(untitled) (Advances in Applied Mathematics and Mechanics, 2022)
In this paper, we develop a direct-forcing immersed boundary projection method for simulating the dynamics in thermal fluid-solid interaction problems. The underlying idea of the method is that we treat the solid as made of fluid and introduce two virtual forcing terms. First, a virtual fluid force distributed only on the solid region is appended to the momentum equation to make the region behave like a real solid body and satisfy the prescribed velocity. Second, a virtual heat source located inside the solid region near the boundary is added to the energy transport equation to impose the thermal boundary condition on the solid boundary. We take the implicit second-order backward differentiation to discretize the time variable and employ the Choi-Moin projection scheme to split the coupled system. As for spatial discretization, second-order centered differences over a staggered Cartesian grid are used on the entire domain. The advantages of this method are its conceptual simplicity and ease of implementation. Numerical experiments are performed to demonstrate the high performance of the proposed method. Convergence tests show that the spatial convergence rates of all unknowns seem to be super-linear in the 1-norm and 2-norm while less than linear in the maximum norm.