Modelling of compressible flows with highly oscillating initial data by homogenization

Abstract

We introduce a mathematical modelling of slightly compressible viscous flows with two well-separated space scales. We use as mean tool formal mathematical homogenization techniques. In the model derived, there appear closure terms, much as in usual physical turbulence models. Here, the closure terms are computed from the solution of a PDE system that governs the turbulent perturbation. This system is coupled to the mean flow PDE system. Starting from this model, we derive another of the k- ε family, including eddy diffusion terms. We solve this model by an explicit mixed finite volume-finite element technique with upwinding. We test it for a compressible steady mixing layer with different convective Mach numbers. The numerical results show a good qualitative prediction of the relevant mean quantities of the flow for moderately high convective Mach numbers. This is consistent with the theoretical foundations of the model.