The Finite Element Method for Fluid Dynamics

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.

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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.

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 foundin...

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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.

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.

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.

Author Biography
Dedication
List of Figures
List of Tables
Preface
Chapter 1. Introduction to the Equations of Fluid Dynamics and the Finite Element Approximation
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Author Biography
Dedication
List of Figures
List of Tables
Preface
Chapter 1. Introduction to the Equations of Fluid Dynamics and the Finite Element Approximation
Abstract
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
References
Chapter 2. Convection-Dominated Problems: Finite Element Approximations to the Convection-Diffusion-Reaction Equation
Abstract
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
References
Chapter 3. The Characteristic-Based Split (CBS) Algorithm: A General Procedure for Compressible and Incompressible Flow
Abstract
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
References
Chapter 4. Incompressible Newtonian Laminar Flows
Abstract
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
Chapter 5. Incompressible Non-Newtonian Flows
Abstract
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
Chapter 6. Free Surface and Buoyancy Driven Flows
Abstract
6.1 Introduction
6.2 Free surface flows
6.3 Buoyancy driven flows
6.4 Concluding remarks
References
Chapter 7. Compressible High-Speed Gas Flow
Abstract
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
References
Chapter 8. Turbulent Flows
Abstract
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
References
Chapter 9. Generalized Flow and Heat Transfer in Porous Media
Abstract
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
References
Chapter 10. Shallow-Water Problems
Abstract
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
Chapter 11. Long and Medium Waves
Abstract
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
References
Chapter 12. Short Waves
Abstract
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
References
Chapter 13. Fluid–Structure Interaction
Abstract
13.1 Introduction
13.2 One-dimensional fluid–structure interaction
13.3 Multidimensional problems
13.4 Concluding remarks
References
Chapter 14. Biofluid Dynamics
Abstract
14.1 Introduction
14.2 Flow in Human Arterial System
14.3 Image-Based Subject-Specific Flow Modeling
14.4 Concluding Remarks
References
Chapter 15. Computer Implementation of the CBS Algorithm
Abstract
15.1 Introduction
15.2 The Data Input Module
15.3 Solution Module
15.4 Output Module
References
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
References
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
References
Appendix I. Mass-Weighted Averaged Turbulence Transport Equations
I.1 Turbulence models
Author Index
Subject Index
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