A compressible vortex method for viscous gas dynamics and its numerical implementations
Description
A method for solving the equations of compressible viscous gas dynamics is presented. It uses the combination of the random vortex method and the monotone upwind scheme for conservation laws (MUSCL). The random vortex method was designed based on the assumption of incompressibility of the fluid, and the MUSCL scheme only solves conservation laws whereas the combination of these two methods gives rise to a new method which is able to solve compressible fluid flow. The new method also enjoys the advantages of the random vortex method and the MUSCL scheme: capable of dealing with high Reynolds number flow and capable of dealing with gas flow involving strong shocks. The idea of making the combination of two methods is based on Hodge's decomposition theorem. According to this theorem, solutions to the gas dynamics equations can be decomposed into two parts, a divergence free part and a curl free part. The divergence free part, corresponding to incompressible but viscous fluid, is solved by the random vortex method. The curl free part, corresponding to inviscid but compressible fluid, is solved by the MUSCL scheme. The combination of these two methods will be referred to as the compressible vortex method. A detailed description of the random vortex method is given. The vortex sheet method is also presented which is used in conjunction with the random vortex method in creating vorticity along the boundaries of the computational domain. The MUSCL scheme is a second order extension of Godunov method, which is studied by first studying Godunov method for conservation laws. Numerical results are presented by MUSCL for both one and two dimensional flow problems. The compressible vortex method has been applied to test problems whose computational domain is a channel with a step. Results are obtained for both subsonic and supersonic flow problems