Abstract

High-order Flux Reconstruction (FR) schemes can simulate unsteady turbulent flows using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) in the vicinity of complex geometries. The application of FR schemes can be limited by non-linear instabilities, related to oscillatory behaviour of the element-wise numerical solution, causing nonphysical solutions. In this study, filtering is studied for hexagonal element types and solution polynomial of degrees 3, 4, and 5 at different Mach numbers ranging from 0.1 to 0.5. A new exponential filtering function is applied globally to all elements and artificially damps high-frequency oscillations to improve numerical stability. Numerous numerical tests have been performed to investigate different parameters in the exponential filtering function. The optimum set of these parameters is obtained such that the highest solution polynomial modes are damped while the lower ones remain untouched to preserve accuracy. The solution polynomial is filtered after each time step; however, the filtering operator is defined, through the concept of characteristic time, to be independent of the time-step size. The exponential filter has been implemented and rigorously studied to evaluate its accuracy. To verify the order of accuracy, advection of an isentropic vortex has been analyzed. To study accuracy for LES, the Taylor-Green vortex test case is studied as a free turbulent flow. Finally, a previously unstable wall-bounded turbulent channel flow test case along with a NACA0020 airfoil at high angle of attack are considered to study the effects of the proposed filtering on stability and accuracy of practical applications.