Dynamics of large-scale quantities in Rayleigh-B\'enard convection

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 $\mathrm{R}\mathrm{a}$ and Prandtl number $\mathrm{P}\mathrm{r}$. 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 ${\mathrm{R}\mathrm{a}}^{1\mathrm{/}2}$ to the experimentally observed scaling of nearly ${\mathrm{R}\mathrm{a}}^{0.3}$.