Heat transport near the transition to the ultimate state of turbulent Rayleigh-B\'enard convection*

Measurements of the Nusselt number $Nu$ 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). For $Ra \leq 2\times 10^{13}$ the data yielded $Nu = Nu_0 Ra^{\gamma_{eff}}$ with $\gamma_{eff} = 0.312$, consistent with classical RBC.\footnote{G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys. {\bf 81}, 503 (2009)} For larger Ra $\gamma_{eff}$ increased, reaching approximately $\gamma_{eff} = 0.38$ near $Ra = 10^{15}$, consistent with predictions\footnote{R. H. Kraichnan, Phys. Fluids {\bf 5}, 1374 (1962)}$^,$\footnote{S. Grossmann and D. Lohse, Phys. Fluids {\bf 23}, 045108 (2011)} for an ultimate state with turbulent boundary layers.