Flux Reconstruction Discontinuous Spectral Element Method for Compressible Turbulent Flows

This project develops a high-order flux reconstruction discontinuous spectral element method (FRDSEM) solver capable of performing large-scale parallel computation of multicomponent compressible turbulent flows. The scope of this project can be broken down into three stages. First, an existing three-dimensional (3D) code is optimized and scaled up to perform large-scale simulations on the order of billions of solution points. The new 3D solver code demonstrates high efficiency in parallel computations and is successfully executed on approximately 130,000 cores. For the next stage, we develop a high-order numerical scheme that performs superior to the current discontinuous spectral element method (DSEM) code to simulate single-component compressible turbulent flows. A flux reconstruction approach is implemented in the DSEM framework to improve the accuracy and stability of numerical simulations. A new class of correction functions is derived based on a weighted orthogonality condition, leading to Jacobi correction functions' introduction. Jacobi correction functions can construct a broad range of other high-order numerical schemes with various numerical characteristics, such as high numerical dissipation to suppress aliasing-driven errors, super accuracy, and solution boundedness in shock prediction. Performing the under-resolved simulation of Taylor-Green vortex flow demonstrates that the Jacobi correction functions can remove aliasing errors leading to an accurate prediction of the decay rate of turbulent kinetic energy. The third stage involves adapting the FRDSEM to multicomponent flow simulation. The nodal FRDSEM is improved to avoid pressure and velocity oscillations from material interfaces in the solution of multicomponent compressible flows. A novel double flux method is tailored to account for variations in the ratio of the specific heats at material interfaces. The double flux model is combined with a conservative method of calculating the convective flux function to avoid energy conservation error at shock fronts. Local artificial diffusion (LAD) terms are also introduced for capturing sharp discontinuities. The new approach maintains the high convergence rate of the FRDSEM scheme at the smooth regions of the solution while removing Gibbs' oscillations. The scheme enables the use of high polynomial orders along with a high CFL number compared to an existing DG scheme.