A least squares augmented immersed interface method for solving Navier–Stokes and Darcy coupling equations

Abstract

A new finite difference method based on Cartesian meshes and fast Poisson/Helmholtz solvers is proposed to solve the coupling of a fluid flow modeled by the incompressible Navier–Stokes equations and a porous media modeled by the Darcy’s law. The finite difference discretization in time is based on the pressure Poisson equation formulation. At each time step, several augmented variables along the interface between the fluid flow and the porous media are introduced so that the coupled equations can be decoupled into several Poisson/Helmholtz equations with those augmented variables acting as jumps of the unknown solution and some directional derivatives. The augmented variables should be chosen so that the Beavers–Joseph–Saffman (BJS) or Beavers–Joseph (BJ) and other interface conditions are satisfied. It has been tested that a direct extension of the augmented idea in [27] does not work well when the fluid flow is modeled by the Navier–Stokes equations. One of the new ideas of this paper is to enforce the divergence free condition at the interface from the fluid side. In this way, the Schur complement matrix for the augmented variables is over-determined and the least squares solution is used for the coupling problem. The new augmented approach enables us to solve the Navier–Stokes and Darcy coupling efficiently with second order accurate velocity and pressure in the L∞ norm for tested problems. The proposed new idea in enforcing the divergence free condition at the interface from the fluid side has also been utilized to solve the Stokes and Darcy coupling equations and shown to outperform the original method in [27]. In additional to the detailed accuracy check for the present method, some interesting numerical simulations for Navier–Stokes and Darcy coupling have been conducted in this paper as well.