Universal scaling of temperature variance in Rayleigh–Bénard convection near the transition to the ultimate state

Abstract

We report measurements of the temperature frequency spectra $P(\,f, z, r)$ , the variance $\sigma ^2(z,r)$ and the Nusselt number $Nu$ in turbulent Rayleigh–Bénard convection (RBC) over the Rayleigh number range $4\times 10^{11} \underset{\smash{\scriptscriptstyle\thicksim}} { < } Ra \underset{\smash{\scriptscriptstyle\thicksim}} { < } 5\times 10^{15}$ and for a Prandtl number $Pr \simeq ~0.8$ ( $z$ is the vertical distance from the bottom plate and $r$ is the radial position). Three RBC samples with diameter $D = 1.12$ m yet different aspect ratios $\varGamma \equiv D/L = 1.00$ , $0.50$ and $0.33$ ( $L$ is the sample height) were used. In each sample, the results for $\sigma ^2/\varDelta ^2$ ( $\varDelta$ is the applied temperature difference) in the classical state over the range $0.018 \underset{\smash{\scriptscriptstyle\thicksim}} { < } z/L \underset{\smash{\scriptscriptstyle\thicksim}} { < } 0.5$ can be collapsed onto a single curve, independent of $Ra$ , by normalizing the distance $z$ by the thermal boundary layer thickness $\lambda = L/(2 Nu)$ . One can derive the equation $\sigma ^2/\varDelta ^2 = c_1\times \ln (z/\lambda )+c_2+c_3(z/\lambda )^{-0.5}$ from the observed $f^{-1}$ scaling of the temperature frequency spectrum. It fits the collapsed $\sigma ^2(z/\lambda )$ data in the classical state over the large range $20 \underset{\smash{\scriptscriptstyle\thicksim}} { < } z/\lambda \underset{\smash{\scriptscriptstyle\thicksim}} { < } 10^4$ . In the ultimate state ( $Ra \underset{\smash{\scriptscriptstyle\thicksim}} { > } Ra^*_2$ ) the data can be collapsed only when an adjustable parameter $\tilde \lambda = L/(2 \widetilde {Nu})$ is used to replace $\lambda$ . The values of $\widetilde {Nu}$ are larger by about 10 % than the experimentally measured $Nu$ but follow the predicted $Ra$ dependence of $Nu$ for the ultimate RBC regime. The data for both the global heat transport and the local temperature fluctuations reveal the ultimate-state transitions at $Ra^*_2(\varGamma )$ . They yield $Ra^*_2 \propto \varGamma ^{-3.0}$ in the studied $\varGamma$ range.