有限元方法 流体力学 第7版PDF电子书下载

CHAPTER 1 Introduction to the Equations of Fluid Dynamics and the Finite Element Approximation 1

1.1 General remarks and classification of fluid dynamics problems discussed in this book 1

1.2 The governing equations of fluid dynamics 5

1.2.1 Velocity,strain rates,and stresses in fluids 5

1.2.2 Constitutive relations for fluids 6

1.2.3 Mass conservation 7

1.2.4 Momentum conservation:Dynamic equilibrium 7

1.2.5 Energy conservation and equation of state 8

1.2.6 Boundary conditions 10

1.2.7 Navier-Stokes and Euler equations 10

1.3 Inviscid,incompressible flow 12

1.3.1 Velocity potential solution 12

1.4 Incompressible(or nearly incompressible)flows 14

1.5 Numerical solutions:Weak forms,weighted residual,and finite element approximation 15

1.5.1 Strong and weak forms 15

1.5.2 Weighted residual approximation 17

1.5.3 The Galerkin finite element method 18

1.5.4 A finite volume approximation 25

1.6 Concluding remarks 28

References 28

CHAPTER 2 Convection-Dominated Problems:Finite Element Approximations to the Convection-Diffusion-Reaction Equation 31

2.1 Introduction 31

2.2 The steady-state problem in one dimension 34

2.2.1 General remarks 34

2.2.2 Petrov-Galerkin methods for upwinding in one dimension 39

2.2.3 Balancing diffusion in one dimension 43

2.2.4 A variational principle in one dimension 43

2.2.5 Galerkin least-squares approximation(GLS)in one dimension 45

2.2.6 Subgrid scale(SGS)approximation 46

2.2.7 The finite increment calculus(FIC)for stabilizing the convective-diffusion equation in one dimension 47

2.2.8 Higher-order approximations 48

2.3 The steady-state problem in two(or three)dimensions 49

2.3.1 General remarks 49

2.3.2 Streamline(upwind)Petrov-Galerkin weighting(SUPG) 49

2.3.3 Galerkin least squares(GLS)and finite increment calculus(FIC)in multidimensional problems 53

2.4 Steady state:Concluding remarks 54

2.5 Transients:Introductory remarks 54

2.5.1 Mathematical background 54

2.5.2 Possible discretization procedures 55

2.6 Characteristic-based methods 57

2.6.1 Mesh updating and interpolation methods 57

2.6.2 Characteristic-Galerkin procedures 58

2.6.3 A simple explicit characteristic-Galerkin procedure 60

2.6.4 Boundary conditions:Radiation 66

2.7 Taylor-Galerkin procedures for scalar variables 70

2.8 Steady-state condition 71

2.9 Nonlinear waves and shocks 71

2.10 Treatment of pure convection 76

2.11 Boundary conditions for convection-diffusion 78

2.12 Summary and concluding remarks 79

References 80

CHAPTER 3 The Characteristic-Based Split(CBS)Algorithm:A General Procedure for Compressible and Incompressible Flow 87

