Riemann solvers and numerical methods for fluid dynamics : a practical introduction ; with 26 tables

cover.jpg......Page 1 Front Matter......Page 2 References......Page 0 Preface......Page 4 Acknowledgements......Page 7 Preface to the Second Edition......Page 9 Table of Contents......Page 10 1. The Equations of Fluid Dynamics......Page 19 1.1 The Euler Equations......Page 20 1.1.1 Conservation-Law Form......Page 21 1.1.2 Other Compact Forms......Page 22 1.2.1 Units of Measure......Page 23 1.2.2 Equations of State (EOS)......Page 24 1.2.3 Other Variables and Relations......Page 25 1.2.4 Ideal Gases......Page 29 1.2.5 Covolume and van der Waal Gases......Page 31 1.3 Viscous Stresses......Page 33 1.4 Heat Conduction......Page 35 1.5 Integral Form of the Equations......Page 36 1.5.1 Time Derivatives......Page 37 1.5.2 Conservation of Mass......Page 38 1.5.3 Conservation of Momentum......Page 39 1.5.4 Conservation of Energy......Page 41 1.6.1 Summary of the Equations......Page 43 1.6.2 Flow with Area Variation......Page 45 1.6.3 Axi-Symmetric Flows......Page 46 1.6.5 Plain One-Dimensional Flow......Page 47 1.6.6 Steady Compressible Flow......Page 50 1.6.8 Free-Surface Gravity Flow......Page 51 1.6.9 The Shallow Water Equations......Page 53 1.6.10 Incompressible Viscous Flow......Page 56 1.6.11 The Artificial Compressibility Equations......Page 57 2.1 Quasi-Linear Equations: Basic Concepts......Page 59 2.2.1 Characteristics and the General Solution......Page 65 2.2.2 The Riemann Problem......Page 67 2.3 Linear Hyperbolic Systems......Page 68 2.3.1 Diagonalisation and Characteristic Variables......Page 69 2.3.2 The General Initial-Value Problem......Page 70 2.3.3 The Riemann Problem......Page 73 2.3.4 The Riemann Problem for Linearised Gas Dynamics......Page 76 2.3.5 Some Useful Definitions......Page 77 2.4 Conservation Laws......Page 78 2.4.1 Integral Forms of Conservation Laws......Page 79 2.4.2 Non-Linearities and Shock Formation......Page 83 2.4.3 Characteristic Fields......Page 94 2.4.4 Elementary-Wave Solutions of the Riemann Problem......Page 101 3.1.1 Conservative Formulation......Page 104 3.1.2 Non-Conservative Formulations......Page 108 3.1.3 Elementary Wave Solutions of the Riemann Problem......Page 111 3.2 Multi-Dimensional Euler Equations......Page 119 3.2.1 Two-Dimensional Equations in Conservative Form......Page 120 3.2.2 Three-Dimensional Equations in Conservative Form......Page 124 3.2.3 Three-Dimensional Primitive Variable Formulation......Page 125 3.2.4 The Split Three-Dimensional Riemann Problem......Page 127 3.3 Conservative versus Non-Conservative Formulations......Page 128 4. The Riemann Problem for the Euler Equations......Page 131 4.1 Solution Strategy......Page 132 4.2 Equations for Pressure and Particle Velocity......Page 135 4.2.1 Function f_L for a Left Shock......Page 136 4.2.2 Function f_L for Left Rarefaction......Page 138 4.2.3 Function f_R for a Right Shock......Page 139 4.2.4 Function f_R for a Right Rarefaction......Page 140 4.3.1 Behaviour of the Pressure Function......Page 141 4.3.2 Iterative Scheme for Finding the Pressure......Page 143 4.3.3 Numerical Tests......Page 145 4.4 The Complete Solution......Page 149 4.5 Sampling the Solution......Page 152 4.5.2 Right Side of Contact: S = x/t > u_*......Page 153 4.6 The Riemann Problem in the Presence of Vacuum......Page 154 4.6.1 Case 1: Vacuum Right State......Page 156 4.6.2 Case 2: Vacuum Left State......Page 