2 Prandtl- and yleigh-number dependence of Reynolds numbers in turbulent yleigh-bénard convection at high yleigh and small Prandtl numbers Xiaozhou He 1,5, Dennis P. M. van Gils 1,5, Eberhard Bodenschatz 1,2,3,5, and Guenter Ahlers 1,4,5 1 Max Planck Institute for Dynamics and Self Organization, D-37073 Göttingen, Germany 2 Institute for Nonlinear Dynamics, niversity of Göttingen, D-37073 Göttingen, Germany 3 Laboratory of Atomic and Solid-State Physics and Sibley School of Mechanical and Aerospace Engineering, Cornell niversity, Ithaca, New York 14853 4 Department of Physics, niversity of California, Santa Barbara, CA 93106, SA, 5 International Collaboration for Turbulence Research (Dated: October 7, 2013) We report measurements of the large-scale-flow Reynolds number Re and the fluctuation Reynolds number Re in turbulent yleigh-bénard convection in the yleigh-number range 4 10 11 < < 2 10 14 and for a Prandtl number Pr 0.80 for a sample with a diameter to height ratio of 1.00. The yleigh-number dependence of Re is described well by Re = Re 0Pr α exp ζ exp with ζexp = 0.435, and comparison with results for Re in the literature for Pr near five suggests αexp 1.20. These results differ from the extended Grossman-Lohse (GL) model [Phys. Fluids 16, 4462 (2004), Stevens et al., J.Fluid Mech. 730, 295 (2013)], where ζgl 0.33 and αgl 0.49. Surprisingly, our result ζexp = 0.435 is similar to what Grossman and Lohse predict for the effective scaling of Re, i.e. ζgl = 0.441; but the Pr dependence αexp 1.20 differs also from the theoretical αgl 0.67. We found the data for Re to be consistent with the GL model. In a fluid between horizontal parallel plates separated by a distance L and heated from below, turbulent convection, known as yleigh-bénard convection or RBC, will occur when the temperature difference T T b T t between the bottom (T b ) and top (T t ) plates is sufficiently large [1 5]. This system is characterized by a dimensionless measure of T known as the yleigh number and by the ratio of the kinematic viscosity ν to the thermal diffusivity κ known as the Prandtl number Pr. The response of the system to the thermal driving can be expressed in terms of a dimensionless effective thermal conductivity known as the Nusselt number Nu and in terms of dimensionless fluid velocities Re known as Reynolds numbers. In this system there are thin thermal and viscous boundary layers (BL) near the top and bottom plate. The majority, or bulk, of the sample is located between these BLs, and is nearly isothermal in the time average (however, see for instance [6, 7]). Most of the experimental work on this problem was done with cylindrical samples, with the ratio Γ D/L between the diameter D and the height L near one. For that case the bulk contains a large-scale circulation (LSC) consisting of a single convection roll, with up flow near the side wall and down flow near that wall on the opposite side. The azimuthal LSC orientation and strength fluctuate in time [8] and can be described as a system driven stochastically by intense background fluctuations [9 11]. A major theoretical success in this field was the formulation by Grossmann and Lohse [12, 13] (GL) of a mathematical model that gives Nu and the Reynolds number Re = L/ν based on the time average = (t) t of a mean-flow velocity (t) (e.g. the maximum velocity within the LSC) when and Pr are specified, provided five parameters of the model are determined by fitting to experimental data. This fit had been carried out for Γ = 1.00 [13 15]. Later the model was extended [16] to also give Re = L/ν based on the root-mean-square (rms) fluctuation velocity [(t) ] 2 t. In this Letter we present measurements of Re and Re for Pr 0.8 over the range 4 10 11 < < 2 10 14 for Γ = 1.00. While Re agrees quite well with the predictions [15], neither the nor the Pr dependence of Re found from experiment agrees with the theoretical model [16]. The data gave Re ζ exp with ζ exp = 0.435, while the model predicts ζgl 0.33. From our result for Pr = 0.80 and data in the literature for Pr 5 we find that Re Pr α exp with α exp 1.20 while the prediction is αgl 0.49. The dependence of Re on Pr for Pr > 7, and the dependence of Re found by us for Pr = 0.80, surprisingly agree with the prediction for Re [15], but not with that for Re [16]. We find that the Pr dependence of Re is described rather well by the GL model [15] over a very wide range of Pr. Comparison of from different investigations is relatively easy because, at least to a good approximation, and thus Re is uniform throughout the bulk of a sample
