J. Fluid Mech. (2013), . 725, pp. doi:10.1017/jfm.2013.179 ...

J. Fluid Mech. (2013), vol. 725, pp. 664–680. c© Cambridge University Press 2013 664doi:10.1017/jfm.2013.179

Reynolds-number measurements forlow-Prandtl-number turbulent convection of

large-aspect-ratio samples

James Hogg and Guenter Ahlers†

Department of Physics and ITST, University of California, Santa Barbara, CA 93106, USA

(Received 30 December 2012; revised 2 March 2013; accepted 1 April 2013)

We present experimental results for the Reynolds number ReU based on the horizontalmean-flow velocity U and for ReV based on the root-mean-square horizontalfluctuation velocity V for turbulent Rayleigh–Benard convection in a cylindricalsample of aspect ratio Γ = 10.9 over the Prandtl number range 0.18 6 Pr 6 0.88.The results were derived from space–time cross-correlation functions of shadowgraphimages, using the elliptic approximation of He & Zhang (Phys. Rev. E, vol. 73,2006, 055303). The data cover the Rayleigh number range from 3 × 105 to 2 × 107.We find that ReU is nearly two orders of magnitude smaller than the values givenby the Grossmann–Lohse (GL) model (Grossmann & Lohse, Phys. Rev. E, vol. 66,2002, 016305) for Γ = 1.00 and attribute this difference to averaging caused bylateral random diffusion of the large-scale circulation cells in large-Γ samples. Forthe fluctuations we found ReV = R0Pr

αRaη, with R0 = 0.31, α = −0.53 ± 0.11 andη = 0.45 ± 0.03. That result agrees well with the GL model. The close agreement ofthe coefficient R0 must be regarded as a coincidence because the GL model was forΓ = 1.00 and for a mean-flow velocity U.

Key words: Benard convection, convection, turbulent flows

1. IntroductionConvection in a fluid confined between parallel horizontal plates and heated from

below (Rayleigh–Benard convection (RBC)) occurs over a wide range of the Rayleighnumber Ra = αg1TL3/κν and the Prandtl number Pr = ν/κ . Here α is the isobaricthermal expansion, g is the gravitational acceleration, 1T is the temperature differencebetween the plates, L is the spacing between the plates, κ is the thermal diffusivityand ν is the kinematic viscosity. Physically realizable studies use cells with finitelateral extent. Thus, the aspect ratio Γ ≡ D/L of the sample is an additional relevantparameter. Here D is the lateral size of the sample, for instance the diameter of acylindrical cell.

Much of the previous work on RBC can be broadly divided into two areas:pattern formation (Cross & Hohenberg 1993; Bodenschatz, Pesch & Ahlers 2000),characterized by low Ra (. 105) and large Γ (& 5), and turbulent convection (Ahlers2009; Ahlers, Grossmann & Lohse 2009; Lohse & Xia 2010), characterized by

† Email address for correspondence: [email protected]

mailto:[email protected]

Reynolds-number measurements for turbulent convection 665

higher Ra (up to 1017 Niemela et al. (2000), Chavanne et al. (2001) and He et al.(2012)) and generally small Γ (O(1) or less). Increasing L helps to reach higherRayleigh numbers, as Ra ∼ L3, but practical constraints limit the lateral extent ofthe cell, hence the prevalence of smaller aspect ratios in high-Rayleigh-number work.In the present investigation we explore the relatively large Γ ' 11, but still reachrelatively large values of Ra (up to 2×107) where the flow is turbulent. We accomplishthis by using compressed gases at elevated pressures.

The turbulent fluid flow in small-Γ cells for Ra & 106 is highly fluctuating, but inthe time average tends to be organized as a single convection roll or possibly tworolls with one located above the other (see Xi & Xia 2008, Weiss & Ahlers 2011 andreferences therein). The strength and azimuthal diffusion of this large-scale circulation(LSC) display a rich dynamics (Brown, Nikolaenko & Ahlers 2005; Brown & Ahlers2008). A dimensionless parameter related to the fluid flow is the Reynolds number

ReU = UL

ν. (1.1)

Here U is a time-averaged characteristic velocity, for instance the maximum velocityfound in the LSC which occurs near (but not too near) the sidewalls or the plates(Qiu & Tong 2001). Good agreement has been reported between some (but not all)experimental measurements of variously defined Reynolds numbers for Pr ≈ 4 andlarger for some Ra ranges and Γ near one (see, for instance, Qiu & Tong 2001; Lamet al. 2002; Brown, Funfschilling & Ahlers 2007; Xia 2007) and a theoretical modelof Grossmann & Lohse (2002) (the GL model; we shall refer to the Reynolds numbersgiven by that model as ReGL). We note that the GL model contains a parameterwhich was adjusted so that the overall magnitude of ReGL agreed with experimentalresults for ReU measured in a sample with Γ = 1.00 and Pr = 5.4 (Qiu & Tong2001). Although the model is expected to give the right dependence of Re on Pr andRa which is expected to be the same for all Γ , it should not be expected that themagnitude of ReGL will agree with data for very different Γ . Nor can it be expected toreproduce the magnitude of Reynolds numbers based on differently defined velocities,such as the root-mean-square (r.m.s.) fluctuation velocity V for instance or the verticalaverage over the sample that is measured by the shadowgraph method (see § 2.3below).

