Velocity oscillations in turbulent Rayleigh–Bénard convectionX.-L. Qiu, X.-D. Shang, P. Tong, and K.-Q. Xia Citation: Phys. Fluids 16, 412 (2004); doi: 10.1063/1.1637350 View online: http://dx.doi.org/10.1063/1.1637350 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v16/i2 Published by the AIP Publishing LLC. Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Velocity oscillations in turbulent Rayleigh–Be ´nard convectionX.-L. QiuDepartment of Physics, Oklahoma State University, Stillwater, Oklahoma 74078

X.-D. ShangDepartment of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong

P. TongDepartment of Physics, Oklahoma State University, Stillwater, Oklahoma 74078and Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay,Kowloon, Hong Kong

K.-Q. XiaDepartment of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong

~Received 24 February 2003; accepted 5 November 2003; published online 7 January 2004!

A systematic study of velocity oscillations in turbulent thermal convection is carried out in smallaspect-ratio cells filled with water. Local velocity fluctuations and temperature-velocitycross-correlation functions are measured over varying Rayleigh numbers and spatial positionsacross the entire convection cell. These structural measurements reveal how the thermal plumesinteract with the bulk fluid in a closed cell and provide an interesting physical picture for thedynamics of the temperature and velocity oscillations in turbulent convection. ©2004 AmericanInstitute of Physics.@DOI: 10.1063/1.1637350#

I. INTRODUCTION

Turbulent Rayleigh–Be´nard convection is an interestingproblem in nonlinear physics and has attracted much atten-tion in recent years.1–4 Despite its relatively low Reynoldsnumber ~Re!, turbulent convection shares many commonfeatures that are usually associated with high-Re turbulentflows. These features include coherent structures, intermit-tent fluctuations, and anomalous scaling. There are two co-herent structures, which are found to coexist in the convec-tion cell. One is the large-scale circulation that spans theheight of the cell, and the other is intermittent bursts of ther-mal plumes from the upper and lower thermal boundary lay-ers. An intriguing feature of turbulent convection is the emer-gence of a well-defined low-frequency oscillation in thetemperature power spectrum.1,5 This oscillation takes placein the hard turbulence regime when the Rayleigh number~Ra! becomes larger than a critical value Rac (Rac.53107

in the aspect-ratio-one cell!. Recent velocity measure-ments6–8 found a similar oscillation in the velocity powerspectrum. The temperature and velocity oscillations havebeen observed in various convecting fluids, including lowtemperature helium gas,1,5 sulfur hexafluoride gas near itscritical point,6 mercury,9,10 and water.7,8,11 These measure-ments have stimulated considerable theoretical efforts, aimedat explaining the dynamic origin of the temperature oscilla-tion.1,5,10,12

In an attempt to further understand the structure and dy-namics of the thermal plumes, we have recently carried out asystematic study of the temperature and velocity fields inturbulent convection.13,14 Using the techniques of laser Dop-pler velocimetry~LDV !, thermometry, and flow visualiza-tion, we map out the temperature and velocity structures in

the plane of the large-scale circulation. The temperature andvelocity measurements are conducted in convection cellswith different aspect ratios and over varying Rayleigh num-bers and spatial positions across the entire cell. Water is usedas the convecting fluid, in which the temperature and veloc-ity measurements can be made simultaneously with highaccuracy. These local measurements provide a body ofreliable velocity and temperature data and complement theglobal measurements of heat transport in turbulentconvection.1,15–19

In Ref. 14 ~and a rapid communication20!, we have de-scribed the structure of the temperature field in turbulentconvection. Direct measurements of local temperature fluc-tuations and their spatial correlation functions are carried outat various Rayleigh numbers and spatial positions. Thesemeasurements fully characterize the spatial structure of thetemperature oscillation and reveal interesting dynamics ofthe thermal plumes near the conducting surface. A sharptransition from a random chaotic state to a correlated turbu-lent state of finite coherence time is found in the aspect-ratio-one cell when Ra becomes larger than Rac.53107. AboveRac the measured temperature correlation functions show awell-defined oscillation with a finite coherence time. The os-cillation period is found to be twice as large as the measuredtransit time for the thermal plumes to travel between theupper and lower conducting surfaces~cell crossing time!.The measurements demonstrate that the thermal plumes in aclosed cell organize themselves both in space and time,which provides a unique driving mechanism for the tempera-ture oscillation in turbulent convection.

The experiment is in good agreement with Villermaux’stheory,12 which proposed that the temperature oscillation iscaused by a thermal boundary layer instability triggered by

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the incoming thermal plumes from the opposite conductingsurface. These thermal plumes are transported by the large-scale circulation having a mean velocityU. Villermaux’smodel assumed that the unstable modes~i.e., the thermalplumes! in the upper and lower thermal boundary layers in-teract through a delayed nonlinear coupling with a time con-stant equal to the cell crossing timet0.L/U, whereL is thecell height. Because of this delayed coupling, the thermalplumes are excited alternately between the upper and lowerboundary layers with a local frequencyf 051/(2t0).U/(2L).

An important question one might ask is how is the tem-perature oscillation related to the velocity oscillation ob-served in the same convection cell? While Villermaux’smodel explained the dynamic origin of the temperature os-cillation, it did not discuss the consequences of the alternat-ing emission of thermal plumes to the flow field. In thispaper we focus our attention on the correlation between thevelocity and temperature fields in turbulent convection. Si-multaneous measurements of the local flow velocity andtemperature are carried out using the techniques of LDV andthermometry. From these measurements we obtain thetemperature–velocity cross-correlation function over varyingRayleigh numbers and spatial positions across the entire cell.The simultaneous velocity and temperature measurementsfully characterize the spatial structure of the velocity oscilla-tion and allow us to answer some important open questionsthat are related directly to the physical understanding of con-vective turbulence. These questions include: How is the ve-locity oscillation generated and sustained? What is the con-nection between the velocity oscillation and the coherentstructures in the flow, such as the large-scale circulation andthe thermal plumes? The experiment demonstrates that tur-bulent convection in a closed cell is a well organized flowand the structural measurements are essential to the physicalunderstanding of convective turbulence.

The remainder of the paper is organized as follows. Wefirst describe the apparatus and the experimental method inSec. II. Experimental results are discussed in Sec. III. Fi-nally, the work is summarized in Sec. IV.

