Generalized Riemann Problems in Computational Fluid Dynamics

Ben-Artzi M., Falcovitz J.0511066759, 0521772966

This monograph gives a systematic presentation of the GRP methodology, starting from the underlying mathematical principles, through basic scheme analysis and scheme extensions (such as reacting flow or two-dimensional flows involving moving or stationary boundaries). An array of instructive examples illustrates the range of applications, extending from (simple) scalar equations to computational fluid dynamics. Background material from mathematical analysis and fluid dynamics is provided, making the book accessible to both researchers and graduate students of applied mathematics, science, and engineering.

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Table of contents :
Half-title……Page 3
Series-title……Page 5
Title……Page 7
Copyright……Page 8
Contents……Page 9
List of Figures……Page 13
Preface……Page 17
1 Introduction……Page 19
Part I Basic Theory……Page 23
2.1 Theoretical Background……Page 25
Weak Solutions and Jump Conditions……Page 29
Shocks, Rarefaction Waves, and Entropy……Page 35
The Riemann Problem……Page 41
2.2 Basic Concepts of Numerical Approximation……Page 43
Convergence……Page 46
3.1 From Godunov to the GRP Method……Page 54
3.2.1 The Linear Conservation Law……Page 67
First-Order Schemes……Page 68
Second-Order Schemes……Page 70
3.2.2 The Burgers Nonlinear Conservation Law……Page 73
First-Order Computation……Page 74
Second-Order Computation……Page 77
3.3 2-D Sample Problems……Page 81
The Operator-Splitting Method……Page 83
The Linear Conservation Law……Page 84
The Nonlinear Burgers Equation……Page 87
Case A……Page 88
Case C……Page 92
Case D……Page 93
The Guckenheimer Equation……Page 94
4.1 Nonlinear Hyperbolic Systems in One Space Dimension……Page 99
Characteristic Curves and Centered Rarefaction Waves……Page 100
Weak Solutions and Jump Discontinuities……Page 105
Entropy Conditions, Shock Waves, and Contact Discontinuities……Page 109
The Riemann Problem……Page 116
4.2 Euler Equations of Quasi-1-D, Compressible, Inviscid Flow……Page 119
The Flow Equations……Page 120
Eigenvalues and Characteristic Equations……Page 122
Weak Solutions and Jump Conditions……Page 128
Lagrangian Coordinates……Page 131
Shock Waves – Detailed Study of the Jump Condition……Page 134
Centered Rarefaction Waves……Page 140
The Riemann Problem (RP) for Planar Flows……Page 144
Perfect (Gamma-Law) Gas……Page 148
5.1 The GRP for Quasi-1-D, Compressible, Inviscid Flow……Page 153
Structure of the Solution to the GRP……Page 154
The Linear GRP in Lagrangian Coordinates – Setup and Statement of the Main Theorem……Page 157
The Acoustic Case……Page 161
Resolution of a CRW in the Lagrangian Framework……Page 164
Explicit Formulas for the (Lagrangian) GRP in the Gamma-Law Case……Page 170
Concluding the Treatment of the CRW……Page 171
Time Derivatives of p, u on the Interface – Proof of the Main Theorem……Page 172
Conclusion of the Linear GRP in the Lagrangian Case……Page 177
The Linear GRP in the Eulerian Framework……Page 180
5.2 The GRP Numerical Method for Quasi-1-D, Compressible, Inviscid Flow……Page 187
The Godunov Scheme……Page 189
The Basic GRP Scheme……Page 191
The E and L Schemes, Intermediate Schemes, and MUSCL……Page 198
Updating the Slopes……Page 199
Concluding the GRP Algorithm……Page 200
6.1 The Shock Tube Problem……Page 202
6.2 Wave Interactions……Page 207
6.2.1 Shock–Contact Interaction……Page 210
6.2.2 Shock–Shock Interaction……Page 213
6.2.3 Shock–CRW Interaction……Page 221
6.2.4 CRW–Contact Interaction……Page 225
Approximate Analysis of the Interaction……Page 228
Numerical (GRP) Solution……Page 231
6.3 Spherically Converging Flow of Cold Gas……Page 236
6.4 The Flow Induced by an Expanding Sphere……Page 237
6.5 Converging–Diverging Nozzle Flow……Page 240
Nozzle Geometry and Steady Flow……Page 242
The Finite-Difference Solution……Page 248
Part II Numerical Implementation……Page 251
7.1 General Discussion……Page 253
7.2 Strang’s Operator-Splitting Method……Page 255
7.3 Two-Dimensional Flow in Cartesian Coordinates……Page 262
The Linear GRP for a Planar System with Advection……Page 263
The Split Scheme for (7.16)……Page 265
The Split Scheme and Conservation Form……Page 267
8.1 Grids That Move in Time……Page 269
8.2 Singularity Tracking……Page 270
8.3 Moving Boundary Tracking (MBT)……Page 273
8.3.1 Basic Setup……Page 275
8.3.2 Survey of the Full MBT Algorithm……Page 282
8.3.3 An Example: Shock Lifting of an Elliptic Disk……Page 284
9 A Physical Extension: Reacting Flow……Page 287
9.1 The Equations of Compressible Reacting Flow……Page 289
The Characteristic Relations……Page 292
Discontinuities and Centered Rarefaction Waves……Page 293
9.2 The Chapman–Jouguet (C–J) Model……Page 294
9.3 The Z–N–D (Zeldovich–von Neumann–Döring) Solution……Page 299
The Associated Riemann Problem……Page 304
Structure of the Solution to the Linear GRP–The Main Theorem……Page 306
The Acoustic Approximation……Page 309
Resolution of the Centered Rarefaction Wave……Page 310
Conclusion of the Linear GRP……Page 314
The Gamma-Law Case……Page 315
9.5 The GRP Scheme for Reacting Flow……Page 316
10 Wave Interaction in a Duct – A Comparative Study……Page 323
Appendix A Entropy Conditions for Scalar Conservation Laws……Page 331
Appendix B Convergence of the Godunov Scheme……Page 338
Appendix C Riemann Solver for a Gamma-Law Gas……Page 348
Appendix D The MUSCL Scheme……Page 351
Bibliography……Page 355
Riemann and Generalized Riemann Problem……Page 363
Functional Spaces……Page 364
Index……Page 365