The Finite Element Method for Fluid Dynamics - 6th Edition - ISBN: 9780750663229, 9780080455594

The Finite Element Method for Fluid Dynamics

6th Edition

Authors: Olek Zienkiewicz Robert Taylor P. Nithiarasu
eBook ISBN: 9780080455594
Imprint: Butterworth-Heinemann
Published Date: 24th November 2005
Page Count: 400
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Dealing 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 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.

Key Features

  • 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.

Table of Contents

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 1.3 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


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About the Author

Olek 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.

Affiliations and Expertise

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

Robert Taylor

R. L. Taylor is Emeritus Professor of Engineering and Professor in the Graduate School, Department of Civil and Environmental Engineering at the University of California, Berkeley.

Affiliations and Expertise

Emeritus Professor of Engineering, University of California, Berkeley, USA.

P. Nithiarasu

Dr. P. Nithiarasu, Senior Lecturer at the School of Engineering, University of Wales Swansea, has over ten years of experience in the finite element based computational fluid dynamics research. He moved to Swansea in 1996 after completing his PhD research at IIT Madras. He was awarded Zienkiewicz silver medal and prize of the Institution of Civil Engineers, UK in 2002. In 2004 he was selected to receive the European Community on Computational Methods in Applied Sciences (ECCOMAS) award for young scientists in computational engineering sciences. Dr Nithiarasu is the author of several articles in the area of fluid dynamics, porous medium flows and the finite element method.

Affiliations and Expertise

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


It is very difficult to write a book which covers the entire finite element field. ..The authors have made a splendid attempt at a very difficult task. The books remain a tremendous bargain...and are an invaluable guide to the entire field of finite elements. If you are serious about working on finite elements you cannot do without this book. - Mathematics Today, August 2001. "...the publication of the first edition was an epoch making is written by...the greatest theorist of the subject. If you are serious about finite elements, this is a book that you simply cannot afford to be without." - International Journal of Numerical Methods in Engineering. "..the pre-eminent reference work on finite element analysis." - Applied Mechanical Review "...a very good book...presentation is first class...will be of great assistance to all engineers and scientists interested in the method...a very commendable piece of work." - Journal of the British Society for Strain Measurement