On the Equivalence of the Discontinuous One- and Two-Domain Approaches for the Modeling of Transport Phenomena at a Fluid/Porous Interface

Abstract

In the quest (i) to determine the form of the boundary conditions that must be applied at a fluid/porous interface and (ii) to determine the value of the jump parameters that appear in the expression for these boundary conditions, two different approaches are commonly considered: the so-called one-domain and two-domain approaches. These approaches are commonly thought to be different, and they are thus sometimes compared to each other to determine the value of jump parameters. In this article, we show that the two-domain and discontinuous one-domain approaches are actually strictly equivalent, provided that the latter is mathematically interpreted in the sense of distributions. This equivalence is shown in details for a heat conduction problem and for the more classical Darcy-Brinkman problem. We show in particular that interfacial jumps are introduced in the discontinuous one-domain approach through Dirac delta functions. Numerical issues are then discussed that show that subtle discretization truncation errors give rise to large variations that can be mis-interpreted as the sign of the existence of jump parameters.