A 2D compact finite difference immersed boundary method for flow in porous media

We present a compact finite differences method for the calculation of two-dimensional viscous flows in porous media. This is achieved by using body forces that allow for the imposition of boundary conditions that coincide with the computational grid. An implementation of the forcing of Mohd-Yusof is used in order to implement the immersed boundary. A detailed description of the original compact finite difference method used can be found in Ferreira de Sousa et al.. The unsteady, incompressible Navier-Stokes equations are solved in a Cartesian staggered grid with fourth-order Runge-Kutta temporal discretization and fourth-order compact schemes for spatial discretization, used to achieve highly accurate calculations. Special attention is given to the boundary condition implementation on the immersed media. In this paper, two different flows are calculated. First, the flow over a 2D square cylinder located along the centreline of a channel with free-slip boundary conditions. The computed drag coefficient is compared with numerical results available in the literature. The second flow configuration analyzed is the flow over a porous matrix composed of staggered square cylinders. Results for the pressure drop across the porous matrix are presented for a wide range of Reynolds numbers, along with flow visualization