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# The Finite Element Method for Fluid Dynamics

- 6th Edition - November 24, 2005
- Authors: Olek C Zienkiewicz, Robert L. Taylor, P. Nithiarasu
- Language: English
- Hardback ISBN:9 7 8 - 0 - 7 5 0 6 - 6 3 2 2 - 9
- Paperback ISBN:9 7 8 - 1 - 4 9 3 3 - 0 2 9 0 - 1
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 4 5 5 5 9 - 4

Dealing with general problems in fluid mechanics, convection diffusion, compressible and incompressible laminar and turbulent flow, shallow water flows and waves, this is the… Read more

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Request a sales quoteDealing with general problems in fluid mechanics, convection diffusion, compressible and incompressible laminar and turbulent flow, shallow water flows and waves, this is the leading text and reference for engineers working with fluid dynamics in fields including aerospace engineering, vehicle design, thermal engineering and many other engineering applications. The new edition is a complete fluids text and reference in its own right. Along with its companion volumes it forms part of the indispensable

New material in this edition includes sub-grid scale modelling; artificial compressibility; full new chapters on turbulent flows, free surface flows and porous medium flows; expanded shallow water flows plus long, medium and short waves; and advances in parallel computing.

*Finite Element Method*series.New material in this edition includes sub-grid scale modelling; artificial compressibility; full new chapters on turbulent flows, free surface flows and porous medium flows; expanded shallow water flows plus long, medium and short waves; and advances in parallel computing.

- A complete, stand-alone reference on fluid mechanics applications of the FEM for mechanical, aeronautical, automotive, marine, chemical and civil engineers.
- Extensive new coverage of turbulent flow and free surface treatments

Practicing engineers, senior students and researchers in mechanical, automotive, aeronautical and civil engineering. Key topic for applied mathematicians and engineering software developers.

**1 Introduction to the equations of fluid dynamics and the finite element**

approximation

approximation

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

1.2 The governing equations of fluid dynamics

1.3 Inviscid, incompressible flow

1.4 Incompressible (or nearly incompressible) flows

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

1.6 Concluding remarks

1.7 Exercises

References

**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

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

2.4 Steady state -- concluding remarks

2.5 Transients -- introductory remarks

2.6 Characteristic-based methods

2.7 Taylor--Galerkin procedures for scalar variables

2.8 Steady-state condition

2.9 Non-linear waves and shocks

2.10 Treatment of pure convection

2.11 Boundary conditions for convection--diffusion

2.12 Summary and concluding remarks

2.13 Exercises

References

**3 The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow**

3.1 Introduction

viii Contents

3.2 Non-dimensional form of the governing equations

3.3 Characteristic-based split (CBS) algorithm

3.4 Explicit, semi-implicit and nearly implicit forms

3.5 Artificial compressibility and dual time stepping

3.6 ‘Circumvention’ of the Babu¡ska--Brezzi (BB) restrictions

3.7 A single-step version

3.8 Boundary conditions

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

3.10 Concluding remarks

References

**4 Incompressible Newtonian laminar flows**

4.1 Introduction and the basic equations

4.2 Use of the CBS algorithm for incompressible flows

4.3 Adaptive mesh refinement

4.4 Adaptive mesh generation for transient problems

4.5 Slow flows -- mixed and penalty formulations

4.6 Concluding remarks

References

**5 Incompressible non-Newtonian flows**

5.1 Introduction

5.2 Non-Newtonian flows - metal and polymer forming

5.3 Viscoelastic flows

5.4 Direct displacement approach to transient metal forming

5.5 Concluding remarks

References

**6 Free surface and buoyancy driven flows**

6.1 Introduction

6.2 Free surface flows

6.3 Buoyancy driven flows

6.4 Concluding remarks

References

**7 Compressible high-speed gas flow**

7.1 Introduction

7.2 The governing equations

7.3 Boundary conditions -- subsonic and supersonic flow

7.4 Numerical approximations and the CBS algorithm

7.5 Shock capture

7.6 Variable smoothing

7.7 Some preliminary examples for the Euler equation

7.8 Adaptive refinement and shock capture in

Euler problems

7.9 Three-dimensional inviscid examples in steady state

7.10 Transient two- and three-dimensional problems

Contents ix

7.11 Viscous problems in two dimensions

7.12 Three-dimensional viscous problems

7.13 Boundary layer--inviscid Euler solution coupling

7.14 Concluding remarks

References

**8 Turbulent flows**

8.1 Introduction

8.2 Treatment of incompressible turbulent flows

8.3 Treatment of compressible flows

8.4 Large eddy simulation

8.5 Detached Eddy Simulation (DES) and Monotonically

Integrated LES (MILES)

