Heat transfer in cryogenic helium gas by turbulent Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1

We present experimental results on the heat transfer efficiency of cryogenic turbulent Rayleigh–Bénard convection (RBC) in a cylindrical cell 0.3 m in both diameter and height which has improvements with respect to various corrections connected with finite thermal conductivity of sidewalls and plates. The heat transfer efficiency described by the Nusselt number Nu = Nu ( Ra , Pr ) ?> is investigated for the range of Rayleigh number 10 6 < Ra < 10 15 ?> , with the Prandtl number varying such that 0.7 ≦̸ Pr < 15 ?> , using cryogenic 4 ?> He gas with well-known and in situ tunable properties as a working fluid. For 7.2 × 10 6 < Ra < 10 11 ?> our data (both corrected and uncorrected) agree with suitably corrected data from similar cryogenic experiments and are consistent with Nu ∝ Ra 2 / 7 ?> . Up to Ra ≃ 10 12 ?> , our data could be treated as Oberbeck-Boussinesq data. For Ra > 10 12 ?> , the heat transfer efficiency becomes affected by non-Oberbeck–Boussinesq (NOB) effects, causing asymmetry of the top and bottom boundary layers. For 10 12 ≦̸ Ra ≦̸ 10 15 ?> , the Nusselt number closely follows Nu ∝ Ra 1 / 3 ?> if Nu ?> and Ra ?> are evaluated on the basis of the working fluid properties at the directly measured bulk temperature, T c ?> , and suitable corrections are taken into account. In contrast, if the mean temperature is determined as an arithmetic mean of the bottom and top plate temperatures, Nu ( Ra ) ∝ Ra &ggr; ?> displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime. The second step of our analysis, reported here for the first time, is to ignore the NOB effects affecting the top half of the RBC cell. We replace it by the inverted nearly OB bottom half in order to eliminate the boundary layer asymmetry. This leads to the effective temperature difference &Dgr; T eff = 2 ( T b − T c ) ?> , where T b ?> denotes the bottom plate temperature, and to effective Nu eff ?> and Ra eff ?> values. The effective heat transfer efficiency obtained, showing no tendency of crossover to the ultimate regime up to 2 × 10 15 ?> in Ra eff ?> , is reported and discussed.

( ) Nu Nu Ra, Pr is investigated for the range of Rayleigh number < < 10 Ra 10 6 1 5 , with the Prandtl number varying such that ⩽ < 0.7 Pr 15, using cryogenic 4 He gas with wellknown and in situ tunable properties as a working fluid. For × < < 7.2 10 Ra 10 6 1 1 our data (both corrected and uncorrected) agree with suitably corrected data from similar cryogenic experiments and are consistent with ∝ Nu Ra 2 7 . Up to ≃ Ra 10 12 , our data could be treated as Oberbeck-Boussinesq data. For > Ra 10 12 , the heat transfer efficiency becomes affected by non-Oberbeck-Boussinesq (NOB) effects, causing asymmetry of the top and bottom boundary layers. For ⩽ ⩽ 10 Ra 10 12 15 , the Nusselt number closely follows ∝ Nu Ra 1 3 if Nu and Ra are evaluated on the basis of the working fluid properties at the directly measured bulk temperature, T c , and suitable corrections are taken into account. In contrast, if the mean temperature is determined as an arithmetic mean of the bottom and top plate temperatures, ∝ γ ( ) Nu Ra Ra displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime. The second step of our analysis, reported here for the first time, is to ignore the NOB effects affecting the top half of the RBC cell. We replace it by the inverted nearly OB bottom half in order to eliminate the boundary layer asymmetry. This leads to the effective temperature difference Δ = − ( ) where T b denotes the bottom plate temperature, and to effective Nu eff and Ra eff values. The effective heat transfer efficiency obtained, showing no tendency of crossover to the ultimate regime up to × 2 10 15 in Ra eff , is reported and discussed.

