High-order Immersed Boundary Method for Flux Reconstruction Method on Hierarchical Cartesian Grid

We propose new high-order Immersed Boundary Methods for a hierarchical Cartesian grid-based flow solver using the Flux Reconstruction (FR) method. The FR method is a high-order accuracy method for flow simulation. It can apply to unstructured grids suitable for analyzing a flow around complex geometries and can achieve high parallelization efficiency. Also, we can generate the hierarchical Cartesian grid automatically and robustly for complex geometries. However, the accuracy deteriorates near the wall in the Cartesian grid, although the interior domain is calculated using the high-order FR method. Therefore, we developed new high-order Immersed Boundary Methods using a high-order distribution of physical quantities from the FR method. Specifically, the curvature effect of the wall surface is investigated in detail. The accuracy is validated through basic inviscid flow simulations. As a result, we achieved the same accuracy as the order of the FR method (up to 4th-order) by polynomial extrapolation of the physical quantities from the fluid domain or by polynomial interpolation of the physical quantities considering the curvature of the wall surface.