The Finite Element Method for Fluid Dynamics

The Finite Element Method for Fluid Dynamics

7th Edition - November 14, 2013

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  • Authors: Olek Zienkiewicz, Robert Taylor, P. Nithiarasu
  • Hardcover ISBN: 9781856176354
  • eBook ISBN: 9780080951379

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The Finite Element Method for Fluid Dynamics offers a complete introduction the application of the finite element method to fluid mechanics. The book begins with a useful summary of all relevant partial differential equations before moving on to discuss convection stabilization procedures, steady and transient state equations, and numerical solution of fluid dynamic equations. The character-based split (CBS) scheme is introduced and discussed in detail, followed by thorough coverage of incompressible and compressible fluid dynamics, flow through porous media, shallow water flow, and the numerical treatment of long and short waves. Updated throughout, this new edition includes new chapters on: Fluid-structure interaction, including discussion of one-dimensional and multidimensional problems Biofluid dynamics, covering flow throughout the human arterial system Focusing on the core knowledge, mathematical and analytical tools needed for successful computational fluid dynamics (CFD), The Finite Element Method for Fluid Dynamics is the authoritative introduction of choice for graduate level students, researchers and professional engineers.

Key Features

  • A proven keystone reference in the library of any engineer needing to understand and apply the finite element method to fluid mechanics
  • Founded by an influential pioneer in the field and updated in this seventh edition by leading academics who worked closely with Olgierd C. Zienkiewicz
  • Features new chapters on fluid-structure interaction and biofluid dynamics, including coverage of one-dimensional flow in flexible pipes and challenges in modeling systemic arterial circulation


Mechanical, Aerospace, Automotive, Marine, Biomedical, Environmental and Civil Engineers, applied mathematicians and computer aided engineering software developers

Table of Contents

  • 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

    1.3 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


    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

    2.3 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 Nonlinear waves and shocks

    2.10 Treatment of pure convection

    2.11 Boundary conditions for convection-diffusion

    2.12 Summary and concluding remarks


    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

    3.4 Explicit, Semi-Implicit, and Nearly Implicit Forms

    3.5 Artificial Compressibility and Dual Time Stepping

    3.6 “Circumvention” of the Babuška-Brezzi (BB) Restrictions

    3.7 A Single-Step Version

    3.8 Splitting Error

    3.9 Boundary Conditions

    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


    Chapter 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


    Chapter 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


    Chapter 6. Free Surface and Buoyancy Driven Flows


    6.1 Introduction

    6.2 Free surface flows

    6.3 Buoyancy driven flows

    6.4 Concluding remarks


    Chapter 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

    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


    Chapter 8. Turbulent Flows


    8.1 Introduction

    8.2 Treatment of incompressible turbulent flows

    8.3 Treatment of compressible flows

    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


    Chapter 9. Generalized Flow and Heat Transfer in Porous Media


    9.1 Introduction

    9.2 A generalized porous medium flow approach

    9.3 Discretization procedure

    9.4 Forced convection

    9.5 Natural convection

    9.6 Concluding remarks


    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

    10.5 Drying areas

    10.6 Shallow-water transport

    10.7 Concluding remarks


    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)

    11.7 Unbounded Problems

    11.8 Local NonReflecting Boundary Conditions (NRBCs)

    11.9 Infinite Elements

    11.10 Convection and Wave Refraction

    11.11 Transient Problems

    11.12 Linking to Exterior Solutions (or DtN Mapping)

    11.13 Three-Dimensional Effects in Surface Waves

    11.14 Concluding Remarks


    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

    12.7 Refraction

    12.8 Spectral Finite Elements for Waves

    12.9 Discontinuous Galerkin Finite Elements (DGFE)

    12.10 Concluding Remarks


    Chapter 13. Fluid–Structure Interaction


    13.1 Introduction

    13.2 One-dimensional fluid–structure interaction

    13.3 Multidimensional problems

    13.4 Concluding remarks


    Chapter 14. Biofluid Dynamics


    14.1 Introduction

    14.2 Flow in Human Arterial System

    14.3 Image-Based Subject-Specific Flow Modeling

    14.4 Concluding Remarks


    Chapter 15. Computer Implementation of the CBS Algorithm


    15.1 Introduction

    15.2 The Data Input Module

    15.3 Solution Module

    15.4 Output Module


    Appendix A. Self-Adjoint Differential Equations

    Appendix B. Nonconservative Form of Navier-Stokes Equations

    Appendix C. Computing the Drag Force and Stream Function

    C.1 Drag calculation

    C.2 Stream function

    Appendix D. Convection-Diffusion Equations: Vector-Valued Variables

    D.1 The Taylor-Galerkin method used for vector-valued variables

    D.2 Two-step predictor-corrector methods: Two-step Taylor-Galerkin operation


    Appendix E. Integration Formulae

    E.1 Linear triangles

    E.2 Linear tetrahedron

    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

    I.1 Turbulence models

Product details

  • No. of pages: 584
  • Language: English
  • Copyright: © Butterworth-Heinemann 2013
  • Published: November 14, 2013
  • Imprint: Butterworth-Heinemann
  • Hardcover ISBN: 9781856176354
  • eBook ISBN: 9780080951379

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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