Introduction
Natural convection is a complex type of turbulent flow occurring on many length scales across the Universe. At large scales, this buoyancy driven flow controls a number of natural processes such as weather formation through atmospheric [1] and oceanic flows [2], the terrestrial magnetic field and its reversals [3], continental drifts [4], and convection processes in planets and stars, including our Sun [5]. Although the equations describing turbulent flow are known, our ability as regards flow behavior prediction, especially for very intense convection occurring at extremely large scales, is very limited or even absent.
The ideal laterally infinite Rayleigh-Bénard convection (RBC) serves as a very useful model for fundamental studies of buoyancy driven flows. RBC occurs in a fluid layer confined between two laterally infinite, perfectly conducting plates heated from below in a gravitational field, and for an Oberbeck-Boussinesq (OB) fluid it is fully characterized by the Rayleigh numbers, Ra, and the Prandtl numbers, Pr. The main feature of RBC with which we shall be mainly concerned in this paper is that it transfers heat from the heated bottom plate to the cooled top plate. The convective heat transfer efficiency is described by the Nusselt number, via the = ( ) Nu Nu Ra; Pr dependence. Organized features of fluid motion, such as plumes and jets as well as large scale circulation (LSC), are known to exist in RBC and are intensively studied. The LSC, also known as 'wind', of the mean velocity U, has a dimension close to the size of the convective layer, L, and can be characterized by the Reynolds number, Re. The dimensionless numbers describing RBC are defined as follows: From the point of view of a rigorous theoretical standpoint, the problem of = ( ) Nu Nu Ra; Pr scaling (usually expressed in the form of a scaling law ∝ γ β Nu Ra Pr ), is still regarded as open, even for laterally infinite systems [6], although a plethora of exponents have been predicted in the framework of theoretical models developed by various authors. For the early influential works, we refer the reader to table 1 in the review by Ahlers, Grossmann and Lohse [7], where the predicted scaling exponents are conveniently displayed. Here we mention two independent theories, of Castaing [8] and Shraiman and Siggia [9], predicting γ = 2 7, while Priestley [10] and Malkus [11] derived γ = 1 3 in a model where heat transfer is controlled by the heat conduction of marginally stable boundary layers which become thinner with increasing heat flux-transferred heat does not depend on the height L and all the temperature difference ΔT occurs over the (thin in comparison with L) boundary layers, while the central turbulent fluid is effectively mixed and has nearly constant temperature T m .
At very high Ra, RBC ought to enter the 'ultimate', or 'asymptotic', regime (although with great uncertainty in Ra*, the critical Rayleigh number at which this regime should have its onset), as predicted phenomenologically by Kraichnan [12], who assumed very high Ra* indeed, not yet achieved experimentally in any laboratory. He postulated it as the regime in which heat transfer becomes independent of the thermal diffusivity and kinematic viscosity of the working fluid. This should occur at sufficiently large flow velocity, when viscous damping is negligible in comparison with flow inertia. When the convection reaches the ultimate regime, the boundary layers should undergo a laminar-to-turbulent transition and the heat flux would then no longer be controlled by the laminar boundary layers. The predicted scaling law is of the form, for < < 0. 15 1 2 conjecture, see also Spiegel [13].) It is hard to overestimate the utmost importance of this asymptotic regime. Typical values of Ra for convection in the atmosphere, ocean and Sun are of about 10 17 , 10 20 , and 10 30 , respectively. If one wishes to know the heat transport efficiency for such geophysical or astrophysical systems by extrapolation from the currently known experimental or computational data, one may incur uncertainties of up to an order of magnitude, owing to uncertainties in the power law exponent. Experimental, theoretical and computational studies of RBC up to the highest possible Ra are therefore of the utmost importance. For a recent extensive review of this topic, we direct the reader to the excellent work of Chillà and Schumacher [14].
Experimentally, a number of different working fluids have been used to study RBC. The highest Ra values for RBC in water have been achieved by the groups in Hong Kong ( = × Ra 5 10 12 at = Pr 4) [15] and in Lyon ( = × Ra 4 10 12 at = Pr 2) [16,17]. Several gases have been used at ambient temperatures, such as air [18], Ar [19,20], N 2 [19][20][21], He [19,22], ethane C 2 H 6 [23] and sulfur hexafluoride SF 6 [19,21,[24][25][26][27][28][29][30], altogether spanning the range < < 60 Ra 10 15 . In this work, following the 1975 pioneering experiment of Threlfall [31] covering the range < < 60 Ra 10 9 , we utilize the cryogenic helium gas, representing a working fluid with well-known [32,33] and in situ tunable properties. Since then, a number of high Ra cryogenic RBC experiments have been performed by several groups: in Chicago [8,34,35], Grenoble [36][37][38], Eugene [39], Trieste [40][41][42] and Brno [43,44]. In fact, cryogenic helium ( 4 He) offers three outstanding working fluids: besides the cryogenic helium gas also two liquid phases called, for historical reasons, helium I and helium II. Helium II is superfluid and serves as the most common working fluid for experimental studies of quantum turbulence [45]. Liquid helium I represents a classical viscous Navier-Stokes fluid possessing an extremely low kinematic viscosity of the order of − 10 4 cm − s 2 1 . It has also been used to study RBC, especially in the vicinity of the onset of convection [46]. Together with the cold 4 He gas, whose flow properties can conveniently be tuned in the experiment simply by changing the temperature and pressure, helium serves as a very useful classical viscous working fluid, offering a serious medium for classical fluid dynamics research with continuously tunable properties (over at least two orders of magnitude of kinematic viscosity) suitable for generating flows not only with high Reynolds and Rayleigh numbers but also large ranges of these flow parameters. This subject is often called cryogenic fluid dynamics [47][48][49].
The possibility of accessing a large range of Ra up to the highest Ra attainable in RBC laboratory experiments (≈10 17 ) [39] is, however, not the only advantage of using cryogenic helium gas. Cryogenic experiments benefit from an excellent thermal isolation thanks to the deep cryogenic vacuum surrounding the RBC cell. If placed inside a radiation shield thermally anchored at the helium bath, preventing any radiative heat exchange between the cell and the rest of the cryostat, any parasitic heat leak due to the surrounding medium is simply not an issue here. As a consequence, the sidewall is adiabatic. Additionally, favorable cryogenic properties of various construction materials such as copper or stainless steel allow for the design of the RBC cell with minimum influence of its structure on the convective flow studied. As we shall see, this is of utmost importance for experiments at the high end of the attainable range of Ra, where experimental details might become the key issue for the detailed analysis of the convective flow under study.
At this point, however, two very serious warnings are in order. First, in the laboratory experiments the lateral extent of plates is naturally limited, leading to additional dimensionless parameters describing the shape of the RBC cell and, in general, physical properties of walls as well as (except for the already mentioned cryogenic experiments surrounded by radiation shields in a deep vacuum) of the surrounding medium. The efficiency of the convective heat transfer must therefore be generally studied via the functional dependence = … ( ) Nu Nu Ra; Pr; C , , C exp 0 exp N ; here the quantities … C C , , exp 0 exp N are meant to take care of experimental details, leading to various corrections of the raw experimental data. Some of these corrections will be discussed in detail below. In most cases, the RBC experiments take place in cylindrical cells of diameter D and height L; the relevant additional parameter is the aspect ratio defined as Γ = D L, which might be understood as a first approximation in taking into account the shape of the RBC cell. For the heat transfer efficiency we therefore have . It should be emphasized that experimental (as well as theoretical or computational) studies performed on finite aspect ratio cells (especially for Γ < 1) cannot be understood as representing general behavior of laterally infinite RBC even if the container walls are ideal, as a new length scale, the lateral size, D, is introduced to the RBC flow under study.
The remaining experimental parameters could be divided into two groups, which could loosely be called geometrical and physical. The first group includes the actual shape of the cell (e.g., rectangular or cylindrical), possible deviation of the plates from the horizontal position or the surface roughness-i.e., the top and bottom plates are still treated as ideally conducting and the sidewall as ideally insulating. The second group includes the actual physical properties of the working fluid (which are known with limited accuracy and are deduced on the basis of measurements of its temperature and pressure, within certain error bars) as well as those of the RBC cell, such as the thickness, thermal conductivity and heat capacity of plates and walls, the heat conductivity of the electrical leads and, generally, the physical properties of the surrounding medium.
The second serious warning, which we are considering in detail in this paper, is that all of the above considerations implicitly assumed that the working fluid can be treated as an Oberbeck-Boussinesq (OB) fluid. The OB fluid is a fluid with constant physical properties except its density which, moreover, is assumed to linearly depend on temperature. In practice, the OB conditions are never fully satisfied and this might lead, as we shall see, to serious consequences, especially for laboratory experiments performed with gaseous working fluids in the vicinity of their critical points, in order to achieve high Ra.
Results on the heat transfer efficiency obtained in experiments performed for RBC cells with various shapes and aspect ratios, especially for > Ra 10 11 , often appear contradictory. An interesting attempt to unify them is the theoretical model of Grossmann and Lohse [50,51] (see also [7] and references therein for further developments). This approach, very recently updated by Stevens, van der Poel, Grossmann and Lohse [52] 3 , is certainly useful, as it predicts various behaviors of the RBC flow in the (Ra, Pr) parameter plane. The central idea is to split mean dissipation rates into two contributions each, one from the bulk and one from the boundary layers. A Blasius type of velocity (or viscous) boundary layer is assumed with a thickness ∝ − a Re GL 1 2 , where a GL is the parameter of the model which together with an additional four numerical parameters must be extracted by nonlinear fitting of (sometimes apparently contradictory) experimental results. Discussing various modifications of this theory is beyond the scope of this paper; however, to the best of our knowledge, it has not yet been possible to unify all the experimental results (such as various Grenoble cryogenic experiments; see [38]) by using this approach that assumes OB working fluids and we therefore do not discuss it any further.
As for computational studies, let us mention here direct numerical simulations that discretize the RBC equations of motion on a spatio-temporal grid and resolve all features of the turbulent flow down to the smallest physical scale. Currently, DNS for cylindrical cells with Γ = 1 2 can reach = × Ra 2 10 12 [53]. An additional aspect is that the numerical effort grows with Γ 2 when cells of larger aspect ratio are considered, so recent DNS at Γ = 1 (i.e., the aspect ratio of our RBC cell) achieved = × Ra 3 10 10 while requiring almost the same number of grid points [54].
The paper is organized as follows. After this introduction, in section 2 we describe our experimental apparatus and protocol for measuring the raw experimental data and present them in section 3. Various corrections due to physical properties of the RBC cell and the working fluid are discussed in section 4. Section 5 contains a discussion of our results, especially in view of NOB effects and comparison with available published Γ ≃ 1 high Ra cryogenic data. We conclude in section 6. In the appendix, we present our experimental data, tabulated.