Also of interest is the flow behaviour for systems with larger Γ . There the influenceof the walls is less and the experimental systems are better analogues to variousastrophysical and geophysical phenomena (see, for instance, Howard & LaBonte1980; Rahmstorf 2000). For large-Γ systems the LSC takes the form of many cellsdistributed over the horizontal plane (Hartlep, Tilgner & Busse 2005; Bailon-Cuba,Emran & Schumacher 2010; Bosbach, Weiss & Ahlers 2012). Under the influence ofvigorous turbulent fluctuations these cells experience a random lateral dynamics. Thus,we would expect that, in the time average, the value of U measured at any fixedlocation in the sample interior will vanish or become quite small. Here it is interestingto note that the time averages obtained by direct numerical simulation (DNS) stillrevealed a large-scale flow structure (Bailon-Cuba et al. 2010). This is so becausethe time average used in the DNS was sufficiently short to permit the structures tosurvive. We find that the images, averaged over the duration of our experiments, revealvirtually no structure for our sample with Γ ' 11, and that the values for ReU of theaverages are about two orders of magnitude smaller than ReGL. However, a larger r.m.s.fluctuation velocity V characteristic of the turbulent state remains, with an associated

666 J. Hogg and G. Ahlers

Reynolds number

ReV = VL

ν. (1.2)

In the present paper we focus primarily on ReV , where V is a horizontal componentof the r.m.s. velocity fluctuations. Our results for this quantity did not differ verymuch from ReGL. We were surprised by this result because, as mentioned above, theamplitude R0 of the power law Re = R0Ra

η is expected to depend on Γ and on theparticular definition of Re that is investigated. Thus, the approximate agreement of theamplitude of ReGL with our result for the amplitude of ReV for a sample with Γ � 1must be regarded as a coincidence. As implied above, the agreement of η with the GLmodel is of course expected.

Another feature of our work is that we study the nature of the flow for smallerPrandtl numbers than are usually encountered. We used gas mixtures (Liu & Ahlers1997) to access Pr as small as 0.18, much lower than accessible with commonly usedfluids such as pure gases at room temperature (0.67 6 Pr . 1) and water or otherliquids (Pr & 2). Liquids metals have Pr . 0.07 (Lappa 2010), but are not well suitedfor flow visualization. No pure fluids have Pr between about 0.07 and 0.67, so gasmixtures are the only means we know of to access part of this Pr range.

Our measurements of ReU(Ra,Pr) and ReV(Ra,Pr) for 0.18 < Pr < 0.9 andΓ = 10.9 were made by applying the elliptic approximation (EA) of He and Zhang(He & Zhang 2006; Zhao & He 2009) to space–time cross-correlation functions ofshadowgraph images. This technique had not previously been applied to images, asfar as we know. Section 2 describes the experimental apparatus, the working fluids,the shadowgraph images, and the EA applied to the image correlation functionsto obtain the mean velocity U (which, as expected, turns out to be small) andthe horizontal component of the r.m.s. fluctuation velocity V . Section 3 presents theReynolds numbers for the velocities obtained from the image correlations, and theRayleigh- and Prandtl-number dependences of these Reynolds numbers. Finally, § 4summarizes our results. An Appendix discusses the Ra range over which flows may beconsidered turbulent as a function of Pr.

2. Experiment and data analysis2.1. Apparatus

The apparatus was used before and has been described in detail (Xu, Bajaj & Ahlers2000; Ahlers & Xu 2001; Nikolaenko et al. 2005; Funfschilling et al. 2005; Zhong,Funfschilling & Ahlers 2009). For the present work a cylindrical cell had a heightL = 0.93 cm and a diameter D = 10.16 cm. The bottom plate was composed of asapphire disk 0.318 cm thick which was epoxied on top of a 3 cm thick copper plate.The top of the sapphire had a surface coating of gold to produce a mirror finish. Thepower to heat the bottom plate came from a film heater glued to the bottom of thecopper plate. The top plate was a sapphire disk with thickness 1.91 cm, which wascooled by water jets directed onto its top surface and which permitted optical accessto the sample. The walls of the cell were formed by a steel sidewall, and a spacerring which set the distance between the top and bottom plates. Gas entered the cellthrough a small hole in the bottom plate, which connected the cell to a gas-supplymanifold via a steel capillary. The cell was in a can filled with air and foam inside atemperature-controlled recirculating water bath. The water in the bath was regulated atthe desired top plate temperature using a calibrated thermistor in the bath near the top


plate and a heater in a feedback loop. We used a shadowgraph apparatus (de Bruynet al. 1996) for flow visualization.

