An improved unified solver for compressible and incompressible fluids involving free surfaces. Part I. Convection

摘要

An improved numerical solver for the unified solution of flows involving an interface between either compressible or incompressible fluids is proposed. This method is based on the CIP-CUP (Cubic Interpolated Propagation/Combined, Unified Procedure) which is a semi-implicit solver for the Euler equations of fluid flows In the CIP-CUP method, each of convection and acoustic parts of the Euler equations are treated individually by a splitting manner. Namely, the convection part is solved by CIP method and the acoustic part is solved by CUP method. As Part I of this series of articles, we propose an improved scheme for the convection part. The ability of the CIP method to capture interfaces is highly improved by replacing the cubic interpolation function used in the CIP scheme with a quadratic-type extrapolation function only around the interface. With this improvement, oscillation and diffusion in the solution at interfaces (or phase boundaries) are removed, which are the most significant problems on free-surface flow simulations especially in the case where we treat some materials which have quite different properties. By giving some constraints on the extrapolation function, its stability was guaranteed. Furthermore, we propose a simple scheme for recognizing interface location and motion with a color function or a level set function. This scheme is very useful for the extrapolation process. Effectiveness, accuracy and stability of the improved method were demonstrated with some examples.