The experimental apparatus and protocol
Several years ago, the discrepancies among various high Ra cryogenic experiments motivated us to design and build a cryostat, shown in figure 1, containing a cylindrical experimental RBC cell with the height L = 0.3 m and diameter D = 0.3 m, with particular effort taken to minimize the influence of the cell structure and materials on the observed RBC flow [55]. The cell has been designed to withstand pressures up to 3.5 bar in order to cover the range < < 10 Ra 10 6 1 5 with sufficient precision of measurements, aiming to resolve the published controversies as regards the heat transfer efficiency. It is desirable that the top and bottom plates should maintain non-fluctuating constant temperatures T t and T b . The plates of our cell are made of 28 mm thick annealed OFHC copper of very high thermal conductivity (at least 2 kW m −1 K −1 at 5 K). The thermal conductivity of our plates is approximately twice that of the Grenoble and Oregon/Trieste cells; see table 1. The design of the heaters, glued in the spiral grooves milled on the external sides of plates, ensures better than 1 mK temperature homogeneity of the internal side of plates, under the assumption that the heat is uniformly supplied or removed. From the top plate, the heat is removed homogeneously by the heat exchange chamber to the helium bath above. In the experiments, the temperature of the top plate is roughly set by adjusting the pressure in the chamber and stabilized precisely by the heater using the Lake Shore 340 temperature controller.
The parasitic heat fluxes associated with the finite thermal conductivity of the sidewalls are minimized by using very thin (δ = 0.5 mm) stainless steel of relatively low thermal conductivity λ w , as well as by employing a special design of the cell corners; see figure 2.
We also paid attention to assuring good thermal shielding and minimizing other external parasitic heats into the cell which could influence the RBC flow under study. The parasitic heat inputs are suppressed to less than 1% of the lowest heat fluxes used in the experiment, measured within 0.5% accuracy [55].
Four calibrated Lake Shore GR-200A-1500-1.4B Ge temperature sensors (5 mK absolute accuracy guaranteed by the manufacturer) are embedded in the center and near the edge of both Cu plates. Additional home-made equipment for calibration of the Ge temperature sensors allows us to measure the temperature difference ΔT between the plates with uncertainty up to 2 mK [55]. Additionally, we have installed four small Ge sensors [56] in the cell interior, in pairs positioned opposite at half the height of the cell 1.5 cm from the sidewall and 2.5 cm vertically  apart (see figure 1), and calibrated them in situ, within ±1 mK accuracy, against our primary Lake Shore GR-200A-1500-1.4B Ge temperature sensors built into the bottom and top Cu plates. This opened the possibility of measuring the temperature, T c , of the turbulent core of the working fluid directly. Additionally, these small sensors allow measurements of temperature fluctuations and subsequent determination of statistical properties of the convective flow, such as power spectra, mean wind velocity or scaling exponents. A detailed report of our statistical investigation of the RBC flow will be published elsewhere. Two stainless steel tubes are used for venting our RBC cell; their inlets are positioned in the cell sidewall at 2/3 of the cell height (see figure 1). Originally, in order to reduce conductive parasitic heat leak, both tubes were thermally anchored to the liquid helium vessel. Our test measurements, at a deliberately substantially higher mean temperature of the RBC cell T m , revealed a spurious increase in Nu due to convection inside these venting tubes between the cell inlet (at ≈T m ) and the thermal anchor at the liquid helium vessel 140 mm above. We therefore improved the design of the cell [57] and eliminated this 'chimney effect', by mounting electrical resistance heaters and thermometers onto both venting tubes at the level of their thermal anchor to the liquid helium vessel that allow adjusting and controlling the local temperature to ⩾ T T m during measurements.
When testing the newly built apparatus, we encountered problems with the thermal acoustic oscillations in these venting tubes, as mentioned in [55]. This problem was solved by using additional volumes connected to the tubes at room temperature, resulting in suppression of the observed pressure oscillations by at least two orders of magnitude [57].
With the additional improvements described, the apparatus has been found capable of performing experimental research on various aspects of the RBC flow, in particular for studies of open problems at very high Ra. It has also been successfully used to study two-fluid convection; these results and their impact on general thermodynamics have been published elsewhere [58,59].
The experimental protocol is as follows. Cooling the apparatus involves several steps: the evacuation of the cryostat, the evacuation of the experimental cell, the pre-cooling by liquid nitrogen, the filling of the liquid vessel with liquid helium, the filling of the heat exchange chamber with gaseous helium up to the desired pressure, and the filling of the experimental cell with helium from the liquid helium vessel via a home-made cryovalve, operated from the flange of the cryostat. These operations and the thermal stabilization of the whole system take about ten days.
The pressure in the cell is measured with an MKS Baratron 690A (calibration traceable to NIST) with 0.08% reading accuracy. A highly stable temperature of the upper plate is maintained with the Lake Shore 340 temperature controller, which is also used to continuously read the data from the temperature sensors. For each data point, the temperature readings are monitored for sufficiently long time (of the order of an hour, depending on experimental conditions), until the temperature stabilizes, which is additionally checked by several readings averaged typically over intervals of ten minutes.
Helium properties are gained from the NIST database [32,33], based on the actual pressure in the cell and the mean temperature T m assessed either as the arithmetic average of the plate temperatures T t and T b or on the basis of the directly measured bulk temperature T c inside the cell at about half of its height. The temperature T c represents the arithmetic mean from three (in the case where one sensor was in use to provide the temperature fluctuation data) or even all four of these sensors. Let us mention in passing that if (time-averaged over typically 10 min) readings from individual sensors are used instead of their mean value, this does not lead to appreciable changes in the results; the data based on individual sensor readings basically suffer somewhat bigger scatter.
One filling of the cell (one density) allows measurements spanning about one and a half orders of magnitude of Ra. During the measurements, we attempted to choose various working points in the pressure-temperature phase diagram of 4 He, in order to check and eliminate possible uncertainties in the fluid properties, especially closer to the critical point. Figure 3 presents our raw Nusselt versus Rayleigh number data obtained in two long-run experiments, as already reported in our short Letter publications [43,44]. In the first experiment, the deduced Nu and Ra values are based on the measured pressure in the cell, p, four-wire measurements of the heat input to the bottom plate heater,

