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Measured Local Heat Transport in Turbulent Rayleigh-Benard Convection

X.-D. Shang,1 X.-L. Qiu,2 P. Tong,2 and K.-Q. Xia1

1Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China2Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078

(Received 19 September 2002; published 20 February 2003)

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Local convective heat flux in turbulent thermal convection is obtained from simultaneous velocityand temperature measurements in an aspect-ratio-one convection cell filled with water. It is found thatfluctuations of the vertical heat flux are highly intermittent and are determined primarily by thethermal plumes in the system. The experiment reveals a unique mechanism for the heat transport inturbulent convection.

DOI: 10.1103/PhysRevLett.90.074501 PACS numbers: 47.27.Te, 44.25.+f

and the sidewall is a transparent Plexiglas ring with anarrow and long rectangular flat window for velocity

the bottom (z=H � 0) to the cell center (z=H � 0:5). Inthe experiment, the spatial separation between the LDV

An important issue in the study of turbulent Rayleigh-Benard convection is to understand how heat is trans-ported vertically through a convection cell [1–3]. A largenumber of global heat transport measurements have beencarried out in various convecting fluids and under differ-ent experimental conditions. Some of the measurementswere conducted with wide parameter range and greatprecession [4–9]. These measurements have stimulatedconsiderable theoretical efforts, aimed at explaining thefunctional form of the measured Nusselt number (nor-malized heat flux), Nu(Ra,Pr), as a function of the twoexperimental control parameters: the Rayleigh numberRa and the Prandtl number Pr. Like many transport phe-nomena in condensed matter physics, the measured mac-roscopic transport properties can often be explained bytheories with different microscopic mechanisms [1–4]. Amain issue of an unresolved theoretical debate is whetherthe heat transport in turbulent convection is determinedprimarily by thermal plumes, which erupt from the upperand lower thermal boundary layers, or by the large-scalecirculation (LSC) that spans the height of the convectioncell. Direct measurements of the local convective heatflux, therefore, become essential to the understanding ofthe heat transport mechanism in turbulent convection.

In this Letter, we report direct measurements ofthe normalized local convective heat flux, J�r� �hv�r; t��T�r; t�itH=��T, over varying Rayleigh numbersand spatial positions r across the entire cell. Here � is thethermal diffusivity of the convecting fluid, �T is the tem-perature difference across the cell height H (� 20:5 cm),and h. . .it represents an average over time t. In the experi-ment, the local temperature fluctuation �T�r;t��T�r;t� �T0 and the flow velocity v�r;t� are measured simulta-neously. The mean temperature T0 of the bulk fluid iskept at �30�C and the corresponding Prandtl number,Pr�=�, is �5:4. The convection cell is an uprightcylindrical cell of aspect ratio one and is filled with water.Details about the apparatus have been described else-where[10]. The upper and lower plates are made of brass

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measurement. Two silicon rubber film heaters connectedin parallel are sandwiched on the backside of the lowerplate to provide constant and uniform heating. The upperplate is in contact with a cooling chamber, whose tem-perature is maintained constant by circulating cold waterfrom a temperature bath.

Local velocity measurements are conducted using alaser Doppler velocimetry (LDV) system together withan argon-ion laser[11]. The LDV system can simulta-neously measure two components of v�r; t� as a functionof time t and the measuring position r (laser focusingspot) is varied by moving the LDV fiber-optic transceiverprobe, which is mounted on a traversable table. It has beenshown[11] that LDV is capable of measuring the flowvelocity with high accuracy (better than 1%) over theentire convection cell, except near the upper and lowerthermal boundary layers, whose thickness is �1 mm. Thesampling rate of the velocity measurements is �15 Hz,which is much larger than the cutoff frequency of thevelocity power spectrum. Typically, we take two to sevenhour-long time series data (� 5 105 data points) at eachlocation, ensuring that the statistical average of the flowproperties is adequate.

