Heat transport by turbulent Rayleigh-B\'enard convection for $\text{Pra}\simeq 0.8$ and $3\times {10}^{12}\text{alt}\text{Ra}\text{alt}{10}^{15}$: Aspect ratio $\mathrm{\Gamma }=0.50$

We report experimental results for heat-transport measurements, in the form of the Nusselt number \Nu, by turbulent Rayleigh-B'enard convection in a cylindrical sample of aspect ratio $\mathrm{\Gamma }\equiv D\mathrm{/}L=0.50$ ($D=1.12$ m is the diameter and $L=2.24$ m the height). The measurements were made using sulfur hexafluoride at pressures up to 19 bars as the fluid. They are for the Rayleigh-number range $3\times {10}^{12}\text{alt}\text{Ra}\text{alt}{10}^{15}$ and for Prandtl numbers \Pra\ between 0.79 and 0.86. For $\Ra < \Ra^*1 \simeq 1.4\times 10^{13}$wefind$\Nu = N_0 \Ra^{\gamma{eff}}$with$\gamma_{eff} = 0.312 \pm 0.002$, consistent with classical turbulent Rayleigh-B'enard convection in a system with laminar boundary layers below the top and above the bottom plate. For ${\text{Ra}}_1^{\ast }<\text{Ra}<{\text{Ra}}_2^{\ast }$ (with ${\text{Ra}}_2^{\ast }\simeq 5\times {10}^{14}$) ${\gamma }_{eff}$ gradually increases up to $0.37\pm 0.01$. We argue that above ${\text{Ra}}_2^{\ast }$ the system is in the ultimate state of convection where the boundary layers, both thermal and kinetic, are also turbulent. Several previous measurements for $\mathrm{\Gamma }=0.50$ are re-examined and compared with the present results.