A high-order immersed boundary method with ghost cells and flux reconstruction for inviscid flows

We propose a high-order immersed boundary (IB) method for simulating inviscid flows using the flux reconstruction (FR) method, which offers the advantages of a compact stencil, high-order near-wall accuracy, and the ability to handle thin geometries and sharp corners. We utilize the FR method on a Cartesian grid to achieve a compact stencil, which enhances accuracy without requiring surrounding cell information, laying the foundation for future adaptive mesh refinement. Compared to the symmetry wall condition treatment, we combine the curvature corrected symmetry technique with the FR-based IB method to achieve a high-order accuracy near the wall boundary and utilize a third-order polynomial distribution to impose the boundary conditions. We extend the multi-valued ghost-cell method to the present approach to handle thin geometries and sharp corners in the ghost-cell-based IB method. The proposed method can be easily extended to three-dimensional cases. The numerical framework is validated through several two-dimensional simulations of inviscid flows. The approach is first demonstrated for a superposition of an isentropic vortex and a background flow problem with a Mach number of 0.2, which examines the order of accuracy without wall effects in the grid convergence study. Then, inviscid flow around a cylinder and flow through a channel with a bump are tested and compared to body-fitted simulations. Finally, flows around a National Advisory Committee for Aeronautics 0012 airfoil with zero and non-zero angles of attack are demonstrated to show the robustness for the thin geometry calculation. In particular, third-order accuracy is maintained in the near-wall region.