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CHAPTER 1 Introduction to the Equations of Fluid Dynamics and the Finite Element Approximation1

1.1 General remarks and classification of fluid dynamics problems discussed in this book1

1.2 The governing equations of fluid dynamics5

1.2.1 Velocity,strain rates,and stresses in fluids5

1.2.2 Constitutive relations for fluids6

1.2.3 Mass conservation7

1.2.4 Momentum conservation:Dynamic equilibrium7

1.2.5 Energy conservation and equation of state8

1.2.6 Boundary conditions10

1.2.7 Navier-Stokes and Euler equations10

1.3 Inviscid,incompressible flow12

1.3.1 Velocity potential solution12

1.4 Incompressible(or nearly incompressible)flows14

1.5 Numerical solutions:Weak forms,weighted residual,and finite element approximation15

1.5.1 Strong and weak forms15

1.5.2 Weighted residual approximation17

1.5.3 The Galerkin finite element method18

1.5.4 A finite volume approximation25

1.6 Concluding remarks28

References28

CHAPTER 2 Convection-Dominated Problems:Finite Element Approximations to the Convection-Diffusion-Reaction Equation31

2.1 Introduction31

2.2 The steady-state problem in one dimension34

2.2.1 General remarks34

2.2.2 Petrov-Galerkin methods for upwinding in one dimension39

2.2.3 Balancing diffusion in one dimension43

2.2.4 A variational principle in one dimension43

2.2.5 Galerkin least-squares approximation(GLS)in one dimension45

2.2.6 Subgrid scale(SGS)approximation46

2.2.7 The finite increment calculus(FIC)for stabilizing the convective-diffusion equation in one dimension47

2.2.8 Higher-order approximations48

2.3 The steady-state problem in two(or three)dimensions49

2.3.1 General remarks49

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

2.3.3 Galerkin least squares(GLS)and finite increment calculus(FIC)in multidimensional problems53

2.4 Steady state:Concluding remarks54

2.5 Transients:Introductory remarks54

2.5.1 Mathematical background54

2.5.2 Possible discretization procedures55

2.6 Characteristic-based methods57

2.6.1 Mesh updating and interpolation methods57

2.6.2 Characteristic-Galerkin procedures58

2.6.3 A simple explicit characteristic-Galerkin procedure60

2.6.4 Boundary conditions:Radiation66

2.7 Taylor-Galerkin procedures for scalar variables70

2.8 Steady-state condition71

2.9 Nonlinear waves and shocks71

2.10 Treatment of pure convection76

2.11 Boundary conditions for convection-diffusion78

2.12 Summary and concluding remarks79

References80

CHAPTER 3 The Characteristic-Based Split(CBS)Algorithm:A General Procedure for Compressible and Incompressible Flow87

