An implicit integral formulation to model inviscid fluid flows in obstructed media

We focus here on a technique to compute compressible fluid flows in physical domains cluttered up with many small obstacles. This technique, referred to here as the integral formulation, consists in integrating the flow governing equations over the fluid part of control volumes including both fluid and solid zones; doing so, the integral of fluxes over solid boundaries may appear, for which expressions as a function of discrete variables must be provided. The integral formulation presents two essential advantages: first, we naturally recover the standard fluid approach when the mesh is refined; second, fluid/solid interactions may be, to some extent, modelled to recover the singular head losses at the interface between a free and a congested part of the computational domain. We apply here this approach to the Euler equations, using a collocated space discretization and a pressure correction algorithm, preserving the positivity of both the density and the internal energy. Verification test cases are performed, including a Riemann problem in a free domain and a shock wave reflection on a wall, using an equation of state which is suitable for weakly compressible fluid flows. Finally, we address a two-dimensional situation, where a shock wave impacts a set of obstacles; we observe a very encouraging agreement between the integral approach results and a CFD reference solution obtained with a pure fluid approach on a fine mesh.