2 with Γ = 1.0 [17] and thus the precise location of measurement does not matter much. In contradistinction, for Re the comparison between various studies is more difficult because varies throughout the sample, for instance roughly linearly from a large positive to a large negative value along a sample diameter or along the vertical center line (see, e.g. [17]). Thus, in the study for instance of the Pr dependence of Re only results corresponding to equivalent locations within the sample can be used. It is convenient to (and we will) use the maximum speed of the LSC near (but not too near) the sample wall near the horizontal mid plane. For the experiment we used the High Pressure Convection Facility I (HPCF-I) [18], a cylindrical convection cell with L = D = 112 cm (aspect ratio Γ = 1.00), located inside a pressure vessel known as the boot of Göttingen [19]. The boot and HPCF-I can be filled with various gases at pressures up to 19 bars. We used sulfur hexafluoride (SF 6 ) at various pressures to cover the range 4 10 11 2 10 14. The Prandtl number varied slightly with pressure and thus, over the range from 0.786 to 0.860. The Re measurements were made using simultaneous temperature time series from three Honeywell 111-104HAK-H01 thermistors with a bead diameter of 0.36 mm located near the horizontal mid-plane of the sample, at a distance of 1.5 cm from the side wall. The sample was tilted relative to gravity through an angle of 0.76, in order to cause the circulation plane of the LSC to prefer on average an azimuthal orientation Θ LSC t that nearly coincided with the azimuthal location Θ T of the thermistors. The thermistors were separated vertically by distances of 3.5 cm between the first two and 1.5 cm between the second and third. Since, to a very good approximation, temperature in the bulk is a passive scalar [20], temperature and velocity correlation functions are nearly the same and either can be used with the elliptic approximation of He and Zhang [21, 22] to obtain and. This method was used successfully before by various authors [23 27]. The yleigh number was obtained using = (βg T L 3 )/(κν), where β is the thermal expansion coefficient and g the gravitational acceleration. All fluid properties were evaluated at the mean temperature T m = (T b + T t )/2. The results for Re are shown in Fig. 1a, and in the reduced form Re / 0.435 in Fig. 1b, as open squares. The dependence of Pr on is given in the inset of Fig. 1a. Since Re depends both on and Pr, and since comparison with predictions is best accomplished at constant Pr, we divided Re by Pr α to remove the Pr dependence. The value of α is uncertain; but since in any case the Pr α correction is small, we chose initially the value αgl = 0.667 [15] for this reduction because, as we shall see below, that value agrees with measurements of Re for Pr > 7 [28]. The result for Re /Pr α GL is shown in Re or Re / Pr α 10 6 10 5 Pr 0.84 0.78 10 11 10 14 10 12 10 13 10 14 Re / 0.435 or (Re / Pr α ) / 0.435 0.72 0.66 0.6 0.54 10 12 10 13 10 14 FIG. 1: The Reynolds number Re (open squares) and the Prandtl-reduced Reynolds number Re /P r α with α = αgl = 0.667 (solid circles, blue online) and α = αexp = 1.20 (solid squares, red online). : The same data as in, but divided by 0.435. All graphs are on double logarithmic scales. In the upper (horizontal) solid line corresponds to Re 0 = 0.632 and the lower solid line is a power law with ζexp = 0.438, Re 0 = 0.508. The dashed lines (dash-dotted line) are the GL prediction for Re [15] (Re [16]) with the amplitude adjusted to fit the measurements. The inset in shows the variation of Pr with. Fig. 1a and in the reduced form (Re /Pr α GL )/ 0.435 in Fig. 1b as solid circles (blue online). One sees that the slope (which yields ζ) is not influenced very much by the Pr dependence on. Since there is reasonable evidence that the system begins its transition to the ultimate state of RBC near 1 10 13 [18] where the dependence of Re on is expected to change [26, 29, 30], we chose to use only the data for < 10 13 for a quantitative analysis. Over that range Pr varied only from 0.786 to 0.800. A fit to the power law Re /Pr α GL = Re0 ζ exp yielded Re0 = 0.632 and ζexp = 0.435 ± 0.001. A similar fit with Pr = 1 gave ζexp = 0.431, showing that the effect of Pr() on ζexp is not very large. The exponent values derived from the GL model [15] in our and Pr range are ζgl = 0.441 and ζgl 0.33. Surprisingly the experimental result for Re agrees very well with the prediction for Re but disagrees with the theoretical result for Re. Although its relevance is not obvious, the GL prediction for Re (with the amplitude adjusted to match our Re data in the middle of the range) is shown as a dashed line in Fig. 1b. One sees that it has nearly the same slope (ζ value) as the data. Although the agreement between Re and Re GL may be merely a coincidence, we hope that it may provide insight into how to modify the model for Re GL. In Fig. 2a we show Re and Re, and in Fig. 2b we present Re in the reduced form (Re /Pr α GL )/ 0.435, both as a function of on double logarithmic scales. One sees that Re and Re are nearly equal to each
