Heat transport by turbulent Rayleigh-B\'enard convection for $\Pra\ \simeq 0.8$ and $4\times 10^{11} \alt \Ra\ \alt 2\times10^{14}$: Ultimate-state transition for aspect ratio $\Gamma = 1.00$

We report experimental results for heat-transport measurements by turbulent Rayleigh-B\'enard convection in a cylindrical sample of aspect ratio $\Gamma \equiv D/L = 1.00$ ($D = 1.12$ m is the diameter and $L = 1.12$ m the height). They are for the Rayleigh-number range $4\times10^{11} \alt \Ra \alt 2\times10^{14}$ and for Prandtl numbers \Pra\ between 0.79 and 0.86. For $\Ra<\Ra^*_1 \simeq 2\times 10^{13}$ we find $\Nu = N_0 \Ra^{\gamma_{eff}}$ with $\gamma_{eff} = 0.321 \pm 0.002$ and $N_0 = 0.0776$, consistent with classical turbulent Rayleigh-B\'enard convection in a system with laminar boundary layers below the top and above the bottom plate and with the prediction of Grossmann and Lohse. For $\Ra>\Ra_1^*$ the data rise above the classical-state power-law and show greater scatter. In analogy to similar behavior observed for $\Gamma = 0.50$, we interpret this observation as the onset of the transition to the ultimate state. Within our resolution this onset occurs at nearly the same value of $\Ra_1^*$ as it does for $\Gamma = 0.50$. This differs from an earlier estimate by Roche {\it et al.} which yielded a transition at $\Ra_U \simeq 1.3\times 10^{11} \Gamma^{-2.5\pm 0.5}$. A $\Gamma$-independent $\Ra^*_1$ would suggest that the boundary-layer shear transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global $\Gamma$-dependent flow mode. Within the resolution of the measurements the heat transport above $\Ra_1^*$ is equal for the two $\Gamma$ values, suggesting a universal aspect of the ultimate-state transition and properties. The enhanced scatter of \Nu\ in the transition region, which exceeds the experimental resolution, indicates an intrinsic irreproducibility of the state of the system.


Introduction
In this paper we consider turbulent convection in a fluid contained between horizontal parallel plates and heated from below (Rayleigh-Bénard convection or RBC; for reviews written for broad audiences see Refs. [1,2]; for more specialized reviews see Refs. [3,4]). It is now well established experimentally that RBC for Rayleigh numbers Ra (a dimensionless measure of the applied temperature difference ∆T ) below a typical value Ra * 1 is a system with laminar (albeit fluctuating) boundary layers [5,6] (BLs), one below the top and another above the bottom plate. Approximately half of ∆T ≡ T b −T t (T b and T t are the temperatures of the bottom and top confining plate respectively) is sustained by each of these BLs [7,8,9,10,11,12,13,14]). The sample interior, known as the "bulk", is nearly isothermal in the time average (see, however, Ref. [7,15,16,17]), but its temperature and velocity fields are also fluctuating vigorously. This state is known as the "classical" state as it has been studied at great length for nearly a century.
At very large Ra a transition was predicted to take place [18,19,20] from the classical state to the "ultimate" state [21] where the BLs have become turbulent as well because of the shear applied to them by the vigorous fluctuations in the sample interior. Experimentally it was found recently for a cylindrical sample with aspect ratio Γ ≡ D/L = 0.50 (D is the diameter and L the height of the cylindrical sample) and Pr ≃ 0.8 that this transition takes place over a wide range Ra * 1 < ∼ Ra < ∼ Ra * 2 , with Ra * 1 ≃ 1.5 × 10 13 and Ra * 2 ≃ 5 × 10 14 . For a more detailed description of the classical and ultimate state and the transition between them, see for instance Ref. [22] and the review articles [1,2,3].
