2. The experimental cells

We start with a brief description of the experimental cells in which it is claimed that the transition to the ultimate state has been observed.

The Grenoble cells. The first claim of observing the transition (described as ‘possibly corresponding to the asymptotic regime predicted by R. Kraichnan’) by  Chavanne et al. (Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997) – see also Chavanne et al. (Reference Chavanne, Chillà, Chabaud, Castaing and Hébral2001) – results from a cylindrical aspect ratio ${\it\Gamma}=1/2$ cell 20 cm in height. Similar results, displaying the transition to what was later called the ‘Grenoble regime’, have been observed in seven cryogenic convection cells, differing in various details, see figures 2 and 3 of Roche et al. (Reference Roche, Gauthier, Kaiser and Salort2010). All of them are cylindrical, of diameter $D=10~\text{cm}$ and height, $L$ , of 8.8 cm (Short cell), 43 cm (Cigar cell) and 20 cm (Flange, Paper, Screen, Vintage and ThickWall, i.e. all remaining ones), corresponding respectively to aspect ratios of ${\it\Gamma}=1.14$ , 0.23 and 0.50.

Although the transition to the ultimate, or Grenoble, state has been observed in all of these cells, we find it useful to divide them into three groups, simply by their absolute height, $L$ . The common feature of cells with $L=20~\text{cm}$ is that they all displayed the local scaling exponent characterizing the heat transport efficiency via $Nu=Nu(Ra)$ , crossing the value of $1/3$ at approximately $10^{11}{-}10^{12}$ in $Ra$ . The Short cell displays this same feature but at significantly lower $Ra$ . Finally, the scaling exponent measured in the Cigar cell displays an almost constant value of approximately $1/3$ until $Ra$ reaches approximately $5\times 10^{12}$ , where it steeply rises. It should be noted that, if $Ra$ were rescaled assuming the same height (20 cm) for all of the Grenoble cells (i.e. a factor of eight up for the Short cell and the same factor down for the Cigar cell), the observed transitions to the Grenoble state would almost collapse if $Nu$ were plotted versus this rescaled $Ra$ . Let us emphasize that in Grenoble cells the transitions have been observed several times under strictly fixed values of $Pr$  (Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010), i.e. the possible $Nu(Pr)$ dependence cannot be identified as a reason for the observed transition.

The Göttingen cells. Transitions to the ultimate state has been claimed (at $Pr\lesssim 1$ ) in three cylindrical cells of diameter $D=1.12~\text{m}$ : ${\it\Gamma}=1/2$ (i.e. $L=2.24~\text{m}$ ) (Ahlers et al. Reference Ahlers, He, Funfschilling and Bodenschatz2012b ; He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ), ${\it\Gamma}=1$  (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ) and, most recently, ${\it\Gamma}=1/3$  (He et al. Reference He, van Gils, Bodenschatz and Ahlers2014). All of these cells were located in the ‘Uboot of Göttingen’, a pressure vessel of approximately $25~\text{m}^{3}$ volume able to contain up to 2000 kg of the working fluid $(\text{SF}_{6})$ at pressures of up to 19 bars. The sidewall insulated by thermal shields was made of 9.5 mm thick Plexiglas sealed to aluminium top and bottom plates. For most recent experiments cited here (after the authors dealt with the ‘chimney effect’, by sealing the originally existing gap between the plates and the sidewall), at the mid-height of the sidewall, there was a hole connected to a remotely controlled valve, via a tube of 13 mm inner diameter. Before each measurement sequence, the valve was opened and the sample cell and Uboot were filled with $(\text{SF}_{6})$ to the desired pressure. Then, after all pressure and temperature transients had decayed, the valve was closed and the desired measurements were made on a completely closed sample.

We now consider complementary experimental cells reaching very high $Ra$ but displaying no transition to the ultimate state.

