Department of Mathematics - Seminar on Applied Mathematics - 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 higher-order non-equilibrium physics such as non-Newtonian effects. In this work, we present an extension and theoretical analysis of a recent kinetic model for incompressible turbulent flows (Atmos. 14:1109, 2023), which was constructed for unbounded fully developed turbulent flows. The first extension is to reselect a relaxation time such that the turbulent transport coefficients can be obtained more consistently and are in closer agreement with well-established turbulence theory. The Chapman-Enskog analysis of the kinetic model reproduces the traditional linear eddy viscosity and gradient diffusion models for Reynolds stress tensor and turbulent kinetic energy flux at the first order, and nonlinear eddy viscosity and closure models at the second order. The second extension is to enable the model for wall-bounded turbulent flows with preserved near-wall asymptotic behaviors. This involves developing a low-Reynolds number kinetic model that incorporates wall damping effects and explicit viscous diffusion. Comprehensive validation against experimental and DNS data for homogeneous and inhomogeneous shear flows demonstrates excellent agreement in predicting mean velocity, turbulent kinetic energy, and Reynolds stress profiles. It demonstrates that an averaged turbulent flow behaves similarly to a rarefied gas flow at a finite Knudsen number, capturing non-Newtonian effects that cannot be represented by linear eddy viscosity models. The present kinetic model provides a physics-based foundation for turbulence modelling with reduced empirical dependence.