Higher-Order Homogenized Boundary Conditions for Flows Over Rough and Porous Surfaces

A Modification of the Beavers–Joseph Condition for Arbitrary Flows to the Fluid–porous Interface

Physically consistent coupling conditions at the fluid–porous interface with correctly determined effective parameters are necessary for accurate modeling and simulation of various applications. To describe single-fluid-phase flows in coupled free-flow and porous-medium systems, the Stokes/Darcy equations are typically used together with the conservation of mass across the interface, the balance of normal forces and the Beavers–Joseph condition on the tangential velocity. The latter condition is suitable for flows parallel to the interface but not applicable for arbitrary flow directions. Moreover, the value of the Beavers–Joseph slip coefficient is uncertain. In the literature, it is routinely set equal to one that is not correct for many applications, even if the flow is parallel to the porous layer. In this paper, we reformulate the generalized interface condition on the tangential velocity component, recently developed for arbitrary flows in Stokes/Darcy systems, such that it has the same analytical form as the Beavers–Joseph condition. We compute the effective coefficients appearing in this modified condition using theory of homogenization with boundary layers. We demonstrate that the modified Beavers–Joseph condition is applicable for arbitrary flow directions to the fluid–porous interface. In addition, we propose an efficient two-level numerical algorithm based on simulated annealing to compute the optimal Beavers–Joseph parameter.

Article Highlights

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.