Numerical Methods for a One-Dimensional Interface Separating Compressible and Incompressible Flows

Abstract

We develop 1D numerical methods for treating an interface separating a liquid drop and a high speed gas flow. The droplet is an incompressible Navier-Stokes fluid. The gas is a compressible, multi-species, chemically reactive Navier-Stokes fluid (Fedkiw et al., 1996; Fedkiw, 1996). The interface is followed with a marker particle, although the level set method will be used for the eventual 2D extension (Sussman, 1995). Away from the interface, we solve the equations with TVD Runge Kutta schemes in time and conservative finite difference ENO schemes in space (Shu and Osher, 1988). Near the interface, we cannot apply this discretization, since the equations differ in both number and type across the interface. Instead we use the interface location for domain decomposition, and apply a moving control volume formulation nearby. This is done in a conservative framework, compatible with the outer finite difference scheme. Full details are given for a simple forward Euler time stepping scheme, and this has direct, although algorithmically complicated, extensions to 2nd and 3rd order Runge Kutta methods. Future work will focus on the extension to 2D, and simplifications of the higher order time stepping algorithms.