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

Iyer et al. (1) report heat () and momentum () transport results for turbulent Rayleigh-Bénard convection (RBC) for a Prandtl number from direct numerical simulation (DNS) for a cylindrical sample of aspect ratio (diameter /height ) . The data show the classic scaling in the range . The authors emphasize that their data do not reveal a transition Rayleigh number to the RBC ultimate state (2, 3), but neglect to point out that sidewall stabilization, and thus , is expected to increase with decreasing . Here, we point out that experimental values do indeed show a strong dependence with well above for .

Fig. 1 shows the dependence of for and 1.00 (4–6). The data are from experiments using compressed gas with at up to . Each dataset reveals a transition range of . For , we found the classic scaling with (0.321) for (1.00). For , we found for both , consistent with the predicted scaling for the ultimate state (2, 3). Over the transition range, was close to 0.33. One sees that the transition range increased as decreased. The values found for and were confirmed also by Reynolds number measurements (5, 7).

Fig. 1.

Reduced Nusselt number as a function of for (black circles) and 1.00 (blue diamonds). Solid lines denote power laws , with and adjusted to fit the data for for each . Dashed lines are the power-law fits to the data for , with for (Upper) and for (Lower). Vertical dotted lines are and .

Fig. 2 shows the measured and as a function of . While the dependence of is weak, changes by one decade over the data range and can be described by the power law , with and . Extrapolating to indicates that the transition Ra is near for such a slender sample. A similar dependence was found in cryogenic experiments for over the range (8). However, the reported values of (also shown in Fig. 2), a transition Rayleigh number defined by Roche et al. (8), are much lower than our results. Extrapolating them to , one finds that the transition should occur in the range . This disagrees with the DNS result by Iyer et al. (1).

Fig. 2.

(open symbols) and (solid symbols) as a function of . The black dashed line and red solid line represent the power function with the exponent and , respectively. Red stars are the data from ref. 8. The black solid line corresponds to . Vertical black solid line indicates the 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 on . For , a number of experiments (4, 5, 7, 9) have revealed that the transition occurs near , which is consistent with the prediction by Grossmann and Lohse (GL) (3). Note that the GL prediction does not apply for much less than 1, since the parameters in the model are all from experimental data for . Our results show that , leading to a much higher transition for a slender sample. For , our data suggest that the ultimate-state transition will occur near , which is well above the limit of the DNS data in ref. 1. That is why the authors found that “classic 1/3 scaling of convection holds up to .”