II. EXPERIMENT

The experiment is conducted in two up-right cylindricalcells filled with water. The height of the larger cell is 20.4 cmand its inner diameter is 19.0 cm. The smaller cell is 8.2 cmin height and 8.2 cm in inner diameter. The correspondingaspect ratioA (5diameter/height) of these cells is unity. Thesidewall of the cells is made of a transparent Plexiglas ringwith wall thickness 0.6 cm. The top and bottom plates aremade of brass and their surfaces are electroplated with a thinlayer of gold. The thickness of the top plate is 1.0 cm andthat of the bottom plate is 0.85 cm. The Plexiglas ring issandwiched between the two plates and is sealed to the topand bottom plates via two regular rubber O rings. The topand bottom plates together with the sidewall are held to-gether by six Teflon posts evenly spaced on a circle slightlylarger than the outer diameter of the Plexiglas ring. Twosilicon rubber film heaters connected in parallel are sand-

wiched on the backside of the bottom plate to provide con-stant and uniform heating. A dc power supply with a 99.99%long-term stability is used to provide the heating power. Thevoltage applied to the heaters varies from 20 to 90 V, and thecorresponding heating power is in the range between 24 and485 W. The upper side of the top plate is in good contactwith a cooling chamber, whose temperature is maintainedconstant by circulating cold water from a temperate-controlled bath/circulator. The temperature stability of thecirculator is 0.01 °C. Cold water is fed into the chamberthrough two opposing inlets on the side of the chamber andflows out through two outlets on the top of the chamber.

To record the temperature of the conducting plates, weembed two thermistors beneath the surface of each plate.One thermistor is located at the center of the plate and theother is placed half way between the plate center and thesidewall. The whole convection cell~except an optical win-dow! is wrapped in three layers of thermal insulating rubbersheets to prevent heat leakage. In the experiment, the tem-perature differenceDT between the top and bottom platesvaries from 4 to 51 °C depending on the heating power. Byadjusting the temperature of the cooling water, we maintainthe temperature of the bulk fluid at;29 °C for all the mea-surements. The corresponding Prandtl number~Pr! of thefluid is approximately 5.4. The temperature stability of thetop and bottom plates is found to be within 0.1 °C in stan-dard deviation, which is less than 2.5% of the minimumDTused in the experiment.

The two A51 cells are used to extend the accessiblerange of the Rayleigh number Ra5agDTL3/(nk), wheregis the gravitational acceleration,L is the cell height, anda, n,andk are, respectively, the thermal expansion coefficient, thekinematic viscosity, and the thermal diffusivity of the con-vecting fluid ~water!. The larger cell covers the Ra rangebetween 7.53108 and 131010, and the smaller cell coversthe Ra range between 1.753107 and 7.13108. For the high-est Ra (57.13108) in the smaller cell, we haveDT550.9 °C and the non-Boussinesq temperature deviation, de-fined asDc[Tc2(Tb1Tt)/2, is found to beDc52.9 °C,whereTc is the average temperature of the bulk fluid mea-sured at the cell center, andTb andTt are the average tem-perature of the bottom and top plates, respectively. Thus wehaveDc /DT.5.7%. When the Boussinesq approximation isvalid, Dc is expected to be zero.21 In the larger cell at Ra51.53109, we haveDT58.0 °C, Dc50.4 °C, and the cor-responding ratioDc /DT55%. When compared with thenon-Boussinesq temperature deviations in glycerol,22 we findthe values ofDc in water are smaller. However, we have notyet studied any non-Boussinesq effect in our system system-atically. To check the aspect-ratio dependence, we also usetwo other cells withA52 andA50.5, respectively. Thesetwo cells have the same inner diameter of 19.0 cm but theirheight is different; one is 9.5 cm and the other is 40.7 cm.

Figure 1~a! shows the experimental arrangement for thevelocity and temperature measurements. Local velocity mea-surements are conducted using a two-component LDV sys-tem ~TSI Inc.! together with an argon-ion laser~CoherentInnova 90!. A long rectangular flat window (W) made oftransparent Plexiglas with wall thickness 0.6 cm is inserted

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onto the sidewall of the cell to admit the incident laser beamsand observe the scattered light by seed particles. The widthof the optical window is 3 cm for the larger cell and 1.5 cmfor the smaller cell. The length of the optical window re-mains the same as the height of the Plexiglas ring. Two pairsof laser beams of different color~blue and green! comingfrom a LDV fiberoptic transceiver are directed through theoptical window and focused onto a single point inside theconvection cell. The fiberoptic transceiver has a receivingfiber, which collects the scattered light in the backward di-rection and feeds it to two photomultiplier tubes~one detectsthe blue light and the other detects the green light!. The laserfocusing spot has a cylindrical probe volume of 1.31 mm inlength and 0.09 mm in diameter. The two laser beams shownin Fig. 1~a! are in the horizontal plane and are used to mea-sure the horizontal velocityvx in the x direction. The othertwo laser beams~not shown! are in the vertical plane parallelto the laser propagation direction. These two beams are usedto measure the vertical velocityvz along thez axis.

Monodisperse polymer latex spheres of 4.75mm in di-ameter are used as seed particles. Because their density(1.05 g/cm3) matches closely to that of water, the seed par-

ticles follow the local flow well. In the experiment, we mea-sure the local velocity as a function of time and the measur-ing position~laser focusing spot! can be varied continuouslyalong they axis ~horizontal scan! and thez axis ~verticalscan!. This is accomplished by moving the fiberoptic trans-ceiver, which is mounted on a traversable table. A typicalsampling rate of the velocity measurements is 10–15 Hz,which is approximately 10 times larger than the cutoff fre-quency of the velocity power spectrum. Typically, we take7-h-long time series data (;53105 data points! at each po-sition. The data accumulation time is much longer than thetime scale for the large-scale circulation, which is of theorder of 1 min~see Fig. 2 for more details!, ensuring that thestatistical average of the flow properties is adequate.

