2009 M.Sc. Graduated Behzadi Farank

Fully-Implicit Solution of Navier-Stokes Equations

Abstract

In this research a method for numerical solution of the steady incompressible Navier-Stokes equations in primitive variables is proposed. This scheme which is based on the finite difference method employs staggered grid and virtual nodes outside the problem domain. The continuity and momentum equations are discretized in each node of the grid and different boundary conditions are imposed at the virtual nodes. Applying virtual nodes avoids elimination of the motion equations at boundary nodes of the grid. In the presence of virtual nodes using central difference approximation in spatial discretization of the equations at total boundary nodes is possible. A system of equations is obtained in which the velocity and pressure fields are unknown. This system is solved simultaneously to obtain the velocity and pressure fields. The method uses a novel algorithm, which does not require iterating method as done in traditional finite difference schemes. The proposed method is implemented for both linear and non-linear forms of the equations at different Reynolds numbers in laminar flow. The process of the proposed scheme is illustrated through its application to the two-dimensional driven cavity flow. Comparison with other methods reveals the efficiency and accuracy of the present approach. The results demonstrate the success of the method which can be extended to three-dimensional steady incompressible Navier-Stokes equations and the other non-linear equations with different boundary conditions.