Elusive transition to the ultimate regime of turbulent Rayleigh-Bénard convection

By using cryogenic ${}^4\mathrm{He}$ gas as the working fluid in a cylindrical cell 0.3 m in both height and diameter, we study the influence of non-Oberbeck-Boussinesq (NOB) effects on the heat transfer in turbulent Rayleigh-Bénard convection (RBC). We show that the NOB effects increase the heat transfer efficiency when the top plate temperature closely approaches the saturation vapor curve even far away from the critical point. Viewed in this light, our analysis points to the likelihood that the claim of having observed the transition to Kraichnan's ultimate regime, under nominally similar conditions in the experiments with ${\mathrm{SF}}_6$ [Phys. Rev. Lett. 108, 024502 (2012)], is probably an NOB effect and the important issue of the transition to the ultimate state of RBC remains open.

Figure 1

(a) ${}^4\mathrm{He}$ and (b) ${\mathrm{SF}}_6 p-T$ phase diagrams, showing their critical points $(T_{\mathrm{cp}},p_{\mathrm{cp}})$ and SVCs (dashed lines). The color-coded symbols (labeled in the inset) denote the experimental working points $(p,T)$ in ${}^4\mathrm{He}$ and ${\mathrm{SF}}_6$, which are approximately equally distant from the CP in dimensionless phase diagrams. The ${\mathrm{SF}}_6$ data are adapted from Ref. [14]. The data series discussed in the text is traceable via the pairs of corresponding temperatures $T_{\mathrm{b}}$ (red $\times$) and $T_{\mathrm{t}}$ (blue $\times$) as measured in particular experimental runs. The arrows show the series which operate close to SVC.

Figure 2

(a) Compensated $\mathrm{Nu}/{\mathrm{Ra}}^{1/3}$ and (b) $\mathrm{Nu}/{\mathrm{Ra}}^{0.38}$ vs $\mathrm{Ra}$ plots of the Brno data (series A, B, and C) shown together with the Göttingen data (series D and E) [13], obtained under nominally similar conditions. Note the departure from the $\mathrm{Nu}\propto {\mathrm{Ra}}^{1/3}$ scaling for the data points with $T_{\mathrm{t}}$ lying in close vicinity of SVC, marked by arrows in the corresponding $p-T$ phase diagrams shown in Fig. 1. These data points, with $T_{\mathrm{t}}$ still in the gas phase (see Fig. 3), display a “transition” to $\mathrm{Nu}\propto {\mathrm{Ra}}^{0.38}$, claimed as effectively ultimate scaling [13, 33], while the last three data points of series A and the last point of series B, with $T_{\mathrm{t}}$ already in the liquid side, indicate an abrupt increase in heat transfer efficiency due to condensation and evaporation in the boundary layer [34], not present in the ${\mathrm{SF}}_6$ data. In contrast, the controlled Brno series C as well as ${\mathrm{SF}}_6$ data points [13] that correspond to the lowest Ra with $T_{\mathrm{t}}$ farther away from the SVC (marked as series E in Fig. 1) follow the ${\mathrm{Ra}}^{1/3}$ scaling (top). The two lower panels show (c) the Prandtl number and (d) $\alpha \mathrm{\Delta }T$ evaluated at $\alpha \left(T_{\mathrm{m}}\right)$ for the same data.

Figure 3

Temperature dependencies of the relevant physical properties $fp$ evaluated for measurements in ${}^4\mathrm{He}$ [(a) series A] and ${\mathrm{SF}}_6$ [(b) series D; the $T_{\text{t}}$ and $T_{\text{b}}$ temperatures corresponding to the maximum-Ra measurement are highlighted by circles], relative to the $fp$ values at respective $T_{\text{c}}$. The insets show that the scatter of pressure values is (a) $\delta p/p\lesssim 0.05\%$ and (b) $\lesssim 0.5$%, respectively, and confirm that all the Göttingen ${\mathrm{SF}}_6$ data lie in the gaseous side of the SVC [inset (b)], while the top plate temperatures for three ${}^4\mathrm{He}$ data points of series A lie on the liquid side of the SVC [inset (a)]. Due to asymmetry of boundary layers, the calculated $T_{\mathrm{m}}$ temperatures gradually shift with increasing $\mathrm{\Delta }T$, while the directly measured center temperatures $T_{\mathrm{c}}$ stay constant (within experimental accuracy). The directly measured top (bottom) plate temperatures are indicated by red (blue) crosses. The SVC is marked by a jump in the density $\rho$.