A Reynolds-averaged Navier–Stokes closure for steady-state simulations of Rayleigh–Bénard convection

Abstract

A new turbulence model has been developed for a Reynolds-averaged Navier–Stokes (RANS) simulations of buoyancy-driven flows. This study proposes a modification to the buoyancy-related term in the conventional k–ε RANS model's ε equation. Typical two-equation RANS models provide accurate predictions in homogeneous shear flow, decaying turbulence, and log-law regions, but have uncertain effectiveness for buoyancy-driven flows, particularly concerning the buoyancy-related term in the ε equation. They have produced significant errors in natural convection scenarios where the buoyancy-related term dominantly affects the modeling results, such as in the Rayleigh–Bénard (RB) convection. Conventional models are known to inaccurately predict RB convection when treated as a steady-state problem with zero mean velocity, considering only the gravity-directed coordinate as the independent variable. The analysis reveals that the conventional RANS model, along with modeling terms for buoyancy effects, provides not only inaccurate but also divergent turbulent heat fluxes in RB convection at high Rayleigh numbers. The proposed model establishes mathematical conditions that enable steady-state RANS simulations to converge to consistent scaling relations for the Nusselt number across a wide range of Rayleigh and Prandtl numbers in RB convection. This approach algebraically modifies a single term in the ε equation, so that the term vanishes in the absence of buoyancy, so the modification integrates seamlessly with the conventional k–ε RANS model.