A general two-equation macroscopic turbulence model for incompressible flow in porous media

Abstract view of the practical and fundamental importance to heat and mass transfer, we present a two-equation turbulence model for incompressible flow within a fluid saturated and rigid porous medium. The derivation consists of time-averaging the general (macroscopic) transport equations and closing the model with the classical eddy diffusivity concept and the Kolmogorov-Prandtl relation. The transport equations for the turbulence kinetic energy (κ) and its dissipation rate (e) are attained from the general momentum equations. Analysis of the κ-e equations proves that for a small permeability medium, small enough to minimize the form drag (Forchheimer term), the effect of a porous matrix is to damp turbulence, as physically expected. For the large permeability case the analysis is inconclusive as the Forchheimer term contribution can be to enhance or to damp turbulence. In addition, the model demonstrates that the only possible solution for steady unidirectional flow is zero macroscopic turbulence kinetic energy. The implications of this conclusion are far reaching. Among them, this conclusion supports the hypothesis of having microscopic turbulence, known to exist at high speed flow, damped by the volume averaging process. Therefore, turbulence models derived directly from the general (macroscopic) equations will inevitably fail to characterize accurately turbulence induced by the porous matrix in a microscopic sense.