A Homogenization Approach for Turbulent Channel Flows over Porous Substrates: Formulation and Implementation of Effective Boundary Conditions

The turbulent flow through a plane channel bounded by a single permeable wall is considered; this is a problem of interest since a carefully chosen distribution of grains and voids in the porous medium can result in skin friction reduction for the flow in the channel. In the homogenization approach followed here, the flow is not resolved in the porous layer, but an effective velocity boundary condition is developed (and later enforced) at a virtual interface between the porous bed and the channel flow. The condition is valid up to order two in terms of a small gauge factor, the ratio of microscopic to macroscopic length scales; it contains slip coefficients, plus surface and bulk permeability coefficients, which arise from the solution of microscale problems solved in a representative elementary volume. Using the effective boundary conditions, free of empirical parameters, direct numerical simulations are then performed in the channel, considering a few different porous substrates. The results, examined in terms of mean values and turbulence statistics, demonstrate the drag-reducing effects of porous substrates with streamwise-preferential alignment of the solid grains.