Spectral Properties of High-Order Element Types for Implicit Large Eddy Simulation

The use of high-order schemes continues to increase, with current methods becoming more robust and reliable. The resolution of complex turbulent flows using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) can be computed more efficiently with high-order methods such as the Flux Reconstruction approach. We make use of the implicit form of LES, referred to as ILES, in which the numerical dissipation of the spatial scheme passively filters high-frequency modes, and no subgrid-scale turbulence model is explicitly implemented. Therefore, given the inherent three-dimensional behaviour of turbulent flows, it is important to understand the spectral characteristics of spatial discretizations in three dimensions. The dispersion and dissipative properties of hexahedra, prismatic and tetrahedral element types are compared using Von Neumann analysis. This comparison is performed on a per degree of freedom basis to assess their suitability for ILES in terms of computational cost. We observe dispersion relations that display non-smooth behaviour for tetrahedral and prismatic elements. In addition, the periodicity of the dispersion relations in one dimension is generally not observed in three-dimensional configurations. Semilogarithmic plots of the numerical error are presented. We observe that the amount of numerical dissipation and dispersion added by hexahedral elements is the least, followed by prisms and finally tetrahedra. We validate our analysis comparing results obtained on computational domains with comparable computational cost against DNS data. Hexahedral elements have the best agreement with the reference data, followed by prismatic and finally tetrahedral elements, which is consistent with the spectral analysis.