The purpose of the present work was two-fold. First we hoped to determine with high accuracy the dependence of Nu on Ra in the classical state at the largest-possible Rayleigh numbers for a sample of aspect ration Γ = 1.00 and for a Prandtl number Pr ≃ 0.8. Such data make it possible to test in detail the predictions for the classical state by Grossmann and Lohse [23,24] of the relationship between Nu and Ra in a parameter range not explored heretofore. Although in principle these predictions should be applicable to the classical state regardless of Γ, they depend on a number of parameters that had been determined by fitting to experimental data for Γ = 1.00 [25]. This fit was done over the range 4 < ∼ Pr < ∼ 34 and 3 × 10 7 < ∼ Ra < ∼ 3 × 10 9 . Thus, a comparison with new data over the very different Ra and Pr ranges of the present work constitutes a significant test of the prediction. We found that Nu = N 0 Ra γ ef f with N 0 = 0.0776 and γ ef f = 0.321 ± 0.002. This result differs slightly from the case Γ = 0.50 [22] which yielded γ ef f = 0.312 ± 0.002. It is in excellent agreement with the Grossmann-Lohse prediction for the classical state and Γ = 1.00 in our Ra and Pr range.
Second, we hoped to search for the transition to the "ultimate" state of turbulent convection. Experiments searching for this state using Γ = 0.50 had been carried out before [26,27,21,28,29,30,31,32,33,34,35]; results from these searches were reported and/or reviewed in another publication [22]. The transition was found very recently [36,22] to occur over a wide Ra-range, extending from Ra * 1 ≃ 2 × 10 13 to Ra * 2 ≃ 5 × 10 14 . In the present project we focus on the particular case of a cylindrical sample with Γ = 1.00 (D = 1.12 m and L = 1.12 m). This geometry was used in some previous searches for this state [37,38,39,35,40] and thus enables a direct comparison with earlier measurements; but more importantly we chose Γ = 1.00 in order to search for any Γ-dependence of the transition. Earlier a transition in Nu(Ra) had been reported at several Γ values by Roche et al. [35] at Rayleigh numbers Ra U ≃ 1.3×10 11 Γ −2.5±0.5 which those authors attributed to the ultimate-state transition. In contradistinction to this result, we find that the transition occurs at values of Ra that are two orders of magnitude larger than Ra U , and that (for Γ = 0.50 and 1.00) Ra * 1 is independent of Γ within the resolution of the data. A Γ-independent Ra * 1 would suggest that the boundary-layer shear-transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global Γ-dependent flow mode. Within the resolution of the results the heat transport above Ra * 1 is equal for the two Γ values, suggesting a universal aspect of the ultimate-state transition and properties. Unfortunately the necessarily smaller height of the Γ = 1.00 sample (compared to Γ = 0.50) limited our measurement range to Ra < ∼ 2 × 10 14 and prevented us from obtaining data all the way beyond Ra * 2 . Our results were obtained using the High-Pressure Convection Facility (the HPCF, a cylindrical sample of 1.12 m diameter) at the Max Planck Institute for Dynamics and Self-organization in Göttingen, Germany with sulfur hexafluoride (SF 6 ) at pressures up to 19 bars as the fluid. Results for Γ = 0.50 from this work were presented in Refs. [41,42,43,36,22]. A description of the apparatus was given in Ref. [41]. The present paper presents new results obtained for a sample chamber known as HPCF-IV which had a height equal to its diameter.
In Sec. 2 we define the parameters that describe this system. Then, in Sec. 3, we give a brief discussion of the apparatus used in this work. A detailed description of the main features was presented before [41]. Section 4 presents a comprehensive discussion of our results and of the results of others at large Ra for cylindrical samples with Γ = 1.00. We conclude with a Summary in Sec. 5.

The system parameters and data analysis.