157 4.6.3 Case 3: Generation of Vacuum......Page 158 4.7 The Riemann Problem for Covolume Gases......Page 159 4.7.1 Solution for Pressure and Particle Velocity......Page 160 4.7.3 The Complete Solution......Page 163 4.7.4 Solution inside Rarefactions......Page 164 4.8 The Split Multi-Dimensional Case......Page 165 4.9 FORTRAN Program for Exact Riemann Solver......Page 167 5.1 Discretisation: Introductory Concepts......Page 179 5.1.1 Approximation to Derivatives......Page 180 5.1.2 Finite Difference Approximation to a PDE......Page 181 5.2 Selected Difference Schemes......Page 183 5.2.1 The First Order Upwind Scheme......Page 184 5.2.2 Other Well-Known Schemes......Page 188 5.3 Conservative Methods......Page 190 5.3.1 Basic Definitions......Page 191 5.3.2 Godunov's First-Order Upwind Method......Page 193 5.3.3 Godunov's Method for Burgers's Equation......Page 196 5.3.4 Conservative Form of Difference Schemes......Page 200 5.4 Upwind Schemes for Linear Systems......Page 203 5.4.1 The CIR Scheme......Page 204 5.4.2 Godunov's Method......Page 206 5.5.1 Linear Advection......Page 209 5.5.2 The Inviscid Burgers Equation......Page 211 5.6 FORTRAN Program for Godunov's Method......Page 212 6.1 Bases of Godunov's Method......Page 229 6.2 The Godunov Scheme......Page 232 6.3 Godunov's Method for the Euler Equations......Page 234 6.3.1 Evaluation of the Intercell Fluxes......Page 235 6.3.2 Time Step Size......Page 237 6.3.3 Boundary Conditions......Page 238 6.4 Numerical Results and Discussion......Page 241 6.4.1 Numerical Results for Godunov's Method......Page 242 6.4.2 Numerical Results from Other Methods......Page 244 7.1 Introduction......Page 252 7.2 RCM on a Non-Staggered Grid......Page 253 7.2.1 The Scheme for Non-Linear Systems......Page 254 7.2.2 Boundary Conditions and the Time Step Size......Page 258 7.3.1 Review of the Lax-Friedrichs Scheme......Page 259 7.3.2 The Scheme......Page 260 7.4 The RCM on a Staggered Grid......Page 261 7.4.1 The Scheme for Non-Linear Systems......Page 262 7.4.2 A Deterministic First-Order Centred Scheme (FORCE)......Page 263 7.4.3 Analysis of the FORCE Scheme......Page 264 7.5.1 Van der Corput Pseudo-Random Numbers......Page 265 7.5.2 Statistical Properties......Page 267 7.5.3 Propagation of a Single Shock......Page 268 7.6 Numerical Results......Page 270 7.7 Concluding Remarks......Page 272 8.1 Introduction......Page 280 8.2.1 Upwind Differencing......Page 281 8.2.2 The FVS Approach......Page 283 8.3 FVS for the Isothermal Equations......Page 285 8.3.1 Split Fluxes......Page 286 8.3.2 FVS Numerical Schemes......Page 287 8.4.1 Recalling the Equations......Page 288 8.4.2 The Steger-Warming Splitting......Page 290 8.4.3 The van Leer Splitting......Page 291 8.4.4 The Liou-Steffen Scheme......Page 293 8.5.2 Results for Test 1......Page 295 8.5.4 Results for Test 3......Page 296 8.5.6 Results for Test 5......Page 297 9.1 Introduction......Page 307 9.2 The Riemann Problem and the Godunov Flux......Page 308 9.2.2 Sonic Rarefactions......Page 310 9.3 Primitive Variable Riemann Solvers (PVRS)......Page 311 9.4.1 A Two-Rarefaction Riemann Solver (TRRS)......Page 315 9.4.2 A Two-Shock Riemann Solver (TSRS)......Page 317 9.5.1 An Adaptive Iterative Riemann Solver (AIRS)......Page 318 9.5.2 An Adaptive Noniterative Riemann Solver (ANRS)......Page 319 9.6 Numerical Results......Page 320 10. The HLL and HLLC Riemann Solvers......Page 328 10.1 The Riemann Problem