Reynolds numbers and the elliptic approximation near the ultimate state of turbulent Rayleigh–Bénard convection

We report results of Reynolds-number measurements, based on multi-point temperature measurements and the elliptic approximation (EA) of He and Zhang (2006 Phys. Rev. E 73 055303), Zhao and He (2009 Phys. Rev. E 79 046316) for turbulent Rayleigh–Bénard convection (RBC) over the Rayleigh-number range 10 11 ≲ Ra ≲ 2 × 10 14 ?> and for a Prandtl number Pr ≃ 0.8. The sample was a right-circular cylinder with the diameter D and the height L both equal to 112 cm. The Reynolds numbers ReU and ReV were obtained from the mean-flow velocity U and the root-mean-square fluctuation velocity V, respectively. Both were measured approximately at the mid-height of the sample and near (but not too near) the side wall close to a maximum of ReU. A detailed examination, based on several experimental tests, of the applicability of the EA to turbulent RBC in our parameter range is provided. The main contribution to ReU came from a large-scale circulation in the form of a single convection roll with the preferred azimuthal orientation of its down flow nearly coinciding with the location of the measurement probes. First we measured time sequences of ReU(t) and ReV(t) from short (10 s) segments which moved along much longer sequences of many hours. The corresponding probability distributions of ReU(t) and ReV(t) had single peaks and thus did not reveal significant flow reversals. The two averaged Reynolds numbers determined from the entire data sequences were of comparable size. For Ra < Ra 1 * ≃ 2 × 10 13 ?> both ReU and ReV could be described by a power-law dependence on Ra with an exponent ζ close to 0.44. This exponent is consistent with several other measurements for the classical RBC state at smaller Ra and larger Pr and with the Grossmann–Lohse (GL) prediction for ReU (Grossmann and Lohse 2000 J. Fluid. Mech. 407 27; Grossmann and Lohse 2001 86 3316; Grossmann and Lohse 2002 66 016305) but disagrees with the prediction ζ ≃ 0.33 ?> by GL (Grossmann and Lohse 2004 Phys. Fluids 16 4462) for ReV. At Ra = Ra 2 * ≃ 7 × 10 13 ?> the dependence of ReV on Ra changed, and for larger Ra Re V ∼ Ra 0.50 ± 0.02 ?> , consistent with the prediction for ReU (Grossmann and Lohse 2000 J. Fluid. Mech. 407 27; Grossmann and Lohse Phys. Rev. Lett. 2001 86 3316; Grossmann and Lohse Phys. Rev. E 2002 66 016305; Grossmann and Lohse 2012 Phys. Fluids 24 125103) in the ultimate state of RBC.

≲ ≲ × and for a Prandtl number Pr ≃ 0.8. The sample was a right-circular cylinder with the diameter D and the height L both equal to 112 cm. The Reynolds numbers Re U and Re V were obtained from the mean-flow velocity U and the root-mean-square fluctuation velocity V, respectively. Both were measured approximately at the mid-height of the sample and near (but not too near) the side wall close to a maximum of Re U . A detailed examination, based on several experimental tests, of the applicability of the EA to turbulent RBC in our parameter range is provided. The main contribution to Re U came from a large-scale circulation in the form of a single convection roll with the preferred azimuthal orientation of its down flow nearly coinciding with the location of the measurement probes. First we measured time sequences of Re U (t) and Re V (t) from short (10 s) segments which moved along much longer sequences of many hours. The corresponding probability distributions of Re U (t) and Re V (t) had single peaks and thus did not reveal significant flow reversals. The two averaged Reynolds numbers determined from the entire data sequences were of comparable size. For Ra Ra 2 10 1 * 13 < ≃ × both Re U and Re V could be described by a power-law dependence on Ra with an exponent ζ close to 0.44. This exponent is consistent with several other measurements for the classical RBC state at smaller Ra and larger Pr and with the Grossmann-Lohse

