This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
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Reynolds number scaling in cryogenic turbulent Rayleigh-B´ enard convection in a cylindrical aspect ratio one cell Vˇ era Musilov´ a1 †, Tom´ aˇ s Kr´ al´ık1 , Marco La Mantia2 , Michal Macek1 , Pavel Urban1 and Ladislav Skrbek2 1 2
Institute of Scientific Instruments of the CAS, v.v.i., Kr´ alovopolsk´ a 147, Brno, Czech Republic Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague, Czech Republic (Received xx; revised xx; accepted xx)
We perform an experimental study of turbulent Rayleigh-B´enard convection up to very high Rayleigh number, 108 < Ra < 1014 , in a cylindrical aspect ratio one cell, 30 cm in height, filled with of cryogenic helium gas. We monitor temperature fluctuations in the convective flow with four small (0.2 mm) sensors positioned in pairs 1.5 cm from the sidewalls and 2.5 cm vertically apart and symmetrically around the midheight of the cell. Based on one-point and two point correlations of the temperature fluctuations, we determine different types of Reynolds numbers, Re, associated with the large scale circulation (LSC). We observe a transition between two types of Re(Ra) scaling around Ra = 1010 − 1011 , which is accompanied by a scaling change of the skewness of the probability distribution functions (PDFs) of temperature fluctuations. The Re(Ra) dependencies measured near the sidewall at Prandtl number Pr ∼ 1 are consistent with the Ra4/9 Pr−2/3 scaling above the transition, while for Ra < 1010 , the Re(Ra) dependencies are steeper. It seems likely that this change in Re(Ra) scaling is linked to the previously reported change in the Nusselt number Nu(Ra) scaling. This behaviour is in agreement with independent cryogenic laboratory experiments with Pr ∼ 1, but markedly different from the Re scaling obtained in water experiments (Pr ∼ 3.3 − 5.6). We discuss the results in comparison with different versions of the Grossmann-Lohse theory.
1. Introduction The ideal Rayleigh-B´enard convection (RBC), occurring in a horizontal fluid layer confined between two laterally infinite perfectly conducting plates heated from below in a gravitational field, serves as a very useful model for fundamental studies of buoyancy driven flows that exist on many length scales across the Universe [for review, see Ahlers, Grossmann & Lohse (2009); Chill` a & Schumacher (2012)]. For an OberbeckBoussinesq fluid, of constant physical properties except its density which linearly depends α ∆T L3 , and the Prandtl on temperature, it is fully characterised by the Rayleigh, Ra = g νκ numbers, Pr = ν/κ. Here g stands for the acceleration due to gravity, and ∆T = Tb − Tt is the temperature difference between the parallel bottom and top plates separated by the vertical distance L. The working fluid is characterised by the thermal conductivity, λ, and by the combination α/(νκ), where α is the isobaric thermal expansion, ν is the kinematic viscosity, and κ denotes the thermal diffusivity. † Email address for correspondence:
[email protected]
2 Although the equations describing turbulent buoyancy driven flows and the corresponding convective heat transfer have been known for a long time (Tritton 1988), our ability to predict flow behaviour for very intense convection occurring at large scales in the atmosphere (Hartmann, Moy & Fu 2001), ocean (Marshall & Schott 1999), stars or Sun (Cattaneo, Emonet & Weiss 2003) is very limited or even excluded. Experimental investigations of RBC under controlled laboratory conditions are therefore instrumental, especially at very high Ra, and represent a lively field of fluid dynamics. Lateral dimension, D, delimited with vertical sidewalls of the experimental cells, complicates general conclusions on RBC - the shape of the convection cell becomes an additional important parameter. It is usually simplified via the aspect ratio Γ = D/L, used together with Ra and Pr as control parameters of RBC. The ability of RBC to transfer heat, of the total convective heat flux density q, ˙ from the heated bottom plate to the cooled top plate is described by the Nusselt number, Lq˙ Nu = λ∆T , via the Nu = Nu(Ra, Pr; Γ ). This convective heat transfer efficiency and its Ra dependence up to very high Ra values is a key integral property of RBC and the subject of study in numerous experiments, utilizing various working fluids such as air, water, ethane, SF6 , glycerol, or helium. In addition to the Nusselt number, Nu, the second crucial response parameter of the RBC flow is the Reynolds number, Re, characterizing the velocity field. Distinct functional dependencies of Nu(Ra, Pr; Γ ) and Re(Ra, Pr; Γ ) are linked with distinct regimes of convection. Existence and nature of these different regimes, and of the transitions between them, have been subject to intense investigations, both experimental and theoretical [for reviews see Ahlers, Grossmann & Lohse (2009); Chill` a & Schumacher (2012)]. While at relatively low Ra < 1011 the results of various groups on Nu(Ra, Pr; Γ ) and Re(Ra, Pr; Γ ) obtained with various working fluids are at fair quantitative agreement and well understood, above about 1011 in Ra strong differences appear, clearly calling for further detailed integral as well as structural studies of RBC. Organised coherent structures - plumes, jets and large scale circulation (LSC), also known as “wind” (Castaing et al. 1989; Niemela et al. 2001), are known to exist and can be characterised by suitably defined Reynolds numbers, Re. These coherent structures interact with the RBC boundary layers at top and bottom plates and therefore may influence also the heat transfer and thus Nu. The topology and geometry of the prominent structure, the LCS, depend on the aspect ratio, as well as the shape of the cell (Daya & Ecke 2001; Foroozani et al. 2014). In particular, in RBC with aspect ratio one, a single central LSC roll, extending nearly to the size of the convection cell and circulating with mean velocity, U , sets on at a critical value of Ra (Sun, Xia & Tong 2005a). At higher Ra, above 107 − 108 , LSC becomes turbulent, self-organised from thermal plumes (Xi et al. 2004) which are emitted randomly (Xi et al. (2009)) from boundary layers at the top and bottom of the convection container. At Pr ∼ 1 the wind persists at least up to Ra ≈ 1014 (Chavanne et al. 2001; Niemela et al. 2001; He et al. 2015, 2016). Dynamics and structure of LSC, some features of which still remain to be fully understood, have been reviewed in detail by Xia (2013). Let us recall here some of the results which are relevant to our cylindrical aspect ratio Γ = 1 experiment. Roughly linear profiles of mean velocity obtained in the LSC plane suggest a fly wheel-like rotation of the fluid, characterised by a specific turnover time. The 2D maps of the mean velocity field, obtained using particle image velocimetry (PIV) in water experiments show an oval shape of the LSC with its long axis inclined diagonally across the convection container, while the shape becomes more squarish at higher Ra ∼ 1010 – see the maps at Ra = 7×109 in (Sun, Xia & Tong 2005a) and at Ra = 9.5 × 1010 in (Zhou et al. 2011). Indeed, Sun, Xia & Tong (2005a) visualised the velocity field at Ra = 7 × 109 by PIV, in vertical planes at angles 0◦ , 45◦ , and 90◦ with respect to the rotation plane of the LSC which