Results
, and measured temperatures of the top, T t , and bottom plates, T b , leading to the assessment of the mean temperature of the helium gas, T m , as their arithmetic mean (thus assuming the validity of OB conditions); we refer to them here as Ra m and Nu m . In the second experiment [44], the raw = the cell at about half of its height and we refer to them here as Ra c and Nu c . We emphasize that the direct measurement of T c is instrumental for conclusions drawn in this study. We indeed show that, generally, ≠ T T c m . Assessing T m as the fluid temperature (which is inevitable when the temperatures of the plates are the only temperatures measured) has no general physical meaning and, as we show later, may lead to spurious conclusions.
To enable public access to all our raw data, and to allow appreciation of the exact experimental conditions of these measurements, we present them, tabulated, in appendix. In order to allow better appreciation of the scaling as well as the significance of all corrections applied to our raw data, in figure 4 we plotted the compensated data over the entire range of Ra investigated. Figures 3 and 4 represent the main results of our experiment. In the following part of the paper, we discuss them, justify all the corrections made and compare our data with similar available high Ra RBC experiments.

Corrections to raw data
Despite the experiment being surrounded by a deep cryogenic vacuum and the cell having been designed having in mind its minimum influence on the RBC flow studied, various corrections to the raw data are required. In this section we describe them in detail.

Corrections for the parasitic heat leak and the adiabatic temperature gradient
The radiation heat load is reduced by a cell radiation shield thermally anchored to the liquid helium vessel. The shield temperature does not exceed 7 K; thus the total external parasitic heat leak to the cell (both radiative and conductive) is suppressed to less than 1% of the lowest convective heat flux used in the experiment. The design of the cell leads to a correction to the adiabatic temperature gradient of less than 1 mK. Within our experimental protocol (Δ > T 30 mK), this correction is not essential; nevertheless, it has been applied to all data in a standard form, i.e., by withdrawing the quantity Δ α = T g LT c

Sidewall corrections
One way to estimate the influence of the sidewall is via the so-called wall parameter defined by Roche and co-workers [60] as where 0.40 (based on the nominal thickness of the sidewall δ = 3 mm) cryogenic cells are higher, leading to larger sidewall corrections when applying equation (4). We note that the Oregon/Trieste sidewalls (not the Grenoble cells) include thick stainless steel flanges for connection to both Cu plates which makes their effective thickness, δ eff , substantially higher than their nominal thickness δ = 3 mm (see later). Our cell also contains two stainless steel flanges that enable access to the cell interior (and, by changing the middle part of the sidewall, alteration of its shape or aspect ratio)-however, 30 mm above the bottom plate (below the top plate) where there is (almost) no vertical temperature gradient and, consequently, the thickness and heat conductivity of the flange hardly matter.
In our previous publication [43], we have shown that for × ⩽ ⩽ 7.2 10 Ra 10 6 1 1 our sidewall corrected data agree with suitably corrected data from these similar cryogenic experiments and are consistent with ∝ Nu Ra 2 7 . On approaching ≈ Ra 10 11 , our data display a broad crossover to ∝ Nu Ra 1 3 , as predicted by analytical theory [10,11]. We emphasize that the differences in Nu(Ra) scaling observed in similar RBC experiments for ⩾ Ra 10 11 cannot be attributed to sidewall corrections, as they do not play a significant role here.