Introduction
Turbulent thermal convection, where fluid motion is driven by a temperature gradient, is an important process in many fields. It plays a crucial role in climatology (see, e.g., [8]), oceanography (see, e.g., [9]), geophysics (see, e.g., [10]), astrophysics (see, e.g., [11]) and numerous industrial processes (see, e.g., [12,13]). While many of the natural or industrial phenomena involve large systems with complicated boundary conditions (BCs), experimental and numerical investigations of this problem usually use a moderately sized system with well controlled parameters and BCs typically consisting of a fluid layer confined by a horizontal warm plate at a temperature T b from below and a parallel cold plate at T t from above. This idealization is generally known as Rayleigh-Bénard convection or RBC. The properties of RBC are determined by the Rayleigh number 3 β Δ κν ≡ and the Prandtl number
( 2 ) ν κ ≡ Here L is the plate separation, g denotes the gravitational acceleration, T T T b t Δ = − , and β, ν and κ are, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of the convecting fluid at the mean temperature T T T ( )2 . An actual sample realizable in the laboratory has to be laterally confined. In the present study, as well as in many others, this confinement takes the form of a right circular cylinder of diameter D, and the geometry can be specified by the aspect ratio D L.
When Ra is not too large, say Ra 10 14 ≲ or so, this system consists of thermal and viscous boundary layers (BLs) adjacent to the top and bottom plates, and a bulk region well away from the BLs. Most of the temperature drop is then across the thermal BLs, and the bulk is at a nearly constant time-averaged temperature T c which is close to T m (see, however, [19][20][21]). This state is now known as the classical state since it has been studied for many decades (see, for instance, [22][23][24]). However, when Ra is large enough, a transition was expected to occur [25][26][27] to a state in which a large-scale circulation (LSC) or the turbulent fluctuations of the bulk drive the BLs turbulent as well. The new state that evolves thereafter became known as the 'ultimate' state [28] because it is expected to prevail asymptotically as Ra goes to infinity. For a sample with 0.50 Γ = it was found experimentally [29,30] that the transition to the ultimate state occurs over a range of Ra, from Ra Ra 2 10 1 * 13 = ≃ × to Ra 5 10 2 * 14 ≃ × 6 . As we shall see in the present paper, Ra 1 * is nearly the same for 1.00 Γ = ; but Ra 7 10 2 * 13 ≃ × , much smaller than it was for 0.50 Γ = . In this paper we report on determinations of the local velocity and of local velocity fluctuations in a sample with 1.00 Γ = and Pr 0.8 ≃ . The measurements were made using the High-Pressure Convection Facility IV (HPCF-IV), a sample cell with D L 1.12 = = m. This cell was located in the 'Uboot of Goetingen', a pressure vessel of 25 m 3 volume able to contain up to 2000 kg of the working fluid sulfur hexafluoride (SF 6 ) at pressures up to 19 bars. This facility allows measurements over the range Ra 10 2 1 0 11 14 ≲ ≲ × . While the results are primarily for the classical state, they also revealed the transition to and yielded limited results for the ultimate state.
We measured the mean-flow velocity U u t ( ) = 〈 〉 near (but not too near) the side wall where it has a maximum as a function of radial position and at an azimuthal position close to the preferred location of the plane of a LSC which prevailed in the sample (here ... 〈 〉 denotes a time average over a long data sequence and u(t) is the instantaneous velocity). We also determined the root-mean-square (rms) fluctuation velocity V u t U [ ( ( ) ) ] 2 1 2 = 〈 − 〉 . This was accomplished by using the elliptic approximation (EA) of He and Zhang [1,2] which permits the determination of the velocities from velocity space-time correlation functions. The EA is based on a second-order Taylor-series expansion of correlation functions which is valid near the origin of the space-time plane. However, He and Zhang postulated that the validity of the EA extends throughout the inertial range of length and time because of flow self-similarity. Analogous derivations and self-similarity assumptions can be applied to a passive scalar, for instance to the temperature in the bulk of turbulent RBC (see [35] and references therein), and thus we were able to make velocity determinations from measurements of the temperature space-time correlation functions. This procedure was used before for smaller Ra and various Pr on several occasions [29,[36][37][38][39].
In the present paper, after a description of the apparatus and some measurement procedures in section 2, we first discuss in section 3 measurements of the LSC circulation-plane orientation because this orientation influences U. Then, in section 4, we discuss relevant consequences of the EA. While many of these results have appeared already in the literature, they are scattered among several papers and not always easy to find. Our discussion also includes a derivation of the equivalence between the space and the time domain, which remains valid in the presence of fluctuations and replaces the Taylor frozen-flow hypothesis which breaks down in the highly fluctuating RBC system. Since the EA assumes that the field is spatially homogeneous over the locations of the measurement probes, we show in section 5 from measurements of the temperature probability-distribution function and the time auto-correlation function at many different vertical but constant radial locations that the homogeneity assumption in the direction parallel to the cylinder axis is well satisfied in our system (obviously, in the presence of a LSC, the system is not homogeneous in the radial direction).
In section 6 we provide several experimental tests of the applicability of the EA and derive a quantitative criterion for its validity. Then, in section 7, we determine the time dependent velocities U(t) and V(t) by using short time sequences, of 10 s duration, sliding along much longer sequences, with the EA. These results show that in our sample the LSC rarely, if ever, underwent a reversal as reported in a previous study [40].
Finally, in section 8, we present and discuss the main results of our paper, namely the results for U and V in the form of the corresponding Reynolds numbers Re UL . Here we first review the predictions based on the model of Grossmann and Lohse (GL) [3,4,6] in section 8.1. This model predicts Nu (Ra) and Re U (Ra), and has been remarkably successful in that several quantitative experimental measurements (see, for instance, [30,41,42], and several other papers) are in close agreement with it. It predicts [3] [27]. There is to our knowledge no prediction for V ζ in the ultimate state, and we do not know why the prediction U GL ζ should agree so well with V exp ζ ; however, we remind the reader that this was the case also in the classical state. Section 8 finishes with section 8.8 which presents a remark about the GL prediction [6] for Re V . Then the paper concludes with a brief Summary.

Apparatus and experimental procedures 2.1. Apparatus
We used the apparatus described previously [30,42,44], and here we mention only some key features. The RBC sample cell and associated shields were known as the HPCF-IV. HPCF-IV contained a sample chamber with a side wall consisting of a right-circular cylinder with both the diameter D and the height L equal to 1120 ± 2 mm, which yielded the aspect ratio D L 1.000 0.004 Γ ≡ = ± . The side wall was made of 9.5 mm thick Plexiglas which was sealed to aluminum top and bottom plates [44]. HPCF-IV had the same thermal shields as those used for a sample with 0.50 Γ = known as HPCF-II [45], except for a shorter side shield to fit its lesser length. At the mid height of the side wall, there was a hole which was connected to a remotely controlled valve, located outside but near the sample, via a tube of 13 mm inner diameter. After it was assembled, HPCF-IV was put inside the large pressure vessel known as the Uboot of Göttingen. The Uboot and HPCF-IV could be filled with up to two tons of the working fluid SF 6 to reach pressures up to 19 bars. Before each measurement sequence, the valve was opened and the sample cell and Uboot were filled with gas 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. Using separate instrumentation and temperature-controlled water circuits, we were able to take measurements with HPCF-II and HPCF-IV simultaneously.
The two experimental control parameters Ra (equation (4)) and Pr (equation (5)) can be written in the form where ρ is the density, η ρν = the shear viscosity, and C p λ ρ κ = the thermal conductivity (C p is the heat capacity per unit mass). For gases well away from their critical points η and λ are only weakly dependent on pressure [46]. Thus, the Prandtl number Pr is also nearly pressure independent. For an ideal gas one has