The Chicago, Oregon/Trieste and Brno cells have been used in clean cryogenic conditions with cryogenic He gas as working fluid. The largest Chicago cell used in the experiments of Wu (Reference Wu1991) – see also Castaing et al. (Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989) – was identical in shape to the Chavanne cell (Chavanne et al. Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997; Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010) but twice as large ( $L=40~\text{cm}$ ); values of $Ra$ of up to $2\times 10^{14}$ were obtained.

The original Oregon/Trieste cell (Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000), of the same shape as above, is the largest ( $L=100~\text{cm}$ ) cryogenic cell used so far. No sign of transition was observed up to so far the highest $Ra$ ${\approx}10^{17}$ . The least-square fit of the Nusselt number versus Rayleigh number for the original ‘Oregon’ data spanning 11 orders of $Ra$ in turbulent convection up to $10^{17}$ yields a $\text{d}\log Nu/\text{d}\log Ra$ slope of 0.32 (Niemela & Sreenivasan Reference Niemela and Sreenivasan2006). The design of the Oregon cell is close to the Flange Grenoble cell (Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010) – its sidewall consists of two equal halves, allowing a simple change to the ${\it\Gamma}=1$ cell ( $L=50~\text{cm}$ ), which was later used in Trieste (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003, Reference Niemela and Sreenivasan2006, Reference Niemela and Sreenivasan2010).

Our cylindrical ${\it\Gamma}=1$ Brno cell (Urban et al. Reference Urban, Hanzelka, Králík, Musilová, Skrbek and Srnka2010) is slightly smaller ( $L=30~\text{cm}$ ), capable of covering the range $10^{6}<Ra<10^{15}$ with sufficient precision of measurements.

We point out here that in earlier work (Funfschilling, Bodenschatz & Ahlers Reference Funfschilling, Bodenschatz and Ahlers2009) the ${\it\Gamma}=1/2$ Göttingen cell has been used to explore the global heat transport with other working fluids in addition to $\text{SF}_{6}$ ( $Pr=0.79{-}0.84$ ), namely with gaseous He ( $Pr=0.67$ ) and $\text{N}_{2}$ ( $Pr=0.72$ ) at near-ambient temperatures, altogether covering the $10^{9}\lesssim Ra\lesssim 3\times 10^{14}$ range. These measurements did not reveal any sign of the transition to the ultimate regime and were claimed to be roughly consistent with the cryogenic Oregon data but inconsistent with the Grenoble results.

Figure 1.

Growing asymmetry with increasing $Ra$ of relative changes in relevant fluid properties for the Göttingen ${\it\Gamma}=1$  (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ) and ${\it\Gamma}=1/2$  (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ) experiments. Plots of dimensionless combinations of the physical quantities ${\it\alpha}$ , ${\it\nu}$ , ${\it\kappa}$ , ${\it\eta}={\it\alpha}/({\it\nu}{\it\kappa})$ and ${\it\lambda}$ as indicated in the legend (left axis) evaluated at $T_{\mathit{t}}$ , $T_{\mathit{b}}$ and $T_{\mathit{c}}$ versus $Ra_{c}$ (for these Göttingen data sets, the result hardly changes if $T_{\mathit{m}}$ is used instead of $T_{\mathit{c}}$ ) are shown to be closely correlated with the increase in the local power law exponent ${\it\gamma}$ in the observed $Nu=Nu(Ra)$ scaling, plotted here in the compensated form as $Nu/Ra^{1/3}$ (right axis). The vertical dashed lines indicate where the transition to the ultimate state was claimed by He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ).

3. Analysis

In our previous study (Urban et al. Reference Urban, Musilová and Skrbek2011), we have shown that for $7.2\times 10^{6}\leqslant Ra\leqslant 10^{11}$ our sidewall-corrected data agree with suitably corrected data from complementary cryogenic experiments, and are consistent with $Nu\propto Ra^{2/7}$ . On approaching $Ra\approx 10^{11}$ , all cryogenic data display a broad crossover to $Nu\propto Ra^{1/3}$ , as predicted by the theory (Malkus Reference Malkus1954; Priestley Reference Priestley1959). It is mainly above approximately $10^{12}$ in $Ra$ that strong differences appear.