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was locked at 0◦ by a slight tilt of the cell. In the LSC plane the mean velocity reaches its maximum at mid-height, at a distance of about 5% of diameter from the sidewall, in agreement with LDV velocity profiles previously published by Qiu & Tong (2001b). The highest velocity in the 45◦ plane reaches only about 80% of its maximum in the plane at 0◦ . This value correlates well with the observation reported by Brown et al. (2007), where the temperature of the sidewall at its mid-height, and also at heights L/4 and 3L/4, was monitored and the increased or decreased temperature at side wall was interpreted as rising warm or descending cool fluid flow of the LSC. These authors found that the temperature profile along the perimeter is fitted quite well with a cosine function with the period equal to the perimeter length. This results in the temperature value at the 45◦ plane being 70 % of maximum at 0◦ plane and gives thus a rough idea about transverse dimension of LSC, being about two thirds of the cell diameter (Zhou et al. 2009). Azimuthal dynamics of LCS is to a large extent stochastic. In a water experiment, Brown & Ahlers (2006a) studied for 109 < Ra < 1011 the position of the LSC plane by monitoring the sidewall temperature at the horizontal mid-plane. Azimuthal angle of LSC plane meanders around a preferential angle (Xi & Xia 2008; Ahlers et al. 2006), which is set either by Earth’s Coriolis force or by imperfect symmetry of the sample, including its tilt. Brown & Ahlers (2006a) have claimed, that even slight tilting by 0.2 degrees canceled the effect of Coriolis force at Ra = 9 × 109 . Changes of the angle are erratic and follow Poisson distribution, suggesting their mutual independence. Sudden changes in orientation of the wind rotation (change of azimuthal angle by ∼ π) and cessations were observed with very low probability (Brown et al. 2005; Brown & Ahlers 2006b). Ahlers et al. (2006) found that slight tilting of the sample not only sets preferential position of the LSC plane about the tilt direction but, additionally, it dramatically suppresses the azimuthal meandering of the LSC plane and the rotation reversals. This fact has in turn been used in experiments to lock the LSC position (Sun, Xia & Tong 2005a; He et al. 2015). More orderly modes of LCS dynamics have also been identified. Qiu et al. performed first studies on correlation among temperature measurements at several points within the fluid (Qiu & Tong 2002) and 1D velocity measurements by laser Doppler velocimetry (LDV), see Qiu et al. (2004). These studies have revealed a common characteristic oscillation frequency of both the temperature and the velocity fields. Additionally, oscillations of the velocity component perpendicular to the LSC plane have been observed by Qiu et al. (2004). Sun, Xia & Tong (2005a) performed similar, but more detailed PIV study of the velocity field oscillations. In the study of Brown et al. (2007), the frequency of temperature oscillations was found to be independent of the temperature sensor position. Angular oscillations of the LSC velocity vector with opposite phases at the top and the bottom plates were monitored by shadowgraphs in methanol (Ra = 3 × 109 , P r = 6.0) by Funfschilling & Ahlers (2004), and in air experiments at 4 × 1011 6 Ra 6 7 × 1011 (P r = 0.71) by Resagk et al. (2006). By monitoring the sidewall temperatures at three heights (L/4, L/2 and 3L/4) in experiments with water at 9 × 108 6 Ra 6 1.1 × 1010 , the effect of velocity angular oscillations was identified by Funfschilling et al. (2008) as twisting of the LSC around a vertical axis. More detailed analyses by Xi et al. (2009); Zhou et al. (2009) at 109 6 Ra 6 1010 distinguished the twisting oscillation from a horizontal oscillation of LSC plane phase-shifted by π/2—an off-centre movement, called the sloshing mode. Finally, a slightly modified interpretation of temperature records performed by Brown & Ahlers (2009) at Ra = 1.1 × 1010 confirmed the presence of both the twisting and sloshing modes; this report also introduces a physically motivated model of such fluid oscillations. A similar study, at higher Pr=19.4 and 8 × 108 6 Ra 6 2.9 × 1011 , has been
4 done by Xie et al. (2013). Recently also a vertical mode of bulk fluid oscillations was reported by Wei & Ahlers (2016), who also performed a detailed analysis, distinguishing the effects of various coherent structures on the temperature fluctuations in different regions inside the RBC cell. Most of 3D studies on LSC have been performed with water, 4 6 Pr 6 6, at Ra up to 1012 . On the other hand, the highest Ra have been achieved in experiments using pressurized SF6 (He et al. 2012) and cryogenic helium gas (Niemela et al. 2000). Direct visualization techniques have not yet been applied for these experiments and information on flow structure at the highest attainable Ra in laboratory experiments is thus still limited to data collected using temperature sensors positioned within the fluid (Chavanne et al. 2001; Niemela et al. 2001; He et al. 2015, 2016). This paper complements our detailed studies of the heat transfer efficiency Nu(Ra) at very high Rayleigh numbers Ra (Urban et al. 2011, 2012, 2014; Skrbek & Urban 2015) by a statistical investigation of the structure of convective flow in the bulk. We focus on the Reynolds number scaling Re(Ra) associated with the LSC. The Reynolds numbers are determined on the basis of one-point and two-point local temperature fluctuations measurements near the sidewalls, and at approximately mid-height of the cell (details below in §. 2.1), which were collected simultaneously with the measurements of Nu(Ra), reported previously in Urban et al. (2011, 2012, 2014). As well known, discrepancies in measured Nu(Ra) dependencies from different experiments (Roche et al. 2010; Chill` a & Schumacher 2012; Skrbek & Urban 2015; He et al. 2016) at Ra & 1010 are not fully understood. Here, our main aim was to determine the Re(Ra) dependencies, complementary to the Nu(Ra) dependencies, and to compare them carefully with available cryogenic helium experiments on Re(Ra, Pr) scaling, conducted in various laboratories using various cylindrical containers of heights ranging from 8.7 cm to 100 cm (Chavanne et al. 2001; Niemela & Sreenivasan 2003a; Roche et al. 2010; Niemela et al. 2000), emphasizing their common features, as well as the differences between Re scaling observed in cryogenic helium (Pr ≈ 0.7), pressurized SF6 (Pr ≈ 0.8) and water (Pr > 3) experiments. We discuss the results with respect to the Grossmann-Lohse (GL) theory, both in the original (Grossmann & Lohse 2001, 2002) and the updated (Stevens et al. 2013) parameter-fit versions. The results on Re(Ra) are supplemented by analyses of the temperature probability density functions (PDFs), paying attention to the Radependencies of their skewness and flatness (kurtosis) near the sidewall. To facilitate direct comparison with various other experiments, we utilize several types of Reynolds numbers’ definitions: (i) Ref 0 based on the frequency f0 determined from the auto-correlation functions, (ii) Rep determined from cross-correlation functions assuming the Taylor frozen flow hypothesis, and finally (iii) ReU , ReV , and Reeff determined from the more general elliptic approximation of the space-time correlation function (He et al. 2014b, 2015). When determining the frequency-based Ref 0 , we assume—in line with the observations of Brown et al. (2007); Qiu & Tong (2001a) and others—that the frequency f0 is independent of the LSC plane orientation as measured at our sensor positions, and that it can be interpreted as the LSC sloshing mode frequency (Xi et al. (2009); Zhou et al. (2009); Brown & Ahlers (2009)). In order to deduce the most probable azimuthal orientation of the LSC plane in our cryogenic helium sample, we made careful comparisons with two reports of water experiments by Brown et al. (2007) and Zhou et al. (2009), as unlike ours, these experiments measured the position of the LSC plane directly. Despite this limitation, we believe that the quality of the cryogenic conditions reached in our apparatus and especially the large control parameter (Ra) range covered with a single apparatus provide valuable new experimental input for understanding RBC at high Ra. The paper is organised as follows: In §2, we describe our experimental apparatus and
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Figure 1. Scheme of the apparatus: (Left panel) Inner part of helium cryostat (Urban et al. 2010) containing the cylindrical aspect ratio one RBC cell. Four small Ge temperature sensors (named T1 - T4) measure temperature fluctuations in the fluid at opposite sides in the cell interior. Temperatures of the top (Tt ) and bottom (Tb ) plates are measured by four additional temperature sensors embedded in the centre and near the edge of both plates. The top plate temperature is stabilised. Temperature of the vent lines near the cell is controlled with electrical heaters to suppress potential effect of convection inside the lines. (Right panel) Scheme of the azimuthal positions of vent lines, filling tube and sensors in the cell with respect to the cardinal points as seen from top. Small red and blue circles show the prevailing flow direction (warm and cold stream, respectively) observed with the sensors.