In this way, we obtain a complete series of velocity dataat various Rayleigh numbers and spatial locations with thehighest statistical accuracy possible. In the calculation of thevelocity statistics, we use the transit time weighting to cor-rect for the velocity sampling bias.23 The velocity powerspectrum is obtained using a software package provided byTSI Inc. With unevenly sampled velocity time series datav(t), the program calculates the autocorrelation function ofv(t) first and then Fourier-transforms the autocorrelationfunction back to the frequency domain. As a result, the ob-tained velocity power spectrum becomes sensitive to themean flow ~extra-low frequency component!. In the data

FIG. 1. ~a! Schematic diagram of the experimental setup~top view!: T1, T2,temperature probes; 1, 2, two laser beams for LDV; W, optical window.~b!Space coordinates and eight locations for the velocity and temperature mea-surements. The long arrows near the sidewall indicate the direction of thelarge-scale circulation.

FIG. 2. Measured frequency power spectra~a! Px( f ), ~b! Py( f ), and ~c!Pz( f ) for vx , vy , andvz , respectively, at Ra53.73109. ~a! The measure-ments are made along the centralz axis at positions 1 (z59 mm, dottedcurve!, 2 (z542 mm, dashed curve!, and 3 (z5102 mm, solid curve!. ~b!The measurements are made at positions 1~dashed curve!, 3 ~solid curve!,and 5~8 mm away from the sidewall, dotted curve!. ~c! The measurementsare made at positions 1 (z520 mm, dotted curve!, 3 ~solid curve!, and 5~15mm away from the sidewall, dashed curve!.

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analysis to be discussed below, we will ignore this low-frequency tail.

Simultaneous velocity and temperature measurementsare carried out using a multichannel LDV interface module~TSI Datalink! to synchronize the data acquisition. A trigger-ing pulse from the LDV signal processor initiates the acqui-sition of an analog temperature signal. A small movable ther-mistor ~T1 or T2! of 0.2 mm in diameter, 15 ms in responsetime, and 1 mK/V in temperature sensitivity~Thermomet-rics, AB6E3-B10KA202J! is used to measure the local fluidtemperature. To guide the thermistor into the cell, we installa horizontal stainless steel tube on the sidewall. The stainlesssteel tube~Type 304 SS hypodermic tubing! has an outerdiameter of 1.07 mm and wall thickness of 0.19 mm. Thinthermistor wires thread through the stainless steel tube fromthe outside and a small head piece of the sensor tip sticks outof the tube end inside the convection cell. The tube endtogether with the sensor tip is sealed with glue so that theconvecting fluid cannot leak out through the stainless steeltube.

The horizontal tube can be mounted at three differentheights on the sidewall. At a given height the tube can slidein and out so that the local fluid temperatureTi(t) ~herei isused to indicate the probe position! can be measured at vari-ous horizontal positions away from the sidewall. The ther-mistor is connected to an ac bridge as a resistance arm. Thevoltage signals from the ac bridge are fed to the interfacemodule, whose output is connected to the LDV signal pro-cessor. A host computer is used to store each pair of thevelocity and temperature measurements with the same timestamp. All the thermistors used in the experiment are cali-brated individually with an accuracy of 0.01 °C. In the ex-periment, we also vary the spatial separation between thelaser focusing spot and the thermistor tip.

Figure 1~b! shows the space coordinates to be used be-low in the presentation of the velocity and temperature mea-surements. The origin of the coordinate system is chosen tocoincide with the center of the lower conducting surface. Thex andz axes are in the rotation plane of the large-scale cir-culation~LSC! and they axis is perpendicular to the rotationplane. It has been shown24 that the extent of LSC in theydirection may vary between 2/3 and 1/3 of the cell diameterwhen Ra is in the range 53109,Ra,1010 ~in the A51cell!. The eight crosses shown in Fig. 1~b! indicate the loca-tions of the velocity and temperature measurements made inthe rotation plane. Early temperature measurements10

showed that the azimuth of LSC rotates slowly in time whenthe cylindrical cell is levelled perfectly. It was found recentlythat LSC in the levelled cylindrical cell also reverses its ro-tational direction randomly.25,26 The reversal of LSC occurssuddenly in a time span much less than a rotation period,whereas the typical duration between two successive rever-sals remains quite long when compared with the rotationperiod ~more than 10 rotation periods!.

To pin down the azimuthal rotation and the random re-versal of LSC, we tip the cell with a small angle (,1°) byadding a few sheets of paper on one side of the cell bottom.Ciliberto et al.11 have shown that such a small tilt does notaffect turbulent convection very much. When we tip the cell

at position B@see Fig. 1~a!#, LSC is setup in thex–z planeand the direction of rotation is shown in Fig. 1~b!. In thiscase, the LDV setup shown in Fig. 1~a! is capable of mea-suring the two in-plane velocity componentsvx andvz . Onecan also tip the cell at positionA and setup LSC in they–zplane. In this case, we can measure the in-plane velocitycomponent along the mean flow direction and an out-of-plane velocity component perpendicular to the rotation planeof LSC. To avoid confusion, hereinafter we choose thex–zplane to coincide with the rotation plane of LSC regardlessof the actual arrangement of the tilt.

III. RESULTS AND DISCUSSION

A. Spatial structure of velocity oscillations

Figure 2 shows the measured frequency power spectraPx( f ), Py( f ), andPz( f ) for the three velocity componentsvx , vy , andvz , respectively. The measurements are made atvarious locations in the central core region, near the sidewall,and near the lower conducting surface. A sharp peak atf 0

50.01860.002 Hz is found in all the horizontal power spec-tra, Px( f ) andPy( f ), measured at different locations. Fromthe height of the frequency peak we find that the horizontalvelocities at the cell center oscillate with a much larger am-plitude than those around the cell periphery. At the cell cen-ter, the measuredPy( f ) has a larger frequency peak whencompared withPx( f ). Contrary to the horizontal powerspectra, the vertical power spectrumPz( f ) shows a sharpfrequency peak only in the sidewall region. In the centralregion vz fluctuates randomly and no obvious frequencypeak is found in the measuredPz( f ).

Figure 3 shows the time series measurements of thethree velocity componentsvx , vy , andvz at the cell center.

FIG. 3. Time series measurements of velocity fluctuations at the cell center.The measurements are made atR53.73109 and the three velocity compo-nents are~a! vx , ~b! vy , and~c! vz .

