Immersed boundary method for high-order flux reconstruction based on volume penalization

In the last decade, there has been a lot of interest in developing high-order methods as a viable option for unsteady scale-resolving-simulations which are increasingly important in the industrial design process. High-order methods offer the advantages of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. Despite significant advance in high-order solution methods, the general CFD workflow (geometry, CAD preparation, meshing, solution, post-processing) has largely remained unchanged, with mesh generation being a significant bottleneck and often determining the overall quality of the solution. In this work, we aim to combine the numerical advantages of the high-order Flux Reconstruction (FR) method and the simplicity of the mesh generation (or lack thereof) of the Immersed Boundary Method (IBM) for steady and unsteady problems over moving geometries. The volume penalization (penalty-IBM) method is selected for its ease of implementation and robustness. Detailed discussions about numerical implementation, including the boundary representation, mask function, data reconstruction, and selection of the penalization parameter are given. Advantages of combining volume penalization in the high-order framework are shown by various numerical test cases. The approach is firstly demonstrated for the linear advection-diffusion equation by investigating the numerical convergence for the coupled FR-IBM approach. Thereafter, the accuracy of the approach is demonstrated for canonical (static) test cases in 2D and 3D when compared to a standard body-fitted unstructured simulation. Finally, the efficiency of the method to handle moving geometries is demonstrated for the flow around an airfoil with pitching and plunging motions.