Here, we show that the probable reason for this disagreement is a gradual departure from the Oberbeck–Boussinesq conditions upon increasing $Ra$ in each particular cell. To justify that the fluid sample can be treated as Oberbeck–Boussinesq in high- $Ra$ experiments, various phenomenological requirements have usually been applied, such as ${\it\alpha}(T_{b}-T_{t})<C$ , with $C$ typically up to 0.2 (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003). As a result of our experimental study, based on figure 3 in Urban et al. (Reference Urban, Hanzelka, Králík, Musilová, Srnka and Skrbek2012) we concluded, however, that it is hard to set any simple quantitative criterion justifying the validity of Oberbeck–Boussinesq conditions.

The gradual departure from the Oberbeck–Boussinesq conditions with increasing $Ra$ gives rise to asymmetry between the boundary layers on the top and bottom plates. We illustrate this feature in figure 1 for the ${\it\Gamma}=1$  (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ) and ${\it\Gamma}=1/2$ (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ) Göttingen experiments, making use of the known $\text{SF}_{6}$ properties. (To evaluate the $\text{SF}_{6}$ properties we use the same computer program as the Göttingen group, kindly provided to us by G. Ahlers and X. He.) The results of this analysis for other cells mentioned in the previous section look similar. For an additional example of the influence of the non-Oberbeck–Boussinesq conditions on the effectiveness of heat transport, plotted in a different way for our own cryogenic He data, see figure 6 of Urban et al. (Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014).

In order to quantify the effect of boundary layer asymmetry on the heat transfer efficiency, we follow Wu & Libchaber (Reference Wu and Libchaber1991) and use their $X$ parameter, defined as $X=(T_{\mathit{c}}-T_{\mathit{t}})/(T_{\mathit{b}}-T_{\mathit{c}})$ . Where available, we use the experimentally measured temperature $T_{\text{c}}$ of the (almost isothermal) cell interior and evaluate $X=X_{expt}$ . Whenever $T_{\mathit{c}}$ is not available, we use the following simple idea to estimate $X=X_{\mathit{th}}$ . The underlying physics of Malkus’ theory is such that the temperature drop occurs entirely across the two boundary layers, the fluid in the bulk of the apparatus being almost isothermal. The boundary layers, of thicknesses $d_{\mathit{b}}$ and $d_{\mathit{t}}$ , are assumed to be marginally stable, meaning that the bottom and top Rayleigh numbers $Ra_{BL}^{crit}$ , calculated on the basis of the boundary layer thicknesses $d_{\mathit{b}}$ , $d_{\mathit{t}}$ and characteristic fluid properties $[{\it\alpha}/({\it\nu}{\it\kappa})]_{\mathit{b}}$ and $[{\it\alpha}/({\it\nu}{\it\kappa})]_{\mathit{t}}$ , are equal. We therefore write

The heat flux $\dot{q}$ passes both (laminar) boundary layers, and assuming that Fourier’s law holds, we can infer

where we introduced the characteristic thermal conductivities of the boundary layers. This allows us to replace the ratio $d_{\mathit{b}}/d_{\mathit{t}}$ with the combination $({\it\lambda}_{B}/{\it\lambda}_{T})/X$ and estimate the $X$ parameter as

Figure 2.

The asymmetry $X$ parameter plotted versus $Ra$ for various experiments as indicated. We plot both its experimental values, $X=X_{expt}$ , based on the direct measurements of $T_{\mathit{c}}$ (where available, (a)) as well as its theoretically estimated value, $X=X_{\mathit{th}}$ , (3.3) (b). Panel (c) (the meaning of the symbols is as in (b)) shows the observed $Nu(Ra)$ scaling for experiments, with $Nu$ and $Ra$ appropriately corrected, calculated in a conventional way, i.e. based on ${\rm\Delta}T=T_{\mathit{b}}-T_{\mathit{t}}$ and fluid properties evaluated at $T_{\mathit{m}}=(T_{\mathit{t}}+T_{\mathit{b}})/2$ .

