Temperature fluctuations in turbulent Rayleigh- B\'enard convection for Ra up to $2\times 10^{14}$ and Pr $\simeq 0.8$

We report on measurements of temperature space-time cross-correlation functions $C_T(r,\tau)$ in Rayleigh-B\'enard convection (RBC) near the side wall of a cylindrical sample with aspect ratio $\Gamma \equiv D/L = 1.00$ ($D= 1.12$ m was the diameter and $L = 1.12$ m was the height). The results covered the Rayleigh-number range $4\times 10^{11} \leq Ra \leq 2\times 10^{14}$ and the Prandtl-number range $0.79 \leq Pr \leq 0.86$. Our results extend previous measurements for a lower Ra range\footnote{X. He, G. He, and P. Tong, Phys. Rev. E, {\bf 81}, 065303 (2010).} and confirmed the elliptic approximation (EA) of He and Zhang\footnote{G.-W. He and J.-B. Zhang, Phys. Rev. E, {\bf 73}, 055303 (2006).} up to $Ra \simeq 10^{14}$. Using the EA, we determined an effective Reynolds number near the transition to the ultimate state of turbulent RBC.\footnote{X. He, D. Funfschilling, H. Nobach, E. Bodenschatz and G. Ahlers, Phys. Rev. Lett., {\bf 108}, 024502 (2012).}