On the analytical behavior of the $k$--$ω$ turbulence model in buoyant-driven thermal convection
Authors
Da-Sol Joo
Abstract
The representation of buoyancy-driven turbulence in Reynolds-averaged Navier--Stokes models remains unresolved, with no widely accepted standard formulation. A key difficulty is the lack of analytical guidance for incorporating buoyant effects, particularly under unstable stratification. This study derives an analytical solution of the standard $k$--$ω$ model for Rayleigh--Bénard convection in an infinite layer, where turbulent kinetic energy is generated solely by buoyancy. The solution provides explicit scaling relations among the Rayleigh ($\mathit{Ra}$), Prandtl ($\mathit{Pr}$), and Nusselt ($\mathit{Nu}$) numbers that capture the simulation trends: $\mathit{Nu} \sim \mathit{Ra}^{1/3}\mathit{Pr}^{1/3}$ for $\mathit{Pr} \ll 1$ and $\mathit{Nu} \sim \mathit{Ra}^{1/3}\mathit{Pr}^{-0.415}$ for $\mathit{Pr} \gg 1$. This framework quantifies the discrepancies in the conventional buoyancy treatment and clarifies their origin. Informed by this analysis, the buoyancy-related modelling terms are reformulated to recover the measured $\mathit{Nu}$--$\mathit{Ra}$--$\mathit{Pr}$ trends. Only two dimensionless algebraic functions are introduced, which vanish in the absence of buoyancy, ensuring full compatibility with the standard closure. The corrected model is validated across a range of buoyancy-driven flows, including two-dimensional Rayleigh--Bénard convection, internally heated convection in two configurations, unstably stratified Couette flow, and vertically heated natural convection with varying aspect ratios. Across all cases, it provides highly accurate predictions.