3.1 Introduction87

3.2 Nondimensional form of the governing equations89

3.3 Characteristic-based split(CBS)algorithm90

3.3.1 The split:General remarks90

3.3.2 The split:Temporal discretization91

3.3.3 Spatial discretization and solution procedure94

3.3.4 Mass diagonalization(lumping)99

3.4 Explicit,semi-implicit,and nearly implicit forms100

3.4.1 Fully explicit form100

3.4.2 Semi-implicit form100

3.4.3 Quasi-(nearly)implicit form101

3.4.4 Evaluation of time step limits:Local and global time steps101

3.5 Artificial compressibility and dual time stepping103

3.5.1 Artificial compressibility for steady-state problems103

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

3.6 "Circumvention"of the Babu？ka-Brezzi(BB)restrictions106

3.7 A single-step version107

3.8 Splitting error109

3.8.1 Elimination of first-order pressure error110

3.9 Boundary conditions110

3.9.1 Fictitious boundaries110

3.9.2 Real boundaries112

3.9.3 Application of real boundary conditions in the discretization using the CBS algorithm112

3.10 The performance of two-and single-step algorithms on an inviscid problem114

3.11 Performance of dual time stepping to remove pressure error116

3.12 Concluding remarks118

References118

CHAPTER 4 Incompressible Newtonian Laminar Flows127

4.1 Introduction and the basic equations127

4.2 Use of the CBS algorithm for incompressible flows129

4.2.1 The fully explicit artificial compressibility form129

4.2.2 The semi-implicit form129

4.2.3 Quasi-implicit solution139

4.3 Adaptive mesh refinement140

4.3.1 Second gradient(curvature)based refinement143

4.3.2 Local patch interpolation:Superconvergent values145

4.3.3 Estimation of second derivatives at nodes146

4.3.4 Element elongation146

4.3.5 First derivative(gradient)based refinement148

4.3.6 Choice of variables149

4.3.7 An example149

4.4 Adaptive mesh generation for transient problems149

4.5 Slow flows:Mixed and penalty formulations151

4.5.1 Analogy with incompressible elasticity151

4.5.2 Mixed and penalty discretization151

4.6 Concluding remarks153

References155

CHAPTER 5 Incompressible Non-Newtonian Flows163

5.1 Introduction163

5.2 Non-Newtonian flows:Metal and polymer forming163

5.2.1 Non-Newtonian flows including viscoplasticity and plasticity163

5.2.2 Steady-state problems of forming166

5.2.3 Transient problems with changing boundaries169

5.2.4 Elastic springback and viscoelastic fluids174

5.3 Viscoelastic flows177

5.3.1 Governing equations179

5.4 Direct displacement approach to transient metal forming185

5.5 Concluding remarks187

References188

CHAPTER 6 Free Surface and Buoyancy Driven Flows195

6.1 Introduction195

6.2 Free surface flows195

6.2.1 General remarks195

6.2.2 Lagrangian method197

6.2.3 Eulerian methods200

6.2.4 Arbitrary Langrangian-Eulerian(ALE)method210

6.3 Buoyancy driven flows215

6.4 Concluding remarks218

References219

CHAPTER 7 Compressible High-Speed Gas Flow225

7.1 Introduction225

7.2 The governing equations226

7.3 Boundary conditions:Subsonic and supersonic flow227

7.3.1 Euler equation228

7.3.2 Navier-Stokes equations229

7.4 Numerical approximations and the CBS algorithm230

7.5 Shock capture231

7.5.1 Second derivative-based methods232

7.5.2 Residual-based methods233

7.6 Variable smoothing234

7.7 Some preliminary examples for the Euler equation234

7.8 Adaptive refinement and shock capture in Euler problems238

7.8.1 General238

7.8.2 The h-refinement process and mesh enrichment243

7.8.3 h-refinement and remeshing in steady-state two-dimensional problems245

7.9 Three-dimensional inviscid examples in steady state246

7.9.1 Solution of the flow pattern around a complete aircraft253

7.9.2 THRUST:The supersonic car255

7.10 Transient two-and three-dimensional problems256

7.11 Viscous problems in two dimensions260

7.11.1 Adaptive refinement in both shock and boundary layer262

7.11.2 Special adaptive refinement for boundary layers and shocks264

7.12 Three-dimensional viscous problems271

7.13 Boundary layer:Inviscid Euler solution coupling271

7.14 Concluding remarks273

References274

CHAPTER 8 Turbulent Flows283

8.1 Introduction283

8.1.1 Time averaging284

8.1.2 Relation between κ,ε,and vT286

8.2 Treatment of incompressible turbulent flows286

8.2.1 Reynolds-averaged Navier-Stokes286

8.2.2 One-equation models287

8.2.3 Two-equation models288

8.2.4 Nondimensional form of the governing equations289

8.2.5 Shortest distance to a solid wall291

8.2.6 Solution procedure for turbulent flow equations292

8.3 Treatment of compressible flows298

8.3.1 Mass-weighted(Favre)time averaging300

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 remarks306

References306

CHAPTER 9 Generalized Flow and Heat Transfer in Porous Media309

9.1 Introduction309

9.2 A generalized porous medium flow approach310

9.2.1 Nondimensional scales313

