Jump Conditions and Surface-Excess Quantities at a Fluid/Porous Interface: A Multi-scale Approach

Abstract

We present a two-step up-scaling approach that allows to derive the jump conditions that must be imposed at the interface to account for transport phenomena in a fluid/porous domain. This general approach is first applied to a heat conduction problem to illustrate the main steps of the analysis. The heat flux and temperature jump conditions are related to surface-excess quantities, whose values depend on the interface location. Good agreement between the mesoscopic and macroscopic results are obtained, whatever the position of the interface inside the transition region. The approach is then applied to the problem of a laminar flow over a porous medium. The Beavers and Joseph relation is recovered, but only for a particular position of the interface.

The Interface Control Domain Decomposition Method for Stokes--Darcy Coupling

“…In Table 5.8 we report the number of ICDD iterations required to satisfy the stopping test on the residual up to a tolerance ǫ = 10 −9 , the infimum of the cost functional J h attained at convergence, and the norms 8) that measure the gap between Stokes and Darcy solutions on the overlap, normalized with respect to the size of the overlap (which is proportional to δ). We see that the gap on the velocity decays as κ, while that on the pressure is independent of κ. δ is set according to (5.7), by choosing α BJ = 1.…”

“…The BJS condition was derived from experimental observations by Beavers and Joseph [1], then simplified by Saffmann [46] and later justified mathematically by Jäger and Mikelić [29,30] by using homogenization techniques. Other approaches to derive the same condition are based on volume averaging, upscaling, or matched asymptotic expansion techniques (see, e.g., [40,41,31,8]). However, since the BJS condition depends on a coefficient related to the structure of the porous material close to the interface region and to the position of the interface itself, it is not straightforward to be characterized (see, e.g.…”

“…It is known that for isotropic media, δ should be set proportional to the characteristic length of the pores ε or, after nondimensionalization, to the ratio between ε and the characteristic length x s of the Stokes domain [46,40,30]. Using a totally different argument, a similar quantity d/h was introduced in the asymptotic expansion of Chandesris and Jamet in [8], where d is proportional to the pore dimension, while h is the height of the fluid standing above the porous layer.…”

Investigation of the effective permeability of vuggy or fractured porous media from a Darcy-Brinkman approach

“…In that sense, α must be considered as an adjustable parameter while the impact remains small and of second order if the interface varies within the range of the grain size [35]. Partial theoretical justifications of this approach were proposed [9,15,25,53] and were used practically in the [4] and [40] (i); geometry with Couette flow considered in [29] (ii) case of the simple geometry represented in Fig. 2i.…”

Pressure-driven flow in a channel with porous walls

“…James & Davis 2001). Numerical simulations have been carried out in the Stokes regime (Larson & Higdon 1987), for flow over banks of cylinders (Sahraoui & Kaviany 1992;Zhang & Prosperetti 2009) and for arrays of cubes (Breugem, Boersma & Uittenbogaard 2005;Chandesris & Jamet 2009;Valdés-Parada et al 2009). …”