the measurement procedure. In §3, we present the experimental data obtained and discuss them in comparison with available data from other experiments and with the GL theory. Summary and conclusions are given in §4.
2. Methods 2.1. Experimental apparatus We perform our experiments in clean cryogenic environment and utilize the cryogenic helium gas, representing a working fluid with well known (NIST 2000; McCarty 1972) and in situ tunable properties. Our helium cryostat, described in detail by Urban et al. (2010), is schematically shown in figure 1. In short, the convection cell is thermally connected to a liquid helium vessel (4.2 K), via a heat exchange chamber filled with gaseous helium; its main function is to remove, as uniformly as possible, the convective heat flux from the top plate. The cell and the heat exchange chamber are protected against thermal radiation by a copper shield thermally anchored to the liquid helium vessel. All internal parts of the cryostat are suspended on a central thin wall stainless steel neck in high cryogenic vacuum of order 10−6 Pa. The cylindrical experimental cell with the height L = 0.3 m and diameter D = 0.3 m has been designed with particular effort to minimize the influence of its structure and
6 materials on the observed RBC flow (Urban et al. 2010). The cell withstands pressures up to 3.5 bar in order to cover the range 106 < Ra < 1015 with sufficient precision of measurements. The plates are made of 28 mm thick annealed OFHC copper of very high thermal conductivity (at least 2 kWm−1 K−1 at 5 K). 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 the plates, in the case of the top plate also assuming that the heat is uniformly removed by the exchange chamber. In the experiments, the temperature of the top plate is roughly set by adjusting the pressure in the heat exchange 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 sidewalls are minimized by using very thin (δ = 0.5 mm at the boundary layers and δ = 1 mm at the central part) stainless steel of low thermal conductivity, as well as by employing a special design of the cell corners (Urban et al. 2010). Temperature fluctuations inside the fluid volume are monitored using four small Ge sensors [250 µm cubes, type TTR-G, Mitin et al. (2007, 2011)], placed in two pairs at opposite sides near the mid-height of the cell, 1.5 cm from the sidewall and 2.5 cm vertically apart, see the left panel of figure 1. Each sensor (typically of resistance 6 kΩ and sensitivity ≃ 1 kΩ/K) is a component of its own impedance bridge (Wheatstone bridge analog), with both the sensor resistivity and parasitic capacity of its cabling balanced. The temperature fluctuations are evaluated from the voltage signal across the particular compensated bridge by the phase sensitive (lock-in) amplifier Stanford Research Systems SRS830 and recorded by PC via GPIB-USB (National Instruments) interface. The sampling of the four simultaneously measured signals is synchronised using a special data transfer mode (FAST) of the SR830 and a trigger signal of the external generator (Velleman). Temperatures at the top (Tt ) and bottom (Tb ) plates are measured by additional four Lake Shore GR-200A-1500-1.4B germanium temperature sensors (calibrated with 5 mK uncertainty to the ITS-90 International Temperature Scale by the manufacturer) embedded in the centre and near the edge of both Cu plates. We made an additional calibration of the sensors which allows us to measure ∆T = Tb − Tb with uncertainty up to 2 mK (Urban et al. 2010). The pressure in the cell is measured with MKS Baratron 690A (calibration traceable to NIST) with 0.08% reading accuracy. 2.2. Data acquisition and processing Two series of data, labeled “c” and “d”, have been obtained in two independent experimental runs, with the cryostat and the experimental cell dismantled in between. The “c” data contain records from one sensor (T4 only), obtained at 1011 6 Ra 6 3×1014 during the experiment focused on heat transfer efficiency measurement aimed at the maximum Ra values (Urban et al. 2011, 2012, 2014). The sampling frequency is 32 Hz and the lock-in amplifier integration time is 100 ms. Typical length of one record is an hour. The quality of the signal recorded is slightly affected by parasitic capacitances due to the sensor cabling which could not have been fully balanced in the bridge circuit used in this experiment. Thus the signal to noise ratio is only moderately high, of the order of 10, and decreases towards the highest values of Ra. The “d” data consist of synchronous records of temperature signals of four sensors (T1 - T4, see figure 1) obtained for 108 6 Ra 6 3 × 1013 . We typically apply a sampling frequency of 32 Hz or higher (up to 80 Hz), with lock-in amplifier integration times being 100 ms (300 ms in few cases). The lengths of individual records range from 1 to 47 hours. The signal to noise ratio is 100 times larger in comparison with the “c” data and, additionally, the linear trend of the recorded signal is numerically suppressed.
