A second-order, direct forcing, immersed boundary method for conjugate heat transport

We propose a direct forcing method for simulating discontinuous Dirichlet and Neumann conditions at the interface between two materials. The method is motivated by applications to conjugate heat transfer and electrohydrodynamics. We consider a 2D Poisson equation in a square domain in which an object made of one material is immersed in a second material. The Poisson equation is solved in both materials using finite-volume methods on a Cartesian grid. To extend the direct forcing method of Fadlun et al. (2000), we identify pairs of adjacent grid points, called forcing pairs, that lie on opposite sides of the interface. We approximate the fields about the forcing pairs using 2D Taylor polynomials. We show that Neumann conditions generally require a 12 point stencil, with 6 points in each material. The stencils and coefficients are easily implemented using the signed distance function. We verify second-order spatial accuracy by comparing with analytical solutions for two- and three-phase problems of varying geometries. We also explore cases where complex geometries require reduced stencils and/or smoothing of the local interface.