Keywords: Brinkman penalization, complex geometries, compressible Navier-Stokes equations, shock waves.

In recent years a penalization technique introduced by Arquis and Caltagirone [2] and analyzed theoretically by Angot [1] has been considered by various authors in numerical simulations involving solid obstacles in incompressible flows. The physical idea of this Brinkman's type penalization is to model the obstacle as a porous medium with porosity tending to zero. The major advantage is the fact that the Navier-Stokes equations are then solved in an obstacle-free computational domain and no expensive techniques like body fitted grids or coordinate transformations need be involved.

This method requires the addition of a supplementary term in the momentum equations of the Navier-Stokes system with the idea of forcing the velocity to satisfy the no-slip conditions on the boundaries of the obstacle. Theoretical and numerical proofs of the efficiency of this method have been presented in the last decade

In this work, we extend this technique to the case of compressible shocked flows. A penalization term is added to the momentum but also to the energy equations with the aim of enforcing the physical boundary conditions on the obstacles.

The resulting system of equations is solved numerically using a high order shock capturing scheme to compute accurately the evolution of the flow while keeping sharp numerical profiles for shock waves. Thanks to the penalization technique, a regular Cartesian mesh, where such numerical schemes are especially efficients, can be used.

After presenting the details of the method and of the resulting algorithm, we design a series of numerical simulations of strong shock-obstacle interactions, including difficult impulsively started obstacles at high Mach numbers. The results are carefully analyzed and compared with well known theoretical properties of compressible flows. This computational study indicates that the penalization technique combined with a robust shock capturing scheme is able to represent accurately and efficiently the expected behavior of the considered compressible flows.

To the best of our knowledge this approach has not been considered in the literature and we hope this paper will confirm that the penalization method for compressible flows could be efficiently used in the future in numerical simulations involving complex industrial flows.

1

P. Angot, C.-H. Bruneau, P. Fabrie, A penalization method to take into account obstacles in incompressible flows, Numer. Math. (1999)81, pp 497-520. doi:10.1007/s002110050401

2

E. Arquis, J.P.Caltagirone, Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieu poreux: application à la convection naturelle, C.R. Acad. Sci. Paris II, 299 (1984), pp 1-4.

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