Pressure and temperature measurements
Using a Paroscientific Model 745 A pressure gage, pressure measurements with a resolution of about 100 μbars for pressures up to 19 bars were made at time intervals of 5.5 s in conjunction with each set of temperature measurements. The original calibration of the gage was checked against a DH-Budenberg Model 558 deadweight tester. We estimate the accuracy of the pressure measurements to be better than one mbar.
Twenty six thermistors were used to measure temperatures of the top plate, the bottom plate, and numerous thermal shields. Each thermistor was calibrated in a separate calibration facility with a precision of 1 mK against a Hart Scientific Model 5626 platinum resistance thermometer which in turn had a calibration on the ITS-90 temperature scale traceable to standards maintained by the National Institute of Standards and Technology. These 26 thermistors provided temperature signals that were used to control the RBC system. Details of the thermometer locations were described in [44].
Temperatures in the interior of the sample were measured with two types of thermistors. One type (Honeywell 112-104KAJ-B010), to be referred to as thermometer type T 1 , had glass-encapsulated thermistor beads of 1.14 mm outer diameter. Three sets of eight of them were placed in the sample ∼1 mm from the side wall. Each set was located along a horizontal circle, spaced uniformly in the azimuthal direction at intervals of 4 π . The three sets were located at heights L 4, L 2, and L 3 4above the top surface of the bottom plate. These thermistors were read by a digital multimeter together with the 26 calibrated thermistors of the previous paragraph. The data were used to study the strength and orientation of the LSC in the sample, as had been described for instance in [47] and [48].
Another type of thermistor (Honeywell 111-104HAK-H01), to be referred to as T 2 , also was glassencapsulated, was roughly spherical, and had a bead of 0.36 mm diameter and two platinum-iridium leads of 20 μm thickness. T 2 thermistors were used to measure temperature cross-correlation functions C z ( , ) = − is the vertical separation between the two thermistors at x j and x i . T 2 thermistors had a response time that was an order of magnitude shorter than that of T 1 . They could respond without any noticeable attenuation to fluctuations with frequencies up to 0.5 Hz (see [49]). In a separate investigation using HPCF-II at Ra 10 15 = we found, by comparison with even smaller thermistors, that the spectra measured with T 2 type thermometers contained 95% of the total energy of the fluctuations. Figure 2 shows the schematic diagram of the thermistor locations. The thermistors were arranged in a single column with varying vertical separations spanning the range z 0 20 ⩽ ≲ cm roughly centered around the mid height x L 2 = of the sample. They were all located 1.5 cm away from the inner side-wall surface, corresponding to r R 1 0.027 ξ ≡ − = where R D 2 = and r is the radial distance from the vertical centerline of the sample. This column of thermometers was separated azimuthally by π from the hole leading to the valve (see section 2.1). Table 1. The vertical locations x (in cm, measured from the bottom plate) and x L of the type T 2 thermistors used to measure temperature correlation functions. All were located in the fluid at a radial distance R r 15 − = mm from the side wall (R = 560 mm is the sample radius), corresponding to r R 1 0.027 ξ ≡ − = .
No. The T 1 thermistors had sufficiently strong leads to permit inserting them and their leads directly through small holes in the side wall. To guide the T 2 thermistors into the sample, their leads were glued to pairs of 80 m μ thick copper extension wires using electrically conducting silver epoxy. The copper wires were then passed through ceramic rods, each rod containing two holes along its axis. The rods had an outer diameter of 0.8 mm, and the diameters of the holes were 0.13 mm. After insertion through 0.9 mm diameter holes in the side wall, both ends of the rod were sealed by silicone rubber adhesive (E41). As shown in figure 2, the thermistor bead was left deliberately about 1 mm or more away from the glue cap in order to reduce perturbations of the local flow near the bead as much as possible.
We connected each thermistor to an alternating-voltage transformer bridge as one resistor arm and used a fixed reference resistor for a second arm. The reference resistor was chosen so as to match the average thermometer resistance. Two further bridge arms were provided by two equal arms of the secondary transformer coil. A lock-in amplifier (Stanford Research SR850) was used at a working frequency f 0 near one kHz to drive each bridge with a voltage in the range V 0.5 0.2 0 = ± V and to amplify the bridge unbalance (which was proportional to the temperature displacement from a set point). This technique was the same as reported previously for RBC using water as the fluid [50]. Four bridges and lock-in amplifiers were used simultaneously. Each operated at a slightly different frequency; the frequencies were shifted relative to each other by increments of 200 Hz to avoid cross-talk. All seventeen thermometers were measured, albeit in multiple runs at the same Ra with only four in a given run. Thus, not all cross-correlation functions could be calculated because each function required data from the same run for both thermometers. The sampling rate of the temperature measurements was set at 40 Hz. From these measurements, we calculated temperature crosscorrelation functions C z ( , ) i j , τ using various thermometer combinations (i,j) at various separations

Orientation of the LSC plane
The velocity U and the corresponding Reynolds number Re U (see section 8.3 below) may depend on the azimuthal orientation m θ of the LSC circulation plane relative to the measurement location Re θ . Thus, we shall explore first, in this section, the dependence on Ra of the time averaged orientation We also examine the width σ Δ (see equation (8) below) of the probability distribution p ( ) m Δ θ .

LSC orientation measurements
The LSC is a highly fluctuating convection roll which contains warm and cold vertical currents, including plumes, separated by π. It is primarily responsible for the creation of the velocity U. The sample was tilted by a small angle 0.013 0 ϕ = rad in such a way that the down-flow, occurring at an azimuthal angle θ, would tend to pass by the azimuthal T 2 thermistor locations Re θ used for the Re measurements (see section 2.2). Recent unpublished measurements with a leveled sample yielded nearly the same Nu(Ra) and the same Ra 1 * and Ra 2 * . As is well known, m θ undergoes large fluctuations [47]. Since the LSC dynamics can affect local measurements of U, especially near the side wall [51], we monitored m θ simultaneously with the Re measurements to be discussed below in section 8.3.
The LSC azimuthal orientation was determined by fitting the function to a row of eight T 1 thermistors located in the horizontal mid plane at a height L 2. The eight individual thermistors are identified by the index i, which increases in the counter-clockwise direction. Using the leastsquares fit for each time stamp, we were able to obtain the averaged temperature T w m , near the wall, the LSC temperature amplitude m δ , and the azimuthal orientation m θ of the LSC. This method had been reported originally in [47] where water was used as the fluid (Pr 4.4 ≃ ) for 1 Γ ≃ , and was described in more detail elsewhere (see, for instance, [48,[52][53][54]).

Orientation and fluctuations of the large-scale-flow circulation-plane
In figure 3 we show the probability-density functions (pdfs) of the orientation differences m Δ θ for several Ra. Here

