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

• Published in: Computers & fluids Vol. 188; pp. 136 - 163
• Main Authors: Colas, Clément, Ferrand, Martin, Hérard, Jean-Marc, Latché, Jean-Claude, Coupanec, Erwan Le
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 30-06-2019; Elsevier BV; Elsevier
• Subjects: Algorithms; Analysis of PDEs; Barriers; Compressible flows; Compressible fluids; Domains; Equations of state; Euler equations; Euler-Lagrange equation; Finite element method; Finite volumes; Fluid mechanics; Fluxes; Integral formulation; Integrals; Internal energy; Mathematical models; Mathematics; Mechanics; Physics; Porous media; Pressure-correction scheme; Shock wave reflection; Shock waves; Porous media; Pressure-correction scheme; Finite volumes; Euler equations; Compressible flows; Integral formulation; Finite Volumes; integral formulation; compressible flows; porous media; pressure-correction scheme
• ISSN: 0045-7930; 1879-0747
• DOI: 10.1016/j.compfluid.2019.05.014

•We propose an integral formulation of the Euler equations in order to account for obstructed domains.•The implicit numerical method enables to cope with liquids and gases with any EOS.•Positivity results are provided for pressure and density variables.•Verification test cases include reflection of shock wave on a wall.•A 3D validation test case focuses on the interaction of a shock wave with a set of obstacles.
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.