Wave Propagation in Fluids......Page 5
Table of Contents......Page 7
Introduction......Page 17
1.1.1. Hyperbolic scalar conservation laws......Page 21
1.1.2. Derivation from general conservation principles......Page 23
1.1.3. Non-conservation form......Page 26
1.1.4. Characteristic form – Riemann invariants......Page 27
1.2.1. Representation in the phase space......Page 29
1.2.2. Initial conditions, boundary conditions......Page 32
1.3.1. Physical context – conservation form......Page 34
1.3.2. Characteristic form......Page 36
1.3.3. Example: movement of a contaminant in a river......Page 37
1.4.1. Physical context – conservation form......Page 41
1.4.2. Characteristic form......Page 43
1.4.3. Example: propagation of a perturbation in a fluid......Page 44
1.5.1. Physical context – conservation form......Page 48
1.5.2. Non-conservation and characteristic forms......Page 49
1.5.3. Expression of the celerity......Page 51
1.5.4. Specific case: flow in a rectangular channel......Page 54
1.5.5. Summary......Page 55
1.6.1. Physical context – conservation form......Page 56
1.6.2. Characteristic form......Page 59
1.6.3. Example: decontamination of an aquifer......Page 60
1.7.1. Physical context – conservation form......Page 62
1.7.2. Characteristic form......Page 65
1.7.3. Summary......Page 67
1.8.2.1. Exercise 1.1: the inviscid Burgers equation......Page 68
1.8.2.2. Exercise 1.2: the kinematic wave equation......Page 69
1.8.2.3. Exercise 1.3: the kinematic wave equation......Page 70
1.8.2.4. Exercise 1.4: the Buckley-Leverett equation......Page 71
1.8.2.5. Exercise 1.5: linear advection with adsorption-desorption......Page 72
2.1.1. Hyperbolic systems of conservation laws......Page 75
2.1.2. Hyperbolic systems of conservation laws – examples......Page 77
2.1.3. Characteristic form – Riemann invariants......Page 79
2.2.1. Domain of influence, domain of dependence......Page 82
2.2.2. Existence and uniqueness of solutions – initial and boundary conditions......Page 84
2.3.2. Conservation form......Page 85
2.3.3. Characteristic form......Page 88
2.3.4. Physical interpretation......Page 90
2.4.1. Physical context – hypotheses......Page 91
2.4.2.2. Continuity equation......Page 93
2.4.2.4. Simplification – vector form......Page 97
2.4.3. Characteristic form – Riemann invariants......Page 98
2.4.4.1. Treatment of internal points......Page 102
2.4.4.2. Treatment of boundary conditions......Page 104
2.4.4.3. Bergeron’s graphical method......Page 106
2.5.1. Physical context – hypotheses......Page 107
2.5.2.1. Notation......Page 108
2.5.2.2. Continuity equation......Page 109
2.5.2.3. Momentum equation......Page 110
2.5.3.1. Non-conservation form......Page 114
2.5.3.2. Characteristic form......Page 117
2.5.3.3. Expression of the speed of the waves in still water......Page 118
2.5.3.4. Riemann invariants......Page 121
2.5.4.1. The various possible flow regimes......Page 125
2.5.4.2. Treatment of internal points......Page 127
2.5.4.3. Treatment of boundary conditions......Page 129
2.5.4.4. Boundary conditions for a rectangular channel......Page 130
2.6.1. Physical context – hypotheses......Page 132
2.6.2.1. Definitions – notation......Page 134
2.6.2.2. Continuity equation......Page 135
2.6.2.3. Momentum equation......Page 136
2.6.2.4. Energy equation......Page 137
2.6.3. Characteristic form – Riemann invariants......Page 138
2.6.4.1. The various possible flow regimes......Page 142
2.6.4.2. Treatment of internal points......Page 144
2.6.4.3. Treatment of boundary points......Page 145
2.6.5. Summary......Page 146
2.7.1. What you should remember......Page 147
2.7.2.1. Exercise 2.1: the water hammer equations......Page 148
2.7.2.2. Exercise 2.2: the water hammer equations......Page 149