For turbulent RBC in cylindrical containers there are two parameters which, in addition to Γ, are expected to determine its state. They are the dimensionless temperature difference as expressed by the Rayleigh number and the ratio of viscous to thermal dissipation as given by the Prandtl number Here α is the isobaric thermal expansion coefficient, g the gravitational acceleration, κ the thermal diffusivity, ν the kinematic viscosity, and ∆T ≡ T b − T t the applied temperature difference between the bottom (T b ) and the top (T t ) plate. In the present paper we present measurements of the heat transport in the form of the scaled effective thermal conductivity known as the Nusselt number, which is given by Here Q is the applied heat current, A = D 2 π/4 the sample cross-sectional area, and λ the thermal conductivity. The measurements cover the range 5 × 10 11 < ∼ Ra < ∼ 2 × 10 14 and are for Pr ranging from 0.79 at the lowest to 0.86 at the highest Ra. All fluid properties needed to calculate Ra, Pr, and Nu were evaluated at the mean temperature T m = (T t + T b )/2 of the sample. They were obtained from numerous papers in the literature, as discussed in Ref. [44]. A small correction for the nonlinear contribution of the side-wall conductance [45,46] to the heat carried by the sample was no more that 3% and was applied to the data.
In a recent communication [47] it was suggested that the fluid properties should be evaluated at the sample center temperature T c rather than at T m in order to avoid or minimize effects due to departures from the Oberbeck-Boussinesq (OB) approximation [48,49]. We note that this would be contrary to the convention adopted in the usual studies of non-OB effects (see, for instance, [50,51,52]). Nonetheless we explored the importance of the choice between T c and T m for our data. In Fig. 1a we show T c − T m as a function of Ra at the largest Ra of our work where its magnitude is also largest. In One sees that the largest difference, which occurs at the largest Ra, is only about a third of a percent. Such a difference is essentially negligible and does not influence the interpretation of our results.
The data obtained in this study are presented as an Appendix to this paper.

Apparatus
The apparatus was the same as the one described before [41], except that a new sample cell, known as the High-Pressure Convection-Facility IV or HPCF-IV, was constructed. This cell had an internal height L = 1120 ± 2 mm and a diameter D equal to L, yielding an aspect ratio Γ ≡ D/L = 1.000 ± 0.004. It had the aluminum top and bottom plates described in Ref. [41], and a 9.5 mm thick plexiglas side wall. The plates were sealed to the side wall, and a tube of 13 mm diameter entered the HPCF-IV at mid height through the side wall to permit filling the sample cell with gas to the desired pressure. This tube was sealed by a remotely controlled valve after the sample was filled and all transients had decayed, yielding a completely closed sample. All thermal shields were duplicates of those used for another sample with Γ = 0.50 known as HPCF-II [53], except that the side shield was of course shorter. The HPCF-IV was located in a high-pressure vessel known as the Uboot of Göttingen which could be filled with sulfur hexafluoride (SF 6 ) at pressures up to 19 bars. The Uboot could contain HPCF-IV and as well as HPCF-II simultaneously, as shown in the schematic diagram Fig. 2. Completely separate instrumentation and temperature-controlled water circuits enabled simultaneous measurements in the two units. We refer to our previous publications [41,22] for detailed descriptions of all construction details and experimental procedures.

Classical state
In Fig. 3a we show results for Nu as a function of Ra on double logarithmic scales. The data for Γ = 1.00 are shown in black, and previously published results [36,22] for Γ = 0.50 (HPCF-II) are given in red. One sees that, within the resolution of this graph, there is very little difference between the data for the two Γ values. Also shown in this figure, as a vertical dotted line, is the approximate upper limit of the classical regime and the beginning of the transition range to the ultimate state at Ra * 1 = 1.5 × 10 13 as determined from the Γ = 0.50 data and reported in Ref. [36].
