Aspect ratio dependence of the ultimate-state transition in turbulent thermal convection

Iyer et al. (1) report heat ($\text{Nu}$) and momentum ($\text{Re}$) transport results for turbulent Rayleigh-Bénard convection (RBC) for a Prandtl number $\text{Pr}=1$ from direct numerical simulation (DNS) for a cylindrical sample of aspect ratio (diameter $D$/height $H$ ) $\mathrm{\Gamma }=1/10$. The data show the classic scaling $\text{Nu}=0.0525{\text{Ra}}^{0.331}$ in the range $10^{10}\le \text{Ra}\le 10^{15}$. The authors emphasize that their data do not reveal a transition Rayleigh number ${\text{Ra}}^{*}$ to the RBC ultimate state (2, 3), but neglect to point out that sidewall stabilization, and thus ${\text{Ra}}^{*}$, is expected to increase with decreasing $\mathrm{\Gamma }$. Here, we point out that experimental ${\text{Ra}}^{*}$ values do indeed show a strong $\mathrm{\Gamma }$ dependence with ${\text{Ra}}^{*}$ well above $10^{15}$ for $\mathrm{\Gamma }=0.1$.

Fig. 1 shows the $\text{Ra}$ dependence of $\text{Nu}/{\text{Ra}}^{0.331}$ for $\mathrm{\Gamma }=0.50$ and 1.00 (4–6). The data are from experiments using compressed ${\mathrm{S}\mathrm{F}}_6$ gas with $\text{Pr}\simeq 0.8$ at $\text{Ra}$ up to $1.1\times 10^{15}$. Each dataset reveals a transition range ${\text{Ra}}_1^{*}\lesssim \text{Ra}\lesssim {\text{Ra}}_2^{*}$ of $\text{Nu}(\text{Ra})$. For $\text{Ra}\lesssim {\text{Ra}}_1^{*}$, we found the classic scaling $\text{Nu}\propto {\text{Ra}}^{{\gamma }_{e\mathit{ff}}}$ with ${\gamma }_{e\mathit{ff}}=0.312$ (0.321) for $\mathrm{\Gamma }=0.50$ (1.00). For $\text{Ra}\gtrsim {\text{Ra}}_2^{*}$, we found ${\gamma }_{e\mathit{ff}}\simeq 0.37$ for both $\mathrm{\Gamma }$, consistent with the predicted scaling for the ultimate state (2, 3). Over the transition range, ${\gamma }_{e\mathit{ff}}$ was close to 0.33. One sees that the transition range increased as $\mathrm{\Gamma }$ decreased. The values found for ${\text{Ra}}_1^{*}$ and ${\text{Ra}}_2^{*}$ were confirmed also by Reynolds number measurements (5, 7).

Fig. 1.

Reduced Nusselt number $\text{Nu}/{\text{Ra}}^{0.331}$ as a function of $\text{Ra}$ for $\mathrm{\Gamma }=0.50$ (black circles) and 1.00 (blue diamonds). Solid lines denote power laws $Nu=N{u}_0{\text{Ra}}^{{\gamma }_{e\mathit{ff}}}$, with ${\gamma }_{e\mathit{ff}}=0.37$ and $N{u}_0$ adjusted to fit the data for $\text{Ra}>{\text{Ra}}_2^{*}$ for each $\mathrm{\Gamma }$. Dashed lines are the power-law fits to the data for $\text{Ra}<{\text{Ra}}_1^{*}$, with ${\gamma }_{e\mathit{ff}}=0.312$ for $\mathrm{\Gamma }=0.50$ (Upper) and ${\gamma }_{e\mathit{ff}}=0.321$ for $\mathrm{\Gamma }=1.00$ (Lower). Vertical dotted lines are ${\text{Ra}}_2^{*}=8\times 10^{13}$ and $7\times 10^{14}$.

Fig. 2 shows the measured ${\text{Ra}}_1^{*}$ and ${\text{Ra}}_2^{*}$ as a function of $\mathrm{\Gamma }$. While the $\mathrm{\Gamma }$ dependence of ${\text{Ra}}_1^{*}$ is weak, ${\text{Ra}}_2^{*}$ changes by one decade over the data range and can be described by the power law ${\text{Ra}}_2^{*}=a{\mathrm{\Gamma }}^b$, with $a=8.13\hspace{0.17em}\times 10^{13}$ and $b=-3.04$. Extrapolating to $\mathrm{\Gamma }=1/10$ indicates that the transition Ra is near $10^{17}$ for such a slender sample. A similar $\mathrm{\Gamma }$ dependence ${\text{Ra}}^{*}\sim {\mathrm{\Gamma }}^{-2.5\pm 0.5}$ was found in cryogenic experiments for $\text{Pr}\simeq 1.5$ over the range $0.23\hspace{0.17em}\le \hspace{0.17em}\mathrm{\Gamma }\hspace{0.17em}\le \hspace{0.17em}1.14$ (8). However, the reported values of ${\text{Ra}}^{*}$ (also shown in Fig. 2), a transition Rayleigh number defined by Roche et al. (8), are much lower than our results. Extrapolating them to $\mathrm{\Gamma }=1/10$, one finds that the transition should occur in the range $2\times 10^{13}\lesssim {\text{Ra}}^{*}\lesssim 10^{14}$. This disagrees with the DNS result by Iyer et al. (1).

Fig. 2.

${\text{Ra}}_1^{*}$ (open symbols) and ${\text{Ra}}_2^{*}$ (solid symbols) as a function of $\mathrm{\Gamma }$. The black dashed line and red solid line represent the power function $y=a{x}^b$ with the exponent $b=0.40$ and $-3.04$, respectively. Red stars are the ${\text{Ra}}^{*}$ data from ref. 8. The black solid line corresponds to ${\text{Ra}}^{*}\sim {\mathrm{\Gamma }}^{-2.5}$. Vertical black solid line indicates the $\text{Ra}$ range of the DNS data in ref. 1.

Thus, the conclusion by Iyer et al. (1) is incomplete, since they did not consider the strong influence of $\mathrm{\Gamma }$ on ${\text{Ra}}^{*}$. For $\mathrm{\Gamma }\simeq 1$, a number of experiments (4, 5, 7, 9) have revealed that the transition occurs near $\text{Ra}=10^{14}$, which is consistent with the prediction by Grossmann and Lohse (GL) (3). Note that the GL prediction does not apply for $\mathrm{\Gamma }$ much less than 1, since the parameters in the model are all from experimental data for $\mathrm{\Gamma }=1$. Our results show that ${\text{Ra}}_2^{*}\sim {\mathrm{\Gamma }}^{-3.04}$, leading to a much higher transition $\text{Ra}$ for a slender sample. For $\mathrm{\Gamma }=1/10$, our data suggest that the ultimate-state transition will occur near $\text{Ra}=10^{17}$, which is well above the $\text{Ra}$ limit of the DNS data in ref. 1. That is why the authors found that “classic 1/3 scaling of convection holds up to $\text{Ra}=10^{15}$.”