Finite Element Method for Fluid Dynamics (eBook)

Front Cover 1
The Finite Element Method for Fluid Dynamics 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgements 14
Chapter 1. Introduction to the equations of fluid dynamics and the finite element approximation 16
1.1 General remarks and classification of fluid dynamics problems discussed in this book 16
1.2 The governing equations of fluid dynamics 19
1.3 Inviscid, incompressible flow 26
1.4 Incompressible (or nearly incompressible) flows 28
1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 29
1.6 Concluding remarks 41
References 42
Chapter 2. Convection dominated problems– finite element approximations to the convection–diffusion-reaction equation 43
2.1 Introduction 43
2.2 The steady-state problem in one dimension 46
2.3 The steady-state problem in two (or three) dimensions 60
2.4 Steady state – concluding remarks 64
2.5 Transients – introductory remarks 65
2.6 Characteristic-based methods 68
2.7 Taylor–Galerkin procedures for scalar variables 80
2.8 Steady-state condition 81
2.9 Non-linear waves and shocks 81
2.10 Treatment of pure convection 85
2.11 Boundary conditions for convection–diffusion 87
2.12 Summary and concluding remarks 88
References 89
Chapter 3. The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow 94
3.1 Introduction 94
3.2 Non-dimensional form of the governing equations 96
3.3 Characteristic-based split (CBS) algorithm 97
3.4 Explicit, semi-implicit and nearly implicit forms 107
3.5 Artificial compressibility and dual time stepping 110
3.6 'Circumvention' of the Babuška-Brezzi (BB)restrictions 112
3.7 A single-step version 113
3.8 Boundary conditions 115
3.9 The performance of two-step and one-step algorithms on an inviscid problem 118
3.10 Concluding remarks 119
References 120
Chapter 4. Incompressible Newtonian laminar flows 125
4.1 Introduction and the basic equations 125
4.2 Use of the CBS algorithm for incompressible flows 127
4.3 Adaptive mesh refinement 138
4.4 Adaptive mesh generation for transient problems 146
4.5 Slow flows – mixed and penalty formulations 146
4.6 Concluding remarks 151
References 151
Chapter 5. Incompressible non-Newtonian flows 156
5.1 Introduction 156
5.2 Non-Newtonian flows – metal and polymer forming 156
5.3 Viscoelastic flows 169
5.4 Direct displacement approach to transient metal forming 178
5.5 Concluding remarks 180
References 181
Chapter 6. Free surface and buoyancy driven flows 185
6.1 Introduction 185
6.2 Free surface flows 185
6.3 Buoyancy driven flows 204
6.4 Concluding remarks 206
References 208
Chapter 7. Compressible high-speed gas flow 212
7.1 Introduction 212
7.2 The governing equations 213
7.3 Boundary conditions – subsonic and supersonic flow 214
7.4 Numerical approximations and the CBS algorithm 217
7.5 Shock capture 218
7.6 Variable smoothing 220
7.7 Some preliminary examples for the Euler equation 221
7.8 Adaptive refinement and shock capture in Euler problems 227
7.9 Three-dimensional inviscid examples in steady state 232
7.10 Transient two- and three-dimensional problems 241
7.11 Viscous problems in two dimensions 242
7.12 Three-dimensional viscous problems 255
7.13 Boundary layer–inviscid Euler solution coupling 256
7.14 Concluding remarks 257
References 257
Chapter 8. Turbulent flows 263
8.1 Introduction 263
8.2 Treatment of incompressible turbulent flows 266
8.3 Treatment of compressible flows 279
8.4 Large eddy simulation 282
8.5 Detached Eddy Simulation (DES) 285
8.6 Direct Numerical Simulation (DNS) 285
8.7 Concluding remarks 286
References 286
Chapter 9. Generalized flow through porous media 289
9.1 Introduction 289
9.2 A generalized porous medium flow approach 290
9.3 Discretization procedure 294
9.4 Non-isothermal flows 297
9.5 Forced convection 297
9.6 Natural convection 299
9.7 Concluding remarks 303
References 304
Chapter 10. Shallow water problems 307
10.1 Introduction 307
10.2 The basis of the shallow water equations 308
10.3 Numerical approximation 312
10.4 Examples of application 313
10.5 Drying areas 325
10.6 Shallow water transport 326
10.7 Concluding remarks 328
References 329
Chapter 11. Long and medium waves 332
11.1 Introduction and equations 332
11.2 Waves in closed domains – finite element models 333
11.3 Difficulties in modelling surface waves 335
11.4 Bed friction and other effects 335
11.5 The short-wave problem 335
11.6 Waves in unbounded domains (exterior surface wave problems) 336
11.7 Unbounded problems 339
11.8 Local Non-Reflecting Boundary Conditions (NRBCs) 339
11.9 Infinite elements 342
11.10 Mapped periodic (unconjugated) infinite elements 342
11.11 Ellipsoidal type infinite elements of Burnett and Holford 343
11.12 Wave envelope (or conjugated) infinite elements 345
11.13 Accuracy of infinite elements 347
11.14 Trefftz type infinite elements 347
11.15 Convection and wave refraction 348
11.16 Transient problems 350
11.17 Linking to exterior solutions (or DtN mapping) 351
11.18 Three-dimensional effects in surface waves 353
11.19 Concluding remarks 359
References 359
Chapter 12. Shortwaves 364
12.1 Introduction 364
12.2 Background 364
12.3 Errors in wave modelling 366
12.4 Recent developments in short wave modelling 366
12.5 Transient solution of electromagnetic scattering problems 367
12.6 Finite elements incorporating wave shapes 367
12.7 Refraction 379
12.8 Spectral finite elements for waves 387
12.9 Discontinuous Galerkin finite elements (DGFE) 389
12.10 Concluding remarks 393
References 393
Chapter 13. Computer implementation of the CBS algorithm 397
13.1 Introduction 397
13.2 The data input module 398
13.3 Solution module 399
13.4 Output module 402
References 402
Appendix A. Non-conservative form of Navier–Stokes equations 404
Appendix B. Self-adjoint differential equations 406
Appendix C. Postprocessing 407
Appendix D. Integration formulae 410
Appendix E. Convection-diffusion equations: vector-valued variables 412
Appendix F. Edge-based finite element formulation 420
Appendix G. Multigrid method 422
Appendix H. Boundary layer–inviscid flow coupling 424
Appendix I. Mass-weighted averaged turbulence transport equations 428
Author index 432
Subject index 442
Color Plate Section 452