Simultaneous velocity and temperature measurementsare carried out using a multichannel LDV interface mod-ule to synchronize the data acquisition. A triggering pulsefrom the LDV signal processor initiates the acquisition ofan analog temperature signal. A small movable thermis-tor of 0.2 mm in diameter, 15 ms in time constant, and1 mK=� in temperature sensitivity is used to measureT�r; t�. To guide the thermistor into the cell, we install ahorizontal stainless steel tube of 1.1 mm in diameter atthe midheight of the sidewall. The tube can slide in andout so that T�r; t� can be measured at various horizontalpositions along a cell diameter (x axis) from the sidewall(x=H � 0) to the cell center (x=H � 0:5). Similarly, wealso install a vertical stainless steel tube through thecenter of the top plate and measure T�r; t� along thecentral vertical axis of the cell (z axis) from

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focusing spot and the thermistor tip is kept at a minimalvalue of 0:7 0:2 mm. This distance is 3 times largerthan the tip diameter of the thermistor but 20 timessmaller than the correlation length between the tempera-ture and velocity fluctuations [12]. In addition, the LDVfocusing spot is always placed at an upstream position tofurther minimize the disturbance of the thermistor to thevelocity measurement.

We first discuss the measurements of J�r� along the xand z axes; both are in the rotation plane of LSC. For thehorizontal scan, we measure the vertical heat flux Jz�x�and the horizontal heat flux Jy�x� (out of the rotationplane). For the vertical scan, we measure the horizontalheat flux Jx�z� and the vertical heat flux Jz�z�. Figure 1(a)shows the measured Jz�x� (circles) and Jy�x� (triangles) asa function of the normalized horizontal position x=H attwo different values of Ra. It is seen that the heat fluxacross the midheight plane of the cell is predominately inthe vertical direction and the horizontal heat flux is

FIG. 1. (a) Measured vertical flux Jz�x� (circles) and horizon-tal flux Jy�x� (triangles) as a function of x=H. (b) Measuredhorizontal flux Jx�z� (circles) and vertical flux Jz�z� (triangles)as a function of z=H. The measurements in (a) and (b) are madeat Ra � 1:8 109 (open symbols) and 6:0 109 (closed sym-bols). The two insets show, respectively, the normalizedJz�x�=J0 and Jx�z�=J0 at Ra � 1:8 109 (triangles), 3:6 109

(diamonds), and 6:0 109 (circles).

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negligible. The vertical heat flux concentrates in the side-wall region and increases with Ra.While the amplitude ofthe flux profile Jz�x� increases with Ra, its shape remainsinvariant with Ra. This is clearly shown in the inset,which displays the normalized Jz�x� by its peak valueJ0 at three different values of Ra. The vertical heat flux atthe cell center is small but is definitely nonzero. It ac-counts for �5% when compared with the peak value J0near the sidewall.

Figure 1(b) shows the measured Jx�z� (circles) and Jz�z�(triangles) as a function of the normalized vertical posi-tion z=H at two values of Ra. In contrast to the situationnear the sidewall, heat transport near the center of thelower conducting surface is dominated by the horizontalflux along the direction of LSC and the vertical flux isnegligible. When compared with Fig. 1(a), we find thatJx�z� peaks at a location much closer to the lower bound-ary and the width of the peak is narrower. The amplitudeof the flux profile Jx�z� increases with Ra, but its shaperemains unchanged with Ra (see the inset). Another im-portant feature shown in the inset is that the peak posi-tion of Jx�z� scales with the thermal boundary layerthickness � (� 425Ra�0:285 mm [13]). Figure 1 thus re-veals that heat transport in the aspect-ratio-one cell iscarried out mainly along the cell periphery in the direc-tion of LSC.

Our recent temperature and velocity measure-ments[11,14] showed that the spatial distribution of ther-mal plumes in a closed cell is neither homogeneous norisotropic. The thermal plumes organize themselves insuch a way that warm plumes accumulate on one side ofthe cell and cold plumes concentrate on the opposite sideof the cell. Figure 2 shows the plume distribution withwarm rising plumes on the left and cold falling plumes onthe right [15]. The warm and cold plumes, which areseparated laterally in the two opposing sidewall regions,exert buoyancy forces on the fluid and drive the verticalflow near the sidewall. The central core region is‘‘sheared’’ by the rising and falling plumes, resulting ina large-scale circulation across the cell height [11].

