Damping Functions correct over-dissipation of the Smagorinsky Model

TL;DR

This paper provides a mathematical analysis of damping functions in the Smagorinsky model, demonstrating how a modified damping function can prevent over-dissipation and align with Kolmogorov's turbulence theory.

Contribution

The paper introduces a mathematical framework to evaluate energy dissipation in the Smagorinsky model with damping functions and proposes a modified damping that avoids over-dissipation.

Findings

Modified damping reduces over-dissipation in the Smagorinsky model.

Analysis confirms the modified damping aligns with Kolmogorov phenomenology.

The framework allows evaluation of energy dissipation for any damping function.

Abstract

This paper studies the time-averaged energy dissipation rate $\langle \varepsilon_{SMD} (u)\rangle$ for the combination of the Smagorinsky model and damping function. The Smagorinsky model is well known to over-damp. One common correction is to include damping functions that reduce the effects of model viscosity near walls. Mathematical analysis is given here that allows evaluation of $\langle \varepsilon_{SMD} (u)\rangle $ for any damping function. Moreover, the analysis motivates a modified van Driest damping. It is proven that the combination of the Smagorinsky with this modified damping function does not over dissipate and is also consistent with Kolmogorov phenomenology.