Error estimates for the Smagorinsky turbulence model: enhanced stability through scale separation and numerical stabilization

TL;DR

This paper demonstrates that the Smagorinsky turbulence model enhances stability and accuracy in large-scale flow simulations by leveraging spectral gap assumptions and finite element stabilization techniques, especially at high Reynolds numbers.

Contribution

It provides new stability estimates for the Smagorinsky model under spectral gap conditions and shows its effectiveness as a stabilizer in finite element methods for turbulent flow simulations.

Findings

Stability estimates depend only on large scale gradients.

Smagorinsky model acts as an effective stabilizer in finite element methods.

Near-optimal error estimates achieved in high Reynolds number regimes.

Abstract

In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $L^2$-norm for smooth flows in the pre-asymptotic high Reynolds number regime.