The Finite Element Method for Fluid Dynamics - 7th Edition - ISBN: 9781856176354, 9780080951379

The Finite Element Method for Fluid Dynamics

7th Edition

Authors: Olek Zienkiewicz Robert Taylor P. Nithiarasu
Hardcover ISBN: 9781856176354
eBook ISBN: 9780080951379
Imprint: Butterworth-Heinemann
Published Date: 14th November 2013
Page Count: 584
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Table of Contents

Author Biography


List of Figures

List of Tables


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

Author Index

Subject Index


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


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"...this is a book that you simply cannot afford to be without."--INTERNATIONAL JOURNAL OF NUMERICAL METHODS IN ENGINEERING (previous edition)

About the Authors

Olek Zienkiewicz Author

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 Author

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 Author

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