2.2. Working fluidsWe used nitrogen, sulfur hexafluoride, hydrogen–xenon mixtures or helium–sulphurhexafluoride mixtures as the fluid. The fluid pressures were in the range from 10 to 32bars. The pressure was held constant to better than 0.1 % over the course of a givenrun.

The various properties of the gases used were calculated with programs based onexperimental and theoretical results from the literature (Hilsenrath et al. 1960; Gracki,Flynn & Ross 1969; Braker & Mossman 1974; Oda, Uematsu & Watanabe 1983;Hoogland, van den Berg & Trappeniers 1985; Kestin & Imaishi 1985; Boushehri et al.1987; Uribe, Mason & Kestin 1990; Liu & Ahlers 1997). The results from the pure-gas programs were in good agreement with published values from NIST’s REFPROPprogram (Lemmon, Huber & McLinden 2007). The properties of the gas mixtures usedhere were calculated with programs from Liu & Ahlers (1997). For H2–Xe mixturesthe quoted uncertainties, based on comparison with available published experimentalresults, were 1 % for the density, 2 % for the viscosity and thermal conductivity.However, the measured thermal conductivity found by Liu & Ahlers (1997) was up to5 % larger than the calculated value, so the results presented therein used the measuredthermal conductivity for H2–Xe. For the present results, we multiplied the calculatedthermal conductivity by 1.047 to bring it in line with the measured results. For He–SF6

mixtures the quoted uncertainties were estimated to be 2–7 % for the density and 10 %for the thermal conductivity and heat capacity. The measured thermal conductivityfound by Liu & Ahlers (1997) agreed within 3 % or so with the calculated value, sohere we used the calculated values for He–SF6 unchanged.

2.3. Image acquisition and analysisThe shadowgraph tower (de Bruyn et al. 1996) contained a monochrome Retiga 1300(QImaging) CCD camera. The camera was capable of 1280 × 1024 pixels and 12-bitpixel intensities, though to increase the frame rate we used smaller images with 8-bitpixel intensities. We used square images approximately centred in the cell. The imagesize was either 240× 240 or 16× 16 pixels, with frame rates between about 13 and 39images s−1 and a spatial resolution of 1/d = 2.5 to 4.3 pixels mm−1 (d/L = 0.043 to0.025). The results did not depend significantly on the image resolution or frame rate.Either 1024 or 10 240 images were obtained for each run. Background division (deBruyn et al. 1996) was applied to the images to account for non-uniform illuminationof the cell. The background image was the average of all of the images for a givenrun. Two typical examples of divided and rescaled images are shown in figure 1. Theseimages were taken with higher spatial resolution than used for the analysis in order toimprove their visual appearance.

The primary results, presented below in §§ 3.1 and 3.2, are for the background-divided images with no other image processing applied. They contained both lightand dark structures, which are expected to correspond to regions of cold and hotfluid, respectively. However, in § 3.3 we also considered the cases where the imageswere processed to only include either the light or the dark structures, by selectinga threshold value based on the histogram H(I) of pixel intensities I(x, y). Eight-bitimages were used, so the intensity values varied between 0 and the maximum intensityImax in 255 increments. To choose the threshold values, the location Ipeak of themaximum of H(I) was determined first. Then the threshold intensity Ithresh was chosen


(a) (b)

FIGURE 1. Example shadowgraph images: (a) 11 mm square in the centre of the cell, withPr = 0.18, Ra = 1 × 106, 11 pixels mm−1; (b) 94 mm square, with Pr = 0.88, Ra = 2 × 107,5 pixels mm−1. Corresponding movies can be found in supplementary material submitted withthis paper as Movie 1 and Movie 2 available at dx.doi.org/10.1017/jfm.2013.179

such that the area under the histogram between Ipeak and Ithresh equaled some specifiedfraction F of the total area between Ipeak and either Imax or Imin(= 0) as appropriate:

F =

∫ Ithresh

Ipeak

H(I) dI∫ Imax,min

Ipeak

H(I) dI

. (2.1)

We used F = 0.85, although similar values gave essentially equivalent results. Thethresholded image had all points above (below) the threshold value set equal to Ithresh,and the points below (above) the threshold unchanged.