5.89
Re θ = rad (relative to an arbitrary origin used for all azimuthal measurements) is the azimuthal position of the thermometers used for the Re measurements. One sees that the most probable values of m Δ θ were close to zero, which indicates that the LSC orientation was indeed set close to the preference direction Re θ by Δ and σ Δ adjustable parameters that were used to fit each data set. As Ra increased, the fluctuation width σ Δ increased, resulting in a wider range of deviations from Re θ . Measurements of σ Δ for a tilted sample with 1.00 Γ = and as a function of the inclination angle were reported before in [55], albeit at smaller Ra and for Pr = 4.38.
In figures 4(a) and (b) we show results for R 0 Δ and Rσ Δ respectively as a function of Ra. Since R is the sample radius, R 0 Δ describes the average distance along the sample perimeter of the LSC orientation from the thermometer location R Re θ , and Rσ Δ gives the width along the sample perimeter of the fluctuation distribution function of the LSC orientation. From figure 4 (a) it is evident that R 0 Δ changed with Ra. As Ra increased, the absolute value of the most probable displacement R 0 Δ | |gradually increased from 3 cm to 15 cm. However, for Ra Ra 2 10 |started to decrease with further increase of Ra. While one might expect that these changes of R 0 Δ | |would lead to a change of the measured vertical velocity U, this is not necessarily the case. It was found by Sun et al [51] that the azimuthal variation of U is not close to sinusoidal, but rather more nearly like a square wave with period 2π. The changes from U − to U are, of course, somewhat rounded rather than true steps. We would expect them to influence the measurements when 2 m R e θ θ π ≃ ± , i.e. when our sensors are located at an angle of 2 π relative to the LSC orientation. In our experiment this should occur near R 88 0 Δ ≃ cm, which is much further from the location of the thermistors used to determine U than any of the measurements shown in figure 4(a).
The value of Ra 1 * found here is in remarkably good agreement with the value found for the onset of the transition to the ultimate state from measurements of the Nusselt number for 0.50 Γ = [30], and close to a less accurate estimate of Ra 1 * based on Nu measured for the present 1.00 Γ = sample [42]. It is not known to us why the Ra dependence of the LSC orientation should change when the ultimate-state transition-range is entered. Figure 4 (b) shows the width Rσ Δ of the LSC-orientation fluctuations at the circumference as a function of Ra. One can see that Rσ Δ increased from about 15 cm to about 30 cm as Ra increased, without any clear signature at Ra 1 * .
Figures 4 (a) and (b) demonstrate that the statistical properties of the LSC changed over the range of Ra from 10 11 to 10 14 of the present measurements. We do not believe that these changes have a significant influence on the measured U and V when the azimuthal measurement location is near the LSC circulation-plane orientation as was the case in our experiment.

Elliptic approximation 4.1. The approximation and elliptic constant-correlation contours
The EA of He and Zhang [1,2] was first formulated for the velocity space-time correlation functions C z ( , ) v τ of turbulent shear flows. It yields an elliptic form, based on U and V, for the iso-correlation contours near the origin of the z-τ plane. He and Zhang further postulated that the same elliptic form can describe the constantcorrelation contours over a wider range of z and τ because of flow self-similarity in the inertial range of z and τ, and found support for this postulate from an analysis of turbulent channel-flow data obtained by numerical simulation [2].
In the bulk of turbulent RBC temperature has been shown to behave as a passive scalar [56,57] and to exhibit self-similarity scaling in the inertial range [35]; thus similar derivations based on an EA can also be obtained from temperature correlation functions C z ( , ) τ , and one can expect these results to also be valid over a wide range of space and time.
The EA was discussed in detail elsewhere (e.g. in the supplement to [58]) and here we only summarize its most important consequences. The central result of the EA is that, for homogeneous and isotropic turbulence, a systematic second-order Taylor-series expansion of the space-time correlation function x x z τ δ τδ σ σ = + + + can be written as Here ... 〈 〉 denotes the time average, T T t T ( ) δ = −〈 〉, and x σ and x z σ + are the rms values of T δ at the vertical positions x and x z + (which in the homogeneous system are equal to each other). For a statistically stationary process deviations from the EA are expected to be of fourth and higher order since odd derivatives vanish because of homogeneity and stationarity.
An important consequence of equation (10) is that C z ( , ) τ has a constant value whenever z E is constant. From equation (11) one sees that a constant z E corresponds to an elliptic contour in the τ-z parameter space. This is illustrated schematically in figure 5.
We note that, in the absence of fluctuations (i.e. V = 0), equation (11) reduces to which corresponds to the Taylor frozen-flow approximation. This approximation is not generally valid for turbulent RBC because V for many parameter ranges is of the same order of or larger than U (see, for instance, [59][60][61][62]). Thus, for turbulent RBC and any system with significant fluctuations the velocities determined from the EA are more reliable than those obtained using the Taylor approximation.

The equivalence between the space and the time domain
The EA can be used to derive an equivalence between the space and the time domain that is, as all aspects of the EA, a controlled approximation valid in the presence of fluctuations. It is expected to be valid in cases where the Taylor approximation equation (14) fails. This result is important because many experimental measurements are made in the time domain, while often the analogous result in the space domain is needed. The stars in figure 5 show the special cases z ( 0, ) Since the ellipses are constant-correlation contours, we have The determination of 0 λ is similar to the determination of the Taylor microscale from velocity space autocorrelation functions, and corresponds approximately to the smallest lengths of the inertial range of the energy spectrum of turbulent fluctuations. Using equation (15) with equations (18) and (19) gives 0 eff * 0 λ τ = and thus the equivalence in the wavenumber and frequency space. Thus, the normalized spatial and temporal spectra, when scaled by 0 λ and 0 τ respectively as indicated by equations (22) and (23), are equal to each other.

Velocity measurements based on the EA
We can use the EA to determine the velocities U and V from multi-point temperature measurements. This method had been tested previously and applied to RBC temperature data with Pr 5 ≃ and Ra 10 10 ≃ for samples with 1 Γ = [36,37], and it had been used for Pr 0.8 ≃ and 0.50 Γ = for Ra up to 10 15 [29]. The EA had also been applied to particle-image velocimetry data [38], shadowgraph images [39] in turbulent RBC, and velocity data in a wall-bounded shear flow [63] (for a recent review, see [64]). We can examine the consistency of the results for U and V obtained over various displacements z to further test the EA. τ we found the time delay d τ at which  (10)) at constant z will have a maximum at p τ , with p τ given by the value of τ where z ( ) E z τ ∂ ∂ vanishes. The red stars represent two special points on a contour of constant z E and will be used below in section 4.2 to derive an equivalence between spacial and temporal spectra. 11 11,13 τ = as shown in the figure by the horizontal line with double arrows. Since for that time delay the two correlation functions have the same value, they correspond to the same re-scaled space variable z E . Substituting the arguments of each separately into equation (11) and equating the two results, we find z, Using equation (17) this result yields As shown in figure 6, we also found a peak position p τ at which C z ( , ) 11,13 τ reached its maximum. As illustrated in figure 5 at that extremum. Evaluating equation (29) with equation (11), we have Figure 7 shows results for d τ and p τ derived from C z ( , ) i j , τ for different i j , as a function of the corresponding z, obtained from the analysis described above. One sees that, over a range of ±10 cm for z, d τ and p τ are described well by straight-line fits as expected on the basis of equations (25) and (30). For larger z | |the data show deviations from those lines at negative z. We see that the EA is valid over a wide range of z and τ. The range ±10 cm corresponds to L 0.1 ± where L is the sample height (which for 1.00 Γ = is equal to its diameter). It is reasonable to conjecture that the upper limit of the inertial range of z, and thus of the range of applicability of the EA, is of the order of L 10, consistent with this result.
For the particular example shown in figure 7 the fits of equations (25) and (30) to the data yielded  U 11.7 cm s , Although one might attempt to estimate probable errors of U and V from those of 0 α and p α , these errors turn out to be excessively large because the errors of 0 α and p α are correlated, albeit in a way that seems difficult to determine. This is so because the errors of d τ and p τ are not independent of each other. We are thus led to estimate the probable errors of the Reynolds numbers derived from the velocities from their scatter about a fit of a power law to Re U (Ra) and Re V (Ra) (see sections 8.3, 8.4, and figure 17).

