  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.
