Fluid Mechanics Finite Element Applications

Abstract

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.