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Clear oscillations are observed only in the two horizontalcomponents but not in the vertical component. The horizon-tal velocity vy perpendicular to the rotation plane of LSCexhibits the strongest coherent oscillation. It is found that thevelocity oscillations in different directions and at various lo-cations all have the same frequency at a fixed Ra, suggestingthat they are driven by a common mechanism. As will beshown in Fig. 12 below, the velocity oscillation frequencyf 0

varies with both Ra and the cell heightL and its numericalvalue is the same as that of the temperature oscillation.

Figure 4 shows the corresponding velocity histogramsH(vx) ~circles!, H(vy) ~triangles!, andH(vz) ~diamonds! atthe cell center. It is seen that the measuredH(vz) can be welldescribed by a Gaussian function~solid curve! over an am-plitude range of almost 4 decades. The measuredH(vx) andH(vy) are also of Gaussian shape but they exhibit two smallbumps near the zero mean. This is caused by the horizontaloscillations discussed above.

In Ref. 13 we have described the detailed structure of themean velocity field in turbulent convection. Large velocityfluctuations are found both in the central region and near thecell boundary. Despite the large velocity fluctuations, theflow field still maintains a coherent structure. Figure 5 showsa sketch of the velocity field in an aspect-ratio-one cell. Thispicture is derived from that shown in Ref. 4 but is modifiedwith inputs from our recent experiments.13,14 The flow fieldhas a quasi-two-dimensional structure, which undergoes acoherent rotation. The single-roll structure shown in Fig. 5can be divided into three regions in the rotation plane:~i! athin viscous boundary layer around the cell periphery,~ii ! afully mixed central core region of size;L/2, and ~iii ! anintermediate plume-dominated buffer layer of thickness;L/4. The plume-dominated buffer layer is the active regionthat drives the large-scale circulation. Thermal plumes eruptinto the region from the upper and lower thermal boundarylayers and are separated laterally in the two opposing side-wall regions. The warm and cold plumes exert buoyancyforces on the fluid and drive the vertical rising and fallingflows near the sidewall. The central core region is ‘‘sheared’’

by the rising and falling plumes, resulting in a constant meanvelocity gradient in the region.

Figure 6~a! shows how the large-scale flowvz near thesidewall changes with time. It is seen that the large-scaleflow oscillates mainly in one direction, staying at a largevelocity (;18 mm/s) for a while and then falling to a mini-mum velocity~close to 0! momentarily. This asymmetric os-

FIG. 4. Corresponding velocity histogramsH(vx) ~circles!, H(vy) ~tri-angles!, andH(vz) ~diamonds! for the time series data shown in Fig. 3. Thesolid curve shows a Gaussian fit to the diamonds. For clearness, the origin ofthe lower two curves is shifted downwards.

FIG. 5. A schematic drawing of the velocity field and the plume distributionin the aspect-ratio-one cell. The central core region~unshaded! is the fullymixed bulk region and the shaded area shows the plume-dominant region.The thin layers with wiggly lines near the upper and lower surfaces indicatethe thermal boundary layers. Cold plumes~light colored! fall on the left sideof the cell and warm plumes~dark colored! rise on the right side of the cell.The arrows indicate the direction of the large-scale circulation.

FIG. 6. ~a! Time series measurements of the vertical velocityvz at Ra53.73109. The measurements are made at position 5~8 mm away from thesidewall!. ~b! Corresponding velocity histogramH(vz). The solid curve is aGaussian function with mean velocityvz513.0 mm/s and standard devia-tion sz53.8 mm/s.

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cillation gives rise to a large mean vertical velocity(;13 mm/s) in the sidewall region. Figure 6~b! shows thecorresponding velocity histogramH(vz). Contrary to theGaussian distribution at the cell center, the measuredH(vz)near the sidewall shows a bimodal distribution with a domi-nant peak at a positive velocity and a secondary peak at anegative velocity, whose absolute value is comparable to thatof the main peak. It is also seen that the main peak is ad-equately described a Gaussian function~solid curve! and thesecondary peak is approximately 60 times smaller than themain peak.

We find that the measuredH(vz) deviates from theGaussian shape gradually and becomes more and moreasymmetric when the measuring point is moved away fromthe cell center. The histogram is skewed against the meanflow, having the largest skewness at the edge of the velocityboundary layer, at which the mean flow velocity reachesmaximum. The measured histograms at other Rayleigh num-bers and in theA50.5 cell are similar to that shown in Fig.6~b!. Careful examination of the time series data near thesidewall reveals that contributions to the secondary peakcome mainly from occasional negative spikes in the velocitysignals. These negative spikes last only 5–10 s~less than halfof the rotation period! and should not be confused with therandom reversal of LSC. As mentioned in Sec. II, the rever-sal of LSC has a much longer time span~more than 10 rota-tion periods! than these short velocity spikes. We have veri-fied that when the cell is tilted, the direction of the mean flowremains unchanged during the entire experiment. We believethat the negative velocity spikes are caused by a small num-ber of energetic cold plumes, which erupt from the upperconducting surface and move downward against the risingmean flow at position 5. Because thermal plumes erupt alter-nately between the upper and lower boundary layers, thecold plumes fall in the ‘‘quiet’’ time period between twoconsecutive bursts of rising warm plumes at position 5.These events are rare because most cold plumes move down-ward on the other side of the cell~position 7!.

We also study the velocity oscillation in theA52 andA50.5 cells. Figure 7~a! shows the measuredPx( f ) ~uppersolid curve!, Py( f ) ~dashed curve!, andPz( f ) ~lower solidcurve! at the center of theA52 cell. The measurements aremade at Ra54.93108. Clear oscillations are observed in allthree velocity components. Figure 7~b! compares the mea-suredPx( f ) in the A52 cell ~dotted curve! with that in theA51 cell ~solid curve! at a similar Rayleigh number (Ra54.73108). It is seen that the amplitude of the frequencypeak in theA52 cell is approximately two orders of magni-tude larger than that in theA51 cell, suggesting that thevelocity oscillations in theA52 cell are much stronger. Infact, one can observe the velocity oscillations directly fromthe time series data. Note that the frequency peak in theA51 cell can still be identified clearly at this Rayleigh num-ber, as shown in the inset of Fig. 7~b!.

