A global similarity correction for the RANS modeling of natural convection in unstably stratified flows

Abstract

This study proposes a global similarity correction for Reynolds-averaged Navier--Stokes (RANS) modeling of buoyancy effects in unstably stratified flows. Conventional two-equation RANS models (e.g., the $k$-$\varepsilon$ model) lack a clear criterion for incorporating unstable buoyancy effects in their scale-determining equations (e.g., $\varepsilon$-equation). To address this gap, a global correction function is introduced, derived from a generalized algebraic formulation that incorporates available potential energy as an additional parameter. This function reproduces a global similarity law commonly observed in natural convection flows--for instance, the correlation among the Nusselt, Rayleigh, and Prandtl numbers, which can be approximately expressed as a single power law over a wide parameter range. A calibration method is proposed in which an approximate analytical solution for Rayleigh--B\'enard convection is obtained via equilibrium analysis, confirming that the proposed model captures similarity relations not addressed by conventional one-point closures. Numerical results show significantly improved agreement with experimental data, accurately reproducing Nusselt number dependencies over broad ranges of Rayleigh and Prandtl numbers in unstably stratified flows, such as Rayleigh--B\'enard convection and two types of internally heated convection. The method remains fully compatible with standard RANS frameworks and reverts to traditional turbulence treatments in shear-driven flows where buoyant effects are negligible. By introducing only a single, simple, algebraic global function in the conventional $\varepsilon$-equation, this approach significantly enhances the accuracy and robustness of buoyancy-driven turbulence simulations.