Analysis and reformulation of the $k$--$\omega$ turbulence model for buoyancy-driven thermal convection

Abstract

The representation of buoyancy-driven turbulence in Reynolds-averaged Navier--Stokes (RANS) 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 $k$--$\omega$ model for Rayleigh--B\'enard convection in an infinite layer, where turbulent kinetic energy is generated solely by buoyancy. The solution provides explicit scaling relations among the Rayleigh ($Ra$), Prandtl ($Pr$), and Nusselt ($Nu$) numbers that capture the simulation trends: $Nu \sim Ra^{1/3} Pr^{1/3}$ for $Pr \ll 1$ and $Nu \sim Ra^{1/3} Pr^{-0.415}$ for $Pr \gg 1$. This framework quantifies the discrepancies in the conventional buoyancy treatment and clarifies their origin. Informed by this analysis, the buoyancy-related modeling terms are reformulated to recover the measured trends: namely $Nu \sim Pr^{1/8}$ for $Pr \ll 1$ and $Nu \sim Pr^{0}$ for $Pr \gg 1$ at moderate $Ra$. 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--B\'enard convection, internally heated convection in two configurations, unstably stratified Couette flow, and vertically heated natural convection with varying aspect ratios. Across all cases, the corrected model provides significantly improved predictions of mean temperature fields and turbulent heat flux distributions.