The solid black and dash-dotted red lines are fits of the power law to the data with Ra < Ra * 1 . As reported elsewhere [22], the fit to the Γ = 0.50 data yielded γ ef f = 0.312 ± 0.002. The fit to the Γ = 1.00 data for Ra < Ra * 1 gave N 0 = 0.0764±0.0015 and γ ef f = 0.3216±0.0007 where the uncertainties are the standard errors of the parameters. The average value of Pr over the range of the data used in the fit was 0.80. Additional possible systematic errors, primarily due to uncertainties in the side-wall correction, lead us to the best estimate γ ef f = 0.321 ± 0.002 for the Nusselt exponent for Γ = 1.00 and Pr = 0.80. Fixing γ ef f at the value 0.321 let to the amplitude N 0 = 0.0776.
In order to provide a better comparison of these two data sets, we show the results in the form of the reduced Nusselt number Nu/Ra 0.321 as a function of Ra on double logarithmic scales in Fig. 3b. Now the Γ = 1.00 data scatter about the horizontal solid black line, with the scatter corresponding to a standard deviation of 0.21%.
In Fig. 3b the power-law fit to the Γ = 0.50 data is shown again as a red dash-dotted line. One can readily see the positive deviations and enhanced scatter of the Γ = 0.50 data for Ra > Ra * 1 where the transition to the ultimate state is beginning. We note that the enhanced scatter is not due to a sudden increase in experimental scatter, but rather a reflection of the intrinsic irreproducibility of the state of the system. Remarkably, also the Γ = 1.00 data begin to show positive deviations from the horizontal black line and enhanced scatter, suggesting that also the Γ = 1.00 system is undergoing a similar transition to the ultimate state, beginning at about the same Ra * 1 that was found for Γ = 0.50. We shall return to that issue below in Sec. 4.3.
We also show in Fig. 3b, as a short-dashed blue line, the prediction of Grossmann and Lohse [24] (GL) for Nu(Ra) in the classical state with Pr = 0.80. This prediction is based on two coupled equations with several parameters which had been determined   [36,22]). The solid black line is a power-law fit to the data for Γ = 1.00 in the classical state Ra < 1.5 × 10 13 . The fit gave N 0 = 0.0764 ± 0.0015 and γ ef f = 0.3216±0.0007. The red dash-dotted line is the power-law fit to the data in the classical state for Γ = 0.50 which gave N 0 = 0.1404, γ ef f = 0.312, see Ref. [36,22]. The vertical dotted line is the location of Ra * 1 as determined for Γ = 0.50 [36,22]. (b): The reduced Nusselt number Nu/Ra 0.321 as a function of Ra on logarithmic scales. All symbols and lines are as in (a). We added the blue short-dashed line, which is the prediction of Grossmann and Lohse [24].   For the previous work we show data with Pr < 1 in red, data with 1 < Pr < 2 in green, data with 2 < Pr < 4 in blue, and data with Pr > 4 in purple. Open circles: Niemela and Sreenivasan [38]. Open up-pointing triangles: Roche et al. [35]. Open squares: Urban et al. [40]. The short vertical dotted line represents Ra * 1 = 1.5 × 10 13 as determined for Γ = 0.50 [36,22].
by fits to experimental data [25] for Γ = 1.00 over the parameter ranges 4 < ∼ Pr < ∼ 34 and 3 × 10 7 < ∼ Ra < ∼ 2 × 10 9 . One sees that the comparison with the present data up to Ra = 10 13 and for Pr ≃ 0.80 requires a considerable extrapolation. Thus, the excellent agreement is indeed remarkable. Not only does it require a high degree of reliability of the GL equations; it also requires excellent consistency between the experimental data used to determine the free parameters in these equations and the present data.

Comparison with published data
In Fig. 4 we compare our results with other published data for Γ = 1.00 (a detailed comparison with literature data for Γ = 0.50 is being presented elsewhere [22]). Here too we show Nu in Fig. 4a and, for higher resolution, Nu/Ra 0.321 in Fig. 4b. For the literature data we use red symbols for data with Pr < 1, green symbols for data with 1 < Pr < 2, blue symbols for data with 2 < Pr < 4, and purple symbols for data with Pr > 4. Our own data span the range from Pr = 0.79 at the lowest to Pr = 0.86 at the highest Ra.
