Spectral/hp element methods' linear mechanism of (apparent) energy transfer in Fourier space: Insights into dispersion analysis for implicit LES

Abstract

In recent years, different dispersion-diffusion (eigen)analyses have been developed and used to assess various spectral element methods (SEMs) with regards to accuracy and stability, both of which are very important aspects for under-resolved computations of transitional and turbulent flows. Not surprisingly, eigenanalysis has been used recurrently to probe the inner-workings of SEM-based implicit LES approaches, where numerical dissipation acts alone in lieu of a subgrid model. In this study we present and discuss an intriguing linear mechanism that causes energy transfer across Fourier modes as seen in the energy spectrum of SEM computations. Despite its linear nature, this mechanism has not been considered in eigenanalyses so far, possibly due to its connection to the often overlooked multiple eigencurves feature of periodic eigenanalysis. As we unveil the mechanism in the simplified context of linear advection, we point out how its effects might take place in actual turbulence simulations. In particular, we highlight how taking it into account in eigenanalysis can improve dissipation estimates in wavenumber space, potentially allowing for a superior correlation between dissipation estimates and energy spectra measured in SEM-based eddy-resolving turbulence computations.