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The dynamic RBC parameters are calculated based on the fluid properties evaluated at α the mean temperature Tm = (Tb +Tt )/2 in a conventional way, i.e.: Ra = g νκ (Tb −Tt )L3 , (where ∆T = (Tb −Tt ) is corrected with respect to the adiabatic gradient) and Pr = ν/κ. For subsequent statistical analysis, the raw data are processed in the following way: Each raw data record, denoted R(t), sampled at frequency f , consists of N points, where the i-th signal value is recorded at t = i/f . The signal mean Rm is obtained as N N 1 X 1 X i Rm = , (2.1) R(t) = R N i=1 N i=1 f while its standard deviation Rsd is calculated as v u 2 N u1 X i − Rm , R Rsd = t N i=1 f
(2.2)
An example of the raw data records from one of the sensors (T3) can be seen in figure 2. The normalized signal Sr (t) at sensor position r is consequently defined as R(t) − Rm R(i/f ) − Rm i Sr (t) = = = Sr . (2.3) Rsd f Rsd
Cross-correlation functions CCr,s (τ ) of the signals detected at sensor positions r, s are then defined as X X i + m i 1 1 Ss , (2.4) Sr (t + τ )Ss (t) = Sr CCr,s (τ ) = N − |m| i N − |m| i f f where the time-delay τ = m/f , −N < m < N , and the summation index i meets contemporaneous conditions 1 6 i 6 N and 1 6 i + m 6 N . Auto-correlation functions ACr (τ ) at the sensor r are obtained from (2.4) simply by putting r = s, so that ACr (τ ) = CCr,r (τ ). Typical examples of AC and CC functions can be seen in figure 7 (top panel). A further useful characteristic of the signal is the power spectrum Sr (f ), defined in a standard way by the square of the Fourier transform of Sr (t).
3. Results 3.1. Probability distribution functions of temperature fluctuations Temperature records (see example in figure 2) contain characteristic bunches of large|δT | spikes, oriented most of the time towards low temperature at sensors T1 and T2 and towards high temperature at sensors T3 and T4 at the opposite side of the container, as indicated by arrows in figure 3, right. High temperature of the spike bunches at sensors T3 and T4 corresponds to an upward orientation of the mean velocity here, which we deduce from the time delay between signals, read from the cross-correlation function of temperature records, cf. figure 8 further below in § 3.2. Similarly, the prevailing low temperature spikes at sensors T1 and T2 correspond to downward velocity. This type of correlation suggests that the LSC rotates prevalently in the direction from the T1 sensor towards the T2, T3 and T4 sensors. Occasional signal reversals, i.e. sign-changes of the large fluctuation bunches with respect to the mean signal Rm , were detected at all investigated Ra, see figure 2. Figure 3 (left panel) shows typical examples of probability density functions (PDFs) for normalized temperature fluctuations seen at all four sensors at Ra & 1010 . For the T3 and T4 sensors, the PDFs are plotted with respect to −δT /σT , in order to illustrate
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Figure 2. Temperature fluctuations (in arbitrary units) recorded using sensor T3 (upper panel). The lower panel zooms into a detail of the time record. We can see here a typical short-time reversal of the temperature spikes, i.e. sign-changes of the large fluctuation bunches with respect to the mean signal Rm , from prevalent (for T3) positive deviations to negative deviations (see time period between t = 3600 and 3670). These were detected at all investigated Ra, with an overall occurrence of about 10% of time.
Figure 3. Probability density functions (PDFs) of temperature fluctuations δT as measured by all sensors. The sensor positions are shown in the right panel together with the prevailing LSC direction. PDFs obtained with the sensors at opposite side of the RBC cell display opposite skewness (T1, T2 vs. T3, T4), so we reversed the temperature scale in the PDF plots for T3 and T4 to highlight the symmetry. σT stands for the standard temperature deviation. Slight difference between PDFs of the leading sensors (T1 and T3), facing the LSC flow on one hand, and the trailing sensors (T2 and T4) on the other hand can be seen, while members of each pair show mutually almost identical PDFs.
the similarity in shapes. At sensors T3 and T4, the higher shoulders of PDFs at higher temperatures correspond to the prevailing rising warm fluid structures; the skewness (the third moment M3 of the PDF) is positive here. At sensors T1 and T2, a similar but opposite situation occurs, where the higher PDF shoulders at low temperatures reflect the descending cool flow structures, and result in negative skewness M3 < 0. Looking at the evolution of PDFs with Ra in more detail, we see differences between PDFs corresponding to the “leading sensors”, facing the flow (T1 and T3), and the
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Figure 4. Skewness (M3 ) and flatness (M4 ) of temperature fluctuation PDFs. Sensors T1 and T3 are the “leading sensors” facing the mean LSC flow, while T2 and T4 are “hidden” behind them, thus called “trailing sensors”. The values at high end of Ra > 1012 are influenced by lower signal to noise ratio of temperature signals in the experimental run “c”. 0
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Figure 6. Standard deviation of temperature fluctuations σT /∆T , normalised by the temperature difference between the bottom and top plate, and compensated by Ra−1/7 . Our data, measured near the sidewall, are compared with the results of Castaing et al. (1989), measured midway between the sidewall and the centre and Niemela et al. (2000), measured at the cell-centre, and the data-point by He et al. (2014a) near the wall in an SF6 experiment.
“trailing sensors” (T2 and T4), being most apparent in the skewness M3 . First, the absolute value of M3 is larger for the former than for the latter (figure 4, left panels). Second, and more important, M3 breaks for the trailing sensors at about Ra ∼ 1010 from a roughly linear rise (in absolute value) starting from |M3 | ≈ 0, seen at smaller values of Ra, to a value saturated around |M3 | ≈ 0.5 for higher Ra. Figure 5 shows examples of the PDFs corresponding to the low-Ra (“growing skewness”) regime and the high-Ra (“saturated skewness”) regime for one of the leading sensors (left panel), where the differences are marginal, and for a trailing sensor (right panel), where the PDF is considerably more symmetric at the low Ra = 9.4 × 107 (|M3 | ≈ 0.15), than at the high Ra = 4.2 × 1010 (|M3 | ≈ 0.5). At Ra & 1010 , the skewness at the trailing sensors remains still slightly lower compared to the leading sensors, which is related to small difference between the tails of the PDFs, pronounced mainly on the lower-shoulder side, as seen in figure 5. This corresponds to streamwise smearing of the fluctuations (mainly due to decreasing temperature contrast of plumes) away from the top and bottom plates. Concerning the fourth PDF moment (M4 )—the flatness—the leading sensors display slightly lower values of M4 compared to the trailing sensors, see figure 4 (right panels), however no apparent changes in the scaling are seen, in contrast to the behaviour of M3 . The above behavior of the PDF skewness can be interpreted in line with the suggestion of Niemela & Sreenivasan (2003b) (indirect interpretation of a helium experiment) and the measurements of Sun, Xia & Tong (2005a) (direct PIV visualization in water) as a transition in shape of the LSC from an elongated ellipse or oval positioned diagonally across the cell to a more extended squarish or round shape: As the PDF skewness is low in absolute value for the trailing sensors for Ra below ∼ 1010 , we can infer that these sensors are lying further outside of the main LSC flow for Ra < 1010 . Simultaneously, the position of the leading sensors with respect to the main flow remains apparently unchanged for all considered Ra values, as the respective skewness is practically constant. Such a situation presumably corresponds to the elongated shape of the LSC. With increasing Ra, the LSC flow gradually embraces also the trailing sensors, until their skewness saturates at Ra ∼ 1010 , corresponding to the new shape of the LSC, approaching all the cell walls more evenly. Finally, in figure 6, we show the standard deviations normalised by the overall tem-
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Figure 7. Top panel: The auto-correlation functions (AC) together with the cross-correlation function (CC) of temperature fluctuation signals recorded by sensors at opposite sides of the RBC cell. The time T0 derived from the positions of first side maxima corresponds to the main peak frequency f0 = 1/T0 in the power spectra. Bottom panels: Power spectra of temperature fluctuations at mid plane near the sidewall in 2◦ tilted water sample obtained by Xi et al. (2009) at azimuthal position 0◦ (LSC plane) together with 45◦ and 90◦ from LSC plane (Pr = 5.3, Ra = 5 × 109 ) (left panel) and by Niemela et al. (2001) (Pr = 0.8, Ra = 1.5 × 1011 ), Qiu & Tong (2002) (Pr = 5.5, Ra = 1.4 × 109 ), and measured by us (marked as “Brno”) for Pr ∼ 1 and Ra as indicated (right panel). Dependencies compensated with respect to the background power spectra Sbg are shown.