The present results are given as solid black circles. The data of Niemela and Sreenivasan [38] are shown as open circles. For Ra near 10 11 they follow a power law with an exponent near 0.33; but for 3×10 11 < ∼ Ra < ∼ 5×10 13 they rise more steeply, only to level off again for larger Ra to a dependence describable once more by an effective exponent near 0.33. This behavior was attributed by the authors [54] to a special type of non-Boussinesq effect near critical points. Thus the data do not yield reliable parameters of a power law for Nu(Ra) in the classical region that could be compared with the prediction of GL [24]; according to the authors [54] the data also do not yield any evidence for a transition to the ultimate state. The data for the "short cell" of Roche et al. are shown as open up-pointing triangles. They reveal a gradual increase of an effective exponent, starting near Ra = 5 × 10 10 . Although the authors believe that this rise of the exponent is indicative of an ultimatestate transition at Ra U ≃ 10 11 , we do not find the evidence convincing. Particularly troublesome is the low value of Ra U ; it is unlikely that the boundary-layer shear Reynolds-number Re s can be high enough to drive the BLs turbulent at so low a value of Ra [36]. Very recent direct numerical simulations (DNS) for Γ = 1 and Pr = 0.7 [55] suggest that Re s ≃ 65 for Ra = 10 11 , a value much too low to expect a shear instability to turbulence (for the higher Pr values of the experiment Re s would be even lower). On the other hand, we do not have an alternative explanation of the rise of γ ef f indicated by these data.
A third set of data (open squares in Fig. 4) was published recently by Urban et al. [40]. They extend up to Ra ≃ 4 × 10 13 . Although at constant Pr ≃ 0.8 one might have hoped to have reached Ra * 1 at that point, Pr also rose significantly at these large Ra, and one expects that Ra * increases significantly with Pr. In any event, no indication of an ultimate-state transition is seen in these data, nor is one claimed by their authors.
In view of the above it is our view that the ultimate-state transition has not yet been seen in any of the published data for Γ ≃ 1.

Transition toward the ultimate state
Recent measurements [36] for a Γ = 0.50 sample revealed that the transition to the ultimate state for that aspect ratio occurred over the approximate range from Ra * 1 = 1.5 × 10 13 to Ra * 2 = 5 × 10 14 . We show those data in Fig. 5 as open circles and compare them with our new data for Γ = 1.00 (solid circles). The vertical dotted lines in the figure indicate the locations of Ra * 1 and Ra * 2 . In the classical range below Ra * 1 both data sets follow a power law, albeit with the slightly different exponents of 0.312 for Γ = 0.50 and 0.321 for Γ = 1.00. Near Ra * 1 both data sets rise above their respective classical-state power laws, and the enhanced scatter of both data sets reveals the intrinsic irreproducibility of the state of the system in the Ra range of the transition from the classical to the ultimate state. Although in the classical state the two systems had slightly different Nusselt numbers, it is remarkable that in the transition region they display the same Nu values within the resolution allowed by the intrinsic scatter of the two systems. Unfortunately, in view of the smaller height of the Γ = 1.00 sample, our measurements are limited to Ra < ∼ 2 × 10 14 . Thus it is not possible for us to follow the transition all the way beyond Ra * 2 , as was done for Γ = 0.50 with HPCF-II. Finally we note that an extrapolation of the shear Reynolds numbers obtained from DNS [55] for Γ = 1.00 and Pr = 0.7 yields Re s ≃ 250 for Ra = Ra * 1 = 1.5 × 10 13 . This is a reasonable value for the onset of the boundary-layer shear transition to the ultimate state. It is also similar to the value of Re s (Ra * 1 ) deduced from experimental determinations of Re for Γ = 0.50 [36].

