The Galerkin Bem For Solving 2-d Stokes Problem With Open Boundary

Stokes equation is the linearization of Navier-Stokes equation, which is the governing equation of incompressible viscous flow, with small Reynolds number. It forms Stokes problem with the viscous boundary conditions attached. Among so many numerical methods to solve this problem, the boundary element method is an ideal method, since it is easy to handle the continuity equation, or the incompressibility condition,moreover, computing the velocity and computing the pressure can be separated in the solution procedure. In this paper,we solved the Stokes problem in a bounded plane region with non-closed line or curve segment as an open boundary,which is a two-Dimensional screen problem in fluid flow. The difficulty in solving this kind of problems is how to deal with the singularity near the endpoints on the open boundary. We have established the singular element with singular shape function while the element contains endpoint in order to simulate the inherent singularity.We establish the variational formulation based on the Fredholm vector integral equations of the first kind, which are derived by the Green's formula and the fundamental solution corresponding to Stokes operator. Then, we solve the boundary variational equation by Galerkin Boundary Element Method, to get vector density function of the simple layer. The velocity at any point in the flow field is calculated by the discrete form of boundary integral expression.We first set forth the formulation of the Galerkin Boundary Element Method to solve the 2-D Stokes problem in a complex connected region, and devise the computing codes by Fortran program for calculate the velocity field of the flow. The streamlines are drawn by a Matlab program to verify the reliability of the program. Then we focus on the numerical simulation of the problem containing open boundary in a bounded domain. Because by Galerkin boundary element method we need to calculate the double integrals in computing matrix coefficients, we derived the analytical integral formula with shape functions contained singularity or not, and special Gauss integral formula with singular weight function. For the double integrations on singular boundary element, we carry out the first integration by analytical integral formula and the second integration by a Gauss integral formula with a special singular weight function, while the usual Gauss integral formulas are used when there is no singularity on the element. In this paper, several numerical tests simulated the flow of the viscous incompressible fluid in a complex connected region and a bounded region with an open boundary.