2.7.2.3. Exercise 2.3: the water hammer equations......Page 150
2.7.2.5. Exercise 2.5: the Saint Venant equations......Page 151
2.7.2.6. Exercise 2.6: the Saint Venant equations......Page 152
2.7.2.7. Exercise 2.7: the Euler equations......Page 153
3.1.1. Governing mechanisms......Page 155
3.1.2. Local invalidity of the characteristic formulation – graphical approach......Page 158
3.1.3.1. Free surface flow: the breaking of a wave......Page 160
3.1.3.2. Aerodynamics: supersonic flight......Page 161
3.2.1. Shock wave......Page 163
3.2.2. Rarefaction wave......Page 164
3.2.4. Mixed/compound wave......Page 165
3.3.1. Definition and properties......Page 166
3.3.2. Generalized Riemann invariants......Page 167
3.4.1. Definitions......Page 169
3.4.3. Jump relationships......Page 170
3.4.4.1. Example 1: the inviscid Burgers equation......Page 172
3.4.4.2. Example 2: the hydraulic jump......Page 174
3.4.5. The entropy condition......Page 177
3.4.6. Irreversibility......Page 179
3.4.7. Approximations for the jump relationships......Page 180
3.5.1. What you should remember......Page 181
3.5.2.2. Exercise 3.2: the kinematic wave equation......Page 182
3.5.2.4. Exercise 3.4: the Saint Venant equations......Page 183
3.5.2.5. Exercise 3.5: the Euler equations......Page 184
4.1.1. The Riemann problem......Page 185
4.1.2. The generalized Riemann problem......Page 186
4.2.1. The linear advection equation......Page 187
4.2.2. The inviscid Burgers equation......Page 188
4.2.3. The Buckley-Leverett equation......Page 190
4.3.1. General principle......Page 195
4.3.2. Application to the water hammer problem: sudden valve failure......Page 196
4.3.3.1. Introduction......Page 199
4.3.3.2. Wave pattern......Page 200
4.3.3.3. Calculation of the solution......Page 201
4.3.3.4. A specific case: dambreak on a dry bed......Page 204
4.3.4.2. Wave pattern......Page 206
4.3.4.3. Calculation of the solution......Page 209
4.4.1. What you should remember......Page 212
4.4.2.1. Exercise 4.1: the Saint Venant equations......Page 213
4.4.2.2. Exercise 4.2: the Euler equations......Page 214
5.1.1. Scalar laws......Page 215
5.1.2. Two-dimensional hyperbolic systems......Page 217
5.1.3. Three-dimensional hyperbolic systems......Page 219
5.2. Derivation from conservation principles......Page 220
5.3.1.1. The bicharacteristic approach......Page 223
5.3.1.2. The secant plane approach......Page 226
5.3.1.3. Domain of influence, domain of dependence......Page 228
5.3.2. Three-dimensional hyperbolic systems......Page 230
5.4.1.1. Physical context – hypotheses......Page 231
5.4.1.2. Continuity equation......Page 233
5.4.1.3. Equation for the momentum in the x-direction......Page 234
5.4.1.5. Vector form......Page 236
5.4.2.1. Characteristic surfaces......Page 237
5.4.2.2. Derivation of the Riemann invariants......Page 240
5.4.3.1. Domain of influence, domain of dependence......Page 242
5.4.3.2. Calculation of the solution......Page 244
5.5.2.1. Exercise 5.1: the Doppler effect......Page 245
5.5.2.2. Exercise 5.2: visual assessment of the Mach number......Page 246
6.1.1. Discretization for one-dimensional problems......Page 249
6.1.2. Multidimensional discretization......Page 250
6.1.3. Explicit schemes, implicit schemes......Page 251
6.2.1.1. Principle of the method......Page 252
6.2.1.2. Interpolation at the foot of the characteristic: first-order formula......Page 254
6.2.1.3. Interpolation at the foot of the characteristic: second-order formula......Page 258
6.2.1.4. Estimation of the source term......Page 259
6.2.1.5. Treatment of boundary conditions......Page 260
6.2.2.1. Principle of the method......Page 261
6.2.2.2. Application example: the water hammer equations......Page 263