3.1 Introduction 87

3.2 Nondimensional form of the governing equations 89

3.3 Characteristic-based split(CBS)algorithm 90

3.3.1 The split:General remarks 90

3.3.2 The split:Temporal discretization 91

3.3.3 Spatial discretization and solution procedure 94

3.3.4 Mass diagonalization(lumping) 99

3.4 Explicit,semi-implicit,and nearly implicit forms 100

3.4.1 Fully explicit form 100

3.4.2 Semi-implicit form 100

3.4.3 Quasi-(nearly)implicit form 101

3.4.4 Evaluation of time step limits:Local and global time steps 101

3.5 Artificial compressibility and dual time stepping 103

3.5.1 Artificial compressibility for steady-state problems 103

3.5.2 Artificial compressibility in transient problems(dual time stepping) 104

3.6 "Circumvention"of the Babu？ka-Brezzi(BB)restrictions 106

3.7 A single-step version 107

3.8 Splitting error 109

3.8.1 Elimination of first-order pressure error 110

3.9 Boundary conditions 110

3.9.1 Fictitious boundaries 110

3.9.2 Real boundaries 112

3.9.3 Application of real boundary conditions in the discretization using the CBS algorithm 112

3.10 The performance of two-and single-step algorithms on an inviscid problem 114

3.11 Performance of dual time stepping to remove pressure error 116

3.12 Concluding remarks 118

References 118

CHAPTER 4 Incompressible Newtonian Laminar Flows 127

4.1 Introduction and the basic equations 127

4.2 Use of the CBS algorithm for incompressible flows 129

4.2.1 The fully explicit artificial compressibility form 129

4.2.2 The semi-implicit form 129

4.2.3 Quasi-implicit solution 139

4.3 Adaptive mesh refinement 140

4.3.1 Second gradient(curvature)based refinement 143

4.3.2 Local patch interpolation:Superconvergent values 145

4.3.3 Estimation of second derivatives at nodes 146

4.3.4 Element elongation 146

4.3.5 First derivative(gradient)based refinement 148

4.3.6 Choice of variables 149

4.3.7 An example 149

4.4 Adaptive mesh generation for transient problems 149

4.5 Slow flows:Mixed and penalty formulations 151

4.5.1 Analogy with incompressible elasticity 151

4.5.2 Mixed and penalty discretization 151

4.6 Concluding remarks 153

References 155

CHAPTER 5 Incompressible Non-Newtonian Flows 163

5.1 Introduction 163

5.2 Non-Newtonian flows:Metal and polymer forming 163

5.2.1 Non-Newtonian flows including viscoplasticity and plasticity 163

5.2.2 Steady-state problems of forming 166

5.2.3 Transient problems with changing boundaries 169

5.2.4 Elastic springback and viscoelastic fluids 174

5.3 Viscoelastic flows 177

5.3.1 Governing equations 179

5.4 Direct displacement approach to transient metal forming 185

5.5 Concluding remarks 187

References 188

CHAPTER 6 Free Surface and Buoyancy Driven Flows 195

6.1 Introduction 195

6.2 Free surface flows 195

6.2.1 General remarks 195

6.2.2 Lagrangian method 197

6.2.3 Eulerian methods 200

6.2.4 Arbitrary Langrangian-Eulerian(ALE)method 210

6.3 Buoyancy driven flows 215

6.4 Concluding remarks 218

References 219

CHAPTER 7 Compressible High-Speed Gas Flow 225

7.1 Introduction 225

7.2 The governing equations 226

7.3 Boundary conditions:Subsonic and supersonic flow 227

7.3.1 Euler equation 228

7.3.2 Navier-Stokes equations 229

7.4 Numerical approximations and the CBS algorithm 230

7.5 Shock capture 231

7.5.1 Second derivative-based methods 232

7.5.2 Residual-based methods 233

7.6 Variable smoothing 234

7.7 Some preliminary examples for the Euler equation 234

7.8 Adaptive refinement and shock capture in Euler problems 238

7.8.1 General 238

7.8.2 The h-refinement process and mesh enrichment 243

7.8.3 h-refinement and remeshing in steady-state two-dimensional problems 245

7.9 Three-dimensional inviscid examples in steady state 246

7.9.1 Solution of the flow pattern around a complete aircraft 253

7.9.2 THRUST:The supersonic car 255

7.10 Transient two-and three-dimensional problems 256

7.11 Viscous problems in two dimensions 260

7.11.1 Adaptive refinement in both shock and boundary layer 262

7.11.2 Special adaptive refinement for boundary layers and shocks 264

7.12 Three-dimensional viscous problems 271

7.13 Boundary layer:Inviscid Euler solution coupling 271

7.14 Concluding remarks 273

References 274

CHAPTER 8 Turbulent Flows 283

8.1 Introduction 283

8.1.1 Time averaging 284

8.1.2 Relation between κ,ε,and vT 286

8.2 Treatment of incompressible turbulent flows 286

8.2.1 Reynolds-averaged Navier-Stokes 286

8.2.2 One-equation models 287

8.2.3 Two-equation models 288

8.2.4 Nondimensional form of the governing equations 289

8.2.5 Shortest distance to a solid wall 291

8.2.6 Solution procedure for turbulent flow equations 292

8.3 Treatment of compressible flows 298

8.3.1 Mass-weighted(Favre)time averaging 300

8.4 Large eddy simulation(LES) 303

8.5 Detached eddy simulation(DES)and monotonically integrated LES(MILES) 305

8.6 Direct numerical simulation(DNS) 306

8.7 Concluding remarks 306

References 306

CHAPTER 9 Generalized Flow and Heat Transfer in Porous Media 309

9.1 Introduction 309