The shadowgraph intensity I(x, y) is proportional to a vertical average of thehorizontal Laplacian of the refractive index field (Rasenat et al. 1989; de Bruynet al. 1996; Trainoff & Cannell 2002); the refractive index in turn is a linear functionof the temperature. The temperature is a passive scalar and follows the local floweverywhere in the cell except near the thermal boundary layers (see, for instance, He,Ching & Tong 2011 and references therein). Thus, temperature measurements yieldthe same correlation functions as velocity measurements. From very extensive workin the field of pattern formation and chaos (see, for instance, de Bruyn et al. 1996;Bodenschatz et al. 2000) it is well established that the shadowgraph images are anexcellent representation of the corresponding temperature and/or velocity patterns. It isthus reasonable to assume that the shadowgraph images will also behave as a passivescalar and yield the same correlation function as the velocity.

One difference between the shadowgraph method and local temperaturemeasurements remains. The shadowgraph method averages vertically over the sampleheight while previous temperature measurements (He, He & Tong 2010; He & Tong2011; He et al. 2012) have been local. However, lateral displacements of structuressurvive the vertical averaging of the shadowgraph method. Although contributionsfrom all vertical levels enter into the average (see, for instance, figure 1), large

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contributions come from hot and cold plumes while these reside close to the bottomand top plates, respectively, and move laterally with the prevailing flows. To the bestof the authors’ knowledge there have been no precise measurements of the distributionfunction of the vertical location of plumes during that stage of their existence, butthey are likely to reside in the bulk rather than in the boundary layers where theywere born. We believe that this is so because they are free to move with any large-scale flow, as can be seen in the movies submitted with this paper as supplementarymaterial, and as has been amply demonstrated before (see, for instance, Funfschilling,Brown & Ahlers 2008). If the plumes resided in the viscous boundary layer, then onewould expect their lateral motion to be inhibited. In conclusion, we expect the plumesand other excitations to be residing primarily in the bulk of the sample where theymay follow the flow and act as passive scalars. Thus, the shadowgraph correlationfunction CI(r, τ ), temperature correlation function CT(r, τ ), and velocity correlationfunction Cv(r, τ ) should have the same behaviour (He et al. 2010). We will use theminterchangeably below. Experimental support for the validity of this approach comesfrom the agreement between the Reynolds numbers deduced from three differentmethods of image processing which emphasize different vertical portions of the sample(see figures 1 and 7).

The shadowgraph-intensity space–time cross-correlation function is given by

CI(r, τ )= 〈δI(x+ r, t + τ) δI (x, t)〉t,x(σI)1 (σI)2

(2.2)

where 〈· · ·〉t,x denotes a space and time average, δI = I − 〈I〉t, and (σI)1 and (σI)2are the standard deviations of δI at x and x + r, respectively. Assuming translationalinvariance of the sample and sufficiently long time series to yield adequate statistics,one expects (σI)1 and (σI)2 to be equal to each other, but in practice each wasevaluated separately at its location.

The correlation function C(r, τ ) for a given run was calculated by averaging theindividual Cj(r, τ ), j = 1, . . . ,NI , for each of the NI = 1024 or 10 240 images fromthat run. The time shift τ was done by comparing the ith image with the (i + τ f )thimage in the sequence, where f is the frame rate in images per second. The spatialshift r was done by comparing a given pixel at (x1, y1) in each image with pixels at(x1+r, y1) and (x1, y1+r) in a second image, and repeating the procedure for all pixelsin each image. The results for all pixels in all images in a particular run were averagedto obtain a single value of C(r, τ ) for each r and τ . Thus our analysis was in a senseone-dimensional, averaging the components Vx and Vy of V in the x and y directions.An analysis using only one direction showed that Vx and Vy were equal to each otherwithin statistical errors; using their average yielded more precise results. Both r andτ were restricted to some range, generally ±5 or 15 images or pixels. Typically thiscorresponded to a physical range of up to ±6 mm for r and up to ±1.2 s for τ . Theshape of C(r, τ ) closest to the peak at r = 0, τ = 0 was most important, and C(r, τ )decreased quickly from the peak, so a wider range of r and/or τ was not necessary.

2.4. Elliptic approximationWe measured the Reynolds number using space and time cross-correlations ofsuccessive shadowgraph images. The turbulent flow and fluctuations in the cell causedthe visible structures to undergo both spatial translation and deformation. We used therecently developed EA (He & Zhang 2006; Zhao & He 2009) to analyse this dynamics.The EA consists of a systematic Taylor-series expansion of the space–time correlationfunction to second order. This expansion can be used to determine both the mean


velocity U and the r.m.s. fluctuation velocity V . This method was previously appliedto RBC with Pr near 5 using local temperature measurements for samples with Γ = 1and Ra near 1010 (He et al. 2010; He & Tong 2011) and with Pr ' 0.8 and Γ = 1/2for Ra up to 1015 (He et al. 2012). It was also used by Zhou et al. (2011) to analyseparticle-image velocimetry data for a Γ = 1.00 sample and Pr ' 5 for Ra over therange from 6 × 109 to 1011. Here we apply the EA to a large array of image pixels,instead of to a small number of discrete temperature sensors, over a range of Pr from0.18 to 0.9 and Ra from 3× 105 to 2× 107. Details of the computation of C(r, τ ) weregiven in the previous section.

