Study of three-dimensional natural convection in a cube near critical Ra values

Numerical simulation of natural convection found in the literatures were usually performed with the Boussinesq approximation, while a few of the study were performed with the less restrictive low-Mach-number approximation. It is interesting to know whether or not the usual approximations made in formulating the natural convection phenomena, particularly under the condition near the critical Rayleigh number (Ra) is suitable. A three-dimensional natural convection in a cube heated by a vertical side is investigated. The tested Ra values range from 10⁵ to the critical value of O (10⁸). Comparison of the numerical results obtained with the Boussinesq approximation with what obtained without any approximation in the model formulation reveals that the Boussinesq approximation works in the cases with small Ra such as the one at Ra = 10⁵ while the apparent discrepancy appears in the cases at Ra ≥ 2 × 10⁸. It evidences that the natural convection near the critical Ra values has to be solved with the formulation without any approximations.