9.2 A generalized porous medium flow approach 310

9.2.1 Nondimensional scales 313

9.3 Discretization procedure 315

9.3.1 Semi-and quasi-implicit forms 315

9.4 Forced convection 316

9.5 Natural convection 318

9.5.1 Constant-porosity medium 319

9.6 Concluding remarks 323

References 324

CHAPTER 10 Shallow-Water Problems 327

10.1 Introduction 327

10.2 The basis of the shallow-water equations 328

10.3 Numerical approximation 332

10.4 Examples of application 334

10.4.1 Transient one-dimensional problems:A performance assessment 334

10.4.2 Two-dimensional periodic tidal motions 334

10.4.3 Tsunami waves 339

10.4.4 Steady-state solutions 343

10.5 Drying areas 346

10.6 Shallow-water transport 346

10.7 Concluding remarks 349

References 349

CHAPTER 11 Long and Medium Waves 355

11.1 Introduction and equations 355

11.2 Waves in closed domains:Finite element models 356

11.3 Difficulties in modeling surface waves 358

11.4 Bed friction and other effects 358

11.5 The short-wave problem 359

11.6 Waves in unbounded domains(exterior surface wave problems) 359

11.6.1 Background to wave problems 359

11.6.2 Wave diffraction 360

11.6.3 Incident waves,domain integrals,and nodal values 362

11.7 Unbounded problems 362

11.8 Local nonreflecting boundary conditions(NRBCs) 363

11.8.1 Sponge layers or perfectly matched layers(PMLs) 365

11.9 Infinite elements 366

11.9.1 Mapped periodic(unconjugated)infinite elements 366

11.9.2 Ellipsoidal type infinite elements of Burnett and Holford 368

11.9.3 Wave envelope(or conjugated)infinite elements 369

11.9.4 Accuracy of infinite elements 371

11.9.5 Other applications 371

11.9.6 Trefftz-type infinite elements 372

11.10 Convection and wave refraction 372

11.11 Transient problems 374

11.12 Linking to exterior solutions(or DtN mapping) 375

11.12.1 Linking to boundary integrals 376

11.12.2 Linking to series solutions 376

11.13 Three-dimensional effects in surface waves 377

11.13.1 Large-amplitude water waves 379

11.13.2 Cnoidal and solitary waves 381

11.13.3 Stokes waves 381

11.14 Concluding remarks 383

References 383

CHAPTER 12 Short Waves 389

12.1 Introduction 389

12.2 Background 389

12.3 Errors in wave modeling 391

12.4 Recent developments in short-wave modeling 391

12.5 Transient solution of electromagnetic scattering problems 392

12.6 Finite elements incorporating wave shapes 392

12.6.1 Shape functions using products of polynomials and waves 394

12.6.2 Shape functions using sums of polynomials and waves 397

12.6.3 The discontinuous enrichment method 398

12.6.4 Ultra weak formulation 399

12.6.5 Trefftz-type finite elements for waves 401

12.7 Refraction 404

12.7.1 Wave speed refraction 405

12.7.2 Refraction caused by flows 410

12.8 Spectral finite elements for waves 412

12.9 Discontinuous Galerkin finite elements(DGFE) 414

12.10 Concluding remarks 415

References 417

CHAPTER 13 Fluid-Structure Interaction 423

13.1 Introduction 423

13.2 One-dimensional fluid-structure interaction 424

13.2.1 Equations 424

13.2.2 Characteristic analysis 427

13.2.3 Boundary conditions 429

13.2.4 Solution method:Taylor-Galerkin method 430

13.2.5 Some results 433

13.3 Multidimensional problems 435

13.3.1 Equations and discretization 435

13.3.2 Segregated approach 440

13.3.3 Mesh moving procedures 441

13.4 Concluding remarks 446

References 446

CHAPTER 14 Biofluid Dynamics 451

14.1 Introduction 451

14.2 Flow in human arterial system 451

14.2.1 Heart 452

14.2.2 Reflections 458

14.2.3 Aortic valve 458

14.2.4 Vessel branching 460

14.2.5 Terminal vessels 462

14.2.6 Numerical solution 464

14.3 Image-based subject-specific flow modeling 470

14.3.1 Image segmentation 471

14.3.2 Geometrical potential force(GPF) 471

14.3.3 Numerical solution,initial and boundary conditions 472

14.3.4 Domain discretization 472

14.3.5 Flow solution 473

14.4 Concluding remarks 479

References 479

CHAPTER 15 Computer Implementation of the CBS Algorithm 485

15.1 Introduction 485

15.2 The data input module 486

15.2.1 Mesh data:Nodal coordinates and connectivity 486

15.2.2 Boundary data 486

15.2.3 Other necessary data and flags 487

15.2.4 Preliminary subroutines and checks 487

15.3 Solution module 487

15.3.1 Time step 488

15.3.2 Shock capture 488

15.3.3 CBS algorithm:Steps 489

15.3.4 Boundary conditions 489

15.3.5 Solution of simultaneous equations:Semi-implicit form 490

15.3.6 Different forms of energy equation 490

15.3.7 Convergence to steady state 490

15.4 Output module 490

References 490

APPENDIX A Self-Adjoint Differential Equations 493

APPENDIX B Nonconservative Form of Navier-Stokes Equations 495

APPENDIX C Computing the Drag Force and Stream Function 497

APPENDIX D Convection-Diffusion Equations:Vector-Valued Variables 499

APPENDIX E Integration Formulae 509

APPENDIX F Edge-Based Finite Element Formulation 511

APPENDIX G Boundary Layer-Inviscid Flow Coupling 515

APPENDIX H Multigrid Method 519

APPENDIX I Mass-Weighted Averaged Turbulence Transport Equations 521

Author Index 525

Subject Index 539