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

TL;DR

This paper derives a general formula for large-scale velocity in Rayleigh-Bénard convection, linking it to Rayleigh and Prandtl numbers, and explains deviations in heat transfer scaling observed experimentally.

Contribution

It provides a new scaling formula for velocity in Rayleigh-Bénard convection applicable across different parameters, and analyzes the impact of wall effects and correlations on heat transfer scaling.

Findings

Derived a general velocity formula fitting simulation and experimental data.

Identified wall-bounded convection enhances viscous forces.

Explained deviation from theoretical to observed Nusselt number scaling.

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 $\mathrm{Ra}$ and Prandtl number $\mathrm{Pr}$. 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 $\mathrm{Ra}^{0.3}$.