An Immersed Boundary Method for Computing Heat and Fluid Flow in Porous Media

A volume-penalizing immersed boundary (IB) method is presented that facilitates the computation of fluid flow in complex porous media. The computational domain is composed of a uniform Cartesian grid, and solid bodies are approximated on this grid using a series of grid cells (i.e., a “staircase” approximation). Solid bodies are distinguished from fluid regions using a binary phase-indicator function: Taking the value of “1” in the solid parts of the domain and “0” in the fluid parts. The effect of solid bodies on the flow is modeled using a source term in the momentum equations. The source term is active only within solid parts of the domain, and enforces the no-slip boundary condition. Fluid regions are governed by the incompressible Navier-Stokes equations. An extension of the IB method is proposed to tackle coupled fluid-solid heat transfer. The extended IB method is validated for Poiseuille flow, which allows for a direct comparison of the numerical results against a closed analytical solution. We subsequently apply the extended IB method to flow in a structured porous medium and focus on bulk properties such as the gradient of the average pressure and the Nusselt number. Reliable qualitative results were obtained with 16-32 grid points per singly-connected fluid region.