k–ε Macro-scale modeling of turbulence based on a two scale analysis in porous media

In this paper, turbulent flows in media laden with solid structures are considered. Following previous studies, we apply both statistical and spatial averages. The solid matrix action on turbulence is then put forward as a sub-filter production. To model this term, we perform a two-scale analysis that highlights energy transfers between the mean motion, the macroscopic and the sub-filter turbulent kinetic energies. Within this framework, we show that the sub-filter production is an energy transfer between the mean motion kinetic energy and the turbulent kinetic energy. We propose to model this sub-filter production using the wake dissipation and the work performed by the mean macroscopic flow against the mean specific drag. From this analysis, a macroscopic k–ε model is then derived for a stratified porous media that includes a friction factor model accounting for turbulence non-equilibrium. Comparisons between this model and a former model [Nakayama, A., Kuwahara, F., 1996. A macroscopic turbulence model for flow in a porous medium. J. Fluid Eng.-T ASME 121, 427–433] are carried out.