Aspect-ratio dependence of small-scale temperature properties in turbulent Rayleigh-B\'{e}nard convection*

We report measurements of the variance $\sigma^2$, skewness $S$, and kurtosis $K$ of temperature fluctuations in turbulent Rayleigh-B\'{e}nard convection of a fluid with Prandtl number $Pr=12.3$ in cylindrical samples with aspect ratios $\Gamma=D/L$ (D is the diameter and L the height) of $0.50, 1.00$ and $2.00$ in the Rayleigh-number range $6\times10^{9} \leq Ra \leq 2\times 10^{12}$. The measurements were primarily for the radial positions $\xi=1.00$ (along the sample center line) and $\xi=0.063$ (near the side wall) at several vertical locations $z/L$. For all $\Gamma$ we found that $\sigma^2$ could be fitted by $\sigma^2 \sim (z/L)^{-\zeta}$ with $\zeta \simeq 0.7$ near the side wall and $\zeta \simeq 1.0$ along the sample center line ($\xi=1.00$). At the sample center and for $\Gamma=1$, the temperature probability distribution was very close to a Laplace distribution, with $K$ close to 6 independent of Ra. However, for $\Gamma=0.5$ the distribution was intermediate between Gaussian and Laplace, with $K$ close to 4 and also independent of Ra. For $\Gamma=2$ the distribution was close to Gaussian near the peak but had exponential tails, yielding $K$ values that decreased from about 6 to about 4 as Ra increased.