Investigation of finite element schemes for the Navier Stokes equations

Abstract

The main aim of this thesis has been to develop a reliable numerical method got the solution of the two-dimensional Navier-Stokes equations. The numerical method to be used was the finite element method. A literature survey revealed that a limitation common to all finite element methods available to date is that they only produce solutions for low Reynold's numbers. However, for aerodynamics applications, Reynold's numbers of the order of 10⁶ are frequently encountered. At these levels conventional finite element methods break down completely. It was felt that this limitation could be overcome by the use of new types of shape functions. The search for for the new shape functions were carried out in three stages. Firstly a new method is presented for deriving shape functions for a wide class of second order ordinary differential equations with significant first order derivatives. The method is then extended to derive shape functions for a wide class of elliptic partial differential equations with similar properties. Several numerical examples are presented to illustrate the advantages of the new shape functions over the traditional polynomial shape functions. The shape functions developed for partial differential equations are then used to construct a new finite element scheme for the Navier-Stokes equations. the scheme was implemented on a computer and the numerical resultsobtained indicated that the new scheme was more stable than the conventional schemes