8.6 Direct Numerical Simulation (DNS)

8.7 Summary

References

**9 Flow through porous media**

9.1 Introduction

9.2 A generalized porous medium flow approach

9.3 Discretization procedure

9.4 Non-isothermal flows

9.5 Forced convection

9.6 Natural convection

9.7 Summary

References

**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

10.5 Drying areas

10.6 Shallow water transport

10.7 Concluding remarks

References

**11 Long and medium waves**

11.1 Introduction and equations

11.2 Waves in closed domains - finite element models

11.3 Difficulties in modelling 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)

11.7 Unbounded problems

11.8 Local Non-Reflecting Boundary Conditions (NRBCs)

11.9 Infinite elements

11.10 Mapped periodic (unconjugated) infinite elements

x Contents

11.11 Ellipsoidal type infinite elements of Burnett and Holford

11.12 Wave envelope (or conjugated) infinite elements

11.13 Accuracy of infinite elements

11.14 Trefftz type infinite elements

11.15 Convection and wave refraction

11.16 Transient problems

11.17 Linking to exterior solutions (or DtN mapping)

11.18 Three-dimensional effects in surface waves

11.19 Concluding remarks

References

**12 Short waves**

12.1 Introduction

12.2 Background

12.3 Errors in wave modelling

12.4 Recent developments in short wave modelling

12.5 Transient solution of electromagnetic scattering problems

12.6 Finite elements incorporating wave shapes

12.7 Refraction

12.8 Spectral finite elements for waves

12.9 Discontinuous Galerkin finite elements (DGFE)

12.10 Concluding remarks

References

**13 Computer implementation of the CBS algorithm**

13.1 Introduction

13.2 The data input module

13.3 Solution module

13.4 Output module

References

Appendix A Non-conservative form of Navier–Stokes equations

Appendix B Self-adjoint differential equations

Appendix C Postprocessing

Appendix D Integration formulae

Appendix E Convection–diffusion equations: vector-valued variables

Appendix F Edge-based finite element formulation

Appendix G Multigrid method

Appendix H Boundary layer–inviscid flow coupling

Appendix I Mass-weighted averaged turbulence transport equations

Author Index

Subject Index

- No. of pages: 464
- Language: English
- Edition: 6
- Published: November 24, 2005
- Imprint: Butterworth-Heinemann
- Hardback ISBN: 9780750663229
- Paperback ISBN: 9781493302901
- eBook ISBN: 9780080455594

OZ

### Olek C Zienkiewicz

O. C. Zienkiewicz was one of the early pioneers of the finite element method and is internationally recognized as a leading figure in its development and wide-ranging application. He was awarded numerous honorary degrees, medals and awards over his career, including the Royal Medal of the Royal Society and Commander of the British Empire (CBE). He was a founding author of The Finite Element Method books and developed them through six editions over 40 years up to his death in 2009. Previous positions held by O.C. Zienkiewicz include UNESCO Professor of Numerical Methods in Engineering at the International Centre for Numerical Methods in Engineering, Barcelona, Director of the Institute for Numerical Methods in Engineering at the University of Wales, Swansea, U.K.

Affiliations and expertise

Finite element method pioneer and former UNESCO Professor of Numerical Methods in Engineering, Barcelona, SpainRT

### Robert L. Taylor

R.L Taylor is Professor of the Graduate School at the Department of Civil and Environmental Engineering, University of California at Berkeley, USA. Awarded the Daniel C. Drucker Medal by the American Society of Mechanical Engineering in 2005, the Gauss-Newton Award and Congress Medal by the International Association for Computational Mechanics in 2002, and the Von Neumann Medal by the US Association for Computational Mechanics in 1999.

Affiliations and expertise

Emeritus Professor of Engineering, University of California, Berkeley, USAPN

### P. Nithiarasu

Professor Nithiarasu is Director of Research and Deputy Head of the College of Engineering of Swansea University, and also holds a position as Dean of Academic Leadership (Research Impact). Previously, PN served as the Head of Zienkewicz Centre for Computational Engineering for 5 years. He was awarded the Zienkiewicz silver medal from the ICE London in 2002, the ECCOMAS Young Investigator award in 2004 and the prestigious EPSRC Advanced Fellowship in 2006.

Affiliations and expertise

Professor, College of Engineering, University of Wales, Swansea, UK