Figure 8 shows the measured frequency power spectra atthe center of theA50.5 cell. The measurements are made atRa53.131010. Contrary to the results obtained in theA51 and A52 cells ~at somewhat smaller Rayleigh num-bers!, no obvious frequency peak is found in the power spec-

tra measured in theA50.5 cell. This conclusion is furtherconfirmed by the measurements at another Rayleigh number(Ra55.831010) in the same hard turbulence regime. In anearly convection experiment with low temperature heliumgas,27 Wu and Libchaber also reported that the temperatureoscillation is absent in theA50.5 cell.

FIG. 7. ~a! Measured frequency power spectraPx( f ) ~upper solid curve!,Py( f ) ~dashed curve!, andPz( f ) ~lower solid curve! at Ra54.93108. Themeasurements are made at the center of theA52 cell. ~b! Comparisonbetween the measuredPx( f ) in theA51 cell at Ra54.73108 ~solid curve!and in theA52 cell at Ra54.93108 ~dotted curve!. Inset shows a linearplot of Px( f ) vs log f for the same data obtained in theA51 cell at Ra54.73108. All the measurements are made at the cell center.

FIG. 8. Measured frequency power spectraPx( f ) ~lower solid curve!,Py( f ) ~dashed curve!, andPz( f ) ~upper solid curve! at Ra53.131010. Themeasurements are made at the center of theA50.5 cell.

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Villermaux12 pointed out that the underlying mechanismfor the temperature oscillation in a closed convection cellrelies on the interplay between the linear growth of primaryinstability disturbances~i.e., thermal plumes! and the delayof nonlinear saturation due to the finite large-scale velocityU. Therefore, the interaction between the upper and lowerthermal boundary layers is critically dependent on the delaytime t0.L/U. It is assumed in Villermaux’s model that for afixed Ra, t0 does not fluctuate. As a result, the calculatedtemperature oscillation has an infinite coherence time. Asshown in Fig. 6~a!, the measuredU is actually a fluctuatingquantity and thus we havedt0 /t05dU/U.sz / vz , wherevz

andsz are, respectively, the mean and rms velocities near thesidewall. Fluctuations in the delay timet0 will broaden thefrequency peak atf 0 .12

Our recent velocity measurements13 showed that thelarge-scale flow in theA51 andA52 cells is fairly stablebut the large-scale circulation~LSC! in the A50.5 cellwobbles even when the cell is tipped at a larger angle. Thewobbly LSC introduces large fluctuations int0 , which maydestroy the coherent oscillation in theA50.5 cell. The mea-suredsz / vz in the A51 andA52 cells has a typical valueof 0.5, whereas in theA50.5 cell it becomes slightly largerthan 1.13 When the rms velocity becomes larger than themean value, one will not be able to visualize a persistentlarge-scale circulation directly. It was also found13 that thetime-averaged large-scale flow in theA50.5 cell has a singlerole structure similar to that in theA51 cell. Recent velocitymeasurements by Masaki Sano and co-workers showed thatthe velocity field in theA50.5 cell filled with mercury (Pr.0.024) has a similar single roll structure~private commu-nication!. In the tiltedA50.5 cylinder filled with water (Pr.5.4), we did not observe a flow structure with two verti-cally stacked, aspect-ratio-one rolls, as suggested by recentnumerical simulations.28 The numerical study28 was carriedout in a levelledA50.5 cylinder filled with helium gas (Pr50.7). A transition from a one-roll flow to a two-roll flowwas used to explain the observed bimodal behavior of themeasured Nusselt number in anA50.5 cell filled with low-temperature helium gas.29

From the above velocity measurements together with therecent temperature measurements,14,20 we arrive at the fol-lowing physical picture for the velocity oscillation in theA51 cell. As discussed in the above, the temperature oscilla-tion is generated by the alternating emission of cold andwarm plumes between the upper and lower thermal boundarylayers. These cold and warm plumes are separated laterallyin the two opposing sidewall regions and exert buoyancyforces to the surrounding fluid. As a result, the vertical ve-locity oscillation in the sidewall region is coupled directly tothe temperature oscillation. Such a coupling via direct en-trainment is expected to become weaker in the central re-gion, where less thermal plumes are present. This conclusionis supported by the experimental observation that the maxi-mum vertical velocity oscillation occurs in the same sidewallregion as that for the temperature oscillation.

To further understand the spatial correlation between thetemperature and velocity oscillations, especially the interac-tion between the vertical thermal forcing and the horizontal

velocity oscillations in a closed cell, we conduct simulta-neous temperature and velocity measurements and obtain thetemperature–velocity cross-correlation function at variouslocations in the cell. In the next section, we discuss the cor-relation and phase relationship between the temperature andvelocity oscillations.

B. Correlations between the temperature and velocityoscillations

Figure 9~a! shows the measured temperature–velocitycross-correlation function, Cx(t)5^dT(t)dvx(t1t)&/(Trmsvx(rms)), as a function of delay timet. Here the tem-perature fluctuation is defined asdT(t)5T(t)2T, whereTis the mean value of the local temperatureT(t), and thevelocity fluctuation is defined asdvx(t)5vx(t)2 vx , wherevx is the mean value of the horizontal velocityvx(t). ThemeasuredCx(t) is normalized by the product of the twostandard deviations,Trms and vx(rms) . In the experimentthe temperature probe is fixed at position 7 and the velocity

FIG. 9. Measured temperature–velocity cross-correlation functions~a!Cx(t), ~b! Cy(t), and ~c! Cz(t) as a function of delay timet at Ra53.33109. In the experiment the temperature probe is fixed at position 7~15 mmaway from the sidewall! and the velocity measurements are conducted at~a!positions 3 (z5102 mm, open circles! and 4 (z5142 mm, closed triangles!;~b! positions 2 (z544 mm, closed triangles! and 3 ~open circles!; and ~c!positions 4~closed triangles! and 3 ~open circles!. The lines are drawn toguide the eye.

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measurements are made at various locations along the centralz axis ~vertical scan!. Similarly, Figs. 9~b! and 9~c! show themeasuredCy(t) andCz(t), respectively.

Figure 10 shows the measuredCy(t) and Cz(t) whenthe temperature probe is fixed at position 7 and the velocitymeasurements are made at positions 3, 6, and 8. These threepositions are all in the central region of the cell. It has beenshown14,20that the temperature oscillation takes place mainlyin the plume-dominated sidewall region. Using the tempera-ture signals at position 7 as a reference, the cross-correlationfunctions measure the phase relation between the tempera-ture and velocity oscillations. This phase relation is deter-mined by the delay time of the first maximum~peak! orminimum ~valley! position in the cross-correlation function.One can also obtain the phase relation between the velocityoscillations at two different locations by comparing the rela-tive phase difference between the cross-correlation functionsmeasured at the two locations.