From these measurements we conclude that the dynam-ics in turbulent convection is determined primarily by thethermal plumes. The spatial separation of warm and coldplumes and the resulting LSC provide a fast channelalong the cell periphery for the transport of heat. Thisphysical picture explains our finding that the plume-dominated sidewall region (of thickness �H=4) coincideswith the ‘‘peak region,’’ in which both Jz�x� and vz reachmaximum. Clearly, this is a self-organizing process inthat the plume separation and LSC help each other andthey are not independent anymore. Because velocity fluc-tuations in the central region are strong, most thermalplumes are mixed up in the region. Nevertheless, there arestill some unmixed warm and cold plumes remaining,which give rise to a nonzero Jz in the region.

Besides the spatial organization, the warm and coldplumes are also coupled in time. As shown in Fig. 3(a),

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FIG. 3. (a) Time series measurements of the temperaturefluctuation �T�t� (top curve), vertical velocity vz�t� (middlecurve), and instantaneous vertical flux jz�t� (bottom curve)near the sidewall. All the measurements are made at Ra �3:6 109. (b) Corresponding autocorrelation functions C���for vz�t� (top curve), �T�t� (middle curve), and jz�t� (bottomcurve). For clarity, the origin of the top and bottom curves isshifted by 0:2.

FIG. 2 (color). A shadowgraph showing the spatial distribu-tion of thermal plumes in an aspect-ratio-one cell (H � 15 cm)filled with dipropylene glycol (Pr � 596). The picture is takenat Ra � 6:8 108.

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temperature fluctuations near the sidewall (top curve) arehighly skewed toward one direction and the upward goingspikes are associated with the rising warm plumes in theregion. It is seen that the thermal plumes do not arriverandomly; rather, they arrive in groups with a well-defined frequency. In a recent experiment [14], we foundthat such an oscillation is caused by the alternating emis-sion of cold and warm plumes between the upper andlower boundary layers. When the warm plumes move upin the sidewall region, they interact with the surroundingfluid through direct entrainment. As shown in Fig. 3(a),the resulting vertical velocity (middle curve) follows thetemperature oscillation in the region. From the simul-taneous velocity and temperature data, we obtain theinstantaneous vertical flux (bottom curve), jz�r; t� ��vz�r; t��T�r; t��H=��T, with hjz�r; t�it � Jz�r�. In thecalculation of jz�r; t�, we have used the convention thatwarm fluctuations (�T > 0) produce positive flux if theirvelocities are in the upward direction (vz > 0).

Figure 3(b) shows the corresponding autocorrelationfunctions of vz, �T, and jz. It is seen that the instanta-neous heat flux oscillates with the same frequency andphase as those of the temperature and velocity fluctua-tions. Figure 3 thus further confirms that the heat trans-port in turbulent convection is indeed carried out by thethermal plumes. Because of spatial average, the oscilla-tory behavior of the local heat flux is not observed in theglobal measurement of the Nusselt number.

Figure 4(a) shows the measured histograms of the in-stantaneous vertical flux H�jz� (triangles) and horizontalflux H�jy� (circles) near the sidewall. It is seen that fluc-tuations of the horizontal flux jy (out of the rotation plane

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of LSC) are symmetric and their mean value is approxi-mately zero. Fluctuations of the vertical flux jz, on theother hand, are asymmetric with positive fluctuationsbeing much larger than negative ones. The net gain ofthe positive fluctuations gives rise to a mean vertical heatflux. It is interesting to note that the negative flux fluctua-tions are approximately the same in both directions,whereas the positive flux fluctuations are much larger inthe vertical direction parallel to gravity. Figure 4(a) re-veals that the measured jz consists of two types of fluc-tuations: uncorrelated (passive) and correlated (active)fluctuations. The negative fluctuations are approximatelyisotropic and are produced by uncorrelated temperatureand velocity fluctuations. Small positive fluctuations sym-metric to the negative ones are also uncorrelated. Thesefluctuations do not contribute to the heat transport. Onlythe large positive fluctuations are correlated; they areproduced primarily by the thermal plumes and makepositive contributions to the mean flux.

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FIG. 4. (a) Measured heat flux histograms H�jz� (closedtriangles) and H�jy� (open circles) near the sidewall (9 mmfrom the sidewall). (b) Measured H�jz� (closed triangles) andH�jx� (open circles) at the cell center. All the measurements aremade at Ra � 2:6 109.