9.3 Discretization procedure315

9.3.1 Semi-and quasi-implicit forms315

9.4 Forced convection316

9.5 Natural convection318

9.5.1 Constant-porosity medium319

9.6 Concluding remarks323

References324

CHAPTER 10 Shallow-Water Problems327

10.1 Introduction327

10.2 The basis of the shallow-water equations328

10.3 Numerical approximation332

10.4 Examples of application334

10.4.1 Transient one-dimensional problems:A performance assessment334

10.4.2 Two-dimensional periodic tidal motions334

10.4.3 Tsunami waves339

10.4.4 Steady-state solutions343

10.5 Drying areas346

10.6 Shallow-water transport346

10.7 Concluding remarks349

References349

CHAPTER 11 Long and Medium Waves355

11.1 Introduction and equations355

11.2 Waves in closed domains:Finite element models356

11.3 Difficulties in modeling surface waves358

11.4 Bed friction and other effects358

11.5 The short-wave problem359

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

11.6.1 Background to wave problems359

11.6.2 Wave diffraction360

11.6.3 Incident waves,domain integrals,and nodal values362

11.7 Unbounded problems362

11.8 Local nonreflecting boundary conditions(NRBCs)363

11.8.1 Sponge layers or perfectly matched layers(PMLs)365

11.9 Infinite elements366

11.9.1 Mapped periodic(unconjugated)infinite elements366

11.9.2 Ellipsoidal type infinite elements of Burnett and Holford368

11.9.3 Wave envelope(or conjugated)infinite elements369

11.9.4 Accuracy of infinite elements371

11.9.5 Other applications371

11.9.6 Trefftz-type infinite elements372

11.10 Convection and wave refraction372

11.11 Transient problems374

11.12 Linking to exterior solutions(or DtN mapping)375

11.12.1 Linking to boundary integrals376

11.12.2 Linking to series solutions376

11.13 Three-dimensional effects in surface waves377

11.13.1 Large-amplitude water waves379

11.13.2 Cnoidal and solitary waves381

11.13.3 Stokes waves381

11.14 Concluding remarks383

References383

CHAPTER 12 Short Waves389

12.1 Introduction389

12.2 Background389

12.3 Errors in wave modeling391

12.4 Recent developments in short-wave modeling391

12.5 Transient solution of electromagnetic scattering problems392

12.6 Finite elements incorporating wave shapes392

12.6.1 Shape functions using products of polynomials and waves394

12.6.2 Shape functions using sums of polynomials and waves397

12.6.3 The discontinuous enrichment method398

12.6.4 Ultra weak formulation399

12.6.5 Trefftz-type finite elements for waves401

12.7 Refraction404

12.7.1 Wave speed refraction405

12.7.2 Refraction caused by flows410

12.8 Spectral finite elements for waves412

12.9 Discontinuous Galerkin finite elements(DGFE)414

12.10 Concluding remarks415

References417

CHAPTER 13 Fluid-Structure Interaction423

13.1 Introduction423

13.2 One-dimensional fluid-structure interaction424

13.2.1 Equations424

13.2.2 Characteristic analysis427

13.2.3 Boundary conditions429

13.2.4 Solution method:Taylor-Galerkin method430

13.2.5 Some results433

13.3 Multidimensional problems435

13.3.1 Equations and discretization435

13.3.2 Segregated approach440

13.3.3 Mesh moving procedures441

13.4 Concluding remarks446

References446

CHAPTER 14 Biofluid Dynamics451

14.1 Introduction451

14.2 Flow in human arterial system451

14.2.1 Heart452

14.2.2 Reflections458

14.2.3 Aortic valve458

14.2.4 Vessel branching460

14.2.5 Terminal vessels462

14.2.6 Numerical solution464

14.3 Image-based subject-specific flow modeling470

14.3.1 Image segmentation471

14.3.2 Geometrical potential force(GPF)471

14.3.3 Numerical solution,initial and boundary conditions472

14.3.4 Domain discretization472

14.3.5 Flow solution473

14.4 Concluding remarks479

References479

CHAPTER 15 Computer Implementation of the CBS Algorithm485

15.1 Introduction485

15.2 The data input module486

15.2.1 Mesh data:Nodal coordinates and connectivity486

15.2.2 Boundary data486

15.2.3 Other necessary data and flags487

15.2.4 Preliminary subroutines and checks487

15.3 Solution module487

15.3.1 Time step488

15.3.2 Shock capture488

15.3.3 CBS algorithm:Steps489

15.3.4 Boundary conditions489

15.3.5 Solution of simultaneous equations:Semi-implicit form490

15.3.6 Different forms of energy equation490

15.3.7 Convergence to steady state490

15.4 Output module490

References490

APPENDIX A Self-Adjoint Differential Equations493

APPENDIX B Nonconservative Form of Navier-Stokes Equations495

APPENDIX C Computing the Drag Force and Stream Function497

APPENDIX D Convection-Diffusion Equations:Vector-Valued Variables499

APPENDIX E Integration Formulae509

APPENDIX F Edge-Based Finite Element Formulation511

APPENDIX G Boundary Layer-Inviscid Flow Coupling515

APPENDIX H Multigrid Method519

APPENDIX I Mass-Weighted Averaged Turbulence Transport Equations521

Author Index525

Subject Index539