Finite Element Techniques for Fluid Flow

Finite Element Techniques for Fluid Flow describes the advances in the applications of finite element techniques to fluid mechanics. Topics covered range from weighted residual and variational methods to interpolation functions, inviscid fluids, and flow through porous media. The basic principles and governing equations of fluid mechanics as well as problems related to dispersion and shallow water circulation are also discussed. This text is comprised of nine chapters; the first of which explains some basic definitions and properties as well as the basic principles of weighted residual and variational methods. The reader is then introduced to the simple finite element concepts and models, and gradually to more complex applications. The chapters that follow focus on the governing equations of fluid flow, the solutions to potential type problems, and viscous flow problems in porous media. The solutions to more specialized problems are also presented. This book also considers how circulation problems can be tackled using finite elements, presents a solution to the mass transfer equation, and concludes with an explanation of how to solve general transient incompressible flows. This source will be of use to engineers, applied mathematicians, physicists, self-taught students, and research workers.

Table of Contents:
1 Weighted Residual and Variational Methods 1.1 Basic Definitions 1.2 Weighted Residual Methods 1.3 Weak Formulations 1.4 Initial Value Problems 1.5 The Case of Quadratic Functional 1.6 Rayleigh-Ritz Method 1.7 Subsidiary Conditions2 The Finite Element Technique 2.1 Localized Functions 2.2 The Finite Element Technique 2.3 Element Matrices 2.4 System Equations 2.5 Solution of the System 2.6 The General Program3 Interpolation Functions 3.1 Introduction 3.2 First-Order Continuity Functions for Triangular Elements 3.3 First-Order Continuity Functions for Rectangular Elements 3.4 Isoparametric Elements 3.5 Second-Order Continuity Functions for Rectangular Elements 3.6 Second-Order Continuity Functions for Triangular Elements4 Basic Principles and Governing Equations of Fluid Mechanics 4.1 Eulerian and Lagrangian Formulations: Material Derivative 4.2 Deformation Rate Measures 4.3 Equilibrium Equations 4.4 The Energy Equation 4.5 Constitutive Equations—Newtonian Fluid 4.6 Navier-Stokes Equations—Incompressible Newtonian Fluid 4.7 The Principle of Virtual Power 4.8 Turbulence5 Inviscid Fluids 5.1 Basic Principles 5.2 Bernoulli's Principle 5.3 The Wave Equation 5.4 Harmonic Response of Coastal Waters 5.5 Stream Function Formulation 5.6 Cylindrical Coordinates6 Flow Through Porous Media 6.1 Principles of Groundwater Flow 6.2 Confined Seepage Problems 6.3 Problems Involving Free Surfaces 6.4 Transient Free Surface Flow 6.5 Confined Aquifer Analysis 6.6 Unconfined Aquifer Analysis7 Shallow Water Circulation Problems 7.1 Shallow Water Equations 7.2 Finite Element Formulation 7.3 Numerical Integration Schemes 7.4 Lake Circulation8 Dispersion Problems 8.1 Introduction 8.2 The Mass Transfer Equation 8.3 Diffusion Problems 8.4 Diffusion and Convection Problems 8.5 Nonlinear Diffusion9 Viscous Incompressible Flow Problems 9.1 Introduction 9.2 Basic Principles 9.3 Stream Function—Vorticity Approach 9.4 Pressure and Velocities Approach 9.5 Free Surface FlowAppendix Numerical Integration FormulaIndex