The characteristic bottom (top) boundary layer properties can be estimated as arithmetic means of fluid properties evaluated at $T_{\mathit{m}}$ and $T_{\mathit{b}}$ ( $T_{\mathit{t}}$ ); these are available for all cryogenic as well as Göttingen experiments. Figure 2 plots the $X$ parameter evaluated by us for a number of experimental cells using (3.3) as well as, where available, based on the direct measurements of $T_{\mathit{c}}$ . It has to be emphasized that the $X$ parameter is not an analytical function of $Ra$ ; it depends on the size (and generally on the shape) of the convection cell and on the choice of the working point on the $p$ – $T$ phase diagram. For high-Rayleigh-number experiments of the Rayleigh–Bénard type, however, the $X$ parameter serves as a useful measure of how closely the Oberbeck–Boussinesq conditions are satisfied, provided that both $T_{\mathit{t}}$ and $T_{\mathit{b}}$ lie on the same side of the saturated vapour pressure (SVP) curve or critical isochore in the pressure/temperature $(p,T)$ phase diagram of the working fluid used in the particular experiment.

Given the crudeness of our model, one cannot expect exact quantitative agreement between values of the experimentally deduced $X=X_{expt}$ parameter and those predicted by (3.3) ( $X=X_{\mathit{th}}$ ). Figure 2 shows, however, fairly good agreement in behaviour, both quantities departing down from unity with increasing $Ra$ .

As $T_{\mathit{c}}$ was not directly measured in the Grenoble cryogenic experiments (Chavanne et al. Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997; Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010), we rely on this similarity (and on Occam’s razor reasoning) in drawing the following conclusion: for experimental reasons (size of the cell, choice of the working point), the Grenoble claims of observing the transition to the ultimate/Grenoble regime of Rayleigh–Bénard convection are most likely not justified, as the transition values of $Ra$ were reached when the $X$ parameter strongly suggested non-Oberbeck–Boussinesq conditions. Moreover, there are complementary experiments with the $X$ value nearly unity, both He cryogenic (Wu Reference Wu1991; Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000; Urban et al. Reference Urban, Hanzelka, Králík, Musilová, Srnka and Skrbek2012) and $\text{SF}_{6}$ (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ), covering the range of $Ra$ (with nearly constant $Pr$ ${\lesssim}1$ ) where the transition is claimed in the Grenoble experiments, and they show no sign of the transition, at least up to $10^{13}$ in $Ra$ . (An additional supporting argument is that suppression of the large-scale flow, by inserting planar obstacles in the Grenoble ${\it\Gamma}=1/2$ Screen cell of 20 cm in height, hardly affected the observed heat transfer efficiency (Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010). This is a surprising result, as the large-scale flow, sweeping the fluid across plates, is assumed to stimulate the laminar to turbulent transition of boundary layers.)

Here, we have to repeat the warning already expressed in Urban et al. (Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014). The high- $Ra$ ranges of the Trieste and Grenoble experiments have often been investigated using a working point $(p,T_{\mathit{m}})$ in the ⁴He phase diagram close to the critical isochore. For small ${\rm\Delta}T$ , our analysis is not applicable, as both boundary layers will be affected and the above assumption that one of the boundary layers can be treated as Oberbeck–Boussinesq is not justified. Moreover, on the basis of the analysis of Ahlers et al. (Reference Ahlers, Araujo, Funfschilling, Grossmann and Lohse2007) of the non-Oberbeck–Boussinesq effects in gaseous ethane, one may expect that the asymmetry of the boundary layers might be partly cancelled if $T_{\mathit{b}}$ and $T_{\mathit{t}}$ lie on opposite sides of the critical isochore in the $(p,T)$ phase diagram. Further experiments are needed to clarify this issue.

Our analysis assumes that there are not multiple steady stages of high- $Ra$ convective flow, such as those that have been recently observed in turbulent von Karman flows – for details see Thalabard et al. (Reference Thalabard, Saint-Michel, Herbert, Daviaud and Dubrulle2015) and references therein.

Figure 3.

