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

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

1.2 The governing equations of fluid dynamics

1.2.1 Velocity,strain rates,and stresses in fluids

1.2.2 Constitutive relations for fluids

1.2.3 Mass conservation

1.2.4 Momentum conservation:Dynamic equilibrium

1.2.5 Energy conservation and equation of state

1.2.6 Boundary conditions

1.2.7 Navier-Stokes and Euler equations

1.3 Inviscid,incompressible flow

1.3.1 Velocity potential solution

1.4 Incompressible(or nearly incompressible)flows

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

1.5.1 Strong and weak forms

1.5.2 Weighted residual approximation

1.5.3 The Galerkin finite element method

1.5.4 A finite volume approximation

1.6 Concluding remarks

References

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

2.1 Introduction

2.2 The steady-state problem in one dimension

2.2.1 General remarks

2.2.2 Petrov-Galerkin methods for upwinding in one dimension

2.2.3 Balancing diffusion in one dimension

2.2.4 A variational principle in one dimension

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

2.2.6 Subgrid scale(SGS)approximation

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

2.2.8 Higher-order approximations

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

2.3.1 General remarks

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

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

2.4 Steady state:Concluding remarks

2.5 Transients:Introductory remarks

2.5.1 Mathematical background

2.5.2 Possible discretization procedures

2.6 Characteristic-based methods

2.6.1 Mesh updating and interpolation methods

2.6.2 Characteristic-Galerkin procedures

2.6.3 A simple explicit characteristic-Galerkin procedure

2.6.4 Boundary conditions:Radiation

2.7 Taylor-Galerkin procedures for scalar variables

2.8 Steady-state condition

2.9 Nonlinear waves and shocks

2.10 Treatment of pure convection

2.11 Boundary conditions for convection-diffusion

2.12 Summary and concluding remarks

References

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

3.1 Introduction

3.2 Nondimensional form of the governing equations

3.3 Characteristic-based split(CBS)algorithm

3.3.1 The split:General remarks

3.3.2 The split:Temporal discretization

3.3.3 Spatial discretization and solution procedure

3.3.4 Mass diagonalization(lumping)

3.4 Explicit,semi-implicit,and nearly implicit forms

3.4.1 Fully explicit form

3.4.2 Semi-implicit form

3.4.3 Quasi-(nearly)implicit form

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

3.5 Artificial compressibility and dual time stepping

3.5.1 Artificial compressibility for steady-state problems

3.5.2 Artificial compressibility in transient problems(dual time stepping)

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

3.7 A single-step version

3.8 Splitting error

3.8.1 Elimination of first-order pressure error

3.9 Boundary conditions

3.9.1 Fictitious boundaries

3.9.2 Real boundaries

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

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

3.11 Performance of dual time stepping to remove pressure error

3.12 Concluding remarks

References

CHAPTER 4 Incompressible Newtonian Laminar Flows

4.1 Introduction and the basic equations

4.2 Use of the CBS algorithm for incompressible flows

4.2.1 The fully explicit artificial compressibility form

4.2.2 The semi-implicit form

4.2.3 Quasi-implicit solution

4.3 Adaptive mesh refinement

4.3.1 Second gradient(curvature)based refinement

4.3.2 Local patch interpolation:Superconvergent values

4.3.3 Estimation of second derivatives at nodes

4.3.4 Element elongation