perature drop ∆T between the top and bottom plates. We detect a uniform scaling hδT /∆T i = (0.41 ± 0.02) × Ra−0.130±0.002 within 108 6 Ra 6 3 × 1013 . Notice that the exponent is steeper than the Ra−1/7 scaling as well as the cryogenic helium results of Castaing et al. (1989), performed midway between the sidewall and the cell-centre, and Niemela et al. (2000), measured in the cell centre. Also the overall amplitude is about 2× higher in our experiment. Both facts are related to the sensor position near the sidewall. This is supported by the comparison with the data-point by He et al. (2014a), measured in a pressurized SF6 Γ = 1/2 cell at Ra = 1.08 × 1015 ; Pr = 0.8 at the mid-height of the cell at three points near the sidewall, giving mutually very close results, which are consistent with our dependence. 3.2. Power spectra, auto-correlations and cross-correlations of temperature fluctuations The power spectra of temperature fluctuations Sf recorded in this experiment at individual sensors generally display one pronounced characteristic peak together with a weaker secondary one at twofold frequency, as can be seen in figure 7, bottom panels, where the data are compensated by the background temperature fluctuations’ power spectra Sbg (f ). The frequency f0 of the main peak corresponds to the occurrence of pronounced spike-bunches in the raw temperature signal (see figure 2), with an associated time period T0 = 1/f0 . Alternatively, this period T0 can be evaluated from the position of the first side maximum of the auto-correlation function (2.4) as seen in figure 7. The values of T0 measured in our experiment range between 6 to 40 s.
12 AC: T1
AC, CC
1.0
CC: T1, T2
0.8 0
p
0.6
0.4 -1
0
1
(s)
Figure 8. Auto-correlation function (red solid) of temperature fluctuations and cross-correlation function (black dashed) of time records at nearby sensors, distant by d = 2.5 cm. Time delays τp and τ0 define velocities Up , U, V and Ueff used in various definitions of Reynolds number (3.2)-(3.5). Time delay τp between signals at positions distant by d relates to the Taylor hypothesis-based mean flow velocity Up = d/τp . The figure is a zoom into the AC and CC functions shown in the upper panel of figure 7.
The character of measured power spectra allows us to infer on the azimuthal orientation of the LSC plane, when we compare them with the azimuthally resolved spectra obtained by Xi et al. (2009) in a water sample tilted by 2◦ , see the left bottom panel in figure 7. To highlight the proportions between the first and second peak in S(f ), we divide the spectra by a fit to the background spectrum Sbg (f ). Due to the sloshing mode, the peak of the “second harmonics” in the experimental spectra of Xi et al. (2009) is found to be comparable or higher than the first peak when the sensor is in azimuthal position of the LSC plane (at 0◦ ). The spectra shown in the right bottom plot of figure 7 suggest that the position of sensors in our experiments is azimuthally turned by an angle between 45◦ − 90◦ with respect to the LSC plane. While the height of the second normalized peak in our (Brno) measurements is less than one, the height of the first peak varies between 2 and 6, and no tendency with Ra is observed. Thus our sensors are most probably positioned at the edge of the mean flow at all Ra. We shall see in § 3.3, figure 9 (panel d), that our measured velocity is by about 40% lower than the velocity measured by He et al. (2015) close to the LSC plane, which indirectly confirms previous conclusion on sensors positions. We emphasize that the sensor position does not affect the measured period T0 (and frequency f0 ) of the bulk fluid oscillations (Brown et al. 2007), and consequently the frequency based Reynolds number, discussed below. Further, we have evaluated two point statistics by calculating the cross-correlations (2.4) between pairs of sensors. Figure 7 (top) shows examples of CCs between a pair suspended 2.5 cm above one another (T1, T2) and a pair of sensors on the opposite sides of the cell (T1, T3). Shift of the main maximum of the CC with respect to that of the AC (i.e. τ = 0), and the shapes of the respective peaks, both shown in detail in figure 8, can be used to estimate the mean and fluctuating turbulent-flow velocities, as discussed in the following section.