The $p$ – $T$ phase diagram of $\text{SF}_{6}$ , showing the working points used in the Göttingen experiments.

Let us now examine the Göttingen $\text{SF}_{6}$ data. Figure 3 shows where in the $p$ – $T$ phase diagram the ${\it\Gamma}=1/2$  (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ) and ${\it\Gamma}=1$  (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ) $\text{SF}_{6}$ data lie. We see that $T_{\mathit{m}}$ was kept roughly constant (except for four data points that are not significant for the following discussion); higher values of $Ra$ were obtained by increasing the pressure (vertically, towards the dashed SVP line) and then by increasing ${\rm\Delta}T$ . The horizontal ‘error bars’ show the span of the internal cell temperature for each particular data point – their left (right) ends correspond to $T_{\mathit{t}}$ ( $T_{\mathit{b}}$ ). The critical point (red symbol) far away does not play any significant role here; however, we see that in some cases (encircled) the $T_{\mathit{t}}$ values are dangerously close to the SVP line (representing the equilibrium first-order liquid–gas phase transition where due to fluctuations condensation/evaporation could take place), and these are precisely those data points that display the reputed $Nu=Nu(Ra)$ transition to the ultimate state! We can further strengthen our argument by pointing out the work of Zhong, Funfschilling & Ahlers (Reference Zhong, Funfschilling and Ahlers2009), showing that the heat transfer efficiency would be considerably enhanced if condensation/evaporation processes were to take place in the vicinity of the SVP line at the top plate of the $\text{SF}_{6}$ cell.

Figure 4.

(a,c) The $Nu/Ra^{1/3}$ versus $Ra$ plots of the Göttingen data (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ,Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers b ) (a) and the Brno cryogenic He data (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014) (c); $Nu$ and $Ra$ are calculated based on the $\text{SF}_{6}$ and He properties at $T_{\mathit{m}},T_{\mathit{c}}$ , ${\rm\Delta}T$ and ${\rm\Delta}T_{eff}$ as indicated. The vertical dashed lines indicate where the transition to the ultimate state was claimed (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ). (b,d) The same $Nu$ and $Ra$ Göttingen data (b) and Brno cryogenic He data (d), but calculated using the experimentally measured and tabulated $T_{\mathit{b}}$ and $T_{\mathit{t}}$ only, with ${\rm\Delta}T_{eff}^{\mathit{th}}=(T_{\mathit{b}}-T_{\text{c}}^{\mathit{th}})$ , where $T_{\text{c}}^{\mathit{th}}$ is evaluated for all available experimental data points using the theoretically estimated value $X_{\mathit{th}}$ , (3.3). It should be noted that for the $\text{SF}_{6}$ experiments there are more data points in (b). The reason is that this approach does not require experimental values of $T_{\mathit{c}}$ (which for (a) we calculated using Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012a )), and we used all $T_{\mathit{b}},T_{\mathit{t}}$ data points tabulated in Ahlers et al. (Reference Ahlers, He, Funfschilling and Bodenschatz2012b ). The theoretically estimated $Nu=Nu(Ra)$ scalings, for both the nearly constant $Pr$ $\text{SF}_{6}$ Göttingen data (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ,Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers b ) (a,b) and the Brno cryogenic He data where $Pr$ increases at the high- $Ra$ end (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014) (c,d), indicate no transition to the ultimate state.

Upon increasing $Ra$ , the top half of the cell becomes affected by the non-Oberbeck–Boussinesq effects, while the bottom half, if lying sufficiently far away from the SVP curve, does not. This scenario is confirmed by the $X$ parameter behaviour as discussed above, see figure 2. In this situation, we can apply the approach that we just described in detail to our own He cryogenic data (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014). In short, one can avoid the non-Oberbeck–Boussinesq effects in the top half of the cell, by replacing it with the inverted (with respect to $T_{\mathit{c}}$ ) nearly Oberbeck–Boussinesq bottom half, thus eliminating the boundary layer asymmetry. (It should be noted that for $X=1$ (fully Oberbeck–Boussinesq case), such an operation does not change the observed $Nu=Nu(Ra)$ scaling.) This leads to effective values ${\rm\Delta}T_{eff}=2(T_{\mathit{b}}-T_{\mathit{c}})$ , $Nu_{eff}$ and $Ra_{eff}$ (evaluated at $T_{\mathit{c}}$ ), changing both $Nu$ and $Ra$ . It is important to emphasize that not only the Nusselt number scaling, but the scaling of any other independently measured quantity, such as the Reynolds number $Re=Re(Ra)$ , that might have displayed ‘phase transitions’ spuriously interpreted as independent confirmation of the transition to the ultimate regime, will change, as well.

