Effects of time-filtering the Navier–Stokes equations

Abstract

The underlying premise of temporal large eddy simulation (TLES) is that the attenuation of high-frequency content also attenuates the high-wavenumber content. Yet, to date, the effect in wavenumber space of removing high-frequency oscillations by time-domain filtering is not well understood. In this work, we numerically investigate the relationship between the frequency and wavenumber with particular attention to the role of the temporal residual-stress in TLES. Moreover, since under-resolved simulations that use high-order, non-dissipative numerical methods require some measure of artificial dissipation for stabilization, we also discuss the regularization term with practical relevance to under-resolved applications of TLES. Specifically, we analyze the effects of Eulerian time-domain filtering with a causal exponential filter on homogeneous isotropic turbulence. The data are generated by direct numerical simulation of the Navier–Stokes equations, which are driven to maintain an average Reynolds number (Reλ) of 200. A priori, Fourier transformations of the velocity fields were performed in order to compute the unfiltered and filtered energy and dissipation spectra in both wavenumber space and wavenumber–frequency space. Furthermore, the amount of unresolved dissipation of an insufficiently resolved simulation was approximated in an attempt to estimate the required additional artificial dissipation. The results indicate that the numerically motivated stabilization term can be reduced due to temporal filtering. Moreover, it has been shown that a sharp cutoff in the frequency domain does not translate into a sharp cutoff in the wavenumber space. Thus, a hybrid model that combines temporal filtering for the residual-stress and spatial filtering for stabilization might be advantageous.