Summary
In this paper we presented new data for heat transport, expressed as the Nusselt number Nu, by turbulent Rayleigh-Bénard convection in a cylindrical sample of aspect ratio Γ = 1.00 over the Ra range 4 × 10 11 < ∼ Ra < ∼ 2 × 10 14 . We note that the Prandtl number was nearly constant for our work, varying only from about 0.79 at our smallest to about 0.86 at our largest Ra. This stands in contrast to other measurements [38,35,40] which were made near the critical point of helium, where Pr typically varied from about 0.7 to about 4 over the same Ra range. Maintaining a constant Pr is important in the search for the ultimate-state transition because the transition range is expected to shift to larger Ra as Pr increases, approximately in proportion to Pr 1.6 [56].
In the classical regime for Rayleigh numbers Ra < ∼ Ra * 1 = 1.5 × 10 13 we found that our measurements are in remarkably good agreement with the predictions of Grossmann and Lohse [24] (GL). We note that this agreement not only implies excellent reliability of the prediction. It also indicates consistency of the new data for Pr ≃ 0.8 and 5 × 10 11 < ∼ Ra < ∼ 1.5 × 10 13 with measurements [25] made a decade ago, using very different experimental techniques and organic fluids rather than compressed gases, since these older data for 4 < ∼ Pr < ∼ 34 and 3 × 10 7 < ∼ Ra < ∼ 2 × 10 9 were used to fix the free parameters of the equations derived by GL.
We compared the Γ = 1.00 results with previous measurements for Γ = 0.50. In the classical regime we found that the two geometries yielded slightly different effective exponents of the power laws that describe Nu(Ra). For Γ = 0.50 we reported elsewhere [22] that γ ef f = 0.312 ± 0.002. For Γ = 1.00 we now find that γ ef f = 0.321 ± 0.002, in excellent agreement with the GL result γ ef f = 0.323 in our parameter range.
In the classical range Ra < ∼ Ra * 1 = 1.5 × 10 13 the data had very little scatter, with root-mean-square deviations from the power-law fit as small at 0.2%. At larger Ra the scatter increased, indicating an intrinsic irreproducibility of the state of the system from one data point to another. Further, most of the points for Ra > Ra * 1 fell well above the power-law extrapolation from the classical state. Both of these phenomena were seen as well at the beginning of the transition to the ultimate state for Γ = 0.5 [36]. Indeed, for Ra > Ra * 1 the Γ = 1.00 data agree quite closely with the Γ = 0.50 data. Thus we believe that we observed the onset of the transition to the ultimate state also for Γ = 1.00, and that Ra * 1 for Γ = 1.00 is very nearly the same as it is for Γ = 0.50. Earlier measurements by Roche et al. [35] had revealed a transition in Nu(Ra) at several Γ values at Rayleigh numbers Ra U ≃ 1.3 × 10 11 Γ −2.5±0.5 which those authors attributed to the ultimate-state transition (for a detailed discussion of some of those data, see Ref. [22]). In contradistinction to this result, the transitions found by us for Γ = 0.50 and 1.00 are, within the resolution of the data, independent of Γ. We believe that a Γ-independent Ra * 1 suggests that the boundary-layer shear-transition is induced by fluctuations on a scale less than the sample dimensions rather than by a global Γdependent flow mode. Above Ra * 1 any difference between the heat transport for the two Γ values is too small to be resolved, suggesting a universal aspect of the ultimate-state transition and properties. Unfortunately the smaller height of the Γ = 1.00 sample, compared to Γ = 0.50, limits the accessible range to Ra < ∼ 2 × 10 14 . Thus, for this case, we were able to cover only a little more than the lower half of the transition range to the ultimate state.
SFB963: "Astrophysical Flow Instabilities and Turbulence". The work of G.A. was supported in part by the U.S National Science Foundation through Grant DMR11-58514. We thank Andreas Kopp, Artur Kubitzek, and Andreas Renner for their enthusiastic technical support. We are very grateful to Holger Nobach for many useful discussions and for his contributions to the assembly of the experiment.