Figures 9 and 10 reveal several interesting features ofthe velocity oscillation in a closed cell. First, while the tem-perature signals at position 7 show clear oscillations,14,20 thevertical velocityvz in the central region fluctuates randomlyand no obvious frequency peak is found in the measuredPz( f ) @see Figs. 2~c! and 3~c!#. Therefore, one expects thatthere will be little correlation betweendT(t) at position 7and dvz(t) in the central region. This is indeed the case asshown in Figs. 9~c! and 10~b!. A small correlation is foundonly whenvz is measured at a location close to position 7, atwhich the temperature measurement is made@solid diamondsin Fig. 10~b!#.

Second, we find that the measured cross-correlationfunctions for a given horizontal velocity component~eithervx or vy) but at different locations coincide with each other,

suggesting that the velocity oscillations at different locationshave the same phase relative to that of the temperature os-cillation. This is possible only when the central region oscil-lates together as a whole. Third, the relative phase differencebetween the measuredCx(t) andCy(t) is found to be zero,suggesting that the velocity oscillation in the central region islinearly polarized. Because the rms values ofvx or vy areapproximately the same@see Fig. 13~b! for more details#, theoscillation direction should be along the bisectional line be-tween thex andy axes.

Finally, we discuss the phase relation between the tem-perature oscillation and the horizontal velocity oscillation.From the recent temperature and velocity measurements,13,14

we learned the mixing dynamics of the thermal plumes. Asillustrated in Fig. 5, when the falling cold plumes arrive atthe lower surface, heat exchange takes place mainly on theleft side of the lower plate (;3L/4 in size!. At ~almost! thesame time, the impact of the cold plumes triggers off aninstability of the lower thermal boundary layer, resulting inan eruption of warm plumes on the right-hand side of thelower plate (;L/4 in size!. Because the strong mixing effectcan locally stabilize the thermal boundary layer, the emissionof warm plumes is suppressed in the region where the fallingcold plumes are annihilated. A similar process takes placenear the upper conducting surface after a delay timet0

5L/U.This dynamic process determines the phase relation be-

tween the cold plumes at position 7 and the resulting hori-zontal velocity fluctuation~say,dvx). Suppose that the bulkfluid is pushed to the right when the cold plumes erupt fromthe upper-left corner of the cell. To calculate the relativephase lets further assume that the horizontal force producedby the eruption of the cold plumes takes a simple formFx(t);sin(2pf0t), wheref 051/(2t0) is the frequency of thetemperature oscillation. The temperature signals at position 7then have the formdT(t);2sin(2pf0t2e). Here a minussign is introduced for the cold plumes (dT(t),0) and theinitial phasee (.2p f 0t0/25p/2) takes care of the delaytime for the cold plumes to travel from the upper surface toposition 7. The velocity signals in the central region can beobtained via dvx(t)5*Fx(t8)dt8;2cos(2pf0t). Finally,the cross-correlation function can be written asCx(t);^dT(t)dvx(t1t)&;^sin (2pf0t2e) cos@2pf0 (t1t)#&52cos (2pf0t).

It is seen from Figs. 9 and 10 that the calculated phaseagrees with the measurements. Figures 9 and 10 thus suggestthat the thermal plumes in the sidewall region interact withthe fluid in the central region via the hydrodynamic forces,which introduce a phase shift to the horizontal velocity. Wenotice that the minimum~valley! position of the measuredCx(t) andCy(t) is not exactly att50 as calculated but isshifted by a small amount to the left. However, this shift isvery small (;6 s) when compared with the oscillation pe-riod 1/f 0.56 s at Ra53.33109. In the above calculation ofthe initial phasee, we have ignored the time delay betweenthe arrival of cold plumes and the emission of warm plumesat the lower conducting surface~and the similar delay time atthe upper conducting surface!. This small delay time has

FIG. 10. Measured temperature–velocity cross-correlation functions~a!Cy(t) and ~b! Cz(t) as a function of delay timet at Ra53.33109. In theexperiment the temperature probe is fixed at position 7 and the velocitymeasurements are conducted at~a! positions 3 ~open circles! and 8 (x543 mm, solid triangles!; and ~b! positions 3~open circles!, 8 ~open tri-angles!, and 6 (x5247 mm, solid diamonds!. The lines are drawn to guidethe eye.

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been detected in a recent experiment14 and may contribute tothe small shift shown in Figs. 9 and 10.

The thermal plumes may also interact with the surround-ing fluid through direct entrainment.30,31 In this case,dT(t)and dvz will be in phase. Figure 11 shows the measuredCx(t) ~open diamonds!, Cy(t) ~open circles!, and Cz(t)~solid triangles! when the velocity measurements are made atpositions 1, 5, and 7. These are three typical locations aroundthe cell periphery, in which the thermal plumes concentrate.It is seen from Fig. 11~c! that the vertical velocity near thesidewall indeed oscillates with the same phase as that of thetemperature oscillation. Because of the direct entrainment bythe cold plumes (dT(t),0) at position 7, the nearby fluidhas a downward velocity (dvz,0). Consequently, the mea-suredCz(t) peaks att50 with a large positive amplitudeCz(0).0.54. The correlation betweendT(t) anddvz decaysas the measuring position ofdvz is moved to the centralregion. At position 6, which is 2.8 cm away from position 7,the measuredCz(0) is reduced to 0.094@see Fig. 10~b!#. It isalso found from Fig. 11~c! that the horizontal velocitydvy atposition 7 has the same phase asdT(t) ~anddvz). The mea-suredCy(t) peaks att50 but the correlation amplitude issmall @Cy(0).0.066#.