Sample homogeneity
In order for the EA to be applicable, the system under investigation must be homogeneous at least in the direction of the desired velocity measurements. In this section we show that our sample satisfies this requirement.
From the temperature time-sequences T i (t) at each location i (see We also determined many of the temperature space-time cross-correlation functions C z ( , ) i j , τ (see equation (9)) from the measurements at the discrete locations x i and x j with z x x j i = − and with i and j two of the thermometer numbers listed in table 1. Figure 8(a) shows pdfs of T i i δ σ for different vertical positions x i . All pdfs were normalized. Over an amplitude range of more than four decades, and over a vertical spacial range of 24.5 cm, from x L 0. 47 = to x L 0.69 = , there are only small deviations in the tails of the pdf data from a unique curve. Since the sample was tilted slightly with the azimuthal orientation of the tilt-angle chosen so that the down-flow of the LSC on average was close to the azimuthal thermistor location, the pdf shapes were all skewed towards negative values.  (see table 1). As mentioned above, spatial homogeneity is a necessary condition for the validity of the EA.

Tests of the EA
In this section we provide several experimental tests of the applicability of the EA to our measurements in the range of Ra up to 10 14 . We also illustrate in more detail the methods by which velocities (and thus Reynolds numbers) are determined from the correlation functions. 6.1. Qualitative tests of the EA Figure 9(a) shows the three-dimensional surface of C z ( , ) i j , τ as a function of z and τ. Figure 9(b) shows several constant-correlation contours in the z-τ plane. As expected, C z ( , ) i j , τ reached its maximum at z 0 τ = = , and from there decayed monotonically in all directions. This ensured that the constant-correlation contours were closed curves. From the inner to the outer curve, the contour amplitude varied from 0.99 to 0.90 in steps of 0.01. Over the range of z and τ studied, the contours all had similar elliptic shapes within experimental error, as expected from equation (11). This result indicates that the EA is valid for temperature correlations measured near the side wall where the flow structure is homogeneous in the axial direction but anisotropic.

Further tests of the EA
To further test the EA, we show many C z ( , ) i j , τ , measured with various i j , and the corresponding z, as a function of the re-scaled length z E on logarithmic scales in figure 10(a). The values of z E were calculated using equation (11) with the above values of U and V. As seen in the figure, all correlation functions collapse onto a single master curve C z ( , 0) as a function of z E , except for the two with the largest z | |. The ones showing deviations from the universal curve are i j ( , ) (17, 11) = and (17, 7), corresponding to z 14.00 = − and −19.50 cm respectively. The reason for these deviations at large z is not obvious to us. We do not think that they are due to an actual variation of the physical velocities U and V with vertical position, as these have been found to have unique values when smaller separations were used over much of the 26.5 cm range covered by the thermometers listed in table 1. One possible reason is that the flow self-similarity assumption, and thus the validity of the EA, breaks down when z exceeds the largest length scale of the inertial range.  In view of the above analysis, one needs measurements for at least two different locations x i and x j , separated by a distance z. The measurements yield C C (0, ) (0, ) τ which can be compared as shown in figure 6 and which will yield values of p α and 0 α . These results in turn will yield U and V. This method was used to measure Re for the 0.50 Γ = sample HPCF-II [29], using a spatial separation z = 3.0 cm. There the authors examined the validity of those results by comparing the two C z ( , 0) . These two curves were found to coincide within experimental scatter for z 10 cm E ≲ .

A quantitative criterion for the validity of the EA
Deviations of experimental data from the EA can arise for several reasons. On the one hand, the spatial separation may be too large for the EA to remain valid. As discussed above, we expect this to happen only when z exceeds perhaps 10 cm or so. On the other, the physical velocities U and V may vary with position over the spatial range included by z when z is large. We gave arguments above in section 5 why we think that this is not a problem for z 10 ≲ cm or so. Lastly, the time sequences used may represent an inadequate sample of the statistical properties of the process under investigation if it is too short. While this is not a problem for our long time sequences of many hours' duration, it becomes an issue below where we investigate velocity fluctuations by analyzing short-time moving-average sections of the long time sequences. For all of the above reasons, we consider now a quantitative criterion that can be used to determine whether the EA is valid for a given data set and thus whether the derived velocities are trustworthy.
The difference between a cross-correlation function C z ( , ) i j , τ and the corresponding auto-correlation functions C (0, ) τ can be described quantitatively by , m a x 1 , , (see equation (17) and (11)). For C i j , , where z 0 | | > , z E is computed from equation (11) using U and V. The number of data points N is the total τ as a function of the re-scaled length z E (see equation (11)) for ten different thermistor pairs (see table 1 number of experimentally measured points of C z in the range of z E up to an arbitrary cutoff z E,max . In order to explore how C z ( ) ,m a x δ depends on z E,max for a case where z is sufficiently large to yield reasonable accuracy for U and V but small enough so that there is no question about the validity of the EA, we first chose the case i j , 9, 11 = (see table 1) where z = 3.50 cm. Using only those two positions, we found U = 11.0 and V = 10.9 cm, consistent within error estimates with the results equations (32) and (33) for Ra 1.25 10 14 = × . With those velocity values, we obtained the results in table 2. Considering that the two correlation functions have values not far below unity over the range of comparison, one sees that they agree with each other to better than a percent or two of their average values over a remarkably large range of z E .
In figure 11 we show four comparisons of C z ( , 0) sample [29] and for the present measurements for 1.00 Γ = the value of C i j , δ based on typical long time sequences of ten hours or so was generally less than 0.003.

