The Reynolds number near the transition to the ultimate state of turbulent Rayleigh-B\'enard convection*

Measurements of a Reynolds number $Re$ for Rayleigh-B\'enard convection (RBC) of a cylindrical sample over the Rayleigh-number range $2\times 10^{12} \leq Ra \leq 2\times 10^{15}$ and the Prandtl-number range $0.79 \leq Pr \leq 0.86$ are presented. The aspect ratio $\Gamma \equiv D/L$ was 0.50 ($D= 1.12$ m was the diameter and $L = 2.24$ m was the height). We used the elliptic approximation of He and Zhang\footnote{G.-W. He and J.-B. Zhang, Phys. Rev. {\bf 73}, 055303 (2006)}$^,$\footnote{X. He, G. He, and P. Tong, Phys. Rev. {\bf 81}, 065303 (2010)} to determine an effective Re. For $Ra \geq 3 \times 10^{14}$ the data yielded $Re = Re_0 Ra^{\alpha_{eff}}$ with $\alpha_{eff} \simeq 0.50$. This result is consistent with predictions for the ultimate state, where the boundary layers are turbulent.\footnote{R. H. Kraichnan, Phys. Fluids {\bf 5}, 1374 (1962)}$^,$\footnote{S. Grossmann and D. Lohse, Phys. Fluids {\bf 23}, 045108 (2011)}