3 Re and Re 10 6 10 5 10 11 10 12 10 13 10 14 (Re / Pr 0.667 ) / 0.435 0.8 0.6 0.4 10 11 10 12 10 13 10 14 FIG. 2: : The large-scale-flow Reynolds-number Re (solid circles, blue online) and the fluctuation Reynolds number Re (open black squares). : The reduced Reynolds number (Re /Pr α GL )/ 0.435 with αgl = 0.667 as a function of on logarithmic scales. other. However, Re does not perfectly follow a unique power law, as revealed by the curvature of the data in. Independent determinations of the orientation of the LSC plane Θ LSC based on the azimuthal temperature variation in the horizontal mid plane [8, 33 35] showed that Θ LSC t depended somewhat on, and one would expect Re to depend on Θ LSC t Θ T. Thus, our determination of Re is not suitable for an accurate determination of ζ since the measurement location (relative to the LSC orientation) is changing with. Since we would like to have Re for the case Θ LSC t = Θ T where we expect it to be a maximum, our best estimate is (Re /Pr α GL )/ 0.435 0.64 as indicated by the dotted horizontal line in the figure. In order to facilitate a comparison of Re with measurements by others, we estimated its value at = 10 10 from the above result and show it in Fig. 3a as a solid circle (blue online). Also shown are data from Refs. [17, 28, 31, 32] extrapolated to or interpolated at = 10 10. The prediction of Re GL, which had been adjusted [15] to fit the data of [17], is shown as a dashed line. Since the model gives Re only to within a multiplicative constant, we also show 1.7 Re GL as a solid line. It comes close to our data and those of Refs. [28, 31, 32]. We conclude that the GL model describes the Pr dependence of Re quite well over the very wide range from Pr = 0.7 to 300. We note that the Re value of [17] was based on a velocity equal to the radial derivative of in the sample interior multiplied by D/2, which would be equal to the maximum velocity of the LSC near the side wall if varied linearly with radial position. However, data indicate [17] that the maximum velocity tends to be several percent larger than this linear estimate, thus explaining at least in part why the Re value of [17] falls below the solid line. In view of all the data we suggest that it would be better to base the GL model coefficients [15] on the solid line in the figure. The model with those parameters would provide an even better basis for estimating Re Re 10 4 10 3 10 2 10 4 10 3 10 2 1 10 100 Pr FIG. 3: Estimates (by interpolation and/or extrapolation) of : Re and : Re for = 10 10 as a function of Pr. Open diamonds (purple online): from [31]. Solid diamonds (purple online): from [28]. Square (red online): from [17]. Black up-pointing triangle (in only): from [32]. Downpoiting triangle (green online, in only): from [25]. Circle (blue online): this work. The black dashed line in is Re GL [15]. The solid line (purple online) in [] is 1.7Re GL [0.373Re GL]. The dash-dotted line (purple online) in is a fit of Re GL = Re 0Pr α GL with α GL = 0.49 to the three points of [28], adjusting only Re 0. The dotted line (red online) in is a power law with α = 1.20. the shear Reynolds numbers for the viscous BLs which in turn could lead to a better estimate [14, 15] of the ultimate-state transition [18]. In Fig. 3b we show results for Re, likewise interpolated at or extrapolated to = 10 10. The solid line is the GL prediction for Re [15], multiplied by a constant factor of 0.373. It fits the data for Re quite well for Pr > 7, but does not fit our new result for Pr 0.80. The experimental value is larger than the model predic-
