Model of incompressible turbulent flows via a kinetic theory

TL;DR

This work refines a kinetic-theory turbulence model based on a Klimontovich-type VDF with a BGK collision term, deriving macroscopic closures via Chapman–Enskog expansion. By selecting a relaxation time $τ=\frac{K}{7ε}$, the authors obtain linear eddy-viscosity and gradient-diffusion closures with coefficients matching conventional $K$-$ε$ schemes, and reveal second-order nonlinear corrections including a Burnett-like TKE flux contribution. The study extends the framework to wall-bounded flows with a low-Reynolds-number (LR-BGK) model and a high-Reynolds-number (HR-BGK) wall-function approach, each paired with appropriate boundary conditions, and validates against turbulent plane Couette flow data. Non-equilibrium velocity-distribution functions demonstrate significant departures from local Gaussian equilibrium for $Φ_2$ and $Φ_3$, enabling the model to capture non-Newtonian, anisotropic stress features beyond linear closures. Overall, the kinetic approach provides a physics-based, less empirically dependent foundation for turbulence modelling with accurate mean and Reynolds-stress predictions, and offers a path toward more comprehensive descriptions of near-wall and non-equilibrium turbulence phenomena.

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.

Figures (13)

• Figure 1: Sketch of turbulent plane Couette flow between two infinite parallel plates separated by $2h$. The plates move in opposite directions with constant velocities $\pm U_w$ in the $x$-direction, generating a monotonic mean velocity profile $U_x(y)$.
• Figure 2: Mean velocity profiles scaled in (a) outer and (b) wall units of the turbulent Couette flow. The solid and dash‑dot lines represent the predictions of the LR-BGK model and the HR-BGK model at $\text{Re} = 1300$, respectively. The diamonds denote the experimental data of bech1995investigation at $\text{Re} = 1260$, and the circles denote the DNS data of bech1995investigation at $\text{Re} = 1300$. The dashed line represents the classical logarithmic law profile $\left(\ln y^+\right)/0.41+6.5$.
• Figure 3: Mean velocity profiles scaled in (a) outer and (b) wall units of turbulent Couette flow. The solid and dash‑dot lines represent the predictions of the LR-BGK model and the HR-BGK model at $\text{Re} = 10133$, respectively. The diamonds denote the experimental data of robertson1959study at $\text{Re} = 10000$, and the circles denote the DNS data of pirozzoli2014turbulence at $\text{Re} = 10133$. The dashed line represents the classical logarithmic law profile $\left(\ln y^+\right)/0.41+7$.
• Figure 4: Variation of skin friction coefficient with Reynolds number. The solid and dash‑dot lines represent the predictions of the LR-BGK model and the HR-BGK model, respectively. Open symbols refer to experimental data by kitoh2005experimental (triangles), el1980velocity (diamonds), reichardt1956geschwindigkeitsverteilung (right‑pointing triangles), robertson1959turbulent (pluses). Filled symbols refer to DNS data by bech1995investigation (left‑pointing triangles), tsukahara2006dns (squares), pirozzoli2014turbulence (circles).
• Figure 5: Reynolds shear stress profiles scaled in wall units of turbulent Couette flow at (a) $\text{Re} = 1300$ and (b) $\text{Re} = 10133$. The solid and dash‑dot lines represent the predictions of the LR-BGK model and the HR-BGK model, respectively. The circles and the diamonds denote the DNS data of bech1995investigation and pirozzoli2014turbulence, respectively.