Model of incompressible turbulent flows via a kinetic theory

TL;DR

This paper extends a kinetic theory model for incompressible turbulent flows, enabling better alignment with turbulence physics and applicability to wall-bounded flows, validated by experimental and DNS data.

Contribution

It introduces a refined kinetic model with improved turbulence coefficient consistency and wall-bounded flow capabilities, including a novel Chapman-Enskog solution for kinetic energy flux.

Findings

The model reproduces linear and nonlinear eddy viscosity models at different orders.

It accurately predicts mean velocity, skin friction, and Reynolds stress in plane Couette flow.

The flow behaves like a rarefied gas at finite Knudsen number, capturing non-Newtonian effects.

Abstract

Kinetic theory offers a promising alternative to conventional turbulence modelling by providing a mesoscopic perspective that naturally captures non-equilibrium physics such as non-Newtonian effects. In this work, we present an extension and theoretical analysis of the recent kinetic model for incompressible turbulent flows developed by Chen et al. (Atmos. 14(7), 1109, 2023), constructed for unbounded flows. The first extension is to reselect a relaxation time such that the turbulent transport coefficients are obtained more consistently and better align with well-established turbulence theory. The Chapman-Enskog (CE) analysis of the kinetic model reproduces the traditional linear eddy viscosity and gradient diffusion models for Reynolds stress and turbulent kinetic energy flux at the first order, and yields nonlinear eddy viscosity and closure models at the second order. Particularly, a previously unreported CE solution for turbulent kinetic energy flux is obtained. The second extension is to enable the model for wall-bounded turbulent flows with preserved near-wall asymptotic behaviours. This involves developing a low-Reynolds number kinetic model incorporating wall damping effects and viscous diffusion, with boundary conditions enabling both viscous sublayer resolution and wall function application. Comprehensive validation against experimental and DNS data for turbulent plane Couette flow demonstrates excellent agreement in predicting mean velocity profiles, skin friction coefficients, and Reynolds stress distributions. It reveals that an averaged turbulent flow behaves similarly to a rarefied gas flow at a finite Knudsen number, capturing non-Newtonian effects inaccessible to linear eddy viscosity models. This kinetic model provides a physics-based foundation for turbulence modelling with reduced empirical dependence.