4 tion by a factor of about four. Of course we do not know how to properly formulate this strong Pr dependence; but if we choose to make a power-law fit to the data for Pr < 7, we find Re Pr 1.20 which is shown as a dotted line (red online). We note that the data by others for 4 < Pr < 7 [17, 25, 31] also are consistent with such a large exponent and scatter more or less randomly about the dotted line. Shown as well in Fig. 3, as a dashed line (purple online), is the GL prediction for Re [16] with the pre-factor adjusted to provide a best fit to the data of [28]; it does not fit any of the data very well, and our result for Pr = 0.80 is larger by a factor of about eight. If we use α exp = 1.20 to reduce Re, we get the results shown as squares (red online) for Re /Pr αexp in Fig. 1 and for (Re /Pr αexp )/ 0.435 in Fig. 1. The solid line passing through the squares in Fig. 1 is a fit of a power law to the data for < 10 13, which yielded ζexp = 0.438, somewhat closer to the theoretical ζgl = 0.441 which is shown as a dashed line. Although not central to the topic of the present paper, we note that the solid squares (red online) in Fig. 1b clearly reveal a transition, which we take to be the onset of the transition to the ultimate state [18, 36], near 1 10 13. Finally we consider phenomenologically what our result Re Re [see Fig. 2] implies within the framework of the extended GL model. From Eq. (56) of [16] we have 3 L ν 2 l 2 (1) where l is a length which, within the GL model, was taken to be a similarity variable η defined by Eq. (16) of [16] and which initially we shall leave undefined. Solving Eq. 1 for l and applying the experimental result / 1, we find l LRe 1/2. (2) We note that, within the GL model, the viscous BL has a thickness λ ν = alre 1/2 [14, 16] and that a fit of the model to experimental results for Re gave a 1 [15]. Thus, we see that one way to achieve agreement with the experimental result Re /Re 1 is to assume that the relevant length in Eq. 1 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 and 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 Eq. (22) of [16] would also be consistent with Eq. 1 and the experiment since also in this case we have λ θ Re 1/2, albeit with a Pr-dependent pre-factor. For Pr = O(1) the pre-factor is close to unity, and thus on purely phenomenological grounds we can not distinguish between the two options for l. One might be tempted to invoke the experimentally determined Pr dependence of Re to distinguish between the relevance of λ ν or λ θ ; but this dependence is not a simple power law for Pr 1 and thus we do not pursue that option at this point. In this Letter we presented new data for Re and Re over the range 4 10 11 < < 2 10 14 and for Pr 0.8. For Re this parameter range had not been studied before. By comparing with the measurements of others [17, 25, 28, 31, 32] we find that the GL model [12 14] does an excellent job of describing the Pr dependence of Re over the wide range 0.7 < Pr < 300. It was a surprise to find that, within a multiplicative factor, its prediction also fits the results for Re (Pr) very well for Pr > 7 while a subsequent extension of the GL model to Re [16] does not fit the Re data over any range. Our new Re results for Pr 0.80 show a large departure, by a factor of about four, from the model prediction for Re, and a larger deviation by a factor of about eight from the prediction for Re. A power-law fit to the Re data for Pr < 7 ([17, 25, 31] and our point for Pr = 0.80) yields an exponent αexp 1.20, but of course we do not know whether a power law is the appropriate representation of these small-pr data. The exceptionally large value of Re found from our measurements, and the departures of Re data over the entire Pr range from the GL prediction for Re, suggests to us that the GL model [12 14] and its extension [16] do not contain all the relevant physics responsible for the generation and/or attenuation of velocity fluctuations. We presented a phenomenological modification of the extended GL model [16] which demonstrates that using either the thermal or the viscous BL thickness at the side wall instead of the similarity length η of the GL model [16] yields agreement with the experimental result Re /Re 1. Acknowledgements: We are grateful to the Max-Planck-Society and the olkswagen Stiftung, whose generous support made the establishment of the facility and the experiments possible. We thank the Deutsche Forschungsgemeinschaft (DFG) for financial support through SFB963: Astrophysical Flow Instabilities and Turbulence. The work of G.A. was supported in part by the.s National Science Foundation through Grant DMR11-58514. We thank Andreas Kopp, Artur Kubitzek, Holger Nobach, and Andreas Renner for their enthusiastic technical support, and Siegfried Grossmann and Detlef Lohse for illuminating discussions. [1] L. P. Kadanoff, Phys. Today 54, 34 (2001). [2] G. Ahlers, Physics 2, 74 (2009). [3] G. Ahlers, S. 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