Energy Dissipation in the Smagorinsky Model of Turbulence

TL;DR

This paper analyzes the energy dissipation of the Smagorinsky turbulence model, showing that without boundary layers, it does not over dissipate, contrary to previous beliefs.

Contribution

The study provides new dissipation estimates for the Smagorinsky model in boundary layer-free flows, demonstrating it aligns with Navier-Stokes behavior when boundary effects are absent.

Findings

Model dissipation rate matches Navier-Stokes estimates without boundary layers.

Over dissipation is primarily due to boundary layer effects, not small-scale nonlinearity.

Smagorinsky model does not inherently over dissipate in idealized conditions.

Abstract

The Smagorinsky model, unmodified, is often reported to severely overdiffuse flows. Previous estimates of the energy dissipation rate of the Smagorinsky model for shear flows reflect a blow up of model energy dissipation as Re increases. This blow up is consistent with the numerical evidence and leads to the question: Is the over dissipation due to the influence of the turbulent viscosity in boundary layers alone or is its action on small scales generated by the nonlinearity through the cascade also a contributor? This report develops model dissipation estimates for body force driven flow under periodic boundary conditions (and thus only with nonlinearity generated small scales). It is proven that the model's time averaged energy dissipation rate satisfies the same upper bound as for the NSE plus one additional term that vanishes uniformly in the Reynolds number as the Smagorinsky length scale decreases. Since this estimate is consistent with that observed for the NSE, it establishes that, without boundary layers, the Smagorinsky model does not over dissipate.