Model of incompressible turbulent flows via a kinetic theory

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 kinetic model for incompressible turbulent flows developed by Chen et al. ( Atmosphere , 2023, vol. 14(7), p. 1109), constructed for unbounded flows. The first extension is to reselect a relaxation time such that the turbulent transport coefficients are obtained consistently and better align with well-established turbulence theory. The Chapman–Enskog (CE) analysis of the kinetic model reproduces the 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. In particular, 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 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 direct numerical simulation data for turbulent Couette flow demonstrates excellent agreement in predicting mean velocity profiles, skin friction coefficients and Reynolds shear-stress distributions, although the near-wall-normal stress anisotropy is underestimated. The results show that averaged turbulent flow behaves similarly to rarefied-gas flow at finite Knudsen number, capturing non-Newtonian effects beyond linear eddy-viscosity models. This kinetic model provides a physics-based foundation for turbulence modelling with reduced empirical dependence.