Remarks on corrections for the finite heat conductivity and heat capacity of plates
Unlike the sidewalls, the bottom and the top plates of thickness a are expected to influence the observed convection at high values of Ra, due to their finite heat conductivity and heat capacity.
First of all, our primary Ge thermometers are placed vertically in the middle of the top and bottom plates. Thus, assuming for simplicity a one-dimensional problem, there is a finite temperature drop across the copper of height × = ( ) a a 2 2 and thermal conductivity λ p , as well as across the RBC cell of height L and effective heat conductivity λ Nu . This would lead to negligible corrections to the measured ΔT for all data points measured; however, the problem is much more complex.
On the basis of numerical simulations, Verzicco found a criterion based on the following parameter: λ plotted for the cryogenic Γ ≃ 1 high RBC experiments in the left panel of figure 5. According to his DNS numerical study [61], Verzicco claims that the Nusselt number is not influenced by non-ideal plates if X p lies above a critical threshold, estimated as ≃ R R 300 f p . We see that nearly all the cryogenic Γ ≃ 1 data satisfy this criterion. It is clear, however, that as X p is inversely proportional to Nu, this criterion will have to be taken into account once a higher range of Ra becomes attainable.
An additional criterion in scrutinizing the quality of plates is based on the consideration of whether the plate is 'fast enough' to supply enough heat for successive plumes. Castaing and co-workers [8] estimated the non-dimensional time between two successive emissions of plumes as τ ≈

( )
Ra Pr 4Nu f 2 ; it therefore decreases with increasing Ra, since Nu increases faster than Ra 1 4 while Pr does not change significantly. In the same spirit, as shown by Verzicco [61], the corresponding time for plates becomes τ λ λ ≈ ( ) ( ) a L Ra Pr p 2 p . In the right panel of figure 5, we plot the ratio τ τ p f for our RBC cell as well as for the similar Grenoble and Trieste Γ ≃ 1 cryogenic experiments-we see that all of them satisfy this dynamical criterion very well.
On the basis of the numerical study of Verzicco [61], the experiments of Funfschilling et al with water [62] and those of Ahlers et al with SF 6 [21], Chillà et al [63] derived approximate criteria for accounting for the 'plates effect'. According to their 'thin plates criterion' (which for our plates is more stringent than their 'thick plates criterion'), the condition means a negligible 'plates effect' in RBC experiments with cryogenic helium up to about ≈ Ra 10 12 . We have discussed this criterion in our previous publication [57], where we plotted the Ra dependence of the parameter Cr evaluated for our cell as well as for the Oregon/Trieste and the Grenoble cryogenic cells of aspect ratio Γ ≃ 1; for the same Ra the values of Cr are similar in magnitude to but slightly lower than ours. This strongly suggests that the observed differences between the cryogenic RBC experiments at high > Ra 10 11 cannot be interpreted as the 'effect of plates'. On the basis of these considerations, we conclude that our cell corrections for the physical properties of plates are hardly needed. Therefore we have not applied them to our data in any form. In the vicinity of the critical point, the physical properties of helium gas (as well as of other gases) vary with temperature more rapidly than far away from it and this might influence the deduced heat transfer efficiency significantly, especially in laboratory experiments aiming to reach high Ra. In order to allow appreciation of this, we plot in figure 6 the compensated Nu as a function of Ra,on the basis of the fluid properties evaluated for our data at four different temperatures: Calculating the Nusselt and Rayleigh numbers on the basis of T b and T t is instructive, as it indicates the absolute gate where the =

( )
Nu Nu Ra data might lie. Note that this gate is, rather surprisingly, clearly defined, despite very different ΔT (also plotted in figure 6) having been used to measure any particular data point.
On the basis of this calculation, for our cryogenic RBC cell with this particular choice of the working points in the p-T phase diagram of 4 He, we can conclude the following: (i) up to about ≃ Ra 10 12 , the particular value of the absolute temperature at which the fluid properties are evaluated (we assume here that the corresponding pressure is known accurately) could be taken anywhere within ΔT (which is assumed to be known with sufficient precision on the basis  As we already stated in section 1, the OB conditions are never fully satisfied in practice; therefore an experimental criterion has been suggested (see e.g. [42]): α Δ < ( ) T 0.1 0.2 4 assuming that a real working fluid satisfying it is regarded as a fully OB fluid. This criterion, however, cannot be accepted in general, especially in the vicinity of the critical point. As an example, all the data in figure 6 satisfy the experimental criterion α Δ < T 0.2 while the NOB effect at the high Ra end is evident.
When NOB effects are present, the mean temperature of the plates, T m , and the temperature of the bulk in the middle of the cell, T c , are different. We follow Wu and Libchaber [64] by stating: 'we redefine Ra based on the fluid properties in the central region of the cell. Although this choice is arbitrary, it seems most reasonable since the central region occupies the majority of the cell volume'.
This physically motivated and from our point of view the most reasonable evaluation of fluid properties for RBC leads to serious consequences for evaluated = …

