Kinetic Theory of Turbulence Modeling: Smallness Parameter, Scaling and Microscopic Derivation of Smagorinsky Model

TL;DR

This paper derives a turbulence model from kinetic theory using a mean-field approach, establishing the tensor-diffusivity model as the only Smagorinsky type model surviving in the hydrodynamic limit, and clarifies the scaling with Reynolds and Knudsen numbers.

Contribution

It provides the first rigorous derivation of a turbulence model from kinetic theory, linking microscopic kinetic equations to macroscopic turbulence modeling.

Findings

Tensor-diffusivity model is the only Smagorinsky type model surviving in the hydrodynamic limit.

Scaling laws for filter-width with Reynolds and Knudsen numbers are established.

First rigorous derivation of turbulence models from kinetic theory.

Abstract

A mean-field approach (filtering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-diffusivity model. Scaling of the filter-width with Reynolds number and Knudsen number is established. This sets the first rigorous step in deriving turbulence models from kinetic theory.