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

We derive a formula for the P\'eclet number ($\mathrm{Pe}$) by estimating the relative strengths of various terms of the momentum equation. Using direct numerical simulations in three dimensions we show that in the turbulent regime, the fluid acceleration is dominated by the pressure gradient, with relatively small contributions arising from the buoyancy and the viscous term, in the viscous regime, acceleration is very small due to a balance between the buoyancy and the viscous term. Our formula for $\mathrm{Pe}$ describes the past experiments and numerical data quite well. We also show that the ratio of the nonlinear term and the viscous term is $\mathrm{Re} \mathrm{Ra}^{-0.14}$, where $\mathrm{Re}$ and $\mathrm{Ra}$ are Reynolds and Rayleigh numbers respectively, and that the viscous dissipation rate $\epsilon_u = (U^3/d) \mathrm{Ra}^{-0.21}$, where $U$ is the root mean square velocity and $d$ is the distance between the two horizontal plates. The aforementioned decrease in nonlinearity compared to free turbulence arises due to the wall effects.