Constructing turbulence models using the kinetic Fokker-Planck equation

This study presents a novel approach for constructing turbulence models using the kinetic Fokker-Planck equation. By leveraging the inherent similarities between Brownian motion and turbulent dynamics, we formulate a Fokker-Planck equation tailored for turbulence at the hydrodynamic level. In this model, turbulent energy plays a role analogous to temperature in molecular thermodynamics, and the large-scale structures are characterised by a turbulent relaxation time. This model aligns with the framework of Pope's generalised Langevin model, with the first moment recovering the Reynolds-averaged Navier-Stokes (RANS) equations, and the second moment yielding a partially modelled Reynolds stress transport equation. Utilising the Chapman-Enskog expansion, we derive asymptotic solutions for this turbulent Fokker-Planck equation. With an appropriate choice of relaxation time, we obtain a linear eddy viscosity model at first order, and a quadratic Reynolds stress constitutive relationship at second order. Comparative analysis of the coefficients of the quadratic expression with typical nonlinear viscosity models reveals qualitative consistency. To further validate this kinetic-based nonlinear viscosity model, we integrate it as a RANS model within computational fluid dynamics codes, and calculate three typical cases. The results demonstrate that this quadratic eddy viscosity model outperforms the linear model and shows comparability to a cubic model for two-dimensional flows, without the introduction of ad hoc parameters in the Reynolds stress constitutive relationship.