocity). At low speeds the ?ow will be deterministic and ordered, i.e., laminar. Laminar ?ows are governed by a balance between the inertial and viscous forces in the ?ow that provides the stability. Flows are often characterized by a dimensionless number known as the Reynolds number, which is the ratio of inertial to viscous forces in a ?ow. Laminar ?ows correspond to smaller Reynolds numbers. Even though laminar ?ows are organized in an orderly manner, the ?ows may exhibit instabilities and bifurcation phenomena which may eventually lead to transition and turbulence. Numerical modelling of suchphenomenarequireshighaccuracyandmostimportantlytogaingreater insight into the relationship of the numerical methods with the ?ow physics.

Autoren/Hrsg.

Drikakis, D.

Rider, W.

Inhaltsverzeichnis

1;Preface by Frank Harlow;7
2;Acknowledgements;9
3;Contents;11
4;1. Introduction;19
5;Fundamental Physical and Model Equations;22
5.1;2. The Fluid Flow Equations;23
5.1.1;2.1 Mathematical Preliminaries;23
5.1.2;2.2 Kinematic Considerations;25
5.1.3;2.3 The Equations for Variable Density Flows;26
5.1.4;2.4 Compressible Euler Equations;32
5.1.5;2.5 Low-Mach Number Scaling;36
5.1.6;2.6 Boussinesq Approximation;39
5.1.7;2.7 Variable Density Flow;39
5.1.8;2.8 Zero Mach Number Combustion;40
5.1.9;2.9 Initial and Boundary Conditions;41
5.2;3. The Viscous Fluid Flow Equations;42
5.2.1;3.1 The Stress and Strain Tensors for a Newtonian Fluid;42
5.2.2;3.2 The Navier-Stokes Equations for Constant Density Flows;46
5.2.3;3.3 Non-Newtonian Constitutive Equations for the Shear- Stress Tensor;48
5.2.4;3.4 Alternative Forms of the Advective and Viscous Terms;53
5.2.5;3.5 Nondimensionalization of the Governing Equations;54
5.2.6;3.6 General Remarks on Turbulent Flow Simulations;57
5.2.7;3.7 Reynolds-Averaged Navier-Stokes Equations ( RANS);58
5.2.8;3.8 Large Eddy Simulation (LES);62
5.2.9;3.9 Closing Remarks;64
5.3;4. Curvilinear Coordinates and Transformed Equations;66
5.3.1;4.1 Generalized Curvilinear Coordinates;66
5.3.2;4.2 Calculation of Metrics;70
5.3.3;4.3 Transformation of the Fluid Flow Equations;72
5.3.4;4.4 Viscous Terms;75
5.3.5;4.5 Geometric Conservation Law;78
5.4;5. Overview of Various Formulations and Model Equations;81
5.4.1;5.1 Overview of Various Formulations of the Incompressible Flow Equations;81
5.4.2;5.2 Model Equations;89
5.5;6. Basic Principles in Numerical Analysis;93
5.5.1;6.1 Stability, Consistency and Accuracy;93
5.5.2;6.2 Fourier Analysis;97
5.5.3;6.3 Modified Equation Analysis;104
5.5.4;6.4 Verification via Sample Calculations;108
5.6;7. Time Integration Methods;112
5.6.1;7.1 Time Integration of the Flow Equations;112
5.6.2;7.2 Lax-Wendroff-Type Methods;113
5.6.3;7.3 Other Approaches to Time-Centering;115
5.6.4;7.4 Runge-Kutta Methods;116
5.6.5;7.5 Linear Multi-step Methods;126
5.7;8. Numerical Linear Algebra;133
5.7.1;8.1 Basic Numerical Linear Algebra;133
5.7.2;8.2 Basic Relaxation Methods;135
5.7.3;8.3 Conjugate Gradient and Krylov Subspace Methods;138
5.7.4;8.4 Multigrid Algorithm for Elliptic Equations;142
5.7.5;8.5 Multigrid Algorithm as a Preconditioner for Krylov Subspace Methods;150
5.7.6;8.6 Newton’s and Newton-Krylov Method;151
5.7.7;8.7 A Multigrid Newton-Krylov Algorithm;152
6;Solution Approaches;156
6.1;9. Compressible and Preconditioned- Compressible Solvers;157
6.1.1;9.1 Reconstructing the Dependent Variables;157
6.1.2;9.2 Reconstructing the Fluxes;166
6.1.3;9.3 Preconditioning for Low Speed Flows;170
6.2;10. The Artificial Compressibility Method;182
6.2.1;10.1 Basic Formulation;182
6.2.2;10.2 Convergence to the Incompressible Limit;183
6.2.3;10.3 Preconditioning and the Artificial Compressibility Method;185
6.2.4;10.4 Eigenstructure of the Incompressible Equations;186
6.2.5;10.5 Estimation of the Artificial Compressibility Parameter;189
6.2.6;10.6 Explicit Solvers for Artificial Compressibility;192
6.2.7;10.7 Implicit Solvers for Artificial Compressibility;193
6.2.8;10.8 Extension of the Artificial Compressibility to Unsteady Flows;197
6.2.9;10.9 Boundary Conditions;199
6.2.10;10.10 Local Time Step;200
6.2.11;10.11 Multigrid for the Artificial-Compressibility Formulation;201