and the Godunov Flux......Page 329 10.2 The Riemann Problem and Integral Relations......Page 330 10.3 The HLL Approximate Riemann Solver......Page 332 10.4 The HLLC Approximate Riemann Solver......Page 334 10.5 Wave-Speed Estimates......Page 336 10.5.1 Direct Wave Speed Estimates......Page 337 10.5.2 Pressure-Velocity Based Wave Speed Estimates......Page 338 10.6.2 The Rusanov Flux......Page 340 10.7 Contact Waves and Passive Scalars......Page 341 10.8 Numerical Results......Page 342 10.9 Closing Remarks and Extensions......Page 344 11. The Riemann Solver of Roe......Page 353 11.1.1 The Exact Riemann Problem and the Godunov Flux......Page 354 11.1.2 Approximate Conservation Laws......Page 355 11.1.3 The Approximate Riemann Problem and the Intercell Flux......Page 357 11.2 The Original Roe Method......Page 359 11.2.1 The Isothermal Equations......Page 360 11.2.2 The Euler Equations......Page 362 11.3.1 The Approach......Page 366 11.3.2 The Isothermal Equations......Page 367 11.3.3 The Euler Equations......Page 371 11.4.1 The Entropy Problem......Page 374 11.4.2 The Harten-Hyman Entropy Fix......Page 375 11.4.3 The Speeds u_* , a_* L, a_* R......Page 378 11.5.1 The Tests......Page 379 11.5.2 The Results......Page 380 11.6 Extensions......Page 381 12. The Riemann Solver of Osher......Page 385 12.1.1 Mathematical Bases......Page 386 12.1.2 Osher's Numerical Flux......Page 388 12.1.3 Osher's Flux for the Single-Wave Case......Page 389 12.1.4 Osher's Flux for the Inviscid Burgers Equation......Page 391 12.1.5 Osher's Flux for the General Case......Page 392 12.2 Osher's Flux for the Isothermal Equations......Page 393 12.2.1 Osher's Flux with P-Ordering......Page 394 12.2.2 Osher's Flux with O-Ordering......Page 397 12.3 Osher's Scheme for the Euler Equations......Page 400 12.3.1 Osher's Flux with P-Ordering......Page 401 12.3.2 Osher's Flux with O-Ordering......Page 403 12.3.3 Remarks on Path Orderings......Page 408 12.3.4 The Split Three-Dimensional Case......Page 411 12.4 Numerical Results and Discussion......Page 412 12.5 Extensions......Page 413 13.1 Introduction......Page 420 13.2.1 Selected Schemes......Page 422 13.2.2 Accuracy......Page 424 13.2.3 Stability......Page 425 13.3.1 The Basic WAF Scheme......Page 427 13.3.2 Generalisations of the WAF Scheme......Page 430 13.4.1 Data Reconstruction......Page 433 13.4.2 The MUSCL-Hancock Method (MHM)......Page 436 13.4.3 The Piece-Wise Linear Method (PLM)......Page 439 13.4.4 The Generalised Riemann Problem (GRP) Method......Page 441 13.4.5 Slope-Limiter Centred (SLIC) Schemes......Page 443 13.4.7 Semi-Discrete Schemes......Page 446 13.4.8 Implicit Methods......Page 447 13.5.1 Monotone Schemes......Page 448 13.5.2 A Motivating Example......Page 451 13.5.3 Monotone Schemes and Godunov's Theorem......Page 453 13.5.4 Spurious Oscillations and High Resolution......Page 456 13.5.5 Data Compatibility......Page 457 14.1 Introduction......Page 459 14.2 CFL and Boundary Conditions......Page 460 14.3.1 The Original Version of WAF......Page 462 14.3.2 A Weighted Average State Version......Page 464 14.3.3 Rarefactions in State Riemann Solvers......Page 465 14.3.4 TVD Version of WAF Schemes......Page 467 14.3.6 Summary of the WAF Method......Page 469 14.4.1 The Basic Scheme......Page 470 14.4.2 A Variant of the Scheme......Page 472 14.4.3 TVD Version of the Scheme......Page 473 14.4.4 Summary of the MUSCL-Hancock Method......Page 476 14.5.1 Review of