Development of an implicit high-order Flux Reconstruction solver for high-speed flows on simplex elements

In this study, a significant emphasis is placed on the extension of high-order Flux Reconstruction (FR) schemes to cater to high-speed flows, specifically those involving high Mach numbers. The paper delves into the development and implementation of these FR schemes on both straight and curved-edged 2D and 3D simplex elements within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The proposed FR solver provides more accurate detection of complex flow features over relatively coarser meshes when compared to their low-order peers (particularly in presence of shock waves), is fully implicit and able to simulate compressible flow problems governed by the Euler or Navier-Stokes equations on triangular and tetrahedral meshes. In order to simulate cases with shocks, a shock-capturing scheme previously developed for quadrilateral elements, was extended to tackle supersonic and hypersonic simulations. Extensive verification of the resulting FR solver (up to 7th order, a.k.a P6, for the solution polynomial and 3rd order a.k.a Q2 for the geometrical representation) has been performed on benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 9.6. The obtained results have been favorably compared to those available in the literature.