At position 5 the velocity fluctuationsdvz and dvy be-come in phase with the warm plumes, whose phase is oppo-site to that of the cold plumes at position 7. BecausedT(t),0, the resulting correlation functionsCz(t) andCy(t) stillpeak att50. This is clearly shown in Fig. 11~b!. At position1 the horizontal velocity fluctuationdvx is defined to bepositive when its direction is the same as the large-scale flowand there is a time delay betweendvx and the cold plumes(dT(t),0) at position 7. Therefore, the measuredCx(t)becomes minimum neart50 with the exact valley positionshifted toward the right in an amount equal to the time forthe cold plumes to travel from position 7 and position 1@seeFig. 11~a!#. Similarly, Cy(t) is found to be in phase withCx(t). Figure 11~a! also reveals that the correlation betweendT(t) and dvz at position 1 is very small withCz(0).0.037, which is at noise level.

C. Rayleigh number dependence

Figure 12 compares the measured velocity oscillationfrequency with that of the temperature oscillation at differentRayleigh numbers. The solid circles show the normalizedtemperature oscillation frequencyf 0L2/k obtained from thetemperature power spectrum near the sidewall. This set ofdata is well described by the power lawf 0L2/k50.167 Ra0.47 ~solid line!. This result is in good agreementwith the recent temperature measurements in low tempera-ture helium gas.25 The open circles are the normalized veloc-ity oscillation frequency obtained from the power spectrumof vx at the cell center. The open triangles are obtained fromthe autocorrelation function ofvy near the sidewall. In thelatter measurements, the bulk fluid temperature varied forless than 6 °C. As a result, the open triangles exhibit some-what larger scatterer at lower Ra. It is seen from Fig. 12 thatthe velocity oscillation frequencies measured at different lo-cations and obtained with different methods can all be super-posed on a single master curve, which coincides with thetemperature oscillation curve. Figure 12 thus demonstratesthat the velocity and temperature oscillations are indeed

FIG. 11. Measured temperature–velocity cross-correlation functionsCx(t)~open diamonds!, Cy(t) ~open circles!, and Cz(t) ~solid triangles! as afunction of delay timet at Ra53.33109. In the experiment the temperatureprobe is fixed at position 7 and the velocity measurements are made atpositions~a! 1 (z514 mm), ~b! 5 (x583 mm), and~c! 7 (x5275 mm).The lines are drawn to guide the eye.

FIG. 12. Normalized oscillation frequencyf 0L2/k obtained from the tem-perature power spectrum near the sidewall~solid circles!, the power spec-trum of the horizontal velocityvx at the cell center~open circles!, and theautocorrelation function of the horizontal velocityvy near the sidewall~opentriangles!. The solid line is a power-law fit,f 0L2/k50.167 Ra0.47, to thesolid circles.

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driven by a common mechanism. As discussed above, boththe velocity and temperature oscillations are generated by thealternating emission of thermal plumes between the upperand lower thermal boundary layers.

Besides the oscillation frequency, we also measure theRa-dependence of the oscillation amplitude of the horizontalvelocities and the vertical velocity fluctuations at the cellcenter. Figure 13~a! shows the normalized vertical rms ve-locity, szL/n, as a function of Ra. Heresz is defined assz5@^(vz2 vz)

2&#1/2. The open circles are obtained at thecenter of theA51 cells and represent the best set of data wehave to date. The time average for each data point runs formore than 8 h and the measurements are repeated manytimes over a period of 6 months. The data points in the Rarange, 33107<Ra<53108, are obtained in the smallerA51 cell. Because the convective flow is slow, the measuredrms values have a relatively larger error bar when comparedwith the mean values. The experimental uncertainties aremainly statistical and the relative error forsz is approxi-mately 15%. This estimate is based on the scatterer of 1-h-long time-averaged rms values obtained from run to run. Thesolid triangles are obtained by Daya and Ecke in a differentexperiment.32 Their data overlap with ours~open circles!well. Because there is no apparent oscillation for the verticalvelocity at the cell center@see Fig. 3~c!#, the measuredsz

represents the typical amplitude of vertical velocity fluctua-tions.

It is seen from Fig. 13~a! that the measuredsz curves upin the entire Ra range, 33107<Ra<93109, and the datacannot be fitted to a single power law. Nevertheless, the ve-locity data measured in the largerA51 cell with Ra.7.63108 can be well described by an effective power law,szL/n54.631023 Rab, with the exponentb50.5560.03~solid line!. Because the data curve up at small Ra, the ob-tained value ofb varies slightly depending on the Ra rangeused. For example, Daya and Ecke fitted their data withb50.5.32 The value ofb becomes slightly larger than 0.5when the data points at larger Ra are included. We also mea-suresz in the sidewall region. It is seen from Fig. 13~a! thatthe measuredsz near the sidewall~open diamonds! is 55%larger than that at the cell center. This result further supportsour conclusion that the vertical velocity oscillation is drivenby the thermal plumes near the sidewall through direct en-trainment. Because the vertical shearing force starts from thesidewall region, the amplitude of the vertical rms velocity isexpected to decay from the sidewall to the cell center.

Figure 13~b! shows the normalized horizontal rms ve-locities,sxL/n ~open circles! andsyL/n ~solid triangles!, asa function of Ra. These velocity measurements are made atthe center of theA51 cells. The open circles have the samestatistical accuracy as those shown in Fig. 13~a!. The soliddiamonds are the measuredsyL/n near the sidewall. Be-cause there are strong oscillations in the horizontal directions~see Fig. 3!, the measuredsx andsy actually represent theoscillation amplitude of the horizontal velocities. It is seenfrom Fig. 13~b! that the measured rms values at differentlocations and along different directions are approximatelythe same, further confirming that the bulk fluid oscillatestogether as a whole in a horizontal direction. As discussed inSec. III B, the horizontal velocity oscillation is linearly po-larized and the polarization direction is along the bisectionalline between thex andy axes, because the rms velocities inthe two horizontal directions are equal (sx.sy).

Similar to the situation forsz , the measuredsx alsocurves up in the entire Ra range and cannot be fitted to asingle power law. Nevertheless, the velocity data measuredin the largerA51 cell with Ra.7.63108 can be describedby an effective power law,sxL/n55.831024 Rab, with theexponentb50.6560.03 ~solid line!. While it varies slightlydepending on the Ra range used in the fitting, the obtainedvalue of b for sx is certainly larger than that forsz (b50.5560.03). Further comparison between Figs. 13~a! and13~b! reveals that the two fitted solid lines forsx and sz

cross each other at Ra.109 and the value ofsx becomeslarger thansz at the high Ra end.