Results for the velocity fluctuations
In an earlier paper Niemela et al [40] obtained cross-correlation functions for Ra up to 10 13 and Pr near one from a sample of 50 cm diameter and 1.00 Γ = . They used two sensors located in the horizontal mid plane 4.4 cm from the side wall. The sensors were separated vertically by 1.27 cm. The authors measured time sequences at a frequency of 50 Hz with an approximate length t 30 avg = s, selected from and sliding along much longer sequences. From the peak positions p τ they computed the velocity z p τ as a function of time. They found frequent reversals of the measured velocity, from positive to negative values (see, however, [65], section 6).
We now know from the EA and equation (31) that the velocity z p τ actually is the combination where U(t) and V(t) are the time averages of the instantaneous velocities u(t) and v(t) over the time interval t avg . We note that this quantity approaches U(t) only in the Taylor frozen-flow approximation V U ≪ (which becomes valid for RBC only in the case of extremely short time sequences which do not permit fluctuations to evolve). In the present section we present a similar analysis using t 10 avg = s, but use the EA to first select only those sequences which reveal sufficiently good statistics to conform to the EA prediction according to the criterion of the previous section. We then interpret the correlation functions in terms of the EA to obtain U(t) and V(t) separately using equations (32) and (33). Our data did not reproduce the frequent reversals found in [40].
From correlation functions C z ( , )  Figure 12 (a) shows the pdf of C i j , δ . Figure 12 (b) gives the integral of the data in (a). One sees that the cutoff at C 0.04 i j , δ = discards about a third of the data, leaving enough points of U(t) and V(t) to reconstruct their time sequences fairly well. A longer averaging time t avg with the same cutoff C 0.04 i j , δ = would have yielded a larger number of useable points for U(t) and V(t), but would have smoothed away more of the rapid time variation.
If no data from the original time sequence had to be discarded, we would expect the rms value to be equal to V where V is the rms velocity obtained from the entire sequence. Since, as illustrated in figure 12, some data had to be discarded in order to assure the validity of the EA, we expect the ratio V U t ( ) σ to be less than one. In figure 13 we show this ratio as a function of t avg for different Ra. As expected, V U t ( ) σ increases as t avg decreases, and approaches 1 as t avg vanishes. Because the velocity spectrum covers a wider frequency range at higher Ra, the cutoff frequency corresponding to t avg discards more rapid fluctuations from the spectrum at high Ra than at low Ra. As a result, one sees at a given t avg that V U t ( ) σ decreases as Ra increases. Figures 14 (a)-(d) show some of the results for U(t) and Here U and V are the long-time averages computed from the entire ten-hour time sequences. For all Ra, U t U ( ) δ showed similar behavior. The data thus imply that the fluctuation amplitudes grew with Ra at about the same rate as U itself. This finding will be confirmed more quantitatively which is shown in figure 15 (a). One again sees the positive mean value of U t U ( ) δ . It indicates that positive fluctuations are slightly larger and/or more frequent than negative ones. There is some broadening with increasing Ra, but not much considering that the data cover nearly three decades of Ra. The data do not show any evidence of bimodality, as would be expected if the LSC at irregular time intervals reversed its direction of circulation and remained in that state for a significant length of time. While such bi-modality was reported in [40], we note that an alternative explanation of that observation had already been offered in [65] (see especially section 6 of that paper). The rare negative spikes reaching values as low as U t U ( ) 2 δ ≃ − are confirmed here as well.   for the classical state when Ra and Pr are specified. An important qualitative prediction of the model [3], soon confirmed by experiment [41,66] for Nu (Ra), is that Nu Ra Pr ( , )and Re Ra Pr ( , ) U do not strictly follow a power law. Instead they are determined by crossover functions that interpolate between different power laws valid asymptotically in various limits (generally not accessible to experiment) of Ra and/or Pr. Consequently a fit of Re data over a limited range of Ra and/or Pr to the function  [43]) which implicitly give Nu GL and Re U GL , but contain five parameters that need to be determined by fitting to experimental data. Four of these parameters are determined entirely from fits to data for Nu Ra Pr ( , )in the classical Ra range. The fifth parameter, designated as 'a' in [5], affects the pre-factor Re U,0 GL , but when an appropriate parameter transformation is used, it does not influence Nu GL and U GL ζ [5]. Recently all five parameters were re-determined [43]   U not only asymptotically but also over the entire range Ra Ra 2 * > [27]. To our knowledge there is no prediction for V ζ and V α in the ultimate state.
8.2. The determination of Re U and Re V At each value of Ra we used three simultaneous temperature time-sequences, taken at a frequency of 40 Hz over approximately ten hours, for the determination of Re U and Re V . The data were taken at locations i 9, 11, = and 12 (see table 1) and the EA velocities and the corresponding Reynolds numbers were calculated using equations (25), (30), (32), and (33). The three thermistor locations had been chosen with non-equal separations, which gave four different values of z: z = 0, 1.5, 3.5 and 5 cm. As is apparent from section 4.3 and figure 7, a larger number of displacements z will lead to better values of 0 α and p α and thus to smaller errors of U and V.
The measurements were made at the average vertical position x L 0.55 ≃  (see table 1), quite close to the horizontal mid plane. The radial position of the measurements was at r R 1 0.027 ξ = − = (see section 2.2). For Pr 1 ≃ the thicknesses of the thermal and viscous BLs are expected to be about equal. Near Ra 4 10 11 ≃ × one can estimate their thickness to be about 1.4 mm or R 0.0025 ; they will be thinner at larger Ra. Thus the measurements are well within the bulk of the sample, but close enough to the wall to correspond to the largest Re U along the sample diameter passing through the measurement position.
The measurements of Re were made at various pressures in order to cover the wide range (see equation (6)) Ra 4 10 1.5 10 11 14 × ≲ ≲ × , and Pr depended slightly on pressure. The variation of Pr with Ra was shown in figure 1, and covered the narrow range from 0.78 to 0.86. Over this Pr range the term Pr U GL α in equation (38) varied by 6%, from 1.174 to 1.106. In the classical state, which extends up to Ra 2 10 13 ≃ × , this term varied by only 1.2%, from 1.174 to 1.160.

