Heat transport by turbulent Rayleigh-B\'enard convection for $\text{Pra}\simeq 0.8$ and $4\times {10}^{11}\text{alt}\text{Ra}\text{alt}2\times {10}^{14}$: Ultimate-state transition for aspect ratio $\mathrm{\Gamma }=1.00$

We report experimental results for heat-transport measurements by turbulent Rayleigh-B'enard convection in a cylindrical sample of aspect ratio $\Gamma \equiv D/L = 1.00$($D = 1.12$misthediameterand$L = 1.12$ m the height). They are for the Rayleigh-number range $4\times10^{11} \alt \Ra \alt 2\times10^{14}$ and for Prandtl numbers \Pra\ between 0.79 and 0.86. For $\text{Ra}<{\text{Ra}}_1^{\ast }\simeq 2\times {10}^{13}$ we find $\Nu = N_0 \Ra^{\gamma_{eff}}$with$\gamma_{eff} = 0.321 \pm 0.002$and$N_0 = 0.0776$, consistent with classical turbulent Rayleigh-B'enard convection in a system with laminar boundary layers below the top and above the bottom plate and with the prediction of Grossmann and Lohse. For $\text{Ra}>{\text{Ra}}_1^{\ast }$ the data rise above the classical-state power-law and show greater scatter. In analogy to similar behavior observed for $\Gamma = 0.50$, we interpret this observation as the onset of the transition to the ultimate state. Within our resolution this onset occurs at nearly the same value of ${\text{Ra}}_1^{\ast }$ as it does for $\mathrm{\Gamma }=0.50$. This differs from an earlier estimate by Roche {\it et al.} which yielded a transition at $\Ra_U \simeq 1.3\times 10^{11} \Gamma^{-2.5\pm 0.5}$\mathrm{.}A$\Gamma$-independent$\Ra^*_1$ would suggest that the boundary-layer shear transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global $\mathrm{\Gamma }$-dependent flow mode. Within the resolution of the measurements the heat transport above ${\text{Ra}}_1^{\ast }$ is equal for the two $\mathrm{\Gamma }$ values, suggesting a universal aspect of the ultimate-state transition and properties. The enhanced scatter of \Nu\ in the transition region, which exceeds the experimental resolution, indicates an intrinsic irreproducibility of the state of the system.