( )
Nu Nu Ra Pr , , scaling. Indeed, as we have shown in our recent Letter [44], NOB conditions lead to asymmetry in boundary layers adjacent to plates. The result is that for > Ra 10 12 , the Nusselt number closely (i.e., within our error bars) follows ∝ Nu Ra 1 3 if the mean temperature of the working fluid-cryogenic helium gas-is measured directly, by small sensors placed inside the cell interior, and all appropriate corrections as described above are taken into account. In contrast, if the mean temperature is determined in a conventional way, assuming full validity of OB conditions, as an arithmetic mean of the bottom and top plate temperatures, then Ra displays a spurious crossover to higher γ that might easily be misinterpreted as a transition to the ultimate Kraichnan regime [12]. Let us, however, discuss the NOB effects in more detail.
As we already mentioned, the NOB effects in high Ra convection of cryogenic helium gas were first studied in some detail by Wu and Libchaber [64]. They confirmed experimentally that the temperature dependence of the fluid properties does indeed lead to a symmetry breaking between the top and the bottom of the sample. In order to describe this effect quantitatively, they introduced the key parameter, x, defined as a ratio of the temperature drop over the top boundary layer to that over the bottom boundary layer: c . Parameter x is plotted versus Ra for their and our data in figure 8. Wu and Libchaber [64] found, perhaps surprisingly, that for their data up to < Ra 10 11 , Nu remains remarkably insensitive to NOB conditions.
A closer examination shows that our results are in fact in fair agreement with those of Wu and Libchaber [64]. Figure 8 shows that for our RBC cell up to < Ra 10 13 the NOB effect is small, thanks to the small changes in the fluid properties evaluated using T c instead of T m , while the data of Wu and Libchaber [64] display a fast drop of the x-parameter already for > Ra 10 10 . If, however, their x-parameter data are recalculated for the RBC cell of our height, taking into account that ∝ L Ra 3 , the renormalized ( )

On the asymmetry of boundary layers in RBC
We have confirmed experimentally the previous results of Wu and Libchaber [64], that in cryogenic helium gas, NOB effects do indeed lead to asymmetry of the boundary layers, which subsequently alters the heat transfer efficiency at high Ra. It is therefore interesting to compare  [64], while the half-filled diamonds represent their data recalculated for our RBC cell height, shifted to higher Ra as indicated by the arrow. The stars stand for the water data of Tisserand and co-workers [17]. For details, see the text. this finding with the situation when the symmetry of boundary layers is broken intentionally, for example, by introducing surface roughness on one plate. The relevant experiment has been performed by Tisserand and co-workers in Lyon [17], with deionized water as the working fluid, in a cylindrical RBC cell 50 cm in diameter for two different heights: L = 1 m and 0.2 m. The sidewall is stainless steel of thickness δ = 2.5 mm. While the top plate is smooth, made of nickel plated copper, the bottom one is rough, made of aluminum. Its roughness consists of a square array of square plots, with 5 mm side and 2 mm height, with a periodicity of 1 cm. The key result is that for > Ra 10 11 the heat transfer efficiency becomes strongly enhanced in comparison with that for the same experiment but with both plates smooth [66], which for > Ra 10 11 displays the usual ∝ γ Nu Ra behavior with γ ≃ 1 3. The observed enhancement in heat transfer efficiency is associated with the strong increase of the parameter x, which is plotted in figure 8 together with our data and the original and rescaled data of Wu and Libchaber [64]. The coincidence of the onset of increased heat transfer efficiency and the departure of the xparameter from a constant value (an increase in this case) is evident.
For our further discussion, it is important to notice that Tisserand and co-workers [17] found, on the basis of comparison with the results for the same RBC cell with smooth plates, that the two boundary layers are independent of each other and that this conclusion could not have been reached solely on the basis of results for a symmetric cell.
An interesting observation concerns the value of the x-parameter before the heat efficiency enhancement occurs. For a strictly OB fluid, it is natural to expect the value of the parameter x to be very close to unity (we are, however, not aware of a rigorous proof of that, especially in view of the adiabatic temperature gradient); all three experiments displayed in figure 8 consistently show otherwise; namely ≃ x 0.8-0.9 already for lower values of Ra where the observed =

( )
Nu Nu Ra scaling still behaves conventionally 5 . This once again confirms that the experimental criterion αΔ < T 0.2 is not sufficient for excluding the NOB effects, which are present but not (yet) causing any appreciable enhancement in the heat transfer efficiency. We speculate that while the change in the case of smooth plates (such as ours) has a gradual onset, in the case of the rough bottom plate, it is triggered when the boundary layer thickness matches the surface roughness, and the transition is then abrupt [17]. Let us mention here that our plates are machined to surface roughness better than μm 1. 6 , while the boundary layer thickness at ≈ Ra 10 14 can be estimated as  . Here we offer a further step forward, based on the following considerations.
In the upper part of the Ra region covered in our experiments, for each =

( )
Nu Nu Ra data point, we have directly measured the pressure 6 and three temperatures T T , b t and T c simultaneously; the temperature T c was evaluated as an average of individual readings from the small in situ calibrated Ge sensors. While most of the RBC cell, the bulk, has temperature very close to T c , in view of the shape of the p-T phase diagram of He 4 and our choice of working conditions there (below the saturated vapor curve and/or critical isochore), the conditions in the bottom (hotter) part of the RBC cell are substantially more OB than in the upper (colder) part. Assuming that the top and bottom boundary layers are independent (this assumption is consistent with findings of Tisserand and co-workers [17] for the intentionally asymmetric case mentioned above as well as with the power scaling law with γ ≈ 1 3 that corresponds to a model where all ΔT occurs across the boundary layers, while in the central turbulent region the working fluid is effectively mixed and heat transfer is therefore controlled by thermal conduction of the boundary layers and the convective heat flux does not depend on L), we replace the top half of the RBC cell by an inverse bottom half. In other words, we ignore the NOB top part of the RBC cell. As a result, we construct an effective, fully symmetric RBC flow, where the temperature difference Δ = − ( ) c . It is plausible to expect that this artificially constructed RBC flow will match the ideal OB conditions much more closely. In  Figure 10 represents the central result of our analysis. It clearly shows the importance of NOB effects at high Ra, taken into account in two steps.
The first step takes into account the actual fluid properties of the bulk of the working fluid, evaluated on the basis of the directly measured temperature of the bulk, T c , instead of T m ; however, the NOB distortion of the (upper half of the) temperature drop, over the upper NOB boundary layer, is ignored. This step leads to the difference between the data sets represented by large circles.
The second step takes into account both the actual fluid properties of the bulk of the working fluid and, additionally, the NOB distortion of the temperature drop over the top NOB boundary layer, by replacing it with the temperature drop over the bottom boundary layer, suffering much less NOB distortion, as the working fluid in the bottom half of the RBC cell satisfies the OB conditions more closely. Figure 10 shows that the NOB effects affect the = ∝ γ ( )