We have already applied this approach to our He experiments (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014), where, however, $Pr$ changes at the high end of attainable $Ra$ , which complicates any definite conclusion of a possible transition to the ultimate state, see figure 4(c). Figure 4(a) displays the nearly constant $Pr$ $\text{SF}_{6}$ original data of He et al. (Reference He, Funfschilling, Bodenschatz and Ahlers2012a ,Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers b ) together with those re-evaluated by us using the data of Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012a ) in order to calculate $T_{\mathit{c}}$ , which is then used to evaluate both ${\rm\Delta}T_{eff}=2(T_{\mathit{b}}-T_{\mathit{c}})$ and the relevant fluid properties. We see (yellow-filled large symbols) that the $Nu_{eff}\propto Ra_{eff}^{1/3}$ scaling is closely followed.

It is interesting to examine the influence of the difference in $Nu=Nu(Ra)$ scaling depending on whether the fluid properties are evaluated conventionally, at $T_{\mathit{m}}$ , or at the directly measured $T_{\mathit{c}}$ . We start with $Ra$ and $Nu$ still based on the total ${\rm\Delta}T=T_{\mathit{b}}-T_{\mathit{t}}$ . We have shown (Urban et al. Reference Urban, Hanzelka, Králík, Musilová, Srnka and Skrbek2012, Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014) that for $10^{12}\lesssim Ra\lesssim 10^{15}$ , $Nu\propto Ra^{1/3}$ within the experimental error, if the properties of the working fluid – cryogenic He gas – are evaluated at $T_{\mathit{c}}$ , while their conventional evaluation at $T_{\mathit{m}}=(T_{\mathit{t}}+T_{\mathit{b}})/2$ lead to spuriously steeper $Nu(Ra)$ scaling, see figure 4(c). Indeed, as the bulk of the Rayleigh–Bénard convection cell has temperature close to $T_{\mathit{c}}$ , we agreed with Wu & Libchaber (Reference Wu and Libchaber1991) that it is physically natural to define $Ra$ , $Nu$ and $Pr$ based on fluid properties evaluated at this temperature rather than $T_{\mathit{m}}$ . Although this question is interesting in its own right, for the reanalysis of the Göttingen $\text{SF}_{6}$ data, the particular choice of temperature – either $T_{\mathit{m}}$ or $T_{\mathit{c}}$ – hardly matters for evaluating fluid properties (figure 4 a; see also He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2013). For $\text{SF}_{6}$ they differ only marginally while for our own data (Urban et al. Reference Urban, Hanzelka, Králík, Musilová, Srnka and Skrbek2012, Reference Urban, Hanzelka, Králík, Musilová, Srnka and Skrbek2013) the influence turns out to be significant.

What appears to be important for both the He and $\text{SF}_{6}$ data is the difference ${\rm\Delta}T=T_{\mathit{b}}-T_{\mathit{c}}$ versus ${\rm\Delta}T_{eff}=2(T_{\mathit{b}}-T_{\mathit{c}})$ . Figure 4(c) shows that, in the latter case, our cryogenic He data, rather than increasing local scaling exponent ${\it\gamma}$ , display slightly less steep than $1/3$ scaling, most likely due to a gradually increasing $Pr$ with $Ra$  (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014). On the other hand, the nearly constant $Pr$ $\text{SF}_{6}$ Göttingen data (He et al. Reference He, Funfschilling, Bodenschatz and Ahlers2012a ,Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers b ) closely follow the $1/3$ scaling, independently of whether $\text{SF}_{6}$ properties are evaluated at $T_{\mathit{m}}$ or at $T_{\mathit{c}}$ , see figure 4(a).