6.3;11. Projection Methods: The Basic Theory and the Exact Projection Method;218
6.3.1;11.1 Grids – Variable Positioning;219
6.3.2;11.2 Continuous Projections for Incompressible Flow;220
6.3.3;11.3 Exact Discrete Projections;222
6.3.4;11.4 Second-Order Projection Algorithms for Incompressible Flow;232
6.3.5;11.5 Boundary Conditions;234
6.4;12. Approximate Projection Methods;245
6.4.1;12.1 Numerical Issues with Approximate Projection Methods;245
6.4.2;12.2 Projection Algorithms for Incompressible Flow;251
6.4.3;12.3 Analysis of Projection Algorithms;252
6.4.4;12.4 Pressure Poisson Equation Methods;258
6.4.5;12.5 Filters;264
6.4.6;12.6 Method Demonstration and Verification;279
7;Modern High-Resolution Methods;301
7.1;13. Introduction to Modern High-Resolution Methods;302
7.1.1;13.1 General Remarks about High-Resolution Methods;302
7.1.2;13.2 The Concept of Nonoscillatory Methods and Total Variation;308
7.1.3;13.3 Monotonicity;310
7.1.4;13.4 General Remarks on Riemann Solvers;312
7.2;14. High-Resolution Godunov-Type Methods for Projection Methods;315
7.2.1;14.1 First-Order Algorithm;315
7.2.2;14.2 High-Resolution Algorithms;322
7.2.3;14.3 Staggered Grid Spatial Differencing;331
7.2.4;14.4 Unsplit Spatial Differencing;333
7.2.5;14.5 Multidimensional Results;346
7.2.6;14.6 Viscous Terms;348
7.2.7;14.7 Stability;349
7.3;15. Centered High-Resolution Methods;352
7.3.1;15.1 Lax-Friedrichs Scheme;353
7.3.2;15.2 Lax-Wendroff Scheme;358
7.3.3;15.3 First-Order Centered Scheme;363
7.3.4;15.4 Second- and Third-Order Centered Schemes;369
7.4;16. Riemann Solvers and TVD Methods in Strict Conservation Form;378
7.4.1;16.1 The Flux Limiter Approach;378
7.4.2;16.2 Construction of Flux Limiters;379
7.4.3;16.3 Other Approaches for Constructing Advective Schemes;387
7.4.4;16.4 The Characteristics-Based Scheme;389
7.4.5;16.5 Flux Limiting Version of the CB Scheme;409
7.4.6;16.6 Implementation of the Characteristics-Based Method in Unstructured Grids;409
7.4.7;16.7 The Weight Average Flux Method;411
7.4.8;16.8 Roe’s Method;414
7.4.9;16.9 Osher’s Method;417
7.4.10;16.10 Chakravarthy-Osher TVD Scheme;419
7.4.11;16.11 Harten, Lax and van Leer (HLL) Scheme;421
7.4.12;16.12 HLLC Scheme;424
7.4.13;16.13 Estimation of the Wave Speeds for the HLL and HLLC Riemann Solvers;425
7.4.14;16.14 HLLE Scheme;426
7.4.15;16.15 Comparison of CB and HLLE Schemes;426
7.4.16;16.16 Viscous” TVD Limiters;429
7.5;17. Beyond Second-Order Methods;434
7.5.1;17.1 General Remarks on High-Order Methods;435
7.5.2;17.2 Essentially Nonoscillatory Schemes (ENO);438
7.5.3;17.3 ENO Schemes Using Fluxes;441
7.5.4;17.4 Weighted ENO Schemes;444
7.5.5;17.5 A Flux-Based Version of the WENO Scheme;449
7.5.6;17.6 Artificial Compression Method for ENO and WENO;452
7.5.7;17.7 The ADER Approach;453
7.5.8;17.8 Extending and Relaxing Monotonicity in Godunov- Type Methods;460
7.5.9;17.9 Discontinuous Galerkin Methods;472
7.5.10;17.10 Uniformly High-Order Scheme for Godunov- Type Fluxes;474
7.5.11;17.11 Flux-Corrected Transport;477
7.5.12;17.12 MPDATA;480
8;Applications;482
8.1;18. Variable Density Flows and Volume Tracking Methods;483
8.1.1;18.1 Multimaterial Mixing Flows;483
8.1.2;18.2 Volume Tracking;494
8.1.3;18.3 The History of Volume Tracking;499
8.1.4;18.4 A Geometrically Based Method of Solution;503
8.1.5;18.5 Results For Vortical Flows;523
8.2;19. High-Resolution Methods and Turbulent Flow Computation;533
8.2.1;19.1 Physical Considerations;533
8.2.2;19.2 Survey of Theory and Models;537
8.2.3;19.3 Relation of High-Resolution Methods and Flow Physics;540
8.2.4;19.4 Large Eddy Simulation: Standard and Implicit;543
8.2.5;19.5 Numerical Analysis of Subgrid Models;547
8.2.6;19.6 ILES Analysis;548
8.2.7;19.7 Computational Examples;556
9;A. MATHEMATICA Commands for Numerical Analysis;560
9.1;A.1 Fourier Analysis for First-Order Upwind Methods;560
9.2;A.2 Fourier Analysis for Second-Order Upwind Methods;561
9.3;A.3 Modified Equation Analysis for First-Order Upwind;562
10;B. Example Computer Implementations;566
10.1;B.1 Appendix: Fortran Subroutine for the Characteristics- Based Flux;566
10.2;B.2 Fifth-Order Weighted ENO Method;571
11;C. Acknowledgements: Illustrations Reproduced with Permission;577
12;References;579
13;Index;616