  The original goal of Large Eddy Simulations of fully developed turbulent
flows was to accurately describe large-scale flow features ${\bf u}(\Delta)$ at
the scales $r\geq \Delta$ where $\Delta$ is a size of computational mesh. The
effect of small-scale velocity fluctuations ($r<\Delta$) was to be accounted
for by effective transport coefficients (subgrid models) in the coarse-grained
Navier-Stokes equations. It is shown in this paper that, due to anomalous
inertial range scaling (intermittency) of the moments of velocity difference,
the existing subgrid models are intrinsically incapable of quantitatively
describing flow features at the scales $r<N\Delta$ with $N\approx 10$. This
increases computational work approximately by a factor $10^{3}-10^{4}$. The
breakdown of the widely used Smagorinsky relation for the subgrid viscosity on
the scales $\Delta/L<1$ is demonstrated and a modification accounting for
intermittency of the filtered out small-scale fluctuations is proposed.