From the above velocity measurements at different loca-tions and in cells of different aspect ratios, we find that themeasured rms velocities at the cell center, where the meanvelocity is zero, are influenced strongly by the large-scalemotion. This is certainly true for the horizontal rms veloci-ties. The bulk fluid in the central region oscillates togetherand thereforesx andsy measure the oscillation amplitude ofthe horizontal velocities. While there is no apparent oscilla-tion found in the vertical velocity at the center of theA51

FIG. 13. ~a! Normalized vertical rms velocityszL/n ~open circles! as afunction of Ra. The solid triangles are obtained by Daya and Ecke~Ref. 32!.These velocity measurements are made at the center of theA51 cells. Thesolid line is a power-law fit,szL/n54.631023 Ra0.55, to the open circles atlarge Ra. The open diamonds are obtained near the sidewall.~b! Normalizedhorizontal rms velocitiessxL/n ~open circles! andsyL/n ~solid triangles! asa function of Ra. The velocity measurements are made at the center of theA51 cells. The solid line is a power-law fit,sxL/n55.831024 Ra0.65, tothe open circles at large Ra. The solid diamonds are the measuredsyL/nnear the sidewall.

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cell, velocity fluctuations in the vertical direction can still beaffected by the overall coherent motion in the region. Evi-dently, strong vertical velocity oscillations are found at thecenter of theA52 cell. These measurements suggest that theoscillation of the velocity field is a response of the bulk fluidin a closed cell to the thermal forcing produced by the peri-odic emission of the thermal plumes near the conductingsurfaces. Thus one expects that the velocity variations can beinfluenced considerably by the geometry of the cell, whichdetermines the large-scale motion. In fact, Daya and Ecke32

have found that the Ra-dependence of the measuredsz at thecell center changes with the shape of the convection cell.

IV. SUMMARY

We have carried out a systematic study of the velocityoscillation in turbulent thermal convection. Simultaneousmeasurements of the local velocity and temperature are con-ducted in small aspect-ratio cells filled with water. Fromthese measurements we obtain the temperature–velocitycross-correlation function and study the velocity oscillationsover varying Rayleigh numbers and spatial positions acrossthe entire cell. These structural measurements provide an in-teresting physical picture for the temperature and velocityoscillations in a closed cylindrical cell. The temperature os-cillation is generated by the alternating emission of cold andwarm plumes between the upper and lower thermal boundarylayers. The cold and warm plumes are separated laterally intwo opposing sidewall regions and exert buoyancy forces tothe bulk fluid. An alternating eruption of the thermal plumes,therefore, gives rise to a periodic impulsive torque, whichdrives the large-scale circulation continuously.

The spatial organization of the thermal plumes producesa unique flow structure, which undergoes a coherent rotation.Careful examination of the measured temperature-velocitycross-correlation functions reveals that the thermal plumesinteract with the convecting fluid in a closed cell in twodifferent modes. First, they directly entrain the surroundingfluid in the sidewall region along the vertical direction. Forexample, when a warm plume is pushed away from the lowerconducting surface by buoyancy, it entrains the surroundingfluid and pulls the nearby plumes towards it.14 This ‘‘hydro-dynamic attraction’’ causes the warm plumes and the sur-rounding fluid to move together, which explains why thevertical velocity oscillation occurs mainly in the plume-dominated sidewall region and its phase remains the same asthat of the temperature oscillation.

Second, the interaction between the thermal plumes andthe bulk fluid in the central region takes place via hydrody-namic forces, which introduce a phase shift to the two hori-zontal velocity components in the region. It is found that thevelocity oscillation in the central region of theA51 cell ispredominantly in the horizontal direction. The horizontal ve-locity oscillation is linearly polarized and its phase is oppo-site to that of the temperature oscillation. The velocity oscil-lation frequencyf 0 is found to be the same as that of thetemperature oscillation and obeys the same power law,f 0L2/k50.167 Ra0.47, in the Ra range between 108 and1010.

The velocity oscillation in the central region of theA52 cell is stronger and is observed in all three velocity com-ponents. Contrary to the situation in theA52 and A51cells, no obvious velocity oscillation is observed in theA50.5 cell. The large-scale flow in theA50.5 cell is notstable, which may destroy the coherent stimulations neededfor the generation of the temperature oscillation.14 The ve-locity measurements conducted at different locations and incells of different aspect ratios clearly show that the large-scale convective flow in a closed cell is influenced stronglyby the cell geometry, such as the shape and the aspect ratioof the cell.

In wind tunnels and other open flow systems, the distur-bance flow produced by the boundaries is confined in thenear-wall region and is quickly discharged to the down-stream. Because of these reasons the flow boundaries usuallydo not perturb the velocity fluctuations in the bulk region.33

This situation is changed completely for flows in a closedcell, in which the disturbances produced by the boundariesare inevitably mixed into the turbulent bulk region. For ex-ample, we find that the measured rms velocities at the cellcenter are coupled directly to the large-scale motion in theclosed convection cell.

While at the moment we do not have a theory to showquantitatively how the thermal plumes near the cell boundaryproduce horizontal velocity oscillations at the cell center, theexperiment clearly reveals that the velocity oscillation is aresponse of the bulk fluid to the periodic emission of thethermal plumes between the upper and lower thermal bound-ary layers. Evidently the cell shape and aspect ratio can in-fluence the spatial distribution and the dynamics of the ther-mal plumes, which may explain the observed geometry-dependence of temperature and velocity fluctuations at thecell center.32 A further theoretical analysis is needed in orderto fully understand the hydrodynamic interaction betweenthe thermal plumes and the horizontal velocity oscillations inthe central region. Clearly, the present experiment providesan interesting example to demonstrate how the thermalplumes organize themselves in a closed cell to generate alarge-scale flow structure, which rotates and oscillates coher-ently in a turbulent environment.

ACKNOWLEDGMENTS

We thank L. Kadanoff, D. Lohse, R. Ecke, Z. Daya, andJ. Niemela for useful discussions and communications. Theassistance of M. Lucas and his team in fabricating the con-vection cells is gratefully acknowledged. P.T. was supportedin part by the National Science Foundation under Grant No.DMR-0071323 and by the Research Grants Council of HongKong SAR under Grant No. HKUST 603003. K.-Q.X. wassupported by the Research Grants Council of Hong KongSAR under Grant No. CUHK4281/00P.

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