Method of Fundamental Solutions for Stokes Problems by the Pressure-Stream Function Formulation

ABSTRACTThe main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.