The Finite Element Method for Fluid Dynamics

The Finite Element Method for Fluid Dynamics

6th Edition - November 24, 2005

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  • Authors: Olek Zienkiewicz, Robert Taylor, P. Nithiarasu
  • eBook ISBN: 9780080455594

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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

    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

    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

    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

    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

    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

    6 Free surface and buoyancy driven flows
    6.1 Introduction
    6.2 Free surface flows
    6.3 Buoyancy driven flows
    6.4 Concluding remarks

    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

    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

    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

    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

    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

    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

    13 Computer implementation of the CBS algorithm
    13.1 Introduction
    13.2 The data input module
    13.3 Solution module
    13.4 Output module

    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

Product details

  • No. of pages: 400
  • Language: English
  • Copyright: © Butterworth-Heinemann 2005
  • Published: November 24, 2005
  • Imprint: Butterworth-Heinemann
  • eBook ISBN: 9780080455594

About the Authors

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. 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, Spain

Robert 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, USA.

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

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