the FORCE Flux......Page 477 14.5.2 A Flux Limiter Centred (FLIC) Scheme......Page 478 14.5.3 A Slope Limiter Centred (SLIC) Scheme......Page 480 14.6.1 Formulation of the Equations and Primitive Schemes......Page 481 14.6.2 A WAF-Type Primitive Variable Scheme......Page 483 14.6.3 A MUSCL-Hancock Primitive Scheme......Page 486 14.6.4 Adaptive Primitive-Conservative Schemes......Page 488 14.7.1 Upwind TVD Methods......Page 489 14.7.2 Centred TVD Methods......Page 490 15.1 Introduction......Page 496 15.2 Splitting for a Model Equation......Page 497 15.3.1 Model Equations......Page 500 15.3.2 Schemes for Systems......Page 501 15.4.1 First-Order Systems of ODEs......Page 502 15.4.2 Numerical Methods......Page 504 15.4.3 Implementation Details for Split Schemes......Page 505 15.5 Concluding Remarks......Page 506 16.1 Introduction......Page 508 16.2.1 Splitting for a Model Problem......Page 509 16.2.2 Splitting Schemes for Two-Dimensional Systems......Page 510 16.2.3 Splitting Schemes for Three-Dimensional Systems......Page 512 16.3.1 Handling the Sweeps by a Single Subroutine......Page 514 16.3.3 The Intercell Flux and the TVD Condition......Page 516 16.4.1 Introductory Concepts......Page 520 16.4.2 Accuracy and Stability of Multidimensional Schemes......Page 523 16.5 A MUSCL-Hancock Finite Volume Scheme......Page 526 16.6.1 Two-Dimensional Linear Advection......Page 528 16.6.2 Three-Dimensional Linear Advection......Page 532 16.6.3 Schemes for Two-Dimensional Nonlinear Systems......Page 535 16.6.4 Schemes for Three-Dimensional Nonlinear Systems......Page 538 16.7.1 Introduction......Page 539 16.7.2 General Domains and Coordinate Transformation......Page 540 16.7.3 The Finite Volume Method for Non-Cartesian Domains......Page 542 17.1 Explosions and Implosions......Page 549 17.1.1 Explosion Test in Two-Space Dimensions......Page 550 17.1.2 Implosion Test in Two Dimensions......Page 553 17.1.3 Explosion Test in Three Space Dimensions......Page 554 17.2 Shock Wave Reflection from Wedges......Page 555 17.2.1 Mach Number M_s = 1.7 and Phi= 25 Degrees......Page 556 17.2.2 Mach Number M_s = 1.2 and Phi = 30 Degrees......Page 559 18.1 Summary......Page 561 18.2.2 Steady Supersonic Euler Equations......Page 562 18.2.5 Compressible Materials......Page 563 18.2.8 Magnetohydrodynamics (MHD)......Page 564 18.3 NUMERICA......Page 565 References......Page 567 B......Page 588 C......Page 589 D......Page 591 F......Page 592 G......Page 593 I......Page 594 L......Page 596 N......Page 597 P......Page 598 R......Page 599 S......Page 600 T......Page 602 V......Page 603 W......Page 604 Z......Page 605 "High resolution upwind and centered methods are today a mature generation of computational techniques applicable to a wide range of engineering and scientific disciplines. Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. The book is designed to provide readers with an understanding of the basic concepts, some of the underlying theory, the ability to critically use the current research papers on the subject, and, above all, with the required information for the practical implementation of the methods. Applications include compressible, steady, unsteady, reactive, viscous, non-viscous and free surface flows."--Jacket This is the eBook version of the printed book. If the print book includes a CD-ROM, this content is not included within the eBook version. 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