Consistent closure modeling in large eddy simulations by direct approximation of the filtered advection term

Abstract

This article addresses the widely overlooked conceptual inconsistency of the large eddy simulation (LES) framework, namely that the commonly used advection term introduces higher wave numbers in the filtered Navier-Stokes equations than consistent with the definition of a filtered equation. It is explained how this inconsistency is the reason that flux limiters, stabilization terms, or dealiasing is often required and that the LES solution is typically mesh dependent. A consistent alternative is the direct approximation of the filtered advection term, for which we derive an exact expression based on an infinite series expansion with terms of increasing order in the filter width. We show that truncating the series expansion after few terms gives an expression that is highly correlated with the filtered advection term and a suitable LES model. A posteriori studies with decaying turbulence and a turbulent shear flow are conducted that reveal that the proposed approximation of the filtered advection term predicts improved kinetic energy spectra and filtered velocity correlations compared to classical LES.

Figures (17)

• Figure 1: Sketch of the filtered and unfiltered kinetic energy spectrum, $E$, as a function of the wave number, $k$. With the classical decomposition of the advection term, the advection term and $\tau_{ij}$ are even nonzero in the highlighted wave number range that lies beyond the maximum wave number of the velocity field, $k_\mathrm{c}$.
• Figure 2: Joint PDF between the diagonal components ($i=j$) of different modeled filtered advection terms and the explicitly filtered advection term at a filter width of $\sigma/\Delta x_\mathrm{DNS}=4$.
• Figure 3: Joint PDF between the off-diagonal components ($i\ne j$) of different modeled filtered advection terms and the explicitly filtered advection term at a filter width of $\sigma/\Delta x_\mathrm{DNS}=4$.
• Figure 4: Joint PDF between the diagonal components ($i=j$) of different modeled filtered advection terms and the explicitly filtered advection term at a filter width of $\sigma/\Delta x_\mathrm{DNS}=8$.
• Figure 5: Joint PDF between the off-diagonal components ($i\ne j$) of different modeled filtered advection terms and the explicitly filtered advection term at a filter width of $\sigma/\Delta x_\mathrm{DNS}=8$.