Reynolds number scaling in cryogenic turbulent Rayleigh-B´enard convection
13
3.3. Reynolds numbers We use several definitions of Reynolds number aiming to compare our data with results published by other authors directly. For a survey of various definitions of Re depending on the method historically used for characterisation of the LSC, see the review of Ahlers, Grossmann & Lohse (2009). In short, in one probe measurement, the characteristic frequency f0 = 1/T0 determined from the power spectrum (or auto-correlation function) is used in definition of the frequency based Reynolds number Ref0 : L2 f 0 . (3.1) ν In two probe measurements, the simplest approach uses the Taylor’s frozen flow hypothesis (Tritton 1988; Kadanoff 2001), assuming that temperature structure is not dissipated between the sensors. The time delay τp between records at two nearby positions (see figure 8) spaced by a distance d determines the velocity Up = d/τp . The corresponding Reynolds number is defined as LUp Rep = . (3.2) ν The Taylor’s hypothesis does not apply when velocity fluctuations are not small in comparison with the mean flow velocity. Then, a more appropriate method is the elliptic approximation, proposed for the general space-time cross-correlation function along the homogeneous flow by He & Zhang (2006) and Zhao & He (2009), and which has been tested in RBC by He et al. (2010) and used frequently since. In this method, Reynolds numbers ReU related to the mean flow velocity, ReV related to velocity fluctuations, and the “effective” Reeff may be defined as: Ref0 = 2
2 τp τp ; U = d 2 = Up ; τ0 τ0 s 2 LV d τp ReV = ; V = ; 1− ν τ0 τ0
LU ReU = ν
(3.3)
(3.4)
p LUeff d ; Ueff = U 2 + V 2 = . (3.5) ν τ0 In our experiment, recording temperature fluctuations at two nearby positioned sensors along the flow enables us to deduce both velocities U and V thus defining the full structure of the space-time cross-correlations. Indeed, from the auto-correlation function (supposed to be identical at both these nearby positions) and the cross-correlation function between the corresponding two records at a leading and a trailing sensor (see figure 8), we obtain the time delays τ0 , τp and, subsequently, velocities U and V . We obtain the values of U and V by fitting the elliptic model to the AC and CC curves, such as those plotted in figure 8. It is evident from the relation for fluctuation velocity V that τ0 > τp and thus Ueff < Up . As a result, Rep > Reeff > ReU ; ReV . In figure 9, we present the one-point and two-point Reynolds numbers Ref 0 , and Rep , ReU , ReV , Reeff determined from our data (labeled “Brno”) and compare them with results of other cryogenic experiments (with gaseous helium as working fluid) performed for 106 < Ra < 1014 in cylindrical Γ = 1 (Niemela et al. 2001; Castaing et al. 1989) and Γ = 1/2 (Chavanne et al. 2001) RBC cells and with the data of He et al. (2015) obtained with pressurized SF6 †. We utilise here one of the theoretically predicted (Grossmann & Reeff =
† Note that both fluids should behave similarly for about Ra < 1013 , due to their very similar
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Figure 9. Prandtl (panel a) and Reynolds numbers (panels b-d) in cryogenic helium and pressurized SF6 experiments: Our results (Brno) are compared with data obtained in aspect ratio one cell of Niemela et al. (2001), Castaing et al. (1989), He et al. (2015) and aspect ratio 1/2 of Chavanne et al. (2001). The Reynolds numbers Ref0 (3.1), Rep (3.2) and the elliptic method based ReU , ReV and Reeff (3.3)-(3.5) are all compensated by the ∼ Ra4/9 Pr−2/3 dependence. The vertical lines indicate the values of Rac (dashed) and the corresponding transition intervals (dotted) for Ref 0 , Rep and Reeff (see text and Tab. 1).
Reynolds number scaling in cryogenic turbulent Rayleigh-B´enard convection Brno
Brno
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)
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Figure 10. Comparison of our experimental values of Reeff with the Grossmann-Lohse theory in the original (Grossmann & Lohse 2001, 2002) and the updated (Stevens et al. 2013) fits. The zoom-in, shown in the right panel, traces in detail the change of scaling associated with the IVl -IVu transition predicted for helium data by the original fit, while no transition is seen in the updated fit, as it places the helium data fully into the IVu region in this Ra range, cf.(Stevens et al. 2013). The vertical lines in the right panel relate to the GL(2002) curve, and their meaning is as in Fig. 9.
Re
/(Ra f0
4/9 -2/3 Pr )
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Niemela et al. 2001 W ater: Brown et al. (2007), Pr=3.26
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Figure 11. Frequency Reynolds numbers versus Ra measured in cryogenic helium (Brno - our data, Niemela et al. (2001)) and water experiments at 3.3 < Pr < 5.7 (Brown et al. (2007), Sun, Xia & Tong (2005a) and Qiu & Tong (2001a)) compensated by Ra4/9 Pr−2/3 , as in figures 9 and 10.
Lohse 2000, 2001, 2002, 2004) scalings Re ∼ Ra4/9 Pr−2/3 , and display the data in a compensated form. This is useful especially in order to compare the data at higher Ra, where the Prandtl numbers Pr in ours and the other experiments differ. In all considered experiments, the temperature probes were positioned at middle height and near the sidewall (Niemela et al. (2001), Brno) or midway the cell radius (Chavanne et al. 2001; Castaing et al. 1989). Values of ReU , ReV and Reeff were not evaluated in the older experiments (Castaing et al. 1989; Chavanne et al. 2001; Niemela et al. 2001; Niemela & Sreenivasan 2003a), and we are thus restricted to comparisons of Ref 0 and Rep . Looking at the cryogenic helium data shown in figure 9 in detail, we notice that the values of Pr in this range of Ra . In particular, according to the original parametrisations of the GL theory (Grossmann & Lohse 2001, 2002), they should undergo the IVl -IVu transition at Ra ∼ 1010 , see also below.
16 differences in scalings are visible for Ra < 1010 . Here, having Pr = 0.686 ± 0.007, the scaling of the frequency number Ref0 ∝ Ra0.47 (see panel b) is less steep than that of all the velocity numbers Rep ∝ Ra0.52 (panel c) and also ReU , ReV , Reeff (panel d). In contrast, for Ra > 1010 (where Pr varies from 0.7 to 10) scalings of both types of Reynolds numbers converge and are consistent with Re ∝ Ra4/9 Pr−2/3 . The prefactors ξ depend on the Re definition, ξ = 0.40 for Ref 0 , from 0.6 to 0.7 for Rep and from 0.3 to 0.4 for ReU and ReV . To capture the above changes in scaling of Re more quantitatively, we searched for an optimal fit of our data by two power laws of the form Re ∝ Raζ P r−2/3 , where we assumed a simple power law dependence on P r†, and twofold dependence on Ra with ζ = ζL or ζR . In case of each Reynolds number type, we split the data into two disjoint sets (denoted L, R), with the optimum splitting determined by the minimal sum of squared errors for the two fits combined. Intersection of these two fits provided us with the critical points Rac of the transition, demarcated in each panel by the dashed vertical lines. The “error of the critical point”, or better the “transitional interval”, was obtained by similar intersections of alternative fits, corresponding to bounds of the 95%confidence level intervals for the respective fit parameters. The results are indicated by vertical lines in Fig. 9 and summarized in Table 1 together with the exponents ζ = ζL and ζ = ζR . We can see that the interval for Rep in panel (c) is very wide, reflecting the not too pronounced change of scaling seen, when assuming the Taylor hypothesis of frozen fluctuations. In contrast, the Re numbers determined by the more advanced elliptic approximation show a much sharper transition, with the interval corresponding to Reeff spanning less than a decade, which is comparable to the interval obtained by the same method for the theoretical curve of the GL theory [the original Grossmann & Lohse (2001, 2002) fit], and slightly narrower than the interval for Ref 0 , which spans about two decades in Ra. The transitional intervals determined here are affected by both the experimental errors of our data, as well as fundamentally by the character of the Re(Ra) dependencies deviating from pure power laws Grossmann & Lohse (2000, 2001, 2002, 2004). Within the precision of the current data, the critical points of the different Re numbers are mutually compatible, and we conclude that the transition takes place roughly between Ra ∼ 1010 − 1011 . In general, the Prandtl number in cryogenic experiments is fairly constant (0.7 < Pr < 1) up to Ra ∼ 1012 (see figure 9, panel a). We can notice a very good agreement between Ref 0 (panel b) and Rep (panel c) from our experiment with the data of Niemela et al. (2001) and Chavanne et al. (2001), especially within the region of nearly constant Pr ≃ 0.7, up to Ra ∼ 1011 . The comparison of our (cryogenic helium) and G¨ottingen data (obtained with SF6 in a considerably larger cell at the same absolute distance of the sensors from the side wall) on ReU , ReV and Reeff in figure 9 (panel d) shows some differences: We see that over the Ra domain of overlap, the G¨ottingen data lay by about 40 % higher, which is in qualitative agreement with the fact that our sensors are azimuthally shifted from the LSC plane, see § 3.2. Additionally, the G¨ottingen data display distinctly different scaling of ReU and ReV , while our data seem to show similar scaling for both of these quantities. We do not see a clear explanation, and apparently, further dedicated experiments are needed to fully resolve this issue. Let us now compare the experimental Re(Ra) dependencies with those obtained from † The Re ∝ P r−2/3 scaling was chosen in accord with the scaling determined from our data at Ra > 1011 within the experimental Ra − P r space for the frequency Reynolds number, yielding Ref 0 ∝ Ra0.437 × P r−0.675 , as well as applying the GL model (Grossmann & Lohse 2001, 2002; Stevens et al. 2013) to our data, yielding similarly Re ∝ Ra0.442 × P r−0.666 .
