Asymptotic Expansion of a Multiscale Numerical Scheme for Compressible Viscous Multiphase Flows

An asymptotic development of a numerical scheme for the simulation of compressible multiphase flows including viscous effects is illustrated. First, a numerical approximation of the Navier-Stokes equations for each phase is provided. Then, an average procedure of this approximation is used to define, in a probabilistic framework, all the interactions between the two phases. This enables an accurate resolution method for all terms. Thus, the proposed scheme is used for the discretization of a seven-equations model, including relaxation terms that allow obtaining the pressure and velocity equilibrium between the two phases. Finally, an asymptotic analysis at the discrete level is performed, where the relaxation terms disappear. The influence of the viscous terms is studied, comparing the results obtained by solving the Navier-Stokes equations (validated with reference solutions given in the literature) with the Euler solutions