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Heat transport by turbulent Rayleigh-Bénard convection for $\Pra\ \simeq 0.8$ and $4\times 10^{11} \alt \Ra\ \alt 2\times10^{14}$: Ultimate-state transition for aspect ratio $Γ= 1.00$

We report experimental results for heat-transport measurements by turbulent Rayleigh-Bénard convection in a cylindrical sample of aspect ratio $Γ\equiv D/L = 1.00$ ($D = 1.12$ m is the diameter and $L = 1.12$ 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 $\Ra < \Ra^*_1 \simeq 2\times 10^{13}$ we find $\Nu = N_0 \Ra^{γ_{eff}}$ with $γ_{eff} = 0.321 \pm 0.002$ and $N_0 = 0.0776$, consistent with classical turbulent Rayleigh-Bénard 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 $\Ra > \Ra_1^*$ the data rise above the classical-state power-law and show greater scatter. In analogy to similar behavior observed for $Γ= 0.50$, 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 $\Ra_1^*$ as it does for $Γ= 0.50$. This differs from an earlier estimate by Roche {\it et al.} which yielded a transition at $\Ra_U \simeq 1.3\times 10^{11} Γ^{-2.5\pm 0.5}$. A $Γ$-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 $Γ$-dependent flow mode. Within the resolution of the measurements the heat transport above $\Ra_1^*$ is equal for the two $Γ$ 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.