Embedding geometries in structured grids allows a simple treatment of complex objects in fluid flows. Various methods are available. The commonly used Brinkman-volume-penalization models geometries as porous media, where in the limit of vanishing porosity a solid object is approximated. In the simplest form, the velocity equations are augmented by a term penalizing the fluid velocity, the body velocity. yielding good results in many applications. However, it induces numerical stiffness, especially if high pressure gradients need to be balanced. Here, we focus on the effect of the reduced effective volume (commonly called porosity) of the porous medium. An approach is derived, which allows to reduce the flux through objects to practically zero with little increase of numerical stiffness. Also, non-slip boundary conditions and adiabatic boundary conditions are easily constructed. The porosity terms allow to keep the skew symmetry of the underlying numerical scheme, by which the numerical stability is improved. Furthermore, a very good conservation of mass and energy in the non-penalized domain can be achieved. The scheme is tested for acoustic scenarios, near incompressible and strongly compressible flows.
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