Speaker
Description
Many applications in engineering and the applied sciences require coupling free incompressible flow with porous media flow. Such multi-physics coupling can be achieved via a split-domain approach, in which numerical techniques are tailored to free flow and porous media flow in the respective domains. This, however, introduces the problem of tracking the interface between subdomains on which explicit boundary conditions have to be formulated. A different approach considers the hydro-mechanical Darcy-Brinkman-Stokes model that formulates mass and momentum balance of an incompressible fluid on a domain of varying permeability. It implicitly contains free flow governed by the classical Navier-Stokes equations and porous media flow governed by the Darcy equation as permeability limits.
While the conceptual simplicity of the Darcy-Brinkman-Stokes model makes it attractive from a mathematical modeling perspective, the numerical solution procedure is typically challenging. Stability requirements depend on the flow regime and the design of accurate, uniformly stable Finite Element discretization hence requires special care. Furthermore, strong nonlinearities in material parameters, such as large localized variations in permeability, adversely affect the conditioning of resulting linear systems. In this presentation, we will introduce a mixed Discontinuous Galerkin (DG) finite element discretization for the Darcy-Brinkman-Stokes model that addresses these challenges and covers both free flow and porous media flow regimes.
The method integrates recently proposed ideas for DG methods that solve incompressible Navier-Stokes equations and Darcy flow to obtain a robust solver in both permeability limits via systematic mass-flux stabilization. The method is also stable for sudden spatio-temporal changes in the permeability pattern. We conduct numerical experiments using the discontinuous generalization of triangular and quadrilateral Taylor-Hood finite elements to demonstrate near-optimal L2 convergence for both Stokes and Darcy regimes, as well as for coupled Stokes-Darcy regimes. We show robustness of the method on a variety of 2D numerical benchmark problems, including problems with discontinuous, anisotropic, and time-dependent permeability fields, and discuss the challenges moving forward.