The Taylor-series expansion of C(r, τ ) has the form

C(r, τ )= C(0, 0)+ ∂C(0, 0)∂r

r + ∂C(0, 0)∂τ

τ

+∂2C(0, 0)∂r∂τ

rτ + 12

[∂2C(0, 0)∂r2

r2 + ∂2C(0, 0)∂τ 2

τ 2

]+ · · · . (2.3)

For a spatially homogeneous and statistically stationary sample the two first derivativesvanish (He & Zhang 2006). The EA consists of truncating (2.3) at second order. Thissecond-order expansion can be written as He & Zhang (2006) and Zhao & He (2009)

C(r, τ )= C(0, 0)+ 12∂2C(0, 0)∂r2

r2E (2.4)

where

r2E = (r − Uτ)2+V2τ 2 (2.5)

with

U =−∂2C(0, 0)∂r∂τ

[∂2C(0, 0)∂r2

]−1

(2.6)

and

V2 = ∂2C(0, 0)∂τ 2

[∂2C(0, 0)∂r2

]−1

− U2. (2.7)

The contribution U is the time-averaged mean-flow velocity which for our caseis expected to be small or zero, and V is dominated by a contribution from ar.m.s. fluctuation velocity v0 with v2

0 = 2∫

E(k) dk where E(k) is the energy spectrumof the velocity (Zhao & He 2009). A small additional contribution to V is proportionalto the local shear times the Taylor length microscale (Zhao & He 2009), which in ourcase is also expected to be negligible or absent. Thus, we expect our measurements ofV to be dominated by the velocity fluctuations.

2.5. Data analysis based on the EAOne way to determine U and V is to use the derivatives of rE(r, τ ). One sees from(2.4) that a constant value of the length rE corresponds to a constant C(r, τ ). Equation(2.5) shows that a constant rE (and, thus, a constant C(r, τ )) corresponds to an ellipsein the r–τ plane. This is illustrated in figure 2(a). The values rp and τp can thusbe obtained from the extrema of rE, i.e. by setting the partial derivatives of rE(r, τ )(equation (2.5)) to zero (He et al. 2010; Zhou et al. 2011):

∂rE

∂r= 0−→ rp = Uτ (2.8)


(a) (b) (c)

FIGURE 2. (Colour online) (a) Sketch of iso-correlation contours. For a given time shiftτ0, the maximum of the correlation function C(r, τ0) occurs at a spatial shift rp(τ0). Thesame procedure (not shown) for a given spatial shift r0 gives the time shift τp(r0) wherethe maximum of C(r0, τ ) occurs. (b) Sketch illustrating the measurement of U and V fromC(r0, τ ) using the two-point method in the case where U is not negligible compared to V .(c) Sketch illustrating the measurement of V from C(r0, τ ) using the two-point method in thecase where U = 0 or negligibly small.

and

∂rE

∂τ= 0−→ τp =

[U

U2 + V2

]r. (2.9)

Values for rp(τ ) are obtained by fitting the Gaussian function

C(r, τ0)= C0 exp[(r − rp)2 /(2σ 2)] + C1 (2.10)

to values of C(r, τ0) separately for each of all available constant τ0. Similarly, thefunction

C(r0, τ )= C2 exp[(τ − τp)2 /(2σ 2)] + C3 (2.11)

is fit to values of C(r0, τ ) separately at each available constant r0 to obtain τp(r).Straight-line fits of (2.8) or (2.9) to these data for rp(τ ) and τp(r) then yield thedesired results for U and U/(U2 + V2). This method works well when both U andV are of comparable size. When, as in our case, U � V , we can still determineU from (2.8), albeit not with very high precision because it is small. However, thedetermination of V from (2.9) leads to errors that are larger than desired. Thus, for thedetermination of V we turn to the slightly different method described next.