Nu Nu Ra
Ra scaling significantly, with a clear tendency to spuriously increase the exponent γ. Although this was apparent already from our Letter [44], this newly presented analysis confirms this tendency even more clearly. An encouraging aspect, suggesting that the presented analysis is meaningful, is the fact that the scatter of the compensated Nu Ra  figure 10 is that it does not show any tendency towards a transition to the ultimate state of RBC convection, which was reported by He and co-workers [26] to take place between the two vertical dotted lines (see figure 10) at * Ra 1 and * Ra 2 in the Γ = 1 2 SF 6 experiment at ambient temperature. This transition was claimed by the Göttingen group, perhaps with somewhat less confidence, also in a similar Γ = 1 SF 6 experiment [29]. This apparent controversy might possibly be due to the difference in Prandtl number of the working fluids used. This is why our compensated Nu Ra increasing helium density at the high Ra end roughly as ∝ Pr Ra 0.5 , as shown for our experiment in figure 3 7 .
The Prandtl number dependence of the Nusselt number (for Ra well below the transition claimed to occur between * Ra 1 and * Ra 2 , assuming the OB working fluid) has been a subject of numerous studies and is conveniently summarized in [7], where the reader can find a number of references to the original works. Generally, for low Prandtl number, < Pr 1, within the accessible high Ra range, these studies show a rather steep increase of Nu; for Pr larger than about 1, a saturation sets in and within a certain range of Pr, the Nusselt number becomes almost Pr independent. A maximum in the Nu(Pr) dependence is indicated at about Pr = 3; above this value, a very gradual decrease is observed. These Nu(Pr) experimental features are reflected by the Grossmann-Lohse theoretical model [50]. On the other hand, we are not aware of any systematic experimental ( ) Nu Pr data for ⩾ Ra 10 13 . Let us recall the lower panel of figure 6 of [7], plotting Pr versus Ra from various high Ra experiments, where the prediction of the Grossmann-Lohse theory for the transition (for OB fluids) to the Kraichnan regime for Γ = 1 experiments is indicated, assuming critical boundary layer Reynolds numbers equal to 220 and 440. As our Pr c versus Ra c (or Ra eff ) data hardly cross the predicted transition line based on the lower critical boundary layer Reynolds number 220 and never closely approach that based on the higher value of 440, we may conclude that our experiment and the prediction of the Grossmann-Lohse theory are formally not in disagreement.
Indeed, the lower panel of figure 11 shows the quantity Nu eff , compensated both by the ultimate scaling law ( ∝ Nu Ra Pr 1 2 1 2 ) as predicted by Grossmann and Lohse [50] and by the ultimate scaling law as predicted by Kraichnan [12]-see equation (2)-plotted versus Ra eff . We see that, within the parameter range explored, our analysis does not support the possible existence of either of these predicted scalings.
Assuming, therefore, that within the parameter range covered the transition to the ultimate state does not take place, for Prandtl number about 1 or slightly greater, the theoretical predictions of Malkus [11] as well as the Grossmann-Lohse model [50]-∝ Nu Ra 1 3 -do not explicitly include any Pr dependence. The upper panel of figure 11 shows that this scaling for Nu eff holds if a weak Prandtl number power law dependence is assumed, of the form β Pr c with β about −0.12. Note also that if we admit a power law dependence of Nu eff on Pr c , the scatter of the data points will increase. This indicates, within the range of parameters investigated, a rather weak Prandtl number dependence of the effective Nusselt number.

