A coupling concept for Stokes-Darcy systems: the ICDD method

TL;DR

The paper introduces the Interface Control Domain Decomposition (ICDD) method to couple Stokes flow in a free-fluid region with Darcy flow in a porous medium via an overlapping transition layer, applicable to arbitrary flow directions at low Reynolds number. It formulates a macroscale Stokes–Darcy model with velocity continuity on $\Gamma_f$ and pressure continuity on $\Gamma_p$, solved in a non-intrusive manner using a Schur-complement approach that leverages existing CFD tooling. A practical rule links the transition-layer thickness to porosity and pore size via $\widehat{\delta}^*(\vartheta)$, enabling robust placement of the interfaces without auxiliary problems. Validation against microscale DNS across three test configurations shows convergence consistent with homogenization theory, with explicit error-rate predictions for velocity and pressure in both the fluid and porous regions, demonstrating the method’s accuracy and practicality for multiscale filtration problems.

Abstract

We present a coupling framework for Stokes-Darcy systems valid for arbitrary flow direction at low Reynolds numbers and for isotropic porous media. The proposed method is based on an overlapping domain decomposition concept to represent the transition region between the free-fluid and the porous-medium regimes. Matching conditions at the interfaces of the decomposition impose the continuity of velocity (on one interface) and pressure (on the other one) and the resulting algorithm can be easily implemented in a non-intrusive way. The numerical approximations of the fluid velocity and pressure obtained by the studied method converge to the corresponding counterparts computed by direct numerical simulation at the microscale, with convergence rates equal to suitable powers of the scale separation parameter $\varepsilon$ in agreement with classical results in homogenization.

Figures (26)

• Figure 1: Schematic representation of the macroscale domain $\Omega$.
• Figure 2: Schematic representation of the microscale domain $\Omega_\varepsilon$.
• Figure 3: Computational domain and velocity field for the configurations with pressure gradient (almost) parallel (Test case #1, left), perpendicular (Test case #2, centre), and oblique (Test case #3, right) to the porous medium.
• Figure 4: Example of computational mesh for the spectral element discretization of \ref{['eq:stokesGlobal']}.
• Figure 5: Test case #1. From left to right: horizontal and vertical components of the dimensionless microscale velocity and pressure with $\ell=\frac{1}{20}$ and circular obstacles with $r=0.2\,\ell$. (Top) The full solution where only the macroscale behaviour is visible; (bottom) solutions plotted with a different colour scale to highlight small variations both in the transition region and in the porous region.

Theorems & Definitions (1)

• Remark 2.1