Book contents
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
- References
10 - Fluid Mechanics Finite Element Applications
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
- References
Summary
Introduction
During the past half century, engineering analysis has relied on the traditional finitedifference method to obtain computer-based solutions to difficult flow problems. The progress and success achieved in these pursuits have been, in many cases, noteworthy. Slow viscous flows, boundary layer flows, diffusion flows, and variableproperty flows are just some examples of areas for which analysts have developed refined calculation procedures based on the finite-difference method.
Yet there remains a number of problems for which the finite-difference methods were proven inaccurate. Problems involving complex geometries, multiplyconnected domains, and complicated boundary conditions always pose quite a challenge. Finite element methods can help in alleviating these difficulties but should not be expected to triumph in every case where the finite-difference methods have failed. Instead, the finite element methods offer easier ways to treat complex geometries requiring irregular meshes, and they provide a more consistent way of using higher-order approximations. In some cases, the finite element approach can provide an approximate solution of the same order of accuracy as the finite difference method but at less expenses. Regardless of the method used, the accurate numerical solution of most of the viscous-flow problems requires vast amounts of computer time and data storage, and of course, problems of numerical stability and convergence can occur with either method.
Only since the early 1970s has the finite element method been recognized as an effective means for solving difficult fluid mechanics problems. Literature on the application of finite element methods to fluid mechanics is rapidly increasing, with contributions being made virtually daily.
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- Publisher: Cambridge University PressPrint publication year: 2013