Energy dissipation in the Smagorinsky model of turbulence

The Smagorinsky model often severely over-dissipates flows and, consistently, previous estimates of its energy dissipation rate blow up as Re→∞Re→∞. This report estimates time averaged model dissipation, 〈εS〉〈εS〉, under periodic boundary conditions as〈εS〉≤2U3L+Re−1U3L+3227CS2(δL)2U3L, where U,LU,L are global velocity and length scales and CS≃0.1,δ<1CS≃0.1,δ<1 are model parameters. Thus, in the absence of boundary layers, the Smagorinsky model does not over dissipate.