A Kinetic Energy Preserving and Entropy Conserving Two-Point Flux for Multi-species Compressible Flow

Abstract

High-order entropy stable (ES) methods have attracted much interest in the computational fluid dynamics (CFD) research community over the past decade, as we seek to simulate practical engineering applications with high accuracy and robustness. As industry delves further into supersonic flight and space travel, we need schemes capable of modelling flows with multiple chemical species that interact and mix within the fluid. As such, there is a considerable amount of effort currently being focused on developing high-order ES schemes for multi-species (MS) flow. A key ingredient for these entropy stable schemes is the two-point flux, which determines the properties of the resulting high-order discretization. As such, the numerical scheme can possess important properties such as entropy conservation and kinetic energy preservation only if the two-point flux also mirrors these properties. At present, in the literature, the two-point fluxes used for high-order ES multi-species solvers are not kinetic energy preserving (KEP). In this short communication we develop a two-point flux that is both entropy conserving (EC) and kinetic energy preserving by adapting Ranocha's single-species methodology for multi-species. The newly proposed EC and KEP flux is tested against existing two-point fluxes in the literature using the density pulse advection and Taylor-Green vortex cases. It is shown that EC and KEP flux retains both properties while the other fluxes only retain one of the desired properties. Since the two-point flux dictates the properties of the resulting scheme, the proposed flux is the ideal option for simulating multi-species compressible flow.