4.3.5 First derivative(gradient)based refinement

4.3.6 Choice of variables

4.3.7 An example

4.4 Adaptive mesh generation for transient problems

4.5 Slow flows:Mixed and penalty formulations

4.5.1 Analogy with incompressible elasticity

4.5.2 Mixed and penalty discretization

4.6 Concluding remarks

References

CHAPTER 5 Incompressible Non-Newtonian Flows

5.1 Introduction

5.2 Non-Newtonian flows:Metal and polymer forming

5.2.1 Non-Newtonian flows including viscoplasticity and plasticity

5.2.2 Steady-state problems of forming

5.2.3 Transient problems with changing boundaries

5.2.4 Elastic springback and viscoelastic fluids

5.3 Viscoelastic flows

5.3.1 Governing equations

5.4 Direct displacement approach to transient metal forming

5.5 Concluding remarks

References

CHAPTER 6 Free Surface and Buoyancy Driven Flows

6.1 Introduction

6.2 Free surface flows

6.2.1 General remarks

6.2.2 Lagrangian method

6.2.3 Eulerian methods

6.2.4 Arbitrary Langrangian-Eulerian(ALE)method

6.3 Buoyancy driven flows

6.4 Concluding remarks

References

CHAPTER 7 Compressible High-Speed Gas Flow

7.1 Introduction

7.2 The governing equations

7.3 Boundary conditions:Subsonic and supersonic flow

7.3.1 Euler equation

7.3.2 Navier-Stokes equations

7.4 Numerical approximations and the CBS algorithm

7.5 Shock capture

7.5.1 Second derivative-based methods

7.5.2 Residual-based methods

7.6 Variable smoothing

7.7 Some preliminary examples for the Euler equation

7.8 Adaptive refinement and shock capture in Euler problems

7.8.1 General

7.8.2 The h-refinement process and mesh enrichment

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

7.9 Three-dimensional inviscid examples in steady state

7.9.1 Solution of the flow pattern around a complete aircraft

7.9.2 THRUST:The supersonic car

7.10 Transient two-and three-dimensional problems

7.11 Viscous problems in two dimensions

7.11.1 Adaptive refinement in both shock and boundary layer

7.11.2 Special adaptive refinement for boundary layers and shocks

7.12 Three-dimensional viscous problems

7.13 Boundary layer:Inviscid Euler solution coupling

7.14 Concluding remarks

References

CHAPTER 8 Turbulent Flows

8.1 Introduction

8.1.1 Time averaging

8.1.2 Relation between κ,ε,and vT

8.2 Treatment of incompressible turbulent flows

8.2.1 Reynolds-averaged Navier-Stokes

8.2.2 One-equation models

8.2.3 Two-equation models

8.2.4 Nondimensional form of the governing equations

8.2.5 Shortest distance to a solid wall

8.2.6 Solution procedure for turbulent flow equations

8.3 Treatment of compressible flows

8.3.1 Mass-weighted(Favre)time averaging

8.4 Large eddy simulation(LES)

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

8.6 Direct numerical simulation(DNS)

8.7 Concluding remarks

References

CHAPTER 9 Generalized Flow and Heat Transfer in Porous Media

9.1 Introduction

9.2 A generalized porous medium flow approach

9.2.1 Nondimensional scales

9.3 Discretization procedure

9.3.1 Semi-and quasi-implicit forms

9.4 Forced convection

9.5 Natural convection

9.5.1 Constant-porosity medium

9.6 Concluding remarks

References

CHAPTER 10 Shallow-Water Problems

10.1 Introduction

10.2 The basis of the shallow-water equations

10.3 Numerical approximation

10.4 Examples of application

10.4.1 Transient one-dimensional problems:A performance assessment

10.4.2 Two-dimensional periodic tidal motions

10.4.3 Tsunami waves

10.4.4 Steady-state solutions

10.5 Drying areas

10.6 Shallow-water transport

10.7 Concluding remarks

References

CHAPTER 11 Long and Medium Waves

11.1 Introduction and equations

11.2 Waves in closed domains:Finite element models

11.3 Difficulties in modeling surface waves