Another way to determine U and V utilizes measurements at only two points inspace (He et al. 2010; He & Tong 2011). This method usually was employed whenmeasurements were available only at two points, but it has merit also in our case. It isillustrated in figure 2(b) for the case where both U and V are of significant size, andin figure 2(c) for the case where U = 0. Its implementation is illustrated in figure 3.As illustrated in figure 2(b), one has

C(r0, 0)= C(0, τ0) (2.12)

where the two points (r0, 0) and (0, τ0) lie on the same elliptic contour. As seenin figure 3, τ0 is given by the time displacement between the time auto-correlationfunction C(0, τ ) when it has the same value as the cross-correlation functionC(r0, τ = 0). This is illustrated by the double arrow in the figure. Having found


0

0.5

1.0

0–0.1 0.1

FIGURE 3. (Colour online) The time auto-correlation function C(0, τ ) (solid circles) and thespace–time cross-correlation function C(r0, τ ) for a spatial displacement r0 = 0.35 mm (solidsquares, red online). The solid and dashed lines are fits of Gaussian functions, (2.11), to thethree points nearest the maxima. For C(0, τ ) τp = 0 was fixed, while for C(r0, τ ) τp wasadjusted in the fit. The data are for H2–Xe, Ra= 3× 105, Pr = 0.18.

τ0, one can form the effective velocity

Veff = r0/τ0. (2.13)

This procedure can be followed for any displacement r0 for which data may beavailable. In our case, one can use any small integer n (in practice only n = 1 or 2)times the distance d between adjacent pixels, and the value of Veff is expected to beindependent of n.

In practice, we fit (2.11) to C(r0, τ ) versus τ at each measured r0. An example wasshown above in figure 3. The fits were done only to those points close to the peak inorder to avoid undue influence from experimental noise in the tails of the distributions.The time increment τ0 between the fits to C(0, τ ) and C(r0, τ ) were then determined.We used r0 = d where d is the spacing between pixels, but the results for V were notsignificantly different when r0 = 2d was used, provided a fast enough frame rate hadbeen used so that C(r0, τ ) decayed over several points away from the peak on the τaxis.

In order to find the meaning of Veff , we consider (2.5). At the point (r0, 0) it givesrE = r0. Similarly, at (0, τ0) (2.5) gives rE =

√U2 + V2τ0. Equating the two evaluations

of rE and dividing by τ0 leads to (He et al. 2010)

Veff = V ∗√

1+ (U/V)2. (2.14)

As discussed above and shown below in figure 4, for our application we found thatU is typically about two orders of magnitude smaller than V . Thus, the part (U/V)2 in(2.14) is much smaller than our experimental errors, and it is well justified to equateVeff with V in our case.


106 107

Ra

10–1

100

101

102

103

FIGURE 4. (Colour online) The Reynolds number of the LSC ReU as a function of Ra:circles (purple online), Pr = 0.18; squares (orange online), Pr = 0.54; diamonds (red online),Pr = 0.74; up-pointing triangles (green online), Pr = 0.81; down-pointing triangles (blueonline), Pr = 0.88; dotted line (purple online), ReGL for Pr = 0.18; dashed line (blue online),ReGL for Pr = 0.88; solid line, a power law with the exponent fixed at the value 0.45 of theGL model and the prefactor fit to the data.

3. Results3.1. Results for ReU

Figure 4 shows ReU determined using (2.8) as a function of Ra on logarithmic scales.The data show considerable scatter because ReU is remarkably small and for thatreason difficult to measure with our techniques. Nonetheless they reveal a trend withRa. For comparison, the dotted and dashed lines give the GL prediction (Grossmann& Lohse 2002) for ReU for Pr = 0.18 and 0.88, respectively, based on the LSC ina Γ = 1.00 cylindrical sample. As expected, in our spatially extended sample ReU ismuch smaller. The solid line is a fit of a power law to the data, keeping the exponentat the value 0.45 appropriate for the GL model and adjusting the prefactor. The resultis that on average ReU is typically two orders of magnitude smaller than it is for aΓ = 1.00 sample (Grossmann & Lohse 2002). In principle, we might expect that ReU

would average to zero for a sufficiently large sample and over a long enough timeinterval because of the random nature of the LSC dynamics that is expected in such asample. It is not clear whether the small surviving value of ReU in our sample is dueto its finite (albeit large) lateral size or to insufficiently long time series.

3.2. Results for ReV

Figure 5 shows ReV calculated using (2.12) and (2.14) (see figure 3) for severalvalues of Pr. A significant Pr dependence is evident. For comparison we give thedotted, solid and dashed lines, which are the results of the GL model (Grossmann& Lohse 2002) for Pr = 0.18, 0.54 and 0.88, respectively. Quite unexpectedly, themodel (which is for Γ = 1.00 and ReU) comes fairly close to the data for Γ = 10.9and the fluctuation-velocity Reynolds number ReV . One would have expected the sameslope (corresponding to the same effective exponent), but the closeness also of themagnitude must be regarded as fortuitous.


