A Simple Diffuse Interface Approach for Compressible Flows around Moving Solids of Arbitrary Shape Based on a Reduced Baer–Nunziato Model

In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer–Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid. We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso et al. (1995)[89], which justifies the use of a path-conservative approach for treating the non-conservative products. The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion. The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies. © 2020 Elsevier Ltd. All rights reserved.