11.4 Bed friction and other effects

11.5 The short-wave problem

11.6 Waves in unbounded domains(exterior surface wave problems)

11.6.1 Background to wave problems

11.6.2 Wave diffraction

11.6.3 Incident waves,domain integrals,and nodal values

11.7 Unbounded problems

11.8 Local nonreflecting boundary conditions(NRBCs)

11.8.1 Sponge layers or perfectly matched layers(PMLs)

11.9 Infinite elements

11.9.1 Mapped periodic(unconjugated)infinite elements

11.9.2 Ellipsoidal type infinite elements of Burnett and Holford

11.9.3 Wave envelope(or conjugated)infinite elements

11.9.4 Accuracy of infinite elements

11.9.5 Other applications

11.9.6 Trefftz-type infinite elements

11.10 Convection and wave refraction

11.11 Transient problems

11.12 Linking to exterior solutions(or DtN mapping)

11.12.1 Linking to boundary integrals

11.12.2 Linking to series solutions

11.13 Three-dimensional effects in surface waves

11.13.1 Large-amplitude water waves

11.13.2 Cnoidal and solitary waves

11.13.3 Stokes waves

11.14 Concluding remarks

References

CHAPTER 12 Short Waves

12.1 Introduction

12.2 Background

12.3 Errors in wave modeling

12.4 Recent developments in short-wave modeling

12.5 Transient solution of electromagnetic scattering problems

12.6 Finite elements incorporating wave shapes

12.6.1 Shape functions using products of polynomials and waves

12.6.2 Shape functions using sums of polynomials and waves

12.6.3 The discontinuous enrichment method

12.6.4 Ultra weak formulation

12.6.5 Trefftz-type finite elements for waves

12.7 Refraction

12.7.1 Wave speed refraction

12.7.2 Refraction caused by flows

12.8 Spectral finite elements for waves

12.9 Discontinuous Galerkin finite elements(DGFE)

12.10 Concluding remarks

References

CHAPTER 13 Fluid-Structure Interaction

13.1 Introduction

13.2 One-dimensional fluid-structure interaction

13.2.1 Equations

13.2.2 Characteristic analysis

13.2.3 Boundary conditions

13.2.4 Solution method:Taylor-Galerkin method

13.2.5 Some results

13.3 Multidimensional problems

13.3.1 Equations and discretization

13.3.2 Segregated approach

13.3.3 Mesh moving procedures

13.4 Concluding remarks

References

CHAPTER 14 Biofluid Dynamics

14.1 Introduction

14.2 Flow in human arterial system

14.2.1 Heart

14.2.2 Reflections

14.2.3 Aortic valve

14.2.4 Vessel branching

14.2.5 Terminal vessels

14.2.6 Numerical solution

14.3 Image-based subject-specific flow modeling

14.3.1 Image segmentation

14.3.2 Geometrical potential force(GPF)

14.3.3 Numerical solution,initial and boundary conditions

14.3.4 Domain discretization

14.3.5 Flow solution

14.4 Concluding remarks

References

CHAPTER 15 Computer Implementation of the CBS Algorithm

15.1 Introduction

15.2 The data input module

15.2.1 Mesh data:Nodal coordinates and connectivity

15.2.2 Boundary data

15.2.3 Other necessary data and flags

15.2.4 Preliminary subroutines and checks

15.3 Solution module

15.3.1 Time step

15.3.2 Shock capture

15.3.3 CBS algorithm:Steps

15.3.4 Boundary conditions

15.3.5 Solution of simultaneous equations:Semi-implicit form

15.3.6 Different forms of energy equation

15.3.7 Convergence to steady state

15.4 Output module

References

APPENDIX A Self-Adjoint Differential Equations

APPENDIX B Nonconservative Form of Navier-Stokes Equations

APPENDIX C Computing the Drag Force and Stream Function

APPENDIX D Convection-Diffusion Equations:Vector-Valued Variables

APPENDIX E Integration Formulae

APPENDIX F Edge-Based Finite Element Formulation

APPENDIX G Boundary Layer-Inviscid Flow Coupling

APPENDIX H Multigrid Method

APPENDIX I Mass-Weighted Averaged Turbulence Transport Equations

Author Index

Subject Index