Average velocity profile between a fluid layer and a porous medium: Brinkman boundary layer

It has been mentioned that, the existence of some terms in Darcy's law are the result of the up-scaling method applied to the Stokes flow problem at the pore-scale. To address this debate, in this work we perform, at the pore-scale, flow simulations in a free fluid/porous medium system using different models of granular porous media. The local velocity obtained from the Stokes equation allows to obtain the Darcy-scale velocity profiles by a direct averaging instead of using the up-scaled model. The results show the existence of a smooth transition zone in the average velocity profiles near the free fluid/porous medium inter-region. The size and shape of such transition zone depend on the size of the averaging domain and they are a result of averaging local quantities and not a result of solving average equations. In this way, we confirm the existence of an average velocity boundary layer (i.e. Brinkman boundary layer); thus the pertinence of considering other terms in Darcy's law can be certainly justified. We have also determined the extension of the influence of the flow in the free fluid inside the porous medium and the perturbation of the flow in porous medium on the flow in the free fluid.