Kinetic closure of turbulence: collision-side modeling beyond the filtered BGK--Boltzmann equation

TL;DR

This paper develops a kinetic closure framework for turbulence modeling that captures subgrid physics on the collision side of the Boltzmann equation, validated through simulations and compared with existing models.

Contribution

It introduces a novel kinetic closure approach that explicitly models non-Markovian collision dynamics and residual subgrid equilibrium, extending turbulence modeling beyond filtered BGK equations.

Findings

Validated kinetic closures via lattice Boltzmann simulations.

Compared new model with Smagorinsky and regularization collision models.

Showed the importance of non-Markovian effects in collision modeling.

Abstract

This article extends a recently introduced kinetic closure of turbulence by developing its theoretical framework, operational realizations, and validation. In contrast with filtered Navier--Stokes formulations, filtering the Boltzmann equation retains subgrid advective transport under the linear streaming operator, so that unresolved physics is concentrated on the collision side. We show that in the dilute-gas LES and RANS regimes, the main limitation of Boltzmann and BGK-type collision models is not the breakdown of molecular chaos, but the retention of a Markovian collision process at a scale where filtering induces finite temporal correlations in the collision product. In a BGK-type framework, the closure problem is dual: one must infer the filtered fine-grained equilibrium, which is not computable from filtered moments alone, and model the non-Markovian collision dynamics generated by the collision-product covariance. The present framework makes this dual structure explicit and represents the resulting collision-covariance source term through a BGK-like closure built from the subgrid equilibrium residual, with the turbulent relaxation frequency given by a first phenomenological realization. The framework relies on a Chapman--Enskog analysis organized by the reference timescale ratio emerging directly from the nondimensionalization of the kinetic equation and performed in the classical sense, thereby avoiding artificial turbulent scale separations. We show that the Chapman--Enskog structure is not a pure one-parameter Knudsen scaling: the primary ordering is set by the kinetic-to-macroscopic timescale ratio, while higher moments retain an additional Mach dependence through the mixed scaling of particle velocity. The resulting kinetic closures are validated through lattice Boltzmann simulations and compared with the Smagorinsky model and regularization-based collision models.