6.2.3.1. The linear advection equation......Page 266
6.2.3.2. The inviscid Burgers equation......Page 268
6.3.1. The explicit upwind scheme (non-conservation version)......Page 270
6.3.2. The implicit upwind scheme (non-conservation version)......Page 272
6.3.3. Conservative versions of the implicit upwind scheme......Page 273
6.3.4. Application examples......Page 275
6.4.1. Formulation......Page 277
6.4.2. Estimation of nonlinear terms – algorithmic aspects......Page 280
6.4.3. Numerical applications......Page 281
6.5.1. The Crank-Nicholson scheme......Page 287
6.5.2. Centered schemes with Runge-Kutta time stepping......Page 288
6.6.1. Definitions......Page 290
6.6.2. General formulation of TVD schemes......Page 291
6.6.3. Harten’s and Sweby’s criteria......Page 294
6.6.4. Traditional limiters......Page 296
6.6.5. Calculation example......Page 297
6.7.1. Principle of the approach......Page 300
6.7.2.1. Explicit upwind scheme for the water hammer equations......Page 303
6.7.2.2. TVD scheme for the water hammer equations......Page 305
6.7.2.3. Calculation example......Page 307
6.8.1. Motivation and principle of the approach......Page 309
6.8.2.1. Roe’s method......Page 310
6.8.2.2. Expression in the base of eigenvectors......Page 312
6.9.1. Explicit alternate directions......Page 313
6.9.2. The ADI method......Page 316
6.9.3. Multidimensional schemes......Page 318
6.10.1. What you should remember......Page 319
6.10.2.3. Exercise 6.3: finite difference methods for hyperbolic systems......Page 321
7.1.1. One-dimensional conservation laws......Page 323
7.1.2. Multidimensional conservation laws......Page 325
7.1.3. Application to the two-dimensional shallow water equations......Page 328
7.2.1. Principle......Page 330
7.2.2.1. Discretization......Page 331
7.2.2.2. Flux calculation at internal interfaces......Page 332
7.2.2.3. Boundary conditions......Page 333
7.2.2.4. Calculation of the liquid discharge at the cell interfaces......Page 334
7.2.2.5. Algorithm......Page 335
7.2.3.2. Flux calculation at internal interfaces......Page 336
7.2.3.3. Boundary conditions......Page 338
7.2.4.1. Discretization......Page 339
7.2.4.2. Flux calculation at internal interfaces......Page 341
7.2.4.3. Treatment of boundary conditions......Page 342
7.3.1.1. Historical perspective......Page 344
7.3.1.2. Reconstruction of the flow variable......Page 345
7.3.1.3. Slope limiting......Page 346
7.3.1.5. Flux calculation and balance......Page 347
7.3.2.2. Slope limiting......Page 348
7.3.2.3. Solution of the generalized Riemann problem......Page 349
7.4.1. What you should remember......Page 350
7.4.2. Suggested exercises......Page 351
A.1. Definitions......Page 353
A.2.2. Multiplication by a scalar......Page 355
A.2.4. Determinant of a matrix......Page 356
A.3.1. Differentiation......Page 357
A.4.1. Definitions......Page 358
A.4.2. Example......Page 359
B.1.2. Principle of a consistency analysis......Page 361
B.1.3. Numerical diffusion, numerical dispersion......Page 363
B.2.1. Definition......Page 365
B.2.2. Principle of a stability analysis......Page 366
B.2.3.1. The linear advection equation......Page 368
B.2.3.2. The diffusion equation......Page 369
B.2.3.3. The advection-dispersion equation......Page 371
B.2.4. Harmonic analysis of numerical solutions......Page 372
B.2.5. Amplitude and phase portraits......Page 375
B.2.6. Extension to systems of equations......Page 377
B.3.2. Lax’s theorem......Page 379
C.1.1. HLL solver......Page 381
C.1.2. HLLC solver......Page 383
C.2. Roe’s solver......Page 386
Appendix D. Summary of the Formulae......Page 389
References......Page 395
Index......Page 399
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