Reynolds number scaling in cryogenic turbulent Rayleigh-B´enard convection
Ref 0 Rep ReU ReV Reeff ReGL2002
Rac
log(Rac )
ζL
ζR
3.6 × 1010 7.2 × 109 1.5 × 1010 2.8 × 1010 2.5 × 1010 2.9 × 1010
10.56 ± 1.01 9.86 ± 1.75 10.19 ± 0.51 10.44 ± 0.17 10.40 ± 0.25 10.46 ± 0.44
0.47 ± 0.01 0.52 ± 0.03 0.59 ± 0.05 0.56 ± 0.03 0.58 ± 0.03 0.447 ± 0.003
0.44 ± 0.01 0.46 ± 0.02 0.44 ± 0.05 0.44 ± 0.03 0.43 ± 0.03 0.441 ± 0.003
17
Table 1. Transitional values of Rac for different types of Reynolds numbers with the scaling exponents ζ = ζL corresponding to Ra < Rac and ζ = ζR corresponding to Ra > Rac . The Prandtl number dependence is assumed to be Re ∝ P r−2/3 (see text).
the theory of Grossmann and Lohse. The calculations for our values of Ra, Pr using both the original fit of the GL model (Grossmann & Lohse 2001, 2002) and the updated version (Stevens et al. 2013) are shown in left panel of figure 10 together with our data for Reeff . We can immediately notice that both theoretical curves, in contrast to the data, show only a relatively weak dependence on Ra. Zooming into the theoretical dependencies (right panel), different behaviors of the two curves can be seen: the GL model in the original parametrisation shows the IVl −IVu (a crossover between the viscous and thermal boundary layers) transition at Ra ∼ 1010 − 1011 (cf. Tab. 1) accompanied by a change of Re(Ra) scaling from a steeper to a flatter, qualitatively corresponding to the data, however being much less pronounced. On the other hand, the updated parametrisation shows no such transition, as the helium data are predicted here to lie fully in the bulk-dominated IVu region in the range of Ra; Pr considered in our experiment. In this restricted sense, the original fit may be considered more appropriate for the helium data (which are mutually consistent, as we can see) than the updated version. The fact that the experimental data often display more sudden transition effects than do the GL predictions is known, as pointed out in Ahlers, Grossmann & Lohse (2009, figure 4 and the concluding §IX). Our current data provide a further evidence that this might be due to the dynamics of the LSC, which is characterized within the GL model only by a single Reynolds number, associated typically with the mean circulation velocity. In particular, as suggested by our measured PDF skewness dependencies on Ra, the change in LSC shape from an ellipse (impacting the top/bottom plates predominantly on one side) to a squarish shape (affecting a more extended area of the plates) may induce a change in the “effective area of contact” between the boundary layers and the LSC flow, and thus a more steep Re(Ra) dependence below the saturation point (seen here at Ra ∼ 1010 ). Figure 11 shows for a comparison the data on the frequency Reynolds numbers Ref 0 from experiments with working fluids of higher Prandtl number values, 3.3 < Pr < 5.7 (Brown et al. 2007; Qiu & Tong 2001a; Sun, Xia & Tong 2005a). Brown et al. (2007) observed a sharp transition, from Ref 0 ∝Ra4/9 Pr−2/3 , to a regime with higher Rascaling exponent; their findings are consistent with the results of water experiments conducted in the other laboratories. This result, obtained for higher Pr sharply contrasts with the scaling observed at Pr ≃ 0.7 in cryogenic experiments, where, with increasing Ra, a steeper scaling transits to the scaling Ref0 ∝Ra4/9 Pr−2/3 . As well known, such qualitatively different type of transition, as compared to the helium gas, is indeed captured by the GL theory (Grossmann & Lohse 2000, 2001, 2002, 2004) for the Pr
18 values corresponding to water. This transition, called IIu − IVu , is characterised by a crossover from the boundary-layer-dominated to the bulk-dominated thermal dissipation. As discussed in Grossmann & Lohse (2002), the theoretical Re(Ra) dependence (not shown here) displays a transition between a flatter dependence for Ra below 1010 to a steeper one above 1010 (opposite to the helium case), qualitatively in line with the experimental dependence shown in figure 11. Nevertheless, the difference in the respective theoretical slopes above and below the transition is smaller than in the experimental data, in analogy to the situation in helium. For water, the discrepancy between the data and the GL description is even more severe, as the approximate scalings in the experiments are steeper on both sides of the transition than in the theory (with Re ∝ Ra4/9 seen above, not below Ra ∼ 1010 ), calling again for a theoretical description incorporating more details of the LSC dynamics, or coherent structures in general. 3.4. Scaling of kinetic energy dissipation Finally, let us examine the relation between the exact formula of reduced mean energy dissipation rate, eU /(ν 3 L4 ) = (Nu − 1)Ra/Pr2 , and calculated Reynolds numbers Ref 0 , as well as Rep and ReU , ReV , Reeff . To appreciate how closely our data agree with other investigators and with the Grossmann-Lohse (GL) theory in both the most recent version (Stevens et al. 2013), as well as the original version (Grossmann & Lohse 2001, 2002), the figure 12 shows the quantity (Nu − 1)Ra/Pr2 compensated by Re3f 0 . First of all, we see that overall, the updated as well as the original GL do not describe the internally consistent cryogenic helium data at Ref0 < 20000 (i.e. at Ra < 1010 ), while for Re > 20000, the experimental and theoretical scaling exponents are consistent. This discrepancy was also observed by Brown et al. (2007) in water experiments conducted for Ra < 1011 (see crosses in figure 12); they therefore attempted to solve this discrepancy by stating that Ref0 is not the appropriate Reynolds number. Indeed, it may appear more natural to relate the single Reynolds number of the GL model to the LCS circulation, rather than to the LSC sloshing. Thus we compare in figure 13 the GL predictions with our (Nu − 1)Ra/Pr2 data plotted against both Ref 0 (left) and Reeff (right). We see that the correspondence with the theory is in both cases problematic for the low Re (and Ra). To summarize, within the range 15000 < Reeff , Ref 0 < 20000, which corresponds to about 1010 in Ra, a transition in scaling of the dissipation rate from an approximately Re5/2 to approximately Re3 (expected for domination of energy dissipation in the bulk), is observed in helium experiments, the latter being in qualitative agreement with the Grossmann-Lohse model. We note that the Nu(Ra) scaling, as discussed in Urban et al. (2012), undergoes a change at somewhat higher Ra, between 1010 and 1011 (where Pr ≈ 0.7, Ref0 ≈ 40000).