Comparison of our results with available corrected Γ ≃ 1 high Ra cryogenic data
As the possible role of the aspect ratio of the RBC cell used in the heat transfer efficiency is at present not accurately known, we limit ourselves with comparing our data with the Trieste [40,42] and Grenoble [38] cryogenic data obtained with cold helium gas in cells of similar aspect ratio (Γ ≃ 1). Figure 12   , where a transition to the ultimate state was claimed in the SF 6 experiment at ambient temperature [26]. Lower panel: the uncorrected effective Nusselt number, compensated by the ultimate scaling law as predicted by Kraichnan [12] and by Grossmann and Lohse [50], plotted against Ra eff . versus Ra. Additionally, for our data, the influence of evaluation based on T c as opposed to T m is shown.
We have already claimed full quantitative agreement among (suitably corrected for the sidewall) Γ ≃ 1 cryogenic experiments for Ra up to about 10 11 . At higher Ra however, all of these sets of data differ considerably. A natural question is that of whether or not the Trieste [40,42] and Grenoble [38] data could possibly also be affected, similarly to our own, by the NOB effects and whether additional analysis, leading to suitable corrections, might result in a collapse of all of them onto a single dependence.
Before answering this question, let us stress that the Grenoble group performed a large series of cleverly designed experiments aiming to find out why the 'Grenoble regime', characterized by a higher exponent γ, is observed 8 , typically above ≃ Ra 10 11 . A detailed account of a number of these controlled experiments was recently published by Roche, Gauthier, Kaiser and Salort [38]. In particular, figure 5 of [38] shows that transitions to the 'Grenoble regime' can be easily spotted by the increase in the local γ-exponent above the 1/3 value. The authors have underlined the variability of the transition in terms of transitional Ra (over nearly two decades 10 11 -10 13 ), although not for cells of the same aspect ratio. A particularly interesting fact is that the authors found that breaking the LSC with screens has only a limited impact on the heat transfer, which clearly indicates that the LSC alone is not a transition trigger, as is often assumed in the literature. In the conclusion, the authors claim that 'there is evidence that Grenobleʼs regime corresponds to Kraichnanʼs prediction and no experimental fact seems incompatible with such an interpretation. Nevertheless, the conditions for the triggering of this regime are obscure and sometimes surprising. In addition, a few experimental facts cannot be directly explained using the genuine Kraichnan model. Further experimental investigations are clearly needed'.
These results apparently contradict a number of similar RBC experiments performed on cells of similar aspect ratio that show no similar transition over the same range of high Ra up to 10 13 , such as those performed in Chicago [8], and in Oregon/Trieste [40], as well as in experiments performed at ambient temperatures in Rehovot [24] and Göttingen [28,29]; however, let us concentrate here on the cryogenic Γ ≃ 1 experiments [38,40] and comparison with our own experiments.
The experimental protocol of the Grenoble experiments differs from our own and the Trieste experimental protocols in that the Grenoble group have worked with constant density (which has been entirely or in part evaluated at room temperature, using a calibrated volume from which the working fluid condenses to the RBC cell and is sealed there using the cryovalve) of the working helium gas, while in Brno and Trieste the pressure in the RBC cell is measured individually for each data point. Additionally, the Grenoble group evaluates the physical properties of the working fluid differently, partly on the basis of their own calibration procedure, while the Trieste and Brno groups use the improved accuracy software package x-Hepak [33], which was developed by Arp and McCarty for the University of Oregon in connection with the Oregon Γ = 1 2 experiment [39].
We have re-analyzed the Grenoble Γ = 1.14 [38] data using our x-Hepak software package. The resulting Nu and Ra values differ from those published, most notably for high densities of the helium gas; however, the slope of the = ( ) Nu Nu Ra dependence hardly changes. We conclude that re-analyzing the data does not lead to any qualitative changes in the observed heat transfer efficiency, and, in particular, the claimed transition to the 'Grenoble regime' remains valid.
Let us now attempt to answer the crucial question of whether or not the differences among these cryogenic Γ ≃ 1 experiments could be caused by NOB effects. The answer can only be qualitative, as the Grenoble and the Trieste groups did not measure the temperature of the bulk directly in their RBC cells and the possible asymmetry of the top and bottom boundary layers is not experimentally known. All we attempt here is to suggest some hypothetical asymmetry for . In view of the different protocols of these experiments mentioned above, we kept either constant density (Grenoble) or pressure (Trieste) of the cryogenic He gas. Without going into the details of this tedious procedure, we have not been able to find any reasonable degree of asymmetry of boundary layers that would lead to the collapse of all the cryogenic Γ ≃ 1 experiments considered.
This, however, cannot be understood as proof that NOB effects are not, at least in part, responsible for the observed differences, as the high Ra ranges of the Trieste and Grenoble experiments have been investigated using a working point ( ) p T , m close to the critical isochore of helium. For small ΔT , our analysis is not suitable, as both boundary layers will be affected and the above assumption that one of the boundary layers can be treated as OB is not justified. Moreover, on the basis of the analysis of Ahlers, Araujo, Funfschilling, Grossmann, and Lohse [67] of the NOB effects in gaseous ethane, one may expect that the asymmetry of the boundary layers might be partly cancelled if T b and T t lie on the opposite sides of the critical isochore in the (p,T) phase diagram. Further experiments are needed to clarify this issue.
In our Letter [43], we have already claimed that distinctly different ( ) Nu Ra scaling as observed in various high Ra experiments can hardly be explained by the difference in Pr and that other tiny experimental details will have to be carefully considered. This statement was in section 1 expressed as Γ = … ( ) Nu Nu Ra; Pr; , C , ,C N exp 1 exp , where the experimental parameters could be divided into two groups, which could loosely be called geometrical and physical. We presented a number of reasons explaining why, under cryogenic conditions, the physical parameters, especially those describing the physical properties of the RBC cells, are less important and easier to handle. Our presented analysis thus can be understood as a clear call for further detailed high Ra cryogenic experiments that could lead to resolving the most important issue: that of the heat transfer efficiency of the RBC flow.

Conclusions
We have presented a detailed experimental study of the heat transfer efficiency in cryogenic helium gas by turbulent Rayleigh-Bénard convection in a cylindrical cell with aspect ratio 1. Our main findings can be formulated as follows.
In the turbulent regime of RBC, the measured Nu (both corrected and uncorrected) values obey, at least approximately, ( ) Nu Ra power law scaling with exponent γ ≈ 2 7 in the region × ⩽ < 7.2 10 Ra 10 6 9 , where < Pr 1. Within the next 2-3 decades of Ra, the power exponent slowly increases and approaches γ ≈ 1 3. By applying suitable sidewall corrections, we have shown full agreement among Γ ≃ 1 cryogenic experiments for Ra up to about 10 11 , while at higher Ra all these sets of data differ considerably. Up to about ≃ Ra 10 12 , our data could be treated as OB, in that the particular value of the absolute temperature at which the fluid properties are evaluated could by taken anywhere within ΔT . Upon increasing Ra above this value, our data gradually become affected by the NOB effects; up to about × 3 10 12 their influence lies within the typical experimental scatter, while above ≃ × Ra 3 10 12 , the fluid properties start to affect the =

( )
Nu Nu Ra scaling appreciably. The directly measured bulk temperature T c differs from the arithmetic mean of the plate temperatures, = + ( ) , confirming a boundary layer asymmetry. As the bulk of the RBC cell has the temperature T c , it is natural to define Ra, Nu and Pr on the basis of this temperature rather than on T m -this leads to relatively small changes in Nu, but significantly larger changes in Ra. This step can be thought of as a first-order correction in taking NOB effects into account. When applied to our data, the observed Nu(Ra) dependence closely displays power law behavior with the exponent of 1/3 up to about 10 14 in Ra. The second step of our analysis, reported here for the first time, is to ignore the NOB effects in the top half of the RBC cell and replace it with the inverted nearly OB bottom half in order to eliminate the boundary layer asymmetry. This leads to an effective temperature difference Δ = − Thanks to a number of listed reasons why cryogenic conditions are advantageous, our article can be understood as a clear call for further detailed high Ra cryogenic experiments with helium gas as a working fluid, that could lead to resolving the issue of utmost importance-the heat transfer efficiency of the RBC flow.

Appendix. Tabulated experimental data
In tables A1 and A2, values of Ra corr and Nu corr represent the corrected data, with corrections applied as discussed in the text. All remaining values of Rayleigh, Prandtl and Nusselt numbers are not corrected in any way.  Table A1. Tabulated experimental data for T m .