On the effect of the Prandtl number on the onset of Bénard convection

This note summarizes the results of a numerical study designed to question (i.e., refute or validate) Chao et al.'s¹ and Bertin and Ozoe's² conclusion that the critical Rayleigh number increases substantially as the Prandtl number becomes very small. The numerical method is based on the finite-difference control volume formulation and the complete equations for two-dimensional (2-D) time-dependent flow. The present results show that the lowest attainable Rayleigh number for numerically simulated convection increases as Pr decreases below 0.1. These results also extend the Prandtl number domain of the observations down to Pr=10^-4 and indicate that the natural shape of a single roll in this Pr range is approximately square. The discrepancy between these observations and the constant Ra_c=1,707.8 of the linear stability analysis is attributed to the extrapolation method on which the numerical convection-onset Ra data^1,2 were based. It is shown that the numerical results agree with the linear stability constant Ra_c=1,707.8 and Schlüter et al.'s⁹ small amplitude perturbation analysis.