Analysis of Mesh Effects on Turbulent Flow Statistics

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.