Analysis of Mesh Effects on Turbulent Flow Statistics

TL;DR

This paper derives bounds on the energy dissipation rate in under-resolved turbulence simulations using the Smagorinsky model, revealing potential over-dissipation and emphasizing the boundary layer's importance.

Contribution

It provides the first analytical upper bounds on resolved-scale energy dissipation for the Smagorinsky model in turbulent shear flow, highlighting over-dissipation issues.

Findings

Upper bounds on dissipation rate are independent of viscosity at high Reynolds numbers.

The analysis indicates over-dissipation for all positive model parameters.

Turbulent boundary layer is more critical than Kolmogorov microscale in shear flow.

Abstract

Turbulence models, such as the Smagorinsky model herein, are used to represent the energy lost from resolved to under-resolved scales due to the energy cascade (i.e. non-linearity). Analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared, but none (yet) study energy dissipation restricted to resolved scales, i.e. after spacial discretization with $h >$ micro scale. We do so herein for the Smagorinsky model. Upper bounds are derived on the \textit{computed} time-averaged energy dissipation rate, $\langle \varepsilon (u^h)\rangle$, for an under-resolved mesh $h$ for turbulent shear flow. For coarse mesh size $ \mathcal{O}(\mathcal{Re}^{-1}) < h < L $, it is proven, $$ \langle \varepsilon (u^h)\rangle\leq \big[ (\frac{C_s\, \delta}{h})^2+ \frac{L^5}{(C_s \delta)^4\,h}+\frac{L^{\frac{5}{2}}}{(C_s\, \delta)^{4}}\, {h^{\frac{3}{2}}}\big]\, \frac{U^3}{L}, $$ where $U$ and $L$ are global velocity and length scale and $C_s$ and $\delta$ are model parameters. This upper bound is independent of the viscosity at high Reynolds number, is in accord with the scaling theory of turbulent. This estimate suggests over-dissipation for any of $C_s>0$ and $\delta>0$, consistent with numerical evidence on the effects of model viscosity (without wall damping function). Moreover, the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the Kolmogorov microscale.