The Reynolds number of the large-scale circulation in turbulent Rayleigh-Benard convection

We measured Reynolds numbers $R_e$ of the large-scale circulation of turbulent Rayleigh-B\'enard convection over the Rayleigh-number range $2\times 10^8 \alt R \alt 10^{11}$ and Prandtl-number range $3.3 \alt \sigma \alt 29$ for cylindrical samples of aspect ratio $\Gamma = 1$. For $R \alt R_c \simeq 3times 10^9$ we found $R_e \sim R^{\beta_{eff}}$ with $beta_{eff} \simeq 0.46 < 1/2$. Here both the $\sigma$- and $R$- dependences are quantitatively consistent with the Grossmann- Lohse (GL) prediction. For $R > R_c$ we found $R_e = 0.106~ sigma^{-3/4} R^{1/2}$, which differs from the GL prediction. The relatively sharp transition at $R_c$ to the large-$R$ regime suggests a qualitative and sudden change that renders the GL prediction inapplicable.