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Figure 4(b) shows the measured H�jz� (triangles) andH�jx� (circles) at the center of the cell. Similar to thesituation near the sidewall, fluctuations of the horizontalflux jx are symmetric with a mean value close to zero.Fluctuations of the vertical flux jz, on the other hand, areasymmetric with negative fluctuations being smaller thanpositive ones.We believe that this asymmetry is generatedby a small fraction of correlated signals (between tem-perature and velocity), which are produced by the re-maining warm and cold plumes in the region. Whiletemperature and velocity fluctuations are symmetric atthe cell center, the correlated signals associated with thewarm and cold plumes produce only positive flux in thevertical direction. As a result, the negative fluctuationsare suppressed and the positive ones are enhanced whencompared with the horizontal flux fluctuations. This isclearly shown in Fig. 4(b).

From Fig. 3(a) we find that the instantaneous verticalflux jz near the sidewall (bottom curve) is highly skewedtoward one direction. Positive fluctuations are superposedon an average base line. These upward going spikes areassociated with the rising warm plumes in the region.While the input heating power is constant (at a fixed Ra)

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and uniform across the entire conducting surface, therelease of this power to the convecting fluid (via jz) ishighly intermittent both in time and space. It is seen fromFigs. 3(a) and 4(a) that there are some energetic thermalplumes, which carry an instantaneous heat flux as large as2000. This value is approximately 25 times larger than themean heating flux (Nu ’ 80). It is also found that themeasured H�jz� is highly non-Gaussian and exhibits anexponential-like tail.

In summary, the experiment clearly reveals that theheat transport in turbulent convection is carried outmainly by thermal plumes and that the local heat flux ishighly intermittent both temporally and spatially. Thesefeatures are associated with the plume dynamics near thethermal boundary layers and thus should not be sensitiveto the large-scale flow structures in the cell. It is alsofound that the heat transport in the aspect-ratio-one celltakes place primarily along the cell periphery in thedirection of the large-scale circulation. This result shouldapply to other small aspect-ratio cells, in which the large-scale flow structure and the plume dynamics are found tobe similar to those in the aspect-ratio-one cell [11]. It iscertainly important to explore turbulent convection inlarge aspect-ratio cells and the present experiment is, infact, the first step toward this direction.

We thank Siu Lam for his work in producing theshadowgraph and L. Kadanoff, D. Lohse, P. Constantin,and E. Ching for useful discussions. This work was sup-ported in part by the NSF under Grant No. DMR-0071323 (P.T.) and by the Hong Kong RGC under GrantNo. CUHK4242/01P (K. Q. X.).

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(2000).[3] L. P. Kadanoff, Phys. Today 54, No. 8, 34 (2001).[4] B. Castaing et al., J. Fluid Mech. 204, 1 (1989).[5] X. Chavanne et al., Phys. Rev. Lett. 79, 3648 (1997).[6] J. J. Niemela et al., Nature (London) 404, 837 (2000).[7] X. Xu et al., Phys. Rev. Lett. 84, 4357 (2000).[8] G. Ahlers et al., Phys. Rev. Lett. 86, 3320 (2001).[9] X.-Q. Xia et al., Phys. Rev. Lett. 88, 065501 (2002).

[10] Y.-B. Du and P. Tong, J. Fluid Mech. 407, 57 (2000).[11] X.-L. Qiu and P. Tong, Phys. Rev. E 64, 036304 (2001);

66, 026308 (2002).[12] The correlation length was obtained by varying the

separation between the temperature and velocity probes.X.-D. Shang et al. (to be published).

[13] S.-L. Lui and K.-Q. Xia, Phys. Rev. E 57, 5494 (1998).[14] X.-L. Qiu and P. Tong, Phys. Rev. Lett. 87, 094501 (2001).[15] The convection cell is filled with a high Prandtl number

fluid: dipropylene glycol (Pr � 596), which has a largeshadowgraphic contrast. Similar plume distribution isalso observed in water, which has a much smaller shad-owgraphic contrast. S. Lam, Master’s thesis, The ChineseUniversity of Hong Kong, 2000.

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