Ra106 107

102

103

FIGURE 5. (Colour online) The Reynolds number ReV of the r.m.s. fluctuations as a functionof Ra: circles (purple online), Pr = 0.18; squares (orange online), Pr = 0.54; diamonds (redonline), Pr = 0.74; up-pointing triangles (green online), Pr = 0.81; down-pointing triangles(blue online), Pr = 0.88; dotted line (purple online), GL model for Pr = 0.18; solid line(orange online), GL model for Pr = 0.54; dashed line (blue online), GL model for Pr = 0.88.

Pr Fluid P (bar) L/ν (s cm) η

0.18 H2–Xe (50:50) 25.0 245.4 0.46±0.040.54 He–SF6 (27:73) 14.8 353.6 0.49±0.020.74 N2 32.1 167.9 0.47±0.010.81 SF6 10.5 376.5 0.42±0.04— All of above — — 0.45±0.03

TABLE 1. Exponents η from the fit of ReV = R0Raη separately to the data sets at a given

Pr . The last row is from a multivariable simultaneous fit to data for all Pr , using theequation ReV = R0Pr

αRaη. The results for the exponents may be compared with that of theGL model ηGL = 0.45 for this Ra range.

We fit the power law of the form ReV = R0Raη, where R0 and η are constants,

to the ReV results separately for each Pr . The fits were restricted to points whereRa & 3 × 105, which is the approximate criterion for the flow to be turbulent for anyof the Pr values (see the Appendix). The values of the exponent η are presented intable 1. They are in good agreement with the predicted exponent of the GL modelηGL = 0.44 to 0.45 in this Ra and Pr range.

Also shown in table 1 is the value of η obtained from a multivariable regression fitof ReV(Ra,Pr) = R0Pr

αRaη simultaneously to all of the data, including all Pr values.The fit gave ReV = (0.31 ± 0.14)Ra0.45±0.03Pr−0.53±0.11. The effective exponents of theGL model vary somewhat over the parameter ranges of our experiment. While η

remains in the range from 0.44 to 0.45, one finds that α varies from about −0.59 to−0.62. Although our α value falls below the range of the GL model, its relativelylarge statistical uncertainty overlaps that range. The experimental result for η is ingood agreement with the model.


106 107

Ra

700

100

FIGURE 6. (Colour online) Plots of ReV/Prα for Pr = 0.18 (circles, purple online),

Pr = 0.54 (squares, orange online), Pr = 0.74 (diamonds, red online), Pr = 0.81 (up-pointingtriangles, green online) and Pr = 0.88 (down-pointing triangles, blue online). The value for αcame from a multivariable fit of ReV = R0Pr

αRaη to all data which gave α = −0.53 ± 0.11and η = 0.45± 0.03. The dashed line is Re/Prα from the GL model using α =−0.62.

Figure 6 shows ReV/Prα for the ReV data and for ReGL (dashed line), using

α = −0.53 for ReV and α = −0.62 for ReGL. We see that the measured ReV pointslargely collapse onto a single line as expected if Prα indeed represents the Prdependence of ReV . ReV/Pr

α agrees reasonably with ReGL/Prα in both magnitude

and Ra dependence. We note again that there was no a priori reason to expect theagreement in magnitude because ReGL was based on U for a Γ = 1.00 sample withPr = 5.4 and not on a r.m.s. fluctuation velocity for large Γ and Pr less than one.

3.3. Reynolds number ReV near the top and the bottom of the sampleWe examined whether the Ra dependence of Re for the hot fluid near the bottomplate (relatively dark structures) differed from that of the cold fluid near the topplate (relatively bright structures). We did so by retaining only the dark or lightstructures in each image, using the image processing procedure described in § 2.3.Examples of these two types of processed images are shown in figure 7, and moviesare available as supplementary material. In figure 8 we show results for ReV for theoriginal images (solid symbols), for the light structures (cold, open blue symbols) andfor the dark structures (hot, open red symbols) in the reduced form of ReV/Ra

0.45

as a function of Ra. Although there are small systematic differences between thethree data sets, the general trends of the data with Ra are very similar. For the lightstructures a simultaneous fit of ReV/Pr

α = R0Raη with α fixed at −0.53 to the data

for all Pr yielded η = 0.46± 0.05, while the dark structures gave η = 0.44± 0.03. Weconclude that the fluctuation velocities are similar near the top and bottom plates.

4. SummaryWe have reported experimental measurements of Reynolds numbers over the range

3 × 105 . Ra . 2 × 107 and for 0.18 . Pr . 0.88 for turbulent convection in acylindrical sample of aspect ratio Γ = 10.9. The range of Pr was achieved by using


(a) (b)

FIGURE 7. Processed shadowgraph images: (a) only the dark (relatively warm) structuresnear the bottom of the sample remain; (b) only the light (relatively cold) structures near thetop of the sample remain. For both images a 94 mm square was used with a resolution of 5pixels mm−1. They are for Pr = 0.88 and Ra = 2 × 107. Corresponding movies can be foundin supplementary material submitted with this paper as Movie 3 and Movie 4.

