Two immersed boundary methods (IBM) for the simulation of conjugate heat transfer problems with complex geometries are introduced: a finite element (IFEM) and a finite volume (IFVM) immersed boundary methods are discussed. In the IFEM a projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations (NSEs) and energy equation in conjunction with the immersed boundary method for solving fluid flow and heat transfer problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body-fitted mesh. Dirichlet boundary constraints are satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the "immersed" boundary) and vice versa, it is possible to impose the extra constraint to the NSEs as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The IFEM approach presented shows third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution.
For the IFVM a ghost-cell approach with sharp interface scheme is used to enforce the boundary condition at the fluid/solid interface. The interpolation procedure at the immersed boundary preserves the overall second order accuracy of the base solver. The developed ghost-cell method is applied on a staggered configuration with the Semi-Implicit Method for Pressure-Linked Equations Revised algorithm. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the IFVM analytical solution.
Computations were performed using the IFEM and IFVM approaches for the two-dimensional flow over a backward-facing step, two-dimensional flow past a stationary circular cylinder, three-dimensional flow past a sphere and two and three-dimensional natural convection in an enclosure with/without immersed body. The numerical results obtained with the discussed IFEM and IFVM were compared against other IBMs available in literature and simulations performed with the commercial computational fluid dynamics code STAR-CCM+/V7.04.006. The benchmark test cases showed that the numerical results obtained with the implemented immersed boundary methods are in good agreement with the predictions from STAR-CCM+ and the numerical data from the other IBMs. The immersed boundary method based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.
