Mathematical Models of Fluid Dynamics

Mathematical Models of Fluid Dynamics......Page 5
Contents......Page 9
Preface to the Second Edition......Page 13
Preface to the First Edition......Page 15
1.1 Modeling by Euler’s Equations......Page 19
1.2 Characteristics and Singularities......Page 28
1.3 Potential Flows and (Dynamic) Buoyancy......Page 32
1.4 Motionless Fluids and Sound Propagation......Page 47
2.1 Generalization of What Will Be Called a Solution......Page 51
2.2 Traffic Flow Example with Loss of Uniqueness......Page 55
2.3 The Rankine–Hugoniot Condition......Page 60
3.1 Entropy in the Case of an Ideal Fluid......Page 67
3.2 Generalization of the Entropy Condition......Page 71
3.3 Uniqueness of Entropy Solutions......Page 77
3.4 Kruzkov’s Ansatz......Page 87
4.1 Numerical Importance of the Riemann Problem......Page 91
4.2 The Riemann Problem for Linear Systems......Page 93
4.3 The Aw–Rascle Traffic Flow Model......Page 95
5.1 The Navier–Stokes Equations Model......Page 97
5.2 Drag Force and the Hagen–Poiseuille Law......Page 103
5.3 Stokes Approximation and Artificial Time......Page 108
5.4 Foundations of the Boundary Layer Theory and Flow Separation......Page 113
5.5 Stability of Laminar Flows......Page 120
5.6 Heated Real Gas Flows......Page 122
5.7 Tunnel Fires......Page 124
6.1 Some Historical Remarks......Page 131
6.2 Reduction to Properties of Operator Sequences......Page 132
6.3 Convergence Theorems......Page 135
6.4 Example......Page 138
7.1 Some General Remarks......Page 145
7.2 Finite Difference Calculus......Page 149
7.3 The CFL Condition......Page 153
7.4 Lax–Richtmyer Theory......Page 154
7.5 The von Neumann Stability Criterion......Page 159
7.6 The Modified Equation......Page 162
7.7 Difference Schemes in Conservation Form......Page 164
7.8 The Finite Volume Method on Unstructured Grids......Page 166
7.9 Continuous Convergence of Relations......Page 169
8.1 The Viscosity Form......Page 173
8.2 The Incremental Form......Page 174
8.3 Relations......Page 176
8.4 Godunov Is Just Good Enough......Page 177
8.5 The Lax–Friedrichs Scheme......Page 182
8.6 A Glimpse of Gas Dynamics......Page 186
8.7 Elementary Waves......Page 189
8.8 The Complete Solution to the Riemann Problem......Page 196
8.9 The Godunov Scheme in Gas Dynamics......Page 202
9.1 Mappings......Page 205
9.2 Transformation Relations......Page 208
9.3 Metric Tensors......Page 210
9.4 Transforming Conservation Laws......Page 211
9.5 Good Practice......Page 214
9.6 Remarks Concerning Adaptation......Page 221
10.1 Difference Methods on Unstructured Grids......Page 223
10.2 Order of Accuracy and Basic Discretization......Page 226
10.3 Higher-Order Finite Volume Schemes......Page 227
10.4 Polynomial Recovery......Page 229
10.5 Remarks Concerning Non-polynomial Recovery......Page 234
10.6 Remarks Concerning Grid Generation......Page 236
Index......Page 239
Suggested Reading......Page 245