Results for Re U
The Re U results are shown on double logarithmic scales in figure 16. There the solid line is the prediction of the GL model [43]. It falls well below the measurements, but for Ra 2 10 12 ≳ × it has (within the resolution of the measurements) the same slope as the experimental data, corresponding to the same effective exponent U ζ . The GL prediction multiplied by 1.38 is shown as a dashed line; it is an excellent fit to the data for Ra 2 10 12 ≳ × . We do not know the reason for the departures from the dashed line in the figure for Ra 2 10 12 ≲ × . One might argue that they are indicative of a transition between two different states of the system, and conjecture that this transition might be related to those observed by Chavanne et al [28] and in subsequent related cryogenic experiments [31,32] (for a recent re-examination of those data, see [30]). To explore the implications of this = × . That value is larger than that found in the above mentioned low-temperature experiments (see Figure12 of [32]) which, over the range Pr 1.5 20% = ± and for 1.14 Γ = , gave an estimated Ra 1.1 10 t 11 = × , a factor of 27 smaller than suggested by our Re U results. Further, the power-law fit indicated by the dash-dotted blue line in the figure yielded an exponent of 0.56, significantly larger than the values expected theoretically for any state of this system [16] and those found by Chavanne et al [31] or Roche et al [32]. Thus we find it unlikely that there is any relationship between the departure of our Re U data from the dashed line in figure 16 on the one hand and the transitions observed in the earlier helium experiments [28,31,32] on the other. In any case, the exponents implied by the power-law fits displayed in the figure are inconsistent with a transition between the classical and the ultimate state since the larger exponent is found at the smaller Ra values, and because the implied low-Ra exponent value of 0.56 is much too large. We also note that measurements of the Nusselt number for the same sample did not reveal any transition near Ra 3 10 12 = × (see figure 3(b) of [42]). We find it more likely that the departures from the dashed line in the figure for Ra 2 10 12 ≲ × are caused by the change in orientation of the LSC discussed in section 3, although we do not see any obvious correlation with the Ra dependence of the mean orientation or the width of the probability distribution of the LSC orientation (see figure 4). The deviations could also be caused by a change of the LSC shape, for instance from a more nearly circular to a more nearly elliptic form (see, for instance, [51]). The GL model does not take such shape changes into consideration. In any case, we limit further quantitative analysis of the data to the range Ra 2 10 12 ⩾ × , and also exclude all data for Ra values larger than our estimate Ra 2 10 1 * 13 = × . The uncertainties of parameter values derived from fits to experimental data depend on the uncertainties of those data. We determined the latter by first fitting the power law

Re
Re Rã , to the Re U data over the range 2 10 12 × to 2 10 13 × (shown in figure 16 as solid red circles). Deviations from that fit are shown in figure 17(a). Over the Ra-range of the fit the rms deviation from the fit was 3.8%. Using this result in a subsequent weighted fit yielded˜0.437 0.019 U exp ζ = ± . In order to remove the small influence of the variation of Pr over the experimental Ra range, we based the analysis on equation (38) and fitted a power law to the experimental data for Re Pr We are not aware of any Re measurements in the Ra and Pr range of our data and for 1.0 Γ = with which we might compare our results. Measurements for 0.5 Γ = at Ra values similar to ours but Pr values which increased more strongly with increasing Ra yielded exponent values of 0.48 ± 0.02 [32] and 0.49 [31].
As discussed above, the magnitude of Re U (as determined by the pre-factor Re U,0 ), but according to the GL model not the Ra dependence (as reflected in U ζ ), depends on where in the sample the measurement is made. The parameter a of the GL model (which determines Re U,0 GL ) was determined from measurements by Qiu and Tong [61] of the horizontal velocity component along the vertical centerline. These authors linearly extrapolated their results in the sample interior to a position near the top or bottom plate. Stevens et al [43] used this extrapolation to determine a. However, direct measurements [61] of the maximum velocity near the plates show that the extrapolated value is lower than this maximum by 22%. Our measurements were of the vertical velocity component near the side wall and correspond closely to the maximum velocity along a diameter nearly in the horizontal mid plane and the LSC circulation plane. It seems reasonable to us that the ratio of 1.38 between the experimental and the GL-model value may be attributable to these differences between measurement locations. We note that the data for Re U do not reveal any obvious anomaly/singularity at the location of Ra 1 * as determined in figure 4(a) (the vertical dotted line in figures 16(a) and 17(a)). This will be discussed further below in section 8.7.