Ra106 107105

0.2

0.5

1.0

FIGURE 8. (Colour online) The reduced Reynolds number ReV/Ra0.45 as a function of Ra

derived from the full images (no threshold, solid symbols, right image of figure 1) and fromthe thresholded light structures (open squares, blue online, right image in figure 7) and darkstructures (open circles, red online, left image in figure 7).

pure gases and binary gas mixtures. We used correlation functions based on time seriesof shadowgraph images and the EA to determine the Reynolds numbers.

The measured Reynolds number ReU was based on the mean flow velocity U. Wefound it to be very small and attribute this to the averaging of the local flow velocitydue to the random diffusion of large-scale flow cells in large-Γ samples.


The Reynolds number ReV based on the r.m.s. fluctuation velocity V had a sizecomparable with ReU predicted by the GL model (Grossmann & Lohse 2002) for asample of unit aspect ratio where there is only a single LSC cell locked in place bythe sidewalls. Both the Ra dependence and the Pr dependence found for ReV agreedwithin the experimental uncertainty with the GL prediction for ReU and Γ = 1.

DNS (Verzicco & Camussi 1999) had suggested that plumes form in the thermalboundary layers adjacent to the top and bottom plates only when Pr & 0.35. Incontrast to this, we found plumes over the entire Pr range of our measurements, withno noticeable difference in their appearance or number (see figure 1a).

AcknowledgementsWe are grateful to S. Weiss and J. Bosbach for their contributions during the early

stages of this work. G.A. is very grateful to X. He for many illuminating discussionsabout the elliptic approximation and its use to determine Reynolds numbers. Thework was supported by the U.S. National Science Foundation through grant numberDMR11-58514.

Supplementary moviesSupplementary movies are available at http://dx.doi.org/10.1017/jfm.2013.179.

Appendix. Onset of turbulent convectionAt relatively small Ra the flow field of RBC forms patterns (Bodenschatz et al.

2000). With increasing Ra the patterns become time dependent; often this timedependence becomes chaotic as Ra increases further (Krishnamurti 1970) (for adetailed example, see for instance the supplementary material provided by Bosbachet al. (2012)). At some point, with increasing Ra, the chaotic state undergoes atransition to turbulence. At least for some Γ and Pr this transition is discontinuous.For the case studied by Bosbach et al. (Pr = 0.73 and Γ = 10) the transitionoccurred at Rat ' 3 × 105. In another case studied in our laboratory (T. Sharf, M.Schreiner, J. Hogg, J.-Q. Zhong, D. Funfschilling, S. Weiss & G. Ahlers, unpublishedobservations) it was also discontinuous and found to occur at Rat = 9 × 106 forPr = 232 and Γ = 1.00.

Whether or not the state immediately above this transition can be regarded asgenuine turbulence is a matter of definition and of some debate. It has been suggestedthat a suitable criterion can be defined in terms a coherence length lcoh = 10ηK

(Grossmann & Lohse 1993; Sugiyama et al. 2007) where

ηK/L= Pr1/2 [Ra(Nu− 1)]−1/4 ∼ Ra−ζeff (A 1)

is the volume-averaged Kolmogorov length (see, for instance, Sugiyama et al. 2007).The coherence length is an estimate of the smallest size of coherent structures, oreddies, in the bulk. It has been suggested by Sugiyama et al. (2007) that the bulkis not fully turbulent until lcoh/L is as small as 0.1. This criterion is represented bythe solid line in figure 9. We believe that it is too restrictive. The solid circle in thefigure represents the measurement of the discontinuous transition from ‘patterns’ tochaotic or turbulent behaviour observed by Bosbach et al. (2012). Those authors notedthat there were no further transitions and no noticeable change of the power lawsdescribing certain features of the system as Ra increased. The dashed line in the figurewhich passes through the experimental point corresponds to lcoh/L = 0.26. We did notuse any data falling below this line in our analysis.








































Pr0.1 0.3 1.0 3.0

104

105

106

107

108

FIGURE 9. (Colour online) Estimates of the Rayleigh number Rat at the transition toturbulence as a function of Pr : solid circle (red online), experimental result for theonset of turbulence given in the supplementary material of Bosbach et al. (2012); solidline, Rat = Ra(lcoh/L = 0.1); dashed line, Rat = Ra(lcoh/L = 0.26); dash-dotted line, Rat =Ra(lcoh/L = 1). The thin vertical dotted lines represent the values of Pr of the present work.The horizontal dotted line represents the smallest value Ra= 3× 105 of Ra for the data of thispaper.

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