4. Summary and conclusions We performed a statistical analysis of local temperature fluctuations in an aspect ratio one cryogenic helium RBC cell in a large range 108 −1014 of Rayleigh numbers Ra, aimed at determining the Reynolds numbers characterizing the large scale circulation (LSC) dynamics. From the temperature fluctuation PDFs and power spectra, we determined the mean wind/LSC orientation and its sloshing mode frequency. Further, we evaluated the signal auto-correlations AC(τ ) at each individual sensor and the cross-correlations CC(τ, x) between the pairs of leading and trailing sensors on each side of the cell. From the former one-point measurements, we determined the sloshing-mode frequency Reynolds numbers Ref 0 , while from the latter two-point measurements, we determined
Reynolds number scaling in cryogenic turbulent Rayleigh-B´enard convection
19
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Figure 12. Comparison of (N u − 1)Ra/P r between our data (the same as in figure 13 left) and other experiments, plotted against the frequency Reynolds number Ref 0 . Solid blue squares correspond to the cryogenic helium data of Niemela et al. (2001); Niemela & Sreenivasan (2003a). Crosses are data from water experiment by Brown et al. (2007). The dashed line represents the GL model fit by Stevens et al. (2013). 10
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Figure 13. Values of reduced mean energy dissipation rate, (N u − 1)Ra/P r2 , in dependence on frequency based Reynolds number Ref0 (left) and the “elliptic” number Reeff (right). The solid line represents the original fit of the Grossmann-Lohse model (Grossmann & Lohse 2001, 2002), while the dashed line is the updated fit by Stevens et al. (2013).
Rep (based on Taylor frozen flow hypothesis) and ReU , ReV , Reeff (based on the elliptic approximation of the space-time correlation function). The first main result of this study is an experimental determination and discussion of these various Reynolds numbers: (i) we address the Re(Ra) dependencies, pointing out a transition in the dynamics of LSC at Ra ≈ 1010 − 1011 , and (ii) we discuss their relation to the measurements of the Nusselt numbers, obtained within the same experiments [see Urban et al. (2011, 2012, 2014)]. Further, we compare our data with other available high Ra experiments. We find that both the one-point (Ref0 ) and two-point Reynolds numbers (Rep in particular), evaluated in the range 108 < Ra < 1014 , agree over this entire range with the Γ = 1 results of the RBC experiment conducted by Niemela et al. (2001). Moreover, Ref0 roughly agrees also with the result of Castaing et al. (1989) obtained in a smaller container (Γ = 1, L=8.7 cm). For Rep , we also used the data from a Γ = 1/2 experiment published by Chavanne et al. (2001). Such a comparison is meaningful, considering the fact that the scaling of Nusselt number observed in numerous Γ = 1
20 and Γ = 1/2 experiments is similar (Stevens et al. 2011). Close similarity of Reynolds number Rep scaling in our experiment and those of Niemela et al. (2001) and Chavanne et al. (2001) is found. The second main result of this study is an observation of a transition between two types of scaling of LSC characteristics around Ra ∼ 1010 − 1011 and Pr ∼ 0.7. Signatures of the transition are: (i) Scaling of Reynolds numbers varies from steeper to less steep Re ∝ Ra4/9 Pr−2/3 scaling. (ii) Mean energy dissipation rate (N u − 1)Ra/P r2 changes sharply from Re5/2 to Re3 . (iii) Skewness of temperature PDFs for the trailing sensors increases with Ra and saturates beyond Ra ≈ 1010 , while for the leading sensors, this quantity is nearly constant throughout the entire studied range 108 − 1014 of Ra. In contrast, the PDF flatness M4 data would require larger statistics to be conclusive about the transition. The transition observed in the PDF skewness suggests a change in geometry of the LSC flow from an ellipse positioned diagonally across the cell to a more squarish shape, in line with the suggestion of Niemela & Sreenivasan (2003b) based on helium experiments and also with the direct PIV visualisation in a water experiment by Sun, Xia & Tong (2005a). From the theoretical perspective, let us note that a qualitatively similar change in Re scaling (i.e. steeper-to-flatter dependence with increasing Ra) at Ra ∼ 1010 − 1011 follows from the GL model, if the original (Grossmann & Lohse 2001, 2002) parametrisation is applied to our measured values of Pr and Ra. In particular, the change corresponds to the IVl -IVu transition related to the crossover between the thermal and viscous boundary layers. However, the experimental effect that we observe here is much more pronounced. Interestingly, an analogous situation can be noticed for the transition in water at Ra ∼ 1010 , where the experimental effect is again considerably more pronounced than the theory prediction. The GL theory predicts here a change from a flatter to a steeper Reynolds number scaling (notice that the “sense” of the change is opposite to helium) associated with the IIu -IVu transition, related to crossover from boundary-layerdominated to bulk-dominated thermal energy dissipation. The fact that the experimentally observed transition signatures are considerably more pronounced than in the established theory (see also Ahlers, Grossmann & Lohse (2009)) in our opinion indicates that the LSC dynamics may considerably modify the effects at the interface of the bulk and the boundary layers. The situation calls for a refined theoretical model combining such LSC dynamics with a GL-like description and further dedicated experiments. We hope that the presented new experimental data and their analysis and comparison with other experiments can be regarded as a subsequent step to deeper understanding of thermally driven turbulent flows.
Acknowledgments We thank many colleagues for stimulating discussions, especially S. Babuin, P. Hanzelka, X. He, M. Jackson, R. du Puits, P.-E. Roche, J. Salort, D. Schmoranzer and K. R. Sreenivasan. We acknowledge the support of this research by the Czech Science Foundation under the project GA17-03572S. M. Macek acknowledges the support by the MEYS CR project LO1212. REFERENCES
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