The Application of FEM and FDM to Flow Separation Pattern ; A Comparison Study

Abstract

With the advent of high speed computer, it has become possible to develop methods for studying theoretically many of the unsteady incompressible fluid flow problems which previously had been hopelessly complicated for analysis. The purpose of this paper is to describe in detail two popular numerical methods namely; the Finite Element Method (FEM) and the Finite Difference Method (FDM). Both methods are applied to solve the flow separation pattern past an obstruction in a two-dimensional flow field. The differential equations that govern the phenomenon are the Helmholtz type vorticity equation and the Poisson type stream function equation. The solution of the two equations will provide the variation of the vorticity and stream function fields with respect to time. The Finite Element Model is developed using two extremum functionals for the Poisson and Helmholtz vorticity equations (Zienckiewicz 1977). A triangular elements are utilized in the solution with 6-nodal elements used for the Poisson solution and 3-nodal elements for the Helmholtz equation. The Finite Difference Model is developed using a square mesh at which the stream function and vorticity values are evaluated at the mesh points. The derivatives of the Helmholtz and the Poisson equations are replaced with central differences both in space and time (Smith 1965). The FEM and FDM are formulated, developed and tested. The solutions for the separation pattern are compared using both models. The results predicted by both models are compared to an experimental separation pattern using pitot-tube measurements.