Results for Re V
While Re U depends on location and can be influenced by the preferred orientation of the LSC, the fluctuation Reynolds number Re V apparently is largely immune to the vagaries of the LSC and to the measurement location within the bulk of the sample [61].
In figure 18 we show the results for Re V . The solid line in that figure is a power law with 0.331 V GL ζ = , corresponding to the GL prediction [6]. Here the pre-factor Re 16.8 V ,0 = was adjusted to provide a fit of the power law to the measured value at Ra 10 13 = . It is apparent that the theoretical prediction does not fit the data. For further analysis only the data in the classical state with Ra Ra 2 10 1 * 13 < = × were used in order to avoid any influence from the ultimate-state transition-region. A fit of equation (39) to the data, with both pre-factor and exponent adjusted, yielded˜0.430 V exp ζ = and Rẽ 0.844 We note that these results are very close to the GL prediction for Re U , but are not aware of why they should agree with that theoretical result.
In the above analysis we neglected any possible influence of changes of Pr with Ra. However, we note that Pr changed only from 0.785 to 0.800 for the data used in the fits. The effect of this small variation will be examined below in section 8.6.
To for Re U in the Ra range of those measurements. In this case we believe that the GL prediction [6] of V GL ζ is not directly relevant because in the experiment there are several contributions to fluctuations. One contribution comes from the intrinsic fluctuations considered by GL [6]. The other comes from the irregular lateral movement of the LSC structure which leads to the vanishing of U in the time average but must be expected to contribute to the fluctuations measured at a fixed location. 8.5. Dependence of Re V on the Prandtl number Although our measurements were only for Pr 0.80 ≃ , they can be compared with others in the literature for 1.00 Γ = to reveal the Pr dependence of Re V . For this purpose we chose the reference value Ra 10 10 = which is within or close to the Ra range of several previous investigations [38,[59][60][61][62]. The interpolated or slightly extrapolated values at Ra 10 10 = are shown in figure 19. The dash-dotted line (purple), is the prediction of GL for Re V [6], with 0.50 V GL α = − and the pre-factor adjusted to provide a best fit to the large-Pr data of [62] (the three purple diamonds). This power law does not fit any of the data very well, and our result for Pr = 0.80 (solid blue circle) is larger by a factor of about eight. Thus, both the Ra dependence (see figure 18) and the Pr dependence predicted for Re V [6] disagree with the experimental results.
The solid line in figure 19 is the GL prediction for Re U [43], with the pre-factor adjusted so as to provide a best fit to the data of [62]. Although we do not know why it should, it fits the data for Re V quite well for Pr 7 ≳ , but does not fit the measurements for Pr 7 ≲ . Our experimental value at Pr = 0.80 is larger than this model prediction by a factor of about four.
Of course we do not know how to properly formulate the strong Pr dependence found for Pr 7 ≲ ; but if we choose to make a power-law fit to the data in this range, then we find Re Pr V 1.20 ∼ − which is shown as a dotted line (red). We note that the data by others for Pr 3 7 ≲ ≲ [38,[59][60][61] also are consistent with such a large exponent and scatter more or less randomly about the dotted line. It should be pointed out, however, that an alternative fit to the data for Pr 3 ≳ was proposed by Shang and Xia [60]. which is shown in figure 19 as a dashed line. Although this function provides a reasonable representation of most of the data, we note that the points near Pr = 7 fall significantly below the line, while our point at Pr = 0.8 is well above this fit. In any case, also this interpretation, with 0.86 V exp α = − , gives an exponent that disagrees with the GL prediction [6] 0.50 It should be noted that the data for   = as a function of Pr. Open diamond (purple): from [59]. Solid diamonds (purple): from [62]. Square (red): from [61]. Stars (red): from [60]. Down-pointing triangle (green): from [38]. where the large uncertainty of the pre-factor is due to the correlation with the exponent.
In the range Ra Ra 2 * ⩾ we have no experimental information about the Pr dependence of Re V . However, all measurements in that range were made at nearly the same pressure P 17.9 ≃ bars and thus Pr was nearly constant (see figure 1) × ≲ ≲ × , the measurements show that Re U = Re V within the experimental scatter as can be seen in figure 21. As discussed in section 8.3, the reasons for the deviations at small Ra of Re U from the power law that fits both Re U and Re V at larger Ra are unknown to us, although we surmise that they might be found in the vagaries of the LSC orientation or shape rather than in something more fundamental. Thus we proceed for now on the basis of the experimental finding that Re Re 1 U V = within the experimental resolution and examine what this implies phenomenologically within the framework of the GL model for fluctuations [6].
Equation (56) of [6] gives for the volume averaged kinetic dissipation rate U,bulk ϵ in the bulk of the fluid. Here l is a length which, within the GL model, was taken to be a similarity variable η defined by equation (16) of [6] and which initially we shall leave undefined. Solving equation (40) for l and applying the experimental result U V 1 ≃ , we find  [5,6] and that a fit of the model to experimental results for Re U gave a 1 ≃ [43]. Thus, we see that one way to achieve agreement with the experimental result Re Re 1 U V ≃ is to assume that the relevant length in equation (40) is λ ν rather than η. This would suggest that the ratio of the Reynolds numbers is determined by the distance over which the velocities U and V decay from their bulk values to zero as the side wall is approached. However, we note that using l λ = θ where λ θ is the thermal BL thickness given by equation (22) of [6] would also be consistent with equation (40) and the experiment since also in this case we have Re U 1 2 λ ∼ θ − , albeit with a Pr-dependent pre-factor. For Pr (1)  = the pre-factor is close to unity, and thus on purely phenomenological grounds we can not distinguish between the two options for l.
While the above phenomenological argument is suggestive, a more detailed and fundamental reexamination of Re Ra Pr ( , ) V clearly is warranted.

Summary
In this paper we reported on Reynolds-number measurements in turbulent thermal convection at large Rayleigh numbers. The results were obtained from high-precision temperature correlation-function measurements using the EA of He and Zhang and the experimentally well supported assumption that temperature is a passive scalar in the bulk of the sample away from the BLs. The data cover the Ra range Ra 10 2 1 0 11 14 ≲ ≲ × and are for Pr 0.8 ≃ and a cylindrical sample of aspect ratio 1.00 Γ = . The measurements were made in or very close to the LSC circulation plane using a set of thermistors positioned at a constant radial position 1.5 cm away from the side wall at several vertical positions close to the horizontal mid plane of the sample. Extensive experimental tests of the validity of the EA were carried out. Both the mean-flow velocity U and the rms fluctuation-velocity V and the corresponding Reynolds numbers Re U and Re V were determined. In comparison to methods based on Taylor's frozen-flow hypothesis, we find that the EA gives more reliable Reynolds numbers and thus is a new substantial method for turbulent RBC and any system with significant fluctuations.
Initially short temperature sequences, of t 10 avg = s duration, which moved along much longer sequences were used to determine the time dependent 'instantaneous' mean-flow velocity U(t) and the corresponding rms fluctuation velocity V(t). The probability distribution-function of U(t) had a single peak, indicating that the LSC rarely, if ever, underwent a reversal of its flow direction. The rms value U t ( ) σ of the deviation of U(t) from its mean value was somewhat smaller than the fluctuation velocity V determined from an analysis of entire long time sequences, but analyses using many values of t avg showed that the ratio V U t ( ) σ tended toward one as t avg tended toward zero.
Time-averaged results U and V based on the entire data sequences covering many hours were used to determine the Ra dependences of the corresponding Reynolds numbers Re U and Re V . For this purpose only data for Ra Ra 2 10 ζ should be so remarkably close to the GL prediction for the Re U exponent U GL ζ , one may hope that this finding may point the way toward an improved theoretical analysis.
In view of the excellent representation of the Re V data by a power law, it was possible to extrapolate this function with reasonable confidence to the smaller Ra 10 10 = where comparison with measurements by others at larger Pr values was possible. The comparison suggested that at constant Ra Re Pr V 1.2 ∝ − for Pr 8 ⩽ or so, while for larger Pr the data are consistent with Ra Pr 0. 67 ∝ − . The exponents in both of these regimes disagree with the GL predictions for fluctuations [6] which yields an exponent value of −0.50 for Re V . For Pr 8 ≳ the exponent value -0.67 for Re V agrees well with the GL prediction of −0.67 for the Pr dependence of Re U [3][4][5]43], but here also we do not know why this agreement should prevail.
The data for Re V were precise enough to yield new information about the transition to the ultimate state. While they initially followed, albeit with somewhat increased scatter, the power law determined in the classical state as Ra exceeded Ra 1 * , they revealed a transition at