Another independent check that our approach is meaningful is illustrated in figure 4(b,d), where we calculated the $Nu=Nu(Ra)$ scaling for $\text{SF}_{6}$ (b) and cryogenic He (d) theoretically, using the directly measured and tabulated values of $T_{\mathit{t}}$ and $T_{\mathit{b}}$ and calculating ${\rm\Delta}T_{eff}^{\mathit{th}}=(T_{\mathit{b}}-T_{\text{c}}^{\mathit{th}})$ , where $T_{\text{c}}^{\mathit{th}}$ is evaluated for all the available experimental data points using the theoretically estimated value $X_{\mathit{th}}$ , (3.3). Our simple theoretical model clearly confirms that neither the $\text{SF}_{6}$ data of He et al. (Reference He, Funfschilling, Bodenschatz and Ahlers2012a ,Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers b ) nor our cryogenic He data (Urban et al. Reference Urban, Hanzelka, Musilová, Králík, La Mantia, Srnka and Skrbek2014) indicate any transition to the ultimate state, and our theoretical model prediction confirms the tendency seen in the corresponding figures 4(a) and 4(c).

Figure 5.

The $Nu/Ra^{0.312}$ versus $Ra$ plots of the Göttingen data, digitized using figure 1 of He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014). Solid blue symbols represent the data obtained in the ${\it\Gamma}=0.33$ cell, solid orange symbols represent those from the ${\it\Gamma}=1/2$ cell of the same diameter, open blue symbols are the solid blue symbols but shifted by a factor of $(0.33/0.5)^{3}$ down in $Ra$ . The black vertical dashed lines $Ra_{1}^{\ast }$ (left) and $Ra_{2}^{\ast }$ (right) indicate where the transition to the ultimate state was claimed (He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012b ). The black dashed line represents $Nu=0.01035Ra^{0.38}$ , while the blue solid line is the same but shifted by a factor of $(0.33/0.5)^{3}$ down in $Ra$ .

For the ${\it\Gamma}=0.33$ Göttingen cell (He et al. Reference He, van Gils, Bodenschatz and Ahlers2014), we do not have access to the raw data in order to perform a similar detailed analysis. One can predict, however, that under similar experimental conditions (i.e. choosing similar working conditions in the $\text{SF}_{6}$ phase diagram as shown in figure 3) the transition due to the non-Oberbeck–Boussinesq effects will occur at $(0.33/0.5)^{3}\approx 3.4$ times higher characteristic $Ra^{NOB}$ due to the 1.5 times taller cell. This is illustrated in figure 5, which shows the digitized data from the single figure of He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014), together with the same ${\it\Gamma}=0.33$ data shifted down in $Ra$ by a factor of $\cong 3.4$ and consequently reduced by this shifted value of $Ra$ in power 0.312. This operation is somewhat artificial (it is exact only for $1/3$ power law scaling), but it is essentially correct; the open circles cannot be treated in the same way as the experimental data obtained in the ${\it\Gamma}=1/2$ cell, but the purpose of this operation is to show that the non-Oberbeck–Boussinesq effects (assuming similar working points in the $(p,T)$ phase diagram) start to be significant at approximately the same $Ra^{NOB}$ . The situation is therefore similar to the Grenoble cells discussed above. Additionally, figure 5 indicates that the highest- $Ra$ ${\it\Gamma}=0.33$ and ${\it\Gamma}=1/2$ data (i.e. the data points closely following the shown lines of $Nu\propto Ra^{0.38}$ scaling (He et al. Reference He, van Gils, Bodenschatz and Ahlers2014)) nearly collapse if the former are shifted by this factor down in $Ra$ , contrary to the expectation that transition to the ultimate regime in cells of the same diameter would take place at some universal $Ra^{\ast }$ .