Dynamics of large-scale quantities in Rayleigh-Bénard convection

Abstract

In this paper we estimate the relative strengths of various terms of the Rayleigh-B\'enard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity $U$ or the P\'eclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of ${\mathrm{Ra}}^{1/2}$ to the experimentally observed scaling of nearly ${\mathrm{Ra}}^{0.3}$.