A framework for computing effective boundary conditions at the interface between free fluid and a porous medium

Abstract

Interfacial boundary conditions determined from empirical or ad hoc models remain the standard approach to model fluid flows over porous media, even in situations where the topology of the porous medium is known. We propose a non-empirical and accurate method to compute the effective boundary conditions at the interface between a porous surface and an overlying flow. Using a multiscale expansion (homogenization) approach, we derive a tensorial generalized version of the empirical condition suggested by Beavers & Joseph (J. Fluid Mech., vol. 30 (01), 1967, pp. 197-207). The components of the tensors determining the effective slip velocity at the interface are obtained by solving a set of Stokes equations in a small computational domain near the interface containing both free flow and porous medium. Using the lid-driven cavity flow with a porous bed, we demonstrate that the derived boundary condition is accurate and robust by comparing an effective model to direct numerical simulations. Finally, we provide an open source code that solves the microscale problems and computes the velocity boundary condition without free parameters over any porous bed.

Turbulent Drag Reduction Using Anisotropic Permeable Substrates

“…Other macroscopic approaches to model flow within a permeable substrate include the volume-averaged Navier-Stokes equations (VANS) [18] which were used in [19, 20] and homogenisation [21, 22]. In the current work however, we focus on highly connected permeable substrates, for which the Brinkman model is a simple but reasonable approximation.…”

“…As the averaging volumes approach the interface with the free flow, this assumption would eventually break down. This problem is still controversial in the specialised literature [21, 22, 24], and often empirical jump conditions are used as discussed previously. For simplicity, here we will assume that pores are infinitely small, so the continuum hypothesis holds for any vanishingly small volume, and fluid variables are continuous across the interface.…”

A computational continuum model of poroelastic beds

“…Finally, for the material displacement, we impose continuity of stresses at the interface $[\text{boldC}:\frac{1}{2}(\mathrm{bold\nabla }\mathbf{\text{v}}+{false(\mathrm{bold\nabla }\mathbf{\text{v}}false)}^{\mathrm{T}})-\mathit{\alpha }p]\cdot \stackrel{false^}{n}=[-p\mathit{\delta }+\mu (\mathrm{bold\nabla }\mathbf{\text{u}}+{false(\mathrm{bold\nabla }\mathbf{\text{u}}false)}^{\mathrm{normalT}})]\cdot \stackrel{false^}{n}.$ We thus assume, similarly to [28], that the total stress of the free fluid is transferred to the total effective stress of the interior poroelastic medium. In general, the effective elasticity of the composite near the interface could however be different from its value in the interior (it is argued in general to depend on boundary conditions [28]) and, to arrive to a more accurate boundary condition, one could construct an interface cell with an elasticity problem, similarly as done for the velocity boundary condition [42]. One objective of this paper is to understand how the fluid shear stress is transferred to the solid stress (first across the interface then inside the bed) and if the interface correction is necessary; in §4, we will show that approximation of the interface effective stress with the interior parameters is able to capture the transfer of stress reasonably well.…”

“…In this work, we extend the velocity boundary condition derived by Lācis & Bagheri [42] for a rigid porous bed to include poroelasticity. 1 The condition for the tangential interface velocity is $\mathbf{\text{u}}\cdot \stackrel{false^}{\tau }={\mathrm{\partial }}_t\mathbf{\text{v}}\cdot \stackrel{false^}{\tau }+(-\frac{{\text{boldK}}^{\mathrm{normalif}}}{\mu }\cdot \mathrm{\nabla }{p}^{-}+\text{boldL}:[\mathrm{normal\nabla }\mathbf{\text{u}}+{false(\mathrm{normal\nabla }\mathbf{\text{u}}false)}^{\mathrm{normalT}}])\cdot \stackrel{false^}{\tau },$ where the unit vector $\stackrel{false^}{\tau }$ denotes both tangential directions of the surface.…”

“…The slip in the normal direction is essentially the fluid mass transport in and out of the poroelastic material, which has to be equal to the relative velocity from the interior. This same condition also